Essentials of Stochastic Finance: Facts, Models, Theory

ESSENTIALS O F STOCHASTIC FINANCE Facts, Models, Theory

ADVANCED SERIES ON STATISTICAL SCIENCE & APPLIED PROBABILITY

Editor: Ole E. Barndorff-Nielsen Published Vol. 1: Random Walks of Infinitely Many Particles by P. Revesz Vol. 3: Essentials of Stochastic Finance: Facts, Models, Theory by Albert N. Shiryaev Vol. 4: Principles of Statistical Inference from a Neo-Fisherian Perspective by L. Pace and A. Salvan Vol. 5: Local Stereology by Eva B. Vedel Jensen Vol. 6: Eiementary Stochastic Calculus -With by T. Mikosch

Finance in View

Vol. 7 : Stochastic Methods in Hydrology: Rain, Landforms and Floods eds. 0. E. Barndorff-Nielsen et a/.

Forthcoming Vol. 2: Ruin Probability by S. Asmussen

Advanced Series on

Statistical Science & Applied Probability

ESSENTIALS OF STOCHASTIC FINANCE Facts, Models, Theory

Albert N. Shiryaev Steklov Mathematical Institute and Moscow State University

Translated from the Russian by

N. Kruzhilin

orld Scientific New Jersey. London Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 912805 USA oflcet Suite IB, 1060 Main Street, River Edge, NJ 07661

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Contents

Foreword..

Part 1. Facts. Models

1

Chapter I. Main Concepts, Structures, and Instruments. Aims and Problems of Financial Theory and Financial Engineering

2

1. Financial s t r u c t u r e s and i n s t r u m e n t s . . . . . . . ... . 5 la. Key objects and structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 lb. Financial markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 lc. Market of derivatives. Financial instruments . . . . . . . . . . . . . . . . 2 . Financial m a r k e t s under uncertainty. C l a s s i c a l theories of t h e dynamics of f i n a n c i a l indexes, their critics and revision. N e o c l a s s i c a l theories.. ..... 5 2a. R.andoni walk conjecture and concept of efficicnt market . . . . . . . $ 2b. Investment portfolio. Markowitz’s diversification . . . . . . . . . . . . . 3 2c. CAPM: Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . 2d. A P T : Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . $ 2e. Analysis, interpretation, and revision of the classical concepts of efficient market. I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2f. Analysis, interpretation, and revision of the classical concepts of efficient market. I1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Aims and problems of f i n a n c i a l theory, e n g i n e e r i n g , and a c t u a r i a l c a l c u l a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3a. Role of financial theory and financial engineering. Financial risks 5 313. Insurance: a social niechanisni of compensation for financial losses $ 3 ~A. classical example of actuarial calculations: the Lundberg-Cram& theorem . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6 20

35 37 46 51 56 60 65 69 69 71 “rn

vi

Contents

Chapter I1. Stochastic Models . Discrete Time 1. N e c e s s a r y p r o b a b i l i s t i c concepts and s e v e r a l models of t h e d y n a m i c s of market prices ...... ............... l a . Uncertainty and irregularity in the behavior of prices . Their description and representation in probabilistic terms . . . . . 3 l b . Doob decomposition . Canonical representations . . . . . . . . . . . . . 3 l c . Local rnartingalcs . Martingale transformations . Generalized niartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Id . Gaussian and conditionally Gaussian models . . . . . . . . . . . . . . . . 3 l e . Binomial model of price evolution . . . . . . . . . . . . . . . . . . . . . . . . 5 If . Models with discrete intervention of chance . . . . . . . . . . . . . . . . . 2 . L i n e a r stochastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2a . Moving average model MA(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2b . Autoregressive model A R ( p ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2c . Autoregressive and moving average model ARMA(p, q ) and integrated model ARZMA(p,d, q ) . . . . . . . . . . . . . . . . . . . . . 3 2d . Prediction in linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

81

81 89 95 103 109 112 117 119 125

138 142

3 . N o n l i n e a r stochastic c o n d i t i o n a l l y Gaussian m o d e l s .. 53a. ARCH arid GARCH models . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3b. EGARCH, TGARCH, HARCH, and other models . . . . . . . . . . . . 5 3c. Stochastic volatility models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Supplement: d y n a m i c a l chaos m o d e l s .................... 5 4a . Nonlinear chaotic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4b . Distinguishing between ‘chaotic’ and ‘stochastic’ sequences . . . . .

176 176 183

Chapter I11. Stochastic Models . Continuous Time

188

1. N o n - G a u s s i a n m o d e l s of d i s t r i b u t i o n s and processes 3 l a . Stable and infinitely divisible distributions . . . . . . . . . . . . . . . . . 3 l b . Levy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 l c . Stable processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Id . Hyperbolic distributions and processes . . . . . . . . . . . . . . . . . . . . 2 . M o d e l s with s e l f - s i m i l a r i t y . F r a c t a l i t y . . . 5 2a . Hurst’s statistical phenomenon of self-similarity . . . . . . . . . . . . . 5 2b. A digression on fractal geometry . . . . . . . . . . . . . . . . . . . . . . . . 3 2c . Statistical self-similarity . Fractal Brownian motion . . . . . . . . . . . 5 2d . Fractional Gaussian noise: a process with strong aftereffect . . . . . . 3 . Models based on a B r o w n i a n m o t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3a . Brownian motion and its role of a basic process . . . . . . . . . . . . .

152 153 163 168

189 189 200 207 214 221 221 224 226 232

236 236

vii

Contents

§ 3b. Brownian motion: a compendium of classical results . . . . . . . . $ 3 ~Stochastic . integration with respect to a Brownian motion . . . . 3 3d. It6 processes and It6’s formula . . . . . . . . . . . . . . . . . . . . . . . . S 3e. Stochastic differential equations . . . . . . . . . . . . . . . . . . . . . . . $ 3f. Forward and backward Kolmogorov’s equations. Probabilistic representation of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .

.. .. ..

..

240 251 257 264

..

271

4. Diffusion models o f t h e e v o l u t i o n of interest rates, stock and bond prices.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. 3 4a. Stochastic interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4b. Standard diffusion model of stock prices (geometric Brownian motion) and its generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4c. Diffusion models of the term structure of prices in a family of bonds .............. 5. S e m i m a r t i n g a l e m o d e l s . . . . . . . . . . . . . . . . . 3 5a. Sernirriartingales and stochastic integrals . . . . . . . . . . . . . . . . . . . 3 5b. Doob-Meyer decomposition. Compensators. Quadratic variation . 5 5c. It A’s formula for semimartingales. Generalizations . . . . . . . . . . . .

294 294 30 1 307

Chapter IV. Statistical Analysis of Financial Data

314

1. E m p i r i c a l data. P r o b a b i l i s t i c and s t a t i s t i c a l m o d e l s of their description. Statistics o f ‘ t i c k s ’ . ... . . .. . 3 l a . Structural changes in financial data gathering and analysis . . . . . 3 l b . Geography-related features of the statistical data on exchange rates 3 lc. Description of financial indexes as stochastic processes with discrete intervention of chance . . . . . . . . . . . . . . . . . . . . . . Id. On the statistics of ‘ticks’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . S t a t i s t i c s of o n e - d i m e n s i o n a l d i s t r i b u t i o n s . . . . . . . . . . . . . . . . . . . . . $2a. Statistical data discretizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2b. One-dimensional distributions of the logarithms of relative price changes. Deviation from the Gaussian property and leptokurtosis of empirical densities . . . . . . . . . . . . . . . . . . . . 5 2c. One-dimensional distributions of the logarithms of relative price changes. ‘Heavy tails’ and their statistics . . . . . . . . . . . . . . 5 2d. One-dimensional distributions of the logarithms of relative price changes. Structure of the central parts of distributions . . . .

3 . Statistics of v o l a t i l i t y , c o r r e l a t and a f t e r e f f e c t i n prices.. . . . . . . . ... . . . 3 3a. Volatility. Definition and examples . . . . . . . . . . . . . . . . . . . . . . . 3 3b. Periodicity and fractal structure of volatility in exchange rates . .

2 78 278

284 289

315 315 318 321 324 327 327

329 334 340 345 345 351

viii

Contents

5 3c. Correlation properties . . . . . . . . . .

.. . . . . .. . . . . .. . .. . . . .

fj 3d. ‘Devolatixation’. Operational time . . . . . . . . . . . . . . . . . . . . . . . fj 3e. ‘Cluster’ phenomenon and aftereffect in prices . . . . . . . . . . . . . . .

354 358 364

4. S t a t i s t i c a l X I S - a n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 3 4a. Sources arid methods of R/S-analysis . . . . . . . . . . . . . . . . . . . . . 367 3 4b. R/S-analysis of some financial time series . . . . . . . . . . . . . . . . . 376

Part 2. Theory

381

Chapter V. Theory of Arbitrage in Stochastic Financial Models. Discrete Time 1. I n v e s t m e n t p o r t f o l i o on a ( B ,S)-market . . . . . . . . . . .. .. . . .. .. . . 5 la. Strategies satisfying balance conditions , . . . . . . . . . . . . . . . . . fj l b . Notion of ‘hedging’. Upper and lower prices. Complete and incomplete markets . . . . . . . . . . . . . . . . . . . . . . . fj l c . Upper and lower prices in a single-step model . . . . ... . .... . 5 Id. CRR-model: an example of a complete market . . . . . . . . . . . . 2. A r b i t r a g e - f r e e m a r k e t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............ fj 2a. ‘Arbitrage’ and ‘absence of arbitrage’ . . . . . . . . . . ..... .. .. fj2b. Martingale criterion of the absence of arbitrage. First fundamental theorem . . . . . . . . . . . . . . . . . .. 3 2c. Martingale criterion of the absence of arbitrage. Proof of sufficiency . . . . . . . . . . . . . . . . . . . . . . . 5 2d. Martingale criterion of the absence of arbitrage. Proof of necessity (by means of the Esscher conditional transformation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2e. Extended version of the First fundamental t heoreni . . . . . . . . . . . 3 . Construction of m a r t i n g a l e measures by means of an a b s o l u t e l y c o n t i n u o u s change of measure . . . . 3 3a. Main definitions. Density process . . . . , . . . . . . . . . . . . . . . . . . . 5 3b. Discrete version of Girsanov’s theorem. Conditionally Gaussian case 3 3c. Martingale propcrty of the prices in the case of a conditionaIly Gaiissiari arid logarithmically conditionally Gaussian distributions 3 3d. Discrete version of Girsanov’s theorem. General case . . . . . . . . . 3 3e. Integer-valued random measures and their compensators. . Transformation of compensators under absolutely continuous changes of measures. ‘Stochastic integrals’ . . . . . . . . . . . . . . . . . 3 3f. ‘Prcdictable’ crit,eria of arbitrage-free ( B ,S)-niarkcts . . . . . . . . . . I

382

383 383 395 399 408 410 410 413 417

417 424 433 433 439 446 450

459 467

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Contents

4 . C o m p l e t e and p e r f e c t a r b i t r a g e - f r e e markets . . . . . . . . . . . . . . . . . . 5 4a . Martingale criterion of a complete market . Statement of the Second fundamental theorem . Proof of necessity 5 4b. Represcntability of local martingales . ‘S-representability’ . . . . . . 3 4c . R.epresentability of local martingales (‘ir-represeritability’ and ‘(ji-v)-representability’) . . . . . . . . . . . . 5 4d . ‘S-representability’ in the binomial CRR-model . . . . . . . . . . . . . 3 4r . Martingale criterion of a complete market . Proof of necessity for d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4f . Extcndetl version of the Second fundamental theorem . . . . . . . . .

Chapter VI . T h e o r y of Pricing i n Stochastic F i n a n c i a l M o d e l s D i s c r e t e Time

2 . American hedge p r i c i n g on a r b i t r a g e - f r e e markets . . . . . . . . . . . . 5 2a . Optirnal stopping problems. Supermartingale characterization . . . 3 2b . Complete arid incomplete markets . Supermartingale characterization of hedging prices . . . . . . . . . . . 3 2c . Complete and incomplete markets . Main formulas for hedging prices . . . . . . . . . . . . . . . . . . . . . . . . 5 2d . Optional decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.... ................... 3h . Criteria of the absence of asymptotic arbitrage . . . . . . . . . . . . . 3 3c. Asymptotic arbitrage and contiguity . . . . . . . . . . . . . . . . . . . . . . 5 3tl . Some issues of approximation and convergence in the scheme of scries of arbitrage-free markets . . . . . . . . . . . . . . . . . . . . . . . . 4 . E u r o p e a n options on a b i n o m i a l ( B ,S)-market . . . . . . . . . . . . . . . fj4a . Problems of option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 411. Rational pricing and hedging strategies. Pay-off function of the general form . . . . . . . . . . . . . . . . . . . . . fj 4c . Rational pricing and hedging strategies. Markoviari pay-off functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 3a . One model of ‘large’ financial markets

481 483 485 488 491 497

.

1. E u r o p e a n hedge pricing on a r b i t r a g e - f r e e m a r k e t s . . . . . . 3 l a . Risks and their reduction . . . . . . . . . . . . . . . . . .......... fj 1b. Main hedge pricing formula . Complete markets . . . . . . . . . . . . . fj 1c. Main hedge pricing formula. Incomplete markets . . . . . . . . . . . . . 3 Id . Hedge pricing on the basis of the mean square criterion . . . . . . . . 5 l e . Forward contracts and futures contracts . . . . . . . . . . . . . . . . . . .

3 . Scheme o f series o f ‘ l a r g e ’ a r b i t r a g e - f r e e m a r k e t s and asymptotic arbitrage . ....... .....

481

502

503 503 505 512 518 521 525 525 535 538 546 553 553 555 559 575 588 588 590 595

Contents

X

3 4tl. fi 4e.

Standard call and put options . . . . . . . . . . . . . . . . . . . . . . . . . . Option-based strategies (combinations and spreads) . . . . . . . . . .

598 604

5 . A m e r i c a n options on a b i n o m i a l ( B ,S)-market . . . . . . . . . . . . . . . . . 3 5a. American option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5b. Standard call option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . fi 5c. Standard put option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . fi 5d. Options with aftereff’ect. ‘Russian option’ pricing . . . . . . . . . . . .

608 608 611 621 625

Chapter VII. Theory of Arbitrage in Stochastic Financial Models. Continuous Tinie 632 1. I n v e s t m e n t p o r t f o l i o i n s e m i m a r t i n g a l e m o d e l s . . . . . . . . fi l a . Admissible strategies. Self-financing. Stochastic vector integral . . fi l b . Discounting processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 lc. Admissible strategies. Some special classes . . . . . . . . . . . . . . . . .

p o r t u n i t i e s f o r arbit 2. S e m i m a r t i n g a l e m o d e l s witho ........ ........ Completeness . . . . . . . . . . . . . . . . . . . 3 2a. Concept, of absence of arbitrage and its modifications . . . . . . . . . 3 2b. Martingale criteria of the absence of arbitrage. Sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2c. Martingale criteria of the absence of arbitrage. Necessary and sufficient conditions (a list of results) . . . . . . . . . . 3 2d. Completeness in semimartingale models . . . . . . . . . . . . . . . . . . .

3 . S e m i m a r t i n g a l e a n d m a r t i n g a l e measures.. . . . . . . . . . . . . . . . . . . . . 5 3a. Canonical representation of semirnartingales. Random measures. Triplets of predictable characteristics . . . . . . 3 3b. Construction of marginal measures in diffusion models. Girsanov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3c. Construction of martingale measures for L6vy processes. Essclier transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3d. Predictable criteria of the martingale property of prices. I . . . . . . 9: 3e. Predictable criteria of the mart,ingale property of prices. I1 . . . . . 3 3f. Representability of local martingales ( ‘ ( H ” p-v)-representability’) , 3 3g. Girsanov’s theorem for sernimartingales. Structure of the densities of probabilistic measures . . . . . . . . . . . 4. Arbitrage, c o m p l e t e n e s s , and hedge pricing in d i f f u s i o n models of s t o c k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34a. Arbitrage and conditions of its absence. Completeness . . . . . . . . . fi 4b. Price of hedging in complete markets . . . . . . . . . . . . . . . . . . . . . 3 4c. Fundarnental partial differential equation of hedge pricing . . . . . .

633 633 643

646 649 649 651 655 660 662 662 672 683 691 694 698

701 704 704 709 712

Contents

5. A r b i t r a g e . c o m p l e t e n e s s . and hedge p r i c i n g in d i f f u s i o n m o d e l s of bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5a. Models without opportunities for arbitsrage . . . . . . . . . . . . . . . . . fi 5b. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5c. Funtlamental partial differentai equation of the term structure of boncls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter VIII . Theory of Pricing in Stochastic Financial Models . Continuous Time 1. E u r o p e a n options i n d i f f u s i o n (B. S)-stockmarkets . . . . . . . . . . . . . fi 1.2. Bachelier’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11). Black-Scholes forrnula . Martingale inference . . . . . . . . . . . . . . . 5 l c . Black- Scholes formula . Inference based on the solution of the fundamental equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 ltl . Black-Scholes formula . Case with dividends . . . . . . . . . . . . . . .

x1

717 717 728 730

734

735 735 739 745 748

2 . A m e r i c a n options i n d i f f u s i o n (B,S)-stockmarkets . Case of a n i n f i n i t e time h o r i z o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2a . Standard call option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2b . Standard put option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2c . Combinations of put and call options . . . . . . . . . . . . . . . . . . . . 3 2d . Russian option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . A m e r i c a n options in d i f f u s i o n (B,S)-stockmarkets. Finite time horizons . . . 53a . Special features of calculations on finite time intervals . . . . . . . . fi 3b . Optirnal stopping problems and Stephan problenis . . . . . . . . . . 3 3c. Stephan problem for standard call and put options . . . . . . . . . . 5 3d . Relations between the prices of European and American options 4. E u r o p e a n a n d A m e r i c a n o p t i o n s i n a diffusion .. ( B ,P)-bondmarket ................................................ 3 4a . Optlion pricing in a bondmarket . . . . . . . . . . . . . . . . . . . . . . . . 3 4b . European option pricing in single-factor Gaussian models . . . . . fi 4c . American option pricing in single-factor Gaussian models . . . . . Bibliography ...........................................................

792 792 795 799 803

Index ....................................................................

825

I n d e x of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

833

751 751 763 765 767 778 778 782 784 788

Foreword

The author’s intention was: 0

0

to select and eqiose subjects that can be necessary or useful t o those i n terested in stoch,astic calculus and pricing in models of financial m,arkets operating u n d e r m c e r t a i n t y ;

to introduce the reader t o the main concepts, notions, and results

of stochas-

tic financial mathematics; 0

to deiielop applications of these resTults t o various kinds qrhred in, financial e,ngineering.

of calculations re-

The author considered it also a major priority to answer the requests of teachers of financial mathematics and engineering by making a bias towards probabilistic and statistical ideas and thc methods of stochastic calculus in the analysis of m a r k e t risks. The subtitle “Facts, Models, Theory” appears to be an adequate reflection of the text structure and the author’s style, which is in large measure a result of the ‘feedback’ with students attending his lectures (in Moscow, Zurich, Aarhus, . . . ). For instance, a n audicnce of mathematicians displayed always a n interest not only in the mathematical issues of the ‘Theory’, but also in the ‘Facts’, the particularities of real financial markets, and the ways in which they operate. This has induced the author to devote the first chapter t o the description of the key objects and structures present on these markets, to explain there the goals of financial theory and engineering, and to discuss some issues pertaining to the history of probabilistic and statistical ideas in the analysis of financial markets. On the other hand, a n audicnce acquainted with, say, securities markets arid securities trading showed considerable interest in various classes of stochastic processes used (or considered as prospective) for the construction of models of the

xiv

Foreword

dynamics of financial indicators (prices, indexes, exchange rates, tant for calculations (of risks, hedging strategies, rational option prices, etc.). This is what we describe in the second and the third chapters, devoted to stochastic ‘Models’ both for discrete and continuous time. The author believes that the discussion of stochastic processes in these chapters will be useful to a broad range of readers, not only to the ones interested in financial mathematics. We ciiiphasixe here that in the discrete-time case, we usually start in our description of t,lic evolution of stochastic sequences from their Doob decorripo.sition into predictable and martingale components. One often calls this the ‘martingale approach’. Regarded from this standpoint, it is only natural that martingale theory can provide financial mathematics and engineering with useful tools. The concepts of ‘predictability’ and ‘martingality’ permeating our entire exposition are incidentally very natural from economic standpoint. For instance, such economic concepts as investment portfolio and hedging get simple mathematical definitions in terms of ‘predictability’, while the concepts of e f i c i e n q and absence of arbitrage on a financial market can be expressed in the mathematical language, by making references to martingales and martingale measures (the First fundamental theorem of asset pricing theory; Chapter V, 3 2b). Our approach to the description of stochastic sequences on the basis of the Doob decomposition suggests that in the continuous-time case one could turn to the (fairly broad) class of semirriartirigales (Chapter 111, 5 5a). Representable as they are by sums of processes of bounded variation (‘slowly changing’ components) and local martirigales (which can often be ‘fast changing’, as is a Brownian motion, for example), semimartingales have a remarkable property: one can define stochastic integrals with respect to these processes, which, in turn, opens up new vistas for the application of stochastic calculus to the coristriiction of models in which financial indexes are simulated by such processes. The fourth (‘statistical’) chapter must give the reader a notion of the statistical ‘raw material’ that one encounters in the empirical analysis of financial data. Based mostly on currency cross rates (which are established on a global, probably the largest, financial market with daily turnover of several hundred billion dollars) we show that the ‘returns’ (see ( 3 ) in Chapter 11, la) have distribution densities with ‘heavy tails’ and strong ‘leptokurtosis’ around the mean value. As regards their behavior in time, these values are featured by the ‘cluster property’ and ‘strong aftereffect’ (we can say that ‘prices keep memory of their past’). We derrionst,r;ite tht: fractal structure of several characteristic of the volatility of the ‘returns’. Of course, one must take all this into account if one undertakes a construction of a niodel describing thc actual dynamics of financial indexes; this is extremely iriiportant if one is trying to foresee their development in the future. ‘Theory’ in general and, in particular, arbitrage theory are placed in the fifth chapter (discrete t h e ) and the seventh chapter (continuous time).

Foreword

XV

Ceritral points there are the First and the Second fundamental asset pricing theorems. The First theorem states (more or less) that a financial market is arbitrage-free if and only if there exists a so-called martingale (risk-neutral) probability measure such that, the (discounted) prices make up a martingale with respect to it. The Second theorem describes arbitrage-free markets with property of completeness, which ensures that one can build a n investment portfolio of value replicating faithfully any given pay-off. Both theorems deserve the name fundamental for they assign a precise mathematical meaning to the economic notion of a n ‘arbitrage-free’ market on the basis of (well-developed) martingale theory. In the sixth and the eighth chapters we discuss pricing based on the First and the Second fundamental theorems. Here we follow the tradition in that we pay much attention to the calculation of rational prices and hedging strategies for various kinds of (European or American) options, which are derivative financial instruments with best developed pricing theory. Options provide a perfect basis for the understanding of the general principles and methods of pricing on arbitrage-free markets. Of coiirse, the author faced the problem of the choice of ‘authoritative’ data and the mode of presentation. The above description of the contents of the eight chapters can give one a measure for gauging the spectrum of selected material. However, for all its bulkiness, our book leaves aside many aspects of financial theory and its applications (e.g., the classical theories of von Neumann-Morgenstern and Arrow-Debreu and their updated versions considering investors’ behavior delivering the maximum of the ‘utility function’, and also computational issues that are important for applications). As the reader will see, the author often takes a lecturer’s stance by making comments of the ‘what-where-when’ kind. For discrete time we provide the proofs of essentially all main results. On the other hand, in the continuous-time case we often content ourselves with the statements of results (of martingale theory, stochastic calculus, etc.) and refer t o a suitable source where the reader can find the proofs. The suggestion that the author could write a book on financial rnathematics for World Scientific was put forward by Prof. Ole Barndorff-Nielsen a t the beginning of 1995. Although having accepted it, it was not before summer that the author could start drafting the text. At first, he had in mind to discuss only the discretetime case. However, as the work was moving on, the author was gradually coming to the belief that he could not give the reader a full picture of financial mathematics and engineering without touching upon the continuous-time case. As a result, we discuss both cases, discrete and continuous. This book consists of two parts. The first (‘Facts. Models’) contains Chapters I ~ I V The . second (‘Theory’) includes Chapters V-VIII.

xvi

Foreword

The writing process took around two ycars. Several months went into typesetting, editing, and preparing a camera-ready copy. This job was done by I. L. Legostacva, T . B. Tolozova, and A. D. Izaak on the basis of the Information and Publishing Sector of the Department of Mathematics of the Russian Academy of Sciences. Thc author is particularly indebted to them all for their expertise and selfless support as well as for the patience and tolerance they demonstrated each time the author came to them with yet another ‘final’ version, making changes in the already typeset and edited text. The author acknowledges the help of his friends and colleagues, in Russia and abroad; he is also grateful to the Actuarial and Financial Center in Moscow, VW-Stiftung in Germany, the Mathematical Rcscarch Center and the Center for Analytic Finance in Aarhus (Dcnrnark), INTAS, and the A. Lyapunov Institute in Paris and Moscow for their support and hospitality.

Moscow 1995 --- 1997

A. Shiryaev Steklov Mathematical Institute Russian Academy of Sciences and Moscow State University

Part 1

FACTS

MODELS

Chapter I. Main Concepts, Structures, and Instruments. Aims and Problems of Financial Theory and Financial Engineering 1. F i n a n c i a l structures and i n s t r u m e n t s

3

3 la. Key objects and structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 lb. Financial markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fj lc. Market of derivatives. Financial instruments . . . . . . . .

........

2 . F i n a n c i a l markets under u n c e r t a i n t y . C l a s s i c a l t h e o r i e s of t h e d y n a m i c s of f i n a n c i a l i n d e x e s , t h e i r c r i t i c s and r e v i s i o n Neoclassical t h e o r i e s . . .. . ... . . . . . . . .. . . . .. . . . . .. . ... .. ... ... . .. . .

3 2a. Random walk conjecture and concept of efficient market . . . . . . . 3 2b. Investment portfolio. Markowitz’s diversification . . . . . . . . . . . . . 3 2c. CAPM: Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . fj 2d. A P T : Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . Analysis, interpretation, and revision of the classical concepts of efficient market. I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2f. Analysis, interpretation, and revision of the classical concepts of efficient market. I1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 6 20

35

37 46 51 56

5 2e.

3 . A i m s a n d problems of f i n a and a c t u a r i a l c a l c u l a t i o n s

5 3a. Role of financial theory and financial engineering. Financial risks 5 3b. Insurance: a social mechanism of compensation for financial. losses 5 3c. A classical example of actuarial calculations: the Lundberg---Cramkr theorem

..........................................

60 65

69 69 71

77

1. Financial Structures and Instruments

In modern view (see, e.g., [79], [342], and [345]) financial t h e o q and engineering must analyze the properties of financial structures and find most sensible ways t o operate financial resources using various financial instruments and strategies with due account paid to such factors as time, risks, and (usually, random) environ,m,en,t. Time, dynamics, ,uncertainty, stochastics: it is thanks to these elements that probabilistic arid statistical theories, such as, e.g.,

0

the theory of stochastic processes, stochastic calculus, statistics of stochastic processes, st,ocliastic optimization

make up thc rriachincry used in this book and adequate t o the needs of financial theory arid rngineering.

$ l a . Key Objects and Structures 1. Wc call distinguish the following baszc objects and structures of financial theory that drfirir anti explain the specific nature of financial problerris, tlir aims, arid the tools of financial rriatherriatics and engineering: 0

0

indi~iiduals, corporations, in,tcrmediaries, financial markets.

4

Chapter I. Main Concepts, Structurrs, and Instruments

intermediaries

As shown in tlic chart, the theory and practice of finance assign the central role among the above four structures to financial ,markets; they are the structures of :he primary concern for the mathematical theory of finance in what follows. 2. Individuals. Their financial activities can be described in terms of the dilemma ‘ consumption iriimtirrimt’. The ambivalence of their behavior as both consumers (‘consunin inorc now’) arid investors (‘invest now to get more in the future’) brings one to optimization problems formulated in niathcmatical economics as consurription---sawing arid portfolio decision makin,g. In the framework of utility th,eory the first problem is t,reated on the basis of the (von Neumann-Morgenstern) postulates of the rational behavior of individuals under uncertainty. These postulates determine the approaches and methods used to determine pre,ferable strategies by rriearis of a quantitative analysis, c.g., of the mean values of the utility functions. The problem of ‘portfolio decision making’ confronting individuals can roughly be described as the problem of the best allocation (investment) of funds (with due attention to possible risks) among, say, property, gold, securities (bonds, stock, options, futures, etc.), and the like. The idea of diversification (see 5 2b) in building a portfolio is reflected by such well-known adages as “Don’t put all your eggs in one basket” or “Nothing ventured, nothing gained”. In what follows we describe various opportunities (depending on the starting capital) opening for a n individual on a securities market. Corporations (coiripanies, firms), who own such ‘perceptible’ valuables as ‘land’, ‘factories’, ‘machines’, but are also proprietors of ‘organization structures’, ‘patentas’,ctc., organize businesses, maintain business relations, and rrignage manufacturing. To raise funds for the development of manufacturing, corporations occasionally issue stock or bonds (which governments also do). Corporate management must be directed towards meeting the interests of shareholders and bondholders.

1. Financial Structures and Instruments

5

Intermediaries (interrnediate financial structures). These are banks, investment coriqxtiiics (of mutual funds kind), pension funds, insurance companies, etc. Onr can put tierr also stock exchanges, option and futures exchanges, and so on. Arnoiig the world's most renowned exchanges (as of 1997) one must list the USA-bascd NYSE (the New York Stock Exchange), AMEX (the American Stock Exchange), NASDAQ (the NASDAQ Stock Market), NYFE (the New York Futures Exchangc), CBOT (thc Chicago Board of Trade), etc.a 3. Financial markets iricliide money and Forex markets, markets of precious metals, arid markets of financial instruments (including securities). In the market of financial instruments one usually distinguishes

underlying (primary) instruments arid derivative (secondary) instruments; tlie lattcr are hybrids constructed on tlie basis of underlying (more elementary) instruments. The underlying firiaricial instruments include the following securities: bank accounts, bonds, stock.

Thc derivntir,e firiancial instruments include 0

0

0

options, futures contracts, warrants, swaps, combinations, spreads,

We note that financial mgineering is often understood precisely as manipulatioris with derimtive securities (in order to raise capital and reduce risks caused by the u~icertaincharacter of the market situation in the future). We now describe several main ingredients of financial markets. "Throughout the book we turn fairly often to American financial structures and t o financial activities taking place in tlie USA. The main reason is that the American financial markets have deep-root,ed traditions ('the Wall Street'!) and, a t the same time, these are the markets where many financial innovations are put to test. Moreover, there exists an enormous literature describing these markets: periodicals as well as konographs, primers, handbooks, investor's guides, and so on. The reader can easily check through t h e bibliography a t the end of this book.

6

5 lb.

Chapter I. Main Concepts, Structures, and Instruments

Financial Markets

1. Money dates back t o those ages when people learned t o trade ‘things they had’ for ‘t,liirigs they wanted to get’. This mechanism works also these days-we give money iii exchange for goods (and services), and the salesmen, in turn, use it to buy other things. Modern technology means a revolution in the ways of money circulation. Only 8% of all dollars that are now in circulation in the USA are banknotes and coins. The main bulk of payments in retailing and services are carried over by checks and plastic cards (and proceed by wires). Besides its function of a ‘circulation medium’, money plays an important role of a ‘measure of value’ and a ‘saving means’ [log].

2. Foreign currency, i.e., the currency of other nationw (its reserves, cross rates, and so on) is an important indicator of the nation’s well-being and development arid is often a mea’ns of payment in foreign trade. Economic globalization brings into being monetary unions of several nations agreeing to harmonize their monetary and credit policies and regulating exchange rates between their currencies. One example is the well-known Bretton-Woods credit and currency system. In 1944, Bretton-Woods (New Hampshire, USA) hosted a conference of the major participants in the international trade, who agreed to maintain a currency system (the ‘Bretton-Woods system’) in which the exchange rates could deviate from their officially declared levels only in very narrow ranges: 1%in either side. (These parity cross rates were fixed against the USA dollar.) To launch this system and oversee it, the nations concerned established the International Monetary Foundation (IMF). However, with the financial and currency crisis of 1973 affecting the major currencies (the USA dollar, the German mark, the Japanese yen), the Bretton-Woods system was acknowledged to have exhausted its potentialities, and it was replaced by floating exchange rates. In March 1979, the majority of the European Union (EU) nations created the European Monetary System. It is stipulated in this system that the variations of the cross rates of the currencies involved should lie, in general, in the band of &2.25% around the official parity rates. If the cross rates of some currencies are deemed under the threat of leaving this band, then the corresponding central banks must intervene to prevent this course of events and ensure the stability of exchange rates. (This explains why one often calls systems of this kind ‘systems of adjusted floating rates’.) Other examples of monetary unions can be found between some Caribbean, Central American, and South American, countries, which peg exchange rates against some powerful ‘leader currency’. (For greater detail, see [log; pp. 459-4681,) 3. Precious metals, i.e., gold, silver, platinum, and some other (namely, the metals in the platinum group: iridium, osmium, palladium, rhodium, ruthenium)

1. Financial Structures and Instruments

r

played throughout the history (in particular, in the 19th and the beginning of the 20th century) arid still play today a n important role in the international credit and currency system. According t,o [108] the age of gold standard began in 1821, when Britain proclaimed pound sterling convertible into gold. The United States did the same soon afterwards. Th. gold standard came into full blossom in 1880-1914, but it could never recover its status after the World War I. Its traces evaporated completely in 1971, when t,he US Treasury abandoned formally the practice of buying and selling gold at a fixed price. Of course, gold keeps a n important position in the international currency system. For example, governments often use gold t o pay foreign debts. It follows from the above that one can clearly distinguish three phases in the development of the international monetary system: ‘gold standard‘, ‘the BrettonWoods system’, and the ‘system of adjusted floating exchange rates’. 4. Bank account. We can regard it as a security of bond kind (see subsection 5 below) that reduces in effect to the bank’s obligation to pay certain interest on the sun1 put into one’s account. We shall often consider bank accounts in what follows, primarily because it is a convenient ‘unit of measurement’ for the prices of various securities. One usually distinguishes two ways to pay interest: 0 ‘m times a year (simple i n t e r e s t ) , continuously (compound i n t e r e s t ) .

If you open a bank account paying interest m t i m e s a year with i n t e r a t rate r(,rr/,),then on having put an initial capital Bo, in N years you obtain the amount

while in N

+ k / m years’ time (a fractional value, 0 < k < m ) ,your capital will be

In the case of conipound interest with znterest rate r ( w ) the starting capital Bo grows in N years into B N ( W ) = B()er(oo)N.

(2)

Clearly,

BN(m) B N ( W )

+ r(00) and nL + 00. If the compound interest r ( w ) is equal t o T , then the adequate ‘rat$ of interest payable r n times a year’ r ( m ) can be found by the formula as

T(rn)

r ( m ) = m ( e r / m - 1),

(3)

8

Chapter I. Main Concepts, Structures, and Instruments

while the compound rate the formula

T

= r ( m ) corresponding to fixed r ( m ) can be found by

In the particular case of 712 = 1, setting F = r(1) and following conversion formulas for these rates: A

r = ‘e

-

1,

r = In(1 + F).

T

=

T(W)

we obtain the

(5)

Sesides the ‘annual interest rate F ’ , the bank can announce also the value of the ‘annual discount rate F ’ , which means that one must put Bo = Bi(1 - F ) into a bank accoiirit to obtain an amount B1 = &(1) a year later. The relation between F and ;iis straightforward: (1 - @ ) ( 1 +F ) = 1, therefore

One can also ask about the time necessary (under the assumption of continuously payable interest r = a / 1 0 0 ) to raise our capital twofold. Clearly, we can determine N from the relation 2 = e aN/100

1. Financial Structures and Instruments

9

i.e.,

In practice (in the case of interest payable, say, twice a year) one often uses the so-called rule 72: if the interest rate is a/100, then capital doubles in 7 2 / a years. To help the reader make a n idea of the growth of a n investment for various modes of interest payment (mn = 1 , 2 , 3 , 4 , 6 , 1 2 ,w)we present a table (see above) of the values of Bt(rn) (here Bo = 10000) corresponding to t = 0 , 1 , . . . , 10 and ~ ( m=)0.1 for all rri.

5. Bonds are promissory notes issued by a government or a bank, a corporation, a joint stock cornpany, any other financial establishment t o raise capital. Bonds are fairly popular in many countries, and the total funds invested in bonds are larger tJlian the funds invested in stock or other securities. Their main attraction (especially for a conservative investor) is as follows: the interest on bonds is fixed and payable on a regular basis, and the repayment of the entire loan at a specified time is guaranteed. Of course, one cannot affirm beyond all doubts that government or corporate bonds are k-free financial instruments. A certain degree of risk is always here: for instance, the corporation may go bust and default on interest payments. For that reason government bonds are less risky than corporate ones, but the coupon yield on corporate bonds is larger. An investor in bonds is naturally eager to know the riskiness of the corporations he considers for the purpose of bond purchase. Ratings of various issuers of bonds can bc found in scveral publications (“Standard & Poor’s Bond Guide” for one). Corporations considered less risky (i.e., ones with higher ratings) pay lower interest. Accordingly, corporations with low ratings must issue bonds with higher interest rate to attract investors. Wr can characterize a bond issued a t time t = 0 by several numerical indexes ((i)-(vii)): the face value (par value) P(T,T), i.e., the sum payable to the holder of the bond a t the maturity d d e (the year the bond matures) T; (the time to maturity is iisually a year or shorter for short-term bonds, 2 to 10 years for middle-term ones, and T 3 30 years for long-term bonds); the borrd’s in,terest rate (coupon yield) rc defining the dividends, the amount payable to its holder by the issuer, by the formula rcx(faace value); the original price P(0,T) of the bond of T-year maturity issued at time t = 0. If the bond issued at time t = 0 has, say, 10-year maturity, the face value $1000, and the coupon yield T = G / l O O (= G%), then having purchased it, at the original (purc:hase) pricc of $1000 we shall in ten years receive the profit of $600, which is

10

Chapter I. Main Concepts, Structures, and Instruments

the sum of the following terms:

$1000

face value

+

interest for 10 years

1000 x 6% x 10 = $600

-

purchase price

$1000

............................................... -

$600

Of course, the holder of a bond of T-year maturity who bought it a t time t = 0 can keep it all these T years for himself collecting the interest and the face value (at time T ) . On the other hand he may consider it unprofitable t o keep this bond till maturity (e.g., if the inflation rate qnf is greater than rC). In that case the bondholder can use his right to sell the bond (with maturity date T ) a t its (v) market value P ( t , T ) a t time t (in general, t can be an arbitrary instant between 0 and T ) . By definition, P(0, T ) is the original price of the bond at the moment of flotation, while P(T,T) is clearly its face value. (Both are equal to $1000 in the above example.) Although it is in principle possible that the market value P(t, T ) is larger than the face value P(T, T ) ,one typically has the inequality P(t, T ) 6 P(T, T ) . Assume now that the bondholder decides to sell the bond two years after the flotation, and the market value P(2,lO) (here t = 2 and T = 10) of the bond is $800. Then his profit from having purchased the bond and having held it for two years is as follows: market value

+

interest

$800

1000 x 6% x 2 = $120

-

purchase price

$1000

....................................... -

4 8 0

Thus, his profits are in fact negative: his losses amount to $80. On the other hand an investor who buys this bond a t its market value P(2,lO) = $800 will pocket the following return in 8 years: face value

+

interest purchase price

$1000 1000 x 6% x 8 = $480 $800

........................................ -

$680

1. Financial Structures and Instruments

11

We must ernpliasize that the interest rc on the bond, its coupon yield, is fixed over it,s life tirriti, whereas its market value P(t,T) wavers. This is tlie result of the infliiciice of multiple ccoriornic factors: demand and supply, interest payable on other sccurities, spcculators’ activities, etc. Regarding { P ( t , T ) }0, 6 t 6 T , as a random process cvolvirig in time we see that this is a conditional process: the value of P(T, T ) is fixed (arid equal to the face value of the bond). In Fig. 1 we depict possible fluctuations on the time interval 0 6 t 6 T of the rriarkct value P ( t , T ) that takes the prescribed facc value P(T,T) at the maturity date T . A

0

t

T

FIGURE 1. Evolution of the market value P(t,T) of a bond

A frcqucntly used characteristic of a bond is (vi) thc c,u,rrerit yield

tlic ratio of the yearly interest and the (current) market value, which is important for the cornparison of different bonds. (In the above example rc(O,10) = r, = 6 % and r c ( 2 , 10) = 6 % . lOOO/SOO = 7.5 %.) Another, probably the most important, characteristic of a bond, which enables one to estimate the returns from both the final repayment and the interest payments (due for tlic remaining time to maturity) and offers thus yet another opportunity for tlie cornparison of different bonds, is (vii) the yield t o m,aturity (on a percentage basis) or the profitability

(here T - t is thc remaining life time of the bond); its value must ensure that the slim of tlie discounted values of the interest payable on the interval

12

Chapter I. Main Concepts, Structures, and Instruments

( t , T ]and the face value is the current market value of the bond. In other words, p = p(T - t , T ) is the root OF the equation

k=l

(Here we measure time in years, t = 1 , 2 , . . . , T . ) In the case when the market value P ( t , T ) is the same as the face value P ( T , T ) , we see From this definition that p (T - t , T ) = rc. We plot schematically in Fig. 2 a typical ‘yzeld curwe’, the graph of p ( s , T ) as a function of the remaznzng time s = T - t.

5

I I

--

I I

01

S

c

T

FIGURE 2. Profitablity p = p ( s , T ) as a function of s = T

~

t

In the above description we did not touch upon the structure of the market prices { P(t, T)} (regarded from the probabilistic point of view, say). We consider this, fairly complicated, question, further on, in Chapter 111. Here we only note that in discussions of the structure and the dynamics of the prices { P ( t , T ) } from the probabilistic standpoint one usually takes one of the following two approaches: a) direct specification of the evolution of the prices P ( t , T ) (the price-based approach); ti) indirect specification, when in place of the prices P ( t , T ) ,one is given the time structure of the yield { p ( T - t , T ) }or a similar characteristic, e.g., of ‘interest rate’ kind (the term structure approach). We note also that formula ( 6 ) , which is a link between the price of a bond and its yield, is constructed in accordance with the ‘simple interest’ pattern (cf. (1) for ‘rii = 1). It is also easy to firid a corresponding forrriiila in the case of continuously payable interest (‘coinpound interest’; cf. (2)). Wc have already rioted that bonds are floated by various establishments and for various purposes.

1. Financial Structurcs and Instruments

13

Corporate bonds arc issued to accumulate capital necessary for further dcveloprrient, or modernization, to cover operational costs, etc. Moiicy obtained from issuing g o v e m r n e n t borids issued by national govermcnts or municrpal bonds, issued by state governments, city couricils, ctc., is used to carry out various govermental prograrrims or projects (const,riiction of roads, schools, bridges, . . . ) and cover budget deficits. In Arrierica goverrirncnt bonds include Treasury bonds, notes, and bills, which can be purchased through the Federal Reserve Banks or brokers.

Information on corporate bonds arid their properties is available in many publications (e.g., in “New York Stock Exchange”, section “New York Exchange Bonds”, or in “Arnerican Stock Exchange”). This inforriation is organized in tjhe form of a list of quotations of the following foru1: IBM-JJ15-7% of ’01, which means that the bonds in question are issued by IBM, the interest is payableb on January, 15, arid July, 15, with rate rC = 7% arid maturity year 2001. Of course, onc can also find in these publications data on the face value of the t)onds and their current market price. (For details, see, c.g., [469].) 6. Shares (st,oc:k) are also issued by companies and corporations to raise funds. They riiainly bclong to one of the two types: ordinary shares (equities; common stock) arid preference shares (preferred stock), which differ both in riskiness and thc conditions of dividend payments. An owiicr of comniori stock obtains as dividends his share of the profits of the firm, and their anioiirit depends on its financial successes. If the company goos baricriipt, then the shareholder can lose his investment. On the other hand, pr(~f(:rrcdstjock rneans that, the investor’s risk to lose everything is smaller and his tiividcnts are guararitct:tl, although their amount, in general, does not increase with tlic firm’s profits. There also arc other kinds of shares characterized by different degrees of involvenicnt in the rnanagcmerit of the corporation or some peculiarities of dividerit payments, and so on. Wlieri buying stock (or bonds) many investors are attracted not by dividends (or interest 4 i i the case of bonds), but by a n opportunity to make money from fluctuations of st,ockprices, buying cheap ahead of anyone else and selling high (also before the others). According t,o some estimates there are now more than 50m shareholders in the USA. (For iristanre, 2418447 investors had shares in AT&T by the end of 1992, wit,li the tot,al iiurnbcr of shares 1339 916 615; see [357].) bIiit,ercst,is corrirnoiily payable oiice a year in Eiiropc, hut twicc a year, every 6 rnoriths in the USA.

14

Chapter I. Main Concepts, Structures, and Instruments

To buy or sell shares one should address a brokerage house, an investment company that is a niernber of a stock exchange. It should be noted that, although the nurnbcr of individual stockholders increases with time, the share of individuals holding shares directly decreases: individuals are usually not themselves active on the market, but participate through inxtitvtional investors (mutual or pension funds, insurance companies, banks, and others, ‘betting’ on securities markets and, in particular, stockmarkets). Many countries have stock exchanges where shares are traded. Apparently, one of the first was the Amsterdam Stock Exchange (1602), which traded the shares of joint-stock companies. Traditionally, banks exert strong influence on European (stock) exchanges, while the American stock exchanges has been separated from the bank systcm since the 1930s. The two largest American exchanges (as of 1997) are the NYSE (New York Stock Exchange-the name under which it is known since 1817; it had 1366 seats in 1987) and the AMEX (American Stock Exchange, organized on the basis of the New York Curb Exchange founded in 1842). To make its shares eligible for trade on an exchange a company must satisfy certain requirements (concerning its size, profits, and so on). For example, the requirements of the NYSE are as follows: the earinings before taxes must be a t least $2.5 inillion and the number of shares floated must be a t least 1.1 million, of market value at least $ 1 8 million. Hence only the stock of well-known firms may be traded on this exchange (2089 listed corporations by 1993). Trade on the AMEX goes mostly in stock of medium-sized companics; thc number of listed firms there is 841. Thc NASDAQ (National Association of Securities Dealers Automatic Quotations) Stock Market is another major American share-trading establishment; the trade there proceeds by electronic networks. 4700 firms-- -large, medium-sized, arid small but rapidly developing----areregistered here. An impressive number of firms (around 40 000, [357]) are participants of the OTC (Over-the-counter) securities market. This market has no premises or even central office. Deals are made by wire, through dealers who buy and sell shares on their accounts. This trade stretches over most diversified securities: ordinary and preferrerice shares, corporate bonds, the US Government securities, municipal bonds, options, warrants, foreign securities. The main reason why firms that are small, newly established, or issuing small numbers of shares use OTC dealers is that they meet there virtually no (or minimal) restrictions on the size of assets and the like. On the other hand conipanies whose shares are eligible for trade on other exchanges often go to the OTC market if the manner of bargaining and deal-making accepted there seems more convenient than the routine of ‘properly organized’ exchanges. There can be other reasons for dealing through the OTC system, e.g., a firm may be reluctant to disclose its financial state, as required by big exchanges. Of course, it is irriportant for investors, ‘big’ or ‘small’ alike, to have information on the health of firms issuing stock, stock quotations, and the dynamics of prices.

15

1. Financial Structures and Instruments

Inforrriation about the global state of the economy and the markets, as expressed by several ‘composite’, ‘generalized’ indexes, is also of iniportance. As regards indicators of the ‘global’ state of the economy, the most well-known among them are the Dow Jones Averages and Indexes. There are four Averages: the DJIA (Doui Jones Industrial Average, taken over 30 industrial companies); the Dow Jones Transportation Average (over 20 air-carriers, railroad and transportation companies); the Dow Jones Utility Average (over 15 gas, electric, and power companies); the DOIUJones 65 Composite Average (over all the 65 firms included in the above three averages). We note that, say, the DJIA (the D o w ) , which is an indicator of the state of the ‘industrial’ part of the economy and is calculated on the basis of the data on 30 large ‘blue-chip’ companies, is not a mere arithmetic mean. The stock of corporations of higher market value has greater weight in the composite index, so that large changes in the prices of few companies can considerably change the index as a whole [310]. ( O n the backgrounds of the Dow.In 1883, Charles H. Dow began to draw up tables of the average closing rates of nine railroad and two manufacturing companies. These lists gained strong popularity and paved way to “The Wall Street Journal” founded (1889) by Dow and his partner Edward H. Jones. See, e.g., [310].) Alongside the Dow Jones indexes, the following indicators are also widely used: Standard & Poor’s 500 (S&P500) Index, the NYSE Composite Index, the NASDAQ Composite Index, the AMEX Market Value Index, Value Line Index, Russel 2000 Stock Index, Wilshire 5000 Equity Index, Standard & Poor’s 500 Index is (by conrast with the DJ) calculated on the basis of data about many companies (500 in all= 400 industrial companies 20 transportation companies 40 utilities 40 financial companies); the NYSE Index comprises the stock of all firms listed at the New York Stock Exchange, and so on. ( O n the backgrounds of Standard & Poor’s. Henry Varnum Poor started publishing yearly issues of “Poor’s Manual of Corporate Statistics” in 1860, more than 20 years before the Dow Jones & Company’s first publication of the daily averages of closing rates. In 1941, Poor’s Finance Services merged with Standard Statistic Company, another leader in the collection and publishing of financial information. The result of this union, Standard & Poor’s Corporation became a major information service arid publisher of financial statistics (see, for instance, [310]):) Besides various pulications one can obtain information on instantaneous ‘bid’ and ‘ask’ prices of stock from the NASDAQ electronic system (covering shares of

+

+

+

Chapter I. Main Concepts, Structures, arid Instruments

16

about 5 000 firms), Reuters, Bloomberg, Knight Ridder, Telerate. Brokers can at any time learn about prices through dealers and get in direct touch with the ones whose prices seem more attractive. In view of economic globalization, it is important that one knows not only about the positions of national companies, but also about foreign ones. Such data are also available in corresporitling publications. (Note that in the standard American nomenclature the attributes ‘World’, ‘Worldwide’, or ‘Global’ relate to all markets, including tlie USA, while ‘International’ means only foreign markets, outside the USA). One can learii about the activities at 16 major international exchanges from ‘‘The Wall Street Journal” daily, which publishes in its “Stock Market Indexes” section the closing composite indexes of these exchanges anti their realtive and absolute changes from the prcvious day. As everybody knows from numerous publications, even in mainstream newspapers, stock prices arid tlie values of various financial indexes are permanently changing in a tricky, chaotic way. We depict the changes in the DJIA as an example:

I

FIGURE 3 . Dynarriics of the DJIA (Dow Jones Industrial Average). On Oct,ober 19, 1987, the day of crash, the DJIA fell by 508 points

111 the next chart, Fig. 4, we plot the da.ily changes S = (S,) of the S&P500 Index during 1982-88. It, will be clear from what follows that, with an eye to the analysis of the ‘stochastic’components of indexes, it is more convenient to consider the quantities, fir, = In

S,, , rather than the S,,themselves. We can intcrprete S,,, -1

~

thest: quantities as ‘returns’ or ‘logarithmic returns’ (see Chapter 11,’ la). Their behavior is ~ n o r c‘uniform’than that of S = (S,). We plot the corresponding graph of the values of h = (h7,)in Fig. 5 .

1. Financial Structures and Instruments

17

350 --

300 --

250 -.

200 -.

1982

1983

1984

1985

1986

1987

1988

1987

1988

FIGURE 4. T h e S&P500 Index in 1982-88

-0.05 -0.10

-0.15 -0.20

I I

I

t+

1982

1983

1984

1985

1986

FIGURE 5. Daily values of h, = In __ sn for the S&P500 Index; based on -1 the data in Fig. 4

sn

Chapter I. Main Concepts, Structures, and Instruments

18

0.530

1

0.515

t, 0

3

6

9

12

15

18

21

24

FIGURE 6. Round-the-clock dynamics of the averaged (between the ‘bid’ arid ‘ask’ prices) DEM/USD cross rate, August 19, 1993 (The label 0 corresponds to 0:00 GMT)

The dip at the end of 1987, clearly seen in Figs. 3 and 4 is related to the famous October crash, when stock prices fell abruptly and the investors, afraid of losing everything, rushed to sell. Mass sales of shares provoked an ever growing emotional and psychological mayhem and resulted in an avalanche of selling bids. For example, about 300 ni shares changed hands at the NYSE during entire January, 1987, while there were 604m shares on offer on the day of crash, October 19, and this number increased to 608 m on October 20. The opening price of an AT&T share on the day of crash was $30 and the closing price was $23;, so that the corporation lost 21.2% of its market value. On the whole, the D,JIA was 22.6% lower on October 19, 1987, than on December 31, 1986, which means $500 bn in absolute figures. During another well-known October crash, the one of 1929, the turnover of the NYSE was 12.9m shares on October 24 (before the crash) and 16.4 m on the day of crash. Accordingly, DJIA on October 29 was 12.8% lower than on December 31, 1928, which comprised $14 bn in absolute figures. We supplement Figs. 3-5, in which one clearly sees the vacillations of the DJIA and the S&P500 Index over a period of several years, by Fig. 6 depicting the behavior of the DEM/USD cross rate during o n e biisiness d a y (namely, Thursday, October 19, 1993).

1. Financial Structures a n d Instruments

19

The first attempt towards a niathernatical description of the evolution of stock prices S = (St)t>o (on the Paris market) on the basis of probabilistic concepts was made by Louis Bachelier (11.03.1870 28.04.1946) in his thesis “Thhorie tie la spkculation” [ 121 published in Annales Scientifiqucs de 1’Ecole Norrnale Suphrieure, vol. 17, 1900, pp. 21~-86,where be proposed to regard S = (St)t>o as a random (stochastic) process. Analyzing the ‘experimental data’ on the prices Sia,, t = 0, A, 2A,. . . (registered at time intkrvals A) he observed that the differences SiA)-StPA (A) had averages zero (in the statistical scnse) and fluctuations

-

S~~AI

The same properties has, e.g., a random walk Sin,, t

of order

6.

0, A, 2A, . . . with

are itlentically distributed random variables taking two values, where tlic (f) * a f i , with probability Passing to the limit as A + 0 (in the corresponding probabilistic sense) we arrive at the random process

i.

St = so

+ (Twt, t 3 0 ,

whrrc W = (Wt)t>ois just a B r o w n i a n m o t i o n introduced by Bachelier (or a W k n e r process, as it is called after N. Wiener who developed [476] in 1923 a rigorous mathematical theory of this motion; see also Chapter 111, 5 3a). Starting from a Brownian motion, Bachelier derived the forniiila C, = EfT for the expectation (here f~ = ( S , - K ) + ) ,which from the modern viewpoint gives one (under the assumption that the bank interest rate r is equal to zero and Bo = 1) the value of the reasonable (fuir) price (premium) that a buyer of a standard call option must pay to its writer who undertakes to sell stock at the maturity date T at the strike (exercise) price K (see 5 1c below). (If S, > K , then the option buyer gets the profit equal to (ST- K ) - C, because he can buy stock at the price K and sell it promptly at t,he higher price ST,while if S, < K , then the buyer merely does not show thc option for exercise and his losses are equal to the paid premium CT.) Bachelier’s formrile (see Chapter VIII, 3 la)

where

20

Cliapter I. Main Concepts, Strurtures, anti Iristrumerits

was iri effect a forerunner of the famous Black-Scholes f o r m u l a for the rational price of a standard call option in the case where S = ( S t ) can be described by a geometric (or economic) Brownian rnotiori

where W = (Wt)taois a usual Brownian niotion. Under the assumption that the bank interest rate 7' is equal to zero and Bo = I , the Black-Scholes formula gives one the following formula for the rational price @T = € ( S T - K ) + of a standard call option:

where

(As regards the gencral Black-Sclioles formula for r # 0 and Bo ter VIII, 5 Ib). It is worth noting that by ( 7 ) ,if K = SO,then

> 0, see Chap-

which gives m e an itlea of the increase of the rational option price with maturity time T . The problem of an adequate description of the dynamics of various financial indicators S = (St)t>oof stock price kind, is far from being exhausted, and it is the subject of inany studies in probability theory and statistics (we concentrate on these issues in Chiipters 11, 111, and IV). We have already explained (see subsection 4) that a similar (but mayhe even more complicated) problem arises in connection with the tlescription of thc stochastic evolutiori of prices in bond markets P = { P(t, T ) } o < ~ which ~ T , are regarded as random processes with fixed conditions at the 'right-liand end-point' (i.e., for t = 3"). (We discuss these questions below, in Chapter 111.)

3 lc.

Market of Derivatives. Financial Instruments

1. A sharp rise in the interest to securities markets throughout the world goes back to the early 1970s. It would be appropriak to try t,o understand the course of events that gave impetus t o shifts in economy and, in particular, securities markets. In the 6Os, the financial markets--- both the (capital) markets of long-term securitics and the (money) markets of short-term securities-were featured by extremely low rdatility, t,he irit,crest rates were very stable, and the exchange rates fixed. From

1. Financial Structures and Instruments

21

1934 to 1971, tlic USA were adhering to the policy of buying and selling gold at a fixed price of $35 per ounce (= 28.25 grams). The USA dollar was considered t,o be a gold equivalent, ‘as good as gold’. Thus, the price of gold was imposed from out,sid(:, iiot tlet,erniirietl by market forces. This market sit,iiation restricted investors’ initiative and hindered the dcvelopment of new t,echnology in finance. On the other hand, several events in the 1960-70s induced considerable structural changes and the growth of volatility on financial markets. We indicate here the most irnportant, of these events (see also 3 lb.2). 1) The transition from the policy of fixed cross rates (pursued by several groups of countries) to rates freely ‘floating’ (within some ‘band’), spurred by the acute financial and currency crisis of 1973, which affected the American dollar, the German mark, and the Japanese yen. This transition has set, in particular, a n important and interesting problem of the optimal timing and the magnitude of i n t e r v e n t i o n s by a ccntral bank. 2) The tlevaluatiori of the dollar (against gold): in 1971, Nixon’s administration gave up the policy of fixing the price of gold at $ 3 5 per ounce, arid gold shot up: its price was $570 per ounce in 1980, fell to $308 in 1984, and is fluctuating mainly in the interval $300 -$400 since then. 3) The global oil crisis provoked by the policy of the OPEC, which came forward as a major price-maker in the oil market. 4) A decline in stock tmde. (The decline in the USA at this time was larger in real ternis than during the Great Depression of the 1930s.) At that point the old ‘rule of thumb’ and simple regression models became absolutely inadequate to the state of economy and finances. And indeed, the markets responded promptly to new opportunities opened before investors, who gained much more space for speculations. Option and bond futures exchanges sprang up in many places. The first specialist exchange for trade in standard option contracts, the Chicago Board Option Exchange (CBOE), was operied in 1973. This is how the investors responded to the new, more promising opportiinities: 911 contracts on call options (on stock of 16 firms) were sold the opening clay, April 26. A year later, the daily turnover reached more than 20000 contracts, it grew to 100000 three years later, and to 700000 in 1987. Bearing in iriind that one contract inearis a package of 100 shares, we see that the turnover in 1987 was 70m shares each day; see [35]. The same year, 1987, the daily turnover at the NYSE (New York Stock Exchange) was 190 m shares. 1973 was special riot only because the first proper exchange for standard option corit,ract,swas opened. Two papers piiblished that year, The p r i c i n g of o p t i o n s arid c o r y m t e I%abilit%ets by F. Black and M. Scholes ([44]) and The theory.of ,rational option prici,riy by R. C. Merton ([357]), brought a genuine revolution in pricing niethods. It would be difficult to name other theoretical works in the finances literature that could rnatch these two in the speed with which they found applications

22

Chapter I. Main Concepts, Structures, and Instruments

in the practice and became a source of inspiration for multiple studies of more complcx options and other types of derivatives. 2. The most prominent arriorig the derivatives considered in financial engineering are opt,ion arid futures contracts. It is common knowledge that both are highly risky; ricvcrtheless they (anti their various combinations) are used successfully not merely to draw (speculative) profits but also as a protection against drastic changes in prices. Ail Option is a security (contract) issued by a firm, a bank, another financial company and giving its buyer the right to buy or sell something of value (a share, a bond, currency, etc.) on specified ternis at a fixed instant or during a certain period of time in the future. By contrast with option contracts giving a right to buy or sell, a Futures contract is a commitmen,t t o buy or sell something of value (e.g., gold, cereals, foreign currency) at a preset instant of time in the future at a (futures) price fixed at the moment of signing the deal. (One can find a n indispensable source of statistical data on all securities, as well as options and futures, in an American financial weekly “Barron’s”.) ( 2 ) Futures are of practical interest both to sellers and buyers of various goods. For exaniple, a clever farmer worried about selling the future crop at a ‘good’ price and afraid of a drastic downturn in prices would prefer to make a ‘favorable’ agrecrncrit with a miller (baker) on the delivery of (not yet grown) grain instead of waiting the grain to ripe and selling it a t the (who knows what!) market price of the day. Accordingly, the miller (baker) is also interested in the purchase of grain at a ‘suitable’ price and is seeking to forestall large rises in grain prices possible in the future. In the end, both share the same objective of mi,nimiz.ing the risks due to the uncertainty of the future market prices.Thus, a futures contract is a form of a n agreement that can in a way be convenient for both sides. Before a substantial discussion of futures contracts it seems appropriate to consider a related form of agreement: so-called forward contracts. p) As in the case of futures contracts, a Forward contract is also a n agreement to deliver (or buy) something in the future at a specified (forward) price. The difference between futures and forward contracts is as follows: while forwards are usually sold without intermediaries, futures are traded at a specialized exchange, where the seller arid the buyer do not necessarily know each other and where a special-purpose settlement system makes a subsequent renouncement of the contract uneconomical. Usually, thc person willing to buy is said to ‘hold a long position’, while the one undertaking the delivery in question is holding a ‘short position’. A cardinal question of forward and futures contracts is that of the preset ‘for-

23

1. Financial Structures and Instruments

ward’ (‘futures’) price, which, in effect, can turn out distinct from the market price at the time of delivery. Roughly, transactions involving forwards run as follows: money seller

buyer goods

Here we understand ‘goods’ in a broad sense. For example, this can be currency. If you are interested in trading, say, US dollars for Swiss franks, then the quotations that you find may look as follows: Current price 1 USD = 1.20 CHF, (Le., we can buy 1.20 CHF for $1); 30-day forward 1 USD = 1.19 CHF; 90-day forward 1 USD = 1.18 CHF; 180-day forward 1 USD = 1.17 CHF.

This picture is typical in that you tend to get less for $ 1 with the increase of delivery time, and therefore if you need C H F 10000 in 6 months’ time, then you must pay 10 000 = $8547. 1.17 However, if you need CHF 10000 now, then it costs you

10 000 1.20

-= $ 8 333.

Clearly, the actual, market price of C H F 10000 can in 6 months be different from $8547. It can be less or more, depending on the CHF/USD cross rate 6 months later. (On the background of futures. According to N. Apostolu’s “Keys t o investing in options and futures” (Barron’s Educational Series, 1991), “the first organized commodity exchange in the United States was the Chicago Board of Trade (CBOT), founded in Chicago in 1848. This exchange was originally intended as a central market for the conduct of cash grain business, and it was not until 1865 that the first futures transaction was performed there. Today, the Chicago Board of Trade, with nearly half of all contracts traded in the United States, is the largest futures exchange in the world. Financial futures were not introduced until the 1970s. I n 1976, the International Monetary Market (IMM), a subsidiary of the Chicago Mercantile Exchange (CME), began the 90-day Treasury bill futures contract. The following year, the

24

Chapter I. Main Concepts, Structures, and Instruments

CBOT initiated the Treasury bond futures contract. In 1981, the IMM created the Eurodollar fiitiircs contract. A iiiiijor financial futures milestone was reached in 1982, when the Kansas City Board of Trade introduced a stock index futures contract based upon the Value Line Stock Index. This offering was followed in short order by the introduction of the S&P500 Index futures contract on the Chicago Mercantile Exchange and the New York Stock Exchange Composite Index traded on the New York Futures Exchange. All of these contract,s provide for cash delivery rather than delivery of securities. Trading volunie on the futures exchanges has surged in the last three decadesfrom 3.9 inillion contracts in 1960 to 267.4 million contracts in 1989. In their short history, finaric:ial futures have become the doniinant factor in the futures markets. In 1989, 60 percent of all fiitiires contracts traded were financial futures. By far the most actively traded futures contract is the Treasury bond contract traded on the CBOT.”) y) A forward contract, as already mentioned, is a n agreement between two sides and, in principle, there exists a risk of its potential violation. More than that, it is often difficult to find a. supplier of the goods you need or, conversely, a n interested buyer. Apparently, this is what has brought into being futures contracts traded at excharigcs equipped with special settlement mechanisms, which in general can be described as follows. Imagine that on March 1, you instruct your broker to buy some amount of wheat by, say, October 1 (and specify a prospective price). The broker passes your request to a produce exchange, which forwards it to a trader. The trader looks for a suitable price and, if successful, infornis the potential sellers of his wish t o buy a contract at this price. If another trader agrees, then the bargain is made. Otherwise the trader informs the broker and the latter informs you that there arc goods at higher prices, and you mist, take one or another decision. Assiinir: that, finally, the price of the contract is agreed upon and you, the buyer, keep the long position, while the seller keeps the short position. Now, t o make the contract effective each side must put certain amount, called the initial margin, into a special exchange account (this is usually 2%-10% of @o depending on the customers’ history). Next, the settlement mechanism comes into force. Settlements are usually carried out at the end of each day and run as follows. (Here @o is the (futures) pric:e, i.e., both sides agree that the wheat will be delivered on October 1, at the price Q,.) = $ 1 0 000 is the (futures) price of wheat Assunie that a t time t = 0 the price delivery a t the delivery time T = 3 , i.e., in three days. Assume also that a t the end of the next day ( t = 1) the market futures price of the delivery of wheat at the time T = 3 11cc:orncs $ 9 900. In that case the clearing house at the exchange transfers $100 (= 10 000 - 9 900) into the supplier’s (farmer’s) account. Thus, the farmer earns sonie profit and is now in effect left with a futures contract worth $ 9 900 in place of $10 000.

1. Financial Structures arid Instruments

25

Wc note that if the delivery were carried out at the end of thc first, (lay, at the new futures price $ 9 900, then the total revenues of the farmer would be precisely the futures price @O because $100 $ 9 900 = $10 000 = Qo. Of course, the clearing house writes these $100 off the buyer’s account, and he must additionally pay $100 into the margin account. (Sometimes, additional payments are necessary only if the margin account falls below a certain level-the maintenance margin) . Assunie that thc same occurs at time t = 2 (see Fig. 7). Then the farmer takes $200 onto his account and the same sum is written off the buyer’s account.

+

0

1

2

T=3

t

However, if tlie futures price of (instant) delivery (= the market price) becomes $10 100 at time t = 3 , then one must write $300(= 10 100 - 9 800) off the farmer’s account, arid therefore, all in all, he loses $100 (= 300 - 200), while the buyer gains $100. Note that if the buyer decides to buy wheat at time T = 3 on condition of instant delivery, then lie rniist pay $10 100 (= the instant,aneous market price). However, bcaring in mind that $100 have already been transferred t o his account, the delivery actually costs him 10 100 - 100 = $10 000, i.e., it is precisely the same as the contract price Cpo. The same holds for the farmer: the rnoriey actually obtained for the delivery of wheat is precisely $ IOOOO, because he is paid the market price $10 100 at time T = 3, which, combined with $100 written off his account, makes up precisely 10 000 $. This example clearly depicts the role of the clearing house as a ‘watchdog’ keeping track of all the transactions and the state of the margin account, which is all very essential for smooth execution of contracts. We have already noted that one of the cardinal problems of the theory of forward and futures contracts is to determine the ‘fair’ values of their prices.

26

Chapter I. Main Concepts, Structures, and Instruments

We show below, in Chapter VI, 3 l e , how the arguments based on the ‘absence of arbitrage’, conibiried with the martingale methods, enable one to derive formulas for the forward and futures prices of contracts with delivery time T sold a t time

t < T. 6 ) Options. The theory and practice of option contracts has its own concepts and special vocabulary, and it is reasonable to become acquainted with them already a t this early stage. This is all the more important as a large portion of the mathematical analysis of derivatives in what follows is related to options. This has several reasons. First, the mathematical theory of options is the most developed one, and this example is convenient for a study of the main principles of derivatives transactions and, in particular, pricing and hedging (i.e., protecting) strategies. Second, the actual number of options traded in the market runs into millions, therefore, there exists an impressive statistics, which is essential for the control of the quality of various probabilistic models of the evolution of option prices. Although options are long known as financial instruments (see, e.g., the book [346]), L. Bachelier [12] must be the first who, in 1900, gave a rigorous mathematical analysis of option prices and provided arguments in favor of investment in options. Moreover, as already noted, trade in options became institutionalized not so long ago, in 1973 (see 3 lc). ( O n the background of options. According to N . Apostolu’s “Keys to investing in options and futures” (Barron’s Educational Series, 1991), “since the creation of the Chicago Board Options Exchange (CBOE) in 1973, trading volume in stock options has grown remarkably. The listed option has become a practical investment vehicle for institutions and individuals seeking financial profit or protection. The CBOE is the world’s largest options marketplace and the nation’s second-largest securities exchange. Options are also traded on the American Stock Exchange (AMEX), the New York Stock Exchange (NYSE), the Pacific Stock Exchange (PSE), and the Philadelphia Stock Exchange (PHLX). Options are not limited to common stock. They are written on bonds, currencies, and various indexes. The CBOE trades options on listed and over-the-counter stocks, on Standard & Poor’s 100 and 500 market indexes, on U.S. Treasury bonds arid notes, on long-term and short-term interest rates, and on seven different foreign currencies.”) E ) For definiteness, we assume in our discussion of options that transactions can occur at time

n

= 0,1:. . . , N

and all the trade stops after the instant N . Assume also that we discuss options based on stock of value described by the random sequence

s = (srL)o<7Z
1. Financial Structures and Instruments

27

Using the standard vocabulary, we distinguish options of two kinds:

buyer’s options (call o p t i o m ) and seller’s options ( p u t o p t i o n s ) . A call option gives one the right t o buy. A put option gives one the right t o sell. From the standpoint of financial engineering it is important that these two financial instrumentas‘work in opposite directions’: as the gains from one of them increase, the gains from the other drop. This explains the widely used practice of diversification, of operating options in different classes, sometimes combined with other securities. As regards their exercise, options fall into two types, European and American. If an option can be shown for exercise only at some fixed time N , then we call N its maturiky time (option’s expiration date) and our option belongs to the European type. On the other hand, if the option can be shown for exercise at a n arbitrary (e.g., random) instant 7 N , then we call it a n option of A m e r i c a n type. In practice, most options are of American type. This gives the buyer more freedom, allowing him to choose the exercise time 7 . Note that these two types of options are equivalent in certain cases (in the sense that the optimal exercise time 7 of an American-type option is equal to N ; see Chapter VI, 5 5b and Chapter VIII, fj 3c below for detail). We now consider for definiteness a standard call option of European type with maturity time N . It can be characterized by the (strike or exercise) price K (preset at thr: iristarit of writing) at which the buyer will be able to buy, say, shares, whose market price SN at time N may be distinct (sometimes, considerably distinct) from K . If SN > K , then the situation is favorable for the buyer because, by the contract terms, he has the right to buy shares at the price K . Once bought, they can be inimediately sold at the market price SN.In this case his gains from this operation will be equal to SN - K . On the other hand, if SN turns out to be less than K , then the buyer’s right of purchase (at the price K ) is of no value because he will be able to buy shares at a lower price ( S N ) . All combined, we can say that the buyer’s gains fN at time N can be expressed by the formula

<

fN = (SN

-

K)+,

where a+ = max(a, 0). Of course, one must pay certain premium @N for the acquisition of this financial instrument, so that the n e t profit of the buyer of the call option is

Chdptcr I . Main Concepts, Structures, arid Instruments

28

i.e.,

(S, - K ) - CN for S, > K , -CN for SN < K . Accordingly, the writcr. guin is

and

CN for SN

< K.

Hence it, is clear that purchasing a call option one anticipates a vise of stock prices. (Note that, of course, the option premium CN depends not only on N , but also 011 K , arid, clc;trly, the less is K the larger must be CN.)

Remark. Therc exist special names for those acting on the assurriptiori of a ,rise or fall of soiric article. The dealers expecting prices to go up are called huh. A bull opens a long position expecting to sell with profit afterwards, when the markct goes up. Those dealers who expect thc market to move downwards are called bears. A bear tends to sell securities lie has (or even lias not-which is referred to as ‘short selling’). IIc hopes to close his short position by buying the traded iterris afterwards, at lower prices. The difference between the current price and the purchase price in the futurc will be his premium. Depending 011 the relation between the market price SO at t = 0 and K , options can be divided into tlircc classes: options bringin,g a gain (in-the-money), ones with gain zero (&the-money), and options bringing losses (out-of-money). In the case of a call option these classes correspond to the relations SO > K , So = K , and So < K , respectively. We must point out, straight away that there is a n enormous difference between the posit,ions of a buyer and a seller. Tlic buyer purchasing the option can sirriply wait till the maturity date N , watching if lie likes the dynamics of the prices S,, n 3 0. The position of the option writer is much more complicated because he must bear in mind his obligation to meet the terms of the contract, which requires him n 3 0, but to use all to not nic:rc:ly conternplate the changes in the prices S,,,, financial Incans available to him to build a portfolio of securities that ensures the final payrrient of (SN - K )+ . The following two questions are central here: what is the ‘fair’ price CN of buying or selling an option and what must a seller do to carry out the‘contract. In the case of a standmrd p a t option of European type with maturity date N the price K at which the option buyer i s en,titled to sell stock (at time N ) is fixed. ~

1. Financial Structures and Instruments

29

Hence if tlic real price of the stock at time N is SN and SN < K , then selling it at the price K brings K - SN in revenues. His n e t profit, taking into account the premium IPN for the purchase of the option, say, is equal to

( K - S N ) - IPN. On the other hand, if SN > K , i.c., the preset price is less than the market price, then tfliere is no sense in showing the option for exercise. Hence the n e t profit of the b u y e r of a, pwt option is ( K - SN)’ - PN. As in the case of a call option, here we can also ask about a ‘fair’ price PN suitable for both writcr and buyer. 71) As ail illustration we consider a n example of a call option. Assume that we buy 10 contracts on stock. As a rule, each contract involves 100 shares, so we discuss a purchase of 1000 shares. Assume that the market price 5’0of a share is 30 (American dollars, say), K = 35, N = 2, and the premium for the total of 10 contracts is ( $ ) 250. Furthcr, let the market price S2 a t time n = 2 be $40. In this case the option is shown for exercise and the corresponding net profit is positive: (40

-

35) x 1000 - 250

(2) 4 750.

Howevcr, if the market price 5’2 is $35.1, then we again show the option for exercise (since S2 > K = $35), but the corresponding net profit is now negative: (35.1 - 35) x 1000 - 250 = ($)

-

150.

It is clear that our profit is zero if (S2 - K ) . 1000 = ($) 250,

i.e., S2 = $35.25 (because K = $35). Thus, each time the share price S2 drops below $35.25, the buyer of the call option takcs losses. Assume now that we consider a n American-type call option, which gives us the right to exercise it a t time T = 1 or T = 2. Imagine that the share price rises sharply at tlic instant T = 1, so that 5’1= $50. Then the buyer of the call option can show it for exercise at this instant and pocket a huge net profit of (50 - 35) . 1000 - 250 = 15 000 - 250 = ($) 14 750. It is however clear that the premium for such a n option must be considerably larger than $250 t)ecaiise ‘greater opportunities should come niorc expensive’. The actual prices of American-type options are indeed higher than those of European ones. (See Chapter VI below.)

30

Chapter I . Main Concepts, Structures, and Instruments

We have considered the above example from the buyer’s standpoint. We now turn to the writer’s position. In principle, he has two options: to sell stock that he has already (‘writing covered stock’) or to sell stock he has not at the moment (‘writing naked call’). The latter is very risky and can be literally ruinous: if the option is indeed shown for exercise (provided S2 > K ) , then, to meet the terms of the contract, the seller must actiially buy stock and sell it to the buyer a t the price K. If, e.g., S2 = $40, then he must pay $ 4 0 000 for 1000 shares. His premium was only $250, therefore, his total losses are 40 000

-

35 000 - 250 = ($) 4 750.

It must be noted that the writer’s net profit is in both cases a t most $250. This is a purely speculative profit made ‘from swings of stock prices’ and (in the case of ‘naked stocks writing’) in a rather risky way. We can say that here the ‘writer’s profit is his risk premium’.

3. In practice, ‘large’ investors with big financial potentials reduce their risks by an extensive use of diversification, hedging, investing funds in most various securities (stock, bonds, options, . . . ), commodities, and so on. Very interesting and instructive in this respect is G. Soros’s book [451], where he repeatedly describes (see, for instance, the tables in 5 5 11, 13 of [451]) the day-to-day dynamics (in 1968-1993) of the securities portfolio of his Quantum Fund (which contains various kinds of financial assets). For example, on August 4, 1968, this portfolio included stock, bonds, arid various commodities ([451, p. 2431). From the standpoint of financial engineering Soros wrote a masterpiece of a handbook for those willing to be active players in the securities market. 4. In our Considerations of European call options, we have already seen that they can be charactcrized by 1) their maturity date N . 2) the pay-off function fN. For a standard call option we have

For a standard call option with aftereffect we have

where KN = a . rnin(S0, S1,. . . , SN);a = const. For an arithrrietic Asian, call option we have N

1. Financial Structures and Instruments

31

We must note that the quantity K entering, for instance, the pay-off function (SN- K ) + of a standard call option, its strike price, is usually close to So. As a rule, one never writes options with large disparity between SO and K . Pay-off fiirictions for put options are as follows: for a standard p u t option we have fN =

for a standurd p u t option with aftereffect we have f N = (KN

-

KN = a

sN)+,

. max(S0, S1,. . . , S N ) ;u = corist

for an arith,metic A s i a n p u t option we have

There exist inany types of options, some of which have rather exotic names (see, for instance, [414] and below, Chapter VIII, 5 4a). We also discuss several types of option-based strategies (combinations, spreads, etc.) in Chapter VI, 3 4e. One attraction of optlions for a buyer is that they are not very expensive, although the comniission can be considerable. To give a n idea of the calculation of the pyice of a n optiosn (the premium for the option, the non-reimbursable payment for its purchasc), we consider the following, slightly idealized, situation Assume that thc stock price S,, 0 n N , satisfies the relation

< < s ,= so + (El + + r n ) , ' ' '

where So > N is a n integer and (&) is a sequence of independent identically distributed random variables with distribution 1

P(
<

Then, of course, S,,, > 0, 0 6 'ri N . Assume also that we have at our disposal a bank account containing an amount B = ( B n ) o G n Gwhere ~ , B, = 1 (i.e., the interest rate T = 0 and Bo = 1). We consider now a standard European call option with pay-off f N = ( S N - K ) + . We claim that the rational (or fair, mutually appropriate) price CN of such a n option can bc expressed as follows:

i.e., the size of the pre~niurnis equal to the average gain of the hiyer.

32

Chapter I. Main Concepts, Structures, and Iristriimerits

We enlarge on the formal definition of Q,1 and the methods of its calculation below (see Chapter VI), while here we can substantiate the formula @ N = E(SN-K)+ as follows. Assume that the writer’s ask price EN is larger than E(SN - K ) + arid the buyer agrees to -purchase the option a t this price. We claim that the writer has a riskless profit of @N - @N in this case. In fact, the buyer acknowledges that the price must give the writer a possibility to meet the terms of the contract. It is clear that this price cannot be too low. But, understandably, the buyer also would not overpay: he would rather buy a t the lowest price enabling the writer to meet the contract terms. In other words, we must show that the option writer can use the premium CN = E(SN - K ) + to meet the contract terms. For simplicity, we set N = 1 and K = So. Then @1= EE; = We now describe the ways in which the writer can operate with this premium in the securities market. ~e represent as follows:

i.

4

4

i)

Setting Xo = Po . 1 + yo . So (= we can call this premium the (initial) capital of the writer; a part, of it (00)is put into a bank account and another part is invested in (yo) shares. The fact that Po is negative in our case means that the seller merely oiie,rdrnws his account, which, of course, must be repaid. The pair ([&, yo) forms t,he so-called writer’s i n v e s t m e n t portfolio a t time ri = 0. What is this portfolio worth a t the time n = l ? Denoting this amount by X1 we obtain

1 for 0 for

=

1,

= -1.

Since

it is obvious that

XI = f l

(=

(S1-K ) + ) .

In other terms, the portfolio (/3o,yo) ensures that the amount X1 is precisely equal to the pay-off f l , which enables the writer to meet the terms of the contract and repay his debt.

1. Financial Structures and Instruments

For if

<1 =

1, then he gets ;(So

33

+ 1) for the stock. Since

(T--), 1

-1( S o + l ) = l + So

2

2

this nioriey is sufficient to repay the debt of

(Sl- K ) + = (Sl - SO)+ =q= 1.

(48,

and pay the buyer a n amount

-

On the other hand, if(1 = -1, then he obtains ;(SO-1) for the stock. The buyer does not exercise tlie option in this case (because S1 = SO+ E l = SO- 1 < So = K ) , and therefore the writer must merely repay his debt ;(So - l ) , which is precisely equal to tlie rrioricy obtained for the stock (we now have (1 = -1). Hence, if the writer sets > C1 = E(S1 - K ) + , then upon meeting all the terms of the corit,rac:t he gets a riskless profit of - C1. We now claim that if the premium C1 is smaller than Q11 (= ;), then the writer cannot meet the terms (without losses). Indeed, choosing a portfolio (PoBo, 70)we obtain I

xo = Po + roso and

x1 = Po + ro(S0 + E l )

=

xo + roE1.

If 41 = 1,,then, under the terms of the contract, the writer must pay an amount of 1 to the buyer and, additionally, pays -Po. i.e., lie must obtain

for the stock, w l d c if

= -1, then he iniist get

for the stock. All in all, we must set

And yct, for tlicse values of the parameters we have equality XO= l j o 70So is impossible for X O <

+

i.

+ roSo = i, therefore the

Chapter I. Main Concepts, Structures, and Instruments

34

Thus, we have established the formula CN = E(SN

-

K ) + for N = 1 arid

SO = K . One can prove it in the general case by similar arguments. We are not, doing that here because we shall deduce the same formula below (in Chapter VI) from general considerations. Note that if SO = K , then

and

by the Central limit theorem. Hence the option premium increases as with time N . This is perfectly consistent with the result following (in the case of SO= K ) from Bachelier’s formula for CT a t the end of 5 lb:

C T = E (for

= 0,

Bo = 1, (T = I , and So = K ) .

5 . We have already mentioned that there is a great variety of option kinds on the niarket,. Options o n indexes are quite common; for instance, options on the S&P100 and the S&P500 Indexes traded on the CBOE. (Options on S&P100 are of American type, while ones on S&P500 are of European type.) In view of the large volatility, the rriat,iirity time of these options is fairly short. See also Chapter VIII, 4a. In the case of options on futures the role of the prices (S,) is played by the futures prices (an),whereas in the case of options on some index it is the value of this index that plays the same role of the prices ( S t ) .

2. Financial Markets under Uncertainty.

Classical Theories of the Dynamics of Financial Indexes, their Critics and Revision. Neoclassical Theories

Here are just several questions that arise naturally once one has got acquainted with tlie theory and practice of financial markets: how (lo financial markets operate under uncertainty? how are tjhe prices set and how can their dynamics in time be described? what, concepts and theories must one use in calculations? can the future development of the prices be predicted? what are the risks of the use of some or other financial instruments?

In our descriptions of the price dynamics and in pricing derivative financial instruments we shall take the standpoint of a niarket without opportunities for arbitrage. Mothematically, this transparent economic assuniption means that there exists a so-callcd ‘niartingale’ (risk-neutral) probability measure such that the (discounted) prices are niartingales with respect to it. This, in turn, gives one a possibility to use the well-developed machinery of stochastic calculus for the study of the evolution of prices arid for various calculations. Although we do not intend to give here a detailed account of the existing theories and concepts relating to financial markets, we shall discuss those that are closer to our approach, where the main (probabilistic) stress is put on tlie applications of stochastic calculus and statistics in financial mathematics and engineering. (We point out several handbooks and monographs: [79], [83],[112], [117], [151], [240], [268], [284], [332]-[334], [387], [460], where one can firid the discussion of most diverse aspects of financial markets, economical theories and concepts, theories designed for the conditions of certainty or uncertainty, equilibrium models, optimality, utility, portfolios, risks, financial decisions, dividends, derivative securities, etc.) . We note briefly that back in the 1920s, considered to be the birth time of financial theory, its central interests were mainly concentrated around the problems of

36

C h a p k r I. Main Concepts, Structures, and Instruments

managing and raising funds, while its ‘advanced niathernatics’ was virtually reduced to the calculation of compound interest. The further development proceeded along two lines: under the assumption of a complete certainty (as regards the prices, the demand, etc.) and under the assumption of uncertainty. Decisive for the first approach were the results of I. Fisher [159] and F. Modigliani and M. Millcr [356]. [350] who considered the issue of optimal solutions that niust be taken by individual investors and corporations, respectively. Mathematically, this reduces to the problem of maximization under constraints for functions of several variables. In the second field, we must first of all single out H. Markowitz [268] (1952) and M. Kendall [269] (1953). Markowitz’s paper, which established a basis of i n v e s t m e n t portfolio theory, concentrated on the optirnization of i n v e s t m e n t decisions under uncertinity. The corresponding probabilistic method, the so-called mean-variance analysis, revealed a central role of the covariance of prices, which is a key ingredient determining the (unsystematic) risks of the investriient portfolio in question. It was after that paper that one became fully aware of the importance of diversification (in making up a portfolio) to the reduction of unsystematic risks. This idea influenced later two classical theories developed by W. Sharpe [433] (1964) and S. Ross, [412] (1976), CAPM-Capital A s s e t Pricing Model [433] and APT--Arhitrage Pricing Theory [412], These theories describe the components making up the yield from securities (for example, stock), explain their dependence on the state of the ‘global market‘ of this kind of securities (CAPM-theory) and the factors influencing this yield ( APT-theory), arid describe the concepts that must underlie financial calculations. We discuss the central principles of Markowitx’s theory, CAPM, and APT below, in 3 5 2b---2d. It is already clear from this brief exposition that all the three theories are related to the reduction of risks in financial markets. In the discussion of risks,c it should be noted that financial theory usually distinguishes two types of them:

m a t i c risks, which can be reduced by a diversification (they are also called diversifiable, residual, specific, idiosyncratic risks, etc.): i.e., risks that investor can influerice by his actions, and systematic, or market risks in the proper sense of the term (undiversifiable risks). Ail example of a systematic risk is the one related to the stochastic nature of interest rates or stock indexes, which a (‘small’) investor cannot change on his own. This does riot imply, of course, that one canriot withstand such risks. It is in effect to ‘All sorts of risks, incliiding insurance risks (see 3 3 3a; 3b below)

37

2 . Financial Markets under Uncertainty

control possible systematic risks, to work out recommendations for rational financial solutions, to shelter from big and catastrophic risks (e.g., in insurance) that one develops (fairly complicated) systems for the collection and processing of statistical data, arid for the prediction of the development of market prices. This is in fact the ruisolz d ‘Etre of derivative financial instruments, such as futures contracts, options, combinations, spreads, etc. This is the purpose of hed~qing,more complicated techniques than diversification, which were developed for these instruments, take into account random changes of the future prices, and aim a t the reduction of the risks of possible unfavorable effects of these changes. (One can find a detailed discussion of hedging arid tlie corresponding pricing theory in Chapter VI.) M. Keridall’s lecture [269] (1953) at a session of the Royal Statistical Society (London) concentrates around another question, which is more basic in a certain sense: how do m a r k e t prices behave, what stochastic processes can be used t o describe their dynamics? The questions posed by that important work had ultimately brought about, Eficient Capital Market T h e o r y (the ECM-theory), which we discuss in tlie next section, and its many refinements and generalizations.

5 2a.

Random Walk Conjecture and Concept of Efficient Market

1. In several studies made in thc 1930s their authors carried out empirical analysis of various financial characteristics in a n attempt to answer the notorious question: is the m o r i e m e n t of prices, values, and so on predictable? These papers were mostly writt,eii by statisticians; we single out among them A. Cowles [84](1933) and [85] (1944), H. Working [480] (1934), and A. Cowles and H. Jones [86] (1937). A. Cowles considered data from stockmarkets, while H. Working discussed commodity prices. Although onc could find in these papers plenty of statistical d a t a and also interesting---and unexpected-conclusions

sn

that the increments h, = In - of

S,-1

tlie logarithms of the prices S,,, 71 3 1, must be iridependerit, neither economists nor practitioners paid niiich attention to this research. As iiot,ed iri [35; p. 931, one probable explanation can be t h a t , first? economists regartled the price dynamics as a ‘sideshow’ of the economy and, second, there were riot so ~ n a n yeconomists at that tinie who had a suitable background in rnatherriatics and mastered statistical techniques. As regards the practitioners, the coricliision made by these researchers that the 1, H,), = / L I . . . hT,,:had the nature of a r a n d o m wnlk sequence ( H 7 a ) r L 2 where (i.e., was the sum of independent random variables) was in contradiction with the prevailing idea of the time that the prices had their r y t h m s , cycles, trends, . . . , and one coiild predict price m o v e m e n t s by revealing t h e m . There had been in fact no important research in this field since theb arid until 1953, when M. Keiidall published the already mentioned paper [269],which opened up the modern era in the research of the evolution of financial indicators.

+

+

Chapter I. Main Concepts, Structures, and Instruments

38

The starting point of Kendall’s analysis was his intention to detect cycles in the behavior of the prices of stock and commodities. Analyzing factual data (the weekly records of the prices of 19 stocks in 1928-1938, the monthly average wheat prices on the Chicago market in 1883-1934, arid the cotton prices a t the New York Mercantile Exchange in 1816-1951) he, to his own surprise, could find no rhythms, trends, or cycles. More t h a n that, he concluded that these series of data look as if ‘‘ . . . the De,mo,n of Chance drew a rand0.m number . . . and added it to the current price to determine the next . . . price”. In other words, the logarithms of the prices

S

=

( S T Lappear ) to behave as a random walk: setting h, = In

S,

= sOeHn,

n

sn

we obtain

~

Sn-

1

I,

where HT2is the sum of independent random variables h l , . . . , h,. Here it seeins appropriate to recall again (cf. 5 l b ) that in fact the first author (before A . Cowles and H. Working) to put forward the idea to use a random walk to describe the evolution of prices was L. Baclielier [la]. He conjectured that the prices dA) = (Ski))(not logarithms, by the way) change their values at instants A, 2 A , . . . so that

s i t ) = So f (A where values

+ (2A + . + ( k A ,

((in)are independent identically distributed random variables taking the with probability

i. Hence

El ,

t > 0 , and passing formally t o the limit L. Bachelier discovered that the limiting process S = ( S t ) t > ~where , St =

We have already mentioned t h a t , setting k =

lini S(A) (we must understand the limit here in a certain suitable probabilistic

A+O

[i]A

sense), had the following form:

st = so + aWt, where W = (Wt) (Wo = 0, EWt = 0 , EW: = t ) was a process that is usually called now a standard Brownian motion or a Wiener process: a process with indepe,ndent Gaussian (normal) increments and continuous trajectories. (See Chapter 111, $ 5 3a, 3b for greater detail.) 2. The interest to a more thorough study of the dynamics of financial indexes and the construction of various probabilistic models explaining the phenomena revealed by observations (such as, e.g., the cluster property) has significantly ’grown after M. Kendall’s paper. We single out two studies of the late 1950s: [405]by H. Roberts (1959) and [371] by M. F. M. Osborne (1959).

2 . Financial Markets under Uncertainty

39

Roberts’s paper, which drew upon the ideas of H. Working and M. Kendall, was addressed directly to practitioners and contained heuristic arguments in favor of the random walk con,jecture. “Brownian Motion in the Stock Market” by the astrophysicist Osborne grew up, by his own words (see [35; p. 103]), from the desire to test his physical and statistical techniques on such ‘earthly’ items as stock prices. Unacquainted with the works of L. Bachelier, H. Working, and M. Kendall as he was, M. F. M. Osborne came in effect t o the same conclusions; he pointed out, however (and this proved t o be important for the subsequent development), that these were the logarithms of the prices St that varied in accordance with the law of a Brownian motion (with drift), not the prices themselves (which were the main point of Bachelier’s analysis). This idea was later developed by P. Samuelson [420], who introduced into the theory and practice of finance a geometric (or, to use his own term, economic) Br0,wnia.n m o t i o n

where W = ( W t ) is a standard Brownian motion. 3. It would be exaggeration to say that the use of the r a n d o m walk conjecture for t,he description of the evolution of prices was accepted by economists then and there. But it was this conjecture that gave rise to the concept of rational (or, as one often says, efficient) market, whose initial destination was to provide arguments in favor of the use of probabilistic concepts and, in this context, t o demonstrate the plausibility of the rand0.m walk conjecture and of the (more general) martingale conjecture. In few words, ‘efficiency’ here means that the market responds rationally to new information. This implies that, on this market 1) corrections of prices are instantaneous and the market is always in ‘equilibrium’, the prices are ‘fair’ and leave the participants no room for arbitrage, i.e., for drawing profits from price differentials; 2) the dealers (traders, investors, etc.) are u n i f o r m in their interpretation of the obtained information and correct their decisions instantaneously as new information becomes available; 3) the participants are homogeneous in their goals; their actions are ‘collectively rational’. Incidentally, on the form,al side the concept of ‘efficiency’ must be considered as related to arid deperident upon the nature of the information flowing to the market and its participants. One usually distinguishes between three kinds of accessible data: 1” the past values of the prices; 2” the information of a more broad character than the prices contained in generally accesible sources (newspapers, bulletins, T V , etc.); 3” a11 conceiiiable information.

Chapter I. Main Concepts, Structures, and Iristriirnents

40

For a suitable formalization of our concept of ‘information’ we start from the assumption that the ‘uricertainty’ in the market can be described (see 5 l a in Chapter I1 for additional detail) as ‘randomness‘ interpreted in the context of s o m e probability space (R,$,P). As usual, here R = {w} is the space of elementary outcomes 9 is some a-algebra of subsets of R, P is a probability measure o n (R,9 ) .

It is worthwhile to endow the probability space (a, 9, P) with a p o w (filtration) F = ( 9 n ) n 2 of ~ a-subalgebras cFnsuch that 9T,L C .Yn C 9 for m 6 n. We interpret, the events in .Fnas the ‘information’ accessible to a n observer up to the instant

ri.

4. Reniark. In corinection with the concept of an ‘event; observable before time n‘

which is formally a subset in
If we register all the values wl,w2)w g , then, for instance, the set

is an ‘event’ since we cau say with confidence whether ‘w t A’ or ‘ w $ A’. However, if we cannot record the result w2 at time i = 2, i.e., we have no information on this value, then A is not a n ‘event’ any longer: knowing only wl and wg we cannot answer the question whether w = ( w l ; w 2 , w ~ )is in the set A . 5 . In stochastic calciiliis one usually calls spaces (0,9, IF = (cFn,), P) with distinguished flows F = (.FrL)of a-algebras filtered probability spaces. In the framework of financial mathematics we shall also call F = ( 9 n ) a n i n f o r m a t i o n jI0.w. Using this concept we can describe various forms of e&cient markets as follows. Assume that t,liere are three flows of a-algebras

IF’

=

(9,J, 1

F2 = p;).

IF3 = (9;)’

C 9: C 92, and we interpret each of the a-algebras 9:L as in (R,.%, P), where the data of t,he (i)th kind arriving at the instant n.

2. Financial Markets under Uncertainty

41

Following E. Fama [150] (1965) we shall say that a market is weakly efficient if there exists a discounting price B = (Bn),20 (usually, this is a risk-free bank account) and a probability measure locally equivalent to P (i.e., its restrictions P,, = P 1 9;to ,!FAare equivalent to P, = P 1 9,for each n 3 0) such that each price S = (S,) (of a financial instrunlent on this market) satisfies the following condition: the ratio h

h

is a P-martingale, i.c., the variables

s,, are .Fk-rneasurable and

-

B,,

If we have tlir inartingale property with respect to the information flow

F3 = (9:) then we have a stro,ngly efficient market, while in the case of IF2

=

(92)

we have a semi-.strongly efficient market. For simplicity, we shall assume throughout this section that B, = 1 and P = P. Before a discussion of these definitions we point out the following relation between the classes of martingales A1 = ,A(lF1).AX2 = A(lF2)> and AX3 = .A@‘’) with respect to the flows F’, F2,and IF3: A 3

c AX2 c .A1

In fact, the inclusion S = ( S , l ) n 2 ~t At2 (for one) means that the S, are 9:-measurable arid E(S,,+l 1.9;)= S,,.Hence, by the ‘telescopic‘ property of conditional expectations

E(S,+1

I S,t) = E ( E ( S n + l I m 15;)

.9k

and since the S,,, are .Fk-nieasurable (we recall that is generated by all prices up to tiiiic n , including the variables sk, k n ) ,it follows that E(S,+1 19;) = i.e., E A’. If €1, € 2 , . . . is a sequence of independent random variables such that E l & < 00, EEk = 0, k 3 1, % ,: = ( ~ ( € 1 , .. . ,&), 9 6 = (0, and 9: 9;, then, evidently, the sequence S = (S,l),g), where S, = €1 . . . + En, for n 3 1 and So = 0, is a martirigale wit,li respect t o = ( . ~ $ , ) ~and ,~o

<

s

+

a},

s,,

c

c@

Herice the sequence S = (S,L)n20is cltarly a martingale of class A iif for each n the variables C I , + 1 are independent of 9;(in the sense that for each Bore1 set A

42

Chapter I. Main Concepts, Structures, and Instruments

the event {&+I E A } is independent of the events in 9ft).Thus, if we treat [,+I as ‘entirely fresh data’ relative to .9:, then S is in the class A3. It is important for what follows that if X = (X,) is a martingale with respect to the filtration F = (9,and ) X , = xl f . . . x, with xo = 0, then x = (x,) is a martingale diflerence, i.e.,

+

It follows from the last property that, provided that EIx,I2

< 00 for n 3 1, we have

for each n 3 0 and k 3 1 , i.e., the variables x = (x,) are uncorrelated. In other words, square-integrable martingales belong to the class of random sequences with orthogonal increments: E nx,nx,+,+= 0, where A X , = X , - XrL-l = x, and aXn+k = x,+,+. Denoting the class of such sequences by 0 1 2 (here the subscript 2 corresponds to ‘square integrability’) we obtain A; A; 2 A; OI2.

c

c

It follows from the above that, in the final analysis, the e f i c i e n c y of a market is nothing else but the martingal property of prices in it. One example is market where prices are ‘random walks’.d 6. Why do we believe the conjecture of the ‘martingale property’, which generalizes the ‘random walk’ conjecture and is inherent in the concept of ‘efficient market to be quite a natural one? There can be several answers. Arguably, the best explanation can be given in the framework of the theory of arbitrage-free markets, which directly associates the efficiency (or more specifically, the rationality) of a market with the absence of opportunities for arbitrage. It will be clear from what follows that the latter property immediately results in the appearance of martingales. (See Chapter V, 3 2a for fuller detail.) Now, to give an idea of the way in which martingales appear in this context, we present the following elementary arguments. the probability and statistics literature one usually considers a ‘random walk’ to be a walk that can be described by the sum of zndependent random variables. On the other hand, the economists sometimes use this term in another sense, to emphasize merely the random nature of, say, price movement.

2. Financial Markets under Uncertainty

43

Let S = (S71)lL21r where S , is the price of. say, one share at the instant n. Let

(here AS,,, = S , STI,-1)be the relative change in prices (the isnterest r a t e ) and assume that the market is organized in such a way t h a t , with respect to the flow F = (,FT1) of accessible data, the variables S,, are 9rl-nieasurable and (P-almost everywhere) E ( P n I c F T & - l ) = I‘, (1) ~

for sonic corist,arit r . By the last two formulas,

and (assuming that 1 + r f 0)

By assumption,

AS,,= PnS11-1,

71

3 1.

We also assume that, along with our share, we consider a bank account

B = ( B , l ) , L 2such ~ that

where I’ is the 7 n t ~ r ~rate s t of the account and Do > 0. Sincr B,, = Bo(1 + T ) by ~ (4), we can see from ( 3 ) that

is a martingale with respect to

This means precisely that the sequence 7x21

the flow IF = (:Frl),l>l. Our above assuniption E(pn I gT1-l) = r (P-a.e.) seeiris to be fairly natural ‘in an economist’s view’: otherwise (e.g., if E(p, I cF,-l) > r (P-almost everywhere) or E(p, I .!Frl-l) < r (P-almost everywhere) for n 3 1) the investors will find out promptly that it is more profitable to restrict their investment to stock (in the first case) or to the bank account (in the second case). To put it another way, if one security ‘dorriinatcs’ another, then the less valuable one will swiftly disappear, as it should be in a ‘well-organized’, ‘efficient’ market.

Chapter I. Main Concepts, Structures, and Instruments

44

7. We consider now a somewhat more complicated version of our model (2) of the evolution of stock prices. Assliming that a t time n - 1 you buy one share a t a price Sn-l and. a t time n , sell it, (at a price &) your (gross) ‘profit’ (which can be either positive or negative) is AS,, = S,,- Sl,-l. It is, of course, more sensible to measure the ‘profit’ in the relative values

(tJ7 ) ___

\

.”

(as we did it earlier), rather than in the absolute ones, AS,,

-,

paid for the share. i.e., to compare AS, and the money For example, if Sn-l = 20, while S, = 29, then AS, = 9, which is not all that little compared with 20. But if Sn-l = 200 and S,,= 209, then the increment AS, is 9 again; now however, compared with 200, this is not all that much. Thus, pn = 9/20 (= 45%) in the first case, while pn = 9/200 (= 4.5%) in the second. One often calls for brevity these relative profits the returns or the growth coefficients (alongside the already used term ‘(random) interest rate‘). We shall occasionally use this terminology in what follows. In accordance with our interpretat)ion of the increments AS, = S, - S,-1 as the profits from buying (at time n - 1) and selling (at time n ) ,we now assume that we have an additional source of revenue, e.g., dividends on stock, which we assume to be ,F,-nieasurable and equal to 6, at time n. Then our total ‘gross’ profits are AS,, S,,, while their relative value is

+

It would be iritcrcsting to have a notion of the possible .global’ pattern of the behavior of the prices (S,), provided that, ‘locally’, this behavior can be described by the model (5). Clearly, to answer this question we must make certain assumptions about ( p l l ) and (&). With this in mind we now assume, for instance, that

for all

71

3 1. If, iri addition, ElS,,l

< 00,

and E/6,1

< m, then

by (5), where E( . 1 9,,-1) is a conditional expectation. In a similar way,

2. Financial Markets under Uncertainty

45

which in view of ( G ) , brings us to the equality

Continuing, we see that

for each k 3 1 and each 11. 3 1. Hence it is clear that each bounded solution (ISni (for 71 3 1) has the following form (provided that, /E(&+i i 3 1):

< const

for n 3 I) of (6) for n 3 0,

I Fn)l < const

In the economics literature this is called the market fundamental solution (see, e.g., [211]). In the particular case of dividends unchanged with time (6, = 6 = const) and E ( p n I 9 7 1 - 1 ) 5 r > 0 it follows from (8) that the (bounded) prices S,, n 3 1. m i s t also be constant:

8. The class of iiiartingales is fairly wide. For instance, it contains the ‘random walk’ considerctl above. Further, the martingale property

E(Xn I .Yn-l) = X,-1 shows that, as regards the predictions of the values of the increments AX, = X11-X7L-1,the best we can get out of the ‘data’ Sn-1 is that the increment vanishes on, average (with respect to .Fr,,-l).This conforms with our innate perception that the conditional gains E(AX,, I 9 7 L - 1 ) must vanish in a ‘fair’, ‘well-organized’market, which, in turn, can be interpreted as the inipossibility of riskless profits. It is in that coriricction that L. Bachelier wrote (in the English translation): “The matheniatical expectation of the speculator is zero”. (We recall that, in gambling, the system when one doubles one’s stake after a loss and drops out after the first win is called the martingale; the conditional gains for this strategy are E(AX, 1 .Yn-l) = 0; see [439; Chapter VII, 5 11 for detail.) Finally, we point out that, as shows a n empirical analysis of price evolution (Chapter IV, $3c), the autocorrelatiori of the variables

S, h, = In ___ , Sn- 1

n31,

is close to zero, which can be regarded as a n argument (albeit indirect one) in favor of the niartingale conjecture.

46

Chapter I. Main Concepts, Structures, and Instruments

9. The conjecture of efficient markets gave an inipetiis to the development of new financial instruments, suitable for ‘cautious‘ investors adhering t o the idea of diversification (see 32b). Among these iristriinients we m i s t first place a subvariety of ‘Mutual Funds’: the so-called ‘Index Funds’. Tlie peculiarity of these funds is that they invest (their clients’) money in shares of corporations included into one or another ‘Index’ of stock. One of the first such funds was (and still is) Vanguard Index Trust-500 Portfolio (founded in 1976 by Vanguard Group; USA) that operates (buys and sells) shares of the firms included in Standard&Poor’s-500 Index, which comprises the stock of 500 corporations (400 industrial companies, 20 transportation companies) 40 utilities, arid 40 financial corporations). According to the conjecture of an efficient market, prices change (and therefore financial decisions also change-and rather promptly a t that) when the i n f o r m a tion is updated. On the other hand, cornnionplace investors (either individuals or institutions) do riot have sufficient information and usually cannot quickly respond to tlie changes of prices. Moreover, the overheads of a ‘lone trader’ can ‘eat up’ all his profits. For that reason investment in index funds can be attractive for those ‘underinformed’ investors who do not expect ‘prompt-and-big’profits, but prefer (cautious) well-diversified investment in long-term securities instead. Other examples of similar funds (issued by Vanguard) are Vanguard Index Trust--Extended Market Portfolio, Vanguard Index Trust-Small Capitalization Stock Portfolio, and Vanguard Bond Index Fund, based mostly on American securities, and also International Equity Fund-European Portfolio, Pacific Portfolio, and others based on foreign securities.

5 2b.

Investment Portfolio. Markowitz’s Diversification

1. We already rioted in § 2 a that Markowitz‘s paper [332] (1952) was decisive for tlie development of the modern theory and practice of financial management and financial engineering. The niost attractive point for investors in his theory was the idea of the diversification of an i n v e s t m e n t portfolio, because it did not merely demonstrate a theoretical possibility to reduce (unsystematic) investment risks, but also gave recorrinieiidatioris how one could achieve that in practice. To clarify the basics and the main ideas of t,his theory we consider the following single-step investment problem. Assume that the investor can allocate his starting capital L(: a t the instant n = 0 among the stocks A l , . . . , A N a t the prices S o ( A l ) ,. . . , S O ( A N )respectively. , Let X o ( b ) = blSo(A1) . . . ~ N S O ( A Nwhere ) , bi >, 0, i = 1 , .. . ; N . In other words, let

+ +

b = (bl,...,b ~ )

2. Financial Markets under Uncertainty

47

be the investment portfolio, where bi is the number of shares Ai of value So(Ai). We assume the following law governing the evolution of the price of each share Ai: its price S1(Ai) at the instant n. = 1 must satisfy t,he difference equation

or, equivalently,

S l ( A i ) = (1 + P(Ai))SO(Ai), where p ( A i ) is the raxidoni interest rate of Ai, p(Ai) > -1. If the investor has selected the portfolio b = ( b l , . . . , b ~ )then , his initial capital Xo(b) = z becomes

and he would like to make the last value ‘a bit larger’. However, his desire must be weighted against the ‘risks’ involved. To this end Markowitz considers the following two characteristics of the capital X1 ( b ):

EX1 ( b ) , its expectation and

DX1 ( b ) , its variance. Given these parameters, there are several ways to pose an optimization problem of the best portfolio choice depending on the optiniality criteria. For example, we can ask which portfolio b* delivers the maximum value for some performance f = f ( E X l ( b ) D , X l ( b ) )under the following ‘budget constraint’ on the class of admissible portfolios:

B ( z ) = { b = ( b l , . . . , b ~ : bi) 3 0 , Xo(b) = z}, There exists also a natural variational setting: find inf DX1 ( b ) over the portfolios b satisfying the conditions

where m is a fixed constant.

5

> 0.

48

Chapter I. Main Concepts, Structures, and Instruments

d m )

Fig. 8 depicts a typical pattern of the set of points (EXl(b). such that the portfolio b belongs to B ( z ) and, maybe, satisfies also some additional constraints.

01

FIGURE 8. Illusration t o Markowitz’s mean-variance analysis

It is clear from this picture that if we are interested in the m a x i m u m m e a n ,ualue of t,he value of the portfolio with minimi~mvariance, then we must choose portfolios such that the points (EXl(b):JFXm) lie on the thick piece of curve with end-points cy and 0. (Markowitz says that these are e f i c i e n t portfolios and he ternis the almve kind of analysis of mean values and variances Mean-variance analysis.)

2. We now claim that in the single-step optiniization problem for an investment portfolio, in place of the quantities (S1(A1),. . . ,S ~ ( A ~ we J ) can ) directly consider the interest rates ( ~ ( A I. .).,, AN)). This means the following. Let, b E B ( z ) . i.e., let IC = hlSo(Al) . . . b N S O ( A N ) . w e introduce the qiiaritit,ies d = ( d l . . . . , d N ) by the equalities

+

+

N

Since b E B(.r),it follows that d, 3 0 and

1d, = 1. We can represent the portfolio ,=I

value X l ( b ) as the product

Xl(b) = (1 + R ( b ) ) X o ( b ) , and let Clearly.

P ( d ) = dlP(A1) + .

f

dNP(AN).

2. Financial Markets under Uncertainty

49

Thus,

W )= 44, therefore if d = (dl, . . . , d ~and ) b = ( b l , . . . , b N ) satisfy the relations d i = -bi , SO ( A2 ) X

Xl(b) = X ( 1 + P ( 4 )

for b B ( z ) . Hcnce, to solve a n optimization problem for X , ( b ) we can consider the corresponding problem for p ( d ) . 3. We now discuss the issue of diversification as a means of reduction of the (unsystematic) risks to an arbitrarily low level, measured in terms of the variance or the standard error of X , ( b ) . To this end we consider first a pair of random variables (1 and (2 with finite second nionient,s. If c1 and c2 are constants and cri = i = 1 , 2 , then

a,

+ ~ ( 2 )= ( c i r r i

D(~l(1

wherc,

“12

-

~

2

~

+22 ~) 1~~ 2 0 1 ~ 2 ( ~1 +1 2 ) ~

and C O V ( [ ~ , & ) = E[1&

= “1 “2

-

Hence, if c1cr1 = q ( 7 2

E[1E[2.

+

arid “12 = -1, then, clearly, D(clE1 c2<2) = 0. Thus. if t,he variables [I and E2 have negative correlation with coefficient “12 = -1, then we can choose c1 and c2 such that c l c r l = c2cr2 so as to obtain a combination c l [ l c2E2 of variance zero. Of course, we can attain ill this way a fairly sniall mean value E(cl(1 c2<2). (The case of c1 = c2 = 0 is of no interest for optimization sincc b E B ( z ) . ) It becomes clear from these elementary arguments that, given our constraints on (c1,cz) and the class of the variables ((1,&), in the solution of the problem of making E(cl[l c2E2) possibly larger and D(cl(1 ~ ( 2 )possibly snialler we must choose pairs ((1, (2) with covariance as close to -1 as possible. The above phenomenon of negative correlation, which is also called the Mnrkowitz phenomenon, is one of the basic ideas of investment diversification: in building a n investment portfolio one must invest in possibly m a n y negatively correlated securities. Another concept a t the heart of diversification is based on the following idea. be a sequence of uncorrelated random variables with variances Lct (1, &,. . . , DEi 6 C , i = 1,.. . , N , where C is a constant. Then

+

+

+

+

N

N

+ d N ( N ) = Cdz D
Hence, setting, e.g., di = 1/N we see that

i=l

Chapter I. Main Concepts, Structures, a n d Instruments

50

This p h e n o m e n o n of the absence of correlation indicates that, in the case of investment in uricorrelated securities, their number N must he possibly larger to reduce the risk (i.e., the variance D ( d i [ l . . . ~ N [ N ) ) . We return now to the question of the variance Dp(d) of the variable

+ +

+ ...+~ N P ( A N ) .

p(d) = d i p ( A 1 )

We have

N

N

We now set di = l / N . Then

wliere

1 N

-

N

D p ( A i ) is the mean variance. Further,

i=l

c N

d i d j c O v ( p ( ~ i p) , ( ~ j ) = )

1 (-) N

2

( N -~ N ) . COVN,

where

is the mean covariance. Thus, 1 Dp(d) = @?&

and, clearly, if

+ (1 E1 )-COVN, -

< C and GNi cov as N + 00, then Dp(d)

+ cov

cov

as

N

+ 00.

(2)

We see from this formula that if is zero, then using diversification with N sufficiently large we can reduce the investment risk Dp(d) to an arbitrarily low level. Unfortunately, the prices on, say, a stockmarket usually have positive correlation (their change in a fairly coordinated manner, in the same direction), and therefore Z Ndoes n o t approach zero as N + 00. The limit value of COV is just the systematic (or m a r k e t ) risk inherent to the market in question, which cannot be reduced by diversification, while the first term in ( 1 ) describes the unsystematic' risks,which can be reduced, as shown, by a selection of a large number of stocks. (For a more detailed exposition of Mean-variance analysis, see [331]-[333] and [268].)

2 . Financial Markets under Uncertainty

5 2c.

51

C A P M : C a p i t a l Asset Pricing Model

1. To cvaluat,e the optimal portfolio using mean-variance analysis one must know Ep(Ai) and Cov(p(Ai),p(Aj)), and the theory does not explain how one can find

these values. (In practice, one approximates them on the basis of conventional statistical means and covariances of the past data.) CAPM-t,heory (W. F. Sharpe [433] and ,J. Lintner, [301]) and APT-theory that we consider below do not merely answer the questions on the values of E p ( A i ) and Cov(p(Ai),p ( A j ) ) ,but also exhibit the dependence of the (random) interest rates p(Ai) of particular stocks Ai on the interest rate p of the ‘large’ market of Ai. Besides the covariance Cov(p(Ai),p ( A j ) ) , which plays a key role in Markowitz’s mean-variance analysis, CAPM distinguishes another important parameter, the covariance Cov(p(Ai),p)between the interest rates of a stock A and the market interest ratc. CAPM bases its conclusions on the concept of eq?~ilibrzummarket, which implies, in particular, that there are no overheads and all the participants (investors) are ‘uniform’ in that they have the same capability of prediction of the future price movement on the basis of information available to everybody, the same time horizon, and all their decisions are based on the mean values and the covariances of the prices. It is also assumed that all the assets under consideration are ‘infinitely divisible’ and there exists a risk-free security (bank account, Treasury bills, . . . ) with interest rate T . The existence of a risk-free security is of crucial importance because the interest r enters all formulas of CAPM-theory as a ‘basis variable‘, a reference point. It should be pointed out in this connection that long-term observations of the mean values Ep(Ai) of interest rates p(Ai) of risky securities Ai show that Ep(Ai) > 1‘. Table 1, made up of t h e 1926--1985 yearly averages, enables one to compare the nominal and the real (with inflation taken into account) mean values of interest rates. TABLE 1

1

Nominal interest rate

Interest rate in real terms taking account of inflation

12%

8.8%

Corporate bonds

5.1%

2.1%

Government bonds

4.4%

1.4%

Treasury bills

3.5%

0.4%

Security Common stock

2. We now explain the fundamentals of CAPM-theory using an example of a market operating in one step.

52

Chapter I. Main Concepts, Structures, and Instruments

Let S1 = So(l+ p ) be the value of some (random) price on some ‘large’ market (for example, we can consider the values of thc S&P500 Index) a t time n = 1. Let & ( A ) = S,(A)(l + p ( A ) ) be the price of the asset A (some stock from S&P500 Index) with interest rate p(A) at time n = 1. The evolution of the price of the risk-free asset can be described by the formula B1 = Bo(1

+

T).

On the basis of the built-in concept of equilibrium market. CAPM-theory establishes (see, e.g., [268] or [433]) that for each asset A there exists a quantity P(A) (the beta ofthis assete) such that

where

In other words, the mean value of the ‘premium’ p(A) - T (for using the risky asset A in place of the risk-free one) is in proportion to the mean value of the premiiini p - T (of investments in some global characteristic of the market, e.g., the S&P500 Index). Formula (2) shows that the value P(A) of the ‘beta’ is defined by the correlation propertics of the interest rates p and p ( A ) or, equivalently, by the covariance properties of tlic corresponding prices S1 arid S1(A). We now rewrite (1) as follows:

let pLj be the value of the interest rate p(A) of the asset A with P(A) = P. If /3 = 0, then PO = 7-7

while if [j = I , then P1 =

P.

Braring this in mind we see that (3) is the equatron of the CAPM lane EPp =

+ PE(P

~

T),

(4)

eIf tach asset A has its ‘beta’P ( A ) , then one would expect it t o have also ‘alpha’ a ( A ) . This is thc iiame several authors (see, for example [267]) use for the mean value Ep(A).

2. Financial Markets under Uncertainty

53

plotted ill Fig. 9 and depicting the mean returns Epp from assets in their dependence on thc interest rate T , and the mean market interest rate Ep.

a,

r

B

1

FIGURE 9. CAPM line

The value of = P ( A ) plays a n important role in the selection of investment portfolios; this is ‘a measure of sensitivity’, ‘a measure of the response’ of the asset to changes on the market. For definitness we assume t h a t we measure the state of the market in the values of the S&P500 Index and corporation A, whose shares we are now discussing, is one of the 500 firms in this index. If the index changes by 1%and the ’beta’ of the stock A is 1.5, then the change in the price of A is (on the average) 1.5%. In practice, one can evaluate the ‘beta’ of a certain asset using statistical d a t a and conventiorial lincar regression methods (since (3) is a linear relation).

3. For the asset A we now consider the variable

i.e., the variables q ( A ) and p

-

Ep with means zero are uncorrelated. Hence

this brings us by (3) to the following relation between the premiums p ( A ) - r and p

-

r: p(A)

- 7-

= P(A)(P- 7-1

+dA),

(6)

which shows that the premium ( p ( A )- T ) on the asset A is the sum of ‘beta’ (@(A)) times the rriarkct premium ( p - T ) and the variable q ( A ) uncorrelated with p- Ep. By (51, DP(A) = Y 2 ( A ) D P+ D7/(A), (7)

Chapter I. Main Concepts, Structures, and Instruments

54

which ineans that the risk ( D p ( A ) )of a n investment in A is a combination of two, the systematic risk

(p2( A ) D p )

inherent to the market arid corresponding to the given ‘beta’ and the

unsystematic risk

(Dq(A)),

which is the property of the asset A itself. As in the previous section, we claim that here, in the framework of CAPM, unsysteniatic risks can also be reduced by diversification. Namely, assume that there are N assets A l , . . . , A N on the ‘large’ market such that the corresponding variables q(A1),. . . , AN) are uncorrelated: Cov(q(Ai),q ( A j ) )= 0 , i # j . Let d = ( d l , . . . , d N ) be a n investment portfolio with di 3 0,

N

C d i = 1, and i=l

Since

it follows that N

N

i=l

i=7

Hence setting N

N

Consequently, in a similar way to the previous section,

N C C d;Dq(Ai) 6 + 0 as N + x if D q ( A i ) < C and d N .‘

1 -. N Thus, in the framework of CAPM, diversification enables o n e t o reduce nonsystematic risks t o arbitrarily small values by a selection of s u f i c i e n t l y m a n y ( N ) assets.

where D q ( d ) =

-

i=l

-

2. Financial Markets under Uncertainty

55

We can supplement the above table of the mean values of interest rates by the table of their standard errors: TABLE 2

I I

I I

Portfolio Common stocks

I

Standard Deviation 21.2%

Corporate bonds

8.3%

Government bonds

8.2%

I

Treasury bills

3.4%

-

Taking the assumption that p J Y " p , o ) . i.e., we have a normal distribution with expectation p and deviation o-, we obtain that the values of p lie with probability 0.9 in the interval [ p - 1.65a, p 1.65o-] and, with probability 0.67, in the interval [ p - o,p 01. Thus, combining these two tables we obtain the following confidence intervals (on the percentage basis): TABLE 3

+

+

I 1 I

Portfolio Common stocks Corporate bonds

I 1 1

Probability 0.67 [-9.2,33.2] [-3.2,13.4]

Government bonds Treasury bills

1 1 I

Probability 0.9 [-22.98,46.98] [-8.59,118.79]

[-3.8,12.6]

[-9.13,17.93]

[0.1,6.9]

[-2.11,9.11]

1 I I

(Recall that we presented the corresponding mean values in Table 1.) We also present, for a few corporations A, the table of approximate values of their 'betas', the mean expected returns E p ( A ) = ~ + / 3 ( A ) E ( p - r and ) the standard errors (on the percentage basis):

d

m

TABLE 4 Stocks A

I @ ( A )I E d A ) I m I

Exxori

Compaq Computer

20.1

(The data are borrowed from [344] and [55]; the standard errors are calculated on the basis of the data for the period 1981--86; the estimates of /?(A) and Ep(A) are presented as of the beginning of 1987 with T = 5.6% and E(p - T ) taken to be 8.4%.

Chapter I. Main Concepts, Structures, and Instruments

56

3 2d.

A P T : Arbitrage Pricing Theory

1. In the foreground of the CAPM-theory there is the question on the relation between the return from a particular asset on a n ‘equilibrium’ market and the return on it on the ‘large’ market where this asset is traded (see (1) in 5 2c) and of the concomitant risks. Here (see formula (6) in the preceding section) the return (interest) p ( A ) on an asset A is defined by tlie equality P(A) = r

+ P ( A ) b-

+dA).

(1)

A more recent theory of ‘risks and returns’, APT (Arbitrage Pricing Theory; S. A. Ross, R. Roll and S. A. Ross [410]), is based on a multzple-factor model. It takes for granted that the value of p(A) corresponding to A depends on several random factors f l , . . . , f q (which can range in various doniains and describe, say, the oil prices, the interest rates, etc.) and a ‘noise’ term ( ( A )so that P(A) = a o ( A ) + a l ( A ) f l +

+ a q ( A ) f q+ ( ( A ) .

(2)

Here Ef; = 0, Df; = 1, and C o v ( f ; , f j ) = 0 for i # j , while for the ‘noise’ term ( ( A ) we have EC(A) = 0, and it does riot correlate with f l , . . . , f q or with the ‘noise’ terms corresponding to other assets. Comparing (1) arid (2) we see that (1) is a particular case of the single-factor model with factor f l = p. In this sense APT is a generalization of CAPM, although the methods of tlie latter theory remain one of the favorite tools of practitioners in security pricing, for they are clear: simple, and there exists already a well-established tradition of operating ‘betas’, the measures of assets’ responses to market changes. One of the central results of CAPM-theory, based on the conjecture of a n eguilibium market, is formnla (1) in the preceding section, which describes the average premium E ( p ( A )- r ) as a function of the average premium E ( p - r ) . Likewise, the central result of APT, which is based on the conjecture of the abse,nce of asymptotic arbitrage in the market, is the (asymptotic formula) for the mean value E p ( A ) that we present below and that holds under the assumption that the behavior of p ( A ) corresponding to an asset A in question can be described by the multi-factor niodel (2). We recall that p ( A )is the (random) interest rate of the asset A in the sinyle-step model S l ( A ) = So(A)(l+ p ( A ) )(considered above). 2. Assume that, we have a n ‘N-market’ of N assets A1, . . . , AN with q active factors f l , . . . ,f q

such that P ( 4 ) = U O ( A i ) + Ul(Ai)fl

+

+ U q ( A i ) f q + C(Ai),

E f k = 0, EC(A;) = 0, the covariance C o v ( f k ; f i ) is zero for k # 1, Dfk = 1. C o v ( f k , ( ( A ; ) )= 0 , and Cov(C(Ai),C(Aj))= r;j for k , l = 1 , .. . , q and i , j = 1 , . . . ,N .

where

2. Financial Markets under Uncertainty

57

We now consider a portfolio d = ( d l , . . . , d ~ ) The . corresponding return is P((i)= d i p ( A t )

+ ...f ~ N P ( A N )

where aik = ak (Ai) . As shown below, under certain assumptions on the coefficients aik in (2) we can find a norit,rivial portfolio d = ( d l , . . . , d ~ with ) di = d i ( N ) such that

dl+'"fdN=O,

i=l

(4)

i=l

Then, for tlic portfolio 8d = ( B d l , . . . , 8 d ~ (here ) 6' is a constant) we have

d W = @P(d)l and by (2)-(6), p(6'd) = Q

N

N

i=l

i=l

1d: + 8 1d i C ( A i ) .

Hence N p ( 0 d ) = Ep(t)d) = 8 x d:, i=l

N

02(6'd) = Dp(6'd) = O2

C didja;j. ij=l

Wc now set

Then

(7)

Chapter I. Main Concepts, Structures, and Instruments

58

Assuming (for the simplicity of analysis; see, e.g., [240] or [268] for the general case) that g i j = 0 for i # j arid aii = 1, we obtain

Formulas (9) arid (11) play a key role in the asymptotic analysis below. They show N

d:

that if

4 00

as N

+ 00, then

p(Bd)

+ ce and a 2 ( O d ) + 0.

i=l

set So(A1) = . . . = S ~ ( A N=) 1, then by the condition d l t.. . that the initial value of the portfolio Bd is

while its value at time

71 =

+dN

However, if we = 0 we obtain

1 is

Further, if EXl(Hd) = p(Bd) + (xj, while D X l ( B d ) + 0 as N + xj, then for sufficiently large N we have X l ( B d ) 3 0 with large probability and Xi(Qd)>Owith non-zero probability. In other words, starting with initial capital zero and operating on a n 'N-market' with assets A 1 , . . . , A N , N 3 1, one can build a portfolio bringing one (asymptotically) nontrivial profit. This is just what is interpreted in APT as the existence of asymptotic arbitrage. Thus, assuming t,hat the 'N-markets' are asynptotically (as N + 00) arbitragefree we arrive at the conclusiorl that one must rule out the possibility of d: + 00, which would leave space for arbitrage. Of course, this imposes certain constraints on the coefficients of the multi-factor model (2) because the selection of the portfolio d = ( d l , . . . , d ~ ) di, = d i ( N ) , i N , with properties (4)-(6) described below proceeds with an eye to these coefficients. We now consider the matrix

cE1

<

On its basis, we construct another matrix,

3 = d(.&*d)-ld*, which we assume to be well-defined (here '*' signifies transposition)

2. Financial Markets under Uncertainty

59

Let

where I is the identity matrix and a0 is the column formed by a10, . . . , UNO. Then we have the orthogonal decomposition ao=d+e

(15)

and d * l = 0, d * U k = 0, (16) where ak is the colurrln formed by a l k , . . . , a N k and 1 is the column of ones. Forniulas (16) are just the above-discussed relations (4) and (5). By (14) and (15) we also obtain d * n ~= d*d

+ d*e = d*d

which is the required formula (6). We note now that, by (14), the colunin e can be represented as follows: e = A01

+ A l a l + . . . + &aq,

where the numbers XO, . . . , A, satisfy the relation ( A & . . .,A,)*

= (d*d)-1Lii*ao.

Hence k=l

and

N z=1

N

,

a

k=l

/

Of course, for a n ‘N-market’ all the coefficients a i o , . . . , a i k and Ao,. . . ,XI, on the right-hand side of this formula depend on N . Our assumption of the absence of asymptotic arbitrage rules out the possibility

of

(here we write d i ( N ) to emphasize the dependence of the number of shares of one or another asset, on the total number of assets in the market), therefore by (17) the ‘N-rnarket’ coefficients must ensure the inequality

Chapter I. Main Concepts, Structures, and Instruments

60

This relation (deduced from the conjecture of the absence of asymptotic arbitrage) is interpreted in the framework of A P T as follows: i f the n u m b e r N of assets t a k e n i n t o account in the selection of a securities portfolio is suficiesntly large, t h e n for ‘ t h e majority of assets’ we m u s t ensure the ’almost h e a r ’ relation between the coeficients ao(Ai),a l ( A i ) ,. . . , a,(Ai): 4

ao(Ai)

Xo

+

hak(Ai), k=l

where all the i)araables u n d e r consideration depend on N and

Moreover. there exists a portfolio d = ( d l , . . . , d ~ such ) that the varlnnce of the return p ( d ) is sufficiently small (in view of ( l l ) ) which , means that the influence of the noise terms C ( A i ) and individual active factors f j can be reduced by nieans of diversification (provided that there is n o asymptotic arbitrage). One should bear in mind, however, that all the above holds only for large N , i.e., for ‘large’ markets, while for sniall markets calculations of the return expectation Ep(Ai) on the basis of the expression on the right-hand side of (19) can lead t o grave errors. (As regards the corresponding precise assertion, see [231], 12401, and [412]; for a rigorous niatheinatical theory of asymptotic arbitrage based on the concept of contiguity, see [250].[260], [26l], [273], and Chapter VII, $5 3a, b, c).

3 2e.

Analysis, I n t e r p r e t a t i o n , and R e v i s i o n of the Classical Concepts of Efficient M a r k e t . I

1. The centxal idea, the cornerstone of the concept of e f i c i e n t m a r k e t is the assumption that the prices instantaneously assimilate new data and are always set in a way that gives one no opportunity to ‘buy cheap and sell immediately a t a higher price elsewhere’, i.e., as one usually says, there are no opportunities f o r arbitrage. We have already shown that this idea of a ‘rationally’ organized, ‘fair’ market brings one to (normalized) market prices described by martingales (with respect to some measure equivalent to the initial probability measure). We recall that if X = (Xn),>o is a martingale with respect to the filtration ( S n , ) + 0 , then E(X,L+r,L 1 T n )= X,. Hence the optimal (in the mean-square sense) estimator X,,,+7,L;IL for the variable Xn+m based or1 the ‘data’ 9rL is simply the variable X,, because Xn+m;,L is the same as E(X,+, I 9,). Thus, we can say that our martingale conjecture for the prices (X,) corresponds to the (economically conceivable) assumption that, on a ‘well-organized’ market, the best (in the mean-square sense, in any case) projection of the ‘tombrrow’ (‘the day after tomorrow’, etc.) prices that can be made on the basis of the ‘today‘ information is the current price level.

-

2. Financial Markets under Uncertainty

61

In other words, the forecast is trivial. This seemingly rules out any chance of the prediction of ‘the future dynamics of the observable prices’. (In his construction of a Brownian motion as a model for this evolution of prices L. Bachelier started essentially just from this idea of the impossibility of a better forecast of the ‘tomorrow’ prices than a Inere repetition of their current levels.) At the same time, it is well known t h a t market operators (including such experts as ‘fuiidarrientalists’, ‘technicians’, and quantitative analysts, ‘quants’) are still trying to forecast ‘the future price movement’, foresee the direction of changes and tlie future price levels, ‘iuork out’ a timetable for buying and selling the stock of particular corporations, etc.

Rern,ark. ~Fiiritlarrieritalist,s’ make their decisions by looking at the state of the ‘economy at large’, or some its sectors; the prospects of development are of particular interest for them; they build their analysis upon the assumption that the actions of market operators are ‘rational’. Those teriding to ‘technical’ analysis conccntrate on the ‘local’ peculiarities of the market; they emphasize ‘mass behavior’ as a criicial factor. As noted in [385;pp. 15-16], the ‘fundamentalists’ and the ‘technicians’ were the two major groups of financial market analysts in 1920-50. A third group emerged in the 1950s: ’qiiants’, quantitative analysts, who are followers of L. Bachelier. This group is strongor biased towards ‘fundamentalists’ than towards the proponents of the ‘technical’ approach, who attach more weight to market moods than to the ratio,n.nl causes of investors’ behavior. 2. We now return to the problem of building a basis under aspirations and atternptrs to forecast the ‘future’ price movement. Of course. we must start with the anal@s of empirical data, look for explanations of several non-trivial features (for example, the cliistcr property) characteristic for the price movement, and clear up the probabilistic struct,ure of prices as stochastic processes. s71

Let H7, = In - be tlie logarithnis of some (discounted) prices. We can represent

IS, ”,

the H,, as thc sums H,,

= hl

+ . . . + h,

with hh = In -. S k Sk- 1

If the seqiierice ( H T l )is a martingale with respect to some filtration ( S n )then , the variables ( I h , , ) form a m,artingnle-diference (E(h, I 2Jn-l) = O), and therefore the h, (assuming that they are square-integrable) are uncorrelnted, i.c., Eh,+,h, = 0, m 3 1, n 3 1. As is well known, however, this does not m e a n that these variables are independent. It is not iniprohble that, say, hi+,, and h: or Ih,,+,I and are positively correlated. The cinpirical analysis of much financial data shows that this is indeed the case (which does not contradict the assumption of the martingale nature of prices on an ‘efficient’ market!) It is also remarkable that this positive correlation, revealing itself in the behavior of the variables (hn),21 as the phenomenon of their clustering int,o groups of large or small values, can be ‘caught’, understood, by

Chapter I. Main Concepts, Structures, and Instruments

62

means of several fairly simple models (e.g., ARCH, GARCH, models of stochastic volatility, etc.), which we discuss below in Chapter 11. This indicates an opportunity (or a t any rate, a chance) for a more non-trivial (and, in general, non-linear) prediction, e.g., of the absolute values Ihn+ml. There also arises a possibility of making more refined conclusions about the joint distributions of the sequence (/L,),~I. Taking the simplest assumption on the probabilistic nature of the h,,, let

hn = C n E n ,

- +a.

where the crLare independent standard normally distributed variables such that .N(O,l) and the cTLare some constants, the standard deviations of the h,:

E,

0,

=

However, this classical model of a Gaussian random walk has long been considered inconsistent with the actual data: the results of ‘normality’ tests show that the empirical distributiori densities of h, are more extended, more leptokurtotic around the mean value than it is characteristic for a normal distribution. The same analysis shows that the tails of the distributions of the h, are more heavy than in the case of a normal distribution. (See Chapter IV for greater detail.) In the finances literature one usually calls the coefficients crn in the relation fin = ( T ~ the ~ E volatilities. ~ ~ Here it’is crucial that the volatility i s itself volatile: g = (crrl) is not only a function of time (a constant in the simplest case), but it is a r a n d o m variable. (For greater detail on volatility see 5 3a in Chapter IV.) Mathematically, this assumption seems very appealing because it extends considerably the standard class of ( l i n e a r ) Gaussian model and brings into consideration (non-linenr.) conditionally Gaussian models, in which h, = cr,E,.

-

where, as before, the (E,) are independent standard normally distributed variables (E, A ( 0 , l ) ) ,but the cr, = c,(w) are 2Jn_1-nieasurable non-negative r a n d o m variables. Iri other terms,

which means that Law(h,) is a suspension (a ,mixture) of normal distributions averaged over some distribution of volatility (‘random variance’) 0;. It should be noted that, as is well known in mathematical statistics, ‘mixtures’ of distributions with fast decreasing tails can bring about distributions with heavy tails. Since such tails actually occur for empirical data (e.g., for many financial indexes), we niay consider conditionally Gaussian schemes to be suitable probabilistic models. Of course, turning to models with stochastic volatility (and, in particular, of the ‘hTL =0 , ~ ~ kind), ~ ’ it is crucial for their successful application to give an ‘adequate’ description of the evolution of the volatility (on,).

2. Firiancial Markets under Uncertainty

63

Already the first model, ARCH (Autoregressive Conditional Heteroskedasticity), proposed in the 1980s by R. F. Engle [140] and postulating that 2

0,

= a0

2 + alh,-l+

‘.

+ a,h&

enables one to ‘grasp’ the above-mentioned cluster p h e n o m e n o n of the observable data revealed by the statistical analysis of financial time series. The essence of this phenomenon is that large (small) values of h, imply large (respectively, small) subsequent values (of uncertain sign, in general). Of course, this becomes clearer once one makes a reference to the assumption of ARCH-theory about the dependence of 0: on the ‘past’ variables h i p 1 , Thus, in conformity with observations, if we have a large value of Ih model, then we should expect a large value of Ih,+ll. We point out, however, that the model makes n o forecasts of the direction of the price movement, gives no information on the sign of h l L + l . (One practical implication is the following advice to operators on a stochastic market: considering a purchase of, say, an option, one better buy simultaneously a call and a put option; cf. 4e in Chapter VI.) Little surprise that the ARCH model gave birth to many similar models, constructed t o ‘catch on’ to other empirical phenomena (besides the ‘clusters’). The best known among these is the GARCH (Generalized A R C H ) model developed by T. Bollerslcv [48](1986), in which 2

gn

= “0

2 + alh,-l+

+

2 + /310,-1 + . + flpo7”L-,.

(One can judge the diversity of the generalizations of ARCH by the list of their names: HARCH, EGARCH, AGARCH, NARCH, MARCH. etc.) One of thc ‘technical’ advantages of the GARCH as compared with the ARCH models is as follows: whereas it takes one to bring into consideration large values of q to fit the ARCH model to real data, in GARCH models we may content ourselves with small q and p. Having observed that the ARCH and the GARCH models explain the cluster phenomenon, we must also mention the existence of other empirical phenomena showing that tlic relation between prices and volatility is more subtle. Practitioners know very well that if the volatility is ‘small’, then prices tend to long-term rises or drops. On the other hand, if the volatility is ‘large’, then the rise or the decline of prices are visibly ‘slowing’; the dynamics of prices tends to change its direction. All in all, the inner (rather complex) structure of the financial market give one some hopes of a prediction of the price movement as such, or at least, of finding sufficiently reliable boundaries for this movement. (‘Optimists’ often preface such hopes with seritentiae about market prices that ‘must remember their past, after all’, although this thesis is controversial arid far from self-evident.) In what follows (see Chapters I1 and 111) we describe several probakilistic arid statistical models describing the evolution of financial time series in greater detail. Now, we dwell on criticism towards and the revision of the assumption that the traders operate on an ‘efficient’ market.

64

Chapter I. Main Concepts, Structures, and Instruments

3. As said in $ Za, it is built in the ‘efficient market’ concept that all participants are ‘uniform’ as regards their goals or the assimilation of new data, and they make decisions on a ‘rational’ basis. These postulates are objects of a certain criticism, however. Critics maintain that even if all participants have access to the entire information, they do not respond t o it or interpret it uniformly, in a homogeneous ma’nner;their goals can be utterly different; the periods when they are financially active can be various: from short ones for ‘speculators’ and ‘technicians’ to long periods of the central banks’ involvement; the attitudes of the participants to various levels of riskiness can also be considerably different. It has long becn known that people are ‘not linear’ in their decisions: they are less prone to take risks when they are anticipating profits and more so when they confront a prospect of losses [465]. The following dilemma formulated by A. Tversky of the Stanford University can serve an illustration of this quality (see [403]): a) “Would you rather have $85 000 or an 85% chance of $100 OOO?” b) !.Would you rather lose $85 000 or rim an 85% risk of losing $100 000?”

Most people would better take $85 000 and not try to get $100 000 in case a). In the second case b) the majority prefers a bet giving a chance (of 15%) to avoid losses. One factor of prime importance in investment decisions is time. Given an opportunity to obt,ain either $ 5 000 today or $ 5 150 in a month, you would probably prefer money today. However, if the first opportunity will present itself in a year and the second in 13 months, then the majority would prefer 13 months. That is, investors can have different ‘time horizons’ depending on their specific aims, which, generally speaking, conforms badly with ‘rational investment,‘ models assuming (whether explicitly or not) that all investors have the same ‘time horizon’. We noted in 5 2a.3 another inherent feature of the concept of efficient market: the participants must correct their decisions instantaneously once they get acquainted with new data. Everybody knows, however, that this is never the case: people need ccrtain time (different for different persons) to ponder over new information and take one or another decision. 4. As G . Soros argues throughout his book [451], apart from ‘equilibrium’ or ‘close to equilibrium’ state, a market can also be ‘far from equilibrium’. Addressing the idea that the operators on a market do not adequately perceive and interpret information, G. Soros writes: “We may distinguish between near-equilibrium conditions where certain corrective mechanisms prevent perceptions and reality from drifting too far apart, arid far-from-equilibrium conditions where a reflexive double-feedback mechanism is a t work and there is no tendency for perceptions and reality to conic closer together without a significant change in the prevailing conditions, a change of regime. In the first case, classical economical theory applies and the divergence between perceptions and reality can be ignored as mere noise. In the second case; the theory of equilibrium becomes irrelevant and we are confronted with a onedirect,ional historical process where changes in both perceptions and reality are

2. Financial Markets under Uncertainty

65

irreversible. It is important to distinguish between these two different states of affairs because what is normal in one is abnormal in the other.” [451, p. 61. The views of R. B. Olseri 121, the founder of the Research Institute for Applied Economics (“Olsen & Associates”, Zurich), fall in unison with these words: ”There is a broad spectrum of market agents with different time horizons. These horizons range from one minute for short-term traders to several years for central bankers and corporations. The reactions of market agents to an outside event depend on the framework of his/her time horizon. Because time horizons are so different and vary by a factor of one million, economic agents take different decisions. This leads to a ripple effect, where the heterogeneous reactions of the agents are new events, requiring in turn secondary reactions by market participants.” It is apparently too early so far to speak of a rigorous mathematical theory treating finaricial markets as ‘large complex systems’ existing in the environment close to actually observed rather than in the classical ‘equilibrium’ one. We can define the current stage as t,hat of ‘data accumulating’, ‘model refining’. Here new methods of picking and storing information, its processing and assessing (which we discuss below, see Chapter IV) are of prime importance. All this provides necessary empirical data for the analysis of various conjectures pertaining to the operations of securities markets and for the correction of postulates underlying, say, the notion of an e,@cierit market or assumptions about the distribution of prices, their dynamics, etc.

3 2f.

Analysis, Interpretation, and Revision of the Classical Concepts of Efficient Market. I1

1. In this section we continue the discussion (or, rather, the description) of t h e assumptions underlying the concepts of e f i c i e n t market and rational behavior of investors on it. We concentrate now on several new aspects, which we have not yet discussed. The concept of e f i c i e n t m a r k e t was a remarkable achievement at its time, which has played (and plays still) a dominant role both in financial theory and the business of finance. It is therefore clear that pointing out its strong and weak points can help the understanding of neoclassical ideas (e.g., of a ‘fractal’ structure of the market) common nowadays in the economics and mathematics literature devoted to financial markets’ properties and activities.

2. We have already seen that the concept of efficient market is based on the assumption that the ‘today’ prices are set with all the available information completely taken into account and that prices change only when this information is updated, when ‘new’, ‘unexpected’, ‘unforeseen‘ d a t a become available. Moreover, investors on such a market believe the established prices to be ‘fair’ becabse all the participarit,s act in a ‘uniform’, ‘collectively rational’ way. These assumptions naturally make the r a n d o m walk conjecture (that the price

66

Chapter I. Main Concepts, Structures, and Instruments

is a sum of independent terms) and its generalization, the ‘martingale conjecture’ (which implies that the best forecast of the ‘tomorrow’ price is its current level) look quite plausible. All this can be expressed by the phrase ‘the market is a martingale’, i.e., one plays f a i r l y on an ‘efficient market’ (which is consistent with the traditional explanation of the word ‘martingale’;see Chapter 11: 33 l b , c and, for example, [439; Chapter VII, § 11, where more details are given). The reader must have already observed that, in fact, the concept of ‘efficient market’ ( 3 2a) simply postulates that ‘an efficient market is a martingale’ (with respect to one or another ‘information flow‘ and a certain probability measure). The corresponding arguments were not mathematically rigorous, but rather of the intuitive and descriptive nature. In fact, this assertion (‘the market is a martingale’) has an irreproachable mathematical interpretation, provided that we start from the conjecture that (by definition) a ‘fair’, ‘rationally’ organized market is an arbitrage-free m a r k e t . In other words, this is a niarket where no riskless profits are possible. (See 3 2a in Chapter V for the formal definition.) As we shall see below, one implication of this assuniption of the absence of arbitrage is that there exists, generally speaking, an entire s p e c t r u m of (‘martingale’) measures such that the (discounted) prices are martingales with respect to these measures. This means more or less that the market can have an entire range of stable states, which, in its turn, is definitely related to the fact that market operators have various aims and different amounts of time to process and assimilate newly available information. This presence of investors with different interests and potentials is a positive factor rather than a deficiency it may seem a t the first glance. The fact is that they reflect the ‘diversification’ of the market that ensures its liquidity, its capacity to transform assets promptly into means of payment (e.g., money), which is necessary for stability. We can support this thesis by the following well-known facts (see, e.g., [386; pp. 46-47]). The day of President J. F. Kennedy’s assassination (22.11.1963) markets irnrnediately responded to the ensuing uncertainty: the long-term investors either suspended operations or turned to short-term investment. The exchanges were then closed for several days, and when they re-opened, the ’long-term’ investors, guided by ‘finitlamental’information, returned t o the market. Alt,hough the complete picture of the well-known financial crash of October 19, 1978 in the USA is probably not yet understood, it is known, however, that just before that date ‘long-term’ investors were selling assets and switching to ‘short-term’ operations. The reasons lay in Federal Reserve System‘s tightening of monetary policy and the prospects of rises in property prices. As a result, the market was dominated by short-term activity, and in this environment ‘technical” information (based, as it often is in instable times, on hearsay and speculations) came to the forefront.

2. Financial Markets under Uncertainty

67

In both examples, the long-term investors’ ‘run’ from the market brought about a lack of liquidity and, therefore, instability. All this points to the fact that, for stability, a market must include operators with different ‘investment horizons’, there niust hc ‘noriliorriogen~ity’,‘fractionality’ (or? as they put it, ‘fractality’) of the interests of the participants. The fact that financial markets have the property of ‘statistical fractality’ (see the definition in Chapt,er 111, fj 5b) was explicitly pointed out by B. Mandelbrot as long ago as the 1960s. Later on, this question has attracted considerable attention, reinforced by newly discovered phenomena, such as the discovery of the statistical fract,al structurc in the currency cross rates or (short-term) variations of stock and bond prices. As regards our understanding of what models of evolution of financial indicators are ‘correct’ and why ‘stable’ systems must have ‘fractal’ structure, it is worthwhile to compare d e t e r m i n i s t i c and statistical fractal structures. In this connection, we provide a n insight into several interconnected issues related to ‘non-linear dynamic systems’, ‘chaos’, and so on in Chapter 11, 3 4. We have already mentioned that we adhere in this book to a n irreproachable mathematical theory of the ‘absence of arbitrage’. We must point out in this connection that none of such concepts as e f i c i e n c y , absence of arbitrage, fractality can be a substitution for another. They supplement one another: for instance, marly arbitrage-free models have fractal structure, while fractional processes can be martirigalcs (with respect t,o somc martingale measures; and then the corresponding market is arbitrage-free), but can also fail to be martingales, as does, e.g., a fractional Brownian motion (for the Hurst parameter H in (0, ;) U 1)).

(i.

3. In a similar way to $2a, where we gave a descriptive definition of a n ‘efficient’ market, it seems appropriate here, for a conclusion, to sum up the characteristic features of a market with fractal structure (a fractal market in the vocabulary of [386])as follows: 1) at each instant of time prices on such a market are corrected by investors on the basis of the information relevant t o their ‘investment horizons’; investors do not necessarily respond to new information momentarily, h i t can react after it has been reaffirmed; 2) ‘technical information’ and ‘technical analysis’ are decisive for ‘short’ time horizons, while ‘fundamental information’ comes to the forefront with their cxt,cnsion; 3) the prices established are results of interaction between ‘short-term’ and ‘long-term’ investors; 4) the ’high-frequency’ corriporient in the prices is caused by the activities of ’short-term’ investors, while the ‘low-frequency’, ‘smooth’ components reflect the activities of ‘long-term’ ones. 5) the market, loses ‘liquidity’ and ‘stability’ if it sheds investors with various ‘investmelit horizons’ arid loses its fractal character.

68

Chapter I. Main Concepts, Structures, and Instruments

4. As follows from the above we mainly consider in this hook models of (’efficient’) markets lljith, n o opportunities for arbitrage. Examples of such models that we thoroughly discuss in what follows are the Bachelier model (Chapter 111, 4b and Chapter VIII, 5 l a ) , the Black-MertonScholes model (Chapter 111, S 4b and Chapter VII, 3 4c), and the Cox-Ross-Rubinstein model (Chapter 11, 5 l e and Chapter V, 3 I d ) based on a linear Brownian motion, a geomctric Brownian motion, and a geometric random walk, respectively. It seems fairly plausible after our qualitative description of ‘fractal’ markets that there may cxist markets of this kind with opportunities for arbitrage. Simplest examples of such models are (as recently shown by L. R. C. Rogers in “Arbitrage with fractional Brownian motion”, Mathematical Finance, 7 (1997), 95-105) modified models of Bachelier and Black-Merton-Scholes, in which a Brownian motion is replaced by a fractional Brownian motion with W E (0. +), U ( i , 1). See also Chapter VII, fj2c below.

i) (i,

Remark. A fractional Brownian motion with W E (0, U 1) is not a semimartingale, therefore the corresponding martingale measures are nonexistent; see Chapter 111, 3 2c for details. This is a n indirect indication (cf. the First fundamental theorem in Chapter V, §ad) that there may (not necessarily!) exist arbitrage in such models.

3. Aims and Problems of Financial Theory,

Engineering, and Actuarial Calculations

5 3a.

Role of F i n a n c i a l Theory and F i n a n c i a l E n g i n e e r i n g . Financial Risks

1. Our discussion of firiancial structures arid markets operating under uncertainty in the previous sections clearly points out RISKS as one of the central concepts both in Financial theory and Insurance theory. This is a voliiminous notion the contents of which is everybody’s knowledge. For instance, the credit-lending risk is related t o the losses that the lender can incur if the borrower defaults. Operational risk is related to errors possible, e.g., in payments. Iniiestment risk is a consequence of an unsatisfactory study of detail of an investment project, niiscalculated investment decisions, possible changes of the economical or the political situation, and so on. Fiiiancial niathernatics and engineering (in its part relating t o operations with securities) tackles mostly market risks brought about by the uncertainty of the development of market prices, interest rates, the unforeseeable nature of the actions and decisions of market operators, and so on. The steep growth in attention paid by Fzrzariciul theory to mathematical and engineering aspects (particularly ostensible during the last 2-3 decades) must have its explanation. The answer appears to be a simple one; it lies in the radical changes visible in the financial markets: the changes of their structure, greater volatility of prices, emergence of rather sophisticated financial instruments, new technologies used in price analysis, and so on. All this imposes heavy weight on financial theory and puts forward new problems, whose solut,ion requires a rather ‘high-brow’ mathematics. 2. At the first glance at financial rnathernatics and engineering, when one makes a n emphasis on the ‘game’ aspect (‘investor’ versus ‘market’), one can get an impression

70

Chapter I. Main Concepts, Structures, and Iiistrumerits

that it is the rnairi airri of financial mathematics t o work out recommendations and develop firiancial tools enabling the investor to ‘ g a i n over markets’; a t any rate, ‘ n o t t o lose w r y ,much’. However, the role of financial theory (including firiancial mathematics) and financial engineering is much more prominent: they must help investors with the solution of a wide range of problenis relating to rational i n v e s t m e n t by taking account of the risks unavoidable given the random character of the ’economic environment’ and the resulting uncertainties of prices, trading volumes, or activities of market operators. Financial mathematics and engineering are useful and important also in that their recommendations and the proposed financial instruments play a role of a ‘regulat,or’in the reallocation of f u n d s , which is necessary for better functioning of particular sectors arid the whole of the economy. The analysis of the securities market as a ‘large’ and ‘complex’ system calls for complicated, advanced mathematics, methods of data processing, nunierical methods, and corriputing resources. Little wonder, therefore, that the finances literature draws 011 the most up-to-date results of stochastic calculus (on B r o w n i a n m o t i o n , stochastzc diflerentzal equations, local martingales, predictability, . . . ), mathematical statistics (bootstrap, jackknife, . . . ), non-linear dynamics (deterministic chaos, bzfurcations, fractals, ), and, of course, it would be difficult to imagine modern financial markets without advanced computers and telecommunications.

3. Markowitz’s theory was the first significant trial of the strength of probabilistic methods in the minimization of unavoidable commercial and financial risks by building an investment portfolio on a rational basis. As regards ‘risks’, the fundamental economic postulates are as follows: 1) ‘big’ profits m e a n big risks arid, 011 the way to these profits, 2) the risks m u s t be ‘justifiable’, ‘reasonable’, and ‘calculated’. We can say in connection with 1) that ‘big profits are a compensation for risks’; it seems appropriate to recall the proverb: “Nothing ventured, nothing gained”. In connection with 2 ) , we recall that ‘to calculate’ the risks in the framework of Markowitz’s theory amounts to making up an ‘efficient’ portfolio of securities ensuring the maxiiiium average profit that is possible under certain constraints on the size of the ‘risk’ measured in terms of the variance. We point out again that the idea of diversification in building an ‘efficient’ portfolio has gained solid ground in financial mathematics and became a starting point for the development of new strategies (e.g., hedging) and various financial instruments (options, futures, . . . ). It can be expressed by the well-known advice: “Don’t put all your eggs in one basket,. Meaii-variance analysis is based on a ‘quadratic performance criterion’ (it measures the risks in terms of the variance of return on a portfolio) and assumes that the market in question is ‘static’. Modern analysts consider also more general quality

3. Aims and Problems of Financial Theory, Engineering, and Actuarial Calculations

71

functions arid utility functions in their study of optimal investment, consumption, and allocation of resources. It is important to point out here a new aspect: the time dynamics, the need to take solutions ‘successively’, ‘in stages’. Incidentally, decisions must be taken on the basis of generally accessible information and without anticipation. ‘Statics’ means that we are interested in the profits a t time N > 0, while the portfolio is made up a t time n = 0 (i.e., it is 90-measurable). Allowing for ‘dynamics’, we are tracing development in time, so that the returns a t a particular instant 7~ < N are defined by the portfolio built on the basis of information obtained by time n - 1 (inclusive; i.e., the components of the portfolio are 9n-l-measurable), and so on. Clearly, the incorporation of the dynamical aspect in the corresponding problems of financial mathematics calls for concepts and methods of optimal stochastic control, stochastic optimization, dynamical programming, statistical sequential analysis, and so on. In our discussion of the risks relating t o ‘uncertainties’ of all kinds we cannot bypass the issues of ‘risk’ insurance, which are the subject of Insurance theory or, somewhat broader, the Theory of actuarial calculations. In the next section we describe briefly the formation of the insurance business providing a mechanism of compensation for financial losses, the history and the role of the ‘actuarial trade’. We shall not use this information in what follows; nevertheless, we think it appropriate here, since it can give one a n idea of a realm that is closely related to finance. Moreover, it becomes increasingly clear that financial and actuarial mathematics share their ideas and their method.

3 3b.

Insurance: a Social Mechanism of Compensation for Financial Losses

1. An acti~nriusin ancient Rome was a person making records in the Senate’s Acta Publica or an army officer processing bills and supervising military supplies. In its English version, the meaning of the word ‘actuary’ has undergone several changes. First, it was a registrar, then a secretary or a counselor a t a joint stock company. In due course, the trade of an actuary became associated with one who performed mathematical calculations relati,ng t o life expectancies, which are fundamental for pricing life insurance contracts, annuities, etc. In modern usage an actuary is an expert in the mathematics of insurance. They are often called social mathematicians, for they keep key positions it working out the strategy not only of insurance companies, but also in pension and other kinds of funds; government actuaries are in charge of various issues of state insurance, pensions, and other entitlement schemes. Insurance (assurance) is a social mechanism of compensation of individuals and organizations for financial losses incurred by some or other unfavorable circumstances.

72

Chapter I. Main Concepts, Structures, and Instruments

The destination of insurance is t o put certainty in place of the uncertainties of financial evaluations that are due to possible losses in the future. Insurance can be defined as a social instrument that enables individuals or organizations to pay in advance to reduce (or rule out) certain share of the risk of losses. Mankind was fairly quick a t coniprehending that the most efficient way t o lessen the losses from uncertainties is a cooperation mechanism distributing the cost of the losses born by an individual over all participants. Individuals were smart enough to realize that it is difficult to forecast the timing, location, and the scope of events that can have implications unfavorable for their economic well-being. Insurance provided an instrumerit that could help a person t o lessen, ‘tame’ the impact of the uncertain, the unforeseen, and the unknown. It would be wrong to reduce insurance (and the relevant mathematics) to, say, propcrty and life insurance. It must be treated on a broader scale, as risk insurance, which means the inclusion of; e.g., betting in securities markets or investments (foreign as well as domestic). We shall see below that the current state of insurance theory (in the standard sense of the term ‘insurance’) is distinguished by extremely close interrelations with financial theory. One spectacular example here are futures reinsurance contracts floated a t the Chicago Board of Trade in December, 1992. Not all kinds of ‘uncertainty’, not all risks are subject t o insurance. One usually uses here the following vocabulary and classification, which is helpful in outlining the rcalrn of insurance. 2. All ‘uncertainties’ are usually classified with one of the two groups: pure uncertainties and speculative uncertainties. A speculatiue uncertainity is the uncertainty of possible financial gains or losses (generally speaking, one does not sell insurance against such uncertainties). A pure n n c w t a i n i t y means that there can be only losses (for example, from fire); many of thesc: can be insured against. Often, one uses the same word ‘risk’ in the discussion of uncertainties of both types. (Note that, in insurance theory, one often identifies risk and pure uncertainty; on the ot,her harid, a popular finances journal ‘Risk’is mostly devoted to speculative uncertainties.) The sources of risks and of the losses incurred by them are accidents, which are also classified in insurance with one of the two groups: physical accidents and moral accidents. Moral accidents are about dishonesty, unfairness, negligence, vile intentions, etc. With physical accidents one ranks, for instance, earthquakes, economic cycles, weather, various natural phenomena, and so on. There are different ways to ‘combat’ risks: 1) One can deliberately avert risks, avoid them by rational behavior, decisions, and actions.

3. Aims and Problems of Financial Theory, Engineering, and Actuarial Calculations

73

2) One can reduce risks by transferring possible losses to other persons or institutions. 3) One can try to rediice risks by making forecasts. Sthtistical methods are an iinportant weapon of a n actuary in forecasting possible losses. Forecastmaki,riy and accumulation of funds are decisive for a sustained successful insurance business. Althoiigli insuraiice is a logical and in many respects remarkable ‘remedy’ against risks, riot all uncertainties and accompanying financial losses can be covered by it. Insurance risks niust satisfy certain conditions; namely, 1) there must exist a sufficiently large group of ‘uniformly’ insurable participants with ‘characteristics‘ stable in time; 2) insiirance cases should not affect many participants a t one time; 3) these cases and the range of losses should not be coriseqiiences of deliberate actions of those insured: they must be accidental; damages are paid only if the cause of losses can be determined; 4) the potential losses ensuing from the risks in question niust be easily identifiable (‘difficult to counterfeit’); 5) the potential losses must be sufficiently large (‘there is no point in the insurance against small, easily recoupable losses’) ; 6) the probability of losses must be sufficiently small (insurance cases must be rare), so as to makc the insurance economically affordable; 7) statistical d a t a must be accessible and can serve a basis for the calculations of the probabilities of losses (‘representativeness and the feasibility of stat,istic:al estimation’). These and similar requirements are standard, they make up a minimum dose of conditions enabling insurance. Several types (kititis) of insurance are known. Their diversity is easier to sort out if one considers (a) classes of insuretl olljects (1)) unfavorable factors that can lead t o insurance risks (c) modes of payment (premiums, benefits) (d) types of insurance (social versus commercial). Group (a) includes various classes of objects, for instance, life, health, property, tocks of merchandise, and so on. Group (1)) iridutles the above-mentioned physical and moral accidents. The modes of payment (of premiums and benefits) in group (c) vary with irisurance policies; life insiirance is subject to most diversification. Different types of insurance (d)-either free (commercial) or obligatory (social)are based on siinilar principles but differ in their philosophy arid ways of organization.

3. The contmit of insi~mn,ce,its aims and objectives are difficult to understand without a knowledge of both its structure arid history.

74

Chapter I. Main Concepts, Structures, and Instruments

We can say that, in a sense, insurance is as old as the mankind itself. The most ancient forrns of insurance known are the so-called bottomrg and respondencia contracts (for sea carriages) discovered in Babylonian texts of 4-3 millennia BC. B o t t o m r y , which was essentially a pledge contract, was formally a loan (e.g., in the form of goods that must be delivered to some other place) made to the ship owner who could give not only the ship or another ‘tangible’ property as collateral, but also his own life and the lives of his dependents (which means that they could be reduced to slavery). In the case of a respondencia the collateral was always secured by goods. The Babylonians developed also a system of insurance contracts in which the supplier of goods, in case of a risky carriage, agreed to write off the loan if the carrier were robbed, kidnapped for ransom, and so on. The Hammurabe code (2100 BC) legalized this practice. It also provided for damages and compensations (by the government) t o individuals who had suffered from fire, robbery, violence, and so on. Later on, the practices of making such contracts were adopted (through the Phoenicians) by the Greeks, the Romans, and the Indians. It was mentioned in early Roman codes and Byzantine laws; it is reflected by the modern insurance legislation. The origins of life insurance can be traced back to Greek thiasoi and eranoi or to Roman collegia, from 600 BC to the end of the Roman empire (which is traditionally put a t 476 AC). Originally, the collegia (guilds) were religious associations, but with time, they took upon themselves more utilitarian functions, e.g., funeral services. Through their funeral collegia the Romans paid for funerals; they also developed some rudimentary forms of life insurance. The lawyer Ulpian (around 220 AD) made up mortality tables (rather crude ones). The practice of premium insurance has apparently started in Italian city republics (Venice, Pisa, Florence, Genoa), around 1250. The first precisely dated insurance contract was made in Genoa, in 1347. The first ‘proper’ life insurance contract, which related t o pregnant women and slaves, was also signed there (1430). Annuity contracts were known t o the Romans as long ago as the 1 century AD. (These contracts stipulate that the insurance company is paid some money and it pays back periodic benefits during a certain period of time or till the end of life. This is in a certain sense a converse of life insurance contracts, where the insured pays premium on a regular basis and obtains certain sum in bulk.) The text of Rome’s Palcidian law included also a table of the mean values of life expectancy necessary in the calculations of annuity payments and the like. In 225, Ulpian made up more precise tables that were in use in Tuscany through the 18th century. Three hundred years ago, in 1693, Edmund Galley (whose name is associated with a well-known comet) improved Ulpian’s actuarial tables by assuming that the mortality rate in a fixed group of people is a deterministic function of time. In 1756,

3. Aims and Problems of Financial Theory, Engineering, and Actuarial Calculations

75

Joseph Podsoii extended and corrected Galley’s tables, which made it possible to draw up a year-to-year ‘scale of premiums’. With the growth of cities and trade in the medieval Europe, t h e guilds extended their practice of helping their members in the case of a fire, a shipwrecks, a n attack of pirates, etc.; provided them aid in funerals, in the case of disability, arid so on. Following the step of the 14th-century Genoa, sea insurance contracts spread over virtually all European niarine nations. The modern history of sea insurance is primarily ‘the history of Lloyd’s’, a corporation of insurers and insurance brokers founded in 1689, on the basis of Edward Lloyd’s coffee shop, where ship owners, salesmen, and marine insurers used to gather and make deals. It was incorporated by a n act of British Parliament in 1871. Established for the aims of marine insurance, Lloyd’s presently operates almost all kinds of risks. Since 1974, Lloyd‘s publishes a daily Gazette providing the details of sea travel (and, nowadays, also of air flights) and information about accidents, natural disasters, shipwrecks, etc. Lloyd’s also publishes weekly accounts of the ships loaded in British and continental ports and the dates of the end of shipment. General information on the insurance market, can also be found there. In 1760, Lloyd’s gave birth to a society for the inspection and the classification of all sea-going ships of at least 100 tons’ capacity. Lloyd’s surveyors examine and classify vessels according to the state of their hulls, engines, safety facilities, and so on. This society also provides technical advice. The yearly “Lloyd’s Register of British and Foreign Shipping” provides the data necessary for Lloyd’s underwriters to negotiate marine insurance contracts even if the ship in question is thousands miles away. The great London fire of 1666 gave a n impetus to the development of fire insurance. (The first fire insurance company was founded in 1667.) Insurance against break-downs of steam generators was launched in England in 1854; one can insure employers’ liability since 1880, transport liability since 1895, and against collisions of vehicles since 1899. The first fire insurance companies in the United States emerged in New York in 1787 arid in Philadrlphia in 1794. Virtually from their first days, these companies began tackling also the issues of fire prevention and extinguishing. The first life insurance company in the United States was founded in 1759. The big New York fire of 1835 brought to the foreground the necessity to have reserves to pay unexpected huge (‘catastrophic’) damages. The great Chicago fire of 1871 showed that fire-insurance payments in modern, ‘densely built‘ cities can be immense. The first examples of reinsurance (when the losses are covered by several firms) related just to fire damages of catastrophic size; nowadays, this i s standard practice in various kinds of insurance. As regards other first American examples of modern-type insurance, we can point out the following ones: casualty insurance (1864), liability insurance (1880), insurance against burglary (1885), and so on.

76

Chapter I . Main Concepts, Structures, and Instruments

The first Russian joint-stock company devoted to fire insurance was set up in Sibcria, in 1827, and the first Russian firm insuring life and incomes (The Russian Society for Capital and Income Insurance) was founded in 1835. In the 20th century, we are eyewitnesses to the extension of the sphere of insurance related to ‘inland marine’ and covering a great variety of transported items including tourist baggages, express mail, parcels, means of transportation, transit goods, and even bridges, tunnels, and the like. These days one can get insured against any conceivable insurable risk. Such firms as Lloyd’s insure dancers’ legs and pianists’ fingers, outdoor parties against the consequences of bad weather, and so on. Since the end of 19th century one could see a growing tendency of government involvement in insurance, in particular, in domains related to thc protection of eniployees against illness, disability (temporal or perpetual), old age insurance, and unemployment insurance. Germany was apparently a pioneer in the so-called social security (laws of 1883-89). In the middle of this century, a tendency to mergers and consolidations in insurance business became evident. For instance, there was a consolidation of American life and property insurance companies in 1955-65. A new form of ‘merger’ became widespread: a holding company that owns sharcs of other firms, not only insurance conipazliies, but also with businesses in banking, computer services, and so on. The strong point of these firms is their diversification, the variety of their potentialities. The burden of taxes imposed on an insurance firm is lighter if it is a part of a holding company. A holding company can get involved with foreign stock, which is sometimes inipossible for insurance firms. This gives insurance companies greater leverage: they have access to larger resources while investing less themselves. 4. It is impossible to consider now the insurance practices and theory separated from the practices and the theory of finance and investment in securities. It is as good as established that derivative instruments (futures, options, swaps, warrants, straddles, spreads, and so on) will be in the focus of the future global financ:ial systcni. Certainly, the pricing methods used in finance will penetrate the actuarial science ever more deeply. In this connection, it is worthwhile and reasoriable not to separate actuarial and financial problems related, one way or another, to various forms of risks, but to tackle them in one package. In favor of this opinion counts also the following division of the history of insurance mathematics into periods, proposed by H. Buhlmann, the well-known Swiss expert in actuarial science. The first period (‘insurance of the first kind’) dates back t o E. Galley who, as already mentioned, drew up insurance tables (1693) based on the assumption that the mortality rate in a fixed group of people is a deterministic function of time. The second period (‘insurance of the second kind’) is connccted with the introduction of probabilistic ideas and the methods of probability theory and statistics into life insurance and other types of insurance.

3. Aims arid PIoblems of Financial Theory, Engineering, and Actuarial Calculations

77

The third period (‘insurance of the third kind’) can be characterized by the use on a large scale of financial instruments arid financial engineering to reduce insiirance risks. Mathenlatics of the insurance of the second kind is based on the Law of large numbers, the Central limit theorem, and Poisson-type processes. The theory of insurance of the third kind is more sophisticated: it requires the knowledge of stochastic calculus, stochastic differential equations, martingales and related c011cepts, as well as new methods, such as bootstrap, resampling, simulation, neural networking, and so on. A good illustration of the above idea of the reasonableness and advantageousness of an integrated approach to actuarial and financial problems of securities markets is the efficiency of purely financial, ‘optional’ method in actuarial calculations related to reinsiirance of ‘catastrophic events’ [87]. In the global market, one says that a n accident is ‘catastrophic’ if ‘the damages exceed $5111 and a large number of insurers and insured are affected’ (an extract from “Property Claims Services”, 1993). The actual size of the damages in some ‘catastrophic cases’ is such that no single insurer is able (or willing) to insure against such accidents. This explains the fact that insurance against such events becomes not merely a joint but an international undertaking. Here are several exaniples of damages incurred by ‘catastrophic’ accidents. From 1970 through 1993 there occurred on average 34 catastrophes each year, with annual losses arnoiinting to $ 2 . 5 billion. In most cases, a ‘catastrophic’ event broiiglit damages of less t h a n $250m. However, the losses from t h e hurricane ‘Andrew’ (August 1992) are estimated at $ 13.7 billion, of which only about $ 3 billion of damages were reimbursed by insurance. In view of large sums payable in ’catastrophic’ cases: hurricanes, earthquakes, floods, the CBOT (Chicago Board of Trade) started in December of 1992 the futures trade in catastrophe insurance contracts as a n alternative to catastrophe reinsurance. These contracts are easy to sell (‘liquid’), anonymous; their operational costs are low, arid the supervision of all transactions by a clearing house gives one the confidence important in this kind of deals. Moreover, the pricing of such futures turned out in effect to reduce to the calculation of the rational price of arithmetic Asian call options (see the definition in 5 l c ) . See [87]for greater detail.

3 3c.

A Classical Example of Actuarial Calculations: the Lundberg-Cram& Theorem

1. Oddly, around the same time when L. Bachelier introduced a Brownzan motaon to describe share prices and laid in this way the basis of stochastac jinanczal mathematzcs, Ph. Lundberg in Uppsala (Sweden) published his thesis “Approxirnerad franisrollnirig av sannolikhetsfunctionen. Atersforsakring av kollektivrisker” (1903), which became the cornerstone of the theory ofznsurance (of the second kind) and where he developed systematically Poasson processes, which are, together with

78

Chapter 1. Main Concepts, Structures, and Instruments

Brownian motions, central objects of the general theory of stochastic processes. In 1929, on initiative of several Swedish insurers, the Stockholm University established a chair in actuarial mathematics. Its first holder was H. CramQ and this marked the starting point in the activities of the 'Stockholm group', renowned for their results both in actuarial mathematics and general probability theory, statistics, and the theory of stochastic processes. We now forniiilate the classical result of the theory of actuarial calculations, the Lundberg -Crume'r fundamental theorem of risk theory. Rt

t

0

Ti

T2

T3

t

k=l

where 71

is the initial capital,

c is the rate of the collection of premiums, ((k)

is a sequence of independent, identically distributed random variables with distribution F ( z ) = P{& x}, F ( 0 ) = 0, and expectation p =

<

EE1

N

< 00,

= (Nt)tao is a Poisson process:

. . and it where insiiraiice claims are submitted a t the instants Ti,T2,. is assumed that ( T k + 1 - Tk),+>, are independent variables having exponential distribution with parameter A: P{Tk+1

-

Tk

3 t}=

3. Aims and Problems of Financial Theory, Engineering, and Actuarial Calculations

Clearly,

+ (C

ERt = ‘u.

-

Xp)t = u

79

+ pXpt,

where the relative safety loading p = c / ( X p ) - 1 is assumed to be positive (the condition of positive n e t p r o f i t ) . One of the first natural problenis arising in connection with this model is the calculation of the probability P(T < m) of a ruin in general or the probability P(T 6 t ) of a ruin before t i m e t , where

Rt 6 O}. THEOREM (Luiitlberg.~Cram~r). Assume that there exists a constant R > 0 such that T

= inf{t:

Then P(T

< w) 6 e-Ru,

where u is the initial capital. The assumptions in the Lundberg-Cram& model can be weakened. while the model itself can be made more complicated. For instance, we can assume that the risk process has the following form: Nt

Rt

+ (ct + gBt) C &,

= ‘1~

-

k= 1

where ( B t ) is a Brownian motion and ( N t ) is a Cox process (that is, a ‘counting process’ with random intensity; see, e.g., [250]). In conclusion, we briefly dwell on the question of the nature of the distributions F = F ( x ) of insurance benefits. As a matter of a pure convention, one often classifies insurance cases leading to payments with one of the following three types: 0 ‘normal’, ‘extremal’, 0 Lealastrophie’. To describe ‘normal’ events, one uses distributions with rapidly decreasing tails (e.g., an exponential distribution satisfying the condition 1 - F ( z ) e-” as N

x t 00). One describes ‘extremal’ events by distributions F = F ( z ) with heavy tails; e.g., 1 - F ( x ) x-”, a > 0, as x + 00 (Pareto-type distributions) or N

x-p

P

Z>P> 1 - F ( z ) = exp {-(7) } ?

with p E ( 0 , l ) (a Weibull distribution). We note that the Lundberg-Cram&- theorem relates to the ‘normal: case and cannot be applied to large payments. (One cannot even define the ‘Lundberg coefficient’ R in the latter case; for the proof of the Lundberg-Cram6r theorem see, e.g., [439].)

Chapter II. Stochastic Models . Discrete T i m e 1. N e c e s s a r y p r o b a b i l i s t i c concepts and s e v e r a l m o d e l s of t h e dynamics of market p r i c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5 la.

Uncertainty and irregularity in the behavior of prices . Their description and representation in probabilistic terms . . . . . 5 l b . Doob decomposition . Canonical representations . . . . . . . . . . . . . fj l c . Local martingales. Martingale transformations . Generalized mart ingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi and conditionally Gaussian models . . . . . . . . . . . . . . . . 5 l e. Binomial model of price evolution . . . . . . . . . . . . . . . . . . . . . . . . 5 If . Models with discrete intervention of chance . . . . . . . . . . . . . . . . .

81 89 95 103 109 112

2 . L i n e a r stochastic m o d e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

52a . Moving average model MA(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2b . Autoregressive model A R ( p ) . . . . . . . . . . . . . . . . . . . . . . . . . . . fj2c . Autorcgressive and moving average model ARMA(p,y) and integrated model ARlMA(p,d . y) . . . . . . . . . . . . . . . . . . . . . 5 2d . Prediction in linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 125 138 142

3 . N o n l i n e a r stochastic c o n d i t i o n a l l y Gaussian models . . . . . . .

53a . ARCH and GARCH models . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 5 3c . Stochastic volatility models . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 3b . EGARCH, TGARCH . HARCH, and other models

4 . S u p p l e m e n t : dynamical chaos m o d e l s ........................... fj4a . Nonlinear chaotic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54b. Distinguishing between ‘chaotic’ and ‘stochastic’ sequences . . . . .

153 163 168 176 176 183

1. Necessary Probabilistic Concepts and Several Models of the Dynamics of Market Prices

s la. Uncertainty and Irregularity in the Behavior of Prices. Their Description and Representation in Probabilistic Terms 1. Assume that we nicasurc time in days n = 0 , 1 , 2 , . . . , and let

be the market price of a share, or the exchange rate of two currencies, or another finwricial index (of iirilirriited ‘life span’, by contrast to? say, bond prices). An cwipirical study of the S,, n 3 0. shows that they vary in a highly irregular way; tlicy fliict,nate as if (using the words of M. Kendall; see Chapter I, 2a) . . . the D e m o n of Cha,nce drew a random number . . . and added i t t o the current price t o d e t w m i n e the next . . . price”. Beyond all doubts, L. Bachelier was the first to describe the prices (S,),>O using the concepts and the methods of probability theory, which provides a framework for the study of empirical phenomena featured by both statistical uncertainty and stability of statistical frequencies. Taking the probabilistic approach and using A. N. Kolmogorov’s axiomatics of probability theory, which is generally accepted now, we shall assume that all our corisiderations are carried out with respect to some probability space

(a, 3,P), wlicre

R is the space of elementury e,ue,nts

w (’market situations’, in the present context,); SFis some 0-algebra of siibsets of R (the set of ‘observable m e r k e t e v m t s ’ ) ; P is a probability (or probability measure) on 9.

82

Chapter 11. Stochastic Models. Discrete Time

As pointed out in Chapter I, 3 l a , t i m e and dynamics are integral parts of the financial theory. For that reason, it seems worthwhile to define our probability space ( 0 , 9 P) , more specifically, by assuming that we have a pow IF = (9,),>0 of a-algebras such that . . . 9, 9. 90 9 1

c

c

c

c

The point in introducing this flow of nondecreasing n-subalgebras of 9, which is also called a filtration, becomes clear once one have accepted the following interpretation: 9,is the set of events observable through t i m e n. We can express it otherwise by saying that 9nis the ‘information’ on the market situation that is available to an observer up to time n inclusive. (In the framework of the concept of an ‘efficient’ market this can be, e.g., one of the three n-algebras 9:,9:,and 9;; see Chapter I, 0 2a.) Thus, we assume that our underlying probabilistic model is a filtered probability space (a, 9, (9n)n>O,p), which is also called a stochastic basis. In many cases it seems reasonable to generalize the concept of a stochastic basis by assuming that, instead of a single probability measure P, we have an entire family 9 ‘ = {P} of probability measures. (The reasons are that it is often difficult to single out a particular measure P.) Using the vocabulary of statistical decision theory we can call the collection (Q9, (9,),20, 9) afiltered stochastic (statistical) experiment. 2. Regarding 9,as the information that had been accessible to observation through time n. it is natural to assume that

S, is 9,-measurable, or that (using a more descriptive language), prices are formed on the basis of the developnients observable on the market up to time n (inclusive). Bearing in mind our interpretation of S, as the ‘price’ (of some stock, say) a t time n, we shall assume that S, > 0, n 3 0. We now present the two most common methods for the description of the prices

s = (Sn)n>O.

The first method, which is similar to the formula for compound interest (see Chapter I, 5 l b ) uses the representation

s, = S O P ” ,

+ + +

(1)

where I f T L= ho hl . . . h, with ho = 0 and the random variables h, = h 7 1 ( ~ ) r n 3 0 , are 9,-measurable. Hence

Ifn

S7L

= ln-

SO

1. Probabilistic Concepts and Several Models of t h e Dynamics of Market Prices

83

and the 'logarithmic returns' can be evaluated by the formula

(3) where AS,, = S, - Sn-l. We now set h,, = __

and

&,

Hn=

s71-1

(4)

1
Then we can rewrit,e (1) as

l
or, equivalently, as

The representation (5) is just the second method of the description of prices. It is equivalent to the 'simple interest' formula. Let 8(@71be the expression on the right-hand side of (6):

We call the stochastic sequence

defined by this expression the stochastrc exponentral generated by the variables ff = (ff,L),lao,f f o = 1, or the Dole'ans ezponentzal. Thus, we can say that the first method for the description of prices uses the usual exponential S, = SOeHn, A

A

A

while the srcond method involves the stocha.strc exponential:

where

Chapter 11. Stochastic Models. Discrete Time

84

which is equivalent to the following representation:

sn

= Hn

+

( e A H k - A H I , - 1). l
It is also clear from (3) arid (4) that

h

where tirL = A g T L> -1 by our assumption that S,n > 0. I t is worth rioting that the stochastic exponential satisfies the stochastic dzflerence equation

A8(iqT= L qG)n-&n,

(11)

which is an irrirriediat,e consequence of (7).

Remark 1. Formulas (1) and (8) are related to discrete time n = O , l , . . . . In the case when the prices S = (St)t>o evolve in continuous time t 3 0 one m u ally assiinies that the processes H = (Ht)t>O and H = (Ht)t>o are semimartingales (see Chapter 111, 5 5b). Then by the It6 formula (Chapter 111, 5 5c; see also [250; Chapter I, 3 4e]) we obtain h

where A

Ht = Ht

+ -21( H C ) t+ C

(eAHs - 1 - A H s )

(12)

O<s
arid

& ( E )= ( 8 ( f i ) t ) t 3is0 the stochastic

exponential:

satisfying (cf. (11)) the linear stochastic differential equation

(s)

(In (12) and (13) we denote by ( H C )and the quadratic characteristics of the continuous martingale components of the semimartingales H and if; see Chapter 111, 55a. As regards stochastic differential equations in the case when H is a Brownian motion, see Chapter 111, 5 3e.)

1. Piobabilistic Coiicrpts and Several Models of the Dynamics of Market Prices

85

Thus, we have the following representations that are the counterparts of (1) and (8),respectively, in the continuous-time case:

st =

(15)

and

h

where the process H = (Ht)t>o is related to H = (Ht)t>o by (12). The series in (12) is absolutely convergent because, with probability one’ a semimartingale makes only finitely many ‘large’ jumps (lAH,I > on each interval [ O , t ] and lAHSl2 < 00. (See Remark 3 in Chapter 111, 5 5b.)

f)

C O<s<.t

Rernark 2 . By ( 3 ) and (4),

+ h,) h

h,, = ln( 1 and

Clearly, we have

hTLz h, for small valiies of

117b; moreover, h

h,

-

1 h,, = - h i 2

1 + 4; +. 6

3. Wc iiow tiwc*llon the problem of the description of the probability distributions for the sequences S = (S,),>o and H = (H,),ao. Taking the viewpoint of classical probability theory and the well-developed ‘stacould ~ be a Gaussaan tistics of normal distribution’ it would be nice if H = ( H n ) 7 a > (norinally d z s t n h u t e d ) sequence. If we set

then the properties of such a sequence are completely determined by the twodzmenszonal distributions of the sequence h = (h,),>l, which can be characterized by the expectations n 3 1, p, = Ell,,,

Chapter 11. Stochastic Models. Discrete Time

86

and the covariances Cov(h,,

h7n) f E h , h ,

-

rn: n 3 1.

Eh,Eh,,

This assumption of normality can considerably facilitate the solution of many problems relating to the properties of distributions. For instance, the ' Th.eorem o n normal correlation' (see, for example, - [303; Chapter 131) delivers an explicit formula for the conditional expectation h.n+l = E(h,+l 1 h i , . . . h n ) , which is the optimal (in the mean-square sense) estimator for hn+l in terms of h l , . . . , h,. Namely, ~

i=l

where the coefficients ai are evaluated in ternis of the covariance matrix (see [303; Chapter 131 and also [439; Chapter 11, 5 131). Formula (22) becomes particularly simple if h l , . . . , h, are independent. In this case Cov(hn+1, hi) ai = , Dhi and we obtain the estimator

The estimation error

n71+1 = E ( & + l - h,+1)2 can be expressed by the forniula

We note that if

is the density of the normal distribution with parameters ( p , a 2 ) ,then (see Fig. 11)

1. Probabilistic Concepts and Several Models of the Dynamics of Market Prices

87

t

I

-1

1

FIGURE 11. Graph of the density ~ p ( ~ , ~of) (the x )standard normal distribution. The area of the shaded part is approximately 0.6827

In the same way,

/

p+1.650

I.(

’o(p,g2)

/~-1.650

dz

= 0.90.

By the Gaussian property we obtain

-

therefore

P { Ihn+1 - h,+l/

61.65dG)

zz 0.90

by (25). Hrnce we can say that the expected value of hn+l lies in the confidence interval with - probability close to 0.90. This means t h a t , in 90% cases the predicted value S,+l of the market price S,+l (calculated from the observations h l , . . . , h,) lies in

4. However attractive, this conjecture on the ‘normality’ of the distribution of the variables h,, n 3 1, must be taken with caution, because the empirical analysis of much financial data shows (see Chapter IV) that (a) the number of values in a sample that lie outside the ‘confidence’ intervals 1 , [h, - kZ,,h, En]with 5 = 1 , 2 , 3 (here h, = hi is the sample mean and n i=l l?, is the standard deviation:

+

is considerably larger than it should be if this conjecture were true. More geometrically this means that the ‘tails’ of the empirical densities are ‘heavy’, i.e., they decrease a t a much lower rate than it should occur for Gaussian distributions;

Chapter 11. Storhastic Models. Discrete T i m e

88

(here 7ji2 a ~ i d,614 are the empirical second and fourth niornents) is markedly positive (the kurtosis of a riorrnal distribution is equal to zero), which shows that the distribution density has a high peak around the mean value (is leptokurtotic):

30.-

20.-

10.-

0 .-I -0.006

-0.001

0.004

FIGURE12. Empirical density o f the one-dimcnsional distribution of the variables hn, ‘n < 300, governed by the H A R C H ( 1 6 ) model (see 3b). The continuous curve is the density of the corrcsponding normal distribiition N ( m ,c2)wit,li 712 = h300 and u z = Z&

Arguably, the most strong assumption about the general properties of the distributioii of the / i n , apart from the Gaussian property, is the following one: these uariables a m independent and

identically distributed. Under t,his assiirription it is easy to carry out the analysis of the prices S, = SoeHn, where H,, = h 1 h,, by the st,andard methods of probability theory designed expressly for such situations. It is clear, however, that the assumption of the independence of the h , underniines all the hopes that the ‘past’ information could be of any use in the prediction of the ‘fiitiire values’. In practice, t,hc situat,ion is more favorable, since numerous studies of financial time series eriat)le one to say t h a t , as mentioned already, the variablcs ( / i n ) are n o n Gaussian and dependen,t, although they can be iincorrelated arid their dependence can he rathcr weak. The easiest way to demonstrate certain dcgree of depcndence is to corisider t,hc empirical correlations for the Ih,,l or h:,. rathcr then for the h,,,. (In the model of stochastic volatility discussed below we have Cov(h,,, h,) = 0 for n # nb, but the values of C o v ( h i , and Cov(lh,l, Ihm,l)are far from zero; see $ 3 ~ )

+ +

ILL)

1. Probabilistic Concepts and Several Models of the Dynamics of Market Prices

89

Ib. Doob Decomposition. Canonical Representations 1. We shall assume that tlie variables h,, n 3 1, in the model

have finite absolute first moments, i.e., E(h,( < 03 for n 3 1. The Doob decomposition that we discuss later in this section indicates that one should study the sequence H = (H,) in its dependence on t h e properties of the filtration (F,), i.e., of the flow of 'information' gn accessible t o a n 'observer' (of tlie securities market, in the context adopted here). Since E(h,,l < 30 for 71 3 1, the conditional expectations E(hT1I F,-l) are well defined and (Ho = 0, 90= (0,n))

In otlicr words, setting

and

we obtain the following Do00 decomposition for H = ( H r L ) :

where a) the sequence A = ( A n ) l l a where ~, A0 = 0; is predictable, i.e.,

A , are 9,-1-measurable,

n 3 1;

b) the sequence M = ( M T 1 ) Twhere L a ~ ,MO = 0, is a martingale, i.e.,

the M , are .Fn-measurable arid EIM,I

< 00 for n 3 1.

Chapter 11. Stochastic Models. Discrete Time

90

R e m a r k . Assume that, besides the filtration (9,), we have a subfiltration (%,), where %, C_ 9,and 9, Yn+l. Then we can decompose H = (H,) with respect to the flow (%), in a similar way to (5): H n =

n

n

k=l

k= 1

C ~ ( h Ik9 k - 1 ) + C (hk

-

~ ( h Ik9k-l)).

The sequence A = ( A , ) of the variables n

A,

=

c

E ( h k I Yk-1)

k l

is (%,)-predictable (i.e., the A , are Yfn_1-measurable), but M = (M,), where n

Ma =

c

(hk

-

E(hk

I%l))

k=l

i s , generally speaking, n o t a martingale with respect to (gn) since the 11k are measurable with respect to the 9 k , but not necessarily with respect t o the ‘&. It must be pointed out that if, besides ( 5 ) , we have another decomposition,

H , = A:, +MA, where the sequence A’ = (A;, 9,) is predictable (with respect to the flow (s,)), Ah = 0, and M’ = (MA,9,)is a martingale, then A:, = A , and MA = Mn for all n 3 0. For we have A:,+,

-

A;

= (&+I

-

A,)

+ (hfn+l-

M n ) - (MA+i - MA)

Hence, considering the conditional expectations E( . 1 9,) of both sides we see that - A ; = An+l - A , (since the and A,+1 are 9;,-measurable). However, Ah = A0 = 0, therefore A; = A, and MA = M , for all n 3 0. Hence the decomposition (1) with predictable sequence A = ( A , ) is unique. We note also that if E(hk 19k-1) = 0 for k 2 1 in the model under consideration. then the sequence N = (H,) is itself a martrngale by (2). We now present a n example of the Doob decomposition, which shows clearly that, for all its simplicity, it is ‘nontrivial’.

EXAMPLE. Let XO= 0 and X, = (1 Bernoulli variables such that P([,

+.. .+En, = +1) =

n 3 1, where the En are independent 1 2

-.

1. Probabilistic Concepts and Several Models of the Dynamics of Market Prices

Sgn x =

{

91

1, x > 0, 0, x = O , -1, ic < 0.

Thus, the martingale M = (M,L)n21in (5) is now as follows:

Mri =

C

(SgnXk-l)Ax,.

1
Further,

E ( h L I FTL-1) = E(IXn-1+ En1 I xn-1) - IXn-11. The right-hand side vanishes on the set { w : Xn-l = z } with z # 0, while if i = 0, then it is equal to one. Hence n

E(h,i 1 9 i - l ) = N ( 1

< k 6 71: X k - 1

= 0},

i= 1

where N { 1 6 k 6 n : Xk-l = 0} is the number of k , 1 6 k 6 n , such that X k - 1 = 0. Let L,,(O) = N{O 6 k n - 1: XI, = 0} be the number of zeros in the sequence ( X k ) O < k < n - l . Then

<

IXnI =

(SgnXk-lPXk

+L(0)l

l
which is a discrete analogue of the well-known Tnnaka formula for the absolute value of a Brownian motion (see Chapter 111, 5 5c). Hence, in particular.

Xn Since -

fi

-

ELn(0) = EIXnI N(0,l ) , it follows that EIX,,) EL,(O)

N

-

fi.

This is a known result of the average number of zeros in a symmetric random Bernoulli walk (see, e.g., [156]).

Chapter 11. Stochastic Models. Discrete Time

92

2. Let M = (M,l)TL2:1 be a square rntegrable martingale (EM: < 3c. n 3 1) with Mo = 0. Then setting H7) = Ad: in the decomposition ( 2 ) we obtain the following representation:

l
where AM: = M; w e now set

-

MZ-l. ( M ) n=

c

W M ; I ,%-I),

l
m, =

(AM;- E(AM;

(7) 19k-1)).

l
Using this notation we can rewrite (6) as follows:

M: = (M)7L+ mn;

(8)

the (predictable) sequence ( M ) = ( ( M ) n ) n 2 1is called here the quadrutic ch,nructeristic of the niartingale M (cf. Chapter 111, 5 5b). We note that since A4 = (M.,])is a martingale, it follows that

€(AM; I9k-l)

1

= E( (AMk)2 9k-1).

(9)

This property explains why one often calls the quadratic characteristic ( M ) also the predictable quadratic variation of the (square integrable) martingale M . In that case one reserves the term quadratic variation for the (in general, unpredictable) sequence [MI = ([M],),21 of the variables (cf. Chapter 111, 5 5b)

k< n

3. We now assume that the sequence H = ( H n ) is itself a martingale. and even a square integrable one, i.e., E(AHk 1 F k - 1 ) = E ( h k 19k-l) = 0, E h i < a.k 3 1. Then k<71

The variables E(h,; 1 9 k - 1 ) contributing t o the quadratic characteristic ( H ) , define +x the volntilrty of N arid, in large measure, its properties. For example, if with probability one, then the square integrable martingale H satisfies the strong

as n

+ 00. (See [439; Chapter VII] 3 51.)

1. Probabilistic Concepts and Several Models of thc Dynamics of Market Prices

The collection of variables (E(h;

93

I , 9 k - 1 ) ) k >, l will play a n important role in our

subsequeiit analysis of ‘financial’ time series S = ( S , ) with S, = SoeHn. In our considerations of these series, using the vocabulary of the financial theory we shall call the sequence ( E ( l z i I 9 k - 1 ) ) ~the ~ ~stochastic volatility. (See 5 3 for a closer look at volatility.) If the coiiditional expectations E(hi 1 9 ; k - 1) coincide with the unconditional ) a sequence of independent random variables and 9 k - 1 = ones (e.g.. if ( h T Lis a(h1,.. . , h k - l ) is the 0-algebra generated by h l , h n - l ) , then the volatility is = E h i , k 3 1. that are t,he standard measures for a mere collection of variances the dispersion (chmgeabality) of the h k (here we assume that Ehk = 0). ~

02

< 00 for k 3 1. Act,ually, we required this assumpt,ion only to ensure that the conditional expectations E(hk I 9k-1), k 3 1, were well defined. Thus, there arises a natural idea of using the decomposition (2) also in the (more general) case when the conditional expectations E ( h k 1 9 k - l ) are well defined and finite and the condition El& < 00 is not necessarily satisfied. To this end we recall that if Elhk.1 < 00, then the conditional expectation E(hk I . F k - 1 ) is (according to A. N. Kolmogorov) the 9k-l-measurable random variable such that 4. In deducing the Doob decomposition (2) or (5) we assumed that Elhkl

/I

h k dP (13) /A for each A E ,Fk-l. The existence of such a variable is a consequence of the Radon--NikodS;rn theorem (see, e.g., [439; Chapter 11, 5 71). Note, however, that the condition Elhkl < cx, is not at all necessary for the existence of an cFk-l-rneasmable variable E(hk 1 9,t-l)satisfying (13). For instance, if h k 3 0 (P-a.s.), then we can define it without assuming that E h k < 30. Hence the idea of the following definition of a generalized expectation, which we shall also denote by E(hk j.Fk-1). We represent h k as the following sum:

E ( h k I y k - 1 ) dP =

where h: = max(hk, 0) arid h i = - rnin(hk, 0). We assume that we have already defiried E ( h : I 9 - l ) ( u ) and E(hL 19k_l)(w)so that inin{ E ( h c I .?F~-~)(LJ), E ( h i I Fk-l)(w)}

< XI

(14)

for all w E [I. Tlieii wc set

(here and throughout “E”means by definition), and we call E(hk 1 9 k - l ) the generalized conditional expectation,.

Chapter 11. Stochastic Models. Discrete T i m e

94

If ElhkI < 00,then this generalized cxpectat,ion is clearly the same as the usual conditional expectation. IfE(1hkl I9;k-1)(w)
R e m a r k . Proceeding in accordance with the general spirit of probability theory, which usually puts weight on the verification of particular properties ‘for almost all w E f l ’ ? rather then ‘for all’ of them, we can easily construct an ’almost-all’ version of the above definition of the generalized conditional expectation by setting E(hk Ic5Fk-1)(w) to be arbitrary on the zero-probability set where (14) fails. We now consider the representation (2). The right-hand side of (2) is surely well defined if E(lhkl I 9 k - 1 ) < cx) for k 3 1 (and for all or almost all w E 0 ) . In that case we shall say that (2) is a generalized Doob decomposition of the sequence H = (Hn)n>/1. 5 . We now discuss a similar decomposition (or, as we shall also call it, a represent a t i o n ) in the case when the conditional expectations E ( h k I .2J,z-l) (either ‘usual’

or generalized) are n o t defined. Then we can proceed as follows. We represent h k as a sum:

where a is a certain positive constant (one usually sets a = 1) and I ( A ) (we shall also write ZA or I A ( w ) )is the indicator of the set A (i.e., I A ( w ) = 1 if w E A and I A ( w )= 0 otherwise). Now, the variables h k I ( I l i k 1 6 a)have well-defined first absolute moments, therefore

where (A!:fL)),lblis a predictable sequence, (M:L‘a))T121is a martingale, and hkZ(Ilik1

> a ) )n > l is the sequence of ‘large’jumps.

1. Probabilistic Concepts and Several Models of t h e Dynamics of Market Prices

95

Using the vocabulary of the ‘general theory of stochastic processes’ (see Chapter 111, 5 5 and [250; Chapter I, $ 4 ~ 1 we ) call (17) the canonical representation of H . We note that if, besides (17), we have another representation of H in the form

with predictable sequence (AL)and martingale (MA),then, of necessity, A:, = A$a) ( and M:, = MAGa). In other words, there exists a unique representation of the form (18). This justifies the name canonical given to this representation (17).

lc. Local Martingales. Martingale transformations. Generalized Martingales 1. In the above analysis of the sequence H = ( H n ) based on the Doob decomposition ( 5 ) and its generalization these were the concepts of a ‘martingale’ and ‘predictability’ (and, accordingly, the martingale M = ( M n ) and the predictable sequence A = (A71) involved in the representation of H ) that played a key role. This explains why one often calls the subsequent stochastic analysis martingale or stochastic calculus, meaning here analysis in filtered probability spaces, i.e., probability spaces distinguished by a special structure, a flow of a-algebras (Sn) (a ‘filtration’). It is precisely with this structure that stopping times, martingales, predictability, sub- and supermartingales, and some other concepts are connected. Perhaps, an even more important position than that of martingales in the modern stochastic calculus is occupied by the concept of local martingale. Remarkably, local martingales form a wider class than martingales, but, retain many important properties of the latter. We now present several definitions. Let ( s 2 , 9 ,(971), P) be a stochastic basis, i.e., a filtered probability space with discrete time n 3 0. DEFINITION 1. We call a sequence of random variables X = (X,) defined on the stochastic basis a stochastic sequence if X , is .9,-measurable for each n 2 0.

To emphasize this property of measurability one writes stochastic sequences as (X7’,S,), thus incorporating in the notation the a-algebras Snwith respect to which the X , are measurable.

X

=

DEFINITION 2. We call a stochastic sequence X = (X,, 9 : n ) n 2 ~ a martingale, a supermartingale, a submartingale

Chapter 11. Stochastic Models. Discrete Time

96

if EIX,,/ <

00

for each n 3 0 and if (P-as.)

E(X,, 1 & - l ) = Xn-l, E(X,

I 9,-1) < Xn-1,

E(X,

19,-1)

3 Xn-1,

respectively, for all n 3 1. Clearly, EX,, = Const (= EXo) for a martingale, the expectations are nonincreasing (EX, EX,-1) for a supermartingale, and they arc nondecreasing (EX,, 3 EX,,-l) for a submartingale. A classical exaniple of a martingale is delivered by a Le‘vy m a r t i n g a l e X = (X,) with X,, = E([ I 2JTL). where [ is a 9-measurable random variable such that ElEl < M . This is a unifoorndy h t e g r u b l e martingale, i.e., the family {X,,} is uniformly integrable: ~ I I ~ E ( \ X , ~ I (> ~X C)) , , ~+ 0 as C + 00.

<

n

In what follows, we denote by AUI the class of all u n i f o r m l y integrable m a r t i n gales. We denote the class of all martingales by A. In the case when the martingales in question are defined only for n 6 N < 00, the concepts of a niartingale and of a uniformly integrable martingale are clearly the same (.A= h l u ~ ) . Sometimes, when t,here is a need to point out the nieasure P and the flow with respect to which the m a r t i n g a l e property is considered, one also denotes the classes AUIand .dlby . A ~ I ( P(9,)) , and A ( P , (9,)). DEFINITION 3. We call a stochastic sequence z = (z,, martingale diflerence if (here we set 90= {0, a})

E(x,,

19n-l) =

0

(P-a.s.),

9,),21

with E/znl

< cx; a

n 3 1.

Clearly, if rc = (z,,) is such a sequence, then the corresponding sequence of sums (X,,,9,1,),where X , = XO z1 . . . + z, is a martingale. Conversely, with each martingale X = (X,,9,)we can associate the m a r t i n g a l e difference x = (z,, ,%,) with z, = AX,, where A X , = X , - X,-1 for n 3 1 and AX0 = X O for n = 0.

X

=

+

+

DEFINITION 4. We call a stochastic sequence X = (X,,S,) a local m a r t i n g a l e ( s u b m a r t i n g a l e , s u p e m a r t i n g a l e ) if there exists a (localizing) sequence (7k)k21 of Markov tinies (i.e., of variables satisfying the condition { w : ~k n } E 9,, n 3 1; see also Definition 1 in 3 If) such that ~k q.+l (P-a.s.), Q t 00 (P-a.s.) as k + 00, and each ‘stopped’ sequence

<

X T k=

(XTkAi;n,9,)

is a martiiigale (submartingale, supermartingale).

<

1. Protmbilistic Concepts and Several Models of the Dynamics of Market Prices

97

Remark 1. One often includes in the definition of a local martingale the requirement that the sequence X Q be not merely a martingale for each k 3 1, but a uniformly integrable martingale (see, e.g., [250]). We note also that sometimes, intending to consider also sequences X = (X,, 9,) with nonintegrable ‘initial’ random value X o , one defines stopped sequences X Q in a somewhat different way: X7k

=

(XTk*J(Q > 0),9,).

We shall write Aloe or A l o c ( P , (9,)) for the class of local martingales. By Definition 4, each martingale is a local martingale, so that

./dl

c Aloc.

If X E .Alloc and the family of random variables

C = { X 7 : 7 is a finite stopping time} is uriiforrrily integrable (i.e., sup E{ / X T / I ( l X 7 13 C)}

4

0 as C

4

m ) , then

X,€C

X is a martingale ( X E A ) ;moreover, it is a L6vy martingale: there exists a n integrable, 9-measurable random variable X , such that X , = E(X, 1 9,)Thus, . X E 4~1 in this case. (See [250; Chapter 1, 3 le] or [439; Chapter VII, 5 41 for greater detail.) DEFINITION 5. We say that a stochastic sequence X = (Xn,gn),>o is a generalized rri,artiiigale (submartingale, supermartingale) if ElXo 1 < 00, the generalized conditional expectations E(X,, 1 9 , - 1 ) are well defined for each n 3 1 and

E(X,, 1 9?,-1) = XT1-l

(P-as.)

(accordingly, E(X,, 1 .FTL-1)3 Xn-l or E(XT,I 9 : 7 a - 1 ) 6 X,-1).

Remark 2 . By the definition (see 5 l b ) of the generalized expectation E(X, I 9 , - 1 ) and thc ‘martingale’ equality E(X, 1 9,-1) = X,-1 we obtain automatically that E(lX,l I c F T 1 - l )< 00 (P-as.). This means that the conditional expectation E(X, I 9 ,- 1 ) is not merely well defined, but also finite. Hence we can assume in Definition 5 that E(IX,,l 1 &-1) < 00 ( P - a s . ) for ri 3 0. DEFINITION 6. We call a stochastic sequence II: = ( z n ,9,),>1 a generalized martingale difference (,snbmartingale difference, supermartingale difference) if the generalized c:onditiorial expectations E(z, I 9,-1) are well defined for each n 3 1 and

I YTL-1)= 0 (P-as.) 1 cFT,-l)3 0 or E(z, 1 9 , - 1 ) < 0 (P-as.)). E(z,

(respectively, if E(:c,,

Rem.ark 3 . As in Remark 2, it can be requested in the definition of a generalized martingale difference that E(I:cni I .F,-l) < 00 (P-as.) for n 3 1.

98

Chapter 11. Stochastic Models. Discrete Time

DEFINITION 7. Let M = (M,, 9,) be a stochastic sequence and let Y = (Y,, 9,-1) be a predictable sequence (the Y, are 9:,-1-measurable for n 3 1 and YO is &,-measurable). Then we call the stochastic sequence

Y.M= ((Y.Wn,~n), where

( y M):, = YO MO f '

ykahfk,

'

l
the transformation of M by means of Y . If, in addition, M is a martingale, then we call X = Y . M a martingale transformation (of the martingale M by means of the (predictable) sequence Y ) . The following result shows that the concepts introduced by Definitions 4, 5 , and 7, are closely related in the discrete-time case.

THEOREM. Let X = (Xn, 9,),20 be a stochastic sequence with EJXoJ< 00. Then the following conditions are equivalent: (a) X is a local martingale (X E Aloe); (b) X is a generalized martingale (X E G A ) ; (c) X is a martingale transformation (X E A T ) , i.e., X = Y . M for some predictable sequence Y = (Y,, 9,-1) and some martingale M = (M,, 9,). Proof. (c)==+(a). Let X E .AT and let

<

where Y is a predictable sequence and M is a martingale. If IYkl C for k 3 1, then X is clearly a martingale. Otherwise we set rJ = inf{n - 1: IYnl > j}. Since the Yn are 9,-1-measurable1 the rJ are stopping times, rJ I- 00 as j + 00,and the ~ again of the form (1)with bounded Y: = Ykl{k T ~ } 'stopped sequences' X T are Hence X E Aloe. (a)==+(b). Let X E &lOc and let ( r k ) be the corresponding localizing sequence. Then EIXZ+lJ < co and E(JX,+l\ 1 Sn)= E(/XZ+ll 1 5Fn) on the set { T k > n} E SrL. Hence E(IX,+ll 1 9,) < 00 (P-a.s.). In a similar way, on the same set {q> n} we have

<

= X;l" = Xn, E(Xn+119n) = E(X;I"+1 19,)

so that X E G A . (b)==+(c). Let X E G A . We set

A,(k)

= { w : E(IX,+l/ I 9 n )

E [ k , k + I)}.

.

1. Probabilistic Concepts and Several Models of t h e Dynamics of Market Prices

99

Then k2O

n

C

is a .F%,,-nieasiirablerandom variable with E(u,, 1 cF,-l) = 0. Hence Mn = ui is i=l a niartingale (we set A40 = 0) and (1) holds for Y = (Yn). where k>O

so tjhat X E A T . 2. We shall dernoristrate the full extent of the importance of the concepts of a local martingale, a niartingale transformation, and a generalized martingale in financial mathematics in Chapter V. These concepts play a n important role also in stochastic calculus, which can be shown, for example, as follows. Let X = ( X , , .'9T,)n20 be a local submartingale with localizing sequence ( r k ) , where r k > 0 (P-a.s.). Then for each k we obtain the following decomposition for x(Tk) = ( X ~ A T~97,): ~, XnATk= A P )

+M p )

with predictable sequences (A,7'E)),>o ( and martingales ( M ~ ) ) , > o . This decomposition is unique (provided that the sequences ( A P ) ) n > oare predictable), therefore it is easy to see that

Setting A , = A:?) for n 6 r k we see that ( X r L- A,),>O is a local martingale because the 'stopped' sequences

are martingales. Hence if X = (X,, 9 T L ) n > 0is a local submartingale, then

+

X, = A, M,? n 3 0, (2) where A = ( A T 1i)s a predictable sequence with A0 = 0 , and M = ( M n ) is a local martingale. It should be noted that the sequence A = (A,) is increasing (more precisely, nondecreasing) in this case, which is a consequence of the following explicit formula for the A?): AP)= E(AXi19-1),

1

Z
and of the submartingale property E(AXi IFi-1) 3 0. We note also that the decomposition (2) with predictable process A = ( A n )is unique.

Chapter 11. Stochastic Models. Discrete Time

100

DEFINITION 8 . Let X = (X,,9,) be a stochastic sequence admitting a representation X,,, = A , Mn with measurable sequence A = ( A n , S n - l ) and local martingale M = ( M n ,.5Fn). Then we say that X admits a generalized Doob dec0.m.position and A is the compensator (or the predictable compensator, or else the dual predictable projection) of the sequence X . (We call A a ‘compensator’ since it compensates X to a local martingale.)

+

3. For (:oiiclusion, we present a siniple but useful result of [251] describing coriditioris eiisiiririg that a local martingale is a (usual) martingale. LEMMA.1) Let X = (X,,, 9 n ) n 2 ~ be a local martingale such that E/Xol < 00 and either

EX,; < M,

n 3 0,

(3)

or

ExZ<~,

n.30. (4) Theii X = (X,,, c % l r ) n 2 is ~ a martingale. 2) Let X = ( X I L , c F n ) O G be n Ga~ local martingale and assume that N < 00: E / X o l < 00, and either EX; < 00, or EX,$ < 00. Then (3) and (4) hold for each n < N a i d X = (X,, , Y n ) o G nis c ~a martingale. Proof. 1) We claim that each of conditions ( 3 ) arid (4) implies the other and, therefore, the inequality EIX,I < m, n 3 0. For if, say, ( 3 ) holds, then by Fatou lerrirria ([439; Chapter 11, 56]),

EX;

=

E l&X2ATk

< @ EX:,,,&

+ EX,;A,,]

= @[EX,,,,,

k

k

k n

=

6 IEXoI

EX0 +mEX,,,

+C E X i <

00.

i=O

k

Consc:querit,ly, EIX,,I < m, rc 3 0 . n+l

< C

Further, we have IX(,L+l)ATkI

n+l

/Xi1 with E

i=O

C

/Xi1 <

00,

therefore by

i=O

Lebesgue theorem on dominated convergence ([439; Chapter 11, $3 6 and 7 ] ) passing , to the limit (as k: + 00) in the relation E(x~kA(n+l)l(Tk

> O) 1 cqTl)

=XTkAn

we obtain E(X,,+1 I SFTI) = X,, n 3 0. 2) We note that if EX; < 00, then also EX, < cx: for n N , because local martingales are generalized martingales. Hence X , = E(X,,+l I FrL), and therefore X, E(XL+l I Sn)and EX; 6 EX,, EXN for all I% N - 1. Thus, it follows from 1) that the sequence X = ( X n ,S n ) o c n G is amartingale. ~ In a similar way we can corisider the case of EX,$ < m thus completing the proof of the lemma.

<

<

<

<

1. Probabilistic Concepts a n d Several Models of t h e Dynamics of Market Prices

COROLLARY. Eilch local martingale X = (X,),,, 3 C > - m , P - a s ) or above (supX,(w)

101

that is bounded below (inf X,(w)

< C < m, P-as.)

n

is a martingale.

71

4. It is instructive to compare the result of the lernrria with the corresponding result

in the continuous-time case. ( 9 t ) t 2 o ,P) is a filtered probability space with a nondecreasirig family If (n,9, of a-algebras ( 9 t ) t a o (g3C 5Ft C 9,s < t ) , then we call a stochastic process X = (Xt)t2o a martingale (a supermartingale, a submartingale) if the X t are 9t-measural)lc, E/Xtl < 00 for t 3 0. and E(Xt I 9,) = X, (E(Xt 1 9 ~ 3 < )X,9 or E(Xt 1 gS) > X,$) for s < t . We call the process X = (Xt)t>Oa local ,martingale if we can find a nondecreasirig sequence of stopping times (rk)such that r k t x (P-as.) and the stopped processes Xrk = (XtATkI(q> o), . Y t ) are uniformly integrable martingales for all k . Using the same arguments as in the proof of the lemma, which are based on Fatou’s lemma and Lebesgue’s theorem on dominated convergence ([439; Chapter 11, § $ 6 and 71) we can prove the following results. I. Each local martingale X = ( X t ) t > o satisfying the condition EsupX,

< 00,

s
t 3 0,

is a supcrmarfangale. 11. Each local martingale X = ( X t ) t 2 o satisfying the condition

is a m,artingale. 111. Each local niartirigale X = (Xt)t>o satisfying t,he condition E sup IX,(

< 00,

t 3 0,

S
is a unijorrnly ,integrable marti,ngale. 71

It is useful to bear in mind that, in the discrete-time case, if EX, < 3c: for N , then ErriaxX, < 00. However, for continuous time, the inequality

<

EX+ <

n
00, t

< T , does not

in general mean that EsupXt-

<

30.

Essentially,

t
this is the rriain reason explaining why the result of the above lemnia cannot be automatically extended to the case of continuous time. We note also that the local martingale in assertion I can indeed be.a ‘proper’ supermartingale and not a martingale. The following cxaniple is well known.

Chapter 11. Stochastic Models. Discrete Time

102

EXAMPLE. Let 4 = (Bi, B:,B $ ) t p -be ~ a three-dimensional Brownian motion. Let

Then the process Rt (called the Bessel process of order 3 ) has the stochastic differential dt dRt = - d&, Ro = 1, Rt

+

where

= ( [ & ) t 2 0 is a standard Brownian motion (see

3a, Chapter 111, and, e.g.,

[4021). By the It6 formula

(see Chapter 111, 5 3d and also [250], [402]) used €or j ( R t )= 1/Rt (which is possible because the zero value is unattainable for the three-dimensional Bessel process R with Ro = 1) we obtain d dPt

(it) -

R:

or, in the integral form, 1

Rt The stochastic integral the process X = ( X t ) t > owith X t =

is a local martingale (see [402]), therefore

Rt

and Xo = 1 is a local martingale, but not

a martingale. Indeed, by the self-similarity property (see Chapter 111, s3a) of the Brownian motions B1, B 2 , B3 we obtain

where the &, i = 1 , 2 , 3 , are independent standard nornially distributed random variables (& J V ( 0 , l ) ) . On the other hand, if we had the martingale property, then the expectations EXt would be constant. N

1. Probabilistic Concepts and Several Models of t h e Dynamics of Market Prices

5 Id.

103

Gaussian and Conditionally Gaussian Models

1. The concept of an ‘efficient’ market substantiates the ‘martingale conjecture’ for (discounted) prices. This makes ‘martingale’ a central notion in the analysis of the dynamics of prices regarded as stochastic sequences or processes having distributions with specific properties. However, the mere fact that the distributions have the martingale property is not sufficient for concrete calculations; one must know a ‘finer’ structure of these distributions, which brings one to the necessity of a thorough study of most diverse probabilistic and statistical models in order to find ones with distributions better conforming to the properties of the empirical distributions constructed on the basis of statistical data. The rest of this chapter is in fact devoted to achieving this aim. We present models enabling one t o explain some or other properties discovered by the analysis of statistical ‘stock’ provided, in particular, by financial time series. The assumption that the distributions Law(h1,. . . , h,) of the variables h l , . . . , h,, are Gaussian is, of course, most attractive from the viewpoints of both theoretical analysis and the well-developed ‘statistics of the normal distribution’. However, one must take into account the fact (already pointed out in this book) that, as shows the statistical analysis of many financial series, this conjecture does not always reflect the behavior of prices properly. Seeking an alternative to the conjecture that the unconditional distributions Law(h1,. . . , h,) for the sequence h = (hn)n>l are Gaussian and bearing in mind the Doob decomposition, which was introduced in terms of the conditional expectations E(h,,, 1 c97L-1),it seems to be fairly natural t o assume that these are conditional (rather than the unconditional) probability distributions Law(h, I S n - l )that are Gaussian. In other words, Law(h,, I
(1)

for some Sn-1-measurable variables p, = pn(w) and c r i = ~ : ( w ) . ~ More precisely, (1) means that the (regular) conditional distribution P(h,, 6 z 1 9,-1) can be described by the formula

for all x E R and w E R. In view of the regularity, we can (see [439; Chapter 11, 3 71) evaluate the expectations E(hn I gn-l)(w)by mere integration (for each fixed w E 0 ) : E ( h 1 cF,x-~)(w) =

z@(hn

<2I~~-I)(u),

‘To avoid the consideration of trivial cases in which some special explanations are nevertheless necessary we shall assume throughout that o,(w) # 0 for all n and w.

Chaptcr 11. Stochastic Models. Discrete Time

104

which. in our case, brings us to the formula

In a similar way, 2 D ( h n I 9,n-i) = crL.

(3)

Thus, the ‘parameters’ fin and o i have a simple, ‘traditional’ meaning: they are the conditinal expectation and the conditional variance of the (conditional) distribution Law(h, 1 $,-I). The distribution Law(h,,) itself is, therefore, a ‘suspension’ (a ‘mixture’) of the conditionally Gaussian distributions Law(h, 1 .9n-l) averaged over the distribution of p n and o,”,. We note that ‘mixtures’ of normal distributions J ( p , 0 2 )with ‘random’ parameters p = p ( w ) and a’ = a2(w)form a rather broad class of distributions. We shall repeatedly come across particular cases of such distributions in what follows. Besides the sequence h = ( h n ) , it is useful to consider the ‘standard’ conditionally Gaussian sequence E = ( ~ ~1 of~ 9,-nieasurable ) ~ 2 random variables such that Q}. Law(&, 1 9,-1) = .N(O, l ) , where 90= (0, This seqiierice is obviously a martingale difference since E(E,, I Lq,l-l)= 0. More than that, this is a sequence of independent variables with standard normal distribution N ( 0 , l )because

By our assumption above, a,(w) # 0 ( n 3 1, w E n), so that the variables = ( h , - p T L ) / o n ,n 3 1, form a standard Gaussian sequence. Hence we can assume that the conditionally Gaussian (with respect to the flow (2Kn) and the probability P) sequerices h = ( h T Ln) , 3 1, that we consider can be represented as sunis hn 1~n C r n E n , (4) E,

+

where E = (E,) is a sequence of independent 9,-measurable random variables with standard nornial distribution .N(O,l).(As regards the representation of h = (hn) in the general case, when or, may vanish, see [303;Chapter 13, 3 11.) It is clear that to study the probabilistic properties of the sequence h = (h,) (anti therefore of S = (S,)) we must specify further the structure of the variables p n arid u i . This is what we do in the models that follow. We note that in the study of the distributions of h = ( h n ) ,given thkt we prefer to deal with conditionally Gaussian ones, it is often reasonable to consider this property in t,he following framework.

1. Prohatiilist,ic Concepts and Several Models of t h e Dynamics of Market Prices

105

Let (
+ alh,-l + . . . +

(6)

(o > 0 ) .

(7)

ol,E a = Const

Thus, hcre we have

+

+

+ +

h’rl = LLTL aTL&n= no f ( L l h n - 1 aph.n-p CEn. To define the seqiicnce h = ( h n ) n a l . which is called the autoregressive model of order p , we must fix ‘initial values’, the variables h P p ,. . . , hq. If they are constants, then the sequence ( h n ) n 2 1 is not merely conditionally Gaussian but truly Gaussian. In 5 2b, we consider the properties of the aiitoregressive model of order one ( p = 1) in greater detail.

3. Moving Average model ( M A ( q In this model, we fix the ‘initial values’ ( E I - ~ , . . . , E L I , €0) and set 9n =rr(~1,

=

oT1 o =

so that

hn = bo 4. AutoRegressiv We set .8,, = cr(~l, ( h l P P ,... , h-1, h”), = (no

(8)

Const

+ b 1 ~ , - 1 + b2cn-2 + . + bqEn-q +

U E ~ .

(9) oving Average model of order ( p ,q ) ( A R M A ( p ,q ) ) . ,,), define the initial conditions ( E I - ~ , . . . , E L I , € 0 ) and

+ alj~TL-1 + + ~ p h T 1 - p ) + ( h l ~ , L - 1 + b 2 ~ T l - 2 + ‘ . . + b q E n - q )

and

an

(10)

= o = Const.

Hence, in the ARMA(p,q ) model we obtain

+ +

+

+ +

+

h, = ( u 0 + u 1 h 7 , - 1 aphn-p) ( b l ~ z - 1 Cb2~TL-2 ‘ bqEn-q) o E n . (11) All these models, A R ( p ) , M A ( q ) , and ARMA(p,q ) . are linear Gaussian niodels (provided that the ‘initial values’ are, say, constants). We proceed now to certain interesting conditionally Gaussian models that (by contrast to tlic abovc ones) arc nonlinear.

Chapter 11. Stochastic Models. Discrete Time

106

5. AutoRegressive Conditional Heteroskedastic model ( A R C H ( p ) ) . Once again, we assume that the sequence E = is the (only) source of randomness and set 9n= F ( E ~ ., . . ,E,). Now let

and

P

= E(ht

U:

1 .Fn-i)

= cuo

+C ~~ih:~-i,

(13)

i=l

where (YO > 0, cui 3 0, i = 1,.. . , p , 90= (0, a},and h ~ - . .~. ,,ho are certain initial constants. In other words, the conditional variance U: is a function of h t - l , . . . , hnPp. 2 This model, which was introduced (as already mentioned in Chapter I, 0 2e) by R. F. Engle [140] (1982), proved to be rather successful in the explanation of several nontrivial properties of financial time series, such as the phenomenon of the cluster property for the values of the variables h,. Thus, h, = fl,En, n 3 1, (14)

-

where E = (E,) is a sequence of independent, normally distributed random variables, E, .N(O, l ) , arid the F: are defined by (13).

If

p n = a0

in place of (12) and the

+ U l h , - l + + arhn-r

02 satisfy (13), then (4) takes the following form:

Such models are sometimes denoted by A R ( r ) / A R C H ( p ) We now set (assuming that Eht < co) 2

V, = h,

-

2 CT,.

Then 1)

where E(v,

I 9,-1)

= E(h2 I 3,-1)

2 - rn =0

by (13), i.e., the sequence v = (v,) is a martingale diflerence. Thus, we can regard the A R C H ( p ) model as the autoregressive model A R ( p ) for the sequence ( h i ) with ‘noise’ v = (v,) that is a martingale diflerence.

1. Probabilistic Concepts and Several Models of the Dynamics of Market Prices

107

6. Generalized AutoRegressive Conditional Heteroskedastic model ( G A R C H ( pq ) ) . The successes with the application of A R C H ( p ) encouraged the development of various generalizations, refinements, and modifications of this model. The G14RCH(p,q ) model described below and introduced by T. Bollerslev [48] (1986) is one such modification. As before, let p n = E(h, I S n - 1 ) = 0,

but now, in place of (13) we assume that P

4

i=l

j=1

where a 0 > 0, ai,/3j 3 0 and where ( h ~ -. .~. ,,ho), (a&, values’, which we can for simplicity set to be constants. In the GARCH(p,q ) model we set

. ..,a:) arc the ‘initial

where (€1,~ 2. ., . ) is a sequence of independent identically distributed random variables with distribution X ( 0 , 1) and the g: satisfy (19). Now let P

a(L)h:L-l =

1a z h - i , i=l

where L is the lag operator (Lzhi-, = htn-ll-i; see 3 2a.2 below), and

In other terms,

(21)

108

Chapter 11. Stochastic Models. Discrete Time

Hence we can regard the GARCH(p, q ) model as the autoregressive moving atierage model ARMA(max ( p , q ) ,y) for the sequence ( h i ) with ‘noise’ (v,) that is a martingale difference. In particular, setting v,, 3 - a: in ARCH(1) with

hi

h,, = a,,~, and

we see that rL:

= a.

2 an = a0

+ all^^-^

+ ~ ~ 2h , , +, -v,~:

where the ‘noise’ (v,) is a martingale difference. There exist various generalizations of the ARCH and GARCH models (e.g., EGARCH, AGARCH, STARCH, NARCH, MARCH, HARCH) related, in the final analysis, to one or another description of the variables a; = E(hZ 19,-1) as rneasurable functions with respect to the a-algebras 9,-1 = a ( q , . . . ,&,-I). 7. Stochastic volatility model. In the previous models, we had a single source of randomiiess, a Gaussian sequence of independent variables E = (E,). The stochastic iiolatdzty models involve t w o sources of randomness, E = (E,) and 6 = (&). In the most elementary case they are assumed to be independent standard Gaussian sequences, that is, sequences of independent, Jv(O,l)-distributed random variables. Let 9, = o(S1,. . . , 6,). We set

hn = an&,,

(24)

where the a, are ‘$,,-measurable. Then it is clear that Law (h,, I gn) = ~ ( 0a:), ,

(25)

i.e., the Y?,L-conditionaldistribution of h,, is Gaussian, with parameters 0 and a:. We now set an = e $ A n . (26) Then a: = e A n , where the A, are !??,-measurable. There exist highly popular models in which the sequence (A,) is governed by an autoregressive model (we shall write (A,) E AR(p)) as follows: A,, = a0

+ alA,,-l + + ‘

+ c6,.

A natural gcneralizatioii of (24) is the scheme with h, = k n

+ DnEYZ,

(27)

where pLIL and aLIL are Y?LI,-measurable. If E = (E?,) in (27) is a normally distributed stationary scquence with EE,, = 0 and EE:~ = 1, while a = (a,) is independent of E = (E,), then we arrive at the so-called Taylor model. We complete our bmef discussion of several Gaussian and conditionally Gaussian niotlels used in financial mathematics and financial engineering at this point. We study the properties of these models more closely below, in 5 0 2 and 3 .

1. Probabilistic Concepts and Several Models of the Dynamics of Market Prices

5 le.

109

Binomial Model of Price Evolution

1. An exceptional role in probability theory is played by the Bernoulli scheme, the sequence 6 = (61,62,. . . )

of independent identically distributed random variables that take only two values, say, 1 and 0, with probabilities p and q , p q = 1. It was for this scheme that the first limit theorem of probability theory. the Law of large nwmbers, was proved (J. Bernoulli “Ars Conjectandi“, 1713). It states that for each E > 0,

+

(=

)

61 +‘“+S, STl is the frequency of ‘ones’ in the sequence 61,. . . ,6,. where 71. n Likewise, it was for this scheme that several other remarkable results of probability theory (the de Moivre-Laplace limit theorem, the Strong law of large numhers, the Larri of the iterated logaritm, the Aresine law, . . . ) were first proved. Later on, all these results turned out to have a much broader range for applications. The role in financial niatherriatics of the Cox-Ross--Rubinstein binomial model [82] described below is close in this sense to that of the Bernoulli scheme in classical probability theory; very simple as it is, this model enables complete calculations of many financial characteristics and instruments: rational option prices, hedging strategies. and so on (see Chapter VII below). 2. We assume that all the financial operations proceed on a (B,S)-market formed a bank account B = (B,),>0 and some stock of value S = (S,),>O. We can represent the evolution of B and S as follows:

or, equivalently,

LIB,

= rnBn-l>

as, = PnSr1-1 where Bo > 0 and So > 0. The main drstznctron between a bank account and stock is the fact that the bank interest rate r,, is L9Tl-1-measurable, while the ‘market’ interest on the stock pn is s,-measurable,

is thc filtration (information flow) on a given probability space wherc (c9n)

(n,9, P).

Chapter 11. Stochastic Models. Discrete Time

110

In the framework of the Cox-Ross-Rubinstein binomial model (the .CRR-model') of a ( B ,S)-market one assumes that

= T = Const

r,

and that p = (pn),a21 is a Bernoulli sequence of independent identically distributed random variables p1, p2, . . . taking two values, a or b, a < b. We can write each p n as the following sum: Pn =

a f b 2

~

or as pn = a

+-b - a

(3)

En,

+ ( b - a)S,,

(4)

to find out that

Our assumptions that r, = Const, while the pn take only two values, enable us to set the original probability space R from the very beginning to be the space of bznery sequences: 62 = { a , b}"

or

R = {-1,

l}",

or

R = (0, l}".

Comparing this expression and ( 5 ) in 3 l a we see that the pk play the role of the variables x b introduced there. Clearly, we can also represent the S, as follows:

s, = SO& where h,, = ln(1

S"&+.

1

.+h,,

+ p k ) (cf. (1) and (10) in 3 l a ) .

3. We now discuss the particular case of a and b satisfying the relations

for some X > 1. In this case XS,-l

x-'s,-~

if prL = b. if pn = a.

1. Probabilistic Concepts and Several Models of t h e Dynamics of Market Prices

Regarding (3) as the definition of

which is the same as

E,(=

sn -

-

111

& l ) we , can represent the S , as follows:

Soehl+-.+hn

with h,k = ~k In A. This random sequence S = (S,) is called a geometric r a n d o m walk on the set

Eso =

k=O,&l,

...}.

I f S o E E = { A k : k = O , * l , ...}, t h e n E S , = E . I n t h i s c a s e S = ( S , ) i s c a l l e d a Markow walk on the phase space E = {A’: k = O , & l , . . . } . The binomial model in question is a discrete analogue of a geometric Brownian motion S = ( S t ) t 2 0 ,i.e., of a random process having the following representation (cf. (8)): St = Soeo W t + ( p - o 2 / 2 ) t , where W = (Wt)t>ois a standard Wiener process or a standard Brownian motion (see Chapter 111, 53a). It seems appropriate to recall in this connection that it is a n arithmetic random walk S, = Sn-l with a Bernoulli sequence = (&),>I that is a discrete counterpart to usual Brownian motion.

+ <,

<

4. We started our previous discussion with the assuniption that all considerations proceed in the framework of some filtered probability space (R, 9, (g,), P) with probability measure. The question of the properties of this probability measure P and, therefore, of the quantities p = P(p, = b) and q = P(p, = a ) is not so easy in general. It would be more realistic in a certain sense to assume that there is a n entire family 9 = {P} of probability measures in (s2,9 rather ), than a single one, and that the corresponding values of p = P(p, = b ) belong to the interval ( 0 , l ) . As regards the issue of possible generalizations of the binomial model just discussed, it would be reasonable to assume that the variables pn can take all values in the interval [a,b] rather than only the two values a and b. In that case, the probability distribution for pn can be an arbitrary distribution on [a,b]. This is the model we shall consider in Chapter V, 3 l c in connection with the calculations of rational option prices on so-called incomplete markets. In the same section we shall also consider a nonprobabilistic approach based on the assumption that thk p, are ‘chaotic’ variables. (As regards the description of the evolution of prices involving models of ‘dynamical chaos’, see 5 5 4a, b below.)

Chapter 11. Stochastic Models. Discrete Time

112

fj If. Models with Discrete Intervention of Chance 1. In the study of sequences H = (H,),zo, it is often convenient to .embed’ them (in the sensc that we explain below) in certain schemes with continuous time. Given a stochastic basis % = ( n , 9 ,(.F,),~o, P) we consider a n associated basis 9with continuoils time ( t 0) defines as follows: I

-

where .Ft = 9rt]and [t] is the entire part o f t . Given a stochastic sequence H = ( H n , F n ) , we introduce the process H = ( f i t , .%t) (with continuous time) by settingb I

Ht

=

H[t].

T h i s (see Fig. 13), the trajectories - H t -, t 3 0, are piecewise constant arid rightcontinuous, making jumps A H t 3 Ht - Ht- for t = 1,2 , . . . . Moreover; AH,, = AH,, = h,,,. Clearly, tlic converse is also true: a random process H = ( H t ,9 t ) t > o on a probability space .% = (0,.9,( 9 t ) t 2 0 , P) that has piecewise constant right-continuous trajectories iiiakirig jumps for t = 1 , 2 , . . . , is actually a process with discrete time of the above form. I

I

I

-

I

-H3

I I I I I

H1

I I

IH2

I

2

3

I I I

0

1

4

t t

1. Probabilistic Concepts and Several Models of t h e Dynamics of Market Prices

113

S = (St)t20 described by a continuous random process (e.g., the Brownian motion) with contiiiiious time t 3 0. In practice, the statistical analysis (see Chapter IV) of the evolution of actual prices S = (St)t20 shows that they have a hybrid structure in a certain sense. More precisely, this means the following. According to a large stock of data, the trajectories of the prices S = (St)tao look like in the picture below: St

so

0

tI

Lm 71

t

I

I I I I I I I

72

r3

In other words, representing St by the formula St

t

= S0eHt we see that

where r l , r 2 , .. . are the instants of jumps and hk are their amplitudes

HTk - Hrk- = I l k ) . (AH,, Assuming that all our considerations proceed with respect to a stochastic basis (0,,6, ( % t ) t 2 0 , P), it would be natural to impose certain ‘measurability’ conditions on the (random) instants ofjumps T k , k 3 1, and the variables hk, k 3 1, so as to ensure, at any rate, that Ht be 9t-measurable for each t 3 0, i.e., can be defined by the ‘data’ accessible to an observer over the period [O, t ] . 3. To this end we introduce several definitions in which by ‘an extended random variable’ r = T ( W ) we mean a map‘

‘Recall that, in accordance with the traditional probabilistic definitions, a r e d v a l u e d randoni variable can take only finite values, i.e., values in R = (--co,m).

114

Chapter 11. Stochastic Models. Discrete Time

DEFINITION 1. We call a nonnegative random value r a n d o m valriable independent of the ifutiire’ if { w : .(w)

6 t}E 9t

T

= T(W) a

M a r k o v t i m e or a

(2)

for each 1 3 0. Markov times are also called stopping times, though sometimes one reserves the latter term for the Markov times T = T ( W ) such that either T ( W ) < 00 for all w E R, or P ( T ( w ) < m ) = 1. The meaning of condition (2) becomes fairly obvious if we interpret ~ ( was) the instant when one must make certain ‘decisions’ (e.g., to buy or to sell stock). Since 9 t is a a-algebra, (2) is equivalent to the condition { T ( w ) > t } E 9 t , meaning that one’s intention at time t to postpone this ‘decision’ until later is determined by the information 9 t accessible over the period [ O , t ] , and one cannot take into consideration the ‘future’ (i.e., what might happen after time t ) . DEFINITION 2 . Let T = T ( W ) be some stopping time. Then we denote by 9,the collection of sets A E ,F such that

for each t E R+ = [O, m). By Fr- we denote the a-algebra generated by 90 and all the sets A n { w : ~ ( w <) t } , where A E 9 t and t E R+. It is easy to see that cFris a a-algebra. If we interpret .Ft as the collection of all events occurred not later than time t , then it is reasonable to interpret 9,and 9,-as the collections of events observable over the periods [0, T ] and [0, respectively. T ) ?

4. We now turn back to the representation (1) of the process H = (Ht)t2o as

a process with discrete intervention o,f chance (at instants of time rl,rz,. . . ; see Fig. 14). We assume that

for all (or P-almost all) w E R and that 7-k = T ~ ( w is ) a stopping time for each k 3 1, while the variables hk = h k ( w ) are grk-measurable. It follows, in particular, from these assumptions and conditions (2) and (3) that the process H = (Ht)t20 is adapted to the flow ( F t ) t > O , i.e.,

Ht is 9t-measurable for each t 3 0. Thus, in accordance with the above conventions, we can write H = ( H t ,gt)t>o.

1. Probabilistic Concepts and Several Models of the Dynamics of Market Prices

115

The real-valued process H = (Ht)t>O defined by (1) is a particular case of socalled naultivariute point processes ranging in the phase space IE ([250]: in our case we have IE = R), while the term ‘point (or counting) process’ in the proper sense is related to the case of h, = 1, i.e., to the process

Remmrk. Sometimes, one relates ‘point’ processes only to a sequence of instants = ( 7 1 , 7 2 , . . . ) arid reserves the attribute ‘counting’ for the process N = (Nt)t>o,

T

corresponding to this sequence in accordance with formula (4). It is clear that there exists a one-to-one correspondence between 7 and N : we can determine N by 7 in accordance with (4), and we can determine 7 by N because r k = inf{t: Nt =

k}.

We note that, as usual, we set here T ~ ( wt )o be equal to +m if { t : Nt = k } = 0. For instance, considering the stopping times corresponding t o the ‘one-point’ point process with trajectories as in Fig. 15 we can say that 7 2 = g = ... = fm.

01

t

TI

FIGURE 15. ’One-point’ point process 5 . The point processes N = (Nt)t>odefined in (4) can be described in a n equivalent

way using the concept of a random change of time [250; Chapter 11, 5 3b].

DEFINITION3 . We say that a family of random variables CT = (crt)t>o defined ( . 9 r : 1P)Land )n~ ranging ~ , in N = { 0 , 1 , . . . } or on a stochastic basis 3 = (R, 9, N = (0, I , . . . ,a} is a random change of t i m e if

-

(i) the gt are stopping times (with respect to ( . 9 , , L )for n ~all~ t); (ii) (TO = 0; (iii) each trajectory C T ~ ( W )t, 3 0, is increasing, right-continuous, with jumps of amplitude o m . We now associate with

(T

= ( a t ) t > 0 the stochastic basis

B‘

116

Chaptcr 11. Stochastic Models. Discrete Time

with coiitiiiuous time and set

) stopping times with respect to the It is easy to show that the rk = T ~ ( w are flow (%t)t)t>o such that ??t = g o t .Moreover, if rJt < 00 for all t E N,then

It is also cltar that gt =

X I ( % 6 t)’ Ic> 1

i.e., gt is the number of decisions taken over the period [0, t ] .To put it another way, the random change of time = (at)t>Oconstructed by the sequence (1-1,1-2,. . . ) is just thc coirntrng process N = ( N t ) associated with this sequence, i.e., the process

k>l

In all the above formulas t plays the role of real (‘physical’) time, while ut plays the role of operational time counting the ‘decisions’ taken in ‘physical’ time t . (We return to the issue of operational time ~ I Ithe empirical analysis of financial time series in Chapter IV, 3 3d.) We note also that if rJt = [tl, then B‘ = 2, where % is the above-mentioned stochastic basis with discrete time (Q9, (gt)t>o, P), where @t = 9p1. 6. As follows from 8 l b , the Doob decomposition plays a key role in the stochastic analysis of the sequence H = (H,),>o by enabling one to distinguish the ‘martingale’ and the ‘predictable’ components of H in their dependence on the flow (9,),20 of incoming data (on the market situation, in the financial framework). A similar decomposition (the Doob-Meyer decomposition) can be constructed both for the counting process N and for the multivariate point process H . This (as in the discrete-time case) is the main starting point in the stochastic analysis of these processes involving the concepts of ‘marginal’ arid ‘predictability’ (see Chapter 111, 3 5b and, in greater detail, [25O]).

2. Linear Stochastic Models

The empirical analysis of the evolution of financial (and, of course, many other: economic. social. and so on) indexes, characteristics, . . . must start with constructing an appropriak probabilistic, statistical (or some other) model; its proper choice is a rather coniplicated task. The General theory of t i m e series has various ‘standard’ linear models in its store. The first to mention are M A ( q ) (the moving average model of order q ) , AR(p) (the autoregressive model of order p ) , and A R M A ( p , q ) (the mixed autoregressive moving average ,model of order ( p , 4 ) ) considered in 5 Id. These models are thoroughly studied in the theory or time series, especially if one assumes that they are stationary. The reasons for the popularity of these models are their simplicity on the one harid and. on the other, the fact that, even with few parameters involved, these models deliver good approximations of stationary sequences in a rather broad class. However, the class of ‘econometric’ time series is far from being exhausted by stationary series. Often, one can fairly clearly discern the following three ingredients of statistical data: a S ~ 0 7 U hchangi,ng ~ ( e . ~ . ‘inflationary’) , trend component ( 2 ) ; periodic or aperiodic cycles (y), an irregiilar, j h c t i ~ a t i n g(‘stochastic’ or ‘chaotic’) component ( z ) . These components can be combined in the observable d a t a ( h ) in diverse ways. Schematically, we shall write this down as

where the symbol can denote, e.g., addition multiplication‘x’, and so on. There are many monographs devoted to the theory of time series’ and their applications to the analysis of financial d a t a (see, e.g., [62]. [193], [202], [all], [212], [351], or [460].)

Chapter 11. Stochastic Models. Discrete Time

118

In what follows, we discuss several linear (and after that, nonlinear) models with the intention to give an idea of their structure, peculiarities, and properties used in the empirical analysis of financial data. It should be mentioned here that, in the long run, one of the important objectives in the empirical analysis of the statistics of financial indexes is t o p r e d i c t , forecast the ‘future dynamics of prices’:

0 ~

I

the predicted ‘most probable‘ price dynamics

0

I I

01

‘present’

t

FIGURE 16. The region between the curves 1 and 2 is the confidence domain (corresponding to a certain degree of confidence), in which the prices are supposed t o evolve in the future

The reliability of these predictions depends, of course; on a successful choice of a model, the precision in the evaluation of the parameters of this model, and the quality of extrapolation (linear or nonlinear). The analysis of time series that we carry out in Chapter IV is expressive in that respect. We show there how, starting from simple linear Gaussian models, one is forced to ,modify them, make them more complex in order to obtain, finally, a model ‘capturing’ the phenomena discovered by the empirical analysis (e.g., the failure of the Gaussian property, the ‘cluster’ property, and the effect of ‘long memory’ in prices). Recalling our scheme ‘ h = z * y * a ’ of the interplay between the three ingredients x, y, and z of the prices h, we can say that most weight in our discussion will be put on the last, ‘fluctuating’ component z , and then (in the case of exchange rates) on the periodic (seasonal) component y. We shall not discuss the details of the analysis of the trend component z, but we point out that it is often responsible for the ‘nonstationary’ behavior of the models under consideration. A classical example is provided by the so-called A R I M A ( p ,d, q ) models ( AutoRegressiiie Integrated M o v i n g Average models; here d is the order of integration) that are widely used and promoted by G. E. P. Box and G. M. .Jenkins ( [ 5 3 ] ;see 5 2c below for greater detail).

2. Linear Stochastic Models

119

fj 2a. Moving Average Model M A ( q )

1. It is assumed in all the models that follow (either linear or nonlinear) that we have a certain ‘basic’ sequence E = (E,), which is assumed to be white noise (see Fig. 17) in the theory of time series and is identified with the source of randomness responsible for the stochastic behavior of the statistical objects in question. 0.25

0.15

0.05

-0.05

-0.15

-0.25 0

10

20

30

40

50

60

70

80

90

100

-

FIGURE 17. Computer simulation of ‘white noise’ h, = OE, with o = 0.1 and E, A ( O , I)

Alternatively, it is assumed (in the ‘L2-theory’) that a sequence E = ( ~ 00, and noise in the wide sense, i.e., EE, = 0, E E <

E ~ is )

white

E E , E ~ ~= 0 for all n # m. (It is convenient to assume a t this point that the time parameter n can take the values 0, k l , f 2 , . . . .) In other words, whzte nozse in the wide sense is a square integrable sequence of uncorrelated random variables with zero expectations. If we require additionally that these variables be Gaussian (normal), then we call the sequence E = (E,) white noise in the strict sense or simply white noise. This is equivalent to the condition that E = (E,) is a sequence of independent normally J ’ ( 0 , a ; ) ) random variables. In what follows we assume that distributed ( F , u: _= 1. (One says in this case that E = (E,) is a standard Gaussian sequence; it is instructive to compare this concept with that of a fractal Gaussian noise, which is also used in the financial data statistics, see Chapter 111, 3 2d.) N

2. In the moving average scheme MA(q) describing the evolution of the sequence h = (h,), one assumes that the h, can be constructed from the ‘basic’ sequence E

Chapt,er 11. Stochastic Models. Discrete Time

120

as follows: h,, = ( p

+ blE,,-l +

'

.

'

+ bqE,-q) + bOE,.

(2)

Here q is a paranieter describing the degree of our dependence on the 'past' while 'updates' the information contained in the a-algebra 9,-1 = C T ( E , - ~ , ~ ~ - 2 . ,. ). ; see Fig. 18. E~

-1 0

10

20

30

50

40

60

70

80

90

100

FIGURE 18. Computer simulation a sequencr h = (h,) governed by the MA(1) inodcl with 11, = / l + 6 1 ~ ~ - 1+boa,, where ,LL = 1, 61 = 1, bo = 0.1, and E n J V ( 0 , l ) N

For brevity, it is coiivenient to introduce the lag operator L acting on number sequences z = (z?,) by the formula L x , = 5,-1. Since L(Lz,,) = LZ:,,-~= z,-2, operator

(3)

it is natural to use the notation L2 for the

L22, = x,,-2.

In general. we set L ~ z , = , Xn-k for all IC 3 0. We point out the following useful, albeit simple, properties of the operator L (here c, c1, arid c2 are arbitrary constants): L(cx,,) = CL5,'

L(zn + ? h a ) = Lx,,+ LY,, (ClL

+ c2L2

)Z,,

=

+ c2L2xn = c1z,,t-1 + c'25,,-2, ( X I + X 2 ) 2 , - 1 + (XlX2)2,-2.

clLxn

(1 - X1L)(1- X 2 L ) Z , = 2,

-

2. Linear Stochastic Models

121

Using L we cat1 write (2) in the following compact form:

+ O(L)En,

h,, = I-1

where

p ( L ) = bo

(4)

+ b l L + . . + bqLq.

3. We now c:onsider the question on the probabilistic characteristics of the sequence IL = ( / I , , , ) .

Let q = 1. Then hn =

/L

+ b O E n + blEn-1,

(5)

and we immediately see that

The last two properties mean that h = (h?,)is a sequence with correlated consecutive elements (h,,, arid !1,,~,+1), while for k > 2 the correlation between the h, and the JiTL+k is zero. Wc note also that if bob1 > 0, then the elements h,, and h,,+l have positive correlation, while if bob1 < 0, then their correlation is negative. (We come across a similar situation in Chapter IV, 5 3c, when explaining the phenomenon of negative correlation in the case of exchange rates.) By (6) arid (7), the elements of h = (h,) have mean values, variances, and covariances independent of n. (Of course, this is already incorporated into our assumption that E = (E,) is white noise in the wide sense with EE, = 0 and DE;, = 1 and that the coefficients in ( 5 ) are independent of n.) Thus, the sequence h = (&) is (just by definition) stationary in the wide sense. If we assume in addition that E = ( E ? ~ )is a Gaussian sequence, then the sequence h = (h,) is also Gaussian and, t,herefore. all its probabilistic properties can be expressed in terms of expectmation,variance, arid covariance. In this case, the sequence h = (h,n)is stationary in the strict sense, i.e.,

for all

'11

3 1. i l , . . . , i,,, and arbitrary k .

4. Let a trajcct,ory ( h l ,. . . , hTL)be a result of observations of the variables h k at

iristants of time k: = 1, . . . , n and let -

-

1

n be the time average.

n

k:=l

Chapter 11. Stochastic Models. Discrete Time

122

From the statistical viewpoint, the point of considering the ‘statistics’ h, lies in the fact that belief that it is a natural ‘choice’ for the role of an estimator of the expectation p . If we measure the quality of this estimator by the value of the standard deviation A: = E I&, - p i 2 , then the following test is useful: as n -+ 00 we have

+ 0 ++

A n

R(k) 4 0 ,

(9)

where R(k) = Cov(hn,hn+h) and h = ( h n ) is an arbitrary sequence that is stationary in the wide sense. Iiideed, let (for simplicity) ,LL = Eh, = 0. Then

therefore the implication On the other hand.

in (9) holds.

n 1-1

=n2

We now choose 6

1=1 k=O

1 R ( k ) - -R(O). n

> 0 and we choose n = n(6) such that

for all 13 n(6). Then

2. Linear Stoc,hastic Models

for

ri

3 n ( b ) , arid since n ( S ) < 00 and IR(O)/

123

< Const, it follows that

Since b > 0 was chosen arbitrary, this proves t,he implication S. - Thus, the model MA(1) is ‘ergodic’ in the following sense: the t i m e averages /in converge (in L 2 ) to the mean value p , or, as it is alternatively called, to the ensemble average. (As regards the general concepts of ‘ergodicity’, ‘mixing’, ‘ergodic theorems’, see, e.g., [439; Chapter V].) The importance of this ‘ergodic’ result from the statistical viewpoint is completely clear: it substantiates the estimation of mean values in terms of time averages calculated from the observable values of h i , hp, . . . . It is clear that ‘ergodicity’ plays a key role in the substantiation of the statistical methods of estimation from samples not only in the case of mean values, but also for other characteristics: moments of various orders, covariances, and so on. 5 . We recall that, given a covariance Cov(h,,

tim),

we can define the correlation

Corr(h,, hmL)by the formula

<

By the Cauchy-Schwartz-Bunyakovskii inequality, I Corr(h,, h m ) / 1. In the stationary case, Cov(h,, h,+k) is independent of n. We denote this value by R(k) and set R(k) p ( k ) = Corr(h,, &+k) = __ (11) R(O). Then we obtain by ( 7 ) that, in the model MA(l),

f 1,

k = 0,

It is interesting that if we set 81 = b l / b o , then

Chapter 11. Stochastic Models. Discrete Time

124

6. Wc. now considtr the niodcl M A ( q ) : /I,, =

+ [ ~ ( L ) Ewhere ~~,

P ( L ) = bo

+ blL + . . . + b,L4

It is easy to s w that

and

It is clear from (14) that schemes of type M A ( q ) can be used to simulate (by varying the c:o&icients h i ) the behavior of sequences h, = ( h l l )with no correlation between the variables ir,,, arid hll+k for k > q . Remark. I n our discussion of the adjustment of one or another model to empirical data, or, as we put it above, 'the simulation of the behavior of the sequences h = ( h r L ) w(: ' , should inention that the general pattern here is as follows. Given sainple values h l , h2,. . . , we determine first certain empirical characteristics, e.g., the sample mean

the sample variance -

n

t,he samplc correlation (of order k )

the sariiple partial correlations, and so on. After that, using the formulas for the corresponding theoretical characteristics (as, e.g., (12) or (14)) in the models that we are approximating, we vary the parameters (such as. for instance, the hi in (12) and (14)) to adjust the theoretical characteristics to the empirical. Finally, a t the last stage, we estimate the quality of this adjustment using our knowledge about the distributions of the empirical characteristics and their deviations from the theoretical distributions.

2. Linear Stochastic Models

125

7. The next natural step after the models M A ( q ) is to consider MA(oo), i.e., models with J=O

Of course, we must inipose certain conditions on the coefficients to ensure that the slim in (15) is convergent. If we require that x

xbj” < m, j=O

than the series in (15) converges in mean square. Under this assumption, Eh,, = p,

Dh,, =

1bj”,

(17)

j=O

and

Considering the representation (15) for h = ( h T l it ) is usually said in the theory of stationary stochastic processes that the h is the ’output of a physically realizable filter with impulse response b = ( b j ) and with input E = ( E ~ ) ’ . It is reniarkable that, in a way, each ‘regular’, stationary (in the wide sense) sequence h = ( h7%)can be represented as a sum (15) with property (16). As regards the precise forrnulation and all the totality of the problems relating to the Wold expansion of a stationary sequence in a sum of ‘singular’ and ‘regular’ components such that the latter are representable as in (15), see § a d below and, in greater detail, e.g., [439; Chapter VI, 8 51.

5 2b.

Autoregressive Model A R ( p )

1. By the definition in 5 Id, a sequence h = ( h , j n > 1 autoregresszve model (scheme) A R ( p ) of order p if hT, = pn

where ,UTa = a0

is said to be governed by the

+ UE,,

(1)

+ alh,-l + + a p h T 1 k p . ‘

(2)

We can put it another way by saying that h = (h,) satisfies the dtfference eqnation of order p

h, = a0

+ Ulh,-l + +

+

UEn,

(3)

Chapter 11. Stochastic Models. Discrete Time

126

which, using the operator L introduced in (1

~

u1L

-

‘

-

5 2a. can be rewritten as

a,Lp)h, = a0

+

(4)

OE,,

or, in a more compact form,

4 L ) h n = wn, (5) where a ( L ) = 1 - a l L - . . . apLP, wTL= a0 O E ~ . As already mentioned in 3 Id, for a complete description of the evolution of the sequence h = (h,) governed by the difference equation (3) we must also set ‘initial values’ (hl-,, h2-,,. . . , h 0 ) . One often sets hl-, = . . . = ho = 0, or one can assume that these are random variables independent of the sequence ~ 1~ ,2 . ., . . In ‘ergodic’ cases the asymptotic behavior of the h, as n --t 00 is independent on the ‘initial’ conditions, and in this sense, the initial data are not that important. Still, we describe precisely all our assumptions concerning the ‘initial’ conditions in what follows. We present the results of a computer simulation of the autoregressive niodel A R ( 2 ) in Fig. 19.

+

~

0.4

-0.3

Y

t 0

b

100 200

300

400

500

600

700

800

900 1000

FIGURE 19. Coinputcr simulation of a sequence h = (h,) (2 6 n 6 1000) governed by the A R ( 2 ) model with h, = a o + a ~ h , - l + a 2 h n - 2 + m n , where a0 = 0, a1 = -0.5, a2 = 0.01, 0 = 0.1, and ho = hl = 0

2. First, we consider in detail the simple case of p = 1, where

h, = a0

+ Ulh,-l +

OE,.

(6)

It can be distingiiished from the general class of models A R ( p ) by the property that, of all the ‘past-related’ variables h,-1, hn-2,. . . , hn-p involved in (3), the contribution in h,, is made only by the closest in time variable hn-l.

127

2. Linear Stochastic Models

If E = ( E ~ , ) ~ ~ is > I a sequence of independent random variables and JQ is independent of E , then the sequence h = (h,),>l becomes a classical example of a constructavely defined Markov cham. From (6) we find recursively that

Hence the properties of the sequence h essentially depend on the values of the parameter al. We must distinguish between the three cases of lull < 1, /all= 1, and lull > 1, where the case of lull = 1 plays the role of a 'boundary' of sorts, which we explain in what follows. By (7) we obtain

forn-k31. It is clear from these relations that if lull < 1 and Elhol

as n

+ 00 and

(if Dho

< cx), then

< ca)

Hence, the sequence h = (h,),>o approaches in this case (la11 < 1) a steady state as n + 00. Moreover, if the initial distribution (the distribution of ho) is normal, i.e.,

then h = (h7L)T7,20 is a stationary Gaussian sequence (both in the wide and the strict sense) with

Chapter 11. Stochastic Models. Discrete Time

128

and

We recall that a sequence is stationary in the strict sense if Law(h0, h l , . . . , h,)

= Law(hk, h , l + k , . . . , hm+k)

for all adrnissihle values of ‘mi and I;, while it is stationary in the wide sense if Law(h,, h3) = Law(h,+k, ! L ~ + ~ ) .

If

then we obtain by (8) and (9) that

for ( a l l < 1, i.e., the correlation between the variables h, and hn+k decreases geometrically. Comparing the representation (7) and forniula (2) in the preceding section, we observe that for each fixed n the variable h, in the A R ( 1 ) model can be treated as its counterpart h,, in the M A ( q ) model with q = n - 1. Slightly abusing the language one says sonietimes that ‘the A R ( 1 ) model can be regarded as the M A ( m ) model’. In A R ( 1 ) ,the case lull = 1 corresponds to a classical r a n d o m walk (cf. Chapter I, $ 2 a , where we discuss the random walk conjecture and the concept of an ‘efficient’ niarket). For example, if a1 = 1, then h,, = a071

+ ho + C ( E 1 + + E n ) .

and

D/L, =a2,+

00

as

n+

00.

The case of 1011 > 1 is ’exploszve’ (in the sense that both the expectation Eh, and variance Dh, increase ezponmtrally with n ) .

2 . Linear Stc--.astic Models

129

3. We now consider the case p = 2:

Using the above-rrieiitioned operator L we can rewrite this difference equation as follows: (1 - CllL - U 2 L 2 ) h T 1= a0 P E n . (12)

+

0, then the corresponding equation

If a2

(1 - UlL)h, = a0

+

PE,

(13)

fits into the case AR(1) we have already discussed. We now set w , = uo ml,. Then (13) assumes the following form:

+

(1 - q L ) h , = w,.

(14)

It is natural to look for a .conversion' of this relation enabling one to determine the hTLfrom the 'input' (wl,). Based on the properties of L (see fj 2a.2) we can see that

+ UlL + U f L 2 + + aFL"(1 a1L) = (1 .!+1L'C+l 1. (15) We now apply the operator 1 + a l L + a:L2 + . . . + a i L k to both sides of (13). In (1

' ' '

~

~

view of ( I s ) , 11,

Settirig here k = 71

II,,

= (a0

+ a1L + *?L2 + . + .!Lk)WlL + .f+lL"+lh,. 1 arid w, = a0 + P E , ~ we obtain + al(a0 + + . . . + a ? - l ( ~ g+ P E ~ )+ ~ ; " h o ,

= (1

+

-

' '

g ~ n )

OE,,-I)

(16)

(17)

which is precisely the representation (7) above. By (16), bearing in mind (14) with k = n 1 we obtain ~

/I,,,,

If lull

= (1

< 1 and

71

h,

+ n.lL + a:L2 + . . . + aY-'Ln-')(l

-

alL)h,

+ a;"ho.

(18)

is sufficiently large, then we have the approximate equality M

(1

+ a1L + alL2 + . + un-l 1 L ' '

-

)(1- a l L ) h n .

(19)

This suggests that a natural way to define the inverse operator (1 a l L ) - l is to consider t,he limit (in an appropriate sense) of the sequence of operators 1 ulL u!L2 . . . u;"LrLas n + 00. (Cf. the algebraic representation (1 - z1-l = 1 z z 2 . . . for /zI < 1.)

+

+

+ + + + +

Chapter 11. Stochastic Models. Discrete Time

130

These heuristic arguments can be formalized, e.g., as follows. We consider a stat,ionary sequence h = ( h n ) ,where -

I

M

j =0

-

(the series converges in mean square). It is easy to see that (h,) is a solution of (14). We claim that this is the unique stationary solution with finite second moment. Let ti = ( h n ) be another stationary solution. Then

j=O

by (16), therefore

ask-tco. Hence we obtain by (20) that equation (13) has a unique stationary solution (wit,h finite second moment). 4. The above arguments show the way to the 'conversion' of difference equation (12)

enabling one to determine the elements of the sequence (h,) by the elements of (wn). Since (1 - X1L)(1- X2L) = 1 - (A, X2)L X1X2L2 (23)

+

for any A 1 and X p , it follows that if we choose

A1

+

and A2 such that

then 1 - UlL

-

a2L2 = (1 - X1L)(1- A&).

It, is clear from (24) t h a t A 1 and X2 are the roots of the quadratic equation

(25)

131

2. Linear Stochastic Models

To put it another way, X 1 = 2;' and X2 = z;', where z1 and 22 are the roots of the equation 1 - a1z - a 2 2 = 0, (27) while the polynomial 1 - (LIZ - a2z2 if the result of the substitution L operator expression 1 - a l -~a 2 ~ ~ . In view of (25), we can rewrite (12) as follows:

+z

in the

(1 - X1L)(1- X2L)hn = wn.

After the formal multiplication of both sides by (1 - X2L)-l(l that h,, If A 1

#

~

(1 - x2L)-l(1- x1L)-lw,.

XiL)-l we see (28)

X2, then wc formally obtain

therefore

If l X i l < 1, i

= 1 , 2 , i.e., if the roots of the characteristic equation (27) lie outside the unit disc. then

(1 - XzL)-l = 1 +X;L+X?L2+ . . .

,

i = 1,2,

(31)

arid therefore, in view of (30), the stationary solution (with finite second moment) of (12) has the following form:

where

(The uniqueness of the stationary solution to (12) can be proved in the same way as for (13).)

Chapter 11. Stochastic Models. Discrete Time

132

5 . Finally, we proceed to general A R ( p ) models, in which

h, = a0 that is (setting uill = a0

+ alhri-1 + + aphn,--p + g & n ,

(33)

' ' '

+ mTL),

(1 - a1L

-

a2L2

-

" '

-

a p L p )h 7 1 = w,.

(34)

Using the same niethod as for p = 1 or p = 2, we consider the factorization 1 - a1L

-

a2L2

-

. . ' - a,LP = (1- X1L)(1 - X2L) ' . (1 - A&) '

(35)

and assiinie that the X i , i = 1, . . . , p , are all distinct. If l X i l < 1, i = I, . . . , p , then we can express the stationary solution of (34), which is t,hc unique solution with finite second moment starting at --3o in time, as follows: Irn = ( l - A I L ) - l . " ( l - A p L ) - ~ w ~ , , . (36) We note that tjhe numbers X i , XP

-

X2,

. . . , A, are the roots of the equation

alXP-l

- . . . - ap-lX

-

ap = 0

(37)

(cf. (26)). Putting that another way, we can say that X i = z i ' , where the zi are the roots of the equation 1 - u1z - a2z2 . . . - u,zP = 0 (cf. (27)). Hence we can obtain a stationary solution h = (h,) if all the roots of this equation lie outside the unit disc. To obtain a representation of type (30) or (32) we consider the following analogue of the expansion (29): -

where c1, . . . , cp are constants to be determined. Multiplying both sides of (38) by (1 - X l z ) . . . (1- A+), must have 2)

1=

x.2 n i=l

we see that for all z we

(1 -A@).

This equality must hold, in particular, for z = A T 1 , . . . , z = ,A'; the following values of cl , . . . , cp:

(We note that c1

+ . . . + cp = 1.)

(39)

l
which gives us

2. Linear Stochastic Models

133

By (36), ( 3 8 ) , and (40), M

z=o which is a generalization of ( 3 2 ) to the case p 3 2. The representlation (41) enables one to evaluate various characteristics of the seqiience h = (&) such as the moments E h i , the covariances, the expectations E ( h r L + k I .a,,)(where 9,= o(.. . , w-1: P U O , . . . , w r 1 ) )and , so on. Assiiniirig that we have a stationary case it is easy to find the nionierits Eh, 5 p directly from ( 3 3 ) : P=

*,0

1 - ("1

+

'

'.

+

UP)

.

(42)

As regards the covariances R ( k ) = Cov(hll,h7L+k),we see easily from ( 3 3 ) that

for k = 1 , 2 , . . . . If k = 0, then R(0) = alR(1)

+ . . . + a p R ( p ) + 02.

(44)

The correlathi functions p ( k ) , k 3 0, satisfy the same equations (43) and (44) (they are called the Yule- Walker equations, after G. U. Yule and G. T. Walker; [271]). 6. One of the central issues in the statistics of the aiitoregressive schemes A R ( q ) is the estirnntion of the parameters 0 = ( a g , a l , . . . , a P ,.-) involved in (1) arid (a), where we now assume for definiteness that ho, h - 1 , . . . are known constants. If we assiiiiic that the white noise E = ( E ~ is~ Gaussian, ) then the most important estimation tnols is the mazim.um lrkelihood method in which the quantity A

OT1 = argmaxpg(Ii1, ha.. . . , h,) 8

is the cstiriiator for the parameter 0 defined in terms of the observations h l , 1 1 2 , . . . , liT1 (here p e ( h 1 , h a , . . . , h,) is the joint probability density of the (Gaussian) vector ( h l , h2,. . . , h n ) ) . We sh,ill illustrate the gcneral principle pertaining t o the maximum likelihood method by thr c~xarnplrof the AR(1) model with h, = a0 + alh7,-1 + (TE,; here we regard 0 > 0 as a known parameter, ho = 0, and n 3 1. Since 6' = (ao, a l ) , it follows that

Chapter 11. Stochastic Models. Discrete Time

134

It is easy to see that the value of the estimator 0 = (%~,21) can be found from the niinirnality condition for the function n

?ii(ao,ad =

Z ( h k - a0

-

alhk-1)

2

.

k=l

Letting a0 E JR and a1 E JR, we obtain

Solving this linear system one finds the values of the estimators 20 and 21. We now concentrate on the properties of the estimators 21=21(hl,. . . , h n ) , n 3 1, for a l . We assume for simplicity that a0 is a known parameter (a0 = 0) and a=l. Under these assumptions, f i n = alhn-l E~ and

+

by (45). Hence

We now set

n

k=I

The sequencc M = ( M n ) is (for each value of the parameter al) a martingale with quadratic characteristic k=l

Hence if

a1

is an actual value of this unknown parameter, then

From this representation we see that the maximum likelihood estimator iil is strongly consistent: 21 --t a1 as n --t cm with probability one because ( M ) , + 00 (P-as.), and by the strong law of large numbers for square integrable martingales (see (12) in 5 l b and [439; Chapter VII, 5]),

2. Linear Stochastic Models

135

Calculating in our case (a0 = 0, c = 1) the Fisher information

where E,,

is the expectation with respect to the measure Pul = Law(h1,. . . , h, 10 = al),

we obtain

Using the above formulas for the Ehk-1 and Dhk-1, we see that

We sce that la11 = 1 is some distinguished, ‘boundary’ case, when h = (h,)

is a random walk (cf. Chapter I, §2a “Random walk conjecture and concept of efficient market”). For lull # 1the corresponding sequence h = (h,) is Markovian. Moreover, if lull < 1, then the sequence h = (h,) ‘approaches a steady state’ as n + 00. This fact is responsible for the considerable attention that is paid in the statistics of the sequences h = (h,) to the solution of the following problem: which conjecture of the two, H o : lull = 1 or H I : lull > 1, is more likely? In the econometrics literature and, in particular, in the literature on financial models the range of issiies touching upon the validity of the equality la11 = 1 is called ‘the unit root problem’. We now present (without proofs; referring instead t o [424], [445], and the literature cited there for greater detail) several results concerning the limit distribution for the deviations 21 - a1

THEOREM 1. As n

+ 30

w e have

< 1; = 1,

> 1,

Chapter 11. Stochastic Models. Discrete Time

136

L..

p(o,l) ( w ) d y is the standard normal distribution. Ch(z) is the 1 and Ha, (z) is the distribution of the Cauchy distribution with density

where @(x) =

+ .2)

*

Wy1)

-

7r( 1

random viiriable a1

1

1

2fik

W2(s)ds '

where (W ( s ) ) , 1~ is ~ a standard Wiener process (Brownian motion). It is iritcrcsting that if l u l l # 1, then the densities of the limit distributions are syrnrrretric with respect to the origin. However, the density of the distribution H,,(J:) is a s y m m e t r i c if / a l l = 1. (This is an easy consequence of the observation that P(W2(1) - 1 > 0) # The next, result shows that, considering a random, ,normalization, of the deviatio,n (GI- u 1 ) (which means that we use the stochastic Fisher information ( A 4 ) n in place of the Fisher i7iformation I n ( a l ) )we can obtain only two distinct limit distributions rather than t h w e as in (48).

i.)

THEOREM 2. As

71

+x

we have

where H6,(z) is the prohability distribution of the random variable W2(1)

1

a12/F' -

Finally, the sequential maximum likelihood estimators together with random normalization bring us to a unique limit distribution.

THEOREM 3 . Assume that H > 0 , let ~ ( 0= ) inf{n 3 1: ( M ) , and let

for each

wl E

W.

Q},

(50)

2. Linear Stochastic Models

137

We now coniparc this result with assertion (48). If n was our original time parameter, then we can regard 0 as ‘new’: ‘operational’ time defined in terms of stochastic Fisher information ( M ) . We note that if Fisher information changes only slightly on (large) time intervals, then they correspond to small intervals of ‘new time’ U and conversely. Thus, using this ‘new’, ‘operational‘ time we make the flow of information more uniform, homogeneous, the incoming data are now of ‘equal worth’, and are ‘identically distributed’ in a sense for all the values of a l . Eventually, this results in the uniqueness and normality of the limit distribution. We come across problems related t o this ‘new‘ time below, in Chapter IV, 3 3d, where we explain in close detail one such change of time aimed at ‘flattening’ the statistical data on currency cross rates, in the dynamics of which ‘geographic’ component,s of the periodic nature are clearly visible. 7 . We now present several additional results on the properties of maximum likelihood estimators (see [258], [424], and [445]). First, sup Ea, 121 - n l I + 0 as n + 00. a 1 €W

Second; let, U ( a 1 ) be the class of estimators E l with bias b,, ( E l ) satisfying the conditions

E

E,,

(El

-

al)

(The maximurn likelihood estimators 21 are in the class U ( n l ) if In11 # 1.) If lull # 1, then the maximum likelihood estimators 21 are asymptotically efficient in U ( a 1 ) in the following sense: for ?il E U ( a 1 ) we have

If la11 < 1 (t,hc ‘stationary’ case), then the estimators E l are also asymptotrcally ejjiczent in the usual sense, i.e., for all Zl E U ( a l ) ,

Sequential m a z i m u m likelihood estimators have also the propert,y of asymptotic uniformity: as 0 + m. we have

Chapter 11. Stochastic Models. Discrete Time

138

$ 2 ~ Mixed . Autoregressive Moving Average Model ARMA(p,q ) and Integrated Model ARIMA(p, d, q ) 1. These models combine the properties of the already considered M A ( q ) and A R ( p ) models, thus often providing a fair opportunity to find a model that can well explain the probabilistic backgrounds of statistical 'stock'. As above, we consider a filtered probability space (a, 9, (9,) P).; It is now convenient to assume that 9,= B ( . . . , E L I , E O , ~ 1 ,. .. , E ~ ) where , E = (F,) is 'white noise' (in the strict sense). By definition, (see 3 Id), a sequence h = (h,) is governed by the model ARMA if h, = P n C E n , (1) where

+

+

p n = (a0 alhn-1

+ '.. + a p h - p ) + ( b l ~ n - 1+ b 2 ~ n - 2+ ... + b q E n - q ) .

Without loss of generality we can assume that the value of equal to one: c = 1. Then by (1) and (2) we obtain hn

-

(01hn-l

B

(2)

is known to be

+ . . . + a p h n - p ) = + [ ~+nb l ~ n - 1+ b2~,-2 + . . . + bq&n-q]

(3)

or

(4) where

(5) and

P ( L ) = 1 + blL + . . . + bqLq.

We note that if q = 0, then a(L)hn = w n ,

+

where w, = a0 E,, i.e., we arrive at the A R ( p ) model (cf. (5) in 3 2b). On the other hand, if p = 0, then (3) takes the following form: h, = a0 + P ( L ) E , ,

i.e., we obtain the M A ( q ) model (cf. (4) in 3 2a). Considering a formal conversion of (4) we see that

(under the assumption that a1

+ . . . + up # 1).

(7)

2 . Linear Stochastic Models

139

We iiow consider the question on the existence of a stationary solution of equation (3) (in the class L 2 ) . By (8) and in view of our previous discussions (in 5 2b.5), the answer to this question depends on the properties of the operator n ( L ) , that is, of the autoregressive components of our model ARMA(p,4 ) . If all the roots of equation (37) in 5 2b are smaller than one in absolute values (and in this case a1 +. . .+a, # l ) ,then this model has a unique stationary solution h = (/ijr,,) (in the class L ~ ) . For this stationary solution, Eh, =

a0

l-(al+...+up)

by (8) (cf. (42) in 52b). It is easy to conclude from ( 3 ) that the covariance R ( k ) = Cov(h,, & + k ) satisfies for k > q the same relations

as in the A R ( p ) case (cf. formula (43) in § 2b). If k < q , then the corresponding representation of R ( k ) has a more complicated form than ( l l ) , since we must also take into account the correlation dependence

between

E,,-k

and h n - k .

2. As an illust,ration, we consider t h e model ARMA(1, l ) ,which is a combination of A R ( 1 ) arid MA(1):

We assume that la11 < 1 (the ‘stationary’ case). Then

and (8) can be writtcri as follows:

Chapter 11. Stochastic Models. Discrete Time

140

-2

-3

t 0

100 200

300

400

500

600

700

800

900 1000

FIGURE 20. Computer simulation of a sequence h = ( h l L )governed by the ARMA(1,l) model with h , = a0 alh,-l + ~ I ~ E , - I + mn (0 < 'ri < 1000), where a0 = -1, a1 = 0.5, b = 0.1, u = 0.1, arid

+

It0

=0

HCXY we ininiediatcly see that the covariance R ( k ) = Cov(h,, h n + k ) satisfies the following relations: R(k) = alR(k - l ) , k

2 2,

+bl, R(0) = alR(1) + (1 + aibl + b:)> R(1) = alR(0)

and therefore R(0) = Dh, =

1

+ 2albl + b: 1 - 0,:

and

It should be mentioned that the correlation decreases geometrically as k + 30 for (all < 1. This must be taken into account in the adjustment of the A R M A ( 1 , l ) models (or the more general A R M A ( p ,q ) models) to particular statistical data. 3. The above-considered models A R M A ( p ,q ) are well understood and used mostly in the description of stationary time series. On the other hand, .if a time series z = ( z T Lis) nonstationary, then the consideration of the differences AxTL= 2 , - zrL-l or the d-order differences Adz,, brings one to the (occasionally) 'more' stationary sequence Adz = (Adz,).

141

2 . Linear Stochastic Models

There exist special expressions describing this situation: one says that a sequence is governed by the ARIMA(p, d , q ) model if Adz = (Adz,,) is governed by the model ARMA(p,q ) . (In a symbolic form, we write AdARIMA(p,d, q ) =

x

= (z,)

A R M A b q).) For a decper insight iiito the meaning of these models we consider the particular case of ARIMA(O,l, 1). Her? AT,, = h,, where (h,) is a sequence governed by M A ( 1 ) , i.e., Ax,,= p (bo blL)E,.

+ +

Let S 1 ) ~the operator of summation (‘integration’) defined by the formula or, equivalently,

s=

s = 1 + L + L2 +

= (1 - L ) -

1

.

Then we can formally write ZT,

+

+

+

=

(Sh),,

+

where h,, = p (00 b l L ) ~ = , p ~ O Eh 1 ~~ , - 1 . Herice z = (z;,) can be regarded as the result of the ‘integration’ of some sequence 12. = (h7,)governed by the MA(1) model, which explains the name ARIMA = A R I M A . (Cf. Fig. 18 and Fig. 21.)

+ +

100

1

0

10

20

30

40

50

60

70

80

90

100

FIGURE 21. Computer siniulation of a sequence z = (z,) governed by the A R I M A ( O , l , 1) model with Ax, = ,u b l E n - l bOEnr where ,u = 1, bl = 1, bo = 0.1, and 20 = 0

+

+

We have already nieritioned that these models are widely used in the BoxJenkins theory [ 5 3 ] . For information about their applications in financial data statistics, see, e.g., [351].

Chapter 11. Stochastic Models. Discrete Time

142

3 2d.

Prediction in Linear Models

1. We pointed out in the introduction to this section that the construction of probabilistic and statistical models (based on the ‘past’ data) is not an end in itself; it is necessary, in the long run, to predict the ‘future’ price movements. It is very seldom that one can give a n ’error-proof’ forecast using the ‘past’ data. (This is characteristic, e.g., of the so-called singular stationary sequences; see subsection 4 below and, in more detail, e.g.: [439; Chapter VI].) The typical situation is, of course, the one when making a forecast we are inevitably making an error, the size of which determines the risks involved in the solutions based on this forecast. 2. In the case of stationary linear models there exists a well-developed (and beautiful) theory of the coristructiori of optimal linear estimators (in the mean-square sense), which is mainly due to A. N. Kolmogorov and N . Wiener. We have already seen that many of tlie sequences h = (h,) considered can be represented as on,e-sided moving aiierages

k=O cz

where

C lakI2

< ce and

E = (E,)

is some ’basic’ sequence, white noise. (See

k=O

formula (15) in 5 2a for the lllA(q) and M A ( m ) models, formula (41) in 3 2b for the A R ( p ) models, anti (8) in fj 2c for ARMA(p, q ) . ) To describc results on the extrapolation of these sequences we require certain concepts and notation. If [ = (&,) is a stochastic sequence, then let 9 2 = o(.. . , [,-I. Cn) be the 0algebra generated by the ‘past’ {&, k < n } , let 9, E- = be the 0-algebra

v92

generated by all the variables Ell, let H i = F(. . . , [,-I; En) be the closed (in L 2 ) linear niaiiifold spanned by the variables { [ k , k n } , and let f& = p(&., 5 < ce) be the closed linear manifold spanned by all the variables En. Let 1) = r / ( w ) be a random variable with finite second moment E q 2 ( w ) . We now forniulate the problem of finding an estimate of rl in terms of our observations of the sequence (. The following two approaches are most widely used here. If these are the variables (. . . , & I , En) that we must observe, then in the framework of tlie first approach, we consider the class of all 92-rneasurable estimators TrL and the one said to be tlie best ( o p t i m a l ) is the estimator qn delivering the smallest mean-square deviation, i.e.,

<

El71 - qnl 2 = iEf Elrl 7ln

-

;i71I2.

(2)

2. Linear Stochastic Models

143

As is well known, the estimator optimal in this sense must have the following form: irn = E(17

ISL

(3)

where E( . 1 . ) is the conditional expectation, which is, generally speaking, a nonlinear function of the observations (. . . , (,-I, I,). (The issues of the nonlinear optimal filtering, extrapolation, and interpolation for fairly wide classes of stochastic processes are considered, e.g., in [303].) In the framework of the other approach, on which we concentrate now, one considers only the set of linear estimators together with its closure (in L 2 ) , i.e., functions of (. . . , En-1, En) belonging to H,.t In a similar way to (2), by the best ( o p t i m a l ) linear estimator of the variable 77 we mean A, E H i such that

In this case we use the following notation for

where

xn (cf. ( 3 ) ) :

z(.1 . ) is called conditional expectation in the wide sense.

3. We now return to the issue of (linear) prediction on the basis of the ‘information’ available by time 0 for the variables in a sequence h = (h,), n 3 1, described by (1). This problem can be solved relatively easy if the ‘information’ in question means all the ‘past’ generated by E = ( E , ) ~ < O (rather than by h = ( h n ) n ~ Oi.e., ),

H,E = 9(. . .,E - 1 ,

EO).

Indeed,

since A

E(E, j H i ) = E E ~(= 0)

for 1: 3 1, which follows easily from the orthogonality of the components of the white noise E = (E,).

Chapter 11. Stochastic Models. Discrete Time

144

Moreover, it is not difficult to find the extrapolation error

k=O

<

Clearly, D,”,

(T:+~

and

Hence the role of the ‘past data’ H,E = T(.. . , ~ ~ - €0) 1 ,in our prediction of the values of hn diminishes with the growth of n and, in the limit (as n + m), we should take the mere expectation (i.e., 0 in our case) as the best linear estimator. Of course, the problem of extrapolation on the basis of not necessarily znfinztely m a n y variables { E ~ k, 6 O}, but finztely many ones (e.g , the variables { ~ k 1, 6 k 6 0)) is also of interest. It should be noted, however, that this latter problem is technically more complicated that the one under consideration now. The representation k=n

so obtaincd solves the extrapolation problem for the liTL on the basis of the (linear) ‘information‘ H i rather than the ‘information’ H k contained in y(. . . , h-1, ho). Clearly, if we assume that

H,E= H $ ;

(8)

then using (7) wc can ( i nprinciple, a t any rate) express the corresponding estimator x

h

E(h,,iIHgh)in the form of a sum

Ekhn-k. k=n

4. In view of our assumptions (1) and (8), it would be useful to recall several general principles of the theory of stationary stochastic sequences. Let = (<.,,,) be a stationary sequence (in the wide sense). We can represent each element rl E H ( < )as a sum of two orthogonal components:

<

n,,

whew S([)= H i is the (linear) iriforrnatiori contained in the ‘infinitely remote , q E H(<). past’. Let, I?(<) he the set of elements of the form 7 - g ( q I S ( < ) )where Then we see that H ( < ) can itself be represented as an orthogonal sum: H ( < ) =

S ( 0 @ NO. We say that a stationary sequence singrilnr if H ( ( ) = S ( < ) .

< = (Ell)

is reyular if H ( < ) = R(<)and

2 . Linear Stochastic Models

145

The rneaning of t,hc condition H ( $ ) = S ( ( ) is fairly clear: this indicates that all the inforinat.ion providcd by the variables in the sequence $ ‘lies in t h e infinitely remote past’. This explains why singular sequences are also said to be purely or completely deterministic. If S ( $ ) = 0 (i.e., H ( $ ) = R ( [ ) ) then . one says that tlie sequence ( is purely or completely nondeterministic. The following result slieds light on the concepts that we have just introduced.

PROPOSITION 1. Each nondegenerate stationary (in the wide sense) random se(En) has a unique decomposition

quence [ =

=

Fn

,<; +

where $” = (EL,) is a rcgular sequence and $” = ([i) is a singular sequence; moreover, (” anti [” arc orthogonal (Cov($G, [&) = 0 for all n and m ) . (See [ A N ; Chapt,cr IV. 3 51 for greater detail.) We now introduce another important concept. DEFINITION. We call a sequence a)

E

=

( c T La)n innovation sequencc for $ = (En) if 0, EE,~E,, = 0 for

= ( E , ~ , ) is ‘white noise’ in the wide sense, i.c., EcTL= ri # ‘ r n , arid EE), = 1;

E

b) H,f = H i for each n. The iricaiiing of the trrni ‘innovation’ is clear: the elements of the sequence are ort,hogonal. therefore we can regard as (hew‘) iriforniation that ‘updates’, ’innovates’ the data in Hg and enables one to construct HE+1. Since

E

=

H$ = H: for all n,, E T L + l updates also the inforniation in H,.E (Cf. Cliapter I, 5 2a where we consider the ‘random walk’ conjecture and discuss the concept of ‘efficient market .) Tlic next, result, cstablislies a connection between the concepts introduced above. PROPOSITION 2 . A mondc:generate sequence [ = (&) is regular if and o d y if there CYI

exist an innovation sequence E =

(E,)

and nrirribers

(ak)k.O ,such tliat

C

lakI2

< 00

k=O

and (P-as.)

k=O

(See the proof, e.g . in [439; Chapter VI, $51 ) The following result is an immediate ronsequence of Propositions 1 and 2

PROPOSITION 3 . If Wold expansion

E

=

(&) is a nondegeneratr stationary sequence. then it has a

k=O

146

Chapter 11. Stochastic Models. Discrete Time 00

where

C a;

< 00 and E

= ( E ~ is ) some innovation sequence (for E‘).

k=O

5 . Assuming that the sequencp h = (h,) has a representation (1) with innovation sequence (for h ) E = ( E ~ we ) obtain

Hence

co

M

k=n

k=n

by (6), where the last equality is a consequence of the relation H: = H,h holding for all 71. Formula (11) solves the problem of optimal linear extrapolation of the variables h,, on t,he basis of the ‘past’ information { h k , k 0 } in principle. However, the following two questions come naturally: when does a sequence h = (h,) admits a representation (1) with innovation sequence E = (E,) and how can one find the coefficients i i k in (1I)? The answer to the first question is contained in Proposition 2: the sequence h = ( h T Lm ) i s t be regular. It is here that the well-known Kolmogorov test is working: if h = (h7,)i s a stationary sequence in the wide sense with spectral density f = f ( A ) such that

<

then this sequenm i s regular (see, e.g., [439; Chapter VI, 3 51).

6. We recall that each stationary sequence in the wide sense [ = (&) with EEn = 0 has a spectral repre%sentation

where z = z(A) (with A E % ( [ - T , T ] ) ) is a complex-valued stochastic measure with orthogonal values: Ez(Al)%(A2)= 0 if A1 n A2 = 0 (see, e.g., [439; Chapter VI, 531). In addition, the covariance function R ( n )= C O V ( & + ~ ~E,l ; ) has the representation

1, ir

R(n) =

eiXnF ( d A )

with spectral measure F ( A ) = E(t(A)l2. If there exists a (nonnegative) function f = f ( A ) such that

(14)

147

2 . Linear Stochastic Models

then we call it tlie spectral d e n s i t y , while the function F ( X ) = F ( ( - m , A]) is called the spectral fiinction. Hence if we know a priori that the initial sequence h = (h,) is stationary in the wide sense with spectral density f = f ( X ) satisfying condition (12). then this sequence is rqiilar, and admits the representation (9) with innovation sequence E

= (En).

7. In the linear models considcrcd above we did not define the sequence h = ( l z r L ) in ternis of spectral representations; we used instead the series (1) witaha n orthonormal sequence E = ( c r Lsuch ) that EE, = 0 and E ~ i c = j b i j , where bij is the Kronecker delta: 6 i j = 1 for i = j and 6ij = 0 for i # j. Clearly. the sequence

E = ( E ~ is )

stationary arid we have

for itjs correlation (= covariance) function and 1

fE(X) =

G,

-7r

<X <

7r

(17)

for the spectral density. x

If

lirl

=

C

x

nk:c,,,-k

and

k=O

fiinctiori of the seqiierice

11

C

lakI2 < m, then it is easy to see t,hat the covariance k=O = (h,) can be defined as follows:

and k=O

We now assume that the power series (20) has convergence r a d i u s T > 1 and does not 7mnish f o r IzI 1. Under this assumption we can answer the question on tjhc values of t,he coefficients Zk in the representation (11) for the optirnal linear prediction h, as follows (see, e.g., [439; Chapter VI, fj 61 for greater detail). Let 4,

<

Chapter 11. Stochastic Models. Discrete T i m e

148

where 03

k=n

We expaiid

Gr,(A)

iii tlic Fourier series:

Then the coeffirients Z$;L), ,-.

...

are prerzsely those delivering the optimal linear

prediction h , of hrL: x

k=n

Thiis, we have a i n e t h d for constructing a prediction of h,,rLon the basis of the ‘past’ variables {. . . . h - 1 , h o } , provided t,hat we can find the representation ( 2 3 ) . 8 . To illustrate the above method we consider now several examples for the stationary models: M A ( q ) , A R ( p ) , and -4RMA(p. q ) . We shall indicate explicit formulas for the predictions only for several values of p arid q. As regards general formulas, see, c.g., [CD;Chaptcr IV, 5 61.

EXAMPLE 1 (The I

I J O M ~ A ~ ( q ) ) . Let,

Cornparing this representation with (1) we see that a k = bk for 0 a h = 0, k > q . Consrqnt?nt ly, 4

and


and

149

2. Linear Stochastic Models

Hence

<

ri (I and CIL(A)= 0 for n > q. Thus, if 71 > y, then ZTL= 0 for all the coefficients in the expansion ( 2 3 ) . therefore the optiirial prediction in this case is h,, = 0. This is far from surprising. for the corrclation between h, and each of ho, h - 1 . . . . is equal to zero for n > q. If q = 1. tllcll

for

h

+blEn-l

hrL= b o E 7 ~ arid

Of course. we can assume from the very beginning that bo = 1 arid can set bl = 8, wliere 101 < 1, which ensures that the function a(.) = 1 Qz does not vanish for /zI 1. Tlieri

+

<

Comparing this with (23) we see that, for q = 1 arid n = 1,

... ... Hence we obtairi the following formula for the prediction of ti1 on the basis of the 'past' information (. . . h-1, ho) in the model MA(1): h

hl = Z p h O = 6(h,

+Z

-

pLl+

0h-1

+ ZY)h1-k + ' . . + . + (-1)k-lQk-lhI-k +

' ' '

+ 62h-2

' '

' '

.)

I;=O

It is now clear that the largest contribution to out prediction of hl is the contribution of the 'most recent past', the variable ho; while the contribution from the 'preceding' variables decreases geometrically ( h - k enters the formula with coefficient d k , where 181< 1).

Chapter 11. Stochastic Models. Discrete Time

150

EXAMPLE 2 (The rriodel A R ( p ) ) . We assume that (cf. (33) in 11, = nlhn,-l aphn,-p + E n

+ + ' ' '

3 2h) (29)

for -00 < n < 00;moreover, all the A; (i = 1 , .. . , p ) in the factorization (35) (3 2h) are distinct and / A i l < 1. According to forrnula (41) in 5 2b, the stationary solution of (29) can be represented as follows:

i.e., in the representatiori ( I ) we have k

Uk = C I A 1

+ + cpA,.k ' ' '

(31)

In principle, the prediction ^h, of h, on the basis of the variables h,k, k < 0, is described by (24), where the coefficients 2p) can be found from the Fourier expansion of @,,!,(A) (see (21)-(23)) by taking into account formula (31) for the coefficients n k in the definition of &(A). Fairly niariy various methods described in the literature produce formulas for the predictions h,, that are convenient in applications (see, e.g., [ 2 l l ] , where one can find a description of a recursion procedure enabling one to construct h, from h,l,. . . , 1 1 , , - ] ) . We restrict ourselves to a n illustration by carrying out a calculation for the stationary model A R ( 1 ) : h,, = olLn-l E, (32) with 101 < 1. We recall that R ( n ) = Cov(1ik. J L ~ + ~ , can ) be defined in this case by the formula h

A

+

Hence it. is easy to see that the spectral function f = f ( A ) is well defined arid 1 1 f(A) = - . (33) 27r 11 - 8 e - y Coriiparirig ( 3 3 ) arid (19) we see that

a(.)

1

= --

1 - oz

k=O

Hence

@,(A)

= 8"

by (21), and therefore (see (23) and (24)) A

h,, = O7'h0,

n 3 1, i.e., for a predic:tiori of h,, on the basis of { h k , k 0) one requires only the value of ho. This is riot surprising, given that h = (h,,) is a Markov sequence.

<

2. Linear Stochastic Models

151

EXAMPLE 3 (Thr, niotlcl A R M A ( p , q ) with parameters p = q = 1). Setting a0 = 0 in equation (12) (5 2c) we find the following formula for a stationary solution: h, = (-1

+ bl)

c

af-bn-k

+

En

k=l

whcrc (all

< 1 . Hence the coefficients in the representation h,,

as follows:

cc

=

C E k ~ ~ - lare ; k=O

-

a0 = 1 ,

-

ak = (q

+ h1)a;y.

(34)

Consequently,

for

ri

3 1 . In view of (21) and (22), setting z

=

we obtain (for lbll

< 1)

k=O

and (see (23))

@ = aT-l(a1 + b1)(-l)"f Hence

by (24).

Remark. As regards the prediction formulas for the general models A R M A ( p ,q ) and ARIMA(p,d,q), see, e.g., [211].

3. Nonlinear Stochastic Conditionally Gaussian Models

The interest, in nonlinear models has its origin in the quest for explanations of several phenomena (apparent both in financial statistics and economics as the whole) such as the ’cluster property’ of prices, their ‘disastrous’ jumps and downfalls, ‘heavy tails’ of the distributions of the variables h,, = In

sn , ‘the long sn- 1

~

nieniory‘ in priccs, and some other, which cannot be understood in the framework of linear models On the other hand, there is no unanimity as to which of the nonlinear models-stochastic, chaotic (‘dynamical chaos’), or other-should be used. Plenty of advocators are adducing arguments in favor of one or another approach. There can be no doubt that economic indicators, including financial indexes, are prone to fluctuate. Many niacroeconomic indicators (the volumes of production, consumption, or investment, the general level of prices, interest rates, government reserves, and so on); which describe the state of the economy ‘on the average’, ‘at large’, do fluctuate, and so do also microeconomic indexes (current prices, the volume of traded stocks, and so on). Moreover, these fluctuations can have a very high frequency or be extreniely irregular. This is known to occur also in stochastic and chaotic models: hence the researchers’ attempts to describe fluctuation dynamics, abrupt transitions, ‘catastrophic’ outbursts, the grouping of the values (the cluster property), and so on, by nieaiis of these models. Considering thc behavior of many economic indexes (production volumes, the sizes of particular populations, or government reserves) ‘on the average’ one can discern certain trends, but this movement can accelerate or slow down: the growth can occur in cycles of sorts (periodic or aperiodic). Thus, a n analyst of statistical d a t a relating to the economy, finances, or some area of natural sciences or technology, finds himself in front of a rather nontrivial problem of the choice of a ‘right,’ model.

3. Nonlinear Stochastic Conditionally Gaussian Models

153

Below we describe several nonlinear stochastic and chaotic models that are popular in fimncial mathematics and financial statistics. We are making no clainis of a cornpreliensive exposition, but are willing instead to present an ‘introduction’ into this range of problerns. (As regards some nonlinear models, we can recommend, e.g., the monographs [193], [202], [461]: and [462].) fj3a. A R C H and G A R C H Models 1. Lct (a, 9, P) be the original probability space and let E = ( E , ) , ~ I be a sequence of independent, normally distributed random variables ( E ~ N(0,l))simulating the ‘raiidomness’, ‘uncertainty’ in the models that we consider below. By gTL we shall mean the a-algebra o ( ~ 1 ,. .. , E ~ ) we ; set 9 0 = (0; n}. We shall interpret S,, = S,,(w) as the price (of a share or a bond, or a n exchange rate, etc.) at time rt = 0 . 1 , . . . . Here time can be measured in years, months, . . . , minutes, and so on. As already rncritioned (5 Id), to describe the evolution of the variables

-

Sn , h,, = In __ Sn-

1

R. Engle [140] considered the conditionally Gaussian model with h, = uTL&,. Here the volatilities aILare defined as follows: a; = a0

+

c 2,

a;h;,_i,

(3)

i=l

where 00 E

> 0, ai 3 0, and ho

= ho(w) is a raritlom variable independent of

= ( ~ , ~ ) , , 2 1 .(One often sets ho t o be a constant or a random variable with

expectation of its square chosen so that the expectations Eh:; ri 3 0, are constant.) We see from (3) that the ‘T,,~are (predictable) functions of h ; L - l , .. . , h:-p. Moreover, it is clear that large (small) values of the h2-i imply large (respectively, small) values of a:. On the other hand, if 1 ~ : turns out to be large, while the values of the ‘preceding’ variables h:,-l, . . . , h:-p are small, then this can be explained by a large value of E ~ This . gives a n insight into how (nonlinear) models (1)-(3) can explain such phenomena as the ‘cluster property’, i.e., the grouping of the values of the h,,into batchcs of ‘large’ or ‘small’ values. These considerations justify (as already nientioncd in 5 Id) the name of ARCH(p) ( AutoRegressive Conditional Heteroskedastic model) given to this model by R. Engle [140]. The conditional variance (‘volatility’) o: behaves in this model in an ext,reriiely uneven way because, by ( 3 ) , it depends on the ‘past’ variables h2T L - l , h : L - p .

Chapter 11. Stochastic Models. Discrete Time

154

2. We now consider several properties of sequences h = (hTL),>1,governed by the

A R C H ( p ) model. For simplicity, we restrict ourselves t o the case p = 1. (See, e.g., [202], [193], and [393] for a thorough study of the properties of the A R C H ( p ) niotlels and their applications; we present one result concerning the occurrence of 'heavy tails' in such models in 5 3c (subsection 6 ) . ) For p = 1 (see Fig. 22) we have 2

on = a0 The following simple properties of the

0

20

10

30

40

2 + a1fb-1,

tirL = g,~,

50

60

(4)

are obvious:

70

80

90

100

FIGURE 22. Corriputer siniulation of a sequence h = ( h n ) governed by the ARCH(1) model with h,, = Ju.0 culhi-l en (0 n loo), where cro = 0.9, a1 = 0.2; ho = 3

+

< <

If 0 < 0 1 < 1, then the recursion relation (6) has the unique stationary solution

therefore setting h2 -

~

a" we obtain forniula (8) for the Eh:, n 3 1. 1 - rr1

3. Nonliricar Stochastic Conditionally Gaussian Models

155

Next, a simple calculation shows that

Hence, assunling that 0 ‘stationary’ solution (Eh:

<

~1

< 1 and 3,:

= Const): Ehi =

< 1, we can obtain the following

3 c ~ i (-t l 01)

(1 - a1)(1- 3 4 )

By (8) and (lo), the ‘stationary‘ value of the kurtosis is

It is positive, which means that the density of the ‘steady-state’ distribution of the variables h,, has a peak around the mean value (and the larger a:, the more distinguished is this pcak). We recall that K = 0 for a normal distribution.

Remark. The empirical value K N of the kurtosis can be calculated from the values of h l , 1 1 2 , . . . , 1 1 by ~ thc formula

Gold

1975 -82

K N = 8.4

Silver

1970-74

K N = 8.4

GBP/USD

1974-82

K N = 5.4

General Motors

1966--76

K N = 4.2

Chapter 11. Stochastic Models. Discrete Time

156

3. For 0 < < 1 the sequence h = (h,) with h, = o T L is ~n a square integrable niartingalc difference, therefore it is a sequence of orthogonal variables:

Of course, this does riot nicari that 11, arid Itm are independent, because, as we see, their joint, d i s t r i h t i o n Law(h,,, hm) is riot Gaussian for 01 > 0. One can get a n idea of the character of the dependence between 11, and lim by considering the correlation dependence between their squares h i and or their absolute valiies I Lr, 1 and 1 ltm 1. A siniplc c:alculation shows that

hk

Dh?,=

2 2 ( A) ,

1-3af

1 -a1

and

Further,

for k < 7 1 , which, gives the following simple recursion relation for p ( k ) Cov(h;,> q - k ) , in the 'stationary case':

so that p ( k ) = (2:

=

3. Nonliriear Stochastic Coriditionally Gaussian Modcls

157

3 Id that the A R C H ( p ) models are intimately connected with (geiieral) autoregressive schemes A R ( p ) . For assume that we are considering the A R C H ( p ) model arid let vn = h:l - 0:. If Eh:& < 30, then the sequence v = ( 7 / 7 1 ) is and it follows from ( 3 ) that a martiigale difference (with respect to the flow (Tn)) the variables IC, = ti: are governed by the autoregressive niodel A R ( p ) : 4. We pointed out in

ZTL

+ CvlX,L-l + + apICn--p+ v,.

= NO

(14)

' ' '

with noisc I / = (v,,) t,liat is a martingale &ifference. If p = 1, tl1en

x,, = a0

+ CulZn-1 + v,,

therefore thc above forrmila (13) is the same as (10) in

fi 2b.

5 . The A X C H ( p ) riiodels are also closely connected with autoregressive models roith

mridmm coeficie,nts used in tlie description of 'random walks in a random rriedium'. To give an idea, we restrict ourselves again to p = 1. Tlicri it follows from the relations hrl = CT,E,& and m: = a0 C Y ~ / I , : , _ ~ that

+

ll,, = &O

2 + altlTL-1

En.

(16)

We now corisitler the following first-order autoregressive model wit,h rando'm coefficients: IC71 = B17/71x,L-l + Bo6'Tl, (17) where (v,~) and (&) are two independent standard Gaussian sequences. From thc standpoint of finite-dirnensional distributions the sequence z = (IC,,) where we (for definitcncss) set ICO = 0, has the same structure as the sequence 2 = (R:,,) with

5%

=

d-Fn,

ICO = 0,

(18)

and E = (E,!,) a standard Gaussian sequence. Comparing (16) antl (18) we see that h = (hr,) antl rC = ( Z r I )are constructed following the same pattern. Herice if Bi = n o arid L?; = a1.then the probabilistic laws governing tlie sequences ti = ( h r L )arid IC = (?En) with ho = 20 = 0 are the same. 6. Wc now rnakt: the A R C H ( 1) model slightly more complicated by assuming that the relatioils lwtwcen the variables lr,, n 3 1 arc as follows:

h r , = ljo

2 + i i j l ~ l + y'ao + alh,,,_l En.

(19)

In t,liis casc, it is said that ti = (h,,) is governed by the A R ( l ) / A R C H [ l )model or that 11, = (1,,$,) sat,isfies the autoregressive scheme AR(1) with ARCH(1) noise

(J%+ filIL;-1

Chapter 11. Stochastic Models. Discrete Time

158

This model is conditionally Gaussian, therefore we can represent the density p ~ ( h 1 ,. .. , h,) of the joint distribution Po of the variables h l , . . . , h, for a fixed value of the parameter 19= (ag,a l , Po, P i ) as follows (ho = 0):

As an exarnple of the application of this representation we consider the problem of estzmatzon (using the rnaxirnum likelihood method) of the unknown value of 01 under the assumption that all other parameters, PO, a0, and "1, are known. The maximum likelihood estimator of 01 is the root of the equation

&

In view of (20) and (19), we obtain

and

where

is a martineale and

is its quadratic characteristic (cf. (46) in $ 2b). Here ( M ) ,

+ 00

(P-as.), therefore

MTl ~

(M)n

+0

(P-as.) by the strong law of

large nunibers for square integrable niartingales (see (12) in 5 l b and "439; Chap,-. ter VII, $51). Hence the estimators P1 so obtained are strongly consistent in the following sense: P Q ( @+ ~ P ) = 1 for Q = ( ' Y O , "1) [jo, P ) with P E R.

,.

3. Nonlinear Stochastic Conditionally Gaussian Models

159

7. We now consider the issue of the predictzon of the price movement in the future under the assumption that the sequence h = (h,) is governed by the model ARCH(p). Since the sequence h = (h,) is a niartingale difference, it follows that E(h,,+nl I %,,) = 0. Hence the optimal (in the mean-square sense) estimator

(here 9; = n ( h 1 , .. . , h,)), therefore it is clear that this question on the prediction of t,hc future values of hi+,, reduces to the problem of the prediction of the 'volatility' c;,+~~!,on the basis of the past observations ho, h l , . . . , 11,. Sirice 2 cn+m =

2

2

g o + ~l~n+m-lEn+,,-l>

it follows by induction that 2

gn+m = 0 0

+W

2

+

2

2

[ ~ ~Ql~n+m-2&n+m-21E,+,-l 0

Hence, considering the conditional expectation E( .\92) and taking into account the independence of the variables in the sequence ( E ~ ~ we ) , obtain

j= 1

As we might expect, the estimators

one) to the 'stationary' value Eh:

E

-

converge as m

a0

-. 1 - a1

+ LY) (with probability

160

Chapter 11. Stochastic Models. Discrete Time

+

+

8. We recall that the intxrvals ( p - 0 , p a ) and ( p 1.650, p 1.650) are (with a certain degree of precision) confidence intervals for the normal distribution N ( p , c 2 ) with confidence levels 68 % and 90 %, respectively. Since Sn+,n = Sn,hn,l+-+hn+??? (26) ~

and

it follows that we can as a first approzinzation take the intervals

as confitierice int,crvals (with confidence levels 68 % and 90 %, respectively). Hcrc: it, m i s t be clear that writing about ‘first approximation’ we mean that the variables I L are ~ not normally distributed in general, and the question of the difference between the actual confidence levels of the above confidence intervals and 68 % (or 90 %) calls for a n additional irivest,igation of the precision of the normal approxirriation. 9. Tlie success of the conditionally Gaussian A R C H ( p ) model, which provided explanations for a variety of phenomena that could be distinguished in the behavior of the financial indexes (the ‘cluster property’, the ‘heavy tails’, the peaks (leptokurt o s i s ) of the distribution densities of thc variables h,,, . . . ) inspired a n avalanche of its generalizations trying to ‘capture‘, explain several other phenoniena discovered by nieans of statistical analysis. Historically, one of the first generalizations of the A R C H ( p ) model was (as already rrientiorietl in fj Id) the generalized ARCH model of T. Bollerslev [48](1986). Characterized by t,wo parameters p arid q , it is called the GARCH(p,q ) model. In this niodel, just as in A R C H ( p ) , we set h, = on&.,,.As regards the ’volatility’ Q ~ , , liowevc:r, we assume that P

4

i=S

j=S

whcre 00 > 0, m z 3 0, and [jj 3 0. the A R C H ( p ) model.)

(If all the

PJ

vanish. then we obtain

3. Nonlinear Stochastic Conditionally Gaussian Models

161

The central advantage of the GARCH(p,q) models, as compared with their forefather, the ARCH(p) model, is the following (experimental) fact: in the adjustments of the GARCH(p,q ) model to the statistical data one can restrict oneself to small values of p and q , while in the framework of the ARCH(p) model we must often consider uncomfortably large values of p . (In [431] the author had to use autoregressive models of order twelve, AR( 12), to describe the monthly averages of S&P500 Index. See also the review paper [141].) One can carry out the analysis of the GARCH(p,y) models with ‘volatility’ orl that is assumed to depend (in a predictable manner) both on the i 6 p , and the g:Tj. j < \ I , in a similar way to our analysis of the ARCH(p) models. Onlittirig the details we present several simple formulas relating to the GARCH(1,l)model, in which It,,,L = ~

T

with 00 > 0. ( ~ 13 0, ailti It is clear that

/j1

arid ~

~

2

E= a0 ~ cT1

2 + aih,-l2 + Pig,-l

(28)

3 0.

2 + (a1+ P1)Ehn_1 and the ‘stationary’ value E11,2, is well defined for a1 + P1 < 1; namely,

Eli: = a0

If 3cv;

+ 2qp1 +

< 1, then we have

a well-defined ‘stationary’ value

and therefore, for the ‘stationary kurtosis’ we obtain

It is also easy to find the ‘stationary’ values of the autocorrelation function p ( k ) (cf. (13)):

+P p p ( l ) ,

p ( k ) = (a1

k > 1.

(33)

Finally, we point out that we can generalize (25) to the case of the GARCH(1,l)models as follows:

- 4+r,l -=

fL:l+m

where y = a1

+ Dl.

=

2

E(an+m

I&)

162

Chapter 11. Stochastic Models. Discrete Time

10. Models in the ARCH family, which evolve in discrete time, have counterparts in the continuous time case. Moreover, after a suitable normalization, we obtain a (weak) convergence of the solutions of the stochastic difference equations characterizing ARCH, GARCH, and other models to the solutions of the corresponding differential equations. For definiteness, we now consider the following modification of the GARCH(1,1) model (which is called the G A R C H ( l , l ) - M model in [15]).

Let A be the time step, and let H(*) = (Hi;'), k

Hi")

with

+ /L,(~) + . . . + h$) EkA

-

=

0 , 1 , . . . , where Hi;) =

and

.N(O,A), a constant c, and

We set the initial condition Hi") = Ho arid ai"' = a0 for all A > 0, where (Ho,ao) is a pair of randoin variables independent of the Gaussian sequences ( E ~ A ) ,A > 0 . of iridependent random variables. We now erribed the sequence (IdA), = (Hi;), in a scheme with continiious time t 3 0 by setting

02))~~~

<

+

kA t < ( k 1)A. In view of the general results of the theory of weak convergence of random processes (see, e.g., [250] and [304]) it seems natural to expect t h a t , under certain conditions on the coefficients in (34) and (35), the sequence of processes (fdA), ,(*)) weakly converges (as A + 0, in the Skorokhod space D ) to some diffusion process for

( H , a ) = (Ht,Qt)t>,O. As shown in [364], for

the liiriit,ing process ( H ,n ) satisfies the following stochastic differential equations (see Chapter 111, 3 3e):

dHt = CO,Z d t do,2 = (CYg

-

+ nt d W ,(1), pa,", d t + an," dW,( 2 ),

(37) (38)

wlicre ( W ( l ) ,W ( 2 ) are ) two independent standard Brownian niotions that are also indcpcndent of the initial values ( H o ,00) = ( H (, A ), a.( A ) ) ,

3. Nonlinear Stochastic Conditionally Gaussian Modcls

163

g3b. EGARCH, TGARCH, HARCH, and Other Models 1. In 1976. F. Black noticed t,he following phenomenon in the behavior of financial indexes: the variables hTl-l and on are negatively correlated; namely. the enipiric covariance Cov(hl,-l, or,) is negative. This phenomenon, which is called the leverage eflect (or also t,he a . s y n i m e t q effect), is responsible for the trend of growth in volatility after a drop in prices (i.e., when the logarithniic returns become negative). This cannot be understood in the framework of ARCH or GARCH models, where the volatility a:, which depends on the s p a r e s of the h:L-i, i 3 1, is indifferent to the signs of the l i n - j , so that in the GARCH models the values 11,-j = a and h7,-j = -a result in the same value of future volatility o,”,. To explain Black’s discovery D. B. Nelson [366]put forward (1990) the so-called EGARCH(p,y) ( E x p o n e n t i a l GARCH(p,4 ) ) model, in which the ‘asymmetry’ was taken into account by means of the replacement of hipi = o,”,-~E,”,-~ in the GARCH models with linear combinations of the variables ~ ~ and ~ I -~ ~i - i l Namely, . we assume again that 1 1 , = ( T , , , E , ~ , but the o,,, must now satisfy the following relations:

(We note that = E(&,,,-iJ.) are the same. Sincc l i r L - i = orl,-icrL-iand o,,-~ 3 0, the signs of hT1-i and : is equal to b ( B + y ) , while Hence if ~ ~ =~b >- 0, ithen t,he corresponding term in o if E,,-,L = -b < 0, then it is equal to b(-8 7).

+

2. The EGARCH models are not unique in capturing the asymmetry effect while retaining the main properties of the GARCH family. Another example is the TGARCH(p,q ) model (‘T’as in ‘threshold’), which was suggested by the threshold models of TAR (Threshold AR) type. In this model, k

11,

=

C ~A,(hn-d)(ab +

CX;}L,~-~+

...

+ abhn-p),

(2)

i=l

where d is a lag parameter and A l . . . . , A k are disjoint subsets of

R such that

i=l

For instance, we can set 0;

llrL = 0;

+ a;h,-l+ aiti,,-2 + ufh,,-l + a;1irL-2

if h,-2

> 0,

if’ hn-2

< 0.

(Such threshold models were thoroughly investigated in the nionograph [461].)

(3)

Chapter 11. Stochastic Models. Discrete Time

164

By definition (see [399]),a sequence h = (h,) is described by the TGARCH(p.q ) model if /in = a 7 a ~where Tl, P

4

and, as usual, z+ = riiax(z, 0) and rc- = - rnin(s, 0). We do not assume in this model that the coefficients (and, therefore, the volatilities a,) are positive, although a: retains its rrieariiiig of the conditional variance E(hA 1 Sirice

9kP1).

h,, = a,,&,, = (a:

-

a,)(.,'

+ a,.]

= [a,+€,+

- E,)

-

[a,&

+ a;&,],

it follows that

+

and

h,: = [a$&$ a,&,]

hi =

[OLE,'+ a:&;].

These relations enable one to rewrite (4) as follows: D*

Q*

i= 1

i=l

wherc p* = rnax(p, q ) arid the functions a i ( ~ ~ -and i ) P i ( ~ ~ - i are ) , linear combinations of ~ , tarid -~ The study of such models runs into certain technical difficulties rooted in the lack of the Markov property. Nevertheless, in simple cases (say, in the case of y = q = 1) one can analyze the properties of these models fairly completely. Indeed, let p = q = 1. Then 07,

= uo

+ [ W k+T L - l+ blh,-,]

+ ["1&

+ dla,l],

(6)

or, equivalently,

where

If

(YO

a0 = no.

= 0, then = (al(E,-l))+&

0 : -

an =

(a1(E7,--I))-&

+ (Pl(&n-l))+a;-p + (fll(&n-l))-fl;-l

(9)

t'he . sequence by (7). Herice it is clear that, with respect to the flow (9,) is Markov, which enables one to study it by usual 'Markovian' methods. (Sce [399] for greater detail.)

3. Nonlitiertr Stochastic Conditionally Gaussian Models

165

3. We now consider another phenornenon, the ‘long memory’ or ‘strong aftereffect‘ in tlic evolution of priccs S = (STL)n>O. Tliere exists several ways to describe the dependence on the ‘past’of the variables in a random sequence. In probability theory one has various measures of this dependeiice: ergodicity coeficients, mixing coeficients, and so on. For instaiice, we can measure the rate at which the dependence on the past in a stationary scqiieiice of real-valued variables X = ( X T , )fades away by the rate of the convergence to zero (as ni + .c) of the supremum

taken over all Bore1 sets A C R. Of coiirsr, the (aiito)c:orrelation furict,ion is the standard measure of this tlependence. It should be riotcd that, as shown by many statistical studies, financial time series exhibit a stronger corrr:lntio,n dependence between the variables in the seqiierices IIrl = (Ilt,llj)rL21 and h2 = (f~?~),>l than the one attainable in the franiework of ARCH or GARCH (not t o mention MA, A R , or ARMA) models. We recall that, by formula (13) in 5 3a,

in the ARCH(1) model, while the autocorrelation function p ( k ) for the GARCH(1.1) model is described by expressions (32) and ( 3 3 ) in the same 5 3a. According to thcse formulas, the correlation in these models approaches zero at geometric rat,e (‘the past is quickly forgotten’). One often says that a stationary (in the wide sense) sequence Y = (Y,) is a sequence with ‘long memor:y’ or ‘strong aftereffect’ if its autocorrelation function p ( k ) approaclies zero at hyperbolic rate, i.e.,

for soriic p > 0. This rate of dccrease is characteristic, e.g. of the autocorrelation function of fractal Gaussian iioise (see Chapter 111, 5 2d) Y = (Y,),>l with elements ~

where X = ( X t ) + > ois a fractal Brownian motion with parameter W. 0 < IHI < 1 (see Chaptcr 111, 6 2c). For this motion we have (sec (3) in Chapter 111. 3 2c) 1

Cov(X,,Xt) = - { /tj2K+ /sl2”- It - S(~’}EX: 2

Chapter 11. Stochastic Models. Discrete Time

166

and (scc ( 3 ) in Chapter 111, 5 2d)

2

+ 1\22?

CoV(Y71rY,,+k) = - { l k 2

-

21k(2W

+ Ik - 112”).

where m2 = DY,,. Hence the ant,ocorrelation fuiictiori p ( k ) = Corr(Y,,. Y,+k) creases hyperbolically as k + w : p(k)

-

de-

w(2w - q k 2 W - 2 .

We not? that

;

< MI < 1. For W = (a usual Brownian motion) the variables Y = (Y,) form Gaussian ‘whitr. nois(\’ with p ( k ) = 0, k: >, 1. On the o t h r hand, if 0 < W < then

for

4,

M

m

hF1

Ic= 1

Re,mark. In Chapter 7 of t,he monograph [a021 one can find a discussion of various rriodels of processes with ‘strong aftereffect’ and much information about the applications of these models in economics, biology, hydrology, and so on. See also [418]. 4. Another i n o d d HARCH(p), was introduced and analyzed in [360] and “1. This is a inotiel from the ARCH family in which the autocorrelation functions for the absolute values and the squares of the variables h, decrease slower than in the case of models of ARCH(p) or GARCH(p,g) kinds. The same ‘long memory’ phcriornrriori is charactjerktic of the FIGARCH models introduced in [15]. By definition, thc HARCH(p) (Heteroge,neous AutoRegressicie Conditional Heteroskedastic) model (of order p ) is defined by the relation

hn = gnEri1 where

with a0 > 0, a p > 0, and aj >, 0, j = I , . . . , p In particular, for p = 1 we have 2

0,

Le., HARCH(1) = ARCH(1).

= a0

-

1.

2 + alhn-l,

3. Nonlinear Stochastic Conditionally Gaussian Models

For p = 2,

2

CTL=

a0

2 + CYlh,_l+

a2(hn-1

+ h,-2)

2

.

167

(12) We now consider several properties of this model. First. we point out that the presence of the term (h,-1 h,-2)2 enables the model to ’capture’ the above-mentioned asymmetry effects. Further, if a0 a1 a 2 < 1, then it follows from (12) that there exists a ‘stationary‘ value

+

+

+

In a similar way, considering EgA and using the equalities Ehn-1hn-2 = Eh,_,h,-2 3 we obtain by (12) that for (a1

=

Eh,-lhZ-,

= 0,

+ 0 2 ) ~+ a; < i. the ‘stationary value’ c

4

Eh, =

3

-

(a1

+ a 2 ) 2 - a;

is well tlefinrd, where

C=

CY;[~

+ 2a2(a1 4-3a2) [1 - (a1

-

+ 2~l’2)~]

(a1

+ 2a2)12

(We note that EaA = iEhA.) We now find the autocorrelation function for ( h i ) . Let R ( k ) = Eh:Lhi-k. Then in the ‘stationary’ case, for k = 1 we have

Consequently, if a 2

< 1, then

Chapter 11. Stochastic Models. Discxete Time

168

Hcnce the autocorrelation function p ( k ) = Corr(hE, h:l-k) clearly satisfies in the stationary case the equation p(k)=A

+ B p ( k - 1) + C p ( k

-

2),

k

2,

(17)

where

A = - -a()EhF1 , Dh?L

B-

(01

+a2)(EeJ2 Dhi

,

c = (a2

l)(Eh;)’ Dhz

-

1

arid

Wc coritiniic our analysis of the ‘strong aftereffect’ in Chapter IV, 33e, while discussing exchange rates.

3c. Stochastic Volatility Models 1. A characteristic feature of these models, introduced in 0 Id, is the existence of kcuo sources of randomness, E = (E,) and 6 = (&), governing the behavior of the ) that sequence 11 = ( 1 1 , ~ ~so

h, where oTl=

(1)

= g71En,

and the sequence (A,?,)is in the class A R ( p ) ,i.e.,

We shall assiiriie that E = (E,) and 6 = (a,) are independent standard Gaussian sequences. Then we shall say that h = (h,) is governed by the SV(p) (Stochastic Volatility) niodcl. We now consider the proptLties of this model in the case of p = 1 and (all < 1. We have 2 In a: = a0 a1 Inan-, ch,,. h,, = aT1cn, (3)

+

Let

s ~= J .(El,.

. . , ET1; 61,. . . , b,&)and let 3;= a ( ~ 1 ., .. , 6),. E(hn 1 3 : ) = CT,~EE, = 0

and

because EcPI= 0.

+

Clearly,

3. Nonlinear Stochastic Conditionally Gaussian Models

169

Hcnce the sequence I L = (hT1)is a martingale tliff’erence with respect to the flow (S:,”’,). (Although riot with respect to (9;) because the 11, are not 9;measurable. ) Further,

E ~ I , : ~= ED: EE:, = ED: = E e A n . We shall assuiiic that

By ( 3 ) , tlir sequence A = (Arc)fits into the autoregressive scheme A R ( 1 ) (i.e., An = 00 ~ i & ~ - - 1 &) and is stationary (see s2b). By ( 3 ) ,

+

where, we (valuate

+

EeArL using the fact that

for each o and a random variable [ In a similar way,

N

,N(O,1)

We now consider the covariance properties of the sequences h

= (h,)

and

1L2 = (11:).

We have EIL1lhn+l =

0

and, more generally, Eh,h,+k for each k 3 1. Herice h = Rh ( k : ) = Eh,, l i T l + k , then

( } i n ) is

=0

a sequence of uncorrelated random variables: if

Chapter 11. Stochastic Models. Discrete Time

170

Further,

Hence

a1

As rriight be expected, the variables h i and a1 < 0. Besides the above formulas

are positively correlated for

> 0 and negatively correlated for

we present also more general ones. Namely, for positive constants T and s we have ra

p

2 .

Ear - ez(l-'d+x.,_h:

,

n-

rs

c2

Ec~,T,~,S,-~ = Eak E g g e These forniulas can be used in the calculations of various moments of the h,. For example,

3 . Nonlinear Stochastic Conditionally Gaussian Models

171

In particular, these relations yield the following expression for the ’stationary’

which shows that tlic ‘stochastic volatility’ models with two sources of randoniness = ( E ~ and ~ ) S = (6,) enable one (in a similar way to the ARCH family) to generate sequences h = (h,) such that with the distribution densities for the h, have peaks around the niean value Eh,, = 0 (the leptokurtosis phenomenon).

E

2. We now dwell on the issue of constructing volatility estimators Gn on the basis of the observations i l l ! .. . , h,. If h, = p 0 7 L ~then 7Lr Eh, = p and

+

It is natural to make this relation the starting point in our construction of the estirriators ii,,of the volatilities on by adopting the formula

with -

h, =

1 -

71

if

/I

C 71

h,k

k l

is unknown and tlie forrriula

if p is known. Another estimation method for the or”, is based on the fact t h a t Eh:, = En:, i.e., on the properties of second-order moments. Clearly, we could choose = 1 ~ ;as ~ an estimator for o:. This is a rionbiased estimator, Init its niean squared error A

02

can be fairly large. Of coiirsc, if the variables cz,k n , are correlated, then we can use not only h i , but also the preceding observations h i p l ,h2-2, . . . to construct estimators for 0;.

<

Chaptcr 11. Stochastic Moclcls. Discrete Time

172

pi, <

It is clear that if the k n , are 7oeakly correlated, then the past variables . . . must enter our foririulas with small, decreasing weights. On thc other hand, if thc k n , are strongly correlated, then the variables h:,pl, . .. ('an be a so~irceof important additional information about o,", (compared with the inforiiiatioii coiitaiiie(1 in h,:!,). This idea brings one to the consideration of exponentially weighted estimators

hi-2.

oi, <

,. = i(1 - A)

co

Ak'hZ-,,

0

< x < 1,

(7)

k=O

in which, as we see, -w, rather than time 0, is the starting point. Making this assumption we obtain the following formula for the stationary solution of the autoregressive recursion relation (2):

where the series convcrgw in mean square. As shown in $2h, this is the unique statioiiarv solution. !x

We note that (1-A)

C Xk

= 1, i.e., the sum of the weighted coefficients involved

k=O

x

in the construction of oi is equal to one. = Eo:, it follows that E a i = En:, Since E1,,:LL-k = x

which means that the

h

1

estiniator o? is unbiased (as is

0;). x

We also point out that the quality of the estimator 02 considerably depends on t,hc c:lioseri value of the parameter A, so t h a t there arises the (fairly coinplicateti) probleni of the 'optimal' choice of A. By (7) we see that, the c r i satisfy the recursion relations x

which is convenient in the coiistruction of estimators by means of statistical analysis and motfel-building.

3. In our corisideratioris of the model h,, = e+ wherc Art = a0

E,,,

(A7L= lnrr:),

+ alAn-l + cfiTL,it might seem reasonable to use

3 . Nonlinear Stochastic Coriditionally Gaussian Models

173

as an estimator for A?,. This estimator would be optimal in the mean square sense. Unfortunatcly, this is a nonlinear scheme, which makes the problem of finding an explicit formula for m,, almost hopeless. The first idea coming naturally t o mind is to ‘linearize’ this problem and, after that, to use the theory of ‘Gaussian linear filtering’ of R. Kalman and R. Bucy (see, e.g., [303]). For example, we can proceed as follows. By (10) we obtain

where ElnE;& = -1.27 and D l n ~ : = 7r2/2 arrive at the following linear system:

= 4.93.

Hence setting x,

where E In E : ~ )

with E&& = 0 and DCn = 1, arid we can regard the above quantity -1.27 as an approximation to E In E,”, . Thus, we can assume that we have a linear system (11)-(12), in which the distributions of the EY, are nonGaussian. This rules out a direct use of the KalmanBucy linear filtering. Nevertheless, we shall consider the Kalman--Bucy filter as if the En, n 3 1, were normally distributed variables, t7, M ( 0 ,l ) , independent of the sequence (6,). Let p, = E(A, I xi,.. . , x,) and let yn = DA,, under this assumption. Then the evolution of the p r Larid Y,! is described by the following system (see, e.g., [303; Chapter VI. 5 7. Theorern 11):

-

Here po = EAo and 70 = DA,. We note that if the parameter c in (11) is large. then A, is the dominating term in the formula for 2 , . Hence one can expect the p, t o be ‘good’ approximations of the rn,.

174

Chapter 11. Stochastic Models. Discrete Time

4. We point, out that if the parameters 0 = (ao, a l , c) of the initial model (11) are unknown, then one often uses the Bayes method to find an estimator Z, for an on the basis of ( h l , .. . , hT1). Its cornerstone is the assumption that we are

given a prior distribution Law(B) of B. Then, in principle, we can find, first, the posterior distribution Law(@,on I h l , . . . , hT,) and then the posterior distributions Law(@I h 1 , . . . , h,) arid Law(c, 1 h 1 , . . . , h,). This would enable us to construct the estimators H,, and ZT1,e.g., as the posterior means or the values delivering maxima of the posterior densities. For a more comprehensive treatment of the Bayesian approach, s t x , for instance, [252] and the corriments to this paper on pp. 395-417 of the same issue (v. 12, no. 4, 1994) of Journal of Business and E c o a o m i c Statistics. 5 . We have described the models in the GARCH family and ‘stochastic volatility’ models in the framework of the conditional approach. The conditional distribution Law(h, I a,) has always been a normal distribution, A ( 0 , (T:,), with (T; dependent on the ‘past’ in a ‘predictable’way. Taking this approach it is natural to ask about the unconditional distributions Law(h,) and Law(h1,. . . , hTz). To give a notion of the results possible here we consider now the following model (see [105]). ~ , , (E,) is again a standard Gaussian sequence, Let, ti,, = O ~ ~ E where 2

2 = a0,-1

+ b&,

-m

< n < ca,

(16)

and (&) is the stquence of nonnegatrve independent stable random variables with exponent ( 2 , 0 < a < 1 (cf. Chapter 111, 3 lc.4). We assume that the sequences (E,) and ( f i l l ) arp indeprndent. If 0 6 a < 1, then we obtain recursively by (16) that

By t,hc sclf-similarity properties of stable distributions, 00

00

k=O

k=O

Coriscqueiitly, if 0 < (Y < 1 and 0 6 a ‘stationary’ solution ( o i ) , wliere

< 1, then (16) has

a (finite) nonnegative

HCIIWwe see from the defiriithn (h, = C T ~ ~ E , )that the stationary one-dimensional distribution Law(h,,) is stable, with stability exponent 2a.

3. Nonlinear Stochastic Conditionally Gaussian Models

175

6. To complete this section, devoted to nonlinear stochastic models and their properties, we dwell on the above-mentioned phenomenon of ‘heavy tails’ observable in these models. (See also Chapter IV, 3 2c.) Wc corisider the ARCH(1)-model of variables h = (h,),>o with initial condition Irg iritlepcndent on the standard Gaussian sequence E = ( E ~ ) ~ > I h, , =

d=c7%

for n 3 1, 010

> 0, and 0 < a1 < 1.

For an appropriate choice of the distribution of ho this model turns out t o have a solution h = ( I L , ~ ) ~ ~ >that O is a strictly stationary process with phenomenon of ‘heavy tails’ (observable for sufficiently small a1 > 0): P(h, > x) c z 9 , where c > 0 arid y > 0.

-

The corresponding (fairly tricky) proof can be found in the recent monograph of

P. Embrechts, C. Klueppelberg, T. Mikosch “Modelling extremal events for insurance and finance”, Berlin, Springer-Verlag, 1997 (Theorems 8.4.9 and 8.4.12). One can also find there a thorough analysis of many models of ARCH, GARCH and related kinds and a large list of literature devoted to these models.

4. Supplement: Dynamical Chaos Models

4a. Nonlinear Chaotic Models 1. So far. in our descriptions of the evolution of the sequences h = (h,) with

h,, = In ~-

s,,, where S, is the level of some ‘price’ a t time n, we were based on the

SIL-1

con,jecture that these variables were stochastic, i.e., the S, = S,,(w) and h,, = hTL(u) were ,runrlom variables defined on some filtered probability space (0.9: (91a)n>lr P) and simulating the statistical uncertainty of .real-life’ situations. On the other hand, it is well known that even very simple nonlinear deterministic systems of the type 2,+1

= f ( % ; A)

(1)

or z,,+l = f(Z,,Z,-l....,Z~,L-~;

(2)

where X is a parameter, can produce (for appropriate initial conditions) sequences with behavior vary similar to that of stochastic sequences. This justifies the following quest,ion: is it likely that many economic, including financial, series are actually realizations of chaotic (rather than stochastic) systems. i.e., systems described by det,erministic nonlinear systems? It is known that such systems can bring about phenomena (e.g., the ‘cluster propert,y’) observable in the statistical analysis of financial d a t a (see Chapter IV). Referring to a rather extensive special literature for the formal definitions (see, e g . , [59], [71], [104], [198], [378],[379], [383],[385],[386]:[428], or [456]), we now present several examples of nonlinear chaotic systems in order t o provide the reader with a notion of their behavior. We shall also consider the natural question as to how one can guess the kind of the system (stochastic or chaotic) that has generated a particular realization.

4. Applications: Dynaniical Chaos Models

177

Discussing the forecasts of the future price movements, the predictability problem is also of considerable interest in nonlinear chaotic models. As we shall see below, the situation here does not inspire one with much optimism, because, all the determiiiisni notwithstailding, the behavior of the trajectories of chaotic systems can considerably vary after a sinall change in the initial data and the value of A. 2. EXAMPLE 1. We consider the so-called logistic map z

4

Tz

= Xz(1-

z)

and the corresponding (one-dimensional) dynamical system

(Apparently, logistic equations ( 3 ) occurred first in the models of population dynamics that imposed constraints on the growth of a population.)

0.2

0.1

FIGURE 23a. Case X = 1

For X 6 1 the solutions zn = z,(X) converge monotonically to 0 as n + 00 for all 0 < 20 < 1 (Fig. 23a). Thus, the state 2, = 0 is the unique stable state in this case, arid it, is the limit point of the z, as ri + 30. For X = 2 we have z,, t (Fig. 23b). Hence there also exists in this case a unique stable state (zm = attract,ing the 2, as n 3 m. We now consider larger values of A. For X < 3 the system ( 3 ) still has a unique stable state. However, an entirely new phenomenon occurs for X = 3: as 71 grows, one can distinguish two states IC, (Fig. 23c), and the system alternates between these states. This pattern is retained as X increases, until something new happens for X = 3.4494.. . : the system has now four distinguished states ,z and leaps from one to another (Fig. 23d).

i i)

Chapter 11. Stochastic Models. Discrete Time

178

0.3

4 0

10

20

30

40

50

60

70

80

90

100

70

80

90

100

FIGURE 23b. Case X = 2

0.3

+ 0

10

20

30

40

50

60

FIGURE23c. Case X = 3

New distinguished states come into being with further increases in A: there are 8 such states for X = 3.5360.. . , 16 for X = 3.5644.. . , and so on. For X = 3.6, there exists infinitely many such states, which is usually interpreted as a loss of stability and a transition into a chaotic state. Now the periodic character of the movements between different states is completely lost; the system wanders over an infinite set of states jumping from one to another. It should be pointed out that, although our system is deterministic, it is impossible in practice to predict its position at some later time because the limited precision in o u r knowledge of the values of t h e x, and X can considerably influence the results. It is clear from this brief description already that the values (A,) of X at which the system ‘branches’, ‘bifurcates’ draw closer together in the process (Fig. 24). As conjectured by M. Feigenbaum and proved by 0. Lanford [294], for all par-

179

4. Applications: Dynamical Chaos Models

0.9

0.8 0.7

0.6 0.5

0.4 0.3 0

1.0

t

U

10

20

30

40 50 60 70 80 FIGURE 23d. Case X = 3.5

90

100

IU

ZU

JU

YU

IUU

I

4U

5U

bU

(U

EU

FIGURE 23e. Case X = 4

abolic systems we have

where F = 4.669201 . . . is a universal constant (the Fergenbaum constant). Tlic valiir, X = 4 is of particular importance for ( 3 ) : it is for this value of the

Chapter 11. Stochastic Models. Discrete Time

180

,

I

-+

/,

parameter that the sequence of observations ( z n )of our (chaotic) system is similar to a realization of a stochastic sequence of ‘white noise’ type. . ,.~ 1 0 0 0 Irideed, let zo = 0.1. We now calculate recursively the values of ~ 1 ~ x 2. , using (3). The (empirical) mean value and the standard deviation evaluated on the basis of these 1000 nurribers are 0.48887 and 0.35742, respectively ( u p to the fifth digit). TABLE 2.

I

I

I

I

I

I

In Table 2 we present the values of the (empirical) correlation function p^(k) calculated from ~ 0 ~ x. 1 . . ,, ~ 1 0 0 0 It . is clearly visible from this table that the values 2, of the logistic map with X = 4 can in practice be assumed to be uncorrelated.

4. Applications: Dynarnical Chaos Models

181

In this srnsc, thc scqucnce (5,) can be called ‘chaotic wh,ite noise’. It is worth noting that the system 2 , = 4x,,-1(l-zTL-1), n 2 1, with 20 E ( 0 , l ) has an invariant distribution P (which nieans that P satisfies the equality P ( T - l A ) = P(A) for each Bore1 subset A of ( 0 :1)) with density 1

dz)= .ir[z(l 4 ] 1 / 2

5

E (0,l).

(4)

~

Thus, assiirriirig that the initial value rco is a r a n d o m variable with density p = p ( z ) of the probability distribution, we can see that the random variables z, n 3 I , have the same distribution as zo. We point out that all the ‘randonmess’ of the resulting stochastic dynamic system ( x ? ~is) re1at)ed t o the ,random initial value 2 0 , while the dynamics of the transitions 5, -+ x,+1 is detevministic and described by ( 3 ) . = If (4) holds, then it is easy to see that Ex0 = $, Ex; = and Dzg = ( 0 . 3 5 3 5 5 . .. )2. (Cf. the valucs 0.48887 and 0.35742 presented above.) As regards the correlation function EZOX~ EXOEZ~

i>

A k )=

d D G

we have

EXAMPLE 2 ( the Bernoulli transformation). We set zTL= 2xTZ-1 (mod I).

20

E (0, I )

Here thc uniform distribution with density p ( z ) = 1, z E (0. 1). is invariant, and 1 and p ( k ) = 2 T k 7 k = 1 . 2 , . . . . we have E r 0 = Ex: = $, DzO= Iz. EXAMPLE 3 ( t h c ‘tent’ m a p ) . We set

i,

Here, as in Example 2, thc uniform distribution on (0, 1) is invariant. Ezo = Ex; = Dzo = arid p ( k ) = 0 for k # 0.

i,

i,

&,

Hence

EXAMPLE4. Let 2,

zo E (-1,l).

= 1- 2 J M r

Then the distribution with density p ( z ) = (1 - z)/2 on (-1, 1) is invariant. For this distribution we have Ex0 = -;, Ex: = and Dzg = 9. 2

i,

Thc behavior of the sequences ( x , ) , < ~ for zo = 0.2 and N = 100 or N = 1000 is depicted in Fig. 25a,b.

Chapter 11. Stochastic Models. Discrete Time

182

It

0

10

20

30

40

50

60

70

80

90

100

) o zn = 1 - 2 FIGURE25a. Graph of the sequence x = ( z ~ ) ~ ~with xo = 0.2 for N = 100

0

100

200

300

400

500

600

700

800

J

m and

900 1000

FIGURE25b. Graph of the sequence x = (xn)nao with zn = 1 - 2 xo = 0.2 for N = 1000

d m and

3. The above examples of nonlinear dynaniical systems are of interest from various viewpoints. First, considering the example of, say, the logistic system, which develops in accordance with a ‘binary’ pattern, one can get a clear idea of fractality discussed in Chapter 111, 5 2. Second, the behavior of such ‘chaotic’ systems suggests one to use them in the construction of models simulating the evolution of financial indexes, in particular, in tames of crashes, which are featured by ‘chaotic’ (rather than ‘stochastic’) behavior.

4. Applications: Dynamical Chaos Models

3 4b.

183

Distinguishing between ‘Chaotic’ and ‘Stochastic’ Sequences

1. The fact that purely deterministic dynamical systems can have properties of ‘stochastic white noise’ is not unexpected. It has been fairly long known, although this still turns out to be surprising for many. The more interesting, for that reason, are the questions on how one can draw a line between ‘stochastic’ and ‘chaotic’ sequences, whether it is possible in principle, and whether the true nature of ‘irregularities’ in the financial data is ‘stochastic’ or ‘chaotic’. (Presumably, an appropriate approach here could be based on the concept of ‘complexity in the sense of A . N. Kolmogorov, P. Martin-Lof, and V. A. Uspenskii’ adapted for particular realizations.) Below we discuss the approach taken in [305] and [448]. In it, the central role in distinguishing between ‘chaotic’ and ‘stochastic’ is assigned to the function

where, given a sequence (z,), i , j 6 N , such that

the function $ ( N , E ) is the number of pairs ( i , j ) , lZi - Z j l

< E.

Besides C ( E ) we , shall also consider the functions

where $,,(N,E) is the number of ( i , j ) , i , j 6 N , such that all the differences between the corresponding components of the vectors (xi,zi+l, . . . , ~ i + ~ - and l ) ( z j , z j + l , . . , ~ j + ~ ~ -are l ) a t most E . (For m = 1 we have $ ~ ( N , E =)$ ( N , E ) . ) For stochastic sequences (z,) of ‘white noise’ type we have

for small E , where the fractal exponent vm is equal to m. Many deterministic systems also have property (2) (e.g., the logistic system ( 3 ) in the preceding section [305]). The exponent u , is also called the correlation dimension and is closely connected with the Hausdorff dimension and Kolmogorov’s information dimension. Distinguishing between ‘chaotic’ and ‘stochastic’ sequences in [305] and [448] is based on the following observation: these sequences have different correlation dimensions. As seen from what follows, this dimension is larger for ‘stochastic’ sequences than for ‘chaotic’ ones.

Chapter 11. Stochastic Models. Discrete Time

184

By [ 4 8 ] arid [305]. the quantities

and

where ~j = ‘ p J arid 0 < p < 1, can be taken as estimators of the correlation dimension ‘on,. In Table 3a one can find the values of the Vm,j corresponding to ‘p = 0.9: TI, = 1 , 2 . 3 , 4 , 5 , 1 0 , anti several values of j in the case of the logistic sequence ( I ; ~ ~ )where ~ ~ ~ NN = , 5 900 and ~j = pJ (these data are borrowed from [305]) TABLE 3a. Values of V,,,j for the logistic system

We now compare the data in this table with the estimates for Gm,j obtained by a simulation of Gniissiun white noise with parameters characteristic of the logistic map ( 3 ) in 5 Cia (Table 3b: the data from [305]): TABLE 3h. Values of the V,,,j for Gaussian white noise

Comparing. these tables we see that it is fairly difficult to distinguish between ‘chaotic’ and ‘stochastic’cases on the basis of the correlation dimension V l , j (corresporitliiig to ‘mi, = 1). However, if m is larger, then a considerable difference between

4. Applications: Dynamical Chaos Models

185

the values of the V,,,,J for these two cases is apparent. This is a rather solid argument in favor of the conjecture on distinct natures of the corresponding sequences (x,), although there is virtually no difference between their empirical mean values, variances. or correlations. 2. To illustrate the problems of distinguishing between ‘stochastic’ and ‘chaotic’ cases in the case of financzal series we present the tables of the correlation di-

mensions calculated for the daily values of the ’returns’ h, = ln

sn

~

Sn-1

:n21,

corresponding to IBM stock price and S&P500 Index (Tables 4a, and 4b are compi!ed from 5903 observations carried out between June 2, 1962 and December 31, 1985; the data are borrowed from [305]): TABLE 4a. Values of Gm,3 for IBM stock

Am

1

2

TABLE 4b. Values of

N m

1

2

3

Cm,j

3

4

5 1 0

for the S&P500 Index 4

5

10

Comparing these tables we see, first, that the estimates of the ‘correlation dimension’ of the IBM and the S&P500 returns in this two tables are very close. Second, comparing the data in Tables 4a, b and Tables 3a, b we see that the sequences (h7,)corresponding to these two indexes (where hn = In-

s,

s,-1

,n21)

are closer to ‘stochastic white noise’. Of course, this cannot disprove the conjecture that other ‘chaotic sequences’, of larger correlation dimensions, can also have similar properties. (For greater detail on the issue of distinguishing arid for an economist’s commentary, see [305].)

Chapter 11. Stochastic Models. Discrete Time

186

3. We now consider briefly another approach to discovering distinctions between ‘chaotic’ and ‘stochastic’ cases, which was suggested in [17]. Let z = (zn)be a ‘chaotic’ sequence that is a realization of some dynamical system in which zo is a random variable with probability distribution F = F ( z ) invariant with respect to this system. Next, let E = ( Z n ) be a ‘stochastic’ sequence of independent identically distributed variables with (one-dimensional) distribution F = F ( z ) . We consider now the variables

Adn = m a x ( z g , z l , . . . ,z,)

and

-

M,

-

<

= max(EO,El,. . . ,En).

<

Let F n ( z ) = P(Mrl z) and let &(z) = P(Mn x). The approach in [I71 is based on the observation that the maximum is a characteristic well suited for capturing the difference between ‘stochastic’ and ‘chaotic’ sequences. To substantiate this approach, the authors of [17] proceed as follows. In the theory of limit theorems for extremum values there arewell-known necessary arid sufficient conditions ensuring that the variables an(Mn - b,), where a , > 0 and b,, are some constants and n 3 1, have a (nontrivial) limit distribution

-

lirnP(an(M,

-

b,)

71

< x) = G(z)

Referring to [187], [206], [156], [124], and [I371 for details, we now present several examples. If F ( z ) = 1 - z-p, z 3 1, and p > 0, then

If F ( z ) = 1 - (-x)P, -1

< z < 0, and p > 0, then (for x < 0)

If F ( z ) = 1 - e-”, x 3 0, then

-

P(Mn - logn

< z) + exp(-e-”)

for z E R. If F ( z ) = @(z)is a standard normal distribution, then

<

~ ( ( 2 1 o g n ) ~ / -~ b,) ( G ~ z) where we choose the b, such that P(zo

+ (-e-Z),

> bT1) = i. (In this

z

E R:

case b,

-

(210gn)’/~.)

4. Applications: Dynarnical Chaos Models

187

For the Bernoulli transformation (Example 2 in S4a) we have the invariant distribution F ( z ) = 5 , z E ( 0 , l ) . Setting a , = n and b, = 1 - nP1we obtain the liniit distribution

5

-

G(z) = exp(z - I ) ,

z < 1.

In Example 4 from § 4 a we have F ( z ) = 1 - p2(x), where p ( ~ = ) (1 - z)/2, E (-1, l ) ,therefore setting a , = fi and b, = 1 - 2 / f i we see that

For Example 1 in 54a the invariant distribution is F ( s ) =

2 7l

arcsin&

(Example 1 in §4a), and after the corresponding renormalization we obtain

F,(z)

Given the distributions = ( F ( z ) ) , and the limit distribution G(z), it would be reasonable to compare them with the corresponding distributions &(z) and, if possible, with their limits, say, G(z). However, as pointed out in [17], there exists a serious technical problem, because there are no analytic expressions for the F,,(z) in the examples in 3 4a that are convenient for further analysis. For that reason, the approach taken in [17] consists in the numerical analysis of the distributions F,(z) for large n and their comparison with the corresponding distributions For the dynamical systems in 3 4a this analysis shows that, globally, the behavior of the F,(z) (for chaotic systems with invariant distribution F ( z ) ) has a character distinct from the behavior of the (for stochastic systems of independent identically distributed variables with one-dimensional distribution F(rc)). This indicates that, for the models under consideration, the mazimum value is a 'good' statistics for our problem of distinguishing between chaotic and stochastic cases. But of course, this does not rule out the possibility that there exists a chaotic system xTL+l= f(z,,q - 1 , . . . , q - k ; A) with k sufficiently large that is difficult to distinguish from stochastic white noise on the basis of a large (but finite) number of observations.

F,(z).

F,(z)

Chapter 111 . Stochastic Models . Continuous T i m e 1. N o n - G a u s s i a n m o d e l s o f distributions and processes .

3 l a . Stable and infinitely divisible distributions . . . . . . . . . . . . . . . . . 3 l b . LCvy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 l c . Stable processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 I d . Hyperbolic distributions and processes . . . . . . . . . . . . . . . . . . . . 2 . klodels with s e l f - s i m i l a r i t y . Fractality fj 2a . 5 2b . 3 2c . 3 2d .

Hurst’s statistical phenomenon of self-similarity . . . . . . . . . . . . . A digression on fractal geometry . . . . . . . . . . . . . . . . . . . . . . . . Statistical self-similarity. Fractal Brownian motion . . . . . . . . . . . Fractional Gaussian noise: a process with strong aftereffect . . . . .

3 . Models based on a Brownian m o t i o n . . . . . . . . . . . .

3 3a . Brownian motion and its role of a basic process . . . . . . . . . . . . . 3 3b . Brownian motion: a compendium of classical results . . . . . . . . . . 93c . Stochastic integration with respect to a Brownian motion . . . . . . 3 3d . It6 processes arid It6’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3e. Stochastic differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 3 3f . Forward and backward Kolmogorov‘s equations. Probabilistic representation of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 . D i f f u s i o n models o f t h e e v o l u t i o n of interest rates, stock and bond p r i c e s .............

189 189 200 207 214 221 221 224 226 232 236 236 240 251 257 264 271 278

3 4a . Stochastic interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4b . Standard diffusion model of stock prices (geometric Brownian

278

motion) and its generalizations . . . . . . . . . . . . . . . . . . . . . . . . . .

284 289

5 4c . Diffusion models of the term structure of prices in a family of bonds 5 . Semimartingale m o d e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5a. Seniirriartingales arid stochastic integrals . . . . . . . . . . . . . . : . . . . 3 5b. Doob-Meyer decomposition. Compensators . Quadratic variation . 3 5c. It6’s formula for semirnartingales. Generalizations . . . . . . . . . . . .

294 294 301 307

1. Non-Gaussian Models of Distributions and Processes

1 la.

Stable and Infinitely Divisible Distributions

1. In the next chapter we discuss the results of the statistical analysis of prospective models of distribution and evolution for such financial indexes as currency exchange rates, share prices, and so on. This analysis will demonstrate the importance of stable distributions and stable processes as natural and fairly likely candidates in the construction of probabilistic models of this kind. For that reason, we provide the reader in this section with necessary information on both these distributions and processes and on more general, infininely divisible ones: without the latter our description arid the discussion of the properties of financial indexes will lack of completeness. In our exposition of basic concepts and properties of stability and inifinite diuisibility we shall (to a certain extent) follow the chronological order. First, we shall discuss one-dimensional stable distributions considered in the 1920s by P. LCvy, G. P d y a , A . Ya. Khintchine, and then proceed to one and several-dimensional infinitely divisible distributions studied by B. de Finetti, A. N. Kolmogorov, P. Levy, and A. Ya. Kliintchine in the 1930s. After that, in 3 l b we introduce the reader to the main concepts and the properties of Levy processes and stable processes. The monographs [156], [188], [418], and [484] belong among widely used textbooks on stable and infinitely divisible distributions arid processes. 2. DEFINITION 1. We say that a nondegenerate random variable X is stable or has a stable distribution if for any positive numbers a, and b there exists a positive c and a nuniber d such that

Law(aX1f bX2) = Law(cX

+ d)

(1)

for independent random copies X1 and X2 of X (this means that Law(Xi) = Law(X), i = 1 , 2 ; we shall assume without loss of generality that all the random variables under consideration are defined on the same probability space (n,9, P)).

190

Chapter HI. Stochastic Models. Continuous Time

It can be proved (see the above-mentioned monographs) that in (1) we necessarily have cff = aff bff (2) for some a E (0,2] independent of a and b. One often uses another, equivalent, definition.

+

DEFINITION 2. A random variable X is said t o be stable if for each n 3 2 there exist a positive number C, and a number D, such that Law(XI+

X2 + . . . + XTL)= Law(C,X

+ D,),

(3)

where XI, X2, . . . , X,, are independent copies of X . If D,, = 0 in (3) for 71 3 2, i.e., L a w ( X I + X2

+ . . . + X T L=) Law(C,X),

(4)

then X is said to be a strictly stable variable. Remarkably,

c,= n u f f

<

in ( 3 ) and (4) for somc a , 0 < a 2, which is. of course, the same parameter as in (2). To emphasize the role and importance of a ; one often uses the term 'a-stability' in place of 'stability'. For completeness, it is reasonable to add a third definition to the above two. It brings forward the role of stable distributions as the only ones that can be the limit distributions for (suitably normalized and centered) sums of independem! identically distributed random variables. DEFINITION 3. We say that a random variable X has a stable distribution (or simply, is stable) if the distribution of X has a donlain of attraction in the following sense: there exist sequences of independent identically distributed random variables (Y,), positive numbers (d,), and real numbers (a,) such that

here

d "+" means convergence

in distribution, i.e.,

+

Law( Y l + . . . Y,, + a n > + Law(X) dn in the sense of ~ ~ e convergence ak of the corresponding measures. This definition is equivalent to the above two because a r a n d o m variable X Y1 + . " + Y , can be the limit in dzstribution of t h e uaraables + a , f o r s o m e sed, quence (Yr,)of independent identically distributed r a n d o m variables if and only i f X is stable (in the sense of Definition 1 or Definition 2; see the proof in [188] or [439; Chapter 111, 5 51).

1. Non-Gaussian Models of Distributions and Processes

191

3. According to a remarkable result of probability theory (P. LCvy, A. Ya. Khintchine) the characteristic function p(0) = EeieX of a stable random variable X has the following representation:

<

<

where 0 < a 2 , 101 1, 0 > 0, and p E R. Here the four parameters ( a ,p, CT, p ) have the following meaning: (which is, of course, the same as in (2) and (4)) is the stability exponent or the characteristic parameter; ,fl is the skewness parameter of the distribution density; 0 is the scale parameter; p is the location parameter. The parameter Q 'controls' the decrease of the 'tails' of distributions. If 0 < a < 2. then Q

lirri x"P(X

> x) = Cff1 +Pa", ~

2

2100

lim P P ( X

< -x)

p,

2

X+oO

where

= (3"- 1 - P

(lmx-ffsinXdx) { 1-a

c,=

-1

=

r(2-Q)cosY 2 > n-

If a = 2 , than by (6) we obtain

which shows that p(0) is the characteristic function of the normal distribution N ( p , 2 a 2 ) ,i.e., of a normally distributed random variable X such that

EX = p

and

DX = 202.

The value of /3 in (6) is not well defined in this case (because if a = 2, then this parameter enters the term p t a n n , which is equal to zero). One usually sets /3 = 0.

192

Chapter 111. Stochastic Models. Continuous T i m e

Clearly, as regards the behavior of the tails of distributions, the case a = significantly differs from a < 2. For instance. if p = 0 and 2g2 = 1, then

Comparing this with (7) and (8) we see that for a < 2 the tails are ‘heavier’: t h decrease in the nornial case is faster. (This is a right place to point out that, a seen in statistical analysis, for many series of financial indexes S = (S,),>o t h STI distributions of the ‘returns’ h, = In have ’heavy tails’. It is natural, for thi ~

Sn-

1

reason, to try the class of stable distributions as a source of probabilistic model governing the sequences h = (h,) .) It irnportarit to note that, as seen from (7) and (8), the expectation ElXl i finite if and only if a > 1. In general, E / X / P < cc if and only if p < a. In connection with asymptotic forniulas ( 7 ) and (8) the Pareto distrihution i worth recalling. Its distribution density is

with Q

> 0 and h > 0, therefore its distribution function Fa,b(x)satisfies the relatio

Comparing with (7) and (8) we see that the behavior a t infinity of stable distri butions is similar to that of the Pareto distribution. One can say that the tails c stable distributions fall within the Pareto type. The skewness parameter /3 E [-1,1] in (6) characterizes the asymmetry of distribution. If p = 0, then the distribution is symmetric. If P > 0, then t h distribution is skewed on the left, and the closer p approaches one, the greater thi skewness. The case of p < 0 corresponds to a right skewness. The paranieter 0 plays tlie role of a scalp coefficient. For a normal distributioi ( a = 2) we have DX = 2cr2 (note that the variance here is 2 2 , not 0’ as in t h standard notation). On the other hand, if a < 2, then DX is not defined. We call p tlie locatron parameter, because for a > 1 we have p = EX (so tha ElXl < m). There is no such interpretation in the general case for the mere reasoi that EX does not necessarily exist. 4. Following an established tradition, we shall dmote the stable distribution wit1 parameters a , p, ~7,and p by

scu

P,

(0,

1. Non-Gaussian Models of Distributions and Processes

193

-

we shall write X S , ( a > y , p ) to indicate t h a t X has a stable distribution with parameters a , ,L3, n, and p . Note that Sa(a,/3, p ) is a s y m m e t r i c distribution if and only if ,O = p = 0. (It is obvious from the formula fur the characteristic function that the constant D, in (3) is equal to zero in that case.) This distribution is s y m m e t r i c relative to p (which can bc a r b i t r a r y ) if and only if p = 0. In the syinnietric case (,! = Ip = 0) one often uses the notation

x

N

SnS.

In this case the characteristic fiinction is

5 . Unfortunately, explicit formulas for the densities of stable distributions are known only for some values of the parameters. Among these distributions are t h e normal distribution, Sz(n,0 , p ) = A ( p , 2a2) with density

1

-

(X-LL)*

402

2afi

;

th,e C a u c h y d i s t r i b u t i o n Sl(0,0 , p ) with density U

7r(

.(

~

p)2

+ n2) '

t h e one-sided stable distribiition (also called the Lilvy or Smirriov distribution) S1/2(a,1,p ) with exponent Q = l / 2 on ( p , cm)with density

-

We point out two interesting and useful particiilar cases of (16) and (17): if X S1(a, 0, O), t h e n

P(X

< x) = 21 +7 1r arctan X0 -

-

-

f o r x > 0; ij

x

-

S1/2((T, I,O), t h e n

for x > 0. As regards the representation of the densities of stable distributions by series, see [156], [225], [418], and [484].

194

Chapter 111. Stochastic Models. Continuous Time

6. We now assume that X I ,X 2 , . . . , X , are independent random variables such that i = 1 , 2 , .. .,n. Sa(0i,Pi,p i ) ,

xi

N

Although, in general, they are n o t identically distributed, the fact that they all have the same stability exponent a indicates (see formula (6) for the characteristic function) that their sum X = X i . . . X , has a distribution of the same type, S,(a, p, p ) , with parameters

+ +

ff=

(o?

+ +o ' ' '

p

+ . + pna: oy + ' . + og p 1 + + pn. lI=

P=

PlCP'

' '

:

'

'

' ' '

7. We now proceed to the more general class of so-called 'infinitely divisible' distribut,ions, which includes stable distributions. DEFINITION 4. We say that a random variable X and its probability distribution are infinitely divisible if for each n 3 1 there exist independent identically distributed d random variables X,1,. . . , X,, such that X = X,1 + . . . + X,,. The practical importance of this class is based on the following property: these and only these distributions can be the limits of the distributions of the sums

5 X,k)

of variables in the scheme of series

( k l

... every row of which consists of independent identically distributed random variables X n l , Xn2,. . . , Xnn. Note that no connection between the variables in distinct rows of ( 2 0 ) is assumed. (For greater detail, see [188] or [439; Chapter 111, $51.) A more restricted class of stable distributions corresponds to the case when the variables X,I, in (20) can be constructed by means of a special procedure from the variables in a fixed sequence of independent identically distributed variables Y1,Y 2 , . . . (see the end of subsection 2):

1. Non-Gaussian Models of Distributions and Processes

195

(We point, out that the class of infinitely divisible distributions includes also the hyperbolic and the Gaussian\\inverse Gaussian distributions discussed below, in 3 Id.) Definition 4 relates to the scalar case ( X E R). It can be immediately extended to the vector-valued case ( X E Rd) with no significant modifications. Let P = P ( d x ) be the probability distribution of an infinitely divisible random vector x E R~ and let

be its characteristic function; let ( 0 , ~ be ) the scalar product of t,wo vectors 0 = (el,.. . , O d ) and z = ( X I , .. . ,Q). The work of B. de Finetti, followed by A. N. Kolmogorov (for EIXI2 < m) and finally P. L6vy and A. Ya. Khintchine in the 1930s resulted in the following LCvy-Khrntchine formula for the characteristic function of X E Rd:

where B E Rd, C = C(d x d ) is a symmetric nonnegative definite matrix, and v = v(dz) is a Le‘vy measure: a positive measure in Rd such that ~ ( ( 0 ) )= 0 and

(Note that both cases .(EXd) < 00 and v ( R d )= 00 are possible here.) It should be pointed out that p(B) is specified by the three characteristics B , C , and v , and the triplet (B,C , v ) in (22) is unambzgously defined.

EXAMPLES. 1. If X is a degenerate random variable ( P ( X = u ) = 1). then B = a. C = 0, v = 0, and p ( ~=) ezoa.

2. If X is a random variable having the Poisson distribution with parameter A, v ( ~ T= ) A l i l ) ( d x ) is a measure concentrated at the point z = 1, B = A, and

then

3 . If X

-

J ( m ,c 2 ) ,then B = m, C = c 2 ,v = 0, and

196

Chapter 111. Stochastic Models. Continuous Time

4. If X is a random variable having the Cauchy distribution with density (16), then $$Q) = e i P Q - 4 ,

5. If X is a random variable with density (17) (a one-sided stable distribution with exponent a = l / 2 ) , then

8 . A representation of the characteristic function p(0) by forrnula (22) (using the traditional, ‘canonical’ truncation function h ( z ) = z I ( lzl 1)) is not unique. For instance, in place of I(1zI 1): we could use a representation with I ( l x / 6 a ) , where u > 0. Of course, the corresponding triplet of characteristics would also be differcnt. Notably, C arid v do not change in that case; these are ‘intrinsic‘ characteristics independent of our choice of the truncation function. Only the first characteristic, B , actually changes. For precise statements, we shall need the following definition.

<

<

DEFINITION 5. We call a bounded function h = h ( z ) . z E Rd,with compact support satisfying the equality h ( z )= z in a neighborhood of the origin a truncation function. Besides (22), the characteristic function p(0) has the following representations (valid for an arbitrary truncation function h = h ( z ) ) :

where C and v are independent of h and are the same as in ( 2 2 ) , while the value of B ( h ) varies with h in accordance with the following equality:

B ( h ) - B(h’) =

L

( h ( l )- h’(z))v(dx).

(25)

We point out that the integrals on the right-hand sides of (22) and (24) are well defined in view of ( 2 3 ) , because the function

is bouri(ieti and it is 0(1xl2)as 1x1 + 0. If we replace (23) by the stronger inequality

1. Non-Gaussian Models of Distributions and Processes

197

then we can set h ( z )= 0 in (24), so that

The constaiit B ( 0 ) in this representation is called the drift component of the random variable X. On the othcr hand, if we replace (23) by the stronger inequality

then (24) liolds for h ( x ) = z, i.e.,

i( 0,

E)

-

1 2

-(0, C Q )f

(ez(Q1z) - 1 - i ( Qz ,))~ ( d z )

-

(the center) is in fact the expectation B = EX In this case the parameter We note that the condition EIX/ < 00 is equivalent to the inequality

1 . 1 v(dz) < 30. 9. We have already mentioned that there are only three cases when stable distributions are explicitly known. These are (see subsection 5):

normal distribution ( a = 2): Cauchy distrihutron ( a = I ) , Le'vy-Smirnov distribution ( a = 1/2) The class of infinitely divisible distribution is much broader; it covers: in addition, the following distributions (although this is not always easy t o prove): Poisson distribution, gamma-distribution, geometric distribution, negative binomial distribution, t-distribution (Student distribution), F-distribution (Fisher distribution), log-normal distribution logistic distribution, Pareto distribution, two-sided exponential (Laplace) distribution, hyperbolic distribution, Gau,~sian\\i,nverse Gaussian distribution . . .

198

Chapter 111. Stochastic Models. Continuous Time

TABLE 1

(discrete distributions)

F=

Probabilities p,+

Distribution

e-Xxk

Pozsson

k!

, k = O , l , ...

Geometric

Parameters

I

""

I Negative binomial c i p k q n - IC,

Bznominl

k = 0,1, . . . , n

However, many well-known distributions are n o t infinitely divisible. These include: banomial and uniform. distributions, each nondegenerate distribution with finite support, distrib7~tionswith densities j ( z ) = Ce-l"la, where CY > 2. Some of these distributions are discrete, while other have distribution densities. For a complete picture and the convenience of references we list these distributions explicitly in Tables 1 and 2. 10. The notion of 'stable' random variable can be naturally extended to the vectorvalued case (cf. Definitions 1 and 2 ) .

DEFINITION 6. We call a random vector X = (Xl, x2,

' ' '

, X,)

a stable r a n d o m vector in Rdor a vector with stable d-dimensional distribution if for each pair of positive numbers A, B there exists a positive number C and a vector D E Rd such that Law(AX(')

+ S X ( 2 ) )= Law(CX + D ) ,

(30)

where X(l) and X(') are independent copies of X.

It can be shown (see, e.g., [418;p. 581) that a nondegenerate raudonl vector X = (XI,X 2 , . . . , X,) is stable if and only if for each n 3 2 there exist CY E (0,2] and a vector D,, such that Law ( ~ ( 1+) x(')

+ . . . + x("))= ~

+

a (wn l l a x D ~ ) ,

where X ( l ) ,X ( ' ) , . . . , X(") are independent copies of X

31)

1. Non-Gaussian Models of Distributions and Processes

199

TABLE 2 (distributions with densities) Distribution

UnZfOTm 0 7 1 [a, b]

Density p = p(x)

Parameters

1 a<x
a,bER, a < b

~

Normal (Gaussian)

P E R , u>O

Gamma

a>O, p > 0

(I-- dzstrzbution) Exponential (gamma distrzbution with cu = 1, p = 1/X)

X>O

t -distribution (Student distribution)

n = 1 , 2 , .. .

Beta

T > O ,

(0-distribution) Two-sided exponential (Laplace distribution) Chi-squared x2 drstributzon; garnrna distribution with cu = n / 2 . D = 2)

X>O

n = 1,2,..

Cauchy Pareto

S > o

P E E , a>O

cub"

x 3 b

a

> 0,b > 0

Log-normal

P E R , u>O

Logzstzc

QER, P>O

Hype rho lie

Gausszan\\ Gavssian

Z7kUeTSe

see (2) in§ Id

see (14) in

5 Id

Chapter 111. Stochastic Models. Continuous Time

200

If D,,= 0, i.e.,

+ x(2)+ + X'"')

Law (x(1)

' ' '

= Law ( n l l " X ) ,

(32)

then we call X a strzctly stable random vector with exponent a or a strzctly a-stable random vector. d

Rernark. Along with using the notation Law(X) = Law(Y), one often writes X = Y to indicate that the distributions of X and Y are the same; one says that X and Y d coincide 271 distrzbutzon. The notation X" + X or Law(Xn) + Law(X) indicates convergence in distribution (as already pointed out in subsection a ) , i.e., weak convergence of the corresponding measures. (For more detail, see [439; Chapter 1111.) If X = (Xt)t2o and Y = ( % ) t > O are two stochastic processes, then we write d

{ X t , t 3 O} = { % , t 3 0)

or Law(Xt, t 3 0) = Law(%, t 3 0) to denote that all the finzte-damensaonal distributions of X and Y are the same, arid we shall say that X and Y coincide zn dzstrzbutzon.

3 lb.

L6vy Processes

1. The LCvy processes, which we discuss below, are certain stochastic processes with iridependcnt increments. They form one of the most important classes of stochastic processes, which includes such basic objects of probability theory as Brownian motions and Poisson processes. DEFINITION 1. We call a stochastic process X = (Xt)t>o with state space Rd defined on the probability space (CL, 9, P) a (d-dimensional) Le'vy process if 1) Xo = 0 ( P - a s . ) ; 2) for each rh 3 1 and each collection l o , t l , . . . , 0 t o < t l < . ' . < t,, the variables Xt,, Xt, - Xt,, . . . , Xtn - Xt,-, are independent (the property of independent increments); 3) for all s 3 0 and t 3 0,

<

d

Xt+s - x, = xt

-

XO

(the 'homogeneity' property of increments); 4) for all t 3 0 and E > 0, Iim P(jX,

s i t

-

Xtl > E )

(the property of stochastic continuity);

=O

1. Non-Gaussian Models of Distributions and Processes

201

5) for P-almost all w E R the trajectories (X,(w))t>o belong to the space Dd of (vector-valued) functions f = ( f t ) t > o , f t = f ; , . . . , f,") with components f z = ( f t ) t > o , i = 1, . . . , d , that are right-continuous and with limits from the left for all t > 0.

(fi,

R e m a r k 1. If we ask that the process X = (Xt)t>O in this definition have only properties 1)-4), then it can be shown that there exists a modification X ' = ( X ; ) Q ~ of X = ( X t ) t > o (i.e., P(X,' # Xt) = 0 for t 3 0) with property 5). Thus, the process X' is the same as X as regards the properties 1)-4), but its trajectories are 'regular' in a certain sense. For that reason, one incorporates property 5) of the trajectories in the definition of Le'vy processes from the very beginning (without loss of generality). Remar,k 2. Duly interpreting coritlitioris 1)-5) we can reforrnulate the definition of a Lkvy process as follows: this is a stochastically continuous process with homogeneous independent incre,me.nts that starts f r o m the origin and has right-continuous trajectories viith limits from t h e left. A classical example of such a process is a &dimensional Brownian m o t i o n X = ( X ' , X 2 , . . . , X'), the components of which are independent standard Brownian motions Xz = ( X ; ) t 2 0 ,i = 1,.. . , d. It is instructdiveto define a (one-dimensional) Brownian motion separately, outside the general framework of Le'vy processes. DEFINITION 2 . We call a continuous Gaussian random process X = ( X t ) t a o a (standar.d) B,r.onm*icmm o t i o n or a W i e n e r process if X o = 0 and

EXt = 0, EX,Xt = niin(s, t ) . By the Gaussian property and (1) we immediately obtain that this is a process with homogeneous independent (Gaussian) increments. Since

xt - x,

N

.N(O,t - s),

t 3 s,

it follows that ElXt - X,13 = 3jt - sI2, and by the well-known Kolrnogorov test ([470]; see also (7) in 5 2c below) there exists a continuous modification of this process. Herice the Wiener process (the Brownian motion) is a L@vyprocess with an additional important, feature of continuous trajectories. 2. L6vy processes X = ( Xt)t>O have homogeneous independent increments: therefore their tlistritnitions are completely determined by the one-dimensional distributions Pt(d2:) = P(Xt E r l z ) . (Recall that X O = 0.) By the mere definition of these processes, the dist7ibution Pt ( d z ) is infinitely divisible for each t.

Chapter 111. Stochastic Models. Continuous Time

202

Let

be the characteristic function. Then, by formula (22) in same 5 l a ) ,

5la

(see also (24) in the

where Bt E Rd, Ct is a symmetric nonnegative definite matrix of order d x d , and vt(dz) is the LRvy measure (for each t ) with property (23) in 0 la. The increments of LRvy processes are homogeneous and independent, therefore

(The function 1c, = $(Q) is called the cumulant or the c u m u l a n t f u n c t i o n . ) Since the triplets ( B t ,Ct, vt) are unambigously defined by the characteristic function, it follows by ( 5 ) (see, e.g., [250; Chapter 11, 4.191 for greater detail) that

where B = B1,C = C1, and v = v1. Hence it is clear that in ( 5 ) we have

3. The representation (5) with cumulant $(Q) as in (7) is the main tool in the study of the analytic properties of L6vy processes. As regards the properties of their trajectories, on the other hand, the so-called canonical representation (see S 3a, Chapter VI and [250; Chapter 11, S2c] for greater detail) is of importance. It generalizes the canonical representations of Chapter 11, 5 l b for stochastic sequences H = ( H n ) + ~(see (16) in 3 l b and also Chapter IV, 5 3e) to the continuous-time case. 4. We now discuss the meaning of components in the triplet (Bt, Ct, vt)t>o. Figuratively, (Bt)t>ois the trend component responsible for the development of the process X = ( X t ) t > ~o n t h e average. The component (Ct)tbo defines the variance of the continuous Gaussian component of X , while the Le‘vy measwets (vt)tbo are responsible for the behavior of the ‘ju,mp’ component of X by exhibiting the

frequency and magnitudes of ‘jumps’.

1 , Non-Gaussian Models of Distributions and Processes

203

Of course, this (fairly liberal) interpretation must be substantiated by precise results. Here is one such result (see [250; Chapter 11, 2.211 as regards the general case). Let X = ( X t ) t > o be a process with ‘jumps’ lAXt1 1, t 3 0, let X O = 0 , and let (Bt,Ct, vt)t20 be the corresporldirig triplet. Then EX? < m, t 3 0, and the following processes are martzngales (see Chapter 11, 5 lc):

<

a) Mt b) M;

= Xt -

- Bt - X o , t 3 0 ; Ct, t 3 0;

the interval (0, t ] and g = ~ ( zare ) continuous functions vanishing in a neighborhood of the origin. As already pointed out, a standard Brownian motion is a classical exaniple of a continuous Ldvy process (with Bt = 0, Ct = t , and vt = 0). We now consider examples of discontinuous Ldvy processes, which, at the same time, will give us a better insight into the concept of Le‘vy measure v = ~ ( d x ) . 5. The case of a finite Levy measure (v(R)< m ) . A classical example here is, of course, a Poisson process X = (Xt)t>owith X > 0, i.e. (by definition), a Ldvy process with X0 = 0 such that X t has the Poisson distribution with parameter At:

P(Xt = k ) =

e-

( A t )IC

k!

k = 0, 1, .

In this case we have Bt = A t (= E X t ) , Ct = 0 and the Le‘vy measure is concentrated at a single point: v(drc) = A q l ) ( d z ) . The representation (3) takes now the following form:

= exp{ At(e20

-

l)}.

(8)

It is worth noting that, starting form the Poisson process one can arrive at a wide class of purely jnm.p Lkvy processes.

Chapter 111. Stochastic Models. Continuous Time

204

Namely, let N = (Nt)t>O be a Poisson process with parameter X

> 0 and let

< = ([j)j21 be a sequence of independent identically distributed random variables (that must also be independent of N ) with distribution

) 0. where X = u ( R ) < 00 and ~ ( ( 0 ) = We consider now the process X = (Xt)t>o such that X O = 0 and Nt

xt = XCj,

t > 0.

(9)

j=1

Alternatively, this can be written as follows:

j=1

where 0 < 71 < 7-2 < . . . are the tinies of jumps of the process N A direct. calciilation shows that

The process X = ( X t ) t >defined ~ by (10) is called a compound Poisson process. It is easy to see that it is a L6vy process. The ‘standard’ Poisson process corresponds to the case [ j = 1, j 3 1. 6. The case of infinite Lkvy measure (v(R) = m). To construct the simplest example of a L6vy process with .(EX) = 00 we can proceed as follows. Let X = ( X k ) k > l be a sequence of positive numbers and let p = (/3k)kal be a sequence of riorizero real niirnbers such that

k=l

We set

03

1. Non-Gaussian Models of Distributions and Processes

205

and let N ( k ) = ( N t(I,))t>o, k 3 1, be a sequence of independent Poisson processes with parameters Setting

Xk,

k 3 1, respectively.

it is easy to see that for each with L@vymeasure

72

2 1 the process X ( " ) = (X,("))t>ois a L6vy process

c 71

u(7L) (dx)=

XI, I { p k }( d z )

(15)

k=l

and

{ J'

pj'"'(B) = Eez e x ~ ' ' ) = exp t

The limit process

X

=

(eioz - 1 - iHz) ~

}

( (dx) ~ 1.

(16)

(Xt)t>o,

k=l

( X t is the L2-liinit of the X!") as n measure' defined by (13).

+

cm) is also a Lkvy process with 'L6vy

Rema,rk 3 . 111 this case we have property 5) from Definition 1 because the X ( n ) are square integrable rnartingales, for which, by the Doob inequality (see formula (36) in fi 31) and also [25O; Chapter I, 1.431 or [439; Chapter VII, fi 3 ] ) ,we have E inax s
1x.i")

-

+o

~ , / 2

as

n

+ oo.

cc,

Remark 4. Since v(R) =

C

Xk,

~ ( ( 0 ) )= 0 and

k=l

c 00

L ( 2A 1)u ( d z ) ,<

< 00,

k=l

the measure u = u ( d z )in (13) satisfies all the conditions imposed on a Lkvy measure (see formulas (22)-(23) in fi la). 00

If

X,I = 00,but (12) holds, then we have an example of a L6vy process with k=l

L6vy irieasiire v satisfying the relation u(R)= cm. Wc now present another well-known example of a Lkvy process with v(R) = 00. We mean the so-called gamma process X = (Xt)t>o with Xo = 0 and probability distribution P(Xt x) with density

<

(cf. Table 2 in

5 la)

Chapter 111. Stochastic Models. Continuous Time

206

Now, pt(Q) = (1- iep)-t (19) We claim that the characteristic function pt(Q) can be represented as follows:

which yields the equality vt(dz) = t v(dx), where ,-ZIP 4 d X )

= I(O,m)(X)

2 dX.

Clearly, ~ ( 0m) , = 00, however

We now proceed to the proof of the representation (20). We consider the Laplace trarisforni

= exp{ -t I n ( l +

p u ) } = exp

Using analytical continuation in the coniplex half-plane { z = a obtain

+ ib,

a 6 O} we

Setting here

2 = iQ, we arrive a t the required representation (20). 7. Now that we have obtained 'explicit' representations (lo), (14), and (17) for

several (jurrip) LCvy processes, we have at our disposal a method of their sirnulation. We need orily siniulate the random variables [ j and ,Ok and the exponentially distributed variables Ai = ri - ri-1 (the time intervals between two jumps of the Poisson process at instants ri-1 arid ri). For a simulation of infinitely divisible random variables in their turn, the question of their representations as functions of 'elementary', 'standard' random variables becomes now of interest. Look at an example demonstrating opportunities emerging here. Let X and Y be two indeperitlent random variables such that X 3 0 (and X is arbitrary in the rest), while Y has an exponential distribution. Then. as shown by Cli. Goldie, their product X Y is an infinitely divisible random variable. In the next section we shall show how one can 'combine' processes of simple structure and obtain stable processes as result.

1. Non-Gaussian Models of Distributions and Processes

5 lc.

207

Stable Processes

1. We start with the case of a real-valued process X = (Xt)tETwith an (arbitrary) parameter set T , which is defined on some probability space (R, 9, P). Generalizing the definition of a stable random vector (Definition 6 in 3 l a ) we arrive in a natural way to the following concept.

DEFINITION1. We say that a real random process X = (Xt)tET is stable if for each k 3 1 and all t l , . . . , t k in T the random vectors ( x t , , . . . , Xt,) are stable, i.e., all finite-dimensional distributions of X are stable.

If all these finite-dimensional distributions are stable, then it is easy to see from their compatibility that they have the same stability exponent a . This explains the name a-stable that one often gives to such processes when one wants to point out some particular value of a . In what follows, we shall be mainly interested in a-stable processes X = (Xt)tET that are incidentally Lkvy processes (3 l b ) . A suitable name for them is a-stable Le'uy processes. In la.10 we presented two equivalent definitions of stable nondegenerate random vectors (see formulas (30) and (31) in § l a ) . In the case of stable (onedimensional) processes X = (Xt)t>O that are also Lkvy processes, this equivalence brings us to the following result: a nondegenerate (see Definition 2 below) onedimensional Le'vy process X = (Xt)t>O is a-stable ( a € (0,2]) if and only zf f o r each a > 0 there exists a n u m b e r D (depending on a in general) such that {Xat, t 3

o} 2 {al/"Xt + D t } .

(1)

We now present several definitions relating to multidimensional processes

x = (Xt)t>O. 2 . We say that a random process (Xt)taO taking values in Rd is deDEFINITION generate if Xt = y t (P-a.s.) for some y E Rd and all t 3 0. Otherwise we say that the process X is nondegenerate. DEFINITION 3 . We call a nondegenerate random process X = (Xt)t>o taking values in Rd an a-stable LCvy process ( a E (0,2]) if 1) X is a L6vy process and 2) for each a > 0 there exists D E Rd (dependent on a in general) such that d

{Xat, t 3 0} = {al/"Xt or, equivalently,

+ Dt, t > 0 }

Chapter 111. Stochastic Models. Continuous Time

208

DEFINITION 4. We call a n a-stable L6vy process X L k u y process if D = 0 in (2) and ( 3 ) ,i.e.,

{Xatrt 3 o }

= (Xt)t20 a

d { a l l a x t , t 3 o}

strictly a-stable

(4)

or, eqiiivalently, Law(X,t, t

2 0)

=

~ a w ( a l / " ~ tt ,2 0).

(5)

Remark 1 . Sometimes (see, e.g., [423]), a stable vector-valued LCvy random process X = ( X t ) t y is ~ defined as follows: for each a > 0 there exist a number c and D E RTL sucl1 that

{ X a t , t 3 O}

d

= {cXt

+ D t , t 2 O}

(6)

or, equivalently, Law(X,t, t

2 0)

= Law(cXt

+Dt, t

2 0).

(7)

(If D = 0. thcn one talks about strict stability.) It is remarkable t h a t , as in the case of stable variables and vectors, we have c = for nondegenerate Lkvy processes, where n is a universal (i.e., independent of a ) parameter belonging to (0,2]. This result, the proof of which can be found, e.g., in [423], explains the explicit involveincnt of the coefficient a'/" in Definitions 3 and 4.

Remark 2. It is useful to note that the condition

Law(X,t, t

2 0)

= Law(cXt,

t 2 0)

is precisely t,hat of self-slrnilarity. Thus, a-stable Lkvy processes, for which c = a ' / a , are self-siniilar. The quantit,y H = 1/a here is called the Hurst parameter or the Hurst ezpvnen,t. See 5 2c for greater detail.

Rem,ark 3 . If a process X = ( X t ) ~ is o a-stable (0 < a

< 2) and has simultancously

lf-similarity property ~ a w ( ~ , tt 2 , 0) = Law(aHXt, t

2 0),

u

> 0,

but is not, a Ldvy process, then the forniula H = l / a fails. Various pairs ( a .W) can correspond to such processes, provided that

a<1

and

0

< IHI< l/a,

or

(See [418; Corollary 7.1.11 and Figure 7.11.)

1. Non-Gaussian Models of Distributions a n d Processes

209

2. Since a-stable processes are particular cases of Lkvy processes X = ( X t ) t y ~the , characteristic functions of which (pt(0) = Eei('JXt) have the representation pt(Q) = exp{t+(Q)} with cuniulant +(O) defined by formula ( 7 ) in Slb, one may well ask what are the parameters B, C , and v corresponding to these (a-stable) processes. Of a particular interest here is the LCvy measure v = v(dz),which is 'responsible' for the magnitudes of jumps AX, = Xt - X t - of X = ( X t ) t 2 0 . We now present, following [423],the main results obtained in this direction.

THEOREM 1. Let X = ( X t ) t > be ~ a nondegenerate Le'vyprocess in Rd with triplet (B, C, v ). Then 1) the process X is a 2-stable (Le'vy) process if and only if v = 0, i.e., if and only if it is Gaussian; 2) the process X is strictly 2-stable (Lkvy) process if and on1.y if it is a Gaussian process with zero mean ( B = 0). THEOREM 2 . Let X = ( X t ) t 2 o be a nondegenerate Le'vy process in Rd with triplet ( B , C , v ) . Assume that 0 < N < 2. Then X is an a-stable (Le'vy) process if and only if C = 0 and thc L@vymeasure v is as follows:

where X is some nonzero finite measure in S = {x E R d : 1x1 = 1). The cumulant +(O) of such a process has the following representation:

Noteworthy, if 0 < (Y < 2, then the radial part of the LCvy measure is ~ - ( ' + ~ ) d r . With a decrease of a , r - ( l + a ) decreases for 0 < r < 1 and increases for 1 < r < w. Thus, we can say that large jumps of the process trajectories prevail for Q close to zero, but the process is developing in small jumps if a is close to two. Good illustrations to this property can be found in [253].

THEOREM 3. Let X = ( X ~ ) Q be O a nondegenerate Levy process in Rd with triplet ( B ,c, v). 1) Lot Q E ( 0 , l ) . Then X is a strictly a-stable (Le'vy) process if and only if it has the cumulant

with some norizero finite nieasure X in S , i.e., if and only if C = 0 and the 'drift' (see (27) in 5 l a ) is zero.

Chapter 111. Stochastic Models. Continuous Time

210

2) Let a E (1,Z). Then X is a strictly a-stable (Le'vy) process if arid only if C = 0 and the 'center' (see (29) in 5 l a ) is zero (i.e., EX = O ) , or, equivalently, if and only if its cumulant is

with some nonzero finite measure X in S . 3) Let a = 1. Then X is a strictly 1-stable (Levy)process if and only if

for some finite measure X in

S and some

constant B such that

P

and X(S)

+ IB1 > 0.

COROLLARY. Let X = ( X t ) t 2 0 be an a-stable Levy process in Rd.If a # 1, then there always exists y E Rd such that the centered process X = ( X t - Y t ) t > o is strictly a-stable. On the other hand if a = 1 and (13) holds, then the process X = ( X t ) t > o is itself strict1,y 1-stable. For d = 1 we can explicitly calculate integrals in the representations (9)-(11) for cumulants to obtain

< a < 2 (cf. forrnula (6) in l a ) , where p E [-I, 11, c7 > 0, and p E R. A (nonzero) a-stable L6vy process is strictly stable if and only if

for 0

p =0

for

a # 1,

and p=0,

a+lpl>O

for

a=l.

3. We shall now discuss formulas (2) and (3) in the definition of an a-stable process. For t = 1 forrnula (1) can be written as follows: X,

2 ul/"X1 + D a ,

(16) where D, is a constant. Using the representations (14) and (15) for the characteristic function of this process we obtain

1. Nori-Gaussian Models of Distributions and Processes

211

4. We see from the above that operating the distributions of stable process can be difficult because it is only in three cases t h a t we have explicit formulas for the densities (see 3 la.5). However, in certain situations we can show a way for constructing stable processes, e.g., from a Brownian motion, by means of a r a n d o m (and independent on the motion) chnn9e of time. We present an interesting example in this direction concerning the sirnulation of symmetric a-stable distributions using three independent random variables: miformly distributied, Guimiun, a n d exponentially distributed. Let 2 = ( 2 t ) ~ be o a symmetric n-stable Lkvy process with characteristic function e-t/ola pt(0) = Eeiszt > (18)

where 0 < cy < 2. It. will be clear from what follows that the process 2 can be realized as

where B = (Bt)t>ois a Brownian motion with EBt = 0 and €B; = 2 t ; while T = (Tt)t>ois a nonnegative nondecreasing ?-stable random process, which is called a stable subordinator. We say of the process 2 obtained by a transformation (19) that it is constructed from a Brownian motion by means of a r a n d o m chnn,ge of t i m e (subordination) T = (Tt)t>o. The process T = (Tt)tao required for (19) can be constructed as follows. Let U ( " ) = U ( " ) ( w ) be a nonnegative stable random variable with Laplace t ransforrn

= e -xu

x>o,

,

(20)

where 0 < 0 < 1. Note that if U ( " ) and U l , . . . , U, are independent identically distributed random variables, then the sum n

n-lJff

uj

(21)

j=1

has the same Laplace transform as U ( " ) , so that U ( " ) is indeed a stable random variable. Assume that 0 < cy < 2. We now construct a nonnegative nondecreasing pa t a b l c process T = (Tt)t30 such that Law(T1) = L ~ w ( U ( " / ~ ) ) . By the 'self-similarity' property ( 5 ) ,

212

Chapter 111. Stochastic Models. Continuous Time

t licrefore

which delivers the required representation (19). Let y = y ( s ; a ) be the distribution density of the variable U = U ( " ) , where 0 < a < 1. By [238] and [264], this density, which is concentrated on the half-axis T 3 0, has thc following 'explicit' representation:

where

1

sin z

sin a z

As observed by H. Rubiri (see [264; Corollary 4.1]), the density p ( z : a )is the distribution dcrisity of the random variable

<

where (L = a ( z ;0) is as in (25), [ and r/ are independent random variables, is uniformly distributed on [0, T], and rl has t,he exponential distribution with parameter one.

R e r w r k . Tht: fact that p(s;a ) is the distribution density of no tlificult,y. For let h ( z )=

.I'

7r

1 T

C car1 be verified with

a ( z ;a ) exp(-za(z; a ) )d z .

o

Then h = h(.c) is obviously the density for q / a ( < ;a ) . A simple change of variables shows now that p ( x ; a ) is t,he distribution density for C. Thus,

1. Non-Gaussian Models of Distributions and Processes

213

From (27) and ( 2 8 ) , denoting a Gaussian random variable with zero mean and variance 2 by y(0, a),we see that

This representation for Law(& - Zs) shows a way in which, simulating three independerit raiidoiri variables <, 71, and y = y ( 0 , 2 ) , one can obtain an observation saniple for the increments Zt - Z,of a symmetric a-stable random process.

Remark. For general results on the construction of L6vy processes from other L6vy processes (of a simpler structure) by means of subordinators, see [47],[239]: [409], and [483]. The process 2 = (Zt)t>o is used in [327]for the description of the behavior of prices (the Murrdelbrot~Taylormodel). It is worth noting that if t is real, ‘physical’ time, then Tt can be interpreted as ’operational‘ time (see Chapter IV, 53d) or as the randoiri ‘riuinber of transitions’ that occurred before time t . (This, slightly T,

loose, intcrpretation is inspired by siniilarities with the sums C & of a random k=l number T,, of ,random variables
EXAMPLE 1. A standard Brownian motion X = (Xt)t>oin pPd is a strictly 2-stable L6vy process. The corresponding probability distribution P1 = Pl(dz) of X 1 is as follows: P I ( d z ) = (27r-d/2e-lz12/2 ctz, zE (29) The charact,eristic function of X t is

214

Chapter 111. Stochastic Models. Continuous Time

arid one irnniediately sees that (cf. (14))

EXAMPLE 2. A standard Cauchy process in Rd is a strictly 1-stable Lkvy process with PI(&)

=

T) d+l

T - ~ Y (

(1

+ 1z12)-y d z ,

z E

Rd.

(32)

The characteristic function of X t is

pxt (0) = e-tlQI,

(33)

and, clearly (cf. (14)), pX,,(Q) = e - 4 4 = ,-WI

=

P a X t (0).

(34)

EXAMPLE 3 . For a one-sided strictly ;-stable Lkvy process in (0, m ) we have P l ( d Z ) = ( 2 ~ ) - 1 / 2 ~ ( 0 , c o ) ( " ) e 1 / ( 2 ~ ) z - 3 /d2z ,

and p X t (0) = exp{

-tJlei

(1 - i Sgn o)}.

z E

R,

(35)

(36)

We inimediatcly see that Cp,x,(Q)= exp{--(ltJ/61((1

-%no)}

These exaniplcs virtually exhaust all known cases when the one-dimensional distributions of X 1 (arid therefore, of X , ) can be expressed in terms of elementary functions. § Id. Hyperbolic Distributions and Processes 1. In 1977, 0 . Barndorff-Niclsen [21] introduced a class of distributions that is interesting in niariy respects, the so-called generalized hyperbolic distributions. His motivation was the desire to find an adequate explanation for some empirical laws in geology; subsequently, these distributions found applications t o geomorphology, turbulence theory, . . . , and also t o financial mathematics. Generalized hyperbolic distributions are not stable, but they are close to stable ones in that they can be characterized by several purumeters of a close meaning (see subsection 2). We distinguish two distributions in this class that are most frequent in applications: 1) the hyperbolic distribi~tionin the proper sense; 2) the Gnrissiun\\in,,oerse Gaussian distribution.

1. Non-Gaussian Models of Distributions and Processes

215

It should be mentioned that these distributions are m i x t u r e s of Gaussian ones. Hence they are naturally consistent with the idea of the use of conditionally Gaussi a n distributions. At the same time, the distributions in question are infinitely divisible and form a fairly wide subclass of infinitely divisible distributions. Judging by the behavior of their ‘tails’, they are intermediate between stable (with exponent a < 2 ) and Gaussian distributions (with a = 2 ) : their ‘tails’ decrease faster than for stable ( a < a ) , but slower than for Gaussian distributions. 2. The attribute ‘hyperbolic’ is due to the following observation.

For a normal (Gaussian) density

the graph of its logarithm 1ncp(x) is a parabola, while the graph of the logarithm In hl (x)of the density

of a hyperbolic distribution is a hyperbola

with asymptotes

a ( z )= -a/z

-

+ /3(z - p ) .

(4)

The four parameters ( a ,/3, p, 6) in the definition ( 2 ) of a hyperbolic distribution are assumed to satisfy the conditions

The parameters Q and ‘govern’ the form of the density graph, p is the location parameter, and 6 is the scale parameter. The constant Cl has the representation

where K l ( z ) is the modified third-kind Bessel function of index 1 (see [23]). One often uses another set of parameters to describe a hyperbolic distribution. Namely, one sets 1 1 a = -((p+y) and P = - ( p - y ) , (7) 2 2 so that (py= cx2 - p2.

Chapter 111. Stochastic Models. Continuous Time

216

p , 6) expressed in terms of these We shall denote the density h 1 (z) h l ( x ;a.0, new pararrieters by h2 ( x ) = 112 (2; cp, y,p , 6). It has the following representation:

where

with x = (cpy)’I2 and = (cp-’ +y-‘). (The last parameters are used in [289].) It, 15 clear from (8) that if a random variable X has density h2 ( x ;cp, y, p, 6), then the variable Y = ( X - a ) / b , where a E R and b > 0, has the distribution density h 2 ( ~b ;q , by, 6 / b , ( p - a ) / b ) . Thus, the class of hyperbolic distributions is invariant under shafts and scalzng. It is also clear from (8) that h 2 ( x ) > 0 for all 5 E R,and the ‘tails’ of h 2 ( z ) decrease exponmtaally at the ‘rate’ cp as x + -m and at the ‘rate’ y as x + m. As 6 + m, 61% i g 2 , and cp - y i 0, we have

=

whrre ( ~ ( 5 )cp(x;p , (T’) is the normal density. As 6 --f 0, we obtain in the limit the Laplace distribution (which is asymmetric for ‘p # y ) with densit,y

Introducing the parameters

and

we notice that they do not change when one passes from a random variable X with hyperbolic density h z ( x ;cp, y,p, 6) to a random variable Y = X - a with density h 2 ( z ;cp, y,p - a , 6) indicated above. These parameters arid x , which have the meaning of skewness and kurtosis, are good indicators of the deviation of a distribution from riorriiality (see Chapter IV, 3 2b for greater detail). We note t,hat the range of ( x , E ) is the interior of the triangle

<

0 = { ( x d : 0 < 1x1 < E < I}

217

1. Non-Gaussian Models of Distributions and Processes

(see Fig. 26; here A’ corresponds to the nornial distribution, 8 to the exponential, and 2 to the Laplace distribution).

FIGURE 26

The boundary point (0,O) # V corresponds to the normal distribution; the points (-1,l) arid (1,l ) , also lying outside V, to the exponential distribution, and ( 0 , l ) # V to the Laplace distribution. Passing to the limit as x -+ &( we obtain (see [21]-[23], [25], [26])the so-called generalized inverse Gaussian distribution, It was mentioned in [all, [23], [25], [26], [22] that a hyperbolic distribution is a mixture of Gaussian ones: if a random variable X has density hl (x;a , 0, p , 6 ) , then

Law

x = E:~./Y(~ + pu2,c2),

(9)

where by EL2 we mean the result of averaging with respect to the parameter o2 that has the inverse Gaussian distribution with density

here u = a2 - ,kI2 arid b = S2. 3. We now consider another representative of Barndorff-Nielsen‘s class of generalized hyperbolic distributions [21], the so-called Gaussian\\znwerse Gaussian distribution ( GIG-distribution). (Later on, Barndorff-Nielsen started using also the name ‘normal inverse Gaussian distributions’ [22].) In the spirit of symbolical formula (9) we can define the Gavssian\\znzJerse Gaussian distribution of a random variable Y as follows: Law Y

=

E:~N(~

+ pa2,a 2 ) ,

where we consider averaging EZ: of the normal distribution N ( p respect to the inverse Gaussian distribution with density

(11)

+ pu2,

0 2 )with

Chapter 111. Stochastic Models. Continuous Time

218

where a = a2 - p2, b = S2. (The parameters a , 0,p , and 6 must satisfy conditions (5).) It is interesting that if W = (Wt)t20is a standard Brownian motion (a Wiener process) and T ( t )= inf{s 2 0 : W , + 6 s 2 ht} is the first time when the process (Ws+&s)s20 reaches the level v%t then T(1) has just a distribution with density (12). Hence if B = (Bt)taO is a standard Brownian motion independent of W , then Y has the same distribution as the variable

(Cf. formula (19) in 3 Ic.) Now let g(z) = g(z; a , 0, p, 6) be the density of a GIG-distribution. By (11) and (12),

and q ( x ) = Since

m.

it follows that

as / z )+ 03, therefore

The last relation shows that h l ( x ) has ‘heavier’ tails than g(z) 4. A hyperbolic distribution (with density hi($)) has a simpler structure than

a Gaussian\\inverse Gaussian distribution (GIG-distribution) with density g(z). However, one crucial feature of the latter distribution makes it advantageous in certain respects.

1. Non-Gaussian Models of Distributions and Processes

219

Let Y be a random variable with density g(z) = g(z; a , P , p , 6). Then its probability generating function is

4=GT]

+,,A}.

Hence if Y1,.. . , YrrL are independent GIG-distributed variables with the s a m e values of a arid [j, but with (generally speaking) distinct pi and hi, then the sum Y = Y1 + . . . Y,, is also a GIG-distributed variable with the same parameters a and P, hiit with p = 1-11 . . . pLmand 6 = 61 . . . . ,6 In other words, the GIG-distributions are closed (in the above sense) with respect to convolutions. On the other hand, if X has a hyperbolic distribution, then setting for simplicity /3 = 1-1 = 0 we obtain

+

+ +

+ +

Hence the class of hyperbolic distributions is not closed in a way siniilar to GIGdistributions. It is important to point out that both GIG-distribution and hyperbolic distribution are infinitely diwrsible. For GIG-distributions this is clearly visible from (18), while for hyperbolic distribution this is proved in [21]-[23], [25], [26]. By (18) we obtain also the following simple formulae for the expectation EY and the variance DY:

Re,m,arlc. As regards the applications of these distributions to the analysis of financial indexes, see E. Eberlein and U. Keller [127], where one can find impressive results of the statistical processing of several financial indexes (see also Chapter 11, 5 2b). 5. Since hyperbolic distributions are infinitely divisible, we can define a L6vy process (i.e., a process with independent homogeneous increments) with hyperbolic distribution of increments. We restrict ourselves to the case of symmetric centered densities h l ( z ) = hl (z; a , p, p , 6) with parameters 3!, = p = 0. In this case hl (z) has the following renresentation:

Let 2 = (Zt)t>o be the Lbvy process such that Z1 has the distributions with density (20).

220

Chapter 111. Stochastic Models. Continuous Time

It is worth noting that since ElZtl < m , 20= 0, and Z = ( Z t ) has independent increments, this process is a martingale (with respect to the natural flow of 0algebras (9t), 9 t = n(Z,, s t ) ) , i.e.,

<

E(Zt 1 9,) = Z,, t 3 S. (21) (Actually, E l Z l P < 00 for all p 3 1.) Let cpt(8) = Eexp(i8Zt) be the characteristic function of the hyperbolic Levy process Z = (Zt)t>o.Then (cf. formulas (4) and (5) in 5 l b ) c p t ( Q ) = (cpl(Q))t.

By (9) and (10) (with X = 2,)we obtain Law z1= E / ~ ~ J Y (02) o, (because

(23)

= p = O), where

Based on this representation, the following 'LPuy-Khzntrhzne' formula for pt (Q)was put forward in [127] (cf. formulas (22) arid (29) in 5 la):

where t h e L h y measure v has the density p u ( z ) (with respect to Lebesgue measure) with the following (fairly complicated) representation:

Here 51 arid I'\ are the Bessel functions of the first and second kinds, respectively. In view of the asymptotic properties of J1 and Yl (formulas 9.1.7,9 and 9.2.1,2 in [I]),one can show (see [l27]) that the denominator of the integrand in (26) is asymptotically a constant as y 4 0 and is y-'I2 as y + a.Hence 1 pu(x) 2 as z + 0, (27) which shows that the process Z = (Zt)t>omakes infinitely many small jumps on each arbitrarily siriall time interval. Indeed, let

pz(w;A, B)=

C 1(ZS(w) Z,-(w) -

E B),

B t %(a \

{O}),

SEA

be the jump rneasure of 2,i.e., the number of s E A such that the difference ZS(w)- Z,-(w) lies in the set B. Then (see, e g . , [250; Chapter 11, 1.81) E p 2 ( w ; A, B ) =

therefore h o , E j v ( d z = ) 00 and

1

[--E,O)

14 ' 4B),

v(dz) = 00 for each

E

> 0.

2. Models with Self-Similarity. Fractality

It has long been observed in the statistical analysis of financial time series that many of these series have the property of (statistical) self-similarity; namely, the structure of their parts 'is the same' as the structure of the whole object. For instance, if the S,, n 3 0, are the daily values of S&P500 Index, then the empirical and &(x), k > 1, of the distributions of the variables densities

A(.) 11,

(=

hs.') S7L-1

=

In

-),

n 3 1,

Skn

Sk(71-

1)

calculated for large stocks of data, satisfy the relation

where W is some constant (which, by the way, is significantly larger than l/2-by contrast to what one would expect in accordance with the central limit theorem). Of course, such properties call for explanations. As will be clear from what follows, such an explanation can be provided in the framework of the general concept of (statistical) se&similarity, which not only paved way to a fractional B r o w n i a n motion, fractional Gaussian noise, and other important notions) but was also crucial for the development of fractal geometry (B. Mandelbrot). This concept of selfsimilarity is intimately connected with nonprobabilistic concepts and theories, such as chaos or nonlisnear dynamical system.5, which (with an eye to their possible applications in financial mathematics) were discussed in Chapter 11, 3 4.

3 2a.

Hurst's S t a t i s t i c a l Phenomenon of Self-Similarity

1. In 1951, a British climatologist H. E. Hurst, who spent more than 60 years in Egypt as a participant of Nile hydrology projects, published a paper [236]devoted

222

Chapter 111. Stochastic Models. Continuous Time

to his discovery of the following surprising phenomenon in the fluctuations of yearly run-offs of Nile and several other rivers. Let 21,x2,.. . , IC, be the values of n successive yearly water run-offs (of Nile, 71

say, in some its part). Then the value of AX,, where X , = C z k , is a ‘good’ k=l estimate for tiit. expectation of the XI,. The deviation of the cumulative value X I , corresponding to k successive years from tlie (empirical) mean as calculated using the d a t a for n years is

and

(

x

)

rriin X I , - - X T L k
and

niax k
are t,he smallest and the biggest deviat,ions. Let k

be the Lr(mge’characterizing the amplitude of the deviation of the cumulative values X I , from their niean value ;x, over n successive years. However, Hurst did not operate with the values of the R,,themselves; he considered instead the normalized values Qn = R,,/S,, where

is the empirical mean deviation introduced in order to niake the statistics invariant under the change k 3 1. XI, + C(”k m),

+

This is a desirable property because even the expectation and the variance of tlie xk are usually unknown. Based on the large volume of factual d a t a , the records of observations of Nile flows in 622-1469 (i.e., over a period of 847 years), H. Hurst discovered the following behavior of thc statistics 7Zn/S,, for large n:

-2, -cn, w S7l

where c is a certain constant, the equivalence is interpreted in some suitable sense, and the parameter W,which is now called the Hurst parameter or the Hurst exponent, is approximately equal to 0.7. (H. Hurst obtained close values of this

2. Models with Self-Similarity. Fractality

223

parameter also for other rivers.) This was a n unexpected result; he had anticipated that W = 0.5 for the reason that we explain now following a later paper by W. Feller [157]. Let 21, x2,.. . be a sequence of independent identically distributed random variables with Ex, = 0 and Ex: = 1. (This is what Hurst had expected.) Then, as shown by Feller,

for large 7 1 . Since, moreover, S,, + 1 (with probability one) in this case, the values of Q, must increase (on the average, a t any rate) as n1I2 for n large. In a study of the statistical properties of the sequence (z,),>l it makes sense to ask about the structure of the empirical distribution function Law(z1 . . . x,) calculated from a (large) number of samples, say, ( 2 1 ; . . . , xn)% ( ~ , , + 1 , . . , ZZ,), .. . . In the case when tjhe xk are the deviations of the water level from some ‘mean’ value, one finds out (e.g., for Nile again) that

+

where

+

W > 1/2.

2. How and thanks to what probabilistic and statistical properties of the sequence (xn) in (2) can the parameter W be distinct from 0.5? Looking a t formula (4) in 5 l a we see one of the possible explanations for the relation W 0.7: the xj can be independent stable random variables with stability exponent a = 1.48. There exists another possible explanation: relation (2) with W # 1/2 can occur even in the case of normally distributed, but dependent variables 21, x2,.. . ! In that case the stationary sequence ( x T 1 )is necessarily a sequence with strong aftereflect (see 5 2c below.) N

&-

3. Properties (1) and (a), which can he regarded as a peculiar form of self-similarity, can be also observed for many financial indexes (with the h, in place of the xn). Unsurprisingly, the above observation that the x, can be ‘independent and stable’ or ‘dependent and normal’ has found numerous applications in financial mathematics, and in particular, in the analysis of the ‘fractal’ structure of ‘volatility’. Hurst’s results arid the above observations served as the starting point for B. Mandelbrot, who suggested that, in the H u r s t model (considered by Mandelbrot himself) and in many other probabilistic models (e.g., in financial mathematics), one could use strictly stable processes (5 l c ) and fractional B r o w n i a n m o t i o n s (5 2c), which have the property of self-similarity. It should be pointed out that a variety of real-life systems with nonlinear dynam.ics (occurring in physics, geophysics, biology, economics, . . . ) are featured by

224

Chapter 111. Stochastic Models. Continuous Time

self-similarity of kinds (1) and (2). It is this property of self-similarity that occupies the central place in fractal geometry, the founder of which, B. Mandelbrot, has chosen the title ‘ L T h eFractal G e o m e t r y of Nature” for his book [320],so as to emphasize the universal character of self-similarity. We present the necessary definitions of statistical self-similarity and fractional Brownian motion in 2c. The aim of the next section, which has no direct relation to financial mathernatics and was inspired by [104], [379], [385], [386], [428], [456], and books by some other authors, must give one a general notion of self-similarity.

3 2b.

A Digression on Fractal Geometry

1. It is well known that the emergence of the Euclidean geometry in Ancient Greece was the result of an attempt to reduce the variety of natural forms to several ‘simple’, ‘pure’, ‘symmetric’ objects. That gave rise to points, lines, planes, and most simple three-dimensional objects (spheres, cones, cylinders, . . . ). However, as B. Maridelbrot has observed (1984), “clouds are n o t spheres, m o u n taines are n o t co,nes, coastlines are n o t circles: and bark is n o t s m o o t h , n o r does lightning travel in a .straight h e . . . ” Mandelbrot has developed the so-called fractal geometry with the precise aim to describe objects, forms, phenomena that were far from ‘simple’ and ‘symmetric’. On the contrary, they could have a rather complex structure, but exhibited a t the same tinie some properties of self-similarity, self-reproducibility. We have no intention of giving a formal definition of fractal geometry or of a fractal, its central concept. Instead, we wish to call the reader’s attention to the importance of the idea of fractality in general, and in financial mathematics in particular and give for that reason only an illustrative description of this subject. The following ‘working definition’ is very common: “ A fractal i s a n object whose portions have the s a m e structure as t h e whole.” (The word ‘fractal’ introduced by Mandelbrot presumably in 1975 [315] is a derivative of the Latin verb fractio meaning ‘fracture, break up’ [296]). A classical example of a three-dimensional object of fractal structure is a tree. The branches with their twigs (‘portions’)are similar to the ‘whole’, the main trunk with branches. Another graphic example is the Sierpinski gasketa (see Fig. 27) that can be obtained from a solid (‘black’) triangle by removing the interior of the central triangle, arid by removing subsequently the interiors of the central triangles from each of the resulting ‘black’ triangles. The resulting ‘black’ sets converge to a set called the Sierpinski gasket. This limit set is an cxample of so-called attractors. aW. Sierpiiiski (1882-1969), a Polish mathematician, invented the ‘Sierpinski gasket’ and the ‘Sierpinski carpet’ as long ago as 1916.

2. Models with Self-Similarity. Fractality

225

FIGURE 27. Consequtive steps of the construction of the Sierpiliski gasket

Obviously, there are many ‘cavities’ in the Sierpinski gasket. Hence there arises a natural question on the ‘dimension’of this object. Strictly speaking, it is not two-dimensional due to these cavities. At the same time, it is certainly not one-dimensional. Presumably, one can assign some fractal (i.e., fractional) dimension to the Sierpinski gasket. (Indeed, taking a suitable definition one can see that this ‘dimension’is 1.58. . . ; see, e.g. [104], [428], or [456].) The Sierpinski gasket is an example of a s y m m e t r i c fractal object. In the nature, of course, it is ‘asynmietric fractals’ that dominate. The reason lies in the local randomness of their development. This, however, does not rule out its d e t e r m i n i s m on the global level; for a support we refer to the construction of the Sierpihski gasket by means of the following stochastic procedure [386]. We consider the equilateral triangle with vertices A = A(1,2); B = B(3,4),and C = C(5,6) in the plane (our use of the integers 1 , 2 , .. . , 6 will be clear from the construction that follows). In this triangle, we choose an arbitrary point a and then roll a ‘fair’ die with faces marked by the integers 1 , 2 , 6. If the number thrown is, e.g., ‘5’or ‘6’,then we join with the vertex C = C(5,6) and let be the middle of the joining segment. We roll the dice again and, depending on the result, obtain another point, and so on. Remarkably, the limit set of the points so obtained is (‘almost always’) the Sierpihski gasket (‘black’ points in Fig. 27). Another classical, well known to mathematicians example of a set with fractal structure is the C a n t o r set introduced by G. Cantor (1845-1918) in 1883 as an example of a set of a special structure (this is a nowhere dense perfect set, i.e., a closed set without isolated points, which has the cardinal number of the continuum). 1

226

Chapter 111. Stochastic Models. Continuous Time

We recall that this is the subset of the closed interval [0, I] consisting of the numbers representable as

C O0

<

with ~i = 0 or 2 . Geometrically, the Cantor set can be

i=l3

(5, i), i] [i,

obtained from [ O , 1 ] by deleting first the central (open) interval then the central subintervals $) and of the resulting intervals [0, and 11, and so on, ad infinitum. The total length of the deleted intervals is 1; nevertheless, the remaining ‘sparse’ set has the cardinal number of the continuum. The selfsimilarity of the Cantor set (i.e., the fact that ’its portions have the same structure as the whole set,’) is clear from our geometric construction: this set is the union of subsets looking each like a reduced copy of the whole object. Among other well-known objects with property of self-similarity we mention the ‘Pascal triangle’ (after B. Pascal), the ‘ K o c h snowflakes’ (after H. van Koch), the Peano c u r 7 ~(G. Peano), and the Julia sets (G. Julia); see, e.g., [379].

(i,

(6,i)

2. In our above discussion of ‘dimension’ we gave no precise definition (F. Hausdorff, who invented the Hausdorff dimension, pointed out that the problem of an adequate definition of ‘dimension’ is a very difficult one). Referring the reader to literature devoted to this subject (e.g., [104], [428], [456]), we note only that the notion of a ‘fractal dimension’ of, say, a plane curve is fairly transparent: it must show how the curve sweeps the plane. If the curve in question is a realizat i o n X = ( X t ) t > o of some process, then its fractal dimension increases with the proportion of ‘high-frequency’ components in ( X t ) t 2 0 .

3. Let now X = ( X t ) t 2 o be a stochastic ( r a n d o m ) process. In this case, it is reasonable to define the ‘fractal dimension’ for all the totality of realizations, rather than for separate realizations. This brings us to the concept of statistical fractal dimension, which we introduce in the next section.

5 2c.

Statistical Self-Similarity. Fractional Brownian Motion

1. DEFINITION 1. We say that a random process X = (Xt)tao with state space Rd is self-similar or satisfies the property of (statistical) self-similarity if for each a > 0 there exists b > 0 such that

In other words changes of t h e tzme scale ( t --t a t ) produce t h e s a m e results as changes of the phase scale (z + b z ) . We saw in 5 l c that for (nonzero) strictly stable processes there exists a constant W such that b = a’. In addition, for strictly a-stable processes we have

2. Models wit,h Self-Similarity. Fractality

227

In the case of (general) stable processes, in place of (I), we have the property Law(X,t, t 3 0) = Law(a"Xt

+ tD,,

t 3 0)

(3)

(see forniiila (2) in 1 l c ) , which nieans that, for these processes, a change of the time scale produces the same results as a change of the phase scale and a subsequent 'translation' defined by the vector tD,, t 3 0. Moreover, W = 1/a for a-stable processes. 2. It follows from the above, that it would be reasonable t o introduce the following definition.

DEFINITION 2. If b = a' in Definition 1 for each a > 0, then we call X = (Xt)t20 a self-similar process with Hurst exponent H or we say that this process has the property of statistical self-.similarity with Hurst exponent W. The quantity D = is called the statistical fractal di,mension of X.

A classical example of a self-similar process is a Brownian m o t i o n X = (Xt)t>O. We recall that for this (Gaussian) process we have EXt = 0 and EX,Xt = min(s, t ) . Hence EX,,X,t = niin(as, a t ) = a min(s, t ) = E(a1/2XS)(a1/2Xt), so that the two-dimensional distributions Law(X,, Xt) have the property

By the Gaussian property it follows that a Brownian motion has the property of statistical self-similarity with Hurst exponent W = l / 2 . Another example is a strictly a-stable Le'wy m o t i o n X = ( X t ) t > o ,which satisfies the relation X t - x, S a ( ( t - s)1/2,0,0), a E (0,2].

-

For this process with homogeneous independent increments we have

x,t x,, d a -

q x t

-

XS),

so that, the corresponding Hiirst exponent H is l / a and D = a. For a = 2 we obtain a Brownian motion. We emphasize that the processes in both examples have independent increments. The next example relates to the case of processes with dependent increments. 3. Fractional Brownian motion. We consider the function

A ( s , t ) = /sI2'

+ ltI2" - It

-

sI2',

s,t

E

R.

(4)

For 0 < W 6 1 this function is nonnegative definite (see, e.g., [439; Chapter 11, 5 9]), therefore there exists a Gaussian process on some probability space (e.g., on the

Chapter 111. Stochastic Models. Continuous Time

228

space of real functions w = (wt), t E ance function

Iw) that has the zero mean and the autocovari-

1 COV(X,, X t ) = - 4 s , t ) , 2

i.e., a process such that

EX,Xt =

1

-{ 2

+ /tj2"

-

It

-

~1~').

(5)

As in the case of a Brownian motion, whose distribution is completely determined by the two-dimensional distributions, we conclude that X is a self-similar process with Hurst exponent W. By (5), E/Xt - X,I2 = It - ~ 1 ~ ~ . (6) We recall that, in accordance with the Kolmogorov test ([470]), a random process X = (Xt)t>O has u contirivoiis modzfication if there exist constants a > 0 , p > 0, and c > 0 siich that E/Xt - XSI" 6 c It - S /1+0 (7) for all s , t 3 0 . Hence if W > l / 2 , then it immediately follows from (6) (regarded as (7) with = 2 and ,B = 2W-1) that the process X = (Xt)t>ounder consideration has a continuous modification. Further, if 0 < W 6 l / 2 , then by the Gaussian property, for each 0 < k < W we have

with some constant c > 0. Hence we can apply the Kolmogorov test again (with Q = l / k and p = W / k - 1). Thus, our Gaussian process X = (Xt)t20 has a continmous modification for all 0 < 6 1.

w,

w

DEFINITION 3 . We call a continuous Gaussian process X = (Xt)t>o with zero mean and the covariance function (5) a (standard) fractional Brownian motion with Hurst self-similarity exponent 0 < W 6 1. (We shall often denote such a process by Bw = ( B ~ ( t ) ) t > inowhat follows.)

By this definition, a (standard) fractional Brownian motion X = (Xt)t>O has the following properties (which could also be taken as a basis of its definition): 1) Xo = 0 arid EXt = 0 for all t 3 0;

2. Models with Self-Similarity. Fractality

229

2 ) X has homogeneous increments, i.e., Law(Xt+,

-

X , ) = Law(Xt),

s, t

2 0;

3) X is a Gaussian process and

EX;

=

ltI2’,

t 2 0,

where 0 < W < 1; 4) X has continuous trajectories. These properties show again that a fractional Brownian motion has the selfsimilarity property. It is worth pointing out that a converse result is also true in a certain sense ([418; pp. 318 ~3191):i f a nondegenerate process X = (Xt)tao, X o = 0, has finite variame, ho,mogeneous i n c r e m e n t s , a n d i s a self-similar process with Hurst expon e n t W,t h e n 0 < W 1 and th,e autocouariance f u n c t i o n of th,is process sakisfies the equality C O V ( X , ~X, t ) = EX?A(s, t ) , where A ( s ,t ) i s defined by ( 4 ) . Moreover, if

<

< W < 1, then the expectation EXt

d

is zero, and if W = 1, then X t = tX1 (P-as.). We note also that, tiesides Gaussian processes, one knows also of some nonGaussian processes with these properties (see [418; p. 3201).

0

H = 1/2, then a (standard) fractional Brownian motion is precisely a (standard) Brownian motion (a Wiener process). The processes BH so introduced were first considered by A. N. Kolinogorov in [278] (1940), where they were called W i e n e r helices. The name ‘fractional Brownian motion’ was introduced in 1968, by B. Mandelbrot and J. van Ness [328]. By contrast to Kolinogorov, who had constructed the process Ba starting from the covariance function (4), Mandelbrot and van Ness used an ‘explicit’ representation by means of stochastic integrals with respect to a (certain) Wiener process W = (Wt)teswith Wo = 0: for 0 < W < 1 they set

4. If

where the (norirializing) constant

is such that EBi(1) = 1.

Remark 1. As regards various representations for the right-hand side of (8), see [328]. In this paper one can also find a comprehensive discourse on the origins of the term fractional Brownian ,motion and its relation to the fractimnal integral

230

Chapter 111. Stochastic Models. Continuous Time

of Holnigren-Riernanri-Liouville, where also Weyl [475]).

MI can be an arbitrary positive number (see

Remark 2 . Almost surely, the trajectories of a fractional Brownian motion B E , 0 < W < 1, satisfy the Holder condition with exponent p < W.They are incidentally nowhere differentiable and

for each to 3 0. (The corresponding proof can be carried out in the same way as for the standard Brownian motion, i.e., for W = 1/2; see, e.g., [123]).

<

Remark 3. Considering a Holder function Wt (IWt - Wsl c / t - s J a ,a > 0) with values in ( 0 , l ) in place of W in (8) we obtain a random process that is called a mvltifractional Brownian motion. In was introduced and thoroughly investigated in [381]. Remark 4. In the theory of stochastic processes one assigns a major role to semimartingales, a class for which there exists well-developed stochastic calculus (see § 5 and, for more detail, [250] and [304]). It is worth noting in this connection that a fractional Brownian motion Bm, 0 < W 6 1, is not a semimnrtingale (except for the case of W = l / 2 , i.e., the case of a Brownian motion, and IHI = 1). See [304; Chapter 4, 3 9, Example 21 for the corresponding proof for the case of 1 / 2 < W < 1. Remark 5. In connection with Hurst's R/S-analysis (i.e., analysis based on the study of the properties of the range, empirical standard deviation, and their ratio), it could be instructive to note following [328] that if X = ( X t )t > o is a continuous self-similar process with Hurst parameter JHI, X O = 0, and Rt = sup X,5 O<s
inf X,, then Law(Rt) = Law(tHR1), t

o<sgt

(W

= 1/2)

> 0. In the case of

a Brownian motion

W. Feller found a precise formula of the distribution of

R1.

(Its density

00

is 8

C (-l)k-1k2p(kz),

z

3 0, where p(x) = ( 2 ~ ) - ~ / ~ e x p ( - x ' / 2 ) . )

k=l

5 . There exist various generalization of self-similarity property (1). For example, let X ( a ) = (Xt(rw))t>obe a process of Ornstein-Uhlenbeck type with parameter a E R,i.e., a Gaussian Markov process defined by the formula

where W = (Ws)s30is a standard Brownian motion; see 5 3a. (Note that X ( a ) = (Xt(a))t>~ is the solution of the stochastic linear differential equation dXt(cr) = cvXt(a)dt dWt, X o ( a ) = 0.) From (11) it easily follows that

+

Law(X,t(a), t E R) = L a w ( a ' / ' X t ( n a ) , t E R) for each a E R,which can be regarded as a self-similarity (of a sort) of the family of processes { X ( a ) , a E R}.

2. Models with Self-Similarity. Fractality

231

6. We now dwell upon one crucially important method of statistical inference that is based on self-similarity properties. Let X = (Xt)t20 be a self-similar process with self-similarity exponent Assume that A > 0. Then

L ~ w ( X A= ) Law(A'XI),

so that if fA(x) = dP(XA dx then

'

(12)

is the density of the probability distribution of X A , fl

(x)= A'fA (.A").

(13)

It is common in the usual statistical analysis that, based on considerations of the general nature, one can assume X to be self-similar with some, unknown in general, value of the parameter W. Assume that we have managed to find a 'likely' estimate for W. Then the verification of the conjecture that X is indeed a selfsimilarity process with parameter W can be carried out as follows. Assume that, based on independent observations of the values of XI and XA, we can construct empirical densities f l ( x ) and f A ( Z ) for sufficiently many values of A. Then if A

A

A(z) M A G f ~ ( x A " )

(14)

for a wide range of values of x arid A, then we have a fairly solid argument in favor of the conjecture that X is a self-similar process with exponent W. Certainly, if we know the theoretical density fl(x) (which, of course, depends on W), then instead of verifying (14) we should look if the graph of A'~A(xA') is close to that of fl(x) plotted for the Hurst exponent (or for the true value of this parameter if it is known a priori).

7. For a-stable L6vy processes the Hurst exponent W is equal to 1 / a , therefore the estimation of W reduces to the estimation of a. For a fractional Brownian motion Bw = (BM(t))t>owe can estimate W from the results of discrete observations, for instance, as follows. We consider the time interval [0,1] and partition it into n equal parts of length A = I/n. Let

(cf. formula (19) in Chapter IV, 3a). Since

EIBw(t + s)

-

Bw(t)I = ~

it follows that

Chapter 111. Stochastic Models. Continuous Time

232

Hence we infer the natural conclusion: we should take the statistics

as an estimator for H.(It was shown in [380]that

8 2d.

6,+ W with probability one.)

Fractional Gaussian Noise: a Process with Strong Aftereffect

1. In niaiiy domains of applied probability theory a Brownian motion B = ( B t ) t > ~ is regarded as an easy way to obtain white noise. Set t irig ,& = B, - &-I, n 3 1, (1) we obtain a sequence l j = ( & L ) T L1 2of independent identically distributed random Gaussian variables with En, = 0 and E@ = 1. In accordance with Chapter 11, 3 2a, we call such a sequence white (Gaussian) noise; we have used it as a source of randomne.ss in the construction of various random processes, bot,h linear ( M A , A R , A R M A , . . . ), and nonlinear ( A R C H , G A R C H , . . . ) . In a similar way, a fractional Brownian motion Bw is useful for the construction of both stationary Gaussian sequences with strong aftereflect (systems with long 7ne7riory, per.szstent syste,ms), and sequences with intermittency, antzpersis-

tence ( relaxation processes). By analogy with (1) we now set

and we shall call the sequence ,6 = (Pn),21 fractional (Gaussian) noise with Hurst parameter W,0 < IHI < 1. By formula (5) in 5 2c for the covariance function of a (standard) process BMwe obtain that, thc covariance fiinctiori p w ( n ) = Cov(&, &+,) is as follows:

+ co. Thus, if IHI = 1/2, then pa(.) = 0 for 71 # 0, arid (P,),>l is (as already mentioned) a Gaussian sequence of independent random variables. On the other hand if W # l / 2 , then we see from (4) that the covariance decreases fairly slowly (as / T L / - ( ~ - ~ ’ ) ) with the increase of n, which is usually interpreted as a ‘long memory’ or a ‘strong afterrffect’. as

71

2. Models with Self-Similarity. Fractality

233

We should note a crucial difference between the cases of 0 < IHI < l/Z arid 1/2 < W < 1. If O < W < 1/2, then the covariance is negative: p a ( n ) < O for n # 0, moreover, a2

c

n=O

IPWM

< 00.

On the other hand, if 1 / 2 for

71

# 0, arid

< IHI < 1, then the covariance is positive:

p a ( 7 ~ )> 0

m

C

pm(n) =

00.

n=O

A positive covariarice means that positive (negative) values of 0, are usually followed also by positive (respectively, negative) values, so that fractional Gaussian noise with l / 2 < W < 1 can serve a suitable model in the description of the ‘cluster’ pheriorriena (Chapter IV, 3 3e), which one observes in practice, in the empirical ”

An for many financial indexes S analysis of thc returns hn = ln -

SF,- 1

=

(S,),

On the other hand, a negative- covariance means that positive (negative) values are usually followed by negative (respectively, positive) ones. Such s t r m g intermittency (‘up arid down and up . . . is indeed revealed by the analysis of the behavior of volatilities (see 3 5 3 and 4 in Chapter IV). 2. The sequence ,B = (&) is a Gaussian stationary sequence with correlation function pw(n)defined by (3). By a direct verification one can see that the spectral density f ~ q ( X ) of the spectral remesent at ion

(5) can be expressed as follows:

Calculating the corresponding integrals we obtain (see [418] for detail) that

3. The case W = l / 2 . With an eye to the applications o f fractional Brownian motions to thc description of the dynamics of financial indexes, we now choose a unit of time 71 = 0, *l, * 2 , . . . and set

H7, = Bm(n),

h,

= H , - Hn-l.

234

Chapter 111. Stochastic Models. Continuous Time

Clearly, Eh, = 0 and D h i = 1. The corresponding Gaussian sequence h = (h,) of independent identically distributed variables is, as already pointed out in this section, white (Gaussian) noise. 4. The case of 1/2 < W < 1. The corresponding noise h = ( h n ) is often said to be ‘black’. It is featured by a strong afterefleet, long memory (a persistent system). Phenomena of this kind are encountered in the behavior of the levels of rivers, or the character of solar activity, the widths of consecutive annual rings of trees, and, finally (which is the most interesting from our present standpoint), in the values of S, the returns hn = In -, n 3 1, for stock prices, currency cross rates, and other Sn-1

financial indexes (see Chapter IV). If W = l / 2 , then the standard deviation J D ( h l f . . . h,) increases with n as fi,while if W > l / 2 , then the growth is faster, of order n H .To put it otherwise, the dispersion of the values of the resulting variable H , = hl . . . + h, is larger than for white noise (W = 1/2). ) It is instructive to note that if W = 1 then we may take Bw(t) = t B ~ ( 1(P-a.s.) for a fractional Brownian motion. Hence the sequence h = (h,) of increments h, = &(n) - Bw(n - l ) , n 2 1, is trivial in this case: we always have h, E Bw(l), n 3 1, which one could call the ‘perfect persistence’.

+

+

5. The case of 0 < W < l / Z . Typical examples of systems with such values of the Hurst parameter W are provided by turbulence. The famous Kolmogorov’s Law of 2/3 ([276], 1941) says that in the case of an incompressible viscous fluid with very large Reynolds number the mean square of the difference of the velocities at two points lying at a distance T that is neither too large nor too small is proportional to T ~ ’ , where IH[ = 1/3. Fractional noise h = (h,) with 0 < W < l / Z (‘pink noise’) has a negative covariance, which, as already mentioned, corresponds to fast alternation of the values of the h,. This is also characteristic of turbulence phenomena, which (coupled with self-similarity) indicates that a fractional Brownian motion with 0 < W < 1/2 can serve a fair model in a description of turbulence. An example of ‘financial turbulence’, with Hurst parameter 0 < W < 1/2, is provided by the sequence F = (F,) with F, =, In :

on

where

on-1

is the empirical variance (volatility) of the sequence of logarithmic returns h = (h,), Sn h, = ln, calculated for stock prices, DJIA, S&P500 Index, etc. (see ChapSn - 1 ter IV, 3 3a).

2. Mod& with Self-Similarity. Fractality

235

Many authors (see, e.g., [385], [386], and [ISO]) see big similarities between hydrodynamic turbulence arid the behavior of prices on financial markets. This analogy brings, e.g., the authors of [180] to the conclusion that “in a n y case, we have reason t o believe that t h e qualitative picture of turbulence that has developed during the past 70 years will h,elp our underslandiny of t h e apparently remote field of financial markets”.

3. Models Based on a Brownian Motion

5 3a.

Brownian Motion and its Role of a Basic Process

1. In a constructive defiriit,ion of random sequences h, = (h,,) describing the dy-

S,, Sn- 1 at time n ) , one usually assumes, both in linear and nonlinear models, that we have some basic sequence E = ( E ~ ) which , is the ‘carrier’ of randomness and which is assumed to be white ( G a u s s i a n ) noise. generates 11 = ( h l L ) Usually, . E = The choice of such a sequence E = ( E ~ )as a basis reflects the natural wish for building ‘complex’ objects (such as, in general, the variables hn) from ‘simple’ bricks. ~ ) indeed be considered ‘simple’, for it consists of indeThe sequence E = ( E ~ can p e n d m t iden,tically distrihut~drandom, variables with, classical n,ormd ( G a u s s i a n ) distTiblLtio71 “ 0 , l ) . narriics of t,he ‘returns’ h,, = In __ (corresponding, say, to some stock of price S,

2. In the continuous-time case a similar role in the construction of many models of ‘complex’ structure is played by a B r o w n i a n m o t i o n , introduced as a mathematical concept for the first time by L. Bachelier ([12]: 1900) and A. Einstein ([132], 1905). A rigorous rnat,lierriatical theory of a Brownian motion, as well as the corresponding measure in the function space were constructed by N. Wiener ([476], 1923) and one calls them also a W i e n e r process, and the W i e n e r measure. By Definition 2 in 5 l b , the standard B r o w n i a n m o t i o n B = (Bt)t>O is a cont i n i ~ o u sGaussia,n r a n d o m process with homogeneous independent i n c r e m e n t s such that Bo = 0, EBt = 0, and El?: = t . Its covariance function is EB,Bt = min(s,t). We havc already pointed out several times the self-similarity property of a Brownian motion: for each a > 0 we have Law(B,t: t 3 0) = Law(a1/2Bt;t 3 0)

3 . Models Based on a Brownian Motion

iL&Bat

I t follows from this property that the process

237

)t,o

is also a Brownian

motion. In addition, we mention several other transformations bringing about new processes Edi) (i = 1 , 2 , 3 , 4 ) that are also Brownian motions: B,(') = -Bt;

Bj2)= t B l / t fort > 0 with B f ) = 0; B,(")= Bt+,-B, for s > 0; Bi4) = BT-BT-t < t 6 T , T > 0. A multivariate process B = ( B ' , . . . , B d ) formed by d independent standard Brownian motions Bi = (Bi)t,o, i = 1,.. . , d , is called a d-dimensional standard Brownian rnotron.

for 0

Endowed with a rich structure, Brownian motion can be useful in the construction of various classes of random processes. For instance, Brownian motion plays the role of a 'basic' process in the construction of dzffusion Markov processes X = ( X t ) t & o as solutions of stochastic differential equations

d X t = a ( t ,X t ) d t

+ a ( t ,X t ) dBt,

interpreted in the following (integral) sense:

xt

= Xo

+

1 t

a ( s , X,) d s

for each t > 0. The integral

1

+

1

(1)

t

4.5,

X,) dB,

(2)

t

It =

m,

a(.,Xs)

(3)

involved in this expression is treated as a stochastic It6 integral with respect to the Brownian motion. (We c:onsidcr the issues of stochastic integration and stochastic differential equations below, in 5 3c.) An important position in financial mathematics is occupied by a geometric Broumian m o t i o n S = ( St)t20 satisfying the stochastic differential equation

dSt = S ~ (dUt

+

D

dBt)

(4)

with coefficients u E R arid n > 0. Setting an initial value So iridependcnt of the Brownian motion B = (Bt)t,O we can find the explicit solution

of this equation, which can be also written as

(cf. formula (1) in Chapter 11, la) with

Ht

=

(a

-

$)t

+ aBt.

Chaptcr 111. Stochastic Models. Continuous Time

238

We call thc process H = (Ht)t20 a B r o w n i a n motion with local drifl ( u - a 2 / 2 ) arid diffusion 0 2 . We see from (6) that the local drift characterizes the average rate of change of H = ( H t ) t 2 0 ;the diffusion g2 is often called the diflerentiul variance or (in the literature on finances) the volatility. Probably, it was P. Samuelson ([420], 1965) who understood for the first time the importance of a geometric Brownian motion in the description of price dynamics; he called it also economic Brownian motion. We present now other well-known examples of processes obtainable as solutions of stochastic differential equations (1) with suitably chosen coefficients a ( t ,x) and o(t,m ) . A Brownian bridge X = (Xt)O
dXt =

OPxt ~

T-t

dt+dBt,

0

< t < T,

(7)

where B = (Bt)t>ois a Brownian motion. Using, c.g., It8's foririiila (see 5 3d below) one can verify that the process X = (Xt)O
X t = 01 (1 -

+) + pTt + (T t )i;" 2 -

is a solution of (2) (here we treat the integral as a stochastic integral with respect to a Brownian motion). Since this equation is uniquely soluble (see 3 3e), formula (8) defines a Brownian bridge issuing from the point cy at time t = 0 and arriving at /3 for t = T . For a staiitlard Brownian motion, its autocovariance function p(s, t ) is equal to

st

min(s, t ) , arid for a Brownian bridge it is p ( s , t ) = min(s, t )- -. The corresponding

(

expectation is EXt = cy 1 -

3

-

T

+ p-.t

T

It is easy to verify that for each standard Brownian motion W = ( W t ) t 2 0 ,the process W o = ( W f ) o G t < defined ~ by the formula

t wf= wt -wT T -

(9)

st

has the covariance function p ( s , t ) = niin(s, t)--. Hence the process Y = (&)OGt
yt

= (2 (1

~

);

+ B,t + wt",

(10)

has the same finite-dimensional distributions as x (Xt)O
<

<

3. Models Based on a Brownian Motion

239

An Ornstein-Uhlenbeck process ([466], 1930) is a solution of the (linear) stochastic differential equation

Again, using It6's forrnula we can verify that the process X = ( X t ) t 2 0 with

is the (unique; see 3 3e) solution of (11). If the initial value X O is independent of the Brownian motion has finite second moment, then

B = (Bt)taO and

If X o has a nornial (Gaussian) distribution with EX0 = 0 and DXo = 2 / ( 2 a ) , then X = ( X t ) t 2 0 is a stationary Gaussian process with expectation zero and covariance function p(s,t) =

o2

- e-fflt-81.

2cY

(16)

It should be noted in connection with the Ornstein-Uhlenbeck equation (11) that it is a well-posed version of Langevin's equation ([295], 1908)

describing the evolution of the velocity Vt of a particle of mass m put in a fluid and pushed ahead in the presence of friction (-pQ) by clashes with molecules described by a Brownian motion. Written as in (17), this equation has in general no sense if one understands derivatives in the usual sense, because (for almost all realizations) of a Brownian motion the derivative d B t , / d t is nonexistent (see 3 3b.7 below). However, we can assign a precise meaning to this equation if we treat it as the Ornstein-Uhlenbeck equation

P

d& = --& m

dt

o +dBt, m

240

Chapter 111. Stochastic Models. Continuous Time

which has for VO= 0 a solution that can be described, in accordance with ( l a ) , by the equality

The Bessel process of order N > 1 is, by definition, a process X = ( X t ) t 2 o , governed by the (nonlinear) stochastic differential equation

with initial value X O = ic 3 0, where B = (Bt)t>ois a Brownian motion. (This equation has a --unique-strong solution; see 5 3e). If ct = d, where d = 2,3, . . . , then X can be realized [402] as the radial component R = (Rt)t>oof a d-dimensional Brownian motion

with independent standard Brownian motions Bi = (Bi)t>oand x: 1.e..

+ . . .+xi

= x2-

We discuss several other interesting processes governed by stochastic differential equations in 5 4, in connection with the construction of models describing the dynamics of bond prices P(t, T ) (see Chapter I, 5 lb).

5 3b.

Brownian Motion: a Compendium of Classical Results

1. Brownian motion as a limit of random walks. As described by many authors (see, for instance, [201; p. 2541 or [266; p. 47]), circa 1827, a botanist R. Brown discovered that pollen particles put in a liquid make chaotic, irregular movements. (He described this phenomenon in his pamphlet “A Brief Account of Microscopical Observation . . . ” published in 1828.) These movements were called a Brwwnian motion. They, as became clear later, were brought on by clashes of liquid molecules with the immersed particles. The correspondiiig mathematical model of this physical phenomenon was built by A. Einstein ([132], 1905). However, it should be pointed out for fairness sake that even earlier, in 1900, a similar riiotlel had been built by L. Bachelier [la] in connection with the description of the evolution of stock prices and other financial indexes on the Paris securities market. As pointed out in Chapter I (3 l b ) , a Brownian motion arose in Bachelier’s analysis as a (formal) liinit of simplest random walks. Narriely, let (&)k21 be a sequence of independent identically distributed random (the Bernoulli scheme). variable assuming two values, 3 ~ 1 ,with probabilities

4

3. Models Based on a Brownian Motion

241

We consider the half-axis pS+ = (0, m ) , and for each A process dA) = ( A )) t a o ,

> 0 we construct the

(s,

with piecewise constant trajectories. Starting from the processes dA) we can also construct random processes $A) = ( $ A ) ) t a o with con,tin,uous trajectories by setting

By the riiiilt,idirrierisioiial cen,tml limit theorem (see, e.g., [51; Chapter 81 or [439; Chapter VII, r); 81) we can conclude t h a t for all t l , . . . , t k , k 1, the finite-dimensional distributions Law(St(p),. . . , Stk ( A )) and Law($:', . . . ,$f') converge (weakly)

>

to the fiiiitc-diiiiensiorial distributions Law(Btl,. . . , Bt,), where B = (Bt)t>ois a standard Brownian motion. We can actually say even more; namely,

in the sense of weuk convergence of distributions in the (Skorokhod) spaces D (of riglit-co~it,iiiiioiisfunctions having limits from the left) and the space C (of continuous fiinctions); see. e.g., [39] and [250] for greater detail. 2. Brownian motion as a Markov process. Let (n,9, P) be sonie fixed probability space. Let, B = ( B t ( w ) ) t > o be a Brownian motion defined on this space. By 9;= u ( B s ,s 6 t ) we shall nieari the rr-algebra of events generated by the variables B,, s t ; let

<

S>t

be thc cT-algebra of events observable not only on the interval [0, t ] ,but also 'in the infinitesimal fiit,ure' relative to the instant t . We note that, by contrast to ( 3 ; ) t , o , the family ( 9 ; ) t b o has an iniportarit propert,y of right continuity:

n

s>t

c~L:

= 9;.

(4)

Chapter 111. Stochastic Models. Continuous Time

242

Still, the difference between the a-algebras @ and 9 : is not that essentiaI (in the following sense). Let Jv = { A E 9: P(A) = 0} be the set of zero-probability events in 9.Then the a-algebra a ( 9 : u A)generated by the events in 9; and A’coincides with the m-algebra a(.9:,”u A) generated by the events in 9;and Jv:

a ( q u Jv) = a(@

u Jv).

(5)

This is an argument in favor of the introduction of another family of a-algebras 9 t = “(9; U Jv) 5 a ( @ U A)such that each a-algebra is clearly completed with all sets of probability zero, and the whole family is right-continuous, i.e., 9 t= 9,.(One says that the corresponding basis (Q,9, ( 9 t ) t > o , P) satisfies usual conditions; see subsection 3 below.) Assume that T > 0 and let g ( T ) = ( B t ( T ; w ) ) t > obe the process constructed from the Brownian niotiori B = ( B t ( w ) ) t 2 0by the formula ( 9 t ) t 2 0 with

n,

I

& ( T ; w ) = Bt+T(w) - B T ( W ) . We already mentioned in 5 3a.2 that 1) B ( T ) is also a Brownian motion. - Moreover, it is easy to show that 2) the a-algebras 9 : = a(B,, s 6 T ) and 9k(T)= o(E,(T),s2 0) are independent. (As usual, we prove first that the events in the corresponding cylindrical algebras are independent and then use the ‘monotonic classes method’; see, e.g., [439; Chapter 11, a].) It is the combination of these properties that one often calls the Marlcov property of the Brownian motion (see, e.g., [288; ChapterII]) inferring from it: say, the standard Markov property of the independence of the ‘future’ and the ‘past’ for any fixed ‘present’. Namely, i f f = f ( x ) is a bounded Bore1 function and a(&) is the a-algebra ge,nerated by B T , then for each t > 0 we have

E ( f ( B T + t ) 1 g$)= E ( f (BT+t)I a ( B T ) )

(P-a.s.).

(6)

This analytic form of the Markov property leaves place for various generalizaand bounded tions. For instance, one can consider the a-algebra 9~in place of 9$, trajectory function& f (BT+t,t 3 0) in place of f ( B ~ + t )See, . e.g., [123] and [126] for greater detail. The following generalization, which brings one to the strong Marlcou property, relates to the extension of the above Markov properties to the case when, in place of (deterministic) time T, one considers random Markov times T = T ( w ) . To this end we assume that T = T ( W ) is a finite Markov time (relative to the

flow (.%)t>O). By analogy with B ( T ) we now consider the process B ( T )= ( B t ( T ( w ) ; w ) ) t > o , where Bt(.(w);w) = B,+,(,)(4- B,(u)W (7) The most elementary version of the strong Markov property requires that the process g ( r ) be also a Brownian motion and the a-algebras .9T(see Definition 2 in Chapter 11, 3 lb) and @ & ( T ) = o ( $ ~ ( T )U Jv) be independent.

3 . Models Based on a Brownian Motion

243

Tlie analytic property (6) has the following, perfectly natural, generalization: E(f(BT(w)+t(w))

1

t9T)

= E(f(BT(u)+t(w))

I

o(BT))

(8)

(As regards generalizations of this property from the case of a function f to functionals, see, e.g., the books [123], [126], and [ass].) 3. Brownian motion and square integrable martingales. Tlie following properties follow immediately from the definition of the Brownian motion B = (Bt)t>o: for t 3 0

Bt is .Ft-measurable, El%

E(Bt 1 9,s) = B,

(9)

< 00, (P-3,s.) for

s

< t.

(10) (11)

These tjhree properties are precisely the ones from the definition of a martin.gale B = (Bt)t>owith respect t o the flow of c-algebras (.9t) and the probability measure P (cf. Definition 2 in Chapter 11, 5 lc). Further, since E(Bt2 - B,2 I9,) = t - S, (12) the process (B; - t ) t 2 0 is also a martingale. Now let B = (Bt)t>obe some process satisfying (9)-(12). This is a remarkable fact that, in effect, these properties unambiguously specify the probabilistic structure of this process. Namely, let (0,9, ( 9 t ) t 2 0 , P) be a filtered probability space with flow of o-algebras (.Ft)t>o satisfying the usual conditions ([250; Chapter I. 3 11) of right-continuity and corripleteness with respect t o the measure P. (We point out that the 9 t here are not necessarily the same as the earlier introduced algebra o(9: u .A').) Each process B = (Bt)tao with properties (9)-(11) is called a martingale. To emphasize the property of measurability with respect t o the flow ( 9 t ) ~ and o the measure P. one often writes B = ( B t ,9t)or B = ( B t ,9t, P). (Cf. definitions in Chapter II? 5 l c for the discrete-time case.)

THEOREM (P. Lkvy, [298]). Let B = ( B t , 9 t ) t 2 0 be a continuous square inte( S t ) t > o ,P). Assume grable martingale defined on a filtered probability space (0,9, that (12) holds, i.e., (B: - t , 9 t ) t > o is also a martingale. Then B = ( B t ) t g o is a st an ciard BroW Riaii I I I o tion. See the proof in

5 5c below.

4. Wald's identities. Convergence and optional stopping theorems for

uniformly integrable martingales. For a Brownian motion we have

EBt = O

and

EB; = t .

In many problerns of stochastic calculus it is often necessary to find EB, and EB: for Markov times 7 (with respect to the flow ( 9 t ) t y , ) .

Chapter 111. Stochastic Models. Continuous Time

244

Thc following relations are extended versions of Wald’s adentztzes for B (&, S t ) t > o :

=

* EB,=0,

E&<w

+ EB?=Er

Er <m

<

In the particular case when T is a bounded Markov time ( P ( r c ) = 1 for some constant c > O), the equalities EB, = 0 and EB; = E r are ininicdiate consequences of the following result.

THEOREM (J. Doob, [log]). Let X = ( X t ,.Ft)t>o be a uniformly integrable martingale (i.e., a martingale such that s u p E ( / X t / I ( \ X t I> N ) ) + 0 as N + m ) . Then t

1) there exists a n integrable random variable X,

xt + x, ElXt

as t

such that

(P-as.),

- lX ,

+0

+ 00 and E(X,

1st)=

for all t 3 0 ; 2) for all Mxkov timcs a and

7

Xt

(P-as.)

we have

X , A ~= E(X, I cF,) (P-a.s.), where

7

A

~7=

rriin(7, a )

(Cf. Dooh’s c o n ~ i e u ~ e n carid e optional stopping theorems in the discrete-time case in Chapter V, fj3a.) 5. Stochastic exponential. In Chapter 11, 5 l a we gave the definition of a stochastic expomntinl 8(N ) t for processes H = (Ht)t>o that are semimartingales. In the case of H t = A& this stochastic exponential E ( X B ) t can be defined by the equality A

€(XB)t

A

eXBt-Gt.

1

It imniediately follows from It6’s formula stochastic diffcrerit,ial equation

(5 3tl)

dXt = XXt dBt with initial condition Xo = 1. If has the distribution A ( 0 , l ) ,then

<

EeXC-$

= 1.

that X t

(13) G ( X B ) t satisfies the

245

3 . Models Based on a Brownian Motion

By this property arid the self-sirnilarity of a Brownian motion, meaning that Law(ABt) = Law(A&B1), we obtain

In a similar way we can show that for all s


E(@(AB)t 1 9,3) = 8(AB), (P-a.s.). That is, the stochastic exponent &(AB)= ( 8 ( A B ) t ) t > O is a martingale.

(15)

6. Constructions of a Brownian motion. Let E = ( & k ) k 2 o be Gaussian white noise, i.e., a seqiiericc of independent normally (JV(0:1)) distributed random variT (see, e.g., [439; Chapables. Let Hk = H k ( t ) , k 3 0, be the H ~ Ufunctions ter 11, 111) defined on the time interval [0,1], and let

s

& ( t ) = J , ' H k ( S ) ds be the Scha.ude,r f7inctio.n.s. We now sct

n

k=O

It, follows from the results of P. Levy [298] and Z. Ciesielski [76] that the Bin') converge (P-as.) uniform,ly in t E [O, 11 and their P-a.s. continuous limit is a standard Brownian motion An earlier construction of R. Paley antl N. Wiener [374] (1934) is the (uniformly convergent) series

7. Local properties of the trajectories. The following results are well known antl their proofs can he found in many monographs and textbooks (e.g., [124], [245], [2661, [4701). With probability one, the trajectories of a Brownian motion a) satzsfy the Hiilder condition

JBt- B,J < c 11; - sly for each y < f ; b) do not satisfy the Lipschitx condition and therefore are not differentiable at any t > 0; (dB,( = 00. c) have unbounded variation on each interval ( a , 6):

1

(ah)

Chapter 111. Stochastic Models. Continuous Time

246

8. The zeros of the trajectories of a Brownian motion. Let ( B f ( w ) ) t 2be ~ a trajectory of a Brownian rnotiori corresponding to a n elementary outcome w E 0 , arid let %(w) = ( 0 t < 00:& ( w ) = 0}

<

be its zero set. Then we have the following results ([124], [245], [266], [470]): P-almost surely, a) b) c) d)

Lebesgue measure X ( % ( w ) ) is zero; the point, t = 0 is a condensation point of the zeros; there are no isolated zeros in (0, a), so that T ( w ) is dense in itself; the set % ( w ) is closed and unbounded.

9. Behavior at the origin. The Local law of the iterated logarithm states that (P-as.)

By this property, as applied to the Brownian motions (Bt+h- Bt)t>o,we obtain that (P-a.s.) - P t + h - Btl lim =m

4%

hJO

for each t 3 0, i.e., the Lipschitz condition, as already pointed out, fails on Brownian trajectories. 10. The modulus of continuity is a geometrically transparent measure for the oscillations of fiinctioiis, trajectories, and so on. A well-known result of P. L6vy [298] about the modulus ofcont,inuity of the trajectories of a Brownian motion states that, with probability one,

max

- O<s
lirn

-

BsI = 1.

JxGqipJ

hLO

11. Behavior as t

IBt

+ co. With probability Bt

-~

+0

t (Strong law of large numbers). Moreover, Bt

m+0

one,

as

t-

t

+ 30

as

(P-a.s.),

but lirn lBtl

-

~

t+a

v4

=

cx)

(P-as.).

3. Models Based on a Brownian Motion

t

247

The precise asymptotic behavior of the trajectories of a Brownian motion as described by the Law of the iterated logarithm. Namely,

+ x,can be

12. Quadratic variation. Although the trajectories of a Brownian motion have (P-as.) unbounded variation, i.e., ha,bjIdR,l = m, we can assert that - b - Q in a certain sense. jIa.b)

ldBS

-

The corresponding result, which plays a key role in many issues of stochastic calculus (e.g., in the proof of It6's formula; 5 3d). can be stated as follows. Let T ( " ) = (tI;"', . . . , tkn ( n )) be a partitioning of the interval [ u, b] such that

Let

Then we have:

a) if llT(TL)\l t 0 as

TL

+ 00, then

00

b) if

C llT(")il < 00, then we have convergence in (18) with probability one; ri=1

c) if B ( l ) and B ( 2 )are two independent Brownian motions and IIT(n)lIi 0 as 'ri -+ 00, then

In a symbolic form, relations (18) and (19) are often written as follows:

248

Chapter 111. Stochastic Models. Continuous Time

13. Passage times of levels. a) Assiinie that Bt = u } . Then it is clear that

> 0 and let T,

(z

P(T, < t ) = P (sup B, > a > . s
=

inf{t 3 0 :

(21)

Using the reflection principle of D. Andri. we see that P(T,

< t ) = 2P(& 3 a ) .

(See, e.g., [124], [266], arid [439].) Since P(Bt 3 a ) = it follows that P(T,

< t)=

e

~

.i'ta O

therefore the distribution density p,(t) =

J

i

_2t

(22)

dx,

a2

e-23 ds,

S

i)P(T, < t )

at

(24)

.

is defined by the formula

Hence, in particular, P(TcL< x)= 1, ET, = ix), and the corresponding Laplace transform is E~-AT" = e-~fl (26) It should be noted that T = (T,),>o is a process with independent homogeneous increments (by the strong Markov property of a Brownian motion). Moreover, this is a stable process with parameter cr = +, i x . , Law(T,) = ~ a w ( a ~ (cf. ~ 13)la.5). 1)) Assunic that a > 0 arid let S, = inf{t 3 0: lBtl = a } . We claim that ES, = a2 and

<

By Wald's identity, EBiaAt= E(S, A t ) for each t > 0, therefore E(S, A t ) a 2 . Hence ES, = lini E(S, A t ) a2 by the monotone convergence theorem. (Of t+US course, it follows from this that S, < cx, with probability one.) On the other hand, ES, < co, therefore, using Wald's identity again we obtain that EB;, = ES,. Bearing in mind that B;, = a we see that ES, = a2. Bearing in mind that B& = a we sce therefore that ES, = u2. To prove (27) we now consider the martingale X ( " ) = ( X t A s , , st), where

<

3. Models Based on a Brownian Motion

249

<

Since IBtr,~,I a, this is a uniformly integrable martingale and, by Doob’s theorem in subsection 4,

EXtAs, = 1.

(29)

By the theorem on dominated convergence, we can pass to the limit as t in this equalit,y to obtain EXs, = 1,

i 00

i.e.,

Since P(S, < CQ) = 1 arid since P(Bs, = a ) = P(Bs, = - a ) = by symmetry reasons, the required equality (27) follows by (30). c) Let Ta,b = irif{t: Bt = a + b t } , a > 0. If b 0, then P(Ta.b < m) = 1. Using

<

1

the fact that thr process ( e o B t - g t t y l is a martingale, and setting 0 = b + d F + x we can find the Laplace transform

By this formula or directly from Wald’s identity 0 = EBT, obtain a ET,b=--. b

(= a

(Using the trick described after forrriula (27) we can prove that ET,,b If b

> 0, then we consider the martingale 2bBt

-

w} < 2

< ca.)

( e H B t - g t ) t 2 o with 0 = 2b to obtain

u p { 26(a

+ bt)

-

so that this martingale is unifornily integrable. Hence

therefore if a > 0 arid b

+ bET,,b) we

> 0 (or a < 0 and b < O), then

2

Chapter 111. Stochastic Models. Continuous Time

250

14. Maximal inequalities. Let B = (Bt)t>O be a Brownian motion. Then for X > 0, p 3 1, and a finite Markov time T we have

and

for p

> 1.

In particular, if p = 2, then

ErnaxB; t
< 4EB+.

(36)

Inequalities ( 3 3 ) and (35) are called the Kolmogorov-Doob inequalitzes, while (34) and (36) are called Doob’s znequalztzes (see, e.g., [log], [110], [124], [303], [304], or [402]). By (36) we obtain EmaxlBtl t
<2

a

(37)

.

If ET < m , then EB$ = ET, so that

<

~ n l a x j ~ a@. ~l t
As shown in [116], one can refine inequality (38); actually,

where the constant fi is best possible. Setting T = 1 in (39) we obtain

Of course, it would be intcrestirlg to find the precise value of Emax 1Bt1. t
The following argument shows that

251

3. Models Eased on a Brownian Motion

By the self-siniilarity of a Brownian motion, with S1 = inf{t 3 0: jBtl = l}, we find {sup /Btl t
< x}

{sup t
=

1 -

x

IBt 1

< 1} = {sup t
lBt,22

I ,< 1)

sup l B t l g 1 } = { s l > 2 1} = { z < x1}

it
Further, since

for each o > 0 by the properties of the riorrrial integral, setting o = from (27) that

1 ~

we see

which proves the required relation (41).

5 3c.

Stochastic Integration with respect to a Brownian Motion

1. Classical analysis has at its disposal various approaches to the ‘operation of integration’, which bring about such (generally speaking, different) concepts as Riemann, Lebesgue, Rierriann-Stieltjes, Lebesgue-Stieltjes, Denjoy, and other integrals (see A. N. Kolrnogorov’s paper “A study of the concept of integral” in [277]). In stochastic analysis one also considers various approaches to integration of random functions with respect to stochastic processes, stochastic measures, and so on, which brings about, various construction of ‘stochastic integrals’. Apparently, N. Wiener was the first to define the stochastic integral

for smooth deterministic functions f by parts’ (see [375] and [476].)

=

f ( s ) , s 3 0, using the idea of ‘integration

252

Chapter 111. Stochastic Models. Continuous Time

Namely, starting from the ‘natural’ formula d(fB) = fdB definition I t ( f )= f ( t ) B t-

It

f’(S)Bs d s ,

+ Bdf, one sets by (2)

f ’ ( s ) B , d s is treated as the trajectory-wise (i.e., for each ht 12) Ricrnariri integral of the continuous functions f’(s)B,(w), s 3 0.

where the integral w E

2. In 1944, K. It6 [244] made a significant step forward in the extension of the concept of a ’stochastic inkgral’ and laid on this way the foundations of modern stochastic calc?~lu.s,a powerful and efficient tool of the investigation of stochastic processes. ItB’s construction is as follows. Let, (Q, 9, ( 9 t ) t > o ,P) be a filtered probability space satisfying the usual conditions (see 3 3b.2 and details in [250]). Let B = ( B t ,9:t)t>o be a standard Brownian motion and let f = ( f ( t , ~ ) ) ~ be > ~ a random , ~ ~ n function that is measurable with respect to ( t ,w ) anti nonanticipating (independent of the ‘futiire’), i.e.,

f ( t , w ) is 9t-rneasurable for each t 3 0. Such functions f = f ( t , u )are also said to be adapted (to the family of a-algebras ( , Y t ) t > ~ ) . Exarriples are provided by ele,men,tary functions f ( t , W ) = YkJ)I{O)(t),

(3)

where Y ( w ) is a .$o-measurable random variable. Another example is the function

<

(also said to be c l e m m t a r y ) , where 0 7’ < s arid Y ( w ) is a 9‘,-measurable random variable. For fiirictions of type ( 3 ) ,concentrated’ a t t = 0 (with respect to the time variablc) the ‘natural’ value of the stochastic integral

is zero.

If f is a function of type (4), then the ‘natural’ value of I t ( f ) is

Y(u)[BsAt- & A t ] .

For a simple function

3 . Models Based on a Brownian Motion

253

that is a linear combination of elementary functions we set by definition

Rernmrk 1. We point out that one need not assume that B = (Bt)t>ois a Brownian motion in the definition of such integrals of elementary functions. Any process can play this rolc. The pcciiliarities of a Brownian motion, which is the object of our current interests, become essential if one wants t o define a 'stochastic integral' with sirnple properties for a broader store of functions f = f ( t . w),riot merely for elementary ones and their linear combinations, simple functions. Let us now agree to treat integrals (usual or stochastic) as integrals J over the set ( 0 ,t ] . Since thc Y , ( w ) arc 9rt-measurable, it follows that

(04

In a similar way,

Hence if f = f ( t , w)is a simple function, then E

arid

it

f ( s , w)d B s =

0

(7)

t

E ( . / i ; t f ( s , ~ ) d B ~ )=' E L f 2 ( s , w ) d s . In a more coiripact form,

3. Wc shall now describe the classes of functions f = f ( t , ~to) which one can extend 'stochastic integration' preserving 'natural' properties (e.g., (7) and (8)). We shall assume that all the functions f = f ( t , u ) under consideration are defined on Rt x R and are nonanticipating. If

for each t > 0, tlien we say that f belongs t o the class 51.

Chapter 111. Stochastic Models. Continuous Time

254

If, in addition, E

it

f 2 ( s , w)ds

< 00

(10)

for all t > 0, then f is said to belong to J2. It is for these function classes that It6 [244] gave his ‘natural’ definition of a stochastic integral 1,( f ) based on the following observations. Since the trajectories of a Brownian motion (P-as.) are of unbounded variation t

(see 8 3b.7), the integral I t ( f ) = f ( s , w ) dB, cannot be defined as the trajectorywise Lebesgiie-Stieltjes integral. It6’s idea was to define it as the limit (in a suitable probabilistic sense) of integrals I t ( f r L )of simple functions f n , n 3 1, approximating the integrand f . It turned out (see [244] or, for greater detail, [303; Chapter 41) that if f E 5 2 , then there exists a sequence of simple functions fn = f n ( t , w ) such that

for each t 3 0. Hence if f E J 2 , then

as m , n + m. By (8) we obtain the isometry relation

Taken together with ( l a ) , it shows that the random variables { I t ( f n ) } T L 2form 1 a Cauchy sequence in the sense of convergence in mean square (i.e., in L 2 ) . Hence, by the Cauchy criterion for L2-convergence (see, e.g., [439; Chapter 11, lo]) there exists a random variable in L 2 , which we denote by I t ( f ) ,such that I t ( f ) = l.i.ni.It(fn),

i.e., ~ [ ~ t (-f 1) t ( f n ) 1 2+ 0

as

n + 00

This limit, I t ( f ) , which is easily seen to be independent of the choice of an apt

proximating sequence ( f n ) n 2 1 , is also denoted by f ( s , w ) dB, and is called the stochastic integral of the (nonanticipating) function f = f ( s , w)with respect to the Brownian motion B = (B,),>o over the time interval (0, t ] .

3. Models Based on a Brownian Motion

255

We now discuss the properties of the integrals I t ( f ) ( t > 0) so defined for functions f E 5 2 ; the details can be found, e.g., in [123], [250], [ass],or [303]. (a) If f , g E 52 and a and b are constants, then

(b) One can choose modifications of the variables I t ( f ) , t > 0, coordinated so that I ( f ) = ( I t ( f ) ) t > owith l o ( f ) = 0 is a continuous stochastic process (this is precisely the modification we shall consider throughout). Moreover,

(c) If

7

= ~ ( wis)a Markov time such that

T(W)

6 T , then

I T ( f )= h - ( f q O , T ] ) >

(15)

where, by definition, I T ( f )= lT(w)(f). (d) The process I ( f ) = ( I t ( f ) ) t 2 ois a square integrable martingale, i.e.,

I t ( f ) are .Ft-measurable, t > 0;

EI?(f) < 00, t > 0; E(W)

1 9 3 )

=Is(f).

In addition, if f ,y E 5 2 , then

Remark 2. By analogy with the notation used for discrete time (see Definition 7 in Chapter 11, 5 l c ) , one often denotes I t ( f ) also by ( f . B ) t (cf. [250; Chapter I, 3 4dl). We now proceed to the definition of stochastic integrals I t ( f ) for functions in the class 51;we refer to [303;Chapter 4, $41 for more detail. Let f E 51, i.e., let P

(SkE

sequence of functions f ( " )

for each t Since

f2(s,w)ds

.J2, 711

<

00

1

= 1, t

> 0. Then there exists a

3 1, such that

P > 0 (the symbol '+' indicates convergence in probability).

Chapter 111. Stochastic Models. Continuous Time

256

for

E

> 0 ant1 6 > 0, letting first 'm,71 t cc and then E -1 0 we obtain

for each S > 0. Herice t,he sequence I t ( f ( " ) ) , m 3 1, is f u n d a m e n t a l in probability, and by the corresponding Cauchy criterion ([439; Chapter 11, 0 101) there exists a random variable It ( f ) siich that

The variable I t ( f ) (denoted also by (f . B ) t ,

1

f ( s , u )dB,,or

t

f ( s ,u ) d B 3 ) called thr stochastic integral of f over (0, t ] . is Wc now mention several properties of tlie stochastic integrals I t ( f ) ;t > 0 , for f E J1. It can he shown that, again, we can define the stochastic integrals I t ( f ) , in a coordinated nianiier for different t > 0 so that the process I ( f ) = ( I t ( f ) ) t > ohas (P-as.) continuous trajectories. The above-rrientioncd properties (a), (b), ( c ) holding in the case o f f E Jz also persist for f in J1. However, (d) does not hold any more in general. It can be replaced now by the following property: (d') for f E J1 tlir process I ( f ) = ( I t ( f ) ) t z ois a local martingale, i.e., there exists a sequence of Markov times such that r, 30 as n + oc and the 'stopped' processes

I T T L (= f) are martingales for each

TL

(OJI

(ItA~~(f))t>o

3 1 (cf. Definition 4 in Chapter 11; 5 lc.)

(a,

c!F,P) and let ( 9 t ) t g o be the family of a-algebras generated by this process (see fj3b.2; for more precision we shall also denote 9 t by 9:,t 3 0). The following theorem, based on the concept of a stochastic integral, describes the structure of B r o w n i a n functionals.

4. Let B = (Bt)t>obe a Brownian motion defined on a probability space

THEOREM 1. Let X = X ( w ) be a $$-measurable random variable. 1. If EX2 < 00, then there exists a stochastic process f = (ft(u), 9 p ) t c such ~ that

E

iT

fz(w)d t < 00

(17)

and (P-a.s.) (18)

3 . Models Based on a Brownian Motion

257

2. I f E ( X / < 00, then the representation (18) holds for some process f = ( f t (w) )
,.9y

3. Let X = X ( w ) be a positive random variable (P(X>O)= 1) such that EX

Then there exists a process p = ( p t ( w ) , 9 F ) t 6 ~with P(&Tp:(w)dt S I J C ~that (P-as.)

<

CQ)

< a. = 1

From this theorem we derive the following result or1 the structure of Brownian martingales.

THEOREM 2. 1. Let M = ( M t ,5 F ) t G T be a square integrable martingale. Then there exists a process f = ( j t ( w ) , cyp)t
2. Let h ‘ = ( M t ,3 F ) t C T be a local martingale. Then the representation (21) i holds for some process j = ( f t ( w ) , 9 ? ) t G T satisfying (19). 3. Let M = (&It, be a positive local martingale. Then there exists a

.FF)tG~

process cp = ( p t ( w ) , 3 p ) t < T sucfi tliat p(hTp?(w) d t < m ) = 1 and

The proofs of thest theorems, which are niairily due to J. M. C. Clark [77],but werp also in different versions proved by K. It6 [244] and J. Doob [log], [11O], can be found in many hooks; see, e.g., [266], [303], or [402].

3 3d.

It6 Processes and ItB’s Formula

1. The above dcfiriition of a stochastic integral plays a key role in distinguishing the following important class of stochastic processes. We shall say that a stochastic process X = ( X t) t > odefined on a filtered prohability space (O,.!F, ( 9 t ) t a o ,P) satisfying the usual conditions (see 3 3b.3), is an It6 process if there t3xist two nonariticipating processes a = ( a ( t ,w ) ) t > o and b = ( b ( t ,w ) ) t > 0

Chapter 111. Stochastic Models. Continuous Time

258

such that t

P ( l l a ( s , w ) / d s<

and xt = Xo

+

.6”

.)

a ( s , w)ds

t > 0,

= 1,

+

i”

b ( s ,w ) dB,,

(3)

where B = ( B t ,9 t ) t > O is a Brownian motion and XOis a 90-measurable random variable. For brevity, one uses in the discussion of It6 processes the following (formal) differential notation in place of the integral notation (3):

d X t = a ( t ,w ) d t

+ b ( t ,w ) d B t ;

(4)

here one says that the process X = ( X t ) t > o has the stochastic dzflerential (4). 2. Now let F ( t , x ) be a function from the class C1i2 defined in

i

R+ x R i.e., F is \

dF dF a function with continuous derivatives - -, and a t ’ ax

arid let X = (Xt)t>O

be a process with differential (4). Under these assumptions, as proved by It6, the process F has a stochastic differential and

=

( F ( t ,X t ) ) t > o also

More preciscly, for each t > 0 we have the following It6’ s formula (formula of the change of variables) for F ( t ,Xt):

+

i

[% d F + a ( s , w ) d- F ax

+ -21b 2 ( ~ , d2F ~ ) T ] ds + ax

b” Z~(S,U) dB,.

(6)

(The proof can be found in many places; see, e.g., [123], [250; Chapter I, §4e], or [303;Chapter 4, 3 31.)

3. We also present a generalization of (6) to several dimensions. We assunic that B = (B1,. . . , B d ) is a d-dimensional Brownian motion with independent (one-dimensional) Brownian components B Z = (Bl)t>o,i = 1,.. . , d.

3. Models Based on a Brownian Motion

259

We say that, thc process X = ( X ' , . . . , X d ) with X i = ( X j ) ~isoa d-dimensional It6 process if t,licre exists a vector a = ( a 1 , .. . , a d ) arid a matrix b = I/bijll of order d x d with nonanticipating components ui = a i ( t , w ) and entries bzj = b i j ( t , w ) such that

for t

> 0 and dXf = a y t , w)dt

+

c d

b2J( t .w )dBt"

(7)

j=1

for i

=

1 , .. . , d , or, in the vector notation,

dXt = a ( t ,w)d t

+ b ( t ,w ) d B t ,

where n ( t , w ) = ( a l ( t , w ) ,. . . , a " ( t , w ) ) arid Bt = (B;, . . . , B,d) are column vectors. Now let F ( t ," 1 , . . . , xd) be a continuous function with continuous derivatives

@F dF dF and -; at ' dXi ' dZjdZ,

i , j = 1,.. . , d .

Then we have the following d-dimension,al verszon of It6's formula:

F ( t , X , 1 , . . . , X t ) = F ( O , X & . . . , X,d)

For continuous processes X i = (X;)t>o we shall use the notation ( X i , X j ) = ( ( X ' , X J ) t ) , > , , where d

"+

Chapter 111. Stochastic Models. Continuous Time

260

Then (8) can be rewritten in the following compact form:

F ( t . Xt)

=

+

F ( 0 ,X O )

It

g ( s ,X , ) ds

Using differentials, this can bc rewritten again. as

(here F = F ( t , X t ) ) . It is worth noting that if we formally write (using Taylor‘s formula)

and agree that

(CIB;)~ = dt, d B j dt = 0 , d B i d B j = 0,

i

# j,

then we obtain (10) from (11) because

dX,2 d X l = d(XZ.X’)t.

(15) Remark 1. The formal expressions (15) and (14) can be interpreted quite meaningfully, as symbolically written limit relations (2) and ( 3 ) from the preceding section, 3b. We can also interpret relation (13) in a similar way. For the proof of It6’s formula in several dimensions see, e.g., the monograph [250; Chapter I, 3 4e]. As regards generalizations of this formula to functions F @ C1>2,see, e.g., [I661 and [402]. 4. We now present several examples based on I t 6 3 formula

EXAMPLE I . Let F ( z ) = x2 and let X t

= Bt. Then, by ( l l ) ,we fornially obtain

dB; = 2Bt dBt

+ (dBt)2.

In view of (12), we see that

dB; or, in the integral form,

= 2Bt d B t

+ dt,

3 . Models Based on a Brownian Motion

26 1

EXAMPLE 2. Let F ( x ) = e J and X t = Bt.Then

i.e., bearing in mind (11),

In view of (12), we obtain d F ( t , B t ) = F ( t , Bt) d B t . Considering t,he stochastic expmential

g ( B ) t = eot-+t

(19)

(cf. formula (13) in Chapter 11, 5 l a ) , we see that it has the stochastic differential d Q ( B ) t = G ( B ) ,d B t .

('20)

This relation can be treated as a stochastic differential equation (see 5 3a and further on), with a solution delivered by (19).

5 3e

EXAMPLE 3 . Putting the preceding exarriple in a broader context we consider now the urocess

where b = ( b ( t ,w))t>" is a norlanticipating process with P

(Jilt

Sett irig

h2(s,w)d s

1

1

< 30

t

Xt =

= 1,

1

t > 0.

1

h(s,w)dB,3-

b 2 ( s , w )ds

and F ( z ) = e" we can use Itii's formula (5) to see that the process 2 = (Zt)t>o (the 'Girsiinov exponential') has the stochastic differential

dZt = Ztb(t,W ) dBt.

(22)

Chapter. 111. Stochastic Models. Continuous Time

262

EXAMPLE 4. Let X t

=

Bt and let Yt = t . Then d(Bt t ) = t dBt

+ Bt d t ,

or, in the integral form, Bt t =

sdB,

+

B, ds

(cf. N. Wicner's definition of the stochastic integral

/t s d B , , 0

in $ 3 ~ . 1 .

EXAMPLE 5. Let F ( z l r z 2 )= ~ 1 x 2 and let X1 = (X;)t>o. X 2 = (X?)t,o be two processes having It6 differentials. Then we formally obtain

d ( X i X , 2 ) = x;dX,2 + X,2 d X ; + dX,1 dX,2.

(24)

In particular, if

dXf = a z ( t ,W ) d t

+ bi(t, w ) dBi,

i = 1,2,

+ b 2 ( t ,w ) d B t ,

2

then On the othcr hand, if

dX,Z = a z ( ( fW, ) d t

= 1,2,

that

d(X;X?) = X ; d X ? +X,"dXi +b1(t,w)b2(t,w)dt.

(26)

EXAMPLE 6. Let X = ( X l ,. . . , Xd) be a &dimensional It6 process whose components X z have stochastic differentials (7). Let V = V(z) be a real-valued function of z = (zl, . . . , zd) with continuous second derivatives and let

Then the process ( V ( X t ) ) t , o has the stochastic differential dV(Xt)

dV (LtV)(Xt,w)dt -(Xt)b(t,w)d& dz

+

(we use the matrix notation), where

dV -(Xt)b(t,w)dBt dX

d

= ij=l

dV -(Xt)bij(t,w)dBf.

axi

3. Models Based on a Brownian Motion

EXAMPLE 7 . Let V

=

263

V ( t ,x) be a continuous real-valued function in [0, m) x Rd

av -,av

with continuous derivatives -

at

axi

and

~

a2v

axiaxj

. Also, let

where C = C ( t ,w ) is a nonanticipating function with

5 . Remark 2. Let X differential

=

( X t ) t > o be a diffusion Marlcov process with stochastic

dXt

=~

(X t t, ) d t + b ( t , X t ) d B t ,

where htln(s,X,)l ds < 00,h t b 2 ( s , X , )ds < 00 (P-a.s.), and t > 0 (cf. formulas (1) in 5 3a and (9) in 5 3e). If yt = F ( t , X t ) ,where F = F ( t , x ) E C1i2 and tlF > 0, then Y = (yt)t>O is also a diffusion Markov process with

where

and t , x, and y are related by the equality y = F ( t , x ) . These formulas, which describe the transformation of the (local) characteristics a ( t ,x) and b ( t ,x) of the Markov process X into the (local) characteristics a(t,y) and P ( t , y) of the Markov process Y,have been obtained by A. N. Kolmogorov [280;3 171 as long ago as 1931. They are consequences of Itti’s formula ( 5 ) . It would be natural for that reason to call (31) and ( 3 2 ) the Kolmogorov-It6 formulas.

Chapter 111. Stochastic Models. Continuous Time

264

5 3e.

Stochastic Differential Equations

1. Amorig It6 processes X = ( X t ) t a o with stochastic differentials

d X t = ( ~ (Wt ), d t

+P(t,

W)

dBt,

(1)

a major role is played by processes such that the dependence of the corresponding coefficients n ( t , ~arid ) /3(t,u)on w factors through the process X t ( w ) itself, i.e., 4 t , w ) = a ( t ,X t ( w ) ) ,

P ( t , w ) = b(t‘ X t ( w ) ) ,

(2)

where a = a ( t ,x) and b = b ( t , x) are measurable functions on JR+ x For instance, the process

which is callrd a gcomctric (or economic) Brownian motion (see cordance with Itb’s formula) the stochastic differential dSt = aSt d t

R.

5 3a) has

+ aSt d B t .

(in ac-

(4)

It is easy to verify using the same It6 formula that the process

has the differential

d U , = (1+ nU,) dt

+ oU, dBt.

(6) (This process, Y : (Yt)tao, plays an important role in problems of the quickest detection of changes iri the local drift of a Brownian motion; see [440], [441].) If dB, (7) zt = st (Cl - ““2)

ii;”E+“2li; x]

[z,+

for

SOIIK

constants cl and car then, using Itd’s formula again, we can verify that

dZt = (

+

~ 1 a Z t ) dt

+ ( ~ +2 UZt) d B t .

(8)

In the above examples we started from ‘explicit’ formulas for the processes S = ( S t ) ,Y = (yt), and 2 = ( Z t ) arid found their stochastic differentials (4),( 6 ) , arid (8) using Itd’s formula. However, one can change the standpoint; namely, one can regard (4),( 6 ) ,and (8) as stochastic differential equations with respect to unknown processes S = ( S t ) , Y = (yt), Z = ( Z t ) and can attempt to prove that their solutions ( 3 ) , (5); arid (7) are unique (in one or another sense). Of course, we must assign a precise meaning to the concept of ‘stochastic differential equation’, define its ‘solution’,arid explain what the ‘uniqueness’ of a solution means. The above-considered notion of a stochastic integral will play a key role in our introduction of all these concepts.

265

3 . Models Based on a Brownian Motion

2. Let B = (Bt,.F:t)t>o be a Brownian motion defined on a filtered probability space (stochastic basis) ( 1 2 , 9 , ( 9 t ) t a o ,P) satisfying the usual conditions (5 7a.2). Let a = a ( t ,x) and b = b ( t , x) be measurable functions on R+ x R.

DEFINITION 1. We say that a stochastic differential equation

d X t = ~ ( Xt t, ) d t

+ b ( t , X t ) dBt

(9)

with 9o-nieasiirablc initial condition X o has a continuous strong solution (or simply a solution) X = ( X t ) t a o if for each t > 0

X t are ,8-nieasurable,

arid (P-its.)

xt = xo +

it

n ( s ,X , ) ds

t

+

b ( s ,X,) dB,.

(12)

DEFINITION 2 . We say that, two cont,iniious stochastic processes X = ( X t ) t > o and Y = (Yt)t>oare stochastically indisti,nguishable if P sup IX, L t

for each t

-

Ysl > 0)

=0

> 0.

DEFINITION 3 . We say that a rrieasiirable function f = f ( t , x ) on R+ x R satisfies the local Lipschitz con,dition (with respect to the phase variable x) if for each rt~3 1 there exists a qiiaritity K ( ~ L such ) taliat

/ u ( t , z )- .(t,y)I for all t 3 0 a r i d .z, y such that

+jh(t,X) /:I;/

6

71

-

b(t,y)l

and /?y/

< K(TL)/Z

- 7J

(14)

< ri.

THEOREM 1 (K. It6 [242], [243]; see also, e.g., [123; Chapter 91, [288; Chapter V], or [303; Chapter 41). Assririie that the coefficients a ( t ,x) and b ( t , x) satisfy the local Lipschitz condition and the condition of linear g-rowth In(t,z)I

+ /b(t,z)l < K(l)I4,

(14’)

and that the injtjal value X o is 9o-measurable. Then the stochastic differential equation (9) has a (unique, u p to stochastic indistinfi.uistia~,ilify)continuoils solution X = ( X i ,st),which is a Markov process.

266

Chapter 111. Stochastic Models. Continuous Time

This result can be generalized in various directions: the local Lipschitz conditions can be weakened, the coefficients may depend on w (although in a special way), one can consider the case when the coefficients a = a ( t ,X t ) and b = b ( t , X t ) depend on the ‘past’ (slightly abusing the notation, we write in this case a = a ( t ;X,, s t ) and b = b ( t ; X,, s 6 t ) ) . There also exist generalization to several dimensions, when X = ( X I ,. . . , X d ) is a multivariate process, a = a ( t , x ) is a vector, b = b ( t , x ) is a matrix, and B = ( B 1 , . . , B d ) is a d-dimensional Brownian motion; see, e.g., [123], [288],and [303] on this subject. Of all the generalizations we present here only one, somewhat unexpected result of A . K . Zvonkin [485], which shows that the local Lipschitz condition is not necessary for the existence of a strong solution of a stochastic differential equation

<

d X t = a ( t ,X t ) dt

+ dBt;

(15)

the mere measurability with respect to ( t ,z) and the uniform boundedness of the coefficient a ( t ,x) suffice. (A multidimensional generalization of this result was proved by A. Yu. Veretennikov [471].) Hence, for example, the stochastic differential equation

with ‘bad’ coefficient

a(x) =

{

1, x 3 0, -1, x < 0,

has a strong solution (moreover, a unique one). Note, however, that if in place of (16) we consider the equation

d X t = a(&)

dBt,

xo = 0

(18)

with the same o(x),then the situation changes drastically, because, first, this equation is known to have at least two strong solutions on some probability spaces. Second, this equation has no strong solutions whatsoever on some other probability spaces. To prove the first assertion we consider a coordinate Wiener process W = (Wt)t>o in the space of continuous functions w = (wt)t>O on [0, +oo) endowed with Wiener measure, i.e., a process defined as W t ( w ) = w t , t 3 0. By LCvy’s theorem (see $3b.3), if

then B = (Bt)t>ois also a Wiener process ( a Brownian motion). Moreover, it is easy to see that

3. Models Based on a Brownian Motion

267

because 02(z)= 1. Hence the process W = (Wt)t>o is a solution of equation (18) (in our coordinate probability space) with a specially designed Brownian motion B. However, c r - x ) = -o(x),SO that

i.e., the process -W = (-Wt)t>o is also a solution of (18). As regards the second assertion, assume that the equation

has a strong solution (with respect to the flow of cr-algebras ( 9 F ) t a o generated by a Browniari motion B ) . By Levy’s theorem, this process X = ( X t ,9 F ) t > o is a Browniari motion. By Tanaka’s formula (see 5 5c or [402] and compare with the example in Chapter 11, 3 lb)

where 1

rt

is the local tim.e (P. Levy) of the Brownian motion X at the origin over the period [O, tl. Hence (P-as.) t o ( X s )d X s = Ixtl - Lt(O), Bt =

so that 9; C .:7iX‘. The above assumption that X is adapted to the flow g B = ( Y F ) t > o ensures the inclusion 9 : C 9jx1, which, of course, cannot occur for a Brownian motion X . All this shows that an equation does not necessarily have a solution for a n arbztrarily chosen probability space and a n arbitrary Brownian motion. (M. Barlow [20] showed that (18) does not necessarily have a strong solution even in the case of a bounded continuous function o = o ( X ) > 0.)

Chapter 111. Stochastic Models. Continuous Time

268

3. Note that, in fact, the two above-obtained solutions of (18), W arid - W , have the same distribution, i.e., Law(W,, s 3 0) = Law(-VV3, s

0).

This can be regarded as a justification for our introducing below the concept of weak soliitioris of stochastic differential equations.

DEFINITION 4. Lct p be a probability Bore1 measure on the real line R. We say that a stochastic differential equation (9) with initial data Xo satisfying the relation Law(X0) = /I has a weak solution if there exist a filtered probability space 9, ( 9 t ) t a 0 , P), a Brownian niotion B = ( B t ,9 t ) t y ) on this space, and a continuous stochastic process X = ( X t ,2Jt)t>o such that Law(X0 1 P) = p and (12) holds (P-a.s.) for each t > 0.

(a,

It should be pointed out that, by contrast to a strong solution, which is sought on a particular filtered probabiiity space endowed with a particular Brownian motion. we do not fix these objects (the probability space and the Brownian motion) in the definition of a weak solution. We only require that they exist. It is clear from the above definitions that one can expect a weak solution to exist under less restrictive conditions on the coefficients of (9). One of the first results in this direction (see [446], [457]) is as follows.

Wc consider a stochastic differential equation d X t = .(X,) d t

+ b ( X t )dBt

(19)

with initial distribution Law(X0) = p such that / z 2 ( l t ' ) p ( d z ) < 30 for some E > 0. If the coefficients n = a(.) and b = b ( z ) are bovnded continuous functions, then equation (19) has a weak solution. If, in addition, b 2 ( x ) > 0 for 2 E R,then we have the uniqueness (in distribution) of the weak solution.

Reninrk 1. In fact, if b ( z ) is bounded, continuous, and nowhere vanishing, then there exists a unique weak solution even if u ( z ) is merely bounded arid mensurable (see [457]). 4. The above results concerning weak solutions can be generalized in various ways: to the rriultidirrieiisional case, to the case of coefficients depending on the past, and so

011.

One of the most transparent results in this direction is based on Girsanou's theorwn on an absolutely continuous change of measure. In view of the importance of this theorem in many other questions, we present it here. (As regards the applications of this theorem in the discrete-time case, see Chapter V, 3 3 3b and 3tl.)

3. Models Based on a Brownian Motion

269

Let (R, 9, ( s t ) t ) O , P) be a filtered probability space and let B = ( B t ,c F t ) t 2 0 , B = ( B ' , . . . , B d ) ,be a d-dimensional Brownian motion. Let also n = ( a t , LFt), a = ( a ' , . . . , a d ) ,be a d-dimensional stochastic process such that

where (lat(12= (a,')2 + . . . + (a$2 and T < cc. We now construct a process 2 = (Zti s:t)t
where d

( a s ,dB,) =

.

.

a; d B i

(=

u; d B S )

k l

is the scalar product.

If

(the Novikou condition; cf. also the corresponding condition in the discrete-time case in Chapter V, 3 3b), then EZT = 1, (22) so that Z = ( Z t ,9 t ) t G T is a uniformly integrable martingale. - Since 2, is positive (P-as.) and (22) holds, we can define a probability measure PT on (0,9+)by setting

Clearly, if PT = P [ 9~is the restriction of the measure P to

s ~then, PT

N

PT.

THEOREM 2 (I. V. Girsanov [183]). Let

-

-

-

Then B = (Bt,.F:t,PT)t
Chapter 111. Stochastic Models. Continuous Time

270

We now consider the question of the existence of a weak solution of the onedimensional stochastic differential equation

d X t = a ( t ,X ) dt

+ dBt,

(23)

where the coefficient a = a ( t , X ) is in general assumed to depend on the 'past' variables X,, s t . (The case of d > 1 can be considered in a similar way.) Let C be the space of continuous functions z = (zt)t>O,zo = 0 ; let %'t =

<

cr(x: z,, s 6 t ) , arid let %' = cr in (C,'G).

L uO

'6't . Also, let Pw be the Wiener measure

)

We say that a functional a = a ( t , x ) , where t E R+ and ic E C , is measurable if it is a measurable rnap from R+ x C into B,and it is said to be progressively m.easurable if, in addition, for each t > 0 and each Bore1 set A we have the inclusion

We shall consider equation (23) for t 6 T with X O = 0 (for simplicity) and shall assume that: a = a ( t , 3:) is a progressively measurable fiinctional,

and

where W = (Wt(3:))t20is the canonical Wiener process ( W t ( z )= xt) and E w ( . ) is averaging with respect to the measure Pw. In accordance with our definition of a weak solution, we must construct a probability space (0,9, ( F t ) t ~P) , and processes X = ( X t ,9 t ) arid B = (Bt, .9t) on it such that, B is a Brownian motion with respect to the measure P and

Xt = (P-as.) for each t w e now set

lt

Q(S,

X ) ds

+ Bt

< T. 62

= C,

9 =%,

.9t = %'t

and define a measure P in 9~by setting PT(dZ) = Z T ( 2 ) P W ( d z ) ,

3. Models Based o n a Brownian Motion

271

where

The process

B t ( x )= CVt(3) -

it

U(S.

W ( X )ds. )

t


on the probability space (n.Y T ,P T ) is a Brownian motion by Girsanov's theorem. Hence. setting X t ( z )= LV~(I)we obtain

.Y~(.E)=

lil'

U(S.

S(Z)) ds

+ Bt(x).

t 6 T.

which precisely proves the existence of a weak solution to the stochastic differential equation (33) (under assumptions ( a d ) and ( 2 5 ) ) .

Remark 2 . Conditions (24) and ( 2 5 ) hold for sure i f l a ( t , x ) ( 6 c for all t 6 T and x E C. Hence equation (23) has a weak solution in this case. We point out. however. that such an equation does not necessarily have a strong solution. as shows an example suggested by B. Tsirelson (see. for instance. [303; 54.41). In this connection we recall that a n equation d X t = a ( t . 2Yt) d t + dBt with coefficient a(t..Yt) dependent only on the 'present' .Yt,rather than on the entire 'past' X,, s 6 t . (as in ( 2 3 ) ) has not merely a weak solution. but a strong one (see subsection 2 above for a description of A. K. Zvonkin's result [485]). 3f. Forward and Backward Kolmogorov's Equations. Probabilistic Representation of Solutions 1. Below we expose several results and methods of the theory of diffusion Markov processes that have their origin in Al.N. Kolmogorov's fundamental paper .. Uber dze analitzshen !\Jethoden zn der CVar.scheznl2chkeztsrechning" [280] (1931). P. S. Aleksandrov and A . "a. Khintchine [ 5 ] wrote about this work. which revealed close relations of the theory of stochastic processes to mathematical analysis in general and the theory of differential equations (both ordinary and partial) in particular:

-In entzre probabilzty theory of the 20th century zt is difficult t o find another .study as fundamental for the further development of the sczmce . . . . In [280] Kolmogorov does not consider the trajectories of, say, Markov processes studies instead the properties of the transition probabilities

.Y = ( X t ) t a o . He

P(s.z:t..-l) = P(.Yt E ..I1

x,= x),

x E R.

A E ,%(R),

i.e., the probabilitv of the event that a trajectory of X arrives in the set A at time t . provided that ,Y,= x at time s.

Chapter 111. Stochastic Models. Continuous Time

272

The startling point for his analysis is the equation P(s, z; U , d y ) P(u, y; t , A )

(1)

(0 6 s < 71. < t ) , whic:h expresses the Markov property. (OIE usually calls (1) the KoI.m,ogoro.~i~~Cha~r~un equation.) Assiiining that there exist densities

f ( s , z ;t , y ) =

d F ( s ,z; t , y)

aY

where

arid limits

arid imposing certain regularity conditions on the functions in question and also thc: condition

with 6 > 0, Kolrnogorov derives from (1) the following backward parubolic dzffereritin1 rqii(itroris (with respcct to z E R and s < t ) for such dzflusion processes (see [280] or, e.g., [170] and [182] for dehil):

He also obtains tlic fortua,rd parabolic equations (with respect t o

?J

E

R and t > s ) :

He discusses the existence, the uniqueness. and the regularity of the solutions to these equations. (We note that equations (7) had been considered before by A. D. Fokkrr [161]and M. Planck [389] in their analysis of diffusion from a physical standpoint .)

3 . Models Based on a Brownian Motion

273

2. The theory and the methods of Markov processes made the next major step in the 1940s arid the 1950s, in papers by K. It6 [242]-[244], who sought an 'explicit', effective construction of diffusion processes (and also diffusion processes with jumps) with well-defined local characteristics a ( s , x ) and b2(s,x ) (see ( 3 ) and (4)). He siiccceded by the representing the desired processes in terms of some 'basic' Brownian motion B = (Bt)t2o,as solutions of stochastic differential equations

In accordance with [242]-[244] (see also 3e), equation (8) with X O = Const and with coefficients satisfying the local Lipschitz condition and linear growth condition with respect to the phase variable has a strong solution, which, moreover, is unique. If, in addition, the coefficients a ( t ,x ) and b ( t , x ) are continuous in ( t ,z), then X is a Markov diffusion process, which has, in particular, properties (3)-(5). Hence, if we add certain conditions on the smoothness of the transitional densities and the coefficients a ( t ,x ) and b ( t , x ) (see, e.g., 3 14 in [I821 for more detail), then both forward and backward Kolrriogorov equations are satisfied. In a similar way one can treat the case of d-dimensional processes X = ( X l , . . . , X d ) satisfying the equations

Setting

and

(cf. forniula (27) in fj3d) we can write the backward and the forward Kolrriogorou parabolic equatiom in the following form: --

as

-

L ( s , z )f

Chapter 111. Stochastic Models. Continuous Time

274

and

We point out the case when the uz and b2J arc independent of time, i e.,

uz = uz(x) and

bzJ = b Z J ( z )

In this case, for 0 6 s < t we have

f ( ? 2; t , Y) = f (0, x;t

-

s, Y).

Introducing the function g = g ( x ,t ;y) = f ( 0 , x ;t , y) we see from (13) that it satisfies the following parabolic equation with respect to ( t ,z):

3. We proceed now to a discussion of several well-known results showing how one can obtain probabilistic representations (in terms of Brownian motions and solutions of stochastic differential equations) for the solutions of several classical problems of partial differential equations.

A. Cauchy problem Find a continuous function u = u(2,x) in the domain [0, x)x 4 0 ,). = cp().

Bdsuch that (16)

where cp = cp(x) is some fixed function satisfying one of the parabolic equations below (cf. (15)). A l . The heat equation.

where A is the Laplacc operator A =

Cd

i=l

d2

__

ax!

‘

The function

4 2 , ).

=

E,f ( B t )

(17’)

is a solution of the Cauchy problem for (17); it is called the probabilistic solution. By E, in (17’) we mean averaging with respect to the measure P, corresponding to a Brownian motion starting from x; i.e., Bo = z.)

3. Models Based on a Brownian Motion

275

A2. The inhomogeneous heat equation.

3u = -nu+$, 1 at

2

+

where = $ ( t , :I:). The probabilistic solution is as follows:

A3. The Feynman-Kac equation.

8u

-

at

1

= -au+cu,

2

where c = c ( x ) . Its probabilistic solution is

A4. The Cameron-Martin equation.

au at

-

=

1 2

-nu + ( a ,CU),

where a = (a’(.), . . . , a d ( x ) ) and V =

a (dzl

d

Its probabilistic solution is

Rernurk 1. Of course. if one requires that the functions 71(t,x) in (17’)~(20’) be solutions, then one needs first of all that the expectations in these formulas be well defined. To t,his end it suffices, for instance, that the functions p ( z ) ,$ ( t , x ) . c(z), and a(.) be bounded. (These assumptions turn out to be excessively restrictive in many problems; in taliatcase the question should be studied more carefully.) Remurk 2. In our disciission of ‘probabilistic solutions’ we do not mean solutions in some special ‘probabilistic’ class; these solutions are merely defined in probabilistic terms such as averaging with respect to a Wiener measure and so on.

2 76

Chaptrr 111. Storhastic Models. Continuous Time

B. Dirichlet problem One looks for a function 7~ = u ( x ) in the class C 2 in a bounded domain G Rd that satisfies the equation Au=O, XEG (21)

(a liarnioiiic fiinction) aiid tlie condition 142) =

cp(x),

z E

3G.

(22)

Let = inf{t:

T

Bt @ G}.

Then the probabilistic solution is

4.)

=

EZP(B7).

(23)

R e m a r k 3 . Here one must also impose certain conditions on the domain G for T = T ( W ) to be a finite Markov time; it is also necessary that cp(B,) be integrable. 4. Wc now discuss the main steps in the proof that ~ ( x in ) (23) is indeed the

solution of (21) (under several additional assumptions). Let G be a bounded open subset of Rd and let cp = ~ ( 2be ) a bounded fiinction. We shall also assunie that ,u(x) = EZcp(BT) is a C2-function. Then by ItB's formula we obtain (on [O, T [ E { ( w , t ): t < T ( w ) } )

If v satisfies the condition

then the last integral in (24) is a local martingale on [O, By the Markov property of a Brownian motion, EZ((p(BT)

I st)

T[[

(see [250; p. 881).

= v(Bt)

(P-ax) on the set 10, T [ , i.e., ( v ( B t ) )is a martingale

(25) 011

this set.

Hence it follows from (24) that the integral ( h t A v ( B s ) d s ) is a local martingale on 1 0 , ~ " .At the same time it is a continuous process of bounded variation, therefore this is a zero process. (This implication, which is a consequence of the Dooh M e y w decomposition for submartingales, is routinely used when one must verify that a probabilistic solution really satisfies one or another equation; see [25O; Chapter I, $3b] and 55b below.)

3. Models Based on a Brownian Motion

277

Hence we conclude that Aw(x) = 0, z E G, because if this equality fails a t a point zo E G, then Av(z) # 0 in a neighborhood of zo by the continuity of

Au(x), so that, with positive probability, the process (htAv(B,) d s ) on [O,

TI.

t
is nonzero

Remark 4. As regards the unique solvability of Dirichlct problems for elliptic equations and the representation of an arbitrary solution u ( z ) of such a problem as E,cp(B,) see, e.g., [123; 8.51, or [170; vol. 1, Chapter 6, $21. 5 . On the conceptual level, the issue of probabilistic solutions to a Cauchy problem can be treated conceptually much the same way. For exaniple we consider now the heat equation (17); we claim that the function v ( t , x ) = E,cp(Bt) is a solution to this equation (under several additional assumptions) arid satisfies the initial condition u(0,x ) = p(z) Indeed, by the Markov property of a Brownian motion,

E,(f(Bt)

1%)

= vtt

- 3,

Bs).

(26)

Since (E,(f(Bt) I r9,s))s
+

Lt

V v ( t - T , B,) dB,.

(27)

If

for each t , then the last integral in (27) is a local martingale. Hence the process

is indistinguishable (P-as.) from a zero process, because it is both a local martingale and a continuous process of bounded variation. If, in addition cp = cp(x),is a continuous function, then v ( t ,x ) 7' cp(x) as t + 0. In a similar way, using It6's formula, one can prove that. indeed, formulas (17')-(20') describe probabilistic solutions to the problems (17)-(20), respectively.

Remark 5. As regards a niore thorough discussion of the above-stated Cauchy and Dirichlet problems for parabolic equations and also the corresponding problrin for the forward and backward Kolrnogorov equations, see, e.g., [123], [170], [182], and 12881.

4. Diffusion Models of the Evolution

of Interest Rates, Stock and Bond Prices

5 4a.

Stochastic Interest Rates

1. The sirriplest niotlcl where one encounters stochastzc znterest ratps r = (rn),>1, is the model of a bank account B = ( B n ) n 2 0 ,in which (by definition)

If (n,9, (c!K,),)1L20. P) is a stochastic basis ( a filtered probability space) describing the stochastics of a financial market and the available ‘information’ (.9,),20 about this market, then, as already pointed out (Chapter 11, 3 l e ) , it is natural to assume that the state B,, of the bank account is 9,n-1-measurable. Hence both sequences B = (B,),20 and r = ( ~ ~ ~are) predictable ~ 2 l (see Chapter 11, l a ) . This feature explains why one usually imposes the requirement of the predictability of the interest rates r = (r(i!))t20 in the continuous-time case (see $ 5 a and, for greater detail, [250; Chapter I]); this requirement is automatically satisfied if r = ( r ( t ) ) t 2 0is a continuous (or only left-continuous) process. In what follows we consider only models in which the interest rates r = ( r ( t ) ) t a o are diffusion processes and, therefore, have continuous trajectories (so that the additional requirement of predictability becomes superfluous). In the case of continuous time t 3 0 the standard definition of the bank interest rate r = ( r ( t ) ) t 2 0is based on the relation

s

dBt = r ( t ) B t d t , which is a natural continuous analog of (1). Clearly, r ( t ) = (1nBt)’

(2)

(3)

4. Diffusion Models of the Evolution of Interest Rates, Stock arid Bond Prices

279

and

The concept of interest rate (short rate of interest, spot rate, instantaneous interest rate) plays an even more important role in the ‘indirect data’ of the evolution of share prices (see 5 4c below). This explains the existence of a variety of models with interest rates T = (r(t))t>odescribed by diffusion equations

+

d ~ ( t= ) n ( t ,~ ( t dt ) ) b ( t , T ( t ) ) dWt

(5)

or, say, by equations of ‘diffiisiori-with-jurnps’ type

p = p ( d t , drc) is a randorii Poisson measure in Iwt x E x Wd,and v = v(dt,d z ) is its compensator (see [250; Chapter 111, 5 2c] for greater detail).

2. We discuss now several popular models of interest rates T = (r(t))t>ofalling in the class of diffusion niodcls (5) with a standard Wiener process (a Brownian motion) W = (Wt)t>odefined on some stochastic basis ( n , 9 ,(.Ft)t>o,P). The Merton model (R. C . Merton, [346] (1973)):

dr(t)= a d t + y d W t .

(7)

The VusiEek model (0.VasiFek [472] (1977)):

d ~ ( t=) ( a - pT(t))d t

+ 7dWt.

The Dothan model (M. Dothan [ l l l ](1978)): d r ( t ) = CYT(t) dt

+ y r ( t )dWt.

(9)

The Cox-Ingersoll--Ross models (J. C . Cox, J. E. J. Ingersoll, and S. A. Ross [80] (1980) and [81] (1985)): d r ( t ) = p ( ~ ( t ) ) ”dWt, ’~

(10)

and

d r ( t ) = ( a - O r @ ) )d t

+ 7(T(t))1’2

The Ho-Lee model (T. Ho and S. Lee [224] (1986)):

dWt.

(11)

Chapter 111. Stochastic Models. Continuous Time

280

The Black-Dennnn-TOP 7rlOdel (F. Black, E. Derrnan, and W. Toy [42] (1990)): d r ( t ) = a ( t ) r ( t d) t + y(t) dWt. (13) The f f d - W h i t e models (J. Hull and A. White, [234] (1990)):

ddt)=

(4) - P ( t ) r ( t ) )d t + 7 ( t )dWt,

and d r ( t ) = ( a @ )- P ( t ) T ( t ) ) dt

+ y ( t ) ( r ( t ) p d2 W t .

(14) (15)

The Black- Kamsinski model (F. Black and P. Karasinski [43] (1991)):

d r ( t ) = r ( t ) ( a ( t ) p(t) I n r ( t ) ) d t + y ( t ) r ( t )dWt. (16) The Snndmnnn-Son,dermann model (K. Saridrnann arid D. Sondermann [422] (1993)): r ( t ) = ln(1 + E W ) , (17) where @ ( t ) = E ( t ) ( a ( t )dt + 7 ( t ) (18) 3. Introducing these models their authors also explain the motivations behind then1. For instance, the VasiEek model (8) conies naturally with the assumption that the interest rate oscillates near a fixed level a / @ . (It is clear from (8) that the process has a positive drift for ~ ( t<) n/B, and a negative drift for ~ ( t>) a/P; if a = 0, then (8) turns to the Ornstein-Uhlenbeck equation discussed in 3 3a.) It should be noted that, as shown by many empirical studies of bond rates (see, e.g., [69] and [70]), one cannot in general say that interest rates display a trend of returning to some fixed mean value (alp) (mean reversion). This is taken into account in the Hull-White models, in which the variable level a ( t ) / P ( t ) ,t 3 0, replaces the constant cx//3. Moving ever further, we can assume that this variable level is itself a stochastic process. The following model provides a suitable example here. The Cheri nrodd (L. Chen ([70], 1995)): ~

w).

+

h ( t ) = (ff(t)- r ( t ) )dt ( y ( t ) r ( t ) ) l / dw;, 2 wherc a ( t )and y ( t ) are stochastic processes of diffusion type:

dn(t) =

(N -

f f ( t ) )dt

+ ( a ( t ) ) 1 /dW?, 2

+

(19) (20)

d y ( t ) = (y - y ( t ) )d t ( 7 ( t ) p 2 dW?; (21) (liere a a n c i y are constants. wtiile ~ l w2, , and w3 are independent Wiener processes) . In niany of the above models the diffusion coefficient (‘volatility’) depends on the current l e v ~ of l the interest rate r ( t ) . This can be explained, for instance, as follows: if the interest rate strongly increases, then the risks of investing into the assets in questjion are also higher. These risks are reflected by the fluctuation terms (e.g., ( r ( t ) ) l I 2dWt) in the equations for ~ ( t ) .

4. Diffusion Models of the Evolution of Interest Rates, Stock and Bond Prices

281

4. We now present ariotsher (fairly simplified) model of the evolution of interest rates, which was inspired by the following considerations. Plausibly enoiigh, the stochastic process r = ( r ( t ) ) t a ois, to a certain extent, a reflection of the state of t,he ‘economy’, a judgment on this state. Taking this as a starting point, we shall now assume that the state of the ‘economy’ can be siniulatjed by%say, a, homogeneous Markov jump process 8 = (Q(t)),,o that has (for simplicity) only two states, i = 0 and i = 1. Let P(O(0) = 0) = P(O(0) = 1) = and assume that the trarisit,ion probability densities X i j satisfy the equalities X i i = - A arid X i j = X for i # j . Thus, the beco110n1y7switches between the states i = 0 and i = 1, and the time of each stay is distributed exponentially, with parameter A. We also assume that we can judge the state 0 = (O(t)),,o of the ‘economy‘ only by indirect data; narricly. we can observe the process X = (Xt)t>o with differential

dXt

-

O ( t ) dt

=

+dWt,

(22)

-

where W = (Wt)taois some Wiener process. Now let r ( t ) = E(O(t) 1 9 : ) be the optimal (in the niean square sense) estimator for Q ( t )on the basis of the observations X , , s t , (Sf’ = cr(X,, s t ) ) . By general nonlinear filtering theory (see [303: formula (9.86)]),

<

<

d7.(t) = x ( 1 - 2 4 ) ) d t

+r(t)(1

-

r ( t ) ) ( d X t- r ( t )d t ) .

We note that the process W = (Wt)t,o, where rt

is a Wiener process with respect to the flow (9:)t,o (see Chapter VII, fj3b and [303; Theorem 7.121; t,he process W is called an innovation process). Hence (24) can be rewritten as h ( t ) = x(1

~

a+))

dt

+ 7.(t)(I

-T(t))

dWt,

(26)

which is an eqiiation of type (5). It is also interesting that, in view of (22),by (24) we obtain d r ( t ) = a ( r ( t ) ,0 ( t ) )dt b ( r ( t ) )d W t , (27)

+

where

a ( r , O ) = x(1 - 27.) + r ( 1 - r ) ( 8 - T )

(cf. the Chcn motlel (19)).

arid

b(r)= r ( l

~

r)

282

Chapter 111. Stochastic Models. Continuous Time

5 . The above models of the dynamics of interest rates T = (r(t))t>orely on stochastic differential equations that involve some basic Wiener process. Incidentally, many of these equations admit ‘explicit’ solutions that are functionals of this Wiener process. For instance, the solution of the equation (8) in the VasiEek model and of its generalization, equation (14) in the Hiill-White model, has the representation

(because (8) and (14) are linear equations), where

One can see this easily using It6’s formula; cf. formula We now set

T ( t )=

J,”

(g) 2

ds.

Assuming that T ( t )< 00 for all t > 0 arid T ( t )+ 00 as t t 00,we now introduce ‘new’ time 0 by the equality O = T ( t ) (see Chapter IV, 53d for a more detailed discussion of this time as ‘oDerational’). As is well known (see, e.g., [303; Lernma 17.4]), under these assuniptions there exists a (new) Wiener process W* = (W,*)t>osuch that

Hence we obtain the following representation for r ( t ) in (28):

where

If T * ( o )= inf{t: ~ ( t=)O } , then we can pass back from ‘new’ t h e O = T ( t ) to ‘old’ time using the formula t = T * ( Q )This . shows that we can write our process as T * ( O ) = r(T*(O))and the process T * = ( r * ( Q ) ) has ~ o a very simple structure. Namely, T*(O)

=f*(0)

+ g*(O)W,*,

where f*(Q) = f(T*(Q)) arid g * ( Q ) = g(T*(O)).

4. Diffusion Models of the Evo!ution of Interest Rates, Stock and Bond Prices

283

In the Black-Karasinski model (16) we have d l n r ( t ) = ( a ( t )- r2(t)- p(t) l n r ( t ) ) d t

+ y ( t )dWt.

(34)

Defining T ( t )by the same forniiila (30) we see that

r(t)= F ( f N +dtW;(t,) (for some other Wicner process, W * = (W;)t>o)where g ( f ) is as in

(35)

(as),

arid F ( x ) = ez. In a similar way, for the Sandrnanri-Sondermann model (17)-(18) we obtain T(t)

where F ( z ) = ln(1

+ W$(t))>

= F(f(t)

(37)

t + ex), T ( t )=/ r2(s)ds, arid 0

W. Schmidt [42S] has observed that in all the above ‘explicit’ representations the interest, rates r ( t ) have the form (35). This brings one to the following, rather general, model. The Schmidt rriodel (W. Schmidt [426] (1997)):

where W = (Wt)t20is a Wiener process, T ( t )and F ( z ) are continuous nonnegative strictly increasing functions of t 3 0 and z E R,respectively, while f = f ( t ) and g = g ( t ) > O arc continuous functions. Note that ‘new’ time B = T ( t ) in this rnodel is a deterministic function of ‘old’ time t , so that ( X t = f ( t ) g(t)W,(,)) is a Gaussian process. Choosing suitable functions F ( z ) one can obtain various probability distributions of the interest rates r ( t ) .

+

6. The Schmidt ~riodel(38) is also attractive in the following respect: its ‘discrete’ version enables one to build easily dtscrete models of the evolution of interest rates, using one or aiiot her random-walk approximation to a Wiener process. For instance, if

Chapter 111. Stochastic Models. Continuous Time

284

for.

2 1, where i

variables with P(@) ~ ( n=)

= 0 , 1 , . . . arid

(tj’”) is

a sequence of Bernoulli random

= *l/&)

i>then

the (piecewise constant,) process

=

wit11

weakly converges (as n + m) to a Wiener process W = ( W t ) t y ~ . Wr now set

To obtain a discrete-time model of the evolut,ion of the interest rates r:”) ranging in the sets {rj”’(j). j = 0 , f l . . . . , fi},let r t ) = ro and assume that, with probability $, the state ~ i ~ ~ )j ( = j )0,311,. , . . , 3Ii, changes either to rjy\(j to r,i+l(j (71) - 1).

+ 1) or

Clearly, this construction brings one (see [426]) to the binomial model of the evolution of intcrest rates, which can be depicted as follows (for a given value of n = 1 , 2 >. . . ) :

...................................

s4b. Standard Diffusion Model of Stock Prices (Geometric Brownian Motion) and its Generalizations 1. We have already pointed out (Chapter I, 5 l b ) that the first model developed for thc description of the evolution of stock prices S = ( S t ) t 2 0 had been the linear model of L. Bachelier (1900)

st = so + pt + uwt, whcrc W = ( W t ) t y ~was a standard Brownian motion (a Wiener process).

(1)

4. Diffusion Models of the Evolution of Interest Rates, Stock and Bond Prices

285

Although, in principle, it was a major step in the application of probabilistic concepts to the analysis of financial markets, it was clear from the very beginning that the model (1)had many deficiencies. The first of them was that the variables St (that were supposed to represent stock prices) could take negative values. Essential in this connection was the next step, made by P. Samuelson [420]. He suggested to describe stock prices in terms of a geometric (or, as he has put it, economic) Brownian motion

That is, Saniuelson assumed that the logarithms of the prices St (rather then the prices themselves) are governed by a linear model of type (l),so that

(3) By It,6's formula

(5 3d) we immediately obtain that dSt = st ( p d t

+

0

dWt).

(4)

If we rewrite t,his relation in the following (not very rigorous) form

then one can reasonably consider its discrete-time approximation (with time step A; here we set AS, = St - & A ) :

-St-A

-pA+aAWt,

which is very niuch similar to the expression

with 9,,-nieasurable p n , which we have already discussed (see, e.g., Chapter 11, 32a.6) in the description of the evolution of stock prices proceeding in discrete time. It is also instructive to compare (5) with the expression for the (concomitant) bank account B = (Bt)t>owith (fixed) interest rate r > 0, which is governed by the equation d B t = rBt d t . (7)

Chapter 111. Stochastic Models. Continuous Time

286

If the financial market under consideration is formed by a bank account stock of price 5' = (St)t>.o governed by equations ( 7 ) and (4), respectively, then we shall say that we have the standard diffusion ( B ,S ) - m o d e l or a standard diffusion (B. S)-market. This standard diffusion model was considered in the calculations of option prices by F. Black and hl. Scholes [441 and R. Merton [346]. The famous Black-Scholes formula for the rational (fair) price of European call options was invented just in the discussion of this model. (We devote Chapter VIII to these issues.) It is fairly obvious that the standard model is based on assumptions that are not perfectly practical. In fact. it is implied that the interest rate T of the bank account is constant (while it actually fluctuates); the coefficients of volatility and p of growth must also be constant (in fact. they change with time). In the deduction of the Black-Scholes formula one also assumes (see Chapter VIII, 5 5 l b . c below) that the (B. S)-market is .frictionless' (no transaction costs. no dividend payments, no delavs in obtaining information or taking decisions), one can withdraw (or deposit) anv amount from the bank account. and buy or sell any number of shares. A11 this suggests that the standard diffusion (B. S)-market is. of course. a simp1;lication. Nevertheless. it remains one of the most popular models. The following words of F. Black concerning the 'simplicity' of this model ([43]. 1988) are noteworthy in this connection:

B = ( B t ) t 2 0 and

..Yet that weakness zs also zts greatest strength. People lzke the model hecause the71 can easzly understand its assumptzons. T h e model 2s often good as a first approxzmatzon. and zf you can see the holes zn the rissurnptzons you can use the model zn more sophzstzcated ways. '' 2. One of these refinements. immediately suggesting itself. is to consider models w i t h constants I'. p . and o replaced by deterministic or random (adapted to t . 9 t ) t 2 0 ) functions r ( t ) . p ( t ) , and ( ~ ( t ) :

Of course. i n our assessments of more complicated models we must be primarily backed bv facts that have been established experimentally and cannot be explained. 'captured' by the standard (13.S)-model. On the other hand, these more refined models should not becon!. complicated to a point where it is 'no more possible to calculate anvthing'. In this connection we should mention the so-called .smzle eflect'. This is just one fact that the standard (B. S)-model fails to explain; and this failure has brought on various versions of its generalizations and refinements. The smile effect is as Follows.

4. Diffusion Models of the Evolution of Interest Rates, Stock and Bond Prices

287

Assume that the price of a stock is described by equation (4) with SO = 1 (for simplicity) arid let @ = C(g,T ,K ) be the rational price of the standard European call option with payoff f~ = max(ST - K , 0). The Black-~Scholesformula for @ (see formula (9) in Chapter I, 5 l b or, for a more thorough discussion, Chapter VIII, 5 5 l b , c) explicitly describes the dependence of this price on the volatility a,the exercise time T , and the strike price K . However, we can consider the actual market prices of such options with given T and K and cornpare these with the values of @. Let e ( T ,K ) be this (actual) price asked by financial markets. We iiow define l? = Z(T,K ) as the solution of the equation @(a, T ,K ) = e(T, K),

where C(o, T ,K ) is the function given by the Black-Scholes formula. It is precisely this characteristic, Z(T,K ) , which is called the implied volatility, that shows out certain ‘holes’(using the expression of F. Black) in the initial standard model. For these are experimentally established facts that (a) the vahie of Z ( T , K )changes with T for fixed K ; (b) the value of Z ( T , K ) also changes with K for fixed T as a (downwards) convex function (this explains the name ‘smile effect’). To take (a) into account R. Merton proposed ([346], 1973) to regard p and (T in the standard model as functions of time p = p ( t ) arid c = a ( t ) . Schemes of that kind are indeed used in financial analysis, in particular in calculations involving American options. The smile effect, (b) appears more delicate; various modifications of the standard model (‘diffusion with junips’, ‘stochastic volatility’, and others models) have been developed for its explanation. The most trarisparerit of them is the model suggested by B. Dupire [ l a l ] . [122], in which dSt = St(&) d t O(St.t ) d W t ) , (10) where o = a(S,t ) is a function of the state S and time t . In already nientioned papers [la11 and [122] Dupire showed also that accepting the idea of an ‘arbitrage-free complete market’ one can estimate the unknown volatility on the basis of the actual (observed at time t 6 T for the states St = s) prices @,,t(K,T ) of standard European call options with exercise time T and strike price K .

+

3. Conceptually, the process of the refinement of the standard (B,S)-model (4) and (7) has much in coininon with the development of the most simple discretetime models. On agreeing upon this we recall (see Chapter 11, 5 I d ) that in the discrete-time case we started from the representation

s, = SoeH”

(11)

Chapter 111. Stochastic Models. Continuous Time

288

+

with H,, = 161 . . . + lirl arid h, = p n + CT,E,, where prL and al, werc some (nonrandom) numbers and E~~ .N(O,1). If the coefficients p ( t ) and ~ ( tin) (9) are deterministic, then we can write St as follows: St = s o e H t , (12)

-

where

Clearly. Ht has in this case the Gaussian distribution and (12) is a continual analog of the representation (11). Further, in Chapter 11, 3 I d we considered the A R , MA, and ARMA rnodels, in which the volatility cn was assumed t o be constant ( a , = (T = Const), while the p n were defined by the 'past' variables h,-2,. . . and &,-I, ~ ~ - 2 . ,. . Finally. in the ARCH and GARCH models we assumed that h,, = ( T ~ ~ while E ~ ~ , the volatility (T,~, dcpendcd on the 'past' (see, e.g., (19) in Chapter 11, 3 Id). We point out that all tlicse models had a single source of randomness, white ( G a r ~ s s i a nn,oise ) E = ( E ~ ) . As regards the stochastic volatility model (see Chaptcr 11, 3 1d.71, it is assiinicd there that we liavc two sourccs of randomness, two independent white ( Gaussian) n,oi.ses E = ( E ~ ~ ,aiid ) 6 = (&) such that lin = u 7 1 ~ 7where L , C T ~ ,= eian arid

(see Chapter 11, 5 3c). In the sanie way, in the continuous-time case we can consider various countcrparts to ARCH-GARCH models or to models of stochastic volatility kind. In thc sccoritl case we have, e.g., models of the following form:

where At = lno;, while WE = (Wf)t>oand W6 = (W,b')t>oare two independent Wiener processes (cf. (37) arid (38) in Chapter 11, 53a; models of this kind were considered in [235], [364], [432], and [477]). Using the models (15), (16) in the situation when one can observe only the process S = ( S t ) t 2 0 (and not thc volatility a = ( n t ) ~ wc ~ )arrive , at an 'incomplete' market, wlierc one can give no clcar definition of, say, a rational price of an option, so that one mist, resort to a more complicated analysis of upper and lower prices (scc Cliaptcr V, $ Ib).

4. Diffusion Models of the Evolution of Interest Rates, Stock and Bond Prices

289

Diffusion models of ARCH -GARCH type in the continuous-time case seem to be more promising in this respect since one can avail of the well-developed machinery of an ‘arbitrage-free complete’ market. Dupire’s model considered above (10) is one example of such a M a r k o v i a n model. Pursuing the idea of the ‘dependence on the past’ inherent in the ARCHGARCH models it is reasonable t o consider, e.g., the representations

st = sOeHt (17) such that t,lie diffusion process H = ( H t ) t > is ~ a component of a multivariate diflusion process ( H t ,H:, . . . ,H,‘“-l)t>o generated by a single Wiener process. A corresponding example is provided by a stationary Gaussian process with rational spectral density, which can be regarded as a component of a multivariate Markov process satisfying a linear system of stochastic differential equations. (See Theorem 15.4 and the system of equations (15.64) in [303] for details.) 4. We continue the discussion of the diffusion models introduced above and describing the evolution of stock prices in Chapter VII, where we shall consider these models in the context of an ‘arbitrage-free complete market’. We point,ed out in Chapter I that the concept of the absence of arbitrage is precisely the economists’ concept of e f i c i e n t market, t o which we stick in our analysis. In the discrete-time case, the First f u n d a m e n t a l theorem of asset pricing (see Chapter V, 52%) gives one a martingale criterion of whether a particular ( B ,s)-market is arbitrage-free. This property imposes certain constraints on both bank account B = ( B T L ) 7 aand 2 ~ stock price S = ( S n ) n > ~ . In the continuous-time case it is also reasonable t o stay within the realm of arbitrage-free ( B ,5’)-markets. Again, this imposes certain constraints on both dynamics of stock prices and evolution of the bank account (see Chapter VII). It should be noted that (compared with discrete time) the continuous-time case is more delicate. This is primarily due to the technical complexity of the corresponding machinery of stochastic calculus arid the considerable gap between the potentials of continuous and discrete trading. Another property shaping the structure of a (B; 5’)-market is its corr~pleteness (or lack of it,), which means that we can always make up a portfolio with value reproducing our ‘liabilities’ (see details in Chapter V, 3 l b ) . Generally speaking, this desirable property of a market is rather an exception than a rule. It is remarkable, however, that we obtain this property of conipleteness under fairly general assumptions in the case of diffusion (B, s)-markets (see Chapter VII, 3 4a).

5 4c.

Diffusion Models of the Term Structure of Prices in a Family of Bonds

1. We already discussrd bonds as IOU’s and their market prices P(t, T ) in general, in the very beginning of this book (Chapter I, 3 l a ) . We also introduced there

290

Chapter 111. Stochastic Models. Continuous Time

several characteristics of bonds such as the initial price P(O,T),the face value P(T,T ) (which we, for definiteness set to be equal to one), the current interest ,rate, the yield to maturity, and some other. We mentioned that the question on the structure of the prices P ( t , T ) ,0 t < T , regarded as a family of stochastic objects is, in a certain sense, more complicated than the question on the probabilistic structure of stock prices. One difficulty here is that if, for a fixed T we regard P ( t , T ) as a stochastic process for 0 6 t 6 T , then, first of all, this process must be conditional because P ( T , T )= 1. A typical example is a Brownian bridge X = (Xt)tGT considered in §3a and governed by the equation 1dXt = dt dWt

<

xt +

~

T-t It has the property that X t

with X O = Q. + 1 as t t T . In a similar way to Bachelier’s using a linear Brownian motion (see formula (1) in 34b) to sirrnilate stock prices, one could treat the process X = ( X t ) t < as ~ a prospective model of the evolution of the bond prices P(t,T ) ,t 6 T . The complications here are the same as with shares: the variables X t , t > 0, can in general assume any values in R, while for a bond price, by its meaning, we have 0 < P ( t , T )< 1. Another complication that one meets in the construction of models of bond prices relates to the fact that, as a rule, there are bonds with various tinies of maturity T on the market and in investors’ portfolios. Therefore, in sound models of the evolution of bond prices one should not be restricted to the consideration of some particular value of T ; rather, they must be constructed to cover some subset T 2 Iw+ of ternis containing all possible values of the maturity times of the bonds traded on a market. There should also be no opportunities f o r arbitrage on the financial market, i.e., no one should be able to buy a bond and sell it profitably running no risks.

<

2. In our constriiction of models of the t e r m structure of the prices P(t,T ) ,t T , of bonds with maturity date T (we shall call them T-bonds) we shall assiirne that T = R + . In other words, we assume that there exists a (continual) family of bonds with prices P ( t , T ) ,0 6 t 6 T , T E R+. We shall assume in addition that there are no bond interest payments (coupon payments), i.e., we consider only zero-coupon bonds.

3. In dealing with a single T-bond or a f a m i l y of T-bonds, T > 0, one uses several characteristics of bonds that shall be helpful to us over the entire process of model building and analyzing. Namely, let 11s represent the price P(t,T ) by the following expression:

P(t,T ) = e - T ( t > T ) ( T - t ,) t < T , P(t,T ) = e-

:J

f ( t + )d s ,

t 6T,

4. Diffusion Models of the Evolution of Interest Rates, Stock and Bond Prices

291

< < s < T.

with some norincgative fiiiictioris r ( t ,T ) arid f ( t ,s). 0 t Clearly, In P ( t , T) r(t,T) = t < T,

(3)

T-t

arid (provided that P(t, T ) is differentiable with respect t o T , T > t )

<

In the case of one zero-coupon T-boiid, for t T we have defined its yield or t h e yield p(T - t , T ) to the mntu,rity t i m e by the forrniila P ( t . T ) = e -(T--t) l n ( l + p ( T - t , t ) ) (see formula (6) in Chapter I,

5 la).

(5)

Comparing this with (1) we see that

r ( t , T )= I r i ( 1

+ P(T

-

t,t)).

(6)

The quantity r ( t , T )is also often called the yield of the T-bond at time t < T (due for tlie rerriairiirig time T - t ) , and the function t -ri T ( t , T) (t T) is called the yield curoe of the T - b o n d . The quantities r ( t ,T ) are especially useful when one considers various T-bonds with time of maturity T > t . In this case, the function T -+ r ( t , T ) is also called the yield curve of t h e f w n i l y of T - b o n d s (at time t ) . To avoid treating the cases of a single bond arid a bond family separately we shall regard r ( t ,T ) as a function of t w o variables, t and T : assuming t h a t 0 < t 6 T and T > 0. We shall call it the yield again. The quantities f ( t ,T ) are usually called the forward m t e s for the contract sold at time t. A key role in the subsequent analysis is played by the instantaneous rate r ( t ) at time t that can be defined in ternis of the forward interest rate by the equality

<

r ( t )= f ( t , t ) .

(7)

We point out also the following relation between r ( t , T ) and f ( t , T ) : (8)

which is an irrirriediatc consequence of (4) and (1).

3 r ( t ,T ) dT

Hence, if, e g , tlie derivative ____ is finite, then setting T = t we obtain

Cliapt,ex 111. Stochastic Models. Continuous Time

292

4. We consider now the issue of the dynamics of the bond prices P(t, T ) . The two main approaches here are the indirect and the direct ones. (Cf. Chapter I, 5 lb.5.) Taking the first approach, one represents P(t, T ) as the composite

P(t, T ) = F ( t ,7.(t);T ) ,

(10)

where r = ( r ( t ) ) t >is~ some instantaneous rate. In such niotlels the entire term structure of prices is determined by a single factor. r = ( r ( t ) ) t k 0 .For this reason, they are called single-factor models. An important (and treatable by analytic means) subclass of such models is described by fiinctions

F ( t ,T(t)lT ) = e4t>T)--T(t)o(ttT).

(11)

These models are said to be a f i n e , or, sometimes, exponential a f i n e ([117], 5 with some coefficients a ( t ,T ) anti [ j ( t ,T ) . Another wcll-known approach was used in [219]. The corresponding models is called the H.JM-model after its authors (D. Heath, R. Jarrow, and A. Morton). The idea of this approach is to seek the prices P(t, T ) as solutions of certain stochastic differential equations of type [119]) since l n F ( t , x ; T ) = n ( t , T )- zp(t,T ) is a linear function of

d P ( t , T ) = P(t, T ) ( A ( tT, )d t

+ B ( t ,T )d W t ) ,

where A ( T ,T ) = B ( T ,T ) = 0 and P ( T ,T)= 1 (cf. formula (4) in tively, one can consider equations

d f ( t , T ) = a ( t ,T )dt

+ b ( t ,T )d W t

(12)

5 4b).

Alterna(13)

for the forward interest rates f ( t , T ) . This is an appropriate place to recall that, throughout, we assume that we have some filtered probability space (stochastic basis)

(n,9,(.%)t201 p). All the functions in question ( P ( t , T ) ,f ( t , T ) ,A ( t , T ) ,...) are assumed to be 9 t measurable for t T . As usual, W = (Wt,9t)t>ois a standard Wiener process; moreover, we assume that the conditions ensuring the existence of the stochastic integrals in (12) and (13) and the existence of a solution to (12) are satisfied. By relation (2) between the prices P(t, 7‘) and the forward interest rates f ( t , T ) , using Itij’s formula and stochastic Fubini’s theorem (see Lemma 12.5 in [303] or [395]), we can obtain the following relation between the coefficients of the above equations (see [2I9]):

<

4. Diffusion Models of the Evolution of Interest Rates, Stock and Bond Prices

or

1

.T

A ( t ,T) = r ( t ) B ( t ,T ) = -

1’

1(ifT@, s) d s )

a ( t , s) ds + 2

2

,

b ( t , s ) ds.

By (13) we also obtain

R e m a r k 2. In several papers (e.g., in [ 3 8 ] ,[359]. and [421]), in place of the forward interest rate f ( t , T ) ,the authors consider its modification

-

r ( t ,).

= f(t, t

+ .).

They justify this by certain analytic simplification. For instance, after this change of variables equality ( 2 ) acquires a slightly more simple form:

5 . We discuss the structure of bond prices P(t, T), forward rates f ( t , T ) ,and instantaneous r ( t ) in greater detail, in the context of a ‘complete arbitrage-free’ T-bond market, in Chapter VII. Here we point out only that, as with a (B,S)-market (see 5 4b.4), our condition of an aditrage-free T-bond market imposes certain constraints on the structure of coefficients in (12) and (13) and on the coefficients a ( t , T ) ,D ( t , T ) ,and the interest rates in the affine models (11).thus distinguishing some natiiral classes of arbitrage-free models of a bond market.

5. Semimartingale Models

5 5a.

Semimartingales and Stochastic Integrals

1. The abundance of the above-mentioned models designed for the description of the evolution of such financial indexes as, say, the price of a share, poses a natural problem of distinguishing a siifficiently general class of stochastic processes that on the one hand iricliitles many of the processes occurring in these models and, on the other hand, yields to analytic means. From riiany viewpoints, one such class is that of semimartingnles, i.e., of stochastic processes X = ( X t ) t 2 0 representable (not necessarily in a unique way) as sums

Xt

= Xo

+ At + M t ,

(1) where A = (At,.%t)tao is a process of bounded variation ( A E "Y) and M = (Mt,.Ft)t>o is a local martingale (111E d l l o c ) both defined on some filtered probabzlity .space (stochastic basis)

(ft9,( & ) t > O , p) satisfying the usrial conditions, i.e., the a-algebra 9 is P-complete and the .%t, t 3 0, rniist contain all the sets in 9 of P-probability zero (cf. 5 3b.3), and be right =.St+,t 3 0); see, e.g., [250]. continuous (9t We also assume that X = (Xt)t>o is a process adapted to ( 9 t ) t > o arid its trajectories t -+ X t ( w ) , w E R, are right-continuous with limits from the left In French literature a process of this type is called un processus ccidlciy-Coritinu A Droite avec des Liniites Gauche.

A

2. Wc turn to seiriirnartingales for several reasons. First, this class is fairly wide: it contains discrete-time processes X = (X7L)7Lao (for one can associate with such process the continuous-time process X * = ( X * ) t a o with XT = Xit], which clearly belongs to 'U), diffusion processes, diffusion processes with jumps, point processes, processes with independent increments (with several exceptions; see [250; Chapter 11, § 4c]), and many other processes.

5. Semimartingale Models

295

The class of semimartingales is stable with respect to many transformations: absolutely continuous changes of measure, time changes, localization, changes of filtration, and so on (see a discussion in [250; Chapter I, 5 4cl). Second, there exists a well-developed machinery of stochastic calculus of semimartingales, based on the concepts of Markov tames, ,martingales, super- and submartingales, local martingales, optional and predictable a-algebras, and so on. In a certain sense, the crucial factor of the success of stochastic calculus of semimartingales is the fact that it is possible to define stochastic integrals with respect to seminiartingales. Remark 1. One can find an absorbing concise exposition of the central ideas and concepts of the stochastic calculus of semimartingales and stochastic integrals with respect to semimartingales in the appendix to 11391 written by P.-A. Meyer, a pioneer of modern stochastic calculus. An important ingredient of the concept of stochastic bases (S2, (27:t)t20, 9, P), which are definition domains for semimartingales, is a flow of a-algebras ( 9 t ) t 2 0 . As already mentioned in Chapter I, 5 2a, this is also a key concept in financial theory: this is the flow of ‘information’ available on a financial market and underlying the traders’ decisions. It should be noted that, of course, one encounters processes that are not sernimartingales in some ‘natural’ models (from the viewpoint of financial theory), A typical example is a fractional Brownian motion BW = (B?)t>o (see $ 2 ~ ) with an arbitrary Hurst parameter 0 < HI < 1 (except for the case of IHI = 1 / 2 corresponding to a usual Brownian motion). 3. We proceed now to stochastic integration with respect to semimartingales, which nicely describes the growth of capital in self-financing strategies. Let X = (Xt)tao be a semimartingale on a stochastic basis (s2, (9 9 :, t ) t a 0 , P) and let 8 be the class of simple functions, i.e., of functions f ( t > W )=

Y O ( 4 l { O } ( t+ )

cy z ( ~ ) q . & ] ( ~ )

(2)

i

that are linear combinations of finitely many elementary functions

with 9r,-measurable random variables Y , ( w ) ; cf. 3 3c. As with Wiener processes (Brownian motions), a ‘natural’ way to define the stochastac zntegral l t ( f ) of a simple function (2) with respect to a semimartingale X (this integral is also denoted by (f . X ) t ,

t

f ( s , ~d X ) s , or

set

I t ( f )==

%(w)[X3ZAt- Xr,At], 2

where a A b = niin(a, b ) .

/

(OJI

f ( s , w ) d X s )is to (4)

296

Chapter 111. Stochastic Models. Continuous Time

Re,mark 2. The stochastic integral

has a perfectly transparent interpretation in financial theory: if X = (Xt)t>ois, say, the price of a share, and if you are buying x ( w ) shares at time ri (at the price Xrt). then Y , ( w ) [ X S i X r i ] is precisely yoiir profit (which can also be negative) from selling these shares a t time si, when their price is X C T i . We emphasize that,, we need not assume that X is a semimartingale in this definition of the stochastic integrals I ( f ) = ( I t ( f ) ) t > oof simple functions; expression (4) makes sense for a n y process X = ( X t ) t g o . However, our serriirnartingale assumption becomes decisive when we are going to e x t e n d the definition of the integral from simple functions f = f ( t , w ) to broader function classes (with preservation of its ‘natural’ properties). If X = ( X t ) t > o is a Brownian motion, then, in accordance with $ 3 ~we : can define the st,ochastic integral I t ( f ) for each (measurable) function f = ( f ( s , ~ ) ) ~ < t , providd that the f ( s ,w ) are 9:,-measurable and

Tht: key facthr here is that we can a p p r o x i m a t e such functions by simple functions ( f i L , n 3 1) such that the corresponding integrals ( I t ( f n ) ,n 3 1) converge (in probability, at any rate). We have denoted the corresponding limit by I t ( f ) and have called it the stochastic integral o f f with respect to the Brownian motion (over the interval (0, t ] ) . In replacing the Brownian motion by an arbitrary semimartingale, we base the construction of the stochastic integral I t ( f ) again on the approximation of f by simple functions (fn,ri 3 1) with well-defined integrals It ( f T 1 ) and the subsequent passage to the limit as n + 00. However, the problem of approximation is now more complicated, and we need to impose certain constraints on f adapted to the properties of the semimartingale X . 4. As a n illustjration, we present several results that seem appropriate here and have been obtained in the case of X = M , where M = (Mt,.9t)t2o is a square integrable martin.yale in the class ,X2(A4 E X 2 ) Le., , a martingale with

supEM;

< 00.

t>o

Let ( M ) = ( ( M ) t S , t ) t > o be the quadratic characteristic of martingales in X 2 (or in X&), i.e., by definition, a predictable (see Definition 2 below) nondecreasing process such that the process M~ - ( M ) = ( M ; - ( ~ ) Ft t ), t > o is a martingale (see 3 5b and cf. Chapter 11, 3 l b ) .

5. Semimartingale Models

297

The following results are well-known (see, e.g., [303; Chapter 51). A) If the process ( M ) is absolutely continuous ( P - a s . ) with respect t o t: then the set G of sirnplo functions is dense in the space Lf of measurable functions f = f ( t , w)suc:h t,llilt

In other words, for each f in this space there exists a sequence of simple functions ( f 7 l ( t , W ) ) t > O , w E a, 71 3 1, such that

fn =

Note that if B = (Bt)t20 is a standard Brownian motion, then its quadratic characteristic ( B ) t is identically equal to t for t 3 0. B) If the process ( M ) is con,tinvous (P-a.s.), then the set 8 of simple functions is dense in thc space L i of measurable functions f = ( f ( t , w ) ) t > o such that (7) holds and the variables f ( r ( w ) ,w)are .F;,-measurable for each (finite) Markov time 7- = .(w). C) In general, if we do riot additionally ask for the regularity of the trajectories of ( M ) ,then the set 8 of sirriple functions is dense in the space L: of measurable functions f = ( f ( t , w ) , g : t ) ~satisfying " (7) and predictable in the sense that we explain next. First, let X = ( X n ,9 7 1 ) be n 2a ~ stochastic sequence defined on our stochastic basis. Then, in accordance with the standard definition (see Chapter 11, 5 l b ) , by the predictability of X we mean that the variables X , are 9,-1-measurable for all n 3 1. In the corit,inuoiis-tirne case, the following definition of the predictability (with respect to the stochastic basis ( [ I , 9, (.!F:t)t>o, P)) proved to be the most suitable one. Let 9 the sniallest a-algebra in the space R+ x R such that if a (measurable) function Y = ( Y ( t W , ) ) ~ ~ O , + , ~isR 9-measurable for each t 3 0 and its t-trajectories (for each fixed w E 0) are left-continuous, then the map ( t ,w ) -ri Y ( t ,w ) generated by this function is 9-measurable. DEFINITION 1. We call the cr-algebra 9 in sets.

R+ x R the a-algebra

-

of predictable

DEFINITION 2 . We say thnt a stochastic process X = ( X t ( w ) ) t > odefined on a sto( . F F , )P) ~ is ~o predrctabk , if the map ( t , ~ ) X ( t , w ) (= Xt(w)) chastic basis (R,3, i s .!??-measurable.

298

Chapter 111. Stochastic Models. Continuous Time

5. The above results on the approximation of functions f enable one, by analogy with the case of a Brownian motion, to define the stochastic integral

for each M E X 2 by means of an isometry. Then we can define the integrals I t ( f ) for t

> 0 by the formula

It should be pointed out that if we impose no regularity conditions (as in

A) or B)) on ( M ) , then the above discussion shows that for M E X 2 the stochastic integrals I ( f ) = ( I t ( f ) ) t > o are well defined for each predictable bounded function f . The next step in the extension of the definition of the stochastic integral I t ( f ) (or (f . X ) t ; this notation stresses the role of the process X with respect to which the integration is carried out) is to predictable locally bounded functions f and locally square integrable martingales M (martingales in the class XI:,; if X is some class of processes, then we say that a process Y = ( Y t ( w ) ) t a o is in the class XlOc if there exists a sequence (r,),>l of Markov times such that r, t 00 and the 'stopped' processes Y T n= (%ATn.)t2O are in X for each n 3 1; cf. the definition in Chapter 11, 5 lc.) If (7,) is a localizing sequence (for a locally bounded function f and a martingale M E .Ye,:,), then, in accordance with the above construction of stochastic integrals for bounded functions f and M E X 2 ,the integrals f . Mrn = ( ( f. MTn)t)t>oare well-defined. Moreover, it is easy to see that the integrals for distinct n 3 1 are compatible, i.e., (f . Mrn+l)rn = f . Mrn. Hence there exists a (unique, up to stochastic indistinguishability) process f . M = ( ( f . M ) t ) t > osuch that (f . M)rn = f

. Mrn

for each n 3 1. The process f . M = (f . M ) t ,t > 0 so defined belongs to the class 3(e,:, (see [250; Chapter I, 3 4d] for details) and is called the stochastic integral of f with respect to M . 6. The final step in the construction of stochastic integrals f . X for locally bounded predictable functions f = f ( t ,w ) with respect to semimartingales X = ( X t ,9 t ) t > o is based on the following observation concerning the structure of semimartingales. By definition, a semimartingale X is a process representable as ( l ) , where

A = (At,.Ft)taois some process of bounded variation, i.e., LtldA,(w)l < t > 0 and w E R and M = ( M t ,9 t ) is a local martingale.

00, for

5. Semimartingale Models

299

By an important result in general theory of martingales, each local m,artingale M has a (not unique, in general) decomposition

Mt = Mo

+ M; + M:,

t 2 0,

(11)

where M’ = ( M l , 9 t ) t 2 0 and M” = (M;, 9 ) t > o are local martingales, M i = M[ = 0, arid M” has bounded variation (we write this as 111’’ E ”Y), while M’ E Xi& (see [250; Chapter I, Proposition 4.171). Hence each semimartingale X = ( X i ,9 t ) t a o can be represented as a sum

Xt = Xo

+ A: + M i ,

(12)

+

where A‘ = A M” and M‘ E %&. For locally bounded functions f we have the well-defined Lebesgue-Stiel-cjes integrals (for each w E f2)

and if, in addition, f is predictable, then the stochastic integrals ( f . M’)t are also well defined. Thus, we can define the integrals ( f . X ) , in a natural way, by setting

To show that this definition is consistent we must, of course, prove that such an integral is independent (up to stochastic equivalence) of the particular representation ( l a ) , i.e., if, in addition, X t = X O & Mt with 2 E “9‘ and E X&, then f . A’+ f . M’ = f .X+ f .M. (15)

+ +

This is obvious for elementary functions f and, by linearity, holds also for simple functions (f E 8 ) . I f f is predictable and bounded, then it can be approximated by simple functions fil convergent to f pointwise. Using this fact and localization we obtain the required property (15).

Remark 3. As rcgards the details of the proofs of the above results and several other constructions of stochastic integrals with respect to semimartingales, including vector-valued ones, see, e.g., [250]. [248], or [303], and also Chapter VII, 3 l a below.

7. We now dwell on several properties of stochastic integrals of localy bounded functions f with respect to sernirnartingales (see properties (4.33)-(4.37) in [250; Chapter I]): a) f . X is a semimartingale; b) the map f --+ f X is I%n,ear;

Chaptcr 111. Stochastic Models. Continuous Time

300

if X is a local martingale ( X E Aloc)3 then f . X is also a local martingale; for X E ‘V the stochastic integral coincides with the Stieltjes integral; (f . X),= O and f . X = f . (X - X o ) ; A ( f . X ) = f A X , where @X.i = X t - X t - ; g ) if g is a predictable locally bounded function, then g . ( f . X ) = ( g f ) . X . We point out also the following result (sintilar to Lebesgue’s theorem on doniinatetl corivcrgcnce) concerning passages to the limit under the sign of stochastic integral: 11) if gn = g l L ( t ,w ) are predictable processes convergent pointwise t o g = g ( t , w ) arid if l g n ( t , w ) / G ( t , w ) : where G = ( G ( t , w ) ) t > o is a locally bounded predictable process, then gn . X converges to g . X in measure unifornily on each finite interval, i.e., c) d) e) f)

<

sup lgn

x

-

g ‘ XI

P + 0.

s
8. 111 connection with the above construction of stochastic integrals f . X of predictable fiiiictions f with respect to semimartingales X there naturally arises the probleni of their calculation using more simple procedures: e.g., based on the idea of Riemnnn inte,qrcitiorc. Here the case of left-continuous functions f = ( f ( t ,w ) ) t y ) is of interest. To state the corresponding result we give the following definition ([250; Chapter I, 5 .Id]). For each 71. 3 1 let dn) = {T(rL)(m m) ,3 1)

+

<

be a family of Markov times ~ ( ” ) ( ~ r isuch ) that ~ ( ~ ) ( m ~ () “ ) ( m1) on the set

{ 4 1 L ) ( 7 1 1 ) < m}. We shall call T’“’),

3 1, Riemnnn sequences if

71

sup[r(”))7n + 1) A t m

-

~ ( ’ ” ( mA)t ] + 0

for all t E R+ and w E R. We associate now with stochastic integrals ( f . X ) , their imants

(17)

dn) -Riemann apyroz-

It tiirns out that if a process f = ( f ( t ,w ) , 9 t ) t > o , has left-contrnuous trajectories. (f) . X ) converge t o f . X in measure then these T(’L)-Rieniannapproxiniants T ( 7 L uniforriily on each finite tinie interval [0, t ] ,t > 0, i.e., sup I T ( q f X), - ( f X ) , l Z 0 u
5. Semimartingale Models

30 1

The proof is fairly easy. Let

f'"'(t,w) = C f ( T " " ( l ~ b ) , W ) J { ( t , W ) :

r(")(,m)< t

< . r ( n ) ( m +l)}.

(20)

771

Then the functions f ( " ) ( t , w ) are predictable and converge to f pointwise since f is left-continuous. Let Kt = sup I f s ] . Clearly, the process K = ( K t ,9t)is left-continuous, locally S
bounded, arid Hence

\)"'(fI

< K.

by property (h), arid to prove the required result (19) it suffices to observe that T b ) ( fX . )= f ( 7 L ) . X.

3 5b.

Doob-Meyer Decomposition. Compensators. Quadratic Variation

1. In the discrete-time case we have a n important tool of 'martingale' analysis of (arbitrary!) stochastic sequences H = ( H n , S n ) with < x, for n 3 0: the D oo b deco.mpositi mn Hn = H0 An Mn (1)

+ +

where A = (A,,, cFTh-l) is a predictable sequence, and M = ( M n ,SrL) is a rnartingale (see formulas (1)-(5)Chapter 11, in 5 l b ) . In the same way, in the continuous-time case, we have the Doob-Meyer decomposition (of submartingales), which plays a similar role and, together with the concept of a stochastic integral, underlies the stochastic calculus of semirnartingales. Let H = (Ht,.!Ft)t>o be a submartingale, i.e., a stochastic process such that the variables Ht are 9t-measurable and integrable, t 2 0, the trajectories of H are c&dllig (right-continuous and having liniits from the left) trajectories, and (the submartingale propert?/)

E(Ht 1 S3) 3 H,$ (P-a.s.),

s

< t.

(2)

We shall say that an arbitrary stochastic process Y = (Yt)t>obelongs t o the Dirichlet class (D) if

where we take the suprerriiini over all finite Markov times. In other words, the family of random variables

{ Y, : T must be rinzformly antegrable.

is a finite Markov time}

Chapter 111. Stochastic Models. Continuous Time

302

THEOREM 1 (the Doob ~Meyerdecomposition). Each submartingale H in the class (D) has a (uniquc) decomposition Ht = Ho

+ At + M t ,

where A = ( A t ,9 t ) is an increasing predictable process with EAt < A0 = 0, and M = ( M t ,9 t ) is a uniformly integrable martingale.

XI,

(4) t > 0,

It is clear from (4) that each submartingale in the class (D) is a semimartingale such that, in addition, the corresponding process A (belongiiig to the class 7': see 5a.G) is p,redictable. This justifies distinguishing a subclass of speed semimartingales X = ( X t ,.F:t)t20 having a representation (4) with predictable process A = ( A t ,st). In the discrete-time case the Doob decomposition is unique (see Chapter 11, 3 l b ) . Likewise, any two representations (4) with predictable processes of bounded variation of a special semimartingale must coincide. It is instructive to note that each special seniimartingale is in fact a difference of two local submartingales (or, equivalently, of two local supermartingales). R e m a r k 1. The Doob- Meyer decomposition is a 'difficult' results in martingale theory. We do not present its proof here (see, e.g., [103], [248], or [303]); we outline instead a possible approach to the proof based on the Doob decomposition for discrete time. Let X = ( X t ,s t ) t > o be a submartingale and let X(A)= (X,'"', 9t("1 ) t 2 0 be its discrete Lapproximation with

XI") = X[t/A]A and ~9:") = 'F[t/A]A. By the Doob deconipositiori for discrete time,

X i " ) = Xo + AiA) + MjA), where

=

n

/nit/*'"

E(X[s/A]A+A - "[s/A]A

1 9[3/A]A)

ds.

Hence one could probably seek the nondecreasing predictable process A = ( A t ,9 t ) t > o participating in the Doob-Meyer decomposition in the following form:

1

t

At = lim E(X,+a ALoA o

-

X , 1 9,) ds.

Moreover, if this process is well-defined, then it remains only to prove that the compensated process X - i l = (Xt - A t , 9 t ) t > o is a uniformly integrable martingale. It is now clear why one calls the process A = (At,.Ft)t>o in the Doob-Meyer deconiposition the compensator- of the submartingale X.

303

5 . Semimartingale Models

2. The Doob-Meyer decomposition has several useful consequences (see [250; Chapter I, § 3b]). We point out two of them. 1. Each predictable local martingale of bounded variation X = COROLLARY ( X t ,9t)t>o witli X o = 0 is stochastjcaJJy indistinguishable from zero.

2. Let A = (At,.Ft)tao be a process in the class dloc,i.e., locally COROLLARY integrable. Then there exists a unique (up to stochastic indistinguishability) predictable process 2 = ( & , S t ) t > o (the compe,nsator of A) such that A is a local martingale. If, in addition, A is a nondecreasing process, then 2 is also nondecreasing. 3. Now, we consider the concepts of quadratic variation and quadratic covariance for semimartingales, which play important roles in stochastic calculus. (For example, these characteristics are explicitly involved in Itb's formula presented below, in 5 5c.) For the discrete-tirne case we introduced the quadratic variation of a martingale in Chapter 11, 3 It). This definition is extended to the case of continuous time as follows. 1. The quadratic covariance of two semimartingales, X and Y, is the DEFINITION process [X, Y ]= ([X,Y]t, 9 t ) t 2 0 such that

i

t

[ X ,Y ] t = Xtx

-

1

t

X,- d Y , -

Y,- d X ,

-

XoyO.

(5)

(Note that the stochastic integrals in (5) are well defined because the left-continuous processes (Xt-)and (yt-) are locally bounded.) 2 . The quadratic variation of a semirnartingale X is the process DEFINITION [ X ,X ] = ([X, x ] t ,9 : t ) t a o such that

One also writes [XIin place of [X, X ](cf. formula (10) in Chapter 11, 5 Ib). We note that, by definitions (5) and (6) we immediately obtain the polarization formula 1 [X, Y]= ,([X Y , X Y ]- [ X - Y , X - Y ] ) . (7)

+

+

The names 'quadratic variation' and 'quadratic covariance' given to [ X ,Y ]and

[X, X ]can be explained by the following arguments. Let T ( " ) ,71 3 1, be Rieinann sequences (see 5 5a.8) and let

Chapter 111. Stochastic Models. Continuous Time

304

Then sup I S , ~ ' " ) ( Y X ,) - [x, YIJ u
for each t

5o

as

n

+ oo

5o

as

n

+ oo.

> 0; in particular, sup /s,~'")(x, X ) - [x,XI,^ u
To prove (10) it suffices to observe that

S,','")(X,X ) =

x; xi -

-

2~(71)(x.X I ,

(see (6)) arid that, by property (19) in 55a.

T ( , ) ( X _ X ) T ' + ( X - X )u '

'

-

x,- dX,.

Now, relation (9) is a consequence of (10) and polarization formula (7). 4. By (10) we obtain that [ X , X ]is a nondecreasing process. Since it is cAdlg, it follows that [ X , X ]E Y+, therefore (by (7)) the process [ X , Y ] also has bounded variation ([X,Y ]E Y), provided that Y E Y+.

This, together with (5), proves the following result: a product of truo sem,imurtingales i s itself a semimartingale. Rewriting ( 5 ) as

X t y , = XOYi

+

1 t

X,-

dY.5

+

t

K-

dX,

+ [ X ,Ylt,

(12)

where [X,Y ]is the process defined by ( 7 ) ,we can regard this relation as the f o r m u l a of integration b?j parts for semimartingales. Written in the differential form it is even more transparent: d ( X Y ) = X - dY Y- dX d [ X ,Y]. (13)

+

+

Clearly, this is essentially a semimartingale generalization of the classical relation A ( X ~ ~ Y=, )xnP1 AY, y71-l AX, nu, (14) for sequences X = ( X l 1 )and Y = (YT2). A list of properties of the quadratic variation and the quadratic covariance under various assumptions about X arid Y can be found, e.g., in the book [250;Chapter I, §4e]. We point out only several properties, where we always assume that Y E Y + : a) if one of the semimartingales X and Y is continuous, then [ X , Y ]= 0; b) if X is a semiinartingale arid Y is a predictable process of bounded variation, then

+

+ ax,

[ X ,Y ]= AY . X and XY = Y . X +X- .Y; c) if Y is a predictable process and X is a local martingale, then [ X , Y ]is a local martingale; d) A[X, Y] = AX AY.

5. Semimartingale Models

305

5. Besides the processes [ X , Y ] and X , X ] just defined (which are often called ‘square brackets’) an important position in stochastic calculus is occupied by the processes ( X ,Y ) and ( X ,X ) (‘angle brackets’) that we define next. Let M = ( M t ,. 9 t ) t > O be a square integrable martingale in the class 2’.Then, by Doob’s inequality (see, e.g., [303],and also (36) in 3 3b for the case of a Brownian motion), EsupMf 4supEMj < 00. (15)

t

<

t

Hence the process M 2 , which is a submartingale (by Jensen’s inequality), belongs to the class (D). In view of the Doob-Meyer decomposition, there exists a nondecreasing predictable integrable process, which we denote by ( M ,M ) or ( M ) ,such that the difference M 2 - ( M ,M ) is a square integrable martingale. If M E XI:,, then carrying out a suitable localization we can verify that there exists a nondecreasing predictable process (which we define again by ( M ,M ) or ( M ) ) such that M 2 - ( M ) is a locally square integrable martingale. their ‘angle bracket’ ( M ,N ) is defined For two martingales, M and N , in Xl&, by the formula 1 4

( M , N ) =- ( ( M + N , M + N ) - ( M - N , M - N ) ) .

(16)

It is straightforward that ( M ,N ) is a predictable process of bounded variation and M N - ( M ,N ) is a local martingale. It follows from (5) that M N - [ M , N ] is also a local martingale. Hence if the martingales M and N are in the class .?‘Z&, then [ M ,N ] - ( M ,N ) is a local martingale. In accordance with Corollary 2 in subsection 2, one calls the predictable process ( M ,N ) the compensator of [ M ,N ] and one often writes

Remark 2. In the discrete-time case, given two square integrable martingales, M = ( M n , 9 n ) and N = ( N n , 9 n ) , the corresponding sequences [ M , N ] and ( M ,N ) are defined as follows:

and

Chapter 111. Stochastic. Modcls. Contiriuoiis Time

306

Relation (19) suggests the following (formal, but giving one a clear idea) representation for the quadratic covariance in the continuous-time case:

( M ,N ) t =

/

0

t

E(dM3 d N s

l9,3)

(Cf. the forrniila for the At in Remark 3 a t the end of subsection 1.) 6. We now discuss the definition of 'angle brackets' for semimartingales and their connection with 'square brackets'. Let X = ( X t ,S t ) t > o be a semirnartingale with decomposition X = X o + M + A , where M is a local martingale ( M E &lo") and A is a process of bounded variation ( A E 'Y). Besides the already used representation of the local martingale M as the sum M = A40 MI MI' with MI' E "1/ n 4 1 0 c and M' E XI:, n &lo, (see formula (11) in fj5a), each local martingale M can be represented (moreover, in a unique way) as a sum M = Mo M M d , (21)

+ +

+ +

where M C = ( M t ,. q t ) t > o is a continuous local martingale and M" = ( M t ,,Ft)t20 is a p r d y discontini~ouslocal martingale. (We say that a local martingale X is purely discontinuous if X0 = 0 and it is orthogonal to each continuous martingale Y , i.e., X Y is a local martingale; see [250; Chapter I, 5 4b] for detail). Hence each semirnartingale X can be represented as the sum

X

= Xo

+ M" + M d + A.

Remarkably, the continuous martingale component M" of X is unambiguously defined (this is a consequence of the Doob-Meyer decomposition; see [250; Chapter I, 4.18 and 4.271 for a discussion), which explains why it is commonly denoted by Xc. Clearly, [X", X"] = (X", X"); moreover, one can prove ([250; Chapter I, 4.521) that [ X ,XI, = (X", X C ) t+ C ( A X d 2 (22) s
and, for two seniimartingales, X and Y ,

where we set by definition (X, Y ) t = ( X C X"),. ,

Remark 3 . For local martingales M we have

5. Semimartingale Models

307

for each t > 0 arid

W,M E

4 0 ,

(see, e.g., [250; Chapter I, 41). Since processes A of bounded variation satisfy the inequality C lAA,l < 00 (P-a.s.) for each t > 0, it follows that s
+

for the corresponding component of an arbitrary sernirnartingale X = X O 111 + A . Hence s
for each t > 0, arid the right-hand sides of (22) arid (23) are well defined and finite (P-a. s.). 5c. ItB’s Formula for Semimartingales. Generalizations

1. THEOREM (It6’s formula). Let

x = ( X I , . . . ,X d ) be a d-dimensional semimartingale and let F = F ( x 1 , . . . , xd) be a C2-function in R*. Then the process F ( X ) is also a semimartingale and

where DiF =

8F

-,

dXi

d2F DijF = -. axiaxj

The proof can be found in many textbooks on stochastic calculus (e.g., [103], 12481, or [250]).

Chapter 111. Stochastic Models. Continuous Time

308

2. We consider now two examples demonstrating the efficiency of It6’s formula in various issues of stochastic calculus.

EXAMPLE 1 (DolRans’ equation and stochastic exponential). Let X = ( X t , 9 t ) t > o be a fixed semimartingale. We consider the problem of finding, in the class of c&dl&g processes (processes with right-continuous trajectories having limits from the left) a solution Y = (yt, S t ) t > o to DolCans’s equation rt

yt = 1

+ J, Ys- dXs,

or, in the differential form,

dY

= Y-

dX,

Yo = 1.

(3)

By It6’s formula for the two processes

x;= xt xo -

and

X: =

n

-

1 2

-(XC,XC)t

(1 + A X s ) e - A x s ,

X i = 1,

(4)

O<s
and the function F ( x 1 ,x2) = exl . x2, we obtain that the process

1) is a semimartingale; 2) satisfies DolCans’s stochastic equation (2).

Moreover, the process &(X)(the Dol6ans stochastic exponential) is a unique (up to stochastic distinguishability) solution of (2) in the class of processes with c&dl&g trajectories. (See the proof and a generalization to the case of complex-valued semimartingales in [250; Chapter I, 3 4f].)

5 . Semimartingale Models

309

EXAMPLE 2 (Lhy’s theorem; 53b). Let X = ( X t , S t ) t > o be a continuous local martingale with ( X ) t = t . Then X is a Brownian motion. For the proof we consider the function F ( z ) = eiXz and use ItB’s formula (separately for the real and the imaginary parts). Then we obtain

The integral $ eiXxs d X , is a local martingale. We set T~ = inf{t: IXt/ 3 n } . Let E( . ; A ) be averaging over the set A . If A E 9 0 ,then we see from (7) that

Hence, as n t 3 0:

+ 00, we

obtain the following relation for f ~ ( t = ) E(eiA*t;A),

moreover, it is clear that l f A ( t ) l ,< 1 and f~(0) = P(A). The only solution of (9) satisfying these conditions is as follows:

Hence A2

E(eiXXt 19,) =e-Tt.

In a similar way we can show that for all s and t , s 6 t , we have

Consequently, X = ( X t ,9 t ) t a o is a process with independent Gaussian increments, E(Xt - X , ) = 0, and E(Xt - X , ) 2 = t - s. Hence, by our assumption of the continuity of the trajectories X is a Brownian motion (see the definition in 5 lb). 3. It6’s formula, which is an important tool of stochastic calculus, can be generalized in several directions: to nonsmooth functions F = F ( q , . . . , xd) and semimartingales X = ( X I , .. . , X d ) , to smooth functions F = F ( x 1 , . . . , z d ) and nonsemimartingales X = ( X I , .. . , X d ) , and so on. Following [166] and using some ideas from this paper we present here several results in this direction.

310

Chapter 111. Stochastic Models. Continuous Time

a) Let d = 1 and let X be a standard Brownian motion B = (Bt)t20. Let F = F ( z ) he an absolutely continuous function, i.e.,

here we assume that the (measurable) function f = f ( y ) belongs to the class L ~ ~ ~ ( Ri.e., '),

for each K > 0. Note that neither f ( B ) = ( f ( B t ) ) t > onor F = ( F ( B t ) ) t > oare semimartingales in general, therefore we cannot use It6's formula in the form (1). However, it is proved in [166]that we have the following relation:

, is the quadratic covariance of the processes f ( B ) and B defined as where [ f ( B )B] follows:

here the d n )= { t ( " ) ( m )m , 3 l}, n 3 1, are Riemannian sequences of (deterministic) times t ( l L )defined in 3 5a.8. We emphasize that the process f ( B ) is not a sernimartingale in general, therefore, the mere existence of the limit (in probability P) in (15) is nontrivial. One of the results of [166]is the proof of the existence of this limit. COROLLARY

1. Let ~ ( zE) c2.hen

li,

t

[ f ( B ) Blt > =

f ' ( B s ) ds

and (14) coincides with Itb's formula.

COROLLARY 2 . Let F ( z ) = 1x1. Then [ f ( B )B]t , = 2Lt(O), where

5. Semimartingale Models

311

is the local time that the Brownian motion '(spends" a t zero over the period [0, t ] . Hence, from (14) we obtain Tanaka's formula

(cf. the example in Chapter 11, l b and

5 3e in this chapter).

Note that the process (B( = ((Bt/,.Ft)t>o is a submartingale, the stochastic integral is a niartirigale, and L ( 0 ) = ( & ( 0 ) , 9 t ) is a continuous (and therefore, predictable) nondecreasing process. Hence we can regard (17) as an explicit formula for the Doob-Meyer decomposition of the seniimartingale IB(. b) Again, let d = 1, and let X be a fractional Brownian motion BW= (Bi')t20 with Hurst parameter W E I ] ; see $ 2 ~ As . before, let T ( n )= { t ( " ) ( m ) , m 3 I}, n 3 1, be Rieniann sequences of (deterministic) times t(")((m). Since ElABFI2 = lAtI2' (see (6) in 5 2c), it follows that

(4,

and therefore, for W E

(i,1) we have

for the limit in probability.

i,

Remark. If W = then the corresponding limit is equal to t , whereas it is + m for E (0, ;). One usually calls processes with property (18) zero-energy processes (see Chapter IV, 33a.6 and also, e.g., [166]). Let F = F ( z ) be a C2-function and let f ( z ) = F ' ( z ) . For a Brownian motion B = ( B t ) t 2 0 we have

W

1

d F ( B t ) = F ' ( B t )d B t

+ -2F " ( B t ) ( d B t ) 2 ,

(19)

which (on having agreed that ( d B t ) 2 = dt-see §3d for details) gives one the following It6 's for~nrilawritten in the differential form:

d F ( B t ) = F ' ( B t ) dBt

1 + -F"(Bt) dt. 2

(20)

Property (18) holds for a fractal Brownian motion BW = (BF)t>owith parameter < W 1, which makes the following agreement look fairly natural:

<

( d B F ) 2 = 0. (Cf. (12)-(14) in §3d.)

(21)

Chapter 111. Stochastic Models. Continuous Time

312

If we replace B by B" in the expansion ( l Y ) , then, in view of (2l), we arrive (only formally a t this stage) a t the representation (recall that f = F')

d F ( B ? ) = f ( B F )dBF, (22) which, in accordance with the standard conventions of stochastic calculus, must be interpreted in the integral form: for s < t we have

We shall now prove this formula restricting ourselves for simplicity to the case of F E C2 and explaining also on the way how one should interpret the 'stochastic integral' in (23). By Taylor's formula with remainder in the integral form

F ( z )= F ( Y )

+f

( Y ) ( Z - Y)

+

l

f'(.)(.

-

du.

Hence, proceeding. as in [166] and [299], for each dn)-partitioning, n 3 1, we obtain

Clearly, P

sup If'(BF)i < m) = 1 and, in view of (18): ( O < W

The left-hand side of (24) is independent of n and I?;"' exists 'i;IIc

m

P + 0,

f ( B L q n L )( % ) t(7qm+1) - % q m ) )

therefore there (26)

(in the sense of convergence in measure), which we denote by

it

f (B:)dB,"

(27)

and call the stochastic integral with respect to the fractional Brownian motion w -- (B!du ) t > o (WE ( & 11, f E C1). Simultaneously, these arguments also prove required formula (23), which can be regarded as an analogue of It6's formula for a fractional Brownian motion.

5. Semimartingale Models

313

EXAMPLE 3. If F ( z ) = x 2 , then ( 2 3 ) shows that for a fractional Brownian motion BW with < W 6 1 we have

i

d ( B F ) 2 = 2B," d B F .

(28)

Recall that for a Brownian motion B = B ' / ' ,

d ( B t ) 2 = 2Bt dBt

+ dt.

(29)

EXAMPLE 4. If F ( z ) = e", then

d ( e B F ) = eBF d B f

1 + -eBF 2

dt.

For a Brownian motion B = B1I2 we have 1

d ( e B t ) = eBt d B t

+ -eBt 2

dt.

(31)

Our considerations could be extended also to the case of several fractional Brownian motions. For instance, assume that

(xo, x l , . . . , x,) E (

~

BQ 0 , ., . .

~

~ " d ) ,

i

where Wo = (so that BWO is a standard Brownian motion) and < W1 . . . < E d 6 1. If F = F ( t ,zo, 2 1 , . . . x d ) E C 1 ~ ' ~ . . . ~then ', (cf. (11) in 3 3d)

< W2 <

EXAMPLE 5 . We consider the process 02

st = e(/l-+)t+uoBt+ulB:

1

(33)

where B = B112 is a standard Brownian motion and BW is a fractional Brownian motion with $ < W 6 1. Then S = (St)t,o can be regarded as the solution of the stochastic differential equation

dSt = St ( pdt with SO= 1. (Cf. (2) and (4) in 54b.)

+ uo d B t + ~ 7 1dB,W)

(34)

Chapter IV. Statistical Analysis of Financial D a t a 1. E m p i r i c a l data . P r o b a b i l i s t i c and s t a t i s t i c a l m o d e l s of their description . S t a t i s t i c s of ‘ t i c k s ’ .........................

5 l a . Structural changes in financial data gathering and analysis

..... fj l b . Geography-related features of the statistical data on exchange rates fj l c . Description of financial indexes as stochastic processes with discrete intervention of chance . . . . . . . . . . . . . . . . . . . . . . 5 Id . On the statistics of ‘ticks’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Statistics of o n e - d i m e n s i o n a l d i s t r i b u t i o n s . . . . . . . . . . .

....

5 2a . Statistical data discretizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2b . One-diniensional distributions of the logarithms of relative price changes . Deviation from the Gaussian property and leptokurtosis of empirical densities . . . . . . . . . . . . . . . . . . . . fj 2c . One-dimensional distributions of the logarithms of relative price changes . ‘Heavy tails’ and their statistics . . . . . . . . . . . . . . § 2d . One-dimensional distributions of the logarithms of relative price changes . Structure of the central parts of distributions . . . .

3 . Statistics of v o l a t i l i t y , c o r r e l a t i o n dependence, and a f t e r e f f e c t in p r i c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3a .

Volatility . Definition and examples . . . . . . . . . . . . . . . . . . . . . . . fj 3b . Periodicity and fractal structure of volatility in exchange rates . . fj 3c . Correlation properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3d . ‘Devolatization’. Operational time . . . . . . . . . . . . . . . . . . . . . . . 3 3e . ‘Cluster’ phenomenon and aftereffect in prices . . . . . . . . . . . . . . . 4. S t a t i s t i c a l R / S - a n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4a . Sources and methods of R/S-analysis

..................... § 4b . R/S-analysis of some financial time series . . . . . . . . . . . . . . . . .

315 315 318 321 324 327 327

329 334 340 345 345 351 354 358 364 367 367 376

1. Empirical Data. Probabilistic and Statistical Models

of Their Description. Statistics of ‘Ticks’

5 la.

Structural Changes in Financial Data Gathering and Analysis

1. Looking back to the changes in the empirical analysis of time-related financial data, one discovers the following features. In the 1970s and earlier, one mostly operated with data averaged over large time intervals: a year, a quarter, a month, a week. Anong the most widespread probabilistic and statistical models (developed for the description of the behavior of the logarithms of financial indices) a t that time (and nowadays) there were models of random walk type (see Chapter I, §2a), autoregressive, and moving average models, their combinations, and so on (see Chapter 11, 0 Id). It should be noted that most of these models were linear. In the 1980s, in connection with the analysis of daily data, one saw the need to invoke nonlinear models; the ARCH and GARCH models and their various modifications (see Chapter 11, 3 I d ) are the best known examples of these. The analysis of interday data has become feasible in the 1990s. This is primarily a result of the general progress in computer technology and telecommunications accompanied by a sharp improvement (as compared with the ‘paper-based’ technology of data storage and processing) in the methods of the collection, registration, storage, and analysis of statistical information, arriving in a virtually incessant flow. 2. Besides routine news coming even through daily newspapers or TV (for example, the exchange rates, several indexes, the opening and the closing trading prices of major stocks and commodities, and so on), several information agencies (Reuters, Telerate, Knight Ridder, Bloomberg, to mention just a few) deliver on-line megabytes of most diversified financial information to their customers: for instance, one can learn about the recent bid and ask currency prices as well as the name and the location of the bidder.

Chapter IV. Statistical Analysis of Financial Data

316

Below is an example of a message from Reuters concerning the exchange rates (against the US dollar) that is displayed on its customers’ monitors immediately after 7 h 27 GMT (see [204]): 0727 DEM RABO RABOBANK UTR 1.6290/00 DEM 1.6365 1.6270 0727 0726

FRF NLG

BUEX RABO

UECIC RABOBANK

PAR UTR

5.5620/30 1.8233/38

FRF NLG

5.5835 5.5588 1.8309 1.8220

Here 0727 and 0726 is the time of quotation announcements; DEM, F R F , and NLG are the abbreviations for particular currencies (the German mark, the French franc, and the Dutch florin); RABO and BUEX are RABOBANK in Utrecht (UTR) and UECIC bank and Paris (PAR), respectively; 1.6290 is the bid price; 00 following this bid price indicates the ask price, 1.6300; 1.6365 and 1.6270 are the high and the low prices of the last 24 hours. On the third line, 0726 means that RABOBANK announced (or maintained) its florin quotations (1.8233/38) a t 7 h 26 and nobody (including RABOBANK itself) has announced new quotations since. Choosing t o = 7 h 27 as time zero we shall see that, say, the ask price of the dollar against the German mark

,

t 3 to,

behaves as in Fig. 28. (Cf. Fig. 6 in Chapter I, 5 lb.)

-

1.6310 -1.6300

- 4

1.6290 -1.6280 -1.6270 --

I I I I I

I I I

I I I

A

h

I I I 1

k

FIGURE 28. Behavior of the cross rate Sta =

(E),.

t 2 tn

That is, the price S f keeps its value for some time (on the interval [ O , q ) ) ; then, at the instant 71, it drops (one speaks about a ‘tick’ indicating that some bank announced another quotation SF1 a t time T I ) , and so on. This raises two questions: (I) what can be said on the statistics of the lengths of the ‘intertick’ intervals (%+I

-

Tk);

(11) what can be said on the statistics of the changes in price (absolute changes SFk+, - S:k or relative changes S:k+l/S&)?

1. Empirical Data. Probabilistic and Statistical Models

317

Extracting this information from the available data is the p r i m a r y task of t h e statistical analysis of the exchange rates or data on other financial indexes, which often exhibit patterns of dynamics similar to the above one. Clearly, the function of such an analysis is the construction of probabilistic and This is, in statistical models of such processes as ask prices (SF) or bid prices (S,“). the long run, important also for understanding the evolution of financial indexes, the pricing mechanisms, and for predictions of the price development in the future. It should be noted that registering and processing statistical data, storing it in a form allowing an easy recovery is a difficult task, impossible without advanced technology. Still, it is equally clear that an access to the results of statistical analysis packed in a ready-to-use form gives one indisputable advantages in the securities markets as regards building an ‘efficient portfolio’, making rational investment decisions, the assessment of investment projects, securities, and so on.

3. The almost incessant registration and processing of statistical data, which have been made possible by the modern technology, reveals a high-frequency pattern of the behavior of financial indexes, which evolve in a fairly chaotic manner. This pattern is not discernible after discretization (with respect to time or the phase variables). Consequently, the appearance in financial mathematics of problems dealing with ‘high-frequency’ is the result of the new opportunities offered by almost continuous gathering of statistical data. It is this advanced technology of data gathering that enabled one to discover, besides the presence of high frequency components, several other peculiarities of the dynamics of financial indexes. Not plunging into detail herea, we point out, for instance, the nonlinear mode of the formation of financial indexes and the aftereffect, the capacity of many indexes, prices, and so on, to ‘remember’ their past. 4. To give one a notion of the frequency of ticks in currency cross rates, develop-

ing as in the above chart, and also an idea of the bulk of the available statistical information, we now cite the data of Olsen & Associates, an institute mentioned in the footnote on this page; see [go], [91], and also [204]. Over the period January 1, 1987-December 31, 1993, Reuters registered

8 238 532 ticks in the DEM/USD cross rate. Of these, 1466 946 ticks occurred during one year, from October 1, 1992 to September 30, 1993. Over the same period, Reuters registered 570 814 ticks of the exchange rate JPY/USD (see also the table in 3 l b ) . aFor that, see the proceedings [393] of the HFDF-1 conference organized by the Research Institute for Applied Economics Olsen & Associates (March 29-31, 1995, Zurich, Switzerland). The introductory talk “High Frequency Data in Financial Markets: Issues and Applications” by C. A. E. Goodhart and M. O’Hara provides a brilliant introduction into the range of problems arising in connection with ‘high frequencies’, describes new phenomena, peculiarities, and results, their interpretations, and suggests a program of a further research.

Chapter IV. Statistical Analysis of Financial Data

318

These are high-frequency data indeed: on a typical business day, there occurs on the average 4 500 ticks of the DEM/USD exchange rate and around 2 thousand ticks for JPY/USD. In July, 1994, the number of ticks of the DEM/USD rate was close to 9000, that is, 15-20 ticks per minute. (On usual days, there occur on the average 3-4 ticks each minute.) It should be noted that, of all the exchange rates, the rate DEM/USD is in general subject to most frequent changes. It is also worth noting that the quotations (bid and ask prices) cited by various agencies are n o t the real transaction prices. To our knowledge, the corresponding data or the statistics of the volumes of transactions are not easily accessible. 5 . The above example relates to currency exchange rates; however, many other financial indexes behave in a similar way. We can refer to [127; p. 2841 for a chart depicting the behavior of Sieniens shares on the Frankfurt Stock Exchange on March 2, 1992, since the opening at 10 h 30. The pattern is the same as in Fig. 28: the price is flat for sonic time, then (at a random instant) it changes the value.

6. One can find extended statistics of stock prices ‘ticks’ in [217]. Various information relating to the ticks in the prices of all kinds of securities (including shares and bonds) can be obtained, for example, from ISSM

NYSE

-

~

the Institute for the Study of Securities Markets, the New York Stock Exchange.

Berkeley Options Data provides data on the ask and bid prices of options and on the instantaneous prices of CBOE (the Chicago Board of Options Exchange); in the Commodity Futures Trading Commission (CFTC) one can obtain data concerning the American futures market, while the current American stock prices can be, for example, obtained by e-mail, from the MIT Artificial Intelligence Laboratory (http://www.stockmaster.com).

3 lb.

Geography-Related Features of the Statistical Data on Exchange Rates

1. By contrast to, say, bourses, which can trade in stock, bonds, futures and are open only on ‘trading days’, the currency exchange market, the

FX-market (Foreign Exchange or Forex) has several peculiarities that one would like to consider more closely. It should be noted first of all that the FX-market is inherently international. It cannot be ‘localized’, has no premises (of the sort NYSE or CBOE have). Instead, this is an interweaving network of banks and exchange offices, equipped with modern communications, all over the world.

1. Empirical Data. Probabilistic and Statistical Models

319

The FX-market operates continuously round t h e clock; it is more active during the five working days, and less active on week-ends or holidays (for example, on Easter Monday). 2. Judging by the periods of different intensity of currency trade during the day, one usually distinguishes the following three geographic zones (indicated is Greenwich Mean Time, GMT): (1) the East-Asian, with center a t Tokyo, 21:OO-7:OO; (2) the European, with center at London, 7:OO-13:OO; (3) the American, with center a t New York, 13:OO-2O:OO. According to [D4], the E a s t - A s i a n zone includes Australia, Hong Kong, India, Indonesia, Japan, South Korea, Malaysia, New Zealand, and Singapore. The European zone covers Austria, Bahrain, Belgium, Germany, Denmark, Finland, France, Great Britain, Greece, Ireland, Italy, Israel, Jordan, Kuwait, Luxembourg, the Netherlands, Norway, Saudi Arabia, South Africa, Spain, Sweden, Switzerland, Turkey, and United Arab Emirates. The American zone includes Argentina, Canada, Mexico, and the USA. Sometimes one distinguishes four zones in place of the three in our scheme, where the fourth, Pacific, zone is included in the East Asian one. To the already mentioned main centers of trade, Tokyo, London, and New York, one could add also Sydney, Hong Kong, Singapore (the East-Asian zone), Frankfurton-Main, Zurich (the European zone), and Toronto (the American zone). Taking the American dollar (USD) as the basis, the bulk of the trade proceeds between the dollar and the following ‘most important’ currencies: the German mark (DEM), the Japanese yen (JPY), the British pound (GBP), and the Swiss franc (CHF) (we use the abbreviations adopted by the International O ~ q a n i z a t i o n f o r Standartization (IOS), code 4217). These ‘basic’ currencies (including, of course, the American dollar) are traded all over the globe. Other currencies are mostly traded in their geographic zones.

3. To gain some impression of the ‘intensity’ of currency exchange, that is, of the frequency of ‘ticks’ in the cross rates on the FX-market (a global market, as already mentioned) one can consider the following graph (borrowed from [ l S l ] ) of the ‘intensity’ of changes in the DEM/USD exchange rate. The average number of changes, ‘ticks’ occurring every 5 minutes (Fig. 29) or 20 minutes (Fig. 30) is plotted along the vertical axis and time is plotted along the horizontal axis. In these charts one can distinguish cycles, which are clearly related to the rotation of Earth, arid inhomogeneity in the number of changes (‘ticks’). Three peaks of activity, related to the three geographic zones, are discernible. The European and American peaks are fairly similar. The peak of activity in Europe occurs immediately after lunch, when the business day begins in America. The minimum of activity fall precisely at Tokyo lunch time, when it is night in Europe and America.

Chapter IV. Statistical Analysis of Financial Data

320

o + : Mon

Tue

Wed

Thu

Fr i

FIGURE 29. Intensity of the DEM/USD exchange rate from Monday through Friday, over 5-minute intervals

150 100 50 0

Mon

Tue

Wed

Thu

Fri

FIGURE 30. The same as in Fig. 29, over 20-minute intervals

We supplement Fig. 29 and 30 with the following chart from [427], depicting the znterday intensity (the average number of ‘ticks’ occurring at each of the 24 hours) of the DEM/USD cross rate over the period October 5 , 1992-September 26, 1993 (the data of’Reuters). 500 400 300

18

24

FIGURE 31. Average hourly number of ‘ticks’ during one day (24 hours) for the DEM/USD cross rate. Zero corresponds t o Oh00 GMT

1. Empirical Data. Probabilistic and Statistical Models

321

4. The FX-market is the largest financial market. According to Bank for International Settlements in Basel, Switzerland (1993), the daily turnover on this market was approximately $832 bn ( 8 3 2 . 10’) in 1992! One of the largest databases covering the FX-market is to our knowledge the database of Olsen & Associates (see the footnote in Sla.4). The table based on its data gives one an idea of transactions on the FX-market in January 01, 1987December 31, 1993.

I I 1

Exchange rate

Total number of ‘ticks’ registered in the database

Average daily number of ‘ticks’ (52 weeks in a year, 5 business days in a week)

DEM/USD

8.238.532

4500

JPY/USD

4.230.041

2300

GBP/USD

3.469.421

1900

CHF/USD

I

3.452.559

I

1900

FRF/USD

2.098.679

1150

JPY/DEM

190.967

630

FRF/DEM ITL/DEM

I I

132.089 118.114

I I

I

440

I

390

I

GBP/DEM

96.537

320

NLG/DEM

20.355

70

5 . In the above description of the dynamics of exchange rates we were concerned only with the temporal aspect: the frequency (intensity) of changes as a function of time. Looking back a t the classification in 5 la.2 we see that this discussion relates to issue (I), the statistics of ‘intertick’ intervals. However, we have said nothing so far about the character of these changes in the values of prices that occur a t the instants of ticks. We turn to this question (issue (11)) in 3 I d , while in the next section we shall consider the probabilistic and statistical models coming to mind in a natural way in connection with the trajectory behavior of the prices (Sr) and ( S f ) (see Fig. 28).

l c . Description of Financial Indexes as Stochastic Processes with Discrete Intervention of Chance 1. As already said, the monitor of, say, a Reuters customer interested the FX-market shows at each instant t two quotations: the ask price S: and the bid price S t of, say, the American dollar (USD) in German marks (DEM).

322

Chapter IV. Statistical Analysis of Financial Data

The difference Sf - S!, the spread, is an important characteristic of the market situation. It is well known that the spread is positively correlated with the volatilities of the prices (which are understood here as the usual standard deviations). Thus, a rise in volatility (which increases risks, due to the decreasing accuracy of the predictions of the price development) prompts traders to widen the spread as a compensation for greater risks. 2. We now represent the prices SF and

St in the following form:

We also set

Then

Ht

=l

n

d

s

(4)

is the logarithm of the geometric mean and

It is the so-defined prices S = (St)and their logarithms H = ( H t ) that one usually deals with in the analysis of exchange rates, so that the two really existing process ( S f ) and (St)are reduced to a single one, ( S t ) . st in. real The evolution of S = ( S t ) t ) o and H = (Ht)t>o,where Ht = ln-, SO (‘physical’) time t can be fairly adequately (see Fig. 28 in 5 l a ) described by stochastic processes with discrete intervention of chance:

and

where 0 z 70 < 71 < r2 < . . . are the successive instants of ‘ticks’, that is, the instants of price changes, and

1. Empirical Data. Probabilistic and Statistical Models

are S,,-measurable variables (here we set AS,, same holds for AH,,). Clearly,

=

S ,

- S,,-

=

323

S,,

-

S,k-l; the

We already considered processes of types (6) and (7) in Chapter 11, 5 If. We recall several concepts relating to these important processes, which we shall use in the present analysis of the evolution of currency cross rates and other financial indexes. For the simplicity of notation we set
AS,,.

The probability distribution of the process S = (St)t>o (which we denote by Law(S) or Law(&, t >, 0)) is completely defined by the j o i n t distribution Law(r, [) of the sequence (r,O = ( % l k ) k > l , of ‘ticks’ r k and their ‘marks’&. Such a sequence is commonly called a marked point process or a multivariate point process (see, for instance, [250; Chapter 111, 3 lc]). The name point process is usually reserved for the sequence r = ( r k ) alone (see Chapter 11, fj If and, in greater detail, [250; Chapter 11, 5 3c]), which is in our case the sequence of the i n s t a n t s of price j u m p s 7-0 = 0 < r1 < 7-2 < . . . . Setting

Nt =

C I ( r k ,< t)

(8)

k>l

we obtain ~k = inf{t:

Nt = k }

(9)

(here, as usual, ~k = 03 if inf{t: Nt = k } = 0). The process N = ( N t ) t 2 0 is called a counting process, and formulas (8) and (9) establish clearly a one-to-one correspondence N Jr. As regards distributions, the distribntion Law(r, [) is completely defined by the conditional distributions

and

where

TO =

0 and

50 = SO.

324

Chapter IV. Statistical Analysis of Financial Data

3. So far, nothing has been said about r k and [k as ‘random’ variables defined on some probability space. This question is worth a slightly more thorough discussion. To be able to use the well-developed machinery of stochastic calculus in our analysis of stochastic processes (for instance, multivariate point processes that we consider now) and also to take account of the flow of information determining the prices, assume that from the very beginning we have some filtered probability space (stochastic basis) (Q 9, (Tt)t>O> p). Here ($t)t>o is a flow (filtration) of a-algebras 9 t that are the ‘carriers of marketrelated information available on the time intervals [0,t]’. Once we have got this filtered probability space, it comes naturally to regard the 7 k as random variables ( r k = r k ( w ) ) that are Markov t i m e s with respect to ( 9 t ) : (7-


t 3 0.

(12)

In the same way we regard the
<

3 Id.

On the Statistics of ‘Ticks’

1. We now discuss what is known about the unconditional probability distributions Law(r1, 7 - 2 , . . . ). For k 3 1 we set

A,+ = ‘TI,- ‘Tk-1, where TO = 0. Clearly, the knowledge of Law(A1,A,, . . . ) is equivalent to the knowledge of Law(7-1,q ,. . .), so that we can restrict ourselves to the analysis of the distribution of the time intervals between ‘ticks’. Starting from the conjecture that the variables A,, A,, . . . are identically distributed and independent (this assumption enables one to justify, on the basis of the L a w of large numbers, the standard statistical constructions of estimators for parameters, distributions, and so on) one can gain a clear impression of the character of their probability distribution from bar charts for the empirical density @(A) constructed on the basis of the available statistical data. In [145] one can find the following results of such an analysis of 1472240 intervals between ‘ticks’ in the DEM/USD cross rate (the data supplied by Olsen & Associates [22l]). Up to a constant, we have

1. Empirical Data. Probabilistic and Statistical Models

325

where A1

M

0.13 and

A2 M

0.61.

(This is a result of the analysis of the logarithms lnp^(A) as functions of In A; the estimates for A 1 and A2 are obtained by the least squares method.) It is worth recalling in connection with (1) that there is one distribution, wellknown in the mathematical statistics, whose density decreases as a power function: the Pareto distribution with density

(here a > 0, b > 0; see Table 2 in Chapter 111, 3 l a ) . Note that quite often, in particular, in the finances literature, the authors mean by distributions of Pareto type (or simply Pareto distributions) ones whose density decreases as a power function at infinity. Following this trend we can say that, as shows ( l ) ,we have a Pareto distribution with exponent a = A1 on the interval [23 s , 3 m), and a Pareto distribution with exponent a = A2 on the interval [3 m , 3 h). It should be noted that, in the description of the probabilistic properties of a particular index, one can rarely expect (as it is often visible from the statistical analysis of the above kind) to make do with a single ‘standard’ distribution dependent on few unknown parameters. The reason probably lies with the fact that traders on the market have different goals, constraints, and react to risks in different ways. 2. In general, there are no a priori reasons to assume that the variables A,, A,, . . . are independent. Moreover, as shown by empirical analysis, the time when the next ‘tick’ would occur essentially depends on the ‘intensity’, ‘frequency’ of ‘ticks’ in the past. The problem of an adequate description of the conditional distributions Law(& I A l , . . . , Ak-1) is therefore timely. In this connection we present, following [143], an interesting model serving for such a description. This model is ARDM (Autoregressive Conditional Duration Model), and it is related to the ARCH family. Let $k = ‘$k(Ai,. . . , Ak-1) be (sufficient) statistics such that

The simplest conditional distribution for Ak under condition $k that comes to mind is the exponential distribution with density

326

Chapter IV. Statistical Analysis of Financial Data

where the (random) parameters (&.) are defined recursively; namely,

with A0 = 0, $0 = 0 , QO > 0, 01 3 0, and 01 3 0. Clearly, relations (4)-(6) define the conditional distributions

completely and the resulting variables A,,A,, . . . are, in general, dependent. Note the following formula for the conditional expectations in this model:

This first-order autoregressive model ( 6 ) can obviously be generalized t o higher orders (see Chapter 11, 5 I d and [143]).

2. Statistics of One-Dimensional Distributions

2a. Discretizing Statistical Data 1. On gaining some ideas of the character of the ‘intensity’ (frequency) of ‘ticks’ (at any rate, in the example of currency cross rates) and of the one-dimensional distributions of the ‘intertick’ intervals (Q - q - ~ )it , now seems appropriate to consider the statistics of price ‘changes’, that is, of the sequence (STk- S,k-l)k21 STk or of related variables, for instance, h,, = In _ _ .

S,

-1

We point out straight away that ‘daily’, ‘weekly’, ‘monthly’,. . data are different from the ‘interday’ ones. The former can be regarded as data received a t equal, ‘regular’,time intervals A (for instance, A can be one day or one week), while in the analysis of ‘interday’ statistics one must consider figures arriving in a ‘nonregular way’, at random instants 71, 7 2 , . . . with different intervals A,, A,, . . . between them (here Ak = 7 k - 7-k-l). This ‘lack of regularity’ brings forward certain difficulties in the application of the already developed methods of the statistical data analysis. For that reason, one usually carries out some preliminary data processing (‘discretizing’ the data, ‘rejecting’ abnormal observations, ‘smoothing’, separating trend components, and so on). We now dwell on the methods of dascretzzatzon. We fix a ‘reasonable’ interval A of (real, physical) time. It should not be very small: in the case of the statistics of exchange rates we must ensure that this interval is representative, that is, it contains for sure a large number of ‘ticks’ (in other words, A must be considerably larger than the average time between two ‘ticks’). Otherwise, there will be too many ‘blank spaces’ in our ‘discretized data’. For the ‘exchange rates’ of the ‘basic’ currencies it is recommended in [204] to take A not shorter than 10 minutes, which (besides the already mentioned ‘representativeness’) will also enable one to avoid uncertainties arising for small values

Chapter IV. Statistical Analysis of Financial Data

328

of A, when the amplitude of spreads and the range of changes in bid and ask prices are comparable. The simplest discretization method works as follows: having chosen A (say, 10 m, 20 m, 24 h, . . .), we replace the piecewise constant continuous-time process S = (St)t20, t 3 0, by the sequence SA = (St,) with discrete time t k = k A , k = O , l , ... . Using another widespread method one, first, replaces the piecewise constant process

st = so +

c

EkI(Tk


(1)

k> 1

by its continuous modification the values of (ST,):

s

= (st)obtained by linear interpolation between

1.6310 1.6300 1.6290 1.6280 1.6270 I

to

t 71

72

74

73

t

FIGURE 32. Piecewise-constant process ( S t ) and its continuous modification (&)

s

After that, we discretize this modification - = (3,)using the simplest method; that is, we construct the sequence Sa = (st,),where t k = k A , k = 0 , 1 , . . . , and A is the time interval that is of importance to a particular investor or trader (one month, one day, 20 minutes, 5 minutes, . . .). 2. Beside discretization with respect to time, statistical data can also be quantzjied, rounded ofl with respect to the phase variable. Usually, this is carried out as follows. We choose some y > 0 and, instead of the original process S = (St)t>o,introduce a new one, S(y) = (St(y))t20, with variables =Y

[3.

(3)

For example, if y = 1 and St = 10.54, then St(1) = 10; while if y = 3 , then S t ( 3 ) = 9. Hence it is clear that ( 3 ) corresponds to rounding off with error not larger than y.

2. Statistics of One-Dimensional Distributions

329

If y-quantization is carried out first, and A-discretization next, then we obtain from (St) another sequence, Sa(y) or s a ( y ) . Since &(y) + St as y -+ 0, there arises the question how one can consistently choose the values of A and y to achieve that the amount of information contained in the variables St,(y) with t k = k A , k = 0 , 1 , . . . , is ‘almost the same as the information in (St)’. As an initial approach it seems reasonable (following a suggestion of J. Jacod) to find out conditions on the convergence rates A + 0 and y + 0 ensuring the convergence of the finite-dimensional distributions of the processes Sa(y) and s a ( y ) to the corresponding distributions for S.

5 2b.

One-Dimensional Distributions of the Logarithms of Relative Price Changes. Deviation from the Gaussian Property and Leptokurtosis of Empirical Densities

1. We now consider the process of some cross rate (DEM/USD, say). Let

s = (St)t>O, and let s” = (s”t)t,o be the continuous modification of S = (St)t>o obtained by linear interpolation. Further, let s a = ( i ? t k ) k > O , where t k = kn be the ‘&discretization’ of s=(s”t)t>o. We have repeatedly mentioned (see, for instance, Chapter I, 3 2a.4) - that, in the analysis of price changes, it is not their amplitudes A&, 3 St, - St,-, the,mselves I

I

that are of real economic significance, but, rather, the relative changes

~-St‘

st,-,

as,, = ’tk-1

eo1

1. It is understandable for that reason that one is usually not so much

interested in the distribution of We now set -(*) htk

=

-

stk as

in the distribution of H t , = In

AH,, (= Ht,

where t k = k A , k 3 0, arid Ho = 0. Bearing in mind our construction of introduced in

I

-

Htk-l),

k .

(1)

from S and the notation

3 2a, we obtain

where the term irt’,“’ reflects effects related to the end-points of the partitioning intervals and can be called the ‘remainder’ because it is small compared with the sum.

Chapter IV. Statistical Analysis of Financial Data

330

Indeed, if, e.g., A is 1 hour, then (see the table in fjl b . 4 ) the average number of ‘ticks’ of the DEM/USD exchange rate is approximately 187 (= 4500 : 24). Hence the value of the sum in (2) is determined by the 187 values of the h,. At the same time, (F/:)( is knowingly not larger than the sum of the absolute values of the four increments lir corresponding to the ticks occurring immediately before and after the instants t k - 1 and t k . It must be pointed out that the sum in (2) is the sum of a random n u m b e r of r a n d o m variables, and it may have a fairly complicated distribution even if the distributions of its ternis and of the random number of terms are relatively simple. This is a sort of a technical explanation why, as we show below, we cannot assume that the variables have a Gaussian distribution. True, we shall also see that the

xi:’

conjecture of the Gaussian distribution of becomes ever more likely with the decrease of A, which leads to the increase in the number of summands in (2). This is where the influence of the Central limit theorem about sums of large numbers of variables becomes apparent.

R e m a r k . The notation hit’ is fairly unwieldy, although it is indicative of the construction of these variables. In what follows, we shall also denote them by h k for siniplicity (describing their construction explicitly and indicating the chosen value of A). 2. Thus, assume that we are given some value of A > 0. In analyzing the joint distributions Law(lh1, hz, . . . ) of the sequence of ‘discretized’ variables h ~hz, , . . . it is reasonable to start with the one-dimensional distributions, making the assumption that these variables arc identically distributed. (The suitability of this homogeneity conjecture as a first approximation is fully supported by the statistical analysis of many financial indexes; at any rate, for not excessively big time intervals. See also $ 3 below ~ for a description of another construction of the variables hi that takes into account the geography-related peculiarities of the interday cycle.) As already mentioned, there occurred 8.238.532 ‘ticks’ in the DEM/USD exchange rate from January 01, 1987 through December 31, 1993 according to Olsen & Associates. One can get an idea of the estimates obtained for several characteron the basis of istics of the one-dimensional distribution of the variables hi = these statistical data from the following table borrowed from [204]:

lhip’

n

N

Mean hN -2.73. lop7

lop6

K uLto sis

Variance

Skey ness

6 2

SN

KN

2.62.10F7

0.17

35.11

1.45.10F6

0.26

23.55

10 m

368.000

1h

61.200

-1.63.

6h

10.200

-9.84.

9.2O.1Op6

0.24

9.44

24 h

2.100

-4.00. lop5

3.81.10F5

0.08

3.33

2. Statistics of One-Dimensional Distributions

331

In this table, N is the total number of the discretization nodes t l , t 2 , . . . , t N , where t k = kA,

while

is the empirical skewness, and

is the empirical kurtosis. The theoretical skewness S, of the normal distribution is zero. The fact that the empirical coefficient SN is positive means that the empirical (and maybe, also the actual) distribution density is asymmetric, the left-hand side of its graph is steeper than its right-hand side. It is clear from the table that the modulus of the mean value is considerably less than the standard deviation, so that we can set it to be equal to zero in practice. The most serious argument against the conjecture of normality is, of course, the excessively large kurtosis, growing, as we see, with the decrease of A. Since this coefficient is defined in terms of the fourth moment, this also suggests that the distribution of the hk = hi,"' must have 'heavy tails', i.e., the corresponding density p(A)(x) decreases relatively slowly as 1x1 normal density).

+

00

(as compared with the

3. It is not the shape of the bar-charts of statistical densities alone that speaks for the deviation of the variables h k from the normal (Gaussian) property (which is observable not only in the case of exchange rates, but also for other financial indexes, for instance, stock prices). It can be discovered also by standard statistical tests checking for deviations from normality, such as, for example, (1) the quantile method, (2) the x2-test, ( 3 ) the rank tests. We recall the essential features of these methods. The quantile method can be most readily illustrated by the QO-plot (see Fig. 33), where the quantiles of the corresponding normal distribution Y ( p ,a 2 ) with parameters p and a2 estimated on the basis of statistical data are plotted along the horizontal axis a.nd the quantiles of the empirical distribution for the h k are plotted along the vertical axis. (The quantile Q p of order p , 0 < p < 1, of the distribution

Chapter IV. Statistical Analysis of Financial Data

332

5.0 -. 3.01.o ..

-1.0-

-3.0.-5.0 -4.0

-2.0

0.0

2.0

410 Q P

FIGURE 33. QG-quantile analysis of the DEM/USD exchange rate with A = 20 minutes (according t o Reuters; from October 5, 1992 through September 6, 1993; [427]). The quantiles Q p of the empirical

&

I

distribution for the variables = ht(f), t k = kA, k = 1,2,. . . , are plotted along the vertical axis, and the quantiles Q p of the normal distribution are plotted along the horisontal axis

0.3

0.2 0.1

0.0 -3

-2

-1

0

1

2

3

FIGURE - 34.-A typical graph of the empirical density (for the variables hk = hi:), k = 1 , 2 , . . . ) and the graph of the corresponding theoretical (normal) density

of a random variable 6 is, by definition, the value of x such that P(6 6 x) 3 p and P(E 3 x) 3 1 - P . ) In the case of a good agreement between the empirical and the theoretical distributions the set ( Q p , must be clustered close to the bisector. However, this is not

op)

2. Statistics of One-Dimensional Distributions

333

the case for the statistical data under consideration (exchange rates, stock prices, and so on). The (Qp,ap)-graph in Fig. 33 describes the relations between the (normal) theoretical density and the empirical density, which we depict in Fig. 34. 4. The use of K. Pearson’s X2-test in the role of a test of goodness-of-fit is based on the statistics -2

=

k i=l

(Vi -

2

Vi)

npi

1

where the vi are the numbers of the observations falling within certain intervals Ti k

( C 1, = It)

corresponding to a particular ‘grouping’ of the data, and the p i are

i=l

the probabilities of hitting these intervals calculated for the theoretical distribution under test. In accordance with Pearson’s criterion, the conjecture

80: the empirical data agree with the theoretical model

~z-~,~-~,

is rejected with significance level a if 2’ > where x;-l,l-a is the Q I - ~ quantile (that is, the quantile of order 1 - a ) of the X2-distribution with k - 1 degrees of freedom. Recall that the X2-distribution with n degrees of freedom is the distribution of the random variable X n2 = E 2l

+ ... + tn, 2

where [ I , . . . , En are independent standard normally distributed (,N(O,1))random variables. The density fn(x) of this distribution is

In [127] one can find the results of statistical testing for conjecture XOwith significance level a = 1% in the case of the stock of ten major German companies and banks (BASF, BMW, Daimler Benz, Deutsche Bank, Dresdner Bank, Hochst, Preussag, Siemens, Thyssen, VW). This involves the data over a period of three years (October 2, 1989-September 30, 1992). The calculation of g2 and x 2 ~ - ~ (with , ~ Ic- =~22, on the basis of n = 745 observations) shows that conjecture 20must be rejected for all these ten companies. For instance, the values of Xn2 (with pi = l / k , Ic = 22) for BASF and Deutsche Bank are equal to 104.02 and 88.02, respectively, while the critical value of x$-l,l-a for k = 22 and a = 0.01 is 38.93. Hence g2 is considerably larger than

Chapter IV. Statistical Analysis of Financial Data

334

5 2c.

One-Dimensional Distributions of the Logarithms of Relative Price Changes. ‘Heavy Tails’ and Their Statistics

1. In view of deviations from the normal property and the ‘heavy tails’ of empirical densities the researchers have come to a common opinion that, for instance, for the ‘right-hand tails’ (that is, as IC + +m) we must have P ( l p

> x)

N

IC-”L(z),

where the ‘tail index’ 01 = N ( A )is positive and L = L ( z ) is a slowly varying function

(g

t 1 as

x

+ 00 for each y > 0

)

. A similar conclusion holds for ‘left-hand

tails’. We note that discussions of ‘heavy tails’ and kurtosis in the finances literature can be found in [46], [361], [419], and some other papers, and even in several publications dating back to the 1960s (see, e.g., [150], [317]). There, it was pointed out that the kurtosis and the ‘heavy tails’ of a distribution density could occur, for instance, in the case of mixtures of normal distributions. (On this subject, see Chapter 111, I d , where we explain how one can obtain, for example, hyperbolic distributions by ‘mixing’ normal distributions with different variances.) In several papers (see, e.g., [46] and [390]),in connection with the search of suitable distributions for the hit’ the authors propose to use Student’s t-distribution with density

where n is an integer parameter, the ‘number of the degrees of freedom’. Clearly, this is a distribution of Pareto type with ‘heavy tails’. 2. After B. Maridelbrot (see, for instance, [318]-[324]) and E. Fama ([150]),it became a fashion in the finances literature to consider models of financial indexes based on stable distributions (see Chapter 111, 5 l a for the detail). Such a distribution has a stability exponent a in the interval (0,2]. If a = 2, then the stable distribution is normal; for 0 < cr < 2 this is a Pareto-type distribution satisfying (l), and the ‘tail index’ N is just the stability exponent. This explains why the conjecture of a stable distribution with 0 < Q < 2 arises naturally in our search for the distributions of the hk = hjf’: such a distribution is marked by both ‘heavy tails’ and strong kurtosis, which are noticeable in the statistical data. Another argument in favor of these distributions is the characteristic property of self-similarity (see Chapter 111, 5 2b): if X and Y are independent random variables having a stable distribution with stability exponent a , then their sum I

2. Statistics of One-Dimensional Distributions

335

also has a stable distribution with the same exponent (equivalently, the convolution of the distributions of these variables is a distribution of the same type). From the economic standpoint, this is a perfectly natural property of the preservation of the type of distribution under t i m e aggregation. The fact that stable distributions have this property is an additional evidence in favor of their use. However, operating stable distributions involves considerable difficulties, brought on by the following factors. If X is a random variable with stable distribution of index a , 0 < a < 2, then E l X l < 00 only for a > 1. More generally, E I X / P < 00 if and only if p < a . Hence the tails of a stable distribution with 0 < a < 2 are so ‘heavy’ that the second nionient, is infinite. This leads to considerable theoretical complications (for instance, in the analysis of the quality of various estimators or tests based on variance); on the other hand this is difficult to explain from the economic standpoint or to substantiate by facts since one usually has only limited stock of suitable statistical data. We must point out in this connection that the estimate of the actual value of the ‘tail index’ a is, in general, a fairly delicate task. On the one hand, to get a good estimate of a one must have access to the results of sufficiently m a n y observations, in order to collect a stock of ‘extremal’ values, which alone are suitable for the assessment of ‘tail effects’ and the ‘tail index’. Unfortunately, on the other hand, the large number of ‘nonextremal’ observations inevitably contributes to biased estimates of the actual value of a . 3. As seen from thc properties of stable distributions, using them for the description of the distributions of financial indexes we cannot possibly satisfy the following three conditions: stability of the type of distribution u n d e r convolution, heavy tails with index 0 < Q < 2 , and th,e finiteness of the second m o m e n t ( a n d , therefore, of the variance). Clearly, Pareto-type distributions with ‘tail index’ a > 3 have finite variance. Although they are not preserved by convolutions, they nevertheless have an important property of the stability of the order of decrease of the distribution density under convolution. More precisely, this means the following. If X and Y have the same Pareto-type distribution with ‘tail index’ Q and are independent, then their sum X Y also have a Pareto-type distribution with the same ‘tail index’ a. In this sense, we can say that Pareto-type distributions satisfy the required property of the stability of the ‘tail index’ a under convolutions. It is already clear from the above why we pay so much attention to this index a, which determines the behavior of the distributions of the variables at infinity. We can give also an ‘econo-financial’ explanation to this interest towards a . The ‘tail index’ indicates, in particular, the role of ‘speculators’ on the market. If a is large, then extraordinary oscillations of prices are rare and the market is ‘wellbehaved’. Accepting this interpretation, a market characterized by a large value

+

^hit)

Chapter IV. Statistical Analysis of Financial Data

336

of a can be considered as ‘efficient’, so that the value of a is a measure of this ‘efficiency’. (For a discussion devoted to this subject see, for example, [204]). 4. We now consider the estimates for the ‘stability index’ of stable distributions and, in general, for the ‘tail indices’ of Pareto-type distributions. It should be noted from the onset that authors are not unanimous on the issue of the ‘real’ value of the ‘tail index’ a for particular exchange rates, stock prices, or other indexes. The reason, as already mentioned, lies in the complexity of the task of the construction of efficient estimators GN for a (here N is the number of observations). Stating the problem of estimation for this parameter requires on its own right an accurate specification of all details of cooking the statistical ‘stock’ a careful choice of the value of A, and so on. In the finances literature, one often uses the following ‘efficient’ estimators i 3 ~ for a , which were suggested in [152] and [153]:

GN

= 0.827

-Q f

-

Q0.72

Qi-f

-

,

0.95

< f < 0.97,

(3)

QO.28

of

where is the quantile of order f constructed for a sample of size N under the assumption that the observable variables have a symmetric stable distribution. If it were true that the distribution Law(hk) with h k = hb;’ belongs to the class of stable distributions (with stability exponent a ) , then one could anticipate that the GN stabilize (and converge to some value a < 2) with the increase of the sample size N . However, there is no unanimity on this point, as already mentioned. Some authors say that their estimators for certain financial indexes duly stabilize (see, for instance, “31 and [474]). On the other hand, one can find in many papers results of statistical analysis showing that the 6~ do not merely show a tendency to growth, but approach values that are greater of equal to 2 (see, e.g., [27] and [207] as regards shares in the American market, or [127] as regards the stock of major German companies and banks). All this calls for a cautious approach towards the ‘stability’ conjecture, though, of course, there is no contradiction with the conjecture that ‘tails’ can be described in terms of Pareto-type distributions. 5 . Now, following [204], we present certain results concerning the values of the ‘tail

index’ a for the currency cross rates under the assumption that the h k = hi;’ have a Pareto-type distribution satisfying (1). The following values of the ‘tail index’ a = a(A)were obtained in [204] (see the table on the next page). We now make several comments as regards this table. In subsection 6 below we describe estimates of a based on the data of Olsen & Associates (cf. l b ) . For A equal to 6 hours the estimation accuracy is low, due to the insufficient amount of observations.

2. Statistics of One-Dimensional Distributions

337

Analyzing the figures in the table (which rest upon a large database and must be reliable for this reason), we can arrive at an important conclusion that the exchange rates of the basic currencies against the US dollar in the FX-market have (for A = 10 riiiiiut,cs) a Pareto-type distribution with ‘tail index‘ Q = 3.5, which increases with the iricrease of A. Hence it is now very likely that the variance of h k = hie’ niiist be jirlnite (a desirable property!), although we canriot say the same about the fourth irionicnt responsible for the leptokurtosis of distributions. In another papcr [91] by Olseri & Associates one can also find data on the rates XAU/USD arid XAG/USD (XAU is gold and XAG is silver). For A = 10 rriinutes the corrcsporiding cstiiriates for Q are 4.32 f 0.56 arid 4.04 f 1.71, respectively; for A = 30 rriinutes tliesc estimates are 3.88 ?L 1.04 and 3.92 i0.73. 6. In this subsection wo merely outline the construction of estimators 3 for the ’tail index’ cy used for the above table borrowed from [204] (we do not dwell on the ‘bootstrap’ arid ‘jackknife’ methods significant for the evaluation of the bias and the st,aridard deviation of these estimators). We coiisider the P a r e t o distribution with density

where fab(x) = 0 for x For z 3 b we have

< b.

hi f c r b ( x ) = ln Q

+ Q In b

-

(a

+ 1)In IC.

(5)

Chapter IV. Statistical Analysis of Financial Data

338

Hence the maximum likelihood estimator SN constructed on the basis of N independent observations ( X i , . . . , so as to satisfy the condition

x,)

N

Iv

can be dcfined by the formula

Since

for a

> 0 and /3 > 0, it follows that

that is, l / 6 N is an unbiased estimator for l / a , so that the estimator &N for cy has fairly good properties (of course, provided that the actual distribution isexactly-a Pareto distribution with known ‘starting point’ b, rather than a Paretotype distribution that has no well-defined ‘starting point’). There arises a natural idea (see [ 2 2 3 ] ) to use (7) nevertheless for the estimate of cr in Pareto-type distributions, replacing the unknown ‘starting point‘ b by some its suitable estimate. For instance, we can proceed as follows. We choose a sufficiently large number A4 (although it should not be too large in comparison with N ) and construct an estimator for cy using a modification of (7) with b replaced by M and with the sum taken over i 6 N such that X i 3 M. To this end we set

where

2. Statistics of One-Dimensional Distributions

Setting

1

uN,M =

339

lhf(xi),

{iM}

we can rewrite the formula for

yN,M as follows:

and we can consider an estimator i3N.Mfor

Q

such that

We could also proceed otherwise. Namely, we order the sample ( X i ,X z , . . . X N ) to obtain another sample, ( X I ,X:, . . . , X ; ) , with X f 3 X ; 3 . . 3 X;. We fix again some integer M << N as the ‘starting point’ b and set

Then we can consider the estimator

Q L ,for~

Q

such that

The estimator ( Y & ,so ~ obtained was first proposed by B. M. Hill [ 2 2 3 ] , and it is usually called the Hill estimator. Obviously, the ‘good’ properties of this estimator depend on a right choice of M , the number of maximal order statistics generating the statistics However, it is also clear that one can hardly expect to make a ‘universal’ choice of M , suitable for a wide class of slowly changing functions L = L ( z ) describing the behavior of, say, the right-hand ‘tail’ in accordance with the formula

P(Xi > z)

,-.-

z-”L(z).

(13)

Usually one studies the properties of the above estimators and G N , M for a particular subclass of functions L = L ( z ) . For instance, we can assume that L = L ( z ) belongs to the subclass

L , = { L = L ( z ): L ( z ) = 1

+ cz-7 + o(z-Y),

c

> o}

with y > 0. It is shown in [223] under this assumption that if M so that

i.e., the estimators (a>,M)are asymptotically normal.

+ 30

as N

--f 30

340

5 2d.

Chapter IV. Statistical Analysis of Financial Data

One-Dimensional Distributions of the Logarithms of Relative Price Changes. Structure of the ‘Central’ Parts of Distributions

1. And - yet, how (:an one, bearing in mind the properties of the distribution Law(hk), combine large kurtosis and ‘tail index’ a > 2 (as in the case of currency exchange rates)‘! Apparently, one can hardly expect to carry this out by means of a single ‘standard’ distribution. Taking into account the fact that the market is full of traders and investors with various interests and time horizons, a more attractive opt,ion is t,he one of invoking several ‘standard’ - distributions, each valid in a particular domain of the range of the variables l t k . Many authors and, first, of all, Mantlelbrot are insistent in their advertising the use of stable distributions (and some their modifications) as extremely suitable for the ‘ central’ part of this range. (See, for instance, the monograph [352] containing plenty of statistical data, the theory of stable distributions and their generalizations, and results of statistical analysis.) In what. follows we shall discuss the results of [330]relating to the use of stable laws in the description of the S&P500 Index in the corresponding ‘central’ zone. (To describe the ‘tails’ the authors of [330] propose to use the normal distribution; the underlying idea is that the shortage of suitable statistical data rules out reliable conclusions about the behavior of the ‘tails’. See also [464].) As regards other financial indexes we refer to [127], where one can firid a minute statistical analysis of the financial characteristics of ten major German corporations and the conclusion that the hyperbolic distribution is extremely well suited for the ‘central’ zone. In Chapter 111, 3 I d we gave a detailed description of the class of hyperbolic distributions, which, together with the class of stable distributions, provides one with a rather rich armory of theoretical distributions. As both hyperbolic and stable distributions can be described by four parameters, the hopes of reaching a good agreement between ‘theory’ and ‘experiment’ using their combinations seem well based.

2. We consider now the results of the statistical analysis of the data relating to the S&P500 Index that was carried out in [330]. The authors considered the six-year evolution of this index on NYSE (New York Stock Exchange) (January 1984 through December 1989). All in all, 1447514 ticks were registered (the data of the Chicago Mercantile Exchange). On the average, ticks occurred with one-minute intervals during 1984-85 and with 15-second intervals during 1986-87. Since the exchange operates only at opening hours, the construction of the process describing the evolution of the S&P500 Index took into account only the ‘trading time’ t , and the closing prices of each day were adjusted to match the

2. Statist,ics of One-Dimensional Distributions

341

.opening prices’ of the rirxt (lay. Let S = ( S t ) be the resuking process. We shall now consider the changes of this process over some intervals of time A: Astk

= St,

-

St,-,

(1)

I

where t k = k A . The interval A will vary from 1 minute to l o 3 minutes (In [330], A assumes the following values: 1,3,10,32,100,316,and 1000 minutes; the number of ticks corresporidirig to A = 1 m is 493545 arid it is 562 for A = 1000 r n . ) Using the notation of 5 2a, one can observe that

-

-

and thereforc, ASt, M St,-,hjf), sirice St, M St, and the increments ASt, = St, - St,-,are small. This (approximate) equality shows that for independent increnients ( S t , St,_,) ~

<

the distribution P( xjf) x) and the conditional distribution P(ASt,
J

I

0

2

FIGURE 3 5 . Skctches of thc graphs of l o g l o ~ ( * ) ( z )for two distinct values of A

A mere visual inspection of the many graphs of l o g ~ ~ j $ ~ ) for ( z )various values of A shows that the distribution densities are fairly symmetric and are ‘melting’ as A yro’ws. They decrease as IC + f m , but not as fast as it should be for a Gaussian distribution. The uniniodality and the symmetry that are visible here, together with the character of the decrease at infinity of the empirical densities indicate that it might be

Chapter IV. Statistical Analysis of Financial Data

342

reasonable to direct attention to stable symmetric distributions. We recall that the characteristic function q ( Q )= EeieX of a stable random variable X with symmetric distribution is as follows (see formula (14) in Chapter 111, 5 l a ) :

<

where ~73 0 and 0 < a 2. Hence, once one have accepted the 'stability' conjecture one should first of all find an estimate for the parameter 0. Stable distributions have the Pareto type. In the symmetric case (see (7) and (8) in Chapter 111, 3 l a ) , if 0 < (lu < 2, then we have

P(lX1 > z)

-C

~

X

-

as ~

z

+ 00

with some constant Ecy, and we could find the value of a using the techniques of the construction of estimators described in 5 2c. However, as rightly pointed out in [330],this method for the estimation of a is not very reliable due to the insufficient amount of observations; it rcquires many 'extremal' values. For that reason, the approach chosen in [330] is a different one: it takes into account, by contrast, only the results of observations fitting into the 'central' zone. The niaiii idea of this approach is as follows. Assume that the characteristic function

has the following form:

) distribution P(ASt, 6 z) can be represented, by Then the density ~ ( ~ ) (ofzthe the inversion form.ula, as the integral

,-rAlQl" cos Qz do. For z

Hence

=

0 we obtain

2. Statistics of One-Dimensional Distributions

343

Of course, we could obtain the same without resorting to the representation ( 3 ) , by applying directly to the definition of a stable law, which requires that

+

Law(&!%, ASt,

+ . . . +A&,)

and by using the equality

= LaW(Cn(St,

-

St,,)):

to

= 0,

(6)

cn -

-

(see Chapter 111, 3 l a ) . Indeed, since ASt, ASt,

+

Law(St,

+ . . . + A&, = St, So, it follows that - so) = Law(nl/a(St, - so)). -

Hence p'"A'(x) = n-l/cp)(x.L-l/a)' (7) and setting x = 0 we obtain (5). Relation (5) enables one to find an estimate 6 of the 'stability index' Q by considering the empirical densities p^("A)(0) with A equal to 1 niinute and n = 1,3,10,32,100,316,1000,then passing to the logarithms, and using the least squares method. (This choice of n = 1 , 3 , 1 0 , .. . is explained by the fact that the corresponding values of loglo n are approximately equidistant: loglo 3 = 0.477, loglo 10 = 1, loglo 32 = 1.505, . . . .) The value of this estimate of Q obtained in [330] is

6 = 1.40

* 0.05.

(8)

We point out at the onset that there is no contradiction whatsoever between this result and the estimate 3 N 3.5 of the 'tail index' Q in 52c.5. The point is that these estiniates are obtained under different assumptions about the character of distributions. In one case we assume that the distribution has 'stable' type. while in the other-that it is of Pareto type. Moreover (and this can be important), the object of the first research is the currency cross rates and the object of the other is the S&P500 Index. Generally speaking, there are no solid grounds to assume that the behavior of their distributions must be similar, for the defining factors in these two cases are distinct (the state of the world economy in the case of exchange rates and the state of the American national economy in the case of the S&P500 Index). This idea of the different behavior of the exchange rates and financial indexes of S&P500 or DJIA kind is also substantiated by the results of the R/S-analysis described in 5 4b below. We note also that the estimate (8) is based on the 'central' values, while the estimate 2 = 3.5 was found on the basis of the 'marginal' values. Hence this discrepancy is just another argument in favor of the above-mentioned thesis that the financial indexes must be described by different 'standard' distributions in different domains of their values, and that a single 'universal' distribution must be hard to find

344

Chapter IV. Statistical Analysis of Financial Data

3. The conclusion that the erripirical densities j5(7'a) (x)can be well approximated by stable symmetric densities p("a)(x) in the 'central' domain can be substantiated also by the following arguments, based on the se/j'-sindarity property. We consider a sample

of size k , with time step nA, where A is 1 minute. If we proceed now to the sample (n,-

AS ti

,L)

, . . . , 7L-l/aAst(,,,), k

then this new saniple must havt the same distribution as

Hence, in accordance with ( 7 ) ,the corresponding estimates of the one-dimensional densities of the (itlentically distributed) variables n - l / a A S (n) and h S t z must be 'very similar'. A chart in [330] obtained by a 'superposition' (see Chapter 111. 3 2c.6 for information about this method) of' the so transformed empirical densities for the values of A equal to 1, 3, 10, 32, 100, 316, and 1000 minutes strongly supports the stable distribution conjecture (with a z 1.40). One can find an estimate of the coefficient y in (3) on the basis of the empirical density p^(A)(0)and the estimate G = 1.40, by using (4). The corresponding value is = 0.00375.

3. Statistics of Volatility, Correlation Dependence, and Aftereffect in Prices

5 3a.

Volatility. Definition and Examples

1. Argiiably, no c:oncept in financial mathematics if as loosely interpreted and as widely discussed as ‘,uolatility’. A synonym t o ‘changeability’,h ‘volatility’ has rnariy definitions, and is used to denote various measures of changeability. If S, = SoeHn with Ho : 0 and AH,, = a&,,.71. 3 1, where ( E ~ , )is Gaussian white noise ( E , ~ .N(O, l ) ) ,then one means by volatility the natural measure of uncertainty and changeability, the standard deviation a. We recall that, if ( . N ( p ,n 2 ) for a random variable [, then N

-

arid

P(lE - pi 6 1.65a)

M

0.90.

Hence one can expect, in approximately 90 % cases that the result of a n observation of [ deviates from the mean value ,LL by 1.65 a at most. In employing the schenie ‘S,&= S,,-lehnr one usually deals with sniall values of the h,,,, so that S,,, M Sn-l(l h n ) .

+

Hence, if Ii,, = (TE,,, then, given the ‘today’ value of some price that its ‘tornorrow’ value S,, will in ’JO% cases lie in the interval

S,-l we can say

“Random House Webster’s concise dictionary” (Random House, New York, 1993) gives the following explanation of the adjective uolatzle: .‘l.evaporating rapidly. 2. tending or threateriirie to erupt in violence; explosive. 3 . chanqeable; unstable.”

346

Chapter IV. Statistical Analysis of Financial Data

so that it is only in 5 % cases that S, is larger than S,-1(1+1.650) than S,-1(1 - 1.650) in 5 % cases.

and it is smaller

R e m a r k 1. This explains why, in some handbooks of finance (see, for instance, [404]), one measures volatility in terms of v = 1.65 o in place of the standard deviation cr. 2. We have already seen before that the model ‘h, = m n rn 3 1’ is fairly distant from real life. It would be more realistic to use conditional Gaussian models of the kind ‘hTL = o,E, n l’,with a r a n d o m sequence cr = (cr,),>l of 9n-1-measurable variables on and with 9,-measurable E,, where (Sn) is the flow of ‘information’ (on the price values, say; see Chapter I, 3 2a for greater detail). There is an established tradition of calling o = ( C , ) ~ > I (in the above model) the sequence of ‘volatilities’. The random nature of the latter can be reflected by saying that ‘the volatility is volatile on its own’. Note that

>

E(h;

I sFn-1)

2

= o,,

(3)

+

and the sequence H = (H,, .!Fn)TL21 of the variables H , = h l . . . + h,, where Elh,I2 < 00 for 71 3 1, is a square integrable martingale with quadratic characteristic n > 1.

(4)

k=l

In view of (3),

k=l

therefore it is natural to call the quadratic characteristic

the volatility of the sequence H . We note that EH; = E(H),. 3. For ARCH(p) models we have

cr; = (YO

+

c P

i=l

(see Chapter 11, 5 3a). Hence the estimation problem for the volatilities on reduces to a problem of parametric estimation for (YO, ( ~ 1 ., . , a p .

3. Statistics of Volatility, Correlation Dependence, and Aftereffect in Prices

347

There exist also other, e.g., no,npurametric, methods for obtaining estimates of volatility. For instance, if h, = p n for n 3 1, where p = ( p , ) and o = (cr,) are stationary sequences, then a standard estimator for crn is

+

It is worth noting that one can also regard the empiric volatility Z = (i?n)n21 as an index of financial sthkistics and to analyze it using the same methods arid tools as in the case of the prices S = (Sn)la21. To this end we consider the variables

Many authors and numerous observations (see, for example, [386; Chapter lo]) demonstrate that the values of the 'logarithmic returns' ;F'= (Fn)n22 oscillate rather swiftly, which indicates that the variables F, and ?,+I, n 3 2, are negatively correlated. If we consider the example of the S&P500 Index and apply R/S-analysis to the corresponding values of F = (Fn),22 (see Chapter 111, fj 2a and fj 4 of the present chapter), then the results will fully confirm this phenomenon of negative correlation (see [386; Chapter lo]). Moreover, we can assume in the first approximation that the 7, are Gaussian variables, so that this negative correlation (coupled with the property of self-similarity standing out in observations) can be treated as an argument in favor of the thesis that this sequence is a fractional noise with Hurst parameter W < l / 2 . (According to [386],W z 0.31 for the S&P500 Index.) 4. The well-known paper [44] (1973) of Black and Scholes made a significantly

contribution to the understanding of the importance of the concept of volatility. This paper contains a formula for the fair (rational) price CT of a standard call option (see Chapter I, fj l b ) . By this formula, the value of CT is independent of p (surprisingly, at the first glance), but depends on the value of the volatility cr participating in the formula describing the evolution of stock prices S = (St)t20:

") 2

t,

where W = (Wt)t>ois a standard Wiener process. Of course, the assumption made in this model that the volatility cr in (10) is, first, a constant and, second, a known constant, is rather fantastic. Clearly, practical applications of the Black-Scholes formula require one to have at least a rough idea

Chapter IV. Statistical Analysis of Financial Data

348

of the possible value of t,he volatility; this is necessary not only t o find out the fair option prices, but also to assess the risks resulting from some or other decisions in models with priccs described by forniiilas (9) arid (10) in Chapter I, 3 l b . 111 this conriection, we niust discuss another (erripirical) approach t o the concept of ‘volatility’ that uses tlie Black--Scholes formula and the real option prices on securities markets. For this definition, let cCt = Ct( a ;T ) be the (theoretical) value of the ask price a t time t < T of a standard European call-option with f~ = (ST- K ) + and with maturity tirnc T . The price @t is theoretical. What we know in pract,ice is the price Ct that was acttially aririounced a t the instant t , and we can ask for the root of the equation

-

Ct = Q ( a ;T ) .

(11)

This value of CT,denoted by ;it, is called the ‘ i m p l i e d volatility’; it is considered to be a good estimate for the ‘actual’ volatility. It should be noted that, as regards its behavior, the ‘implied volatility’ is similar t o the ‘cnipirical volatility’ clefincd (in the coritimious-tirrie case) by formulae of type (8). Its negative correlation and fractal structure are rather clearly visible (set:. i.g., [38ci; Chapter lo]). 5. We ~iowdiscuss another approach to the definition of volatility, based on the consideration of the variation-related characteristics of the process H = ( H t ) t 2 0 defining the prices S = (St)t20 by the formula St = SoeHi. The results of rnariy statistical observations arid ecoriorriic arguments support the thesis that the processes H = (Ht)t>o have a property of self-similarity, which means, in particular, that tlie distributions of the variables Ht+n - Ht with distinct values of A > 0 are similar in certain respects (see Chapter 111, 5 2 ) . We recall that if H = BH is a fractional Brownian motion, then

for d l A > 0

iLIid

t

0 and

For a strictly a-stable L6vy rnotiou with 0

Hence, setting

< a < 2 we have

IHI = l / a < 1 we obtain

which resembles formula (12) for a fractional Brownian motion.

3. Statistics of Volatility, Correlation Dependence, and Aftereffect in Prices

349

All these formulas, together with arguments based on the Law of large numbers, suggest, that it would be reasonable to introduce certain variation-related characteristics arid to use them for testing the statistical hypothesis that the process H = (Ht)t>o generating the prices S = (St)t>o is a self-sirnilar process of the same kind as a fractional Brownian motion or an a-stable Lkvy motion. It should be also pointed out that, from the standpoint of statistical analysis, different kinds of investors are interested in different t i m e intervals and have different t i m e h,orizo,ns. For instance, the short-term investors are eager t o know the values of the prices S = (St)t>oa t times t k = kA, k 0, with small A > O (several minutes or even seconds). Such data are of little interest t o long-term investors; what they value most are data on t,he price movements over large time intervals (months and even years), inforriiatiori 011 cycles (periodic or aperiodic) and their duration, information on trend phenomena, and so on. Bearing this in mind, we shall explicitly indicate in what follows the chosen time interval A (taken as a unit of time, the .characteristic’ tinie measure of an investor) and also the interval ( a , b] on which we study the evolution and the ‘changeability’ of the financial index in question.

6. On can get a satisfactory understanding of the changeability of a process H = (Ht)t20 011 the tinie interval ( a , b] from the A-variation

<

where the sum is h k e n over all k such that a 6 t k - 1 < t k b, and t k = k A . Clearly, if a particular trajectory of H = (Ht)a 0 is sniall, t,licri the value of Var(,,b](H; A) is close t o the variation

which is by definition the suprerriu~ri

taken over all finite partitiorlings ( t o , . . . , t n ) of the interval ( a ,b] such that a = to < tl < . . < t,, 6 b. In the statistical analysis of the processes H = (Ht)t>o with presumably homogeneous increments, it is reasonable to consider, in place of the A-variations Var(,,b](H; A ) , the riornialized quantities

which we shall call the (empirical) A-volatilities on ( a , b]

Chapter IV. Statistical Analysis of Financial Data

350

It is often iiseful to consider the following A-volatility of order S

> 0:

where and the sunimation proceeds as in (16). We note that for a fractional Brownian motion H = Bw we have

“5”

as A + 0, where is convergence in probability. If H is a strictly 0-stable Lkvy motion with 0 < cy

as A

4

< 2, then

0.

R e m a r k 2. One iisiially calls stochastic processes H = (Ht)t>O with property (23) zero-energy processes (see, i.g., [ISS]). Thus, it follows from (22) and (23) that both fractional Brownian motion with 1 / 2 < W < 1 and strictly 0-stable Lkvy processes with W = 1/a > 1 / 2 are zero-energy processes.

7. The statistical analysis of volatility by means of R/S-analysis discussed below (see 4) enables one to discover several remarkable and unexpected properties, which provide one with tools for the verification of some or other conjectures concerning the space-time structure of the processes H = (Ht)t>o (for models with continuous time) and H = (H,)+o (in the discrete-time case). For instance, one must definitely discard the conjecture of the independence of the variables h,’ n 3 1 (generating the sequence H = (Hn),>0), for many financial indexes. (In the continuous-timo~iecase one must accordingly discard the conjecture that H = (Ht)t>o is a process with illdependent increments.) Simultaneously, the analysis of A-volatility and R/S-statistics support the thesis that the variables h,, n 3 1, are in fact characterized by a rather strong aftereffect, and this allows one to cherish hopes of a ‘nontrivial’ prediction of the price development. The fractal structure in volatilities can be exposed for many financial indexes (stock and bond prices, DJIA, the S&P500 Index, and so on). It is most clearly visible in currency exchange rates. We discuss this subject in the next section.

3. Statistirs of Volatility, Correlation Dependence, and Aftereffect in Prices

5 3b.

351

Periodicity and Fractal Structure of Volatility in Exchange Rates

1. In 5 111 we presented t,he statistics (see Fig. 29 and 30) of the number of ticks occurring during a day or a week. They unambiguously indicate

the i,nterday inhornogeneity

and the presence of daily cycles (periodicity).

st

Representing the process H = ( N t ) t 2 o with Ht = In - by the formula SO

(cf. formula (7) in 5 l c ) , we can say that Fig. 29 and Fig. 30 depict only the part of the development due to the ‘time-related’ components of H , the instants of ticks rk, but give no insight in the structure of the ‘phase’ component, the sequence ( h T k ) I

or the sequence of h k = (see the notation in fj2b). The above-introduced concept of A-volatility, based on the A-variation Var(,,b~(H;A), enables one t o gain a distinct notion of the ‘intensity’ of change in

the processes H and & both with respect to the time and the phase variables. To this end, we consider the A-volatility v ( H ;A) on the interval ( a , b ] . (4 We note on the onset that if a = ( k - l ) A and b = kA, then

(see the notation in 5 2b). We choose as the object of our study the DEM/USD exchange rate, so that

St

=

(DEM/USD)t arid Ht = In -.st

SO We set A to be equal to 1 hour and

t

= 1 ’ 2 , . . . , 24

(hours)

in the case of the analysis of the ‘24-hour cycle’, and we set

t = 1 , 2 , . . . , 168 (hours) in the case of the ‘week cycle’ (the clock is set going on Monday, 0:OO GMT, so that t = 168 corresponds to the end of the week). The inipressive database of Olsen & Associates enables one to obtain quite reli-

-

h

-

able estimates G ( i ( ( k - l ) n , k n ~ (A) H ;= jhk) for the values of V ( ( ~ - ~ ) A , ~ AA) J (=HJ;h k J for each day of the week.

Chapter IV. Statistical Analysis o f Financial Data

352

To this end let, time zero be 0:OO GMT of the first Monday covered by the databasr. If A = 1 h, then setting k = 1 . 2 , . . . ,24, we obtain the intervals ( O , l ] , (1,2],. . . , (23,241 corresponding to the intervals (of GMT) (O:OO, l:OO], (1:00,2:OO], . . . , (23:00,24:00].

As an estiniate for i7((k-llA,kA,(H; A) we take the arithmetic mean of the quan-

IsfA

tities - k[iLl)Al calculated for all Mondays in the database (indexed by the integer j ) . In a similar way we obtain estimates of the B((k-l)A,kAI ( H ;A) for Tuesdays ( k = 25,. . . , 4 8 ) , . . . , and for Sundays ( k = 145,. . . ,168). The following charts (Fig. 36 and Fig. 37) from [427] are good illustrations of the interday inhomogeneity and of the daily cycles -visible all over the week in the behavior of the A-volatility v ( ( ~ - ~ ) A(,H~ ;A A)I = lhkl calculated for the one-hour intervals ( ( k - l ) A , k A ] , k = 1 , 2 , . . . . 0.0020

0.0015

0.0010

0.0005

6

12

18

4

FIGURE 36. A-volatility of the DEM/USD cross rate during one day ( A = 1 hour), according t o neuters (05.10.1992-26.09.1993)

The above-mentioned daily periodicity of the A-volatility is also revealed by the analysis of its correlation properties. We devote the next section t o this issue and to the discussion of the practical recommendations following from the statistical analysis of A-volatility. 2. We discuss now the properties of the A-volatility vt(A) E v(o,tl(H;A ) regarded as a function of A for fixed t . We shall denote by i7t(A) its estimator B(o.t](H; A). Assume that t is sufficiently large, for instance, t = T = one year. We shall now evaluate &(A) for various A. No so long ago (see, i.g., [204], [362], [386], or [427]), the following remarkable property of the FX-market (and some other markets) was discovered: the behavior of the A-volatility is highly regular; namely,

QT(A) % cTAfl

(3)

3. Statistics of Volatility, Correlation Dependence. and Aftereffect in Prices

353

0.0020

0.0015

0.0010

0.0005

0

24

48

72

96

120

144

168

FIGURE 37. A-volatility of the DEM/USD cross rate during one week ( A = 1 hour), according t o Reuters (05.10.1992-26.09.1993). T h e intervals ( 0 , 1 ] , .. . , (167,1681 correspond t o the intervals (0:00, 1:00], . . . , (23:00,24:00] of G M T

-3.0

-4.0 -5.0 -6.0 -7.0 -8.0 6.0 8.0 10.0 12.0 14.0 16.0 FIGURE 3 8 . On the fractal structure of the A-volatility G*(A). The valucs of lni+(A) as a function of I n n are plotted along the vertical axis

with certain constant CT dependent on the currencies in question and with W zz 0.585 for the basic currencies. To give ( 3 ) a more precise form we consider now the statistical d a t a concerning logGT(A) as a function of I n A with A ranging over a wide interval, from 10 minutes (= 600 seconds) to 2 months (= 2 x 30 x 24 x 60 x 60 = 5 184 000 seconds).

354

Chapter IV. Statistical Analysis of Financial Data

The chart in Fig. 38, which is constructed using the least squares method, shows that the empirical data cluster nicely along the straight line with slope IHI 2 0.585. Hence we can conclude from (3) that, for t large, the volatility vt(A) regarded as a function of A has fractal structure with Hurst exponent W 0.585. As shown in 33a, we have E l H a l = J2/,A1/2 for a Brownian motion H = (Ht)t>o,EJHaI = @AW for a fractional Brownian motion H = (Ht)t>o with exponent W,and ElHal = E/H1jA" with W = 1/a < 1 for a strictly a-stable Lkvy process with a > 1. Thus, the experimentally obtained value W = 0.585 > 1 / 2 supports the conjecture that the process H = ( H t ) ,t 3 0; can be satisfactory described either by a frac1 1 tional Brownian motion or by an a-stable L6vy process with a = - M 1.7. W 0.585 Remark. As regards the estimates for MI in the case of a fractional Brownian motion, see Chapter 111, fj2c.6. ~

N

3. We now turn to Fig. 36. In it, the periods of maximum and minimum activity are clearly visible: 4:OO GMT (the minimum) corresponds to lunch time in Tokyo, Sydney, Singapore, and Hong Kong, when the life in the FX-market comes t o a standstill. (This is nighttime in Europe and America). We have already pointed out that the maximum activity (E 15:OO) corresponds to time after lunch in Europe and the beginning of the business day in America. The daily activity patterns during the five working days (Monday through Friday) are rather similar. Activity fades significantly on week-ends. On Saturday and most part of Sunday it is almost nonexistent. On Sunday evening, when the East Asian market begins its business day, activity starts to grow.

5 3c.

Correlation Properties

1. Again, we consider the DEM/USD exchange rate, which (as already pointed out in 3 la.4) is featured by high intensity of ticks (on the average, 3-4 ticks per minute on usual days and 15-20 ticks per minute on days of higher activity, as in July, 1994). The above-described phenomena of periodicity in the occurrences of ticks and in A-volatility are visible also in the correlation analysis of the absolute values of the changes IAHI. We present the corresponding results below, in subsection 3, while we start with the correlation analysis of the values of A H themselves.

st

2. Let St = (DEM/USD)t and let Ht = ln-.

We denote the results of an

-

SO

St appropriate linear interpolation (see 3 2b) by St and Ht = In -, respectively. SO We choose a time interval A: let I

I

3. Statistics of Volatility, Correlation Dependence, and Aftereffect in Prices

355

with t k = kA. (In 32b we also used the notation hi:); in accordance with §3b, I

Ihkt = q t k - l , t k ] ( HA).) ; Let A be 1 minute and let k = 1 , 2 , . . . ,GO. Then hl,hz,.. . , h60 is the sequence of consecutive (one-minute) increments of- H over the period of one hour. We can assume that the sequence h l , ha,. . . , h60 is stationary (homogeneous) on this interval. as a measure of the correlation dependence of stationary sequences - Traditionally, - h = ( h l ,h2, . . . ), one takes their correlation function

(the autocorrelation function in the theory of stochastic processes). The corresponding statistical analysis has been carried out by Olsen & Associates for the data in their (rather representative) database covering the period from .January 05, 1987 through January 05, 1993 (see [204]). Using its results one can plot the following graph of the (empirical) autocorrelation function p ^ ( k ) : 0.0

-0.1

0.0

10.0

20.0

30.0

40.0

50.0

60.0

FIGURE 39. Empirical autocorrelation p^(k) - - function for the sequence of increments h , = H t , - Ht,-l corresponding to the DEM/USD cross rate, with t , = nA and D = 1 Iniriiite

In Fig. 39 one clearly sees a negative correlation on the interval of approximately 4 minutes (?(I) < 0, c ( 2 ) < 0, F ( 3 ) < 0 , F(4) 6 0), while most of the values of the p^(k) with 4 < k < 60 are close to zero. Bearing this observation in mind, we can assume that the variables h, and h,, are virtually uncorrelated for ( n- m( > 4. We note that the p h e n o m e n o n of negative correlation on small intervals (In - 1711 3 minutes) was mentioned for the first time in [189] and [191]; it has been noticed for many financial indexes (see, for instance, [145] and [192]).

<

Chapter IV. Statistical Analysis of Financial Data

356

There exist various explanations in the literature of this negative correlation of the increments A E on sniall intervals of time. For instance; the discussion in [204] essentially reduces to the observation that the traders in the FX-market are riot uniform, their interests can ‘point to different directions’, arid they can interpret available iriforrnation in different ways. Traders often widen or narrow the spread when they are given instructions to ‘get the market out of balance‘. Moreover, many banks overstate their spreads systematically (see [I921 in this connection). A-possible ‘mathematical’ of the phenomenon of negative values of - -explanation C o v ( h I L-h,,,+~,) . = E/r,,h,,+~,-Eh,Eh,+l, for small k can be, e.g., as follows (cf. [481]). Let H , = 111 . .+h, with h, = ~ , , + ( T , , E ~ , where the (T, are 2Fn-1-measurable and ( E ~ is~ )a sequence of independent identically distributed random variables. We can also assume that the p n are 9,_1-measiirable variables. Judging by a large amount of statistical data, the ‘mean values’ pn are much smaller than (T,~ (see, e.g., the table in 3 2b.2) arid can - be set to be equal to zero for all practical purposes. I

+.

S,, are in practice not always knotun precisely; it would be SO niore realistic to assume that what we know are the values of H , = H , + 6,, where I

The values of H I , = In

-

I

(6,) is white noise, the noise component related to inaccuracies in our knowledge about the actual values of prices, rather to these values themselves. We assume that (hI,,)is a sequence of independent random with Ed,,, = 0 - variables arid €6; = C I I

-

I

> 0. Then, considering the sequence h

+

h,,, = A H T L= LIL (&

-

&-I)

=

(L) of the variables

we obtain

and

Herice the covariance function

(provided that En;

=

En;,

7%

3 1) can be described by the formula

- COV&>~i+lc)

=

k = I,

k > 1.

3. Statistics of Volatility, Correlation Drpendcnce, and Aftereffect in Pricps

357

3. To revcd cycles in volatilities by inearis of correlation analysis we proceed as follows. We fix the interval A equal to 20 minutes. Let to = 0 correspond to Monday, 0:OO GMT, Ict t i = A = 20 111, t 2 = 2A = 40 n1. ta = 3A = 1 11, . . . . t j 0 4 = 504 A = 1 week, . . . , t2016 = 2016 A = 4 weeks (= 1 month).

0.3

0.2

0.1

0.0

-0.1

0

504

1008

1512

2016

-

F-I G U R-40. E Empirical autocorrelation function R ( k ) for the sequence Ihr1/= lHtlL- Htn-ll corresponding to the DEM/USD cross rate (the data of Reuters; October 10, 1992-September 26, 1993; 1901, [204]). The value k = 504 corresponds t o 1 week and k = 2016 to 4 weeks

We set

Itla =

-

Ht,,

-

-

H f n - l ; let

-

-

-

be the autocorrclatiori function of the sequence Ihl = (lhll,lh21,. . . ) . The graph of the corresponding empirical autocorrelation function 6(k ) for k = 0 , 1 , . . . 2016 (that is, over the period of four weeks) is plotted in Fig. 40. One clearly sees in it, a periodic component in the - autocorrelation - - function of the with lh,l = jHtn-Ht,-lj, A = t t l L - t T L P 1 . A-volatility oft,lie scquence lh = (jh7L1)7L21 As is known, to demonstrate the full strength of the correlation methods one requires that tlic sequence in question be stationary. We see, however, that A-volaI

Chapter IV. Statistical Analysis of Financial Data

358

tility does not have this property. Hence there arises the natural desire to ‘flatten’ it in one or another way, making it a shtionary, homogeneous sequence. This procedure of ‘flattening’ the volatility is called ‘devolatization’. We discuss it in the next section, where we are paying most attention to the concept of ‘change of time’, well-known in the theory of random processes, and the idea of operational ‘8-time’, which Olsen & Associates use methodically (see [go], [204], and [362]) in their analysis of the data relating to the FX-market.

3 3d.

‘Devolatization’. Operational Time

1. We start from the following example, which is a good illustration of the main stages of devolatization’, a procedure of ‘flattening’ the volatilities. t

Let Ht = J, ~ ( udB,, ) where B = ( B t ) t > o is a standard Brownian motion and (r = (o(t))t>ois some deterministic function (the ‘activity’) characterizing the ‘contribution’ of the dB,,u 6 t , to the value of Ht. We note that

for each n I , where cTL A ( 0 , I ) , 0; = J“ c 2 ( u du, ) and the symbol means n-1 that variables coincide in distribution. Thus, if we can register the values of the process H = (Ht)t>oonly at discrete instants 71 = 1 , 2 , . . . , then the observed sequence h, = Hn - Hn-l has a perfectly simple structure of a Gaussian sequence (u,E,),>~ of independent random variables with zero means and, in general, inhomogeneous variances (volatilities) In the discussion that follows we present a method for transforming the data so that the inhoniogeneous variables (r:, n 3 1, become ‘flattened’. We set N

02.

1 t

T(t)=

2 ( u )du

where 8 3 0. We shall assume that g ( t ) > 0 for each t ensures that the stochastic integral

> 0,

(2)

/ t 02(u)du < 0

00

(this property

1 ~ ( ud B), with respect to the Brownian mot

0

t

tion B = (B,),>o is well defined; see Chapter 111, 3c), and let a 2 ( u )du t 00 ast+m. Alongside physical time t 3 0, we shall also consider new, operational ‘time’ 8 defined by the formula B = T(t). (4)

3 . Statistics of Volatility, Correlation Dependence, and Aftereffect in Prices

359

The ‘return’ from this operational time 0 to physical time is described by the inverse transfoniatiori t = T*(0). (5) We note that 2 ( u )d u = 0

(6)

by (3), i.e., T ( T * ( @ ) )= 8 , so that T * ( O ) = T - ’ ( O ) and T * ( T ( ~ ) = ) t. We now consider the function 0 = ~ ( tperforming ) this transformation of physical time into operational time. Since 02 - 01 =

1;*

2 ( u )du,

(7)

we see that if the activity c 2 ( u )is small, then this transformation ‘compresses’ the physical time (as in Fig. 41).

FIGURE 41. ‘Compression’ of a (large) time interval [ t l , t g ] characterised by small activity into a (short) interval [ e l ,821 of 8-time

On the other hand, if the activity g 2 ( u )is large‘ then the process goes in the opposite direction: short intervals ( t l ta) of physical time (see Fig. 42) correspond to larger intervals (81, 02) of operational time; time is being ‘stretched’. Now, we construct another process,

which proceeds in operational time. Clearly, we can ‘return’ from H* to the old process by the forrriula Ht = H*T ( t ) ’ (9) because

T * ( T ( ~ )= )

t.

Chapter IV. Statistical Analysis of Financial Data

360

FIGURE 42. 'Stretching' a (short) interval [ t l ,t 2 ] characterized by large activity into a (long) interval [el,031 of 8-time

Note that if 81 < 8 2 , then

~(u dBu ) 00

=

I(.*(8,)

< u < .*(O,))

47L)

dB,,

Hence H' is a process with independent increments, H," = 0 , and EH; = 0. Moreover, hy the properties of stochastic integrals (see Chapter 111, § 3c) we obtain

I ( ~ * ( O ,<)

<. * ( ~ , ) ) ~ ~ ( ~ ) d ~

(the last equality is a consequence of (6)). Since H * is also a Gaussian process and has independent increments, zero mean, property ( l o ) , arid continuous trajectories, it is just a standard Brownian m o t i o n (see Chapter 111, 3a), so that,

where o*(11) E 1.

sk

g ( u ) d B , , where C ( U ) $ 1 in general, shows that the transition to operational time has 'flattened' the characteristic of activity ~75 a ( u ) : when measured with respect t o new 'time' 8 , the level of activity is flat, ( a * ( u )= 1).

A comparison with the representation Ht

=

3. Statistics of Volatility, Correlation Dependence, and Aftereffect in Prices

361

We have assumed above that o ( u )is deterministic. However, H i = H7*(8)will be a Wiener process even if the change of time is r a n d o m , defined by formula ( 3 ) with

( ~ ( 7 1 )= O ( U ;w ) ,

provided that

t 0

a2(u;w ) CEIL < x, with probability one and

d u t 00 as t + 00. However, there exists a fundamental distinction between the cases of a deterministic function (T = ( ~ ( uand ) a r a n d o m function (T = a ( 7 ~ ; w )in : the first case we can calculate the change t -+ 8 = r ( t )in advance, also for the ‘future’, which is impossible in the second case, because the ‘random’ changes of time corresponding to distinct realizations (T = a ( u ; w ) , u 3 0, are distinct. Jk.‘(u;w)

2. We consider now a sequence h = ( 1 ~ ~ ~ )where ~ ~ 2 h1 , = with nonflat level of activity crrL, 71 3 1. We shall treat 71 as physical (‘old’) time. We introduce the sequence of times

with positive integer 0 that we treat as operatrorial (‘new’) time Also, let 7 * ( & l ) < k < r * (0)

for 0 = 1 , 2 , . . . , whcrc ~ * ( 0= ) 0. We note that E l i ; = 0 and tlie corresponding variances are

because tlie values of the (T: are usually fairly small (see the table in 3 2b.2). Thus, we can say that thc transition to new ‘time‘ B transfornis the inhomogeneous sequence h = ( / ~ ~ ) into ~ > an 1 (almost) homogeneous sequence h* = ( h r ) ) ~ 1 . If the oTLare random variables, that is nTL= a,(w), and our aim is to calculate the change of time for all instants (including the future), then we can implement the ( w ) by their mean values above-discuss idea of ‘devolatization’ by replacing the Eo;(w) or, in practice, by estimates of these mean values. We see from tlie representation h, = that if the on are .F:,-I-measurable, then E / L ? ~= Ea;. Hence if time 71 of GMT falls, for example, on Monday, then we can take an estimate for E ( T ~equal to the arithmetic mean of the values of hi over all k such that ( ( k - l ) A , kA] corresponds to the same time interval of some other Monday covered by the database. The change of time (2) is defined in terms of o(u)squared, which, of course; is not the only possible way to change time t -.+B = T ( t ) . For example, we could use the values of io(u)/in place of a 2 ( u ) .

02

362

Chapter IV. Statistical Analysis of Financial Data

3. This is just the change of time used by Olsen & Associates ([go], [360]--[362]). They say that this kind of ‘devolatization’ captures periodzczty better and the be/) havior pattern of the autocorrelation function of the ‘devolatized’ sequence (jl* for the DEM/USD exchange rate is more ‘smooth’. Referring to [go] for the details, we shall present only the result of their statistical analysis of the properties of the autocorrelation function of the sequence Ih*I. - As -in $ 3 ~ . 3 ,we assume that A is equal to 20 minutes, t n = nA, and hn = H t , - Ht,-l. In Fig. 42 ( $ 3 ~ we ) plotted the graph of the empirical estimate g ( k ) of the autocorrelation function

in which the periodic structure is clearly visible. In [go], after ‘devolatization’ and the transition to new, operational ‘time’ 6’ the authors plot a graph (see Fig. 43) of the empirical correlation function g*(O), Q 3 0, for the sequence lh*I = (lh;/)021. This graph is rather interesting for further analysis. In the same paper one can find a graph (see Fig. 44) of the transition T*(Q):6’ -+ t from operational time Q to real time t (just in our case of the DEM/USD cross rate; new time is normalized so that a week of physical time corresponds to a week of operational time). It is clear from Fig. 44 that the function t = ~ * ( 6 ’ )is approximately linear during the five business days, while on the week-end, when the transactions in the FX-market fade, large intervals of physical time correspond to small intervals of operational time. (Only the latter are in fact interesting for business trading). 4. It should be noted that the above method of ‘devolatizing’ in order to eliminate

the periodic Component is not the only one used in the analysis of the FX-market. For instance, we can point out the papers [7],[13], and [306], where most diversified techniques, linear and nonlinear regression analysis, Fourier transformation, neural networks are used to find similar patterns in financial time series. We also point out the paper [297] by I. L. Legostaeva and this author, which relates to the same range of problems. In it (in connection with the study of the Wolf numbers characterizing solar activity), to analyze the ‘trend’ component a = ( a k ) of a sequence E = ( E k ) with an additive ‘white noise’ q = ( q k ) (& = ak q k ) , the authors use a minimax approach suitable for the study of a much broader class of ‘trend’ sequences a = ( n k ) than ~ ~ the standard polynomials classes of regression analysis. (See also [45], [338], and [416].)

+

5 . For conclusion we present a graph of the periodic component of the ‘activity’

(see 3b) corresponding to the CHF/USD cross rate, which is isolated by means of ‘devolatization’ (see [go]; cf. also Fig. 37 in $ 3b).

3. Statistics of Volatility, Correlation Dependence, and Aftereffect in Prices

363

0.15

0.10

0.05

0.00 I

0 1000 2000 3000 4000 FIGURE 43. Empirical autocorrelation function g* (0) for the sequence Ill,* 1 = ( I / L ~ i ) s 2 l of the absolute values of ‘devolatized’ variables considered with respect t o operatronal ‘time’ 0, with interval At9 = 20 rn; the case of the DEM/USD cross rate ([go])

”t

FIGURE 44. Graph of the transition t = ~ * ( 0 from ) operational time 0 (plotted along the horizontal axis) to physical time (along the vertical axis). Time is measured in hours; 168 hours form 1 week ([go])

Chapter IV. Statistical Analysis of Financial Data

364

4.0

3.0

2.0

1.o

0.0

24 48 72 96 120 144 168 FIGURE 45. The thick curve is the periodic component of the ‘activity’ corresponding to the CHF/USD cross rate (over the period of 168 hours= 1 week) In Fig. 45 we clearly see the geography-related structure of t,he periodic component (with 24-hour cycle during t,he week) arising from the differences in business time between three major FX-markets: East Asia, Europe, and America. In [D4] one can find an interesting representation of this component as the sum of three periodic components corresponding t o these three markets. This can be helpful if one wants to take the factor of periodicity more consistently into account in the predictions of the evolution of exchange rates.

5 3e.

‘Cluster’ Phenomenon and Aftereffect in Prices

1. We assunicd in our initial scheme t h a t the exchange rates (prices) S = ( S t ) t 2 0

st

and their logarithiris Ht = In - could be described by stochastic processes with SO discrete iritcrvention of chance:

and

-

-

I

After that, we proceeded to their continuous modifications S = (St)t>o, H = (Ht)t?o and, finally, to the variables li,, = Ht, - H t r L p l where , tn - t,-l = A I

I

3. Statistics of Volatility, Correlation Dependence. and Aftereffect in Prices

365

-

(= const,). It was for t,liesc: variables hll arid for A equal-to 1 - rniriiite t h a t we tliscovered the negative autocovariance p ( k ) = Eh,h,+,+ - Eh,, Eh,+k for small values of k ( k = 1 , 2 , 3 , 4 ) ) .For k large the autocovariarice is-close to zero, therefore we can assiirrie for a11 practical purposes that the variables h, and hn+k are uncorrelated for such k . Of course we are f a r f r o m saying that they arc independent our analysis of the empirical aut,ocovariarice function g ( k ) in 5 3c (for the DEM/USD cross rate: as usual) showed that this was riot the case. The next step (‘dcvolatizat,ion’)enables us to flatten the level of ‘activity’ by means of the transition to new, operational time that takes - into account the different intensity of changes in the values of the process H = (fft)t>o on different int,ervals. As is clear from the statistical analysis of the sequence ((h,,l),bl considered with respect, t o operational time 0, the autocorrelation function g*(0) 1) is rottrer large f o r small 8; 2 ) decreases fairly slowly with the growth of 8 . It is rriaintained iri [go] that for periods of about one rriorith the behavior of f i * ( 0 ) can be fairly well described by a power function, rather t h a n an exponential; that is, wc have -

5

I

-

j i * ( ~ )/ i i - a rather tliari

E* (0)

-

exp(-BP)

as

B

as

+ m, B + 30,

(3)

(4)

as could be expected arid which holds in many popular models of financial matherriat,ics (for irist,anc:e, ARCH and GARCH; see [193] and [202] for greater detail). This property of the slow decrease of the empirical autocorrelation function R*(B)has irriportant practical consequences. It means that we have indeed a strong aftereffect in prices; ‘prices remember their past‘, so to say. Herice one may hope t o be able to predict price niovements. To this end, one must, of course, learn to produce scqueiic:es h = ( h11)n21that have a t least correlation properties similar to the ones observctl in practice. (See [89], [360], and Chapter 11, 3 3b in this connection). h

2. Tlic fact that the autocorrelation is fairly strmg for sniall B is a convincing explanation for the cluster property of the ‘activity’ measured in terms of the volatility of I F T L I . This property, known since B. Mandelbrot’s paper [322] (1963), essentially means the kind of behavior when large values of volatility are usually followed also by l i q e iialues and small- values are followed by small ones. That is, if the variation / h 7 , /= /Ht, - Ht,-l/ is large, then (with probability siificicritly close t o one) the next value, Ihr,+ll; will also be large. while if lhnl is small, then (with probability close to one) the next value will also be small. This I

Chapter IV. Statistical Analysis of Financial Data

366

property is clearly visiblr in Fig. 46; it can be observed in practice for many financial indexes. 0 010

0 005

0.000

-0.005 --

-0.010

'

0

504

1008

1512

-

2016

FIGURE 46. Cltister property of the variables h k = zjf) for the DEM/USD cross rat,e (the d a t a of Reuters; October 5, 1992-November 2, 1992; [427]). The interval A is 20 minutes, the mark 504 corresponds to one week, arid 2016 corresponds to four weeks. The clusters of large arid small values of I X , ~are clearly visible.

We note that this cluster property is also well captured by R/S-analysis, which we discuss in the next section.

4. Statistical R/S-Analysis

fj 4a. Sources and Methods of R/S-Analysis

1. In Chapter 111, §2a we described the phenomena of long memory and selfsimilarity discovered by H. Hurst [236] (1951) in the statistical data concerning Nile’s annual run-offs. This discovery brought him to the creation of the so-called R/S-analysis. This method is not sufficiently well-known to practical statisticians. However, Hurst’s method deserves greater attention, for it is robust and enables one to detect such phenomena in statistical data as the cluster property, persistence (in following a trend), strong aftereffect, long m e m o r y , f a s t alternation of successive values (antipersistence), t,he fractal property, the existence of periodic o r aperiodic cycles. It can also distinguish between ‘stochastic’ and ‘chaotic’ noises and so on. Besides Hurst’s keystone paper [236], a fundamental role in the development of R./S-analysis, its methods and applications belongs to B. Mandelbrot and his colleagues ([314], [316]-[319], [321]-[325], [327]-[329]), and also to the works of E. Peters (which include two monographs [385]and [386]),containing large stocks of (mostly descriptive) information about the applications of R/S-analysis in financial markets. 2. Let S = (S7b)7L20 be some financial index and let h, = In

Sn ~

,n>1.

Sn-1

The main idea of the use of R/S-analysis in the study of the properties of h = (hn),21 is as follows. Let H n = h l + . . . + h , , n > 1,andlet

(cf. Chapter 111, 52a).

Chapter IV. Statistical Analysis of Financial Data

368

Hn . The quantity 11, = - is the empirical mean of the sample ( h l ,hz, . . . , h,), I1 k k therefore Hk - -H7, = (hi - En) is the deviation of Hk from the empirical mean

11

k value -H,, I1

C

z=1

R, itself characterizes the range of the sequences

arid the quantity

( h ~. .,. , h,,) and ( H I , .. . , Hr,) relative t o their empirical means.

be the empirical varrance and let

be the norinalized, ‘adjusted range’ (in the terminology of [157]) of the cuniulative sunis H k , k 6 11. It is clear from (1)-(3) that Q, has the iniportant property of invariance under linear trarisforrnations h k + c ( h k m ) , k 3 1. This valuable property makes this statistics n.onpcirametric (at any rate, it is independent of the first two moments of the distributiuiis of h k , k 1).

+

3. In the case when h 1 , h,2,. . . is a sequence of independent identically distributed random variables with Eli,, = 0 and Dh, = 1, W. Feller [157] discovered that

(4) arid D%-

2

7T

( ~ - 2 ) n

(=0.07414 . . . x n )

(5)

for large n. This result can be readily understood using the Dorisker-Prokhorov invariance principle (see, e.g., (391 or [250]) asserting that the asymptotic distribution for R T L / fisi tlie distribution of the range

R* = SUP B: t
-

irif B:

t
of a Brownian, bridge Bo = ( B f ) t ~ i.e., l , of the process

Bf

Bt

-

tB1,

where B = (Bt)t>ois a standard Brownian motion (see Chapter 111, 3 3a).

(7)

4. Statistical R/S-Analysis

369

Indeed, let u s consider the variable

By the very rrieariirig of invariance (i.e., the independence of the particular form of the distribution of the h k ) , in looking for the limit distribution for R n / & as n + m we can assume that the h k have the standard normal distribution X ( 0 , l ) . If B = (Bt)t>ois a standard Brownian motion, then the probability distributions for the collectioris { H k / & , k = 1 , .. . , n} and { l ? k l n , k = 1 , .. . , n } are the same, so that

Rn -

fi

[Bt - t B l ] -

max { t : t=;, ...,+,

d -

niax

BF

min

-

{ t : t=; ,...,+,1}

[Bt - tB1]

min { t : 4, ...,+,

1}

BO t ,

1}

(9)

{ t : t=: ,.._,+,l}

d

where the symbol means coincidrnce in distribution. Hence it is clear that, as n + 00, the distribution of R n / f i (weakly) converges to that of R*. (Note that the functional ( m a x - m i n ) ( . ) is continuous in the spare of right-continuous functions having limits from the left. See, for example, [39;Chapter 61, [250; Chapter VI], and [304]about this property and the convergence of measures in such function spaces.) We know the following explicit formula for the density f*(x) of the distribution function F * ( T )= P(R* x) (see formula (4.3) in [157]):

<

where e(z) = e - 2 2 2 . Using this formula it is easy to see that

We note by the way that E sup lBtl =

J.ir/2;see Chapter 111, 3 3b. Hence the mean

t
values of the statistics R* and sup IBtl are the same. t
Chapter IV. Statistical Analysis of Financial Data

370

4. Assuming that the variables h l , ha,. . . are independent and identically dis-

tributed, with Ehi = 0 and Dhi = 1, we see that S: + 1 with probability one as 71 + co (the Strong law of large numbers). Hence the limit distribution for Qn/fi as 71 + 00 is also the distribution of R*. The distribution of Q, is independent of the mean value and the variance of hk, k ‘71. This ‘rionparanietric’ property brings us to the following criterion, which enables one to reject (with one or another level of reliability) the following hypothesis ,XOlying in the foundation of the classical concept of an eficient market (see Chapter I, 5 5 2a, e): the prices in question are governed by the random walk model. The main idea of this criterion, which is based on the R/S-statistics, is as follows (G. Hurst. [236]; [329], [386]). If hypothesis @ ‘ 3 is valid, then for sufficiently large n the value of X , / S n must

<

(Of course, one can-and must-assign a precise probabilistic meaning to this relation and to relations (13) and (14) below by referring to liniit theorems. We shall not do t,his here, taking the standpoint of a ‘sensible’ interpretation, which is common in practical statistics.) Thus, choosing a logarithmic scale along both axes we see that the values of

2,

In - must ‘cluster’ (provided HOholds) along the straight line a Sn a = In @ (see Fig. 47).

+

I n n with

m

2.0 -1.5 -1.0 --

0.5 -r 0.0 -I b 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Inn FIGURE 47. To R/S-analysis (the case of the validity of hypothesis .Ye,)

This rriakes clear the method of XIS-analysis: given statistical data, we plot the points (lnn,ln

2)

(i.e., we use logarithmic scales); then, using the least squares

4. Statistical R/S-Analysis

371

+

method, we draw a line Er1 b, lrm. If the value of b, ‘significantly’ deviates from 1 / 2 , then hypothesis Xo must be rejected. (Of course, one must master the standard methods of statistical analysis so as t o be able to calculate the significance of the deviation of ZTLfrom 1 / 2 under the hypothesis 20. This is riot easy because the distribution of the statistics R,/S, with finite n is difficult t o find explicitly; we shall touch upon this subject below, in subsection 6.) The principal significance of Hurst’s research lies in his discovery (by means of R/S-analysis) that, in place of the expected (for Nile and other rivers) property

one actually has

R?l Srl with IH considerably larger than 1/2.

-

cn’

5. If an empirical study shows (somewhat unexpectedly) that (14) holds, then one should ask about models governing sequences h = (h,)+l with this property. We must also explain why W > l / 2 in many cases. (As we shall see below, one possible explanation is that h = (h,)TL21is a sequence with long m e m o r y and positive correlation,) In this connection, we discuss now several ideas, which are based, in particular, on numerical computer calculations and several results of R/S-analysis. Doing that we shall mainly follow [316] and [319]. As a rule, if a sequence h = (hn)n)l is characterized by ‘weak dependence’ (as are Markov, autoregressive, and some other sequences) then W (the H w s t paremehas t e r ) is close to l / 2 . One usually says in this case that the system h = (&),>I a ’short R/S-nieniory’. If hrl = & ( n ) - & ( n - l ) , n 3 1, where BH = (B%(t))tao is a fractional Brownian motion (Chapter 111, 5 2b), then the variables

Rn

have a nontrivial SnnH limit distribution as n + 00,which can be indicated as in (14). Hence if a statist,ical study shows t,hat 0 < W < 1 and W # l / 2 , then a fractional Gaussian noise is one possible explanation. We recall (see Chapter 111, 3 2c) that in the case of such a noise, the corresponding correlation is positive if W > 1 / 2 and negative if W < l / 2 . This explain why one talks about persistence in trend-following, a ‘long memory’, a ‘strong aftereffect’ in the first case: if the values have increased, then the odds are that they will increase fiirt her. As pointed out in [386], the (hitherto widespread in the literature) opinion t,hat W must be always larger than 1 / 2 in financial time series is ungrounded. The case MI < l / 2 occurs, for instance, for the returns of volatility (see Chapter 111, 52d.5 and S4b.3 in this chapter), which are featured by strong alternation (‘antipersistence’). ~

Chapter IV. Statistical Analysis of Financial Data

372

6. We have already mentioned that in using R/S-analysis to determine the values of IHI in the conjectural relation (14) we must, of course, determine the degree of agreement between the empirical data and the model corresponding to the chosen value of W. In other words, there arises a standard question on the reliability of statistical inference, and one has to turn to the ‘goodness-of-fit’, ‘significance’, or other tests developed in mathematical statistics. It should be pointed out that the complexity of the statistics

R, leaves no hope

-

s,

for satisfactory formulas describing its probability distributions for various n even if we accept the null hypothesis X o . (Nevertheless, the question of the behavior of the

R, where Eo is averaging under condition Yfo,has been considered

mean value Eo-,

ST,

in [ 8 ] . ) This explains the widespread use of the Monte Carlo method in R/S-analysis (see, for instance, [317], [329], [385], [386]), in particular, to assess the quality of the estimation of the unknown value of IHI.

7. When adjusting theoretical models to real statistical data it is reasonable to start with simple schemes allowing an easy analytic study. For instance, describing the probabilistic structure of the sequences h = (h71),L21one can well assume that this is fractional Gaussian noise with 0 < IHI < 1. If W = l / 2 , then we obtain usual Gaussian white noise, which lies at the core of many linear ( A R , MA, ARMA) and nonlinear ( A R C H , GARCH) models. R/S-analysis produces good results in models where h = (h,),>l is a sequence of a fractional Gaussian noise kind (see [317], [329], [385], - 13861). In dealing with ‘I& -,

other models it, is reasonable to consider, besides Q,

sn

the statistics

A mere visual examination of the latter often brings important statistical conclusions. This method is based on the thesis that, for .white’ noise (W= 1/2) the statistics V,, must stabilize for large n (that is, V,, + c for some constant c, where we understand the convergence in a suitable probabilistic sense). If h = ( h 1 , ) , ) l is fractional Gaussian noise with IHI > 1/2, then the values of VTL must zncrease (with n ) ;by contrast, V , must decrease for MI < l / 2 . Bearing this in mind we consider now the simplest, autoregressive niodel of order onc ( A R ( l ) ) in , which liTL = a0

+ cul h,-1 + E

~

,

n

> 1.

(16)

The behavior of the sequence in this niodel is completely determined by the ‘noibe’ terms E,, and the initial value h o .

373

4. Statistical R/S-Analysis

The corresponding (to these hn,n 3 1) variables V = (V,)n>l increase with ri. However, this does not mean that we have here fractional Gaussian noise with W > 1/2 for the mere reason that this growth can be a consequence of the linear dependence in (16). Hence, to understand the ‘stochastic’ nature of the sequence E = ( E % ) ~ > I in place of the variables h = (hn),21 one would rather consider the linearly adjusted variables ho = ( h : ) , ~ l ,where h: = h, - (a0 alh,-l) and ao, a1 are some estimates of the (unknown, in general) parameters, a0 and ~ 1 . Producing a sequence h = (h,),>l in accordance with (16), where E = ( E ~ ) , > I is white Gaussian noise, we can see that, indeed, the behavior of the corresponding variables V z = V n ( h o ) is just as it should be for fractional Gaussian noise with W = 1/2. The following picture is a crude illustration to these phenomena:

+

A 1.4 --

Vn( h ) 1.2

--

1.0

--

0.8

i

v; = V,(hO)

+

(sr>)

FIGURE 48. Statist,ics V T L ( hand ) V,”,= V n ( h o )for the A R ( 1 ) model The expectation Eo

~

is evaluated under condition X o

If we consider the linear models MA(1) or ARMA(1, l),then the general pattern (see [386; Chapter 51) is the same as in Fig. 48. For the nonlinear ARCH and GARCH models realizations of V,(h) and V,(ho) show another pattern of behavior relative to Eo

(,-:&), ~

n

3 1 (see Fig. 49).

First, the graphs of V,,(h) and V n ( h o )are fairly similar, which can be interpreted as the absence of linear dependence between the h,. Second, if n is not very large, then the graphs of V,,z(h)and V,,(ho) lie slightly above the graph of Eo

(2&) ~

corresponding to white noise, which can be interpreted as a ‘slight persistence’ in the model governing the variables h,, n 3 1. Third, the ‘antipersistence’ effect comes to the forefront with the growth of n.

Chapter IV. Statistical Analysis of Financial Data

374

t

0.8

4 0.5

1.0

1.5

2.0

2.5

3.0

t Inn

FIGURE 49. Statistics V n ( h )and V i = V n ( h o )for the A R C H ( 1 ) niodel

R e m a r k . The terms ‘persistence, ‘antipersistence’, and the other used in this section are adequat,e in the case of models of fractional Gaussian noise kind. However, the ARCH, GARCH, arid kindred models (Chapter 11, 5 5 3a, b) are n e i t h e r fractional nor self-similar. Herice one must carry out a more thorough investigation to explain the phenomena of ‘antipersistence’ type in these models. This is all the more important as both linear and nonlinear models of these types are very popular in the analysis of firiaricial time series and one needs to know what particular local or global features of empirical data related to their behavior in time can be ‘captured’ by these rnodels. 8. As pointed out in Chapter I, 5 2a, the underlying idea of M. Kendall’s analysis of stock and cornniodity prices [269] is t o discover cycles and trends in the behavior of these prices. Market analysts and, in particular, ‘technicians’ (Chapter I, 3 2e) do their research first of all on the premise that there do exist certain cycles and trends in the market, that market dynamics has rhythmic nature. This explains why, in the analysis of financial series, one pays so much attention to finding similar segments of realizations in order t o use these similarities in the predictions of the price development. Statistical X/S-analysis proved to be very efficient not only in discovering the above-meritioncd phenomena of ‘aftereffect’, ‘long memory’, ‘persistence’, and ‘antipersistence’, but also in the search of periodic or aperiodic cycles (see, e.g., [317], [319], [329]. [385], [386].) A classical example of a system where aperiodic cyclicity is clearly visible is solar activity. As is known, there exists a convenient indicator of this activity, the Wolf nu,mbers related to the number of ‘black spots’ on the sun surface. The monthly data for approximately 150 years are available and a mere visual inspection reveals the

4. Statistical RIS-Analysis

375

existence of an 11-year cycle.

FIGURE 50. R/S-arialysis of the Wolf numbers ( n = 1 month, 2.12 = 10g10(12. l l ) , h, = xn - xn-l, zTLare the monthly Wolf numbers)

The results of the X/S-analysis of the Wolf numbers (see [385;p. 781) are roughly as shown in Fig. 50. Estimating the paranieter W we obtain IH = 0.54, which points t o the tendency of keeping the current level of activity (‘persistence’).It is also clearly visible in Fig. 50 A

)I:(

that, the behavior pattern of In

-

changes drastically close t o 11 years: the

R values of 7stabilize, which can be explained by a 11-year cycle of solar activity. 1).

37,

Indeed, if there are periodic or ‘aperiodic’ cycles, then the range calculated for the second, third, and the following cycles cannot be much larger than its value obtained for the first cycle. In the same way, the empirical variance usually stabilizes. All this shows that R/S-analysis is good in discovering cycles in such phenomena as solar activity. In conclusion we point out that the analysis of statistical dynamical systems usually deals with two kinds of noise: the ‘endogenous noise’ of the system, which in fact determines its statistical nature (for example, the stochastic nature of solar activity) and the ‘outside noise’, which is usually an additive noise resulting from measurement errors (occurring, say, in counting ‘black spots’, when one can take a cluster of small spots for a single big one). All this taken into account, we must emphasize that RIS-analysis is robust as concerns ‘outside noises’; this additional feature makes it rather efficient in the study of the ‘intrinsic’ stochastic nature of statistical dynamical systems.

Chapter IV. Statistical Analysis of Financial Data

376

94b. R/S-Analysis of Some Financial Time Series 1. On gaining a general impression of the XIS-analysis of fractal, linear, and nonlinear models and its use for the detection of cycles, the idea to apply it to concrete financial time series (DJIA, the S&P500 Index, stock and bond prices, currency cross rates) comes naturally. We have repeatedly pointed out (and we do it also now) that it is extremely important for the analysis of financial data to indicate the time intervals A at which investors, traders, . . . ‘read off’ information. We shall indicate this interval A explicitly, and, keeping the notation of 3 2b, we shall denote by hi the variable hzA = In

, where St is the value a t time t of the financial index under

~

s(z-l)A

consideration.

Remark. One can find many data on the applications of R/S-analysis to time series, including financial ones, in [317], [323], [325], [327], [329] and in the later books [385] and [386]. The brief outline of these results that is presented below follows mainly [385] and [386]. 2. DJIA (see Chapter I, 3 lb.6; the statistical data are published in T h e Wall Street Journal since 1888). We set A to be 1, 5 , or 20 days. In the following table we sum up thc results of the R/S-analysis: A I

Number of obseravtioris N 12 500

Estimate ,-. WN

Cycles (in days) I

5 days 20 days

As regards the behavior of the statistics V,, one can see that the corresponding values first increase with r i . This growth ceases for n = 52 (in the case of A equal to 20 days; that is, in 1040 days), which indicates the presence of a cyclic component in the data. For the daily data (A is 1 day) the statistics Vn grows till n is approximately 1250. Thereupon, its behavior stabilizes, which indicates (cf. Fig. 49 in 5 4a) a formation of a cycle (with approximately 4-year period; this is usually associated with the 4-year period between presidential elections in the USA). 3. The S&P500 Index (see Chapter I, 3 lb.6; 5 2d.2 in this chapter) develops in accordance with a ‘tick’ pattern. There exists a rather comprehensive database for t,liis index. The analysis of monthly data (A = 1 month) from January, 1950 through July, 1988 shows ([385; Chapter 8]), that, as in the case of DJIA, there exists an approximately 4-year cycle.

377

4. Statistical R/S-Analysis

A more refined R/S-analysis with A

= 1, 5, and 30 minutes (based on the data

from 1989 through 1992; [386; Chapter 91) delivers the following estimates Hurst parameter: 6 = 0.603, 0.590, and 0.653, which indicates a certain ‘persistence’ in the dynamics of S&P500.

-

of the

(1)

,_

s, S,It is worth noting that the transition from the variables h, = In =,

= S,A,

Sn-1

to the ‘linearly adjusted’ ones,

ILi results in srrialler estimates of

-

= h,

-

+ Ulhn-l),

(a0

IHI (cf. (1)). Now,

6 = 0.551,

0.546, and 0.594,

which is close to the mean values E 0 6 (equal to 0.538, 0.540, and 0.563, respectively) calculated under condition .Ye, on the basis of the same observations. All this apparently means that for short intervals of time the behavior of the S&P500 Index can be described in the first approximation by traditional linear models (even by simple ones, such as the AR(1) model). This can serve an explanation for the wide-spread belief that high-frequency interday data have autoregressive character and for the fact that ‘daily’ traders make decisions by reacting to the last ‘tick’ rather then taking into account the ‘long memory’ and the past prices. The pattern changes, however, with the growth of the time interval A determining ‘time sharp-sightedness’ of the traders. In particular, the R/S-analysis carried out for A equal to, e.g., 1 month clearly reveals a fractal structure; namely, the value IHI GZ 0.78 calculated on the 48-month basis is fairly large (the data from January, 1963 through December, 1989; [385; Chapter 81). We have already mentioned that large values of the Hurst parameter indicate the presence of ‘persistence’, which can bring about trends and cycles. In our case, a mere visual analysis of the values of the V , clearly indicates (cf. Fig. 49 in 5 4a) the presence of a four-year cycle. As in the case of DJIA, this is usually explained by successive economic cycles, related maybe to the presidential elections in the USA. It is appropriate to recall now (see 53a) - that the statistical R/S-analysis of A

on

the daily values of the variable F, =, In : where the empirical dispersion 5, is on- 1 calculated by formula (8) in 5 3a on the basis of the S&P500 Index (from January 1, 1928 through December 31, 1989; [386, Chapter lo]) suggests an approximate value of the Hurst parameter of about 0.31, which is much less than 1 / 2 and indicates the phenomenon of ‘antipersistence’. One visual consequence of this is the fast alternation of the values of F,. In other words, if for an arbitrary n the value of 5, is larger than that of sn-1,then the value of 3,+1 is very likely to be smaller than 3,.

378

Chapter IV. Statistical Analysis of Financial D a t a

4. R/S-analysis of stock prices enables one not merely to reveal their fractal

structure and discover cycles, but also to compare them from the viewpoint of the riskiness of the corresponding stock. According to the data in [385; Chapter 81, the values of the Hurst parameter W = W(.) and the lengths @ = C(.) (measured in months) of the cycles for the S&P500 Index and the stock of several corporations included in this index are as follows:

W(S&P500) = 0.78, W(1BM) = 0.72, W(App1e Computer) = 0.75, W(Conso1idated Edison) = 0.68,

C(S&P500) = 46; C(1BM) = 18; C(App1e Computer) = 18; @(Consolidated Edison) = 0.90.

As we see, the S&P500 Index on its own has Hurst parameter larger than the corporations it covers. We also see that the value of this parameter for, say, Apple Computer is rather large (W = 0.75), considerably larger than the parameter characteristic for, say, Consolidated Edison. We note further that if W = 1, then a fractional Brownian motion has the representation B1 ( t ) = t<>where is a normally distributed random variable with expectation zero and variance one. All the ‘randomness’of B1 = (Bl(t))t>ohas its source in <, so that this is the least ‘noisy’ of all fractional Brownian motions with 0 < W 6 1. Clearly, the noise component of the process Bw ’decreases’ as W t 1. In the financial context this means that models based on these processes become less risky. This observation of the decrease of the noise component can be put in the form of a precise statement: the process Bw converges weakly to B1 as W I‘ 1. We note also that the correlation functions pw(n) = Ehkh,+k are positive for W > 1/2 (‘persistence’, the property of keeping the trends of dynamics) and p w ( n ) + 1 for all n as W t 1. As rioted in [385], such an interpretation seems more promising in cases where the processes in question have a large Hurst parameter W,but do not have variance, which lies a t the core of the concept of riskiness. One reasonable explanation of the fact that the value of W corresponding to the S&P500 Index is large, so that securities based on this index are less risky than corporate stock, is diversification (see Chapter I, §2b), which reduces the noise fact or.

<

R e m a r k 1. We emphasize that by ‘lesser risks’ here we mean ‘less noise’, ‘stronger persistence’, showing itself in the tendency to preserve the development trend. It should be pointed out, however, that systems with large W are prone to sharp changes of th,e di~rectionof evolution. A long series of ups can be followed by a long series of downs. Remark 2. Turning back t o the values of W and C for the stock of various corporations, we should also mention the following observation made in [385]: usually, a high level of innovations in a company results in large values of W and short cycles, while a low level of innovations corresponds to small values of W and long cycles.

379

4. Statistical R/S-Analysis

5 . Bonds. The R/S-analysis of the monthly data concerning the American 30-year Treasury Boricls (T-Bonds) over the period from January, 1950 through December, 1989 (see [385; Chapter 81) also discovers a rather high value of the fractality paranieter W M 0.68, with cycles of approximately 5 years.

-

6. Currency cross rates. There exist considerable differences between currency cross rates and such financial indexes as DJIA, the S&P500 Index, or stock and bond prices. The point is that a purchase or a sale of stock has a direct relation to investm e n t s in the corresponding industry. The currency is different: it is bought or sold to facilitate consumption, the expansion of production, and so on. Moreover, large volumes of exchange influence a t least two countries, are determined t o a considerable degree by their political and economical situations, and depend strongly on the policy of the central banks (who make interventions in the nioney markets, change the interest rates, and so on). Of course, these factors influence the statistical properties of cross rates and their dynamics. One important characteristic of the changeability of cross rates is the A-volatility defined in terms of the increments IHka - H ( k - l ) a

1,

st

where Ht = In - for the cross

SO

rate St (see, for example, formula (5) in 3 Ic). We must emphasize in this connection that the above-discussed statistics

R,

Rn and - are also, in effect, characteristics of the changeability, ‘range’of the process S7l

H = ( H t ) t > ~It. is therefore not surprising that many properties of exchange rates already described in the previous sections can be also detected by R/S-analysis. By contrast to such financial indexes as DJIA or the S&P500, the fractal structure (at any rate, for small values of A > 0) arid the tendency towards its preservation in time are clearly visible in the evolution of the exchange rates. R7l This is revealed by a mere analysis of the behavior of the statistics In - as

sn

functions of logn by the least squares method. This analysis shows that the Val-

Rn n 3 1, distinctly cluster along the line S,,

ues of In -,

+ Wlnn, with h

c

h

W consid-

erably larger than 1/2 for most currencies. For instance, G(JRY/USD) % 0.64, $(DEM/USD) M 0.64, ~@GBP/USD)M 0.61. All this means that currency exchange rates have fractal structures with rather large Hurst parameters. It is worth recalling in this connection that, in the case of a fractional Brownian motion, EH? grows as ltI2’. Hence for W > 1/2 the dispersion of the values of lHtl is larger than for a standard Brownian motion, which indicates that the riskiness of exchange grows with time. Arguably, this can explain why high-intensity short-term trading is more popular in the currencies market than long-term operations.

Part 2

THEORY

Chapter V. Theory of Arbitrage in Stochastic Financial Models. Discrete T i m e 1. I n v e s t m e n t p o r t f o l i o on a

3 la.

(B, S)-market

Strategies satisfying balance conditions . . . . . . . . . . . . . . . . . . . . fj l b . Notion of ‘hedging’. Upper and lower prices. and incomplete markets . . . . . . . . . . . . . . . . . . . . . . . 3 lc. Upper and lower prices in a single-step model . . . . . . . . . . . . . . . 3 Id. CRR-model: an example of a complete market . . . . . . . . . . . . . . 2. A r b i t r a g e - f r e e m a r k e t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2a. ‘Arbitrage’ and ‘absence of arbitrage’ . . . . . . . . . . . . . . . . . . . . . 3 2b. Martingale criterion of the absence of arbitrage. First fundamental theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2c. Martingale criterion of the absence of arbitrage. Proof of sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fj 2d. Martingale criterion of the absence of arbitrage. Proof of necessity (by means of the Esscher conditional transformat ion) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2e. Extended version of the First fundamental theorem . . . . . . . . . . . 3 . Construction o f martingale measures by means of a n a b s o l u t e l y c o n t i n u o u s change of measure . fj 3a. Main definitions. Density process . . . . . . . . . . . . . . . . . . . . . . . . fj 3b. Discrete version of Girsanov’s theorem. Conditionally Gaussian case 3 3c. Martingale property of the prices in the case of a conditionally Gaussian and logarithmically conditionally Gaussian distributions 3 3d. Discrete version of Girsanov’s theorem. General case . . . . . . . . . § 3e. Integer-valued random measures and their compensators. Transformation of compensators under absolutely continuous changes of measures. ‘Stochastic integrals’ . . . . . . . . . . . . . . . . . 3 3f. ‘Predictable’ criteria of arbitrage-free ( B ,S)-markets . . . . . . . . . . 4. C o m p l e t e and p e r f e c t a r b i t r a g e - f r e e markets . . . . . . . fj 4a. Martingale criterion of a complete market. Statcnicnt of the Second fundamental theorem. Proof of necessity fj 4b. Representability of local martingales. ‘S-representability’ . . . . . . 3 4c. Representability of local martingales (‘Ir-representability’ and ‘(p-v)-representability’) . . . . . . . . . . . . 5 4d. ‘S-rcpresentability’ in the binomial CRR-model . . . . . . . . . . . . . 3 4e. Martingale criterion of a complete market. Proof of necessity for d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4f. Extended version of the Second fiindarrierital theorem . . . . . . . . .

383 383 395 399 408 410 410 413 417

417 424 433 433 439 446 450

459 467 48 1 48 1 483 485 488 49 1 497

1. Investment Portfolio on a ( B ,S)-Market

3 la.

Strategies Satisfying Balance Conditions

1. We assume that the securities market under consideration operates in the conditions of ‘uncertainty’ that can be described in the probabilistic framework in terms of a filtered probability space

We interpret the flow F = ( 9 7 a ) r Lof2 ~0-algebras as the ‘flow of information’ 9, accessible to all participants up to the instant n , n 3 0. We shall consider a ( B ,S)-market formed (by definition) by d 1 assets:

+

a bank accomt

B (a ‘risk-free’ asset)

and stock (‘risk’ assets)

s = (s’. . . . , sd).

We assume that the evolution of the bank account can be described by a positive stochastic sequence B = (B&O> where the variables B,, are ,F7,-1-measurable for each ‘n 3 1. The dynaniics of the (value) of the i t h risk asset SZ can also be described by a positive stochastic sequence

si= (s;)n>O, where the Sft are 9,,-rneasurable variables for each n 3 0. From these definitions one can clearly see the crucial difference between a bank account and stock. Namely, the 9,-1-measurability of B, means that the state of

384

Chapter V. Theory of Arbitrage. Discrete Time

the bank account at time n is already clear (provided that one has all information) at t i m e n - 1: the variable B, is predictable (in this sense). The situation with stock prices is entirely different: the variables Sg are 9,-measurable, which means that their actual values are k n o w n only after o n e obtains all ‘information’ 9,arriving at time n. This explains why one says that a bank account is ‘risk-free’ while stocks are ‘risk’ assets. Settinn

we can write

where the (interest rates) r,, are Sn-1-measurable and the p i are 9,-measurable. Thus, for n 3 1 we have

and

sf, = s;

(Itpi).

(5)

l
In accordance with our terminology in Chapter 11, 3 l a , we say that the representations (4) and (5) are of ‘simple interest’ kind. 2. Imagine an investor on the (B,S)-market that can a) deposit m o n e y i n t o t h e bank account and borrow from it; b) sell and buy stock. We shall assume that a transfer of nioney from one asset into another can be done with no transaction costs and the assets are ‘infinitely divisible’, i.e., the investor can buy or sell any portion of stock and withdraw or deposit any amount from the bank account. We now give several definitions relating to the financial position of an investor in such a ( B ,S)-market and his actions.

DEFINITION 1. A (predictable) stochastic sequence

1 . lnvestrnent Portfolio on a

( B ,5’)-Market

385

d

where /3 = (,&(w)),>o and y = (yA(w),. . . , 3 ; 1 ( ~ ) ) ~ 2 0with .Fn-l-measurable b,(w) and -yR(w) for all n 3 0 and z = 1 . . . . d (we set .4”-1 = 9 0 ) is called a n investment portfolzo on the ( B ,5’)-market. = 1, then we shall write yn and S, in place of 7; and SA. We must now emphasize several important points. The variables &(w) and y i ( w ) can be positive, equal to zero. or even negative, which means that the investor. in conformity with a ) and b), can borrow from the bank account or sell stock short. The assumption of 9,-1-rneasurability means t h a t the variables &(w) and rb(w) describing t h e financial position of the investor at time n (the amount he has on the bank account. the stock in his possession) are determined by the information available a t time n - 1. not n (the ‘tomorrow’ position is completely defined by the ‘today’ situation). Time n = 0 plays a special role here (as in the entire theory of stochastic processes based on the concept of filtered probability spaces). This is reflected by the fact that the predictability a t the instant n = 0 (formally, this must be equivalent to ‘9-1-measurability‘) is the same as 90-measurability. (The agreement 3-1 = ~90 in the definition is convenient for a uniform approach to all instants n 3 0.) To emphasize the dynamics of a n investment portfolio one often uses the term ‘investment strategy’ instead. We shall also use it sometimes. We have assumed above that n 3 0. Of course. all the definitions will be the same if we are bounded by sorne finite ’time horizon‘ N . In this case we shall assume that 0 < 71 < 1V in place of n 3 0.

If d

DEFINITION 2 . The value of a n znvestment portfolio

7-r

is the stochastic sequence

where

To avoid lengthv formulas we shall use .coordinate-free‘ notation in what follows. denoting the scalar product

386

Chapter V. Theory of Arbitrage. Discrete Time

For two arbitrary sequences a = (an)n2O and b = (bn)nBO we have

Applying this formula (of 'discrete differentiation') to the right-hand side of (7) we see that AX: = [PnABn ir n A S n ] [Bn-lAPn Sn-lAyn]. (9)

+

+

This shows that changes (AX: z -Y: - X Z - , ) in the value of a portfolio are, in general, sums of two components: changes of the state of a unit bank account and of stock prices (PnABn TnAS'n) and changes in the portfolio composition (B n - 1ADn + Sn- 1 A*(,). It is reasonable to assume that real changes of this value the increments A B n and ASn (and not to A&, and Ay,). are always due to Thus. we conclude that the capital gains on the investment portfolio x are described by the sequence G" = (G:)n>o,where GC = 0 and

+

Hence the value of the portfolio a t time

R

is

which brings us in a natural way to the following definition.

DEFINITION 3 . We say that an investment portfolio x is self-financzng if its value .YT = (-Y:)n20 can be represented as the following sum:

That is. self-financing here is equivalent t o the following condition describing 'admissible' portfolios i ~ :

The message is perfectlv clear: the change of the amount on the bank account can be only due to the change of the package of shares and vice versa. We denote the class of self-financing strategies 7-r by SF. We note also that we have shown the equivalence of relations (6) t (13) a (6) + ( 1 2 )

1. Investment Portfolio on a

(B, S)-Market

387

3. It is perfectfly clear that in the operations with the portfolio ~TT one would better reduce the nunit)cr of different assets in it or, at least, simplify their structure. In accordance with one possible (and comnionly used here) approach, 'if we a przorz have B,, > 0, 71 3 1, then we can set B,, = 1'. This is a coriseqiience of the following observation. Along with the ( B ,S)-market we can consider a new market (g, where

s),

and

Then the value

2.

=

(Z:),QO

of the portfolio

7r

= ([J, y) is as follows:

In addition, if ?r is self-financing in the (B,S)-market, then it has this property also on the (B, ?)-market, for (see (13))

for

j7

E SF, or, more cxplicitly,

Thiis, from (14) and (16) we see that the discounted value X T = ~

?r

E SF satisfies the relation

B

(Z)

71>0

of

Chapter V. Theory of Arbitrage. Discrete Time

388

which, for all its simplicity, plays a key role in many calculations below based on the concept of ‘arbitrage-free market’. It follows from the above that (6)

+ (12)

===+

(6)

+ (18).

It is easy to verify that the reverse implication also holds. Thus, we have the following relation, which can be useful in the construction of self-financing strategies and in the inspection of whether a particular strategy is self-financing: (6)

+ (13)

I(6)

+ (12)

-

(6)

+ (18)

We note also that (18) is in a certain sense more consistent from the financial point of view than the equality AX; = &AB, +mAS,,

(19)

following from (12). Indeed, comparing prices one is more interested in their relative values, than in the absolute ones. (We have already mentioned this in Chap-

-

ter I, 2a.4.) This explains why we consider the discounted variables B =

- ( B) s

B - (= 1) B

and S = in place of B and S. Our choice of the bank account as a discount factor is a mere convention; we could take any of the assets S1,.. . , Sd or their combination instead. However, the fact that the bank account is a ‘predictable’ asset simplifies the analysis to a certain extent. For instance, this ‘predictability’ means that the distribution Law

(2 s,,,> ~

~

can be determined from the conditional distribu-

tion Law(Xz 1 9,-1), because the ‘condition’ 9,-1 means that the variable B, is known. Moreover, the bank account plays the role of a ‘reference point’ in economics, a ‘standard’ in the valuation of other securities, which display the same behavior ‘on the average’. 4. The above-discussed evolution of the capital X” in a ( B ,S)-market relates to the case when there are no ‘inflows or outflows of funds’ and the ‘transaction costs’ are negligible. Of course, we can think of other schemes, where the change of capital AX; does not proceed in accordance with (19), but has a more complicated form, where shareholder dividends, consumption, transaction costs, etc. are taken into account. We consider now several examples of this kind. The case of ‘dividends’. Let D = ( D , = ( I l k , . . . , D$)),>o be a d-di6 = 0, and assume that mensional sequence of 9,-measurable variables D k , 0 6; = AD; 2 0. We shall treat D i as the dividend income generated by one share Sk. Then one has a right to assume that the change of the portfolio value X n = ( X ~ ) , > Ocan be described by the formula

AX; = &ABn

+ m(AS, + AD,),

(20)

1. Investment Portfolio on a

( B ,S)-Market

389

while X z is itself the following combination of the bank account, stock prices, and dividends: X z = PnBn + m(sn+ AD,). (21) It is easy to obtain by (20) and (21) the following, suitable for this case, modification of condition (13) of self-financing:

Moreover, (21)

+ (22)

t--r. (21)

+ (20)’

and (18) can be generalized as follows:

The case of ‘consumption and investment’. We assume that C = (Cn)n>o and I = (I,)n20 are nonnegative nondecreasing (AC, 3 0, A I , 3 0 ) processes with Co = 0, 10= 0, and with 5Fn-measurab1e C, and In. Assume that the evolution of the portfolio value X” = (X,”),,o is described by the formula AXE = PnABn m A S n A I n - ACn. (25)

+

+

This explains why we call this the case with ‘consumption and investment‘. The is called process C = (Cn)n20is called the ‘consumption process’ and I = ( I n ) n 2 ~ the ‘investment process’. Clearly, if we write X z = PnBn -ynSn, then (25) brings us to the following ‘admissibility’ constraint on the components of 7 r :

+

Relation (18) has now the following generalization:

Combining both cases (with ‘dividends’and with ‘consumption and investment’) we see that for an ‘admissible’ (i.e., satisfying (26)) portfolio 7r we have

The case of ‘operating expense’ (of stock trading). Relation (13) considered above has a clear financial meaning of a budgetary, balance restriction. It shows

Chapter V. Theory of Arbitrage. Discrete Time

390

that if, e.g., Ay, > 0, then buying stock of (unit) price Sn-l we must withdraw O n the other from the bank account the amount Bn-lA,& equal t o S,-1Ayn. hand, if Ayll < 0 (i.e., we sell stock), then the amount put into the bank account is Bn-lA[jrL= -S,,-lAy, > 0 . Imagine iiow that each stock transaction involves some operating costs. Then, purchasing Ay,rL> 0 shares of share price S n - l , it, is no longer sufficient, due to transaction costs, that we withdraw Sn-lAyn from the bank account; we must with X > 0. withdraw instead an amount (1 X)S,,-lAy, On the other hand, selling stock (Ay, < 0), we cannot deposit all the amount of -Sn-lAyrl into our bank account, but only a smaller sum, say, -(I - p)S,-lAyrl with p > 0. It is now clear that, with non-zero transaction costs, we must replace balance condition (13) imposed on the portfolio 7r by the following condition of admissibility:

+

Bn-lAlMn

+ (1+ X ) S r i - l A ~ n l ( A ~>r c0) + (1 ~

p)Sn-~Aynl(A~ < n0 ) = 0, (29)

which can be rewritten as B?a-lAp7L

+ sn-lAyTl + Asn-l(ArTl)tf ps?L-l (A%)-

= 0,

(30)

where (Ayn)+ = niax(0, Ayll) and (Ayn)- = - min(0, Ayn) If A = p , then (30) takes the fern Bn-lAPn

+ Sn-1

[AT,

+ XIAynI] = 0.

(31)

To find a counterpart to (18) we observe that, in view of (3O),

Hence, if the transaction costs specified by the parameters X and 1-1 are nonzero and the balance requirement (29) holds, then the evolution of the discounted portfolio value

-G with X:

-

B,l

= &BIL

+ ynSn can be described by the relation

1. Investment Portfolio o n a ( B ,S)-Market

391

If A = p , then

We can also regard this problem of transaction costs from another, equivalent, standpoint. e Let S, = mS,,and B,, = &Bn be the funds invested in stock and deposited in the bank account, respectively, for the portfolio T at time n. Assume that h

where L , is the cumulative transfer of funds from the bank account into stock by time n, which has required (due to transaction costs) a withdrawal of the (larger) amount (1 + X)L,,. In a similar way, let M , be the cumulative amount obtained for the stock sold, and let the smaller amount (1 - p)AM,. (again, due to the transaction costs) be deposited into the account. It is clear from (34) that we have the following relation between the total capital assets X z = B,, S,, the strategy T , and the transfers ( L , M ) ,where L = (L,), M = (A&), and Lo = MO = 0: A

+

h

However.

Thus, we see that

Setting here

A L = s7,-1(A~n)+,

=

Sn-l(AYn)-,

we obtain that (35) is equivalent to (30). The general case. We can combine these four cases into one, the case where there are dividends, consuniption, investment, and transaction costs.

Chapter V. Theory of Arbitrage. Discrete Time

392

Namely, using the above notation we shall assume that the evolution of the portfolio value X: PnBn m(sn AD,) (36)

+

+

can be described by the formula

ax; = PnA& + yn(AS, + AD,) + A I n -

ACn

sn-1

-

[A(aYn)+

+CL(bJ-1.

(37)

By (36) and ( 3 7 ) we obtain the following balance condition:

BtL-laP, =

+Sn-l~Yn -[

G

L -u l + Gn-lYn-1

+P s n - l ( h z - +ASn-l(bd+],

(38)

which describes admissible K. Since

(3

a - =

BTL-lMn

+Sn-lkn

I mSn

Bn

B,-1

"in-l&-l

B,-1

+ ma

(Z) ,

it follows in view of (38) that

for an admissible portfolio. If X = p , then

5. We now turn to formulas (18) and (27) and assume that the sequence ~,

S

(I),,,,

is a (d-dimensional) martingale with respect to the basic flow

=

F = ( 9 T L and )n2 the~probability measure

P.

BY (18)7

so that (since the Y k , k 3 1, are predictable) the sequence

($3) n>O

1. Investment Portfolio on a ( B ,S)-Market

393

is a martingale tran,sform and, therefore (see Chapter 11, 3 l c ) , a local martingale. We shall assume that X ; and Bo are constants. Then for each portfolio 7r such that inf->C>-m, (43) n Bn where C is some constant, we see that

x;

for each

7~

3 1, i.e., the local martingale

is bounded below and, by a lemma in Chapter 11, 5 l c is a martingale. Thus, if the discounted value of 7r satisfies (43) (for brevity we shall write T

E

IIc), then this discounted portfolio value

(g)

is a martingale and,

n>O

in narticular.

>

for each rc 0. We can make the following observation concerning this property. Assume that for some ( B ,S)-market the sequence

S

(2)

= is a marB n2O tingale. (Using a term from Chapter I, 3 2a, this market is eficzent). Then there is no strategy 7r E IIc (i.e., satisfying condition (43) with some constant C) that would enablc an invrstor (on an eficaent market) to attain a larger mean discounted -

xi?

x:

portfolio value E - than the initial discounted value -. Bn BO An econoniic explanation is that the strategies 7r E lIc are not sufficiently rrsky: an investor has no opportunity of 'running into large debts' (which would mean that he has an unlimited crcdit). In the same way, if 7r E HC, i.e.,

x; < C < 00

sup n Bn

(P-a.s.)

for some C , then the same lemma (in Chapter 11. 5 l e ) shows that the sequence is a martingale again. and therefore (45) holds (for an e f i c i e n t market).

394

Chapter V. Theory of Arbitrage. Discrete T i m e

It is interesting in many problems of financial mathematics whether property (45) holds after a replacement of the deterministic instant n by a Markov time 7 = ~ ( w ) . This is important, e.g., for American options (see Chapter VI), where a stopping time, the time of taking a decision of, say, exercising the option, is already included in the notion of a strategy. It can be immediately observed that if a stochastic sequence Y = (Yn,Sn)is a martingale, then E(Y, I 9,-1) = Y,,-1 and E/Y,I < 00, so that EY, = EYo, while the property

EY,

= EYo,

(47)

) 00, w E R does not necessarily hold for a random stopping time 7 = ~ ( w < (cf. Chapter 111, 5 3a, where we consider the case of a Brownian motion). It holds if, e.g., 7 is a bounded stopping time ( ~ ( w ) N < 00, w E n). As regards various suficient conditions, we can refer to § 3 a in this chapter and to Chapter VII, $ 2 in [439], where one can find several criterions of property (47). (In a few words, these criterions indicate that one should not allow ‘very large’ 7 and/or EIY,I if (47) is to be satisfied.) We present an example where (47) fails in 3 2b below.

<

6. In the preceding discussion we were operating an investment portfolio with no restrictions on the possible values of the 0, and 7, = (rk,. . . ,&; we set ,& E R and rf E R. In practice, however, there can be various sorts of constraints. For instance, the condition 0, 3 0 means that we cannot borrow from the bank account at time n. If 7, 3 0, then short-selling is not allowed. If we impose the condition

where CY is a fixed constant, then the proportion of the ‘risky’ component the total capital X: should not be larger than a. Other examples of constraints on T are conditions of the types

^ins,in

P(X% E A ) 3 1 - E (for some fixed N , E

> 0, and some set A ) or

for some 9N-measurable functions f N = f N ( W ) . We consider the last case thoroughly in the next section, in connection with hedging.

1. Investment Portfolio on a ( B ,S)-Market

395

l b . N o t i o n of ‘Hedging’. Upper and Lower Prices. C o m p l e t e and I n c o m p l e t e M a r k e t s .

1. We shall assume that transactions in our (B,S)-market are made only at the instants n = 0 , 1 , . . . . N . Let f N = f N ( w ) be a non-negative 9N-nieasurable function (performance) treated as an ‘obligation’, ‘terminal’ pay-off.

DEFINITION 1. An investment portfolio T = (P,?) with p = (On), y = (m), n = 0, I , . . . , N , is called an ’upper (z, fN)-hedge ( a lower (z, f N ) - h e d g e ) for IC 2 0 if X t = z and X & f N (P-a.s.) (respectively, X & < f N (P-a.s.)). We say that an (z, fN)-hedge T is perfect if X $ = x, z > 0, and X & = f N (P-a.s.). The concept of hedge plays an extremely important role in financial mathematics and practice. This is an instrument of protection enabling one to have a guaranteed level of capital and insuring transactions on securities markets. A definition below will help us formalize actions aimed a t securing a certain level of capital. 2. Let

xt = x, x& 2 fN (P-a.s.)}

H * ( x ,f N ; P) = {r: be the class of upper

(IC, fN)-hedges

H * ( x ,f N ; P) =

{T:

and let

xi = z, x)&< f N

(P-a.s.)}

be the class of lower (x>fN)-hedges.

DEFINITION 2. Let

fN

be a pay-off function. Then we call the quantity

@ * ( f N ;P) = inf{z

2 0: H * ( x ,f N ; P) # 0}

the upper price (of hedging against f N ) . The quantity

is called the lower price (of hedging against f

~

)

R e m a r k 1. As usual, we put C * ( f N ;P) = 00 if H * ( x ,f N ; P) = 0 for all z > 0. The set H,(O, f N ; P) is nonempty (it suffices to consider pT1= 0 and yn = 0). If H,(a, f ~P) # ; 0 for all z 0, then @ * (f N ; P) = co.

>

Rem.ark 2. We mention no .balance’ or other constraints on the investment portfolio in the above definitions. One standard constraint of this kind is the condition of ‘self-financing’ (see (13) in the previous section). Of course, one must specify

Chapter V. Theory of Arbitrage. Discrete Time

396

the requirements imposed on ‘admissible’ strategies particular cases.

7r

in one’s considerations of

R e m a r k 3 . It is possible that the classes H * ( z ,fry; P) and H * ( z ,f r y ; P) are empty (for sonic x at any rate). In this case, one reasonable method for comparing the quality of different portfolios is to use the mean quadratic deviation

(see Chapter IV, 5 Id)

3. We now discuss the concepts of ‘upper’ and ‘lower’ prices. If you are selling some contract (with pay-off function f r y ) , then you surely wish to sell it at a high price. However, you must be aware that the buyer wants to buy a secure contract a t a low price. In view of these opposite interests you must determine the smallest acceptable price, which, for one thing, would enable you to meet the terms of the contract (i.e., to pay the amount of f r y a t time N ) , but would give you no opportunity for arbitrage, no riskless gains (no free l u n c h ) , for the buyer has no reason to accept that. On the other hand, purchasing a contract you are certainly willing to buy it cheap, but you should expect no opportunities for arbitrage, no riskless revenues, for the seller has no reasons to agree either. We claim now that the ‘upper’ and ‘lower’ prices @* = @ * ( f ~P); and @* = C*(fry; P) introduced above have the following property: the intervals [0, @*) and (@*, m) are the (maximal) sets of price values that give a buyer or a seller, respectively, opportunities f o r arbit,rage. Assume that the price x of a contract is larger than @* arid that it is sold. Then the seller can get a free lunch acting as follows. He deducts from the total sum x an amount y such that C* < y < x and uses this money to build a portfolio 7r*(y) such that X:*(y)= y and X;(’) 3 fN a t time N. The existence of such a portfolio follows by the definition of @* and the inequality y > @*. (We can describe the same action otherwise: the seller invests the amount y in the ( B ,S)-market in accordance with the strategy 7r*(y) = (D;(y), y ; ( y ) ) ~ ~ ~ ~ r y . ) The value of this portfolio 7r*(y) at time N is X;”), and the total gains from the two trarisactions (selling the contract and buying the portfolio 7r*(y)) are

+ +

Here z X;(’) are the returns from the trarisactions (at time n = 0 and n = N ) and f r y y is the amount payable at t,ime n, = N and, a t time n = 0: for the purchase of the portfolio 7r*(y). Thus. z - g are the net riskless (i.e., arbitrage) gains of the seller. We consider now the opportunities for arbitrage existing for the buyer.

1. Investment Portfolio on a ( B ,S)-Market

fN

397

Assume that the buyer buys a contract stipulating the payment of the amount at a price x < C*. To get a free lunch the buyer can choose y such that

x< y
<

-

.-(-Y)

= (-P*TL(Y)) -"/*lZ(Y))(J
so that (-y) is distributed arnorig bonds and stock in accordance with the formula

The value of T ( - y ) at time

N

is

so that the total gains form the two transactions (buying the contract and investing (-y)) is

which, as we see, are the net (arbitrage) gains of the buyer purchasing the contract at the price x < C*. In the above discussion we considered the investment of a negative (!) amount (-y). What is the actual meaning of this transaction? In fact, we already encountered such short selling-e.g., in the discussion of options in Chapter I, $ 1 ~ . 4 .In the current case of the investment of (-y) this means merely that you find a speculator (cf. Chapter VII, $ l a ) who agrees to pay you the amount y (at time n = 0) in return to your promise of the payment of X;") at t h e n = N ; the latter amount can turn in effect to be more or less than y due to the random nature of prices. Remark. These arguments are (implicitly) based on the assumption that the market is sufficiently liquid: there exists a full spectrum of investors, traders, speculators, . . . , with different interests, views, and expectations of the behavior of prices. All this indicates, incidentally, that for a rigorous discussion one must precisely specify the requirements and constraints on the admissible transactions. The classes of admissible strategies considered below (see, e.g., Chapter VI, l a ) provide examples of such constraints that, in particular, rule out arbitrage.

Chapter V. Theory of Arbitrage. Discrete Time

398

Thus, we have two intervals, [0, C,)and (@*,m), of prices giving opportunities for arbitrage. At the same time, if z E then the buyer and the seller have no such opportunities. This justifies the name ‘interval of acceptable, mutually acceptable prices’ given to [C*,@*]. We emphasize again that a transaction a t a m,utually acceptable price z E [@*, C*] gives no riskless gains to either side. Both can gain or lose due to the random behavior of prices. Hence, as already mentioned, a gain, and especially a big gain, must be regarded as ‘ a compensation f o r t h e risk’

[@*,@.*I,

We sunirnarize our discussion of prices acceptable to buyer and seller in the following figure: Buyer’s preferable price domain

Seller’s preferable price domain /////////////////,

\\\\\\\\\\\\\\\u 0

@*

@*

Domain of prices acceptable for both seller and buyer

FIGURE 51. Donlairis of prices preferable and acceptable for buyer and seller (@, = @ * ( f ~P), ; c* = @*(fN;P)) 4. We consider now more thoroughly the case where, for fixed z and a pay-off f N , there exists a perfect (z, fN)-hedge 7 r , i.e., a strategy such that = z and

xt

XI;

(P-a.s.). The equality X & = f N nieans that the hedge 7r replrcates the contingent claim f N . It is desirable for many reasons that each obligation be replicable (for some value of T). If this is the case, then the classes H*(z,f N ; P) and H * ( z ,f ~P) ; clearly coincide and the upper and lower prices are also the s a m e : fN

In other words, the interval of acceptable prices reduces in this case to a single price c ( f N ; p ) (=@.*(.fN;p)= c * ( . f N ; P ) ) , the rational ( f a i r ) price of the contingent claim f N acceptable for both buyer and seller. As explained above, a deviation from this price will inevitably give one of them rzskless gains! This case, in view of its importance, deserves special terminology.

1. Investment Portfolio on a ( B ,S)-Market

399

DEFINITION3 . A (B,S)-market of securities is said to be N - p e r f e c t or perfect with respect t o t i m e N , if each SN-measurable (finite) pay-off function f N can be replicated, i.e., for some z there exists a perfect (z, fN)-hedge x , a portfolio such

that

XG

=fN

(P-a.s.).

Otherwise the market is said to bc N - i m p e r f e c t or imperfect with respect t o N . In general, the condition that a (B, S)-market be perfect is a fairly strong one and it imposes rather serious constraints on the structure of the market. Incidentally, it is not, necessary in many cases that a perfect hedge exist for all 9N-measurable functions f ~ it ;suffices to consider, for instance, only bounded functions or functions in a certain subclass described by some or other conditions of integrability and measurability. (Sce, nevertheless, the theorem in 4f.) DEFINITION 4. We say that a (B,S)-market of securities is N - c o m p l e t e or complete with respect to time N if each bounded 9N-measurable pay-off is replicable. The question of whether and when a market is perfect or complete is of a major interest for financial mathematics and engineering; t o answer it means to show that it is in principle possible (or impossible) t o make up a portfolio 7r of value X & replicating the pay-off f ~ (In . the case where f N cannot bc precisely replicated one may, as already mentioned, pose the problem of finding a portfolio delivering inf E[X& - fN]’, wherc the infimum is taken over all ‘admissible’ portfolios x: see Chapter VI, I d in this connection.) It seems difficult to answer this question in a satisfactory way in the general case, making no additional assumptions about the structure of t h e market and the probability space underlying this market. However, in the case of so-called arbitragefree markets (see the definitions in § 2a below) this problem of ‘completeness’has an exhaustive solution in terms of the uniqueness of the so-called martingale measures (see Theorem A in 3 2b and Theorem B in 4a). In the next section we consider a single-step model ( N = 1) of a fairly simple (B,S)-market-an example where one can clearly see how martingale ( r i s k neutral) measures enter in a natural way the calculations of the prices C*(fN;P) and C*(fN;P), and one can get acquainted with the general pricing priciples based essentially on the use of these measures.

5 Ic.

Upper and Lower Prices in a Single-Step Model

1. We consider a simple ‘single-step’ model of a (B,S)-market formed by a bank account B = ( B n ) and some stock of price S = ( S n ) ,where n = 0 , l . We assume that the constants Bo and SO are positive and (see l a )

400

Chapter V. Theory of Arbitrage. Discrete Time

where the interest rate r is a constant ( T > -1) and the rate p is a random variable ( P > -1). Since p is the source of all ‘randomness’ in this model, we can content ourselves with the consideration of a probability distribution P = P(dp) on the set { p : -1 < p < m} with Bore1 subsets. 2. We shall assume now that the support of P(dp) lies in the interval [a, b ] , where -1 < a < h < 00. (If P(dp) is concentrated at the two points { u } and { b } , then (1) is the single-step CRR-model introduced in Chapter 11, 5 le.) Let f = f(S1) be the pay-off function and, for the simplicity of notation, let C*(P) = C * ( f ; P ) and @*(P) = @*(f;P). Since Bo > 0, we can assume without loss of generality that B0 = 1. In our single-step model the portfolio T can be described by a pair of nunibers ,B and y that must be specified at time n = 0. In accordance with the definitions in the preceding section,

and

involved in the definition of H * ( P ) and introduce (however artificial it may seem at first glance) the class P ( P ) = {p = P ( d p ) } of distributions on [ a ,h] such that P-P

-

(i.e., the measures P and P must be absolutely continuous with respect to each other: << P and P << p) and

i,”

p ~ ( d p= ) r.

(7)

We assume that Y ( P ) # 0.(This holds surely in the CRR-model considered below, in 5 Id and in detail in 5 3f.)

1. Investment Portfolio on a

( B ,S)-Market

401

We point out the following important property underlying niany calculations in financial mathematics:

-

if a measure P is equii)alent to P; t h e n ‘P-as.’ in inequality (6) can be replaced by ‘P-as.’. I

Consequently, if

(/3,r)E H * ( P ) , then

P(1

+ r ) + rSo(l + P ) 3 f (SO(l+P ) )

F-a.s.),

therefore, integrating both parts of this inequality with respect to taking (7) into account we see that

(8)

pE

9 ( P ) and

From this inequality we can obviously derive the following lower bound for C*(P):

In a similar way we obtain a n upper boiind for

c*(p):

Thus, C*(P) 6 z*6 z*6 @*(PI, hich (provided that 9 ( P ) # 0)proves the inequality C*(P) o vious from the definitions of C*(P) and C* (P). We proceed now to property ( 7 ) ,which can be written as

(12)

< C*(P), not all that

By (l),this is equivalent to

We set 9 0 =

(0,R)and

9 1 = g ( p ) ; then we see that

-

i.e., the sequence

is a martingale with respect t o the measure P.

Chapter V. Theory of Arbitrage. Discrete Time

402

It is because of just this property (which holds also in a more general case) that measures E g ( P ) are usually called martzngale m e a s u r e s in financial mathernatics. We note that their appearance in the calculation of. say, upper and lower prices, can at first seem forcible. because there is already a probability measure on the original basis and, presumably. all the pricing should be based on this measure alone. In effect. this is the case because

dp . where z ( p ) = r p IS the Radon-Nikodym derivative of P with respect to P However. the introduction of martingale measures has a deeper meaning because their existence is in direct connection with the absence of arbitrage on our ( B ,S)market. We shall discuss this issue in 3 2 "Arbitrage-free market". while here we consider the question of equalzty signs in (10) and (11). _I

3. Assume that the function fz = f ( S o ( 1 f z)) is (downwards) convex and continuous on [a?b]. (We recall that each convex function on [a,61 is continuous on the open interval (a, b ) and can make jumps only a t the end-poznts of this interval.)

l

a

I I

I I

r

b

FIGURE 5 2 . Pay-off function fi = f ( S o ( 1

X

+ I))

We plot a straight line y = y ( z ) through the points ( a , fa) and ( b , f b ) . If it has an equation Y(Xc) = PSO(1f ). + u, (16) then. clearly,

Let us introduce the strategy

3'

=

T*

=

U

(,L*,r*) with

- and l f r

-y* = p.

1. Investment Portfolio on a

( B ,S)-Market

403

Since ( 1 + r ) p * + S o ( I + p ) y * = u + p S o ( I + p ) 2 f ( p ) for all p E [ a , b ] ,it follows that x* E H*(P), and therefore

We now make the following assumption about the set 9 ( P ) of martingale measures: (A*):there exists a sequence (Pn)n21 of measures in 9 ( P ) that converges weakly to a measure P* concentrated a t the two points a and b. If ( A * )holds, then we see from the equality I

E- ---=I l+P PnI+r

that 1+p=1 Ep* l+r

Hence the probabilities p* = P*{b} and q* = P*{a} satisfy the conditions p*

+ q* = 1

bp*

+ aq* = r.

and Consequently, p* =

r-a - and b-a

q* =

b-T

-.

(20)

b-a I

Further, using our assumption of the weak convergence of P, to P* again and in view of (19), we obtain

Combined with the reverse inequality (10) this means that

404

Chapter V. Theory of Arbitrage. Discrete Time

Concerning the above discussion it should be noted that our assumption that (So(l+p ) ) is a continuous convex function of p E [a.h] is fairly standard and shoiild encounter no serious objections. Assumption (A*) about the existence of martingale measures weakly convergent to a measure concentrated at a and b is more controversial. However, it is not that ‘dreadful’ provided that there exist non-vanishing P-masses close to n and b (i.e., for each E > 0 we have P[a,a + € ] > 0 and P[b-E. b] > 0). If there exists at least one martingale measure P P with the same properties (i.e., such that P[a, a E ] > 0 ,

f

=f

I

I

N

P [ b - E , b] > 0 for each

I

E

-

> 0, and

1

+

b -

pP(dp) = r ) , then we can construct the I

required sequence { P T 1 } by ‘pumping’ the measure P into ever smaller ( E -1 0) neighborhoods [ a ,a -E ] and - [b - E : b] of the points a and b preserving at the same time the equivalence P, P. In the next section we consider the CRR (Cox-Ross-Rubinstein) model in which the original measure P is concentrated at a and b and the construction of P* proceeds without. complications. (Actually, we already constructed this measure in (20).) Wc now formulate the result on C* obtained in this way.

+

N

THEOREM 1. Assume that the pay-off function f(So(1+ p ) ) is convex and continuous in p on [ u , b] and assume that the weak compactness condition ( A * ) holds. Then the upper price can be expressed by the formula

Moreover, the supremum is attained a t the measure P* and fb fa @.*(P)= r - a + b-r b-a 1+r b-a I + r ’ ~

~

+

whcre f p = f (So(1 p ) ) . 4. We consider now t h t lower price C*. By (3) and ( l l ) ,

@,(P) =

sup (,k+rSo)6 inf E-- fP (A”/W*P) i;€9(P)

(25)

where fP = f(so(1+ P ) ) , P E [ a ,b1. If f p is (downwards) convex on [a, b ] ,then for each T E ( a ?b ) there exists X = X ( r ) such that f (SOP + P I ) 3 f (SOP + r , ) + ( P - r ) X ( r ) (26) for all p E [ a , b ] ,where

is the ‘support line’ (a line through r such that the graph of

fp

lies above it).

(B. S)-Market

1. Investment Portfolio on a

Let

p E 9(P).

405

Then by (26) we obtain

We sct

and X(r) y*=-. SO Then (26) takes the following form for each p E [a.b] we have

f(SO(l+

P I ) 3 @*(I+ 7-1

+r*So(l + PI,

which means that 7r* = (P*,-y*) E H,(P). Following the pattern of the proof of Theorem 1 we assume now that ( A * ) :there cxists a sequence { P 7 L } n 2 1of measures in 9 ( P ) converging weakly to a measure P, concentrated at a szngle point r . Then. assuming that f p is a continuous function. we obtain

= P*

+ SOT* <

sup (O0,Y)€If* (P)

(B + Soy) = C*(P);

which, cornbined with (as), provcs the following result.

THEOREM 2. Let fl, he a continuous, downwards convex furictiori 011 [ a ,b] and assume that weak compactness condition ( A , ) holds. Then the lower price can be expressed b,y the forrnula

Moreover, the infimum is attained a t the measure P, and C*(P) =

_f_r . l + T

Remark. Let, for instance, P(dp) = d p be the uniform measure on [a,b]. Then b-a conditions ( A * ) and ( A , ) are satisfied, so that the upper arid the lower prices can be defined by (24) and (29). ~

Chapter V. Theory of Arbitrage. Discrete T i m e

406

5. In the above 'probabilistic' analysis of upper and lower prices we took it for granted that all the uncertainty in stock prices can be described in probabilistic terms. This was reflected in the assumption that p is a r a n d o m variable with probabilaty distribution P (dp). However, we could treat p merely as a chaotic variable ranging in the interval [ a ,b ] . In that case, in place of H * ( P ) and H,(P) it is reasonable to introduce the classes

and

E* = { (P, Y) : P ( I +

7-)

+ r ~ o ( 1p+) 6 f(so(l+ p))for each P E [ a ,b ~ } ,

(i.e., to replace the condition 'P-a.s.' by the requirement: 'for each p E [ a ,b ] ' ) . Clearly, fi* C H * ( P ) and g,C H,(P) for each probability measure P. We note that even if no probability measure has been fixed a priori, there are no indications against the (maybe, forcible) introduction of a distribution ^P = ^P(dp) on [a,b] such that (cf. (7))

lb

p p ( d p ) = r.

A

The class 9 'of such measures is surely non-empty; it contains the above-discussed measures P', concentrated at a and b (see (20)), and P*, concentrated at a single point r . By analogy with (2) and (3) we now set

and C* =

sup

(P +rSo) (< @.*(PI).

(P,T)EG*

The above arguments (see (S)? (9)' and (10)) show that for each probability measure P we have

1. Investment Portfolio on a

( B ,S)-Market

407

and

We note next that the class @ contains also both ‘two-point’ measure P* and ‘single-point’ measure P, considered above, therefore (cf. (21))

and (33) The strategy i7* discussed in the proof of Theorem 1 belongs clearly to the class @*, and (provided that f p is a (downwards) convex function on [ a ,b ] )

which together with (30) and (32) shows (cf. (23)) that

moreover (cf. (24)),

c*= p * -l ffb r h

+

where f p = f (So(1 p ) ) . In a similar way (cf. (28) and

fa

+q*l+r’

(as)),

moreover,

- fr c,=-. l+r Thus, we have established the following result.

(35)

Chapter V. Theory of Arbitrage. Discretc Time

408

THEOREM 3. If the variable p involved in (1) and ranging in [ a ,b] is ‘chaotic’, then for each downwards convex function f = f(So(1 p ) ) the upper and lower prices and can be defined by formulas (34), ( 3 5 ) and ( 3 6 ) , ( 3 7 ) ,respectively.

+

e* e*

Remark. Comparing Theorems 1, 2, and 3 we see that if the original probability measure P is sufficiently ‘fuzzy’ so that it has masses close to the points a , r , and b, then the class Y ( P ) of martingale measures is as rich as 9 and we have equality signs in the inequalities & 6 6,(P) and 6*(P) 6 A

e*.

5 Id.

CRR Model: an Example of a Complete Market

1. We consider again a ’single-step’ model of a (B.5’)-market. in which we assume that

where p is a random variable taking just two values, a and b, such that -1 < a

< r < b.

(2)

This simple (B,S)-market is called the single-step GRR-model (after J. C . Cox, S. A. Ross, and M. Rubinstein who considered it in [82]). We assume that the initial distribution P of the random variable p is as follows: p = P{b}

> 0,

q = P{a}

> 0.

Then the unique (martingale) measure P* equivalent to P and satisfying property ( 7 ) in the preceding section can be defined as follows:

P*{b} = p * , where (see (20) in

P*{U} = q * ,

3 lc) p* =

and that (see (11) and (12) in

r-a b-a’

~

b-r q*=--, b-a

5 lc)

@*(P) 6 Ep*- f p 6 @*(P). l+r

(4)

In fact, the prices C,(P) and @*(P)are the same in our case for each pay-off function f = f(So(1+ p ) ) , so that their common value is given by the formula

1. Investment Portfolio on a ( B ,S)-Market

409

Actually, all we need for the proof of the equality C*(P) = C*(P) is already contained in the discussion in 5 lc. For let us consider the above-introduced strategy ?r* = (p*,y*)with parameters

p*

=

U

and

~

l+r where u antl p are as in formula (17) in For both p = u antl p = b we have D*(l

y* = p :

lc.

+ ). + r*So(l + P ) = f (SOP+ P ) ) ,

therefore the pay-off function here can be replicated, so that Hence

C*(P) =

(P + 7%) 3 P*

sup

?r*

E H*(P)n H,(P).

+ ?*SO

(io,r)€ H * ( P1

2

inf:

(P+ySo) = C*(P).

(0>7)€ H * (PI Combined with (4), this proved the required equality'of C,(P) and C*(P) and formula (5) for their coinnion value C(P). 2. We point out again the following important points revealed by our exposition here and in the previous section. They are valid also in cases when the martingale methods are used in a more general context of financial calculations relating to a fixed pay-off function: (I) if the class of martingale measures is n o n - e m p t y :

%P) # 0, then C*(P) (by formula (12),

(11)

(6)

< C*(P)

Ic);

if

H * ( P )n H,(P) # 0, i.e., the pay-off f u n c t i o n i s replicable, t h e n

(7)

@*(P) 3 @*(P);

(111) if both ( 6 ) and ( 7 ) hold, t h e n the upper and the lower prices C,(P) and @* (P) are the same. We show in the next section that the non-emptiness of the class of martingale measures 9 ( P ) is directly related to the absence of arbitrage. On the other hand, the non-emptiness of H*(P)n H,(P) (for an arbitrary payoff function f ) is related to the uniqueness of the martingale measure,- i.e., t o the question whether the set 9 ( P ) reduces to a single (martingale) measure P equivalent to P (is P).

-

2. Arbitrage-Free Market

5 2a.

‘Arbitrage’ and ‘Absence of Arbitrage’

1. In a few words, the ‘absence of arbitrage’ on a market means that this market if ‘fair’, ‘rational’, and one can make no ‘riskless’ profits there. (Cf. the concept of ‘efficient market’ in Chapter I, §2a, which is also based on a certain notion of what a ‘rationally organized market’ must be and where one simply postulates that prices on such a market must have the martingale property.) For formal definitions we shall assume (as in la) that we have a filtered probability space p>9, (9:n)n>O,p), and a (B,S)-market on this space formed by d

+ 1 assets:

a bank account B = (B,),~o with 9:,-1-measurable Bnr B,

> 0, and

a d-dimensional risk asset S = (Sl, ...,S d ) ,

Si= (Si,)n20,and the S i are positive and 9:,-measurable. Let X” = (X,”),,o be the value

where

of the strategy

YZ= (rk)n>o.

7r

=

(p,y)

with predictable p =

(pn)nao and

y = (yl,. . . , y d ) ,

2 . Arbitrage-Free Market

If

7r

is a self-financing strategy

(T

E

411

SF), then (see (12) in 3 l a )

and the discounted value of the portfolio

2; =

satisfies the relations

which are of key importance for all the analysis that follows.

2. We fix some N 2 1; we are interested in the value X G of one or another strategy T E SF at this ‘terminal’ instant. DEFINITION 1. We say that a self-financing strategy for arbitrage (at time N ) if, for starting capital

x;

= 0,

T

brings about an opportunity

(3)

we have

XG and X g

0

(P-a.s.)

(4)

> 0 with positive P-probability, i.e., P(X% > 0) > 0

(5)

or, equivalently,

EX;

> 0.

Let SFarb be the class of self-financing strategies with opportunities for arbitrage. If 7r E SF,,b and X $ = 0, then

DEFINITION 2. We say that there exist n o opportunities for arbitrage on a ( B ,S)market or that the market is arbitrage-free if SFarb = 0.In other words, if the starting capital Xg of a strategy 7r is zero, then

Chapter V. Theory of Arbitrage. Discrete Time

412

L

fl #'

1

,4P(XL > 0) = 0 ,,' ,,' , /I'

, , #'

,0'

',

,, I

*',',,

&-*

N

0

,'#'

,

c

-

*c*

_-

.*

I

-7 I I I

P(x; = 0) = 1 t

0

N

arbitrage-free (and X,; 3 0); then the Geometrically, this means that if 7r is chart (Fig. 53a) depicting transitions from X < = 0 to X& must actually be 'degenerate' as in Fig. 53b, where the dash lines correspond to transitions X< = 0 Y Xg of probnbzlati/ zero. In general. if = PoBo f roSo = 0, then the chart of transitions in an arbitrage-free market must be as in Fig. 54: if P(XI; = 0) < 1, then both gains (P(,Yac > 0) > 0) and losses (P(Xl& < 0) > 0) must be possibie. This can also be reformulated as follows: a strategy K (with XR = 0) on an arbitrage-free market must be either trivial (i.e.. P(XE # 0) = 0 ) or risky (i.e., we have both P ( X & > 0 ) > 0 and P ( X & < 0) > 0 ) . Along with the above definition of an arbitrage-free market. several other definitions are also used in finances (see. for instance. [251]). We present here two examples.

Xt

DEFINITION 3 . a ) X (B.S)-market is said to be arbitrage-free zn the weak sense if for each self-financing strategy 7r satisfying the relations X{ = 0 and X: 3 0 ( P - a x ) for all n 6 N we have -Yg= 0 (P-a.s.). b) A ( B .S)-market is said to be arbztrage-free in the strong sense if for each self-financing strategy i~ the relations X,X = 0 and X& 2 0 (P-as.) mean that Xz = 0 ( P - a s . ) for all 7~ < N .

2. Arbitrage-Free Market

413

Remark. Note that in thc above definitions we consider events of the form { X & > 0}, { X s 3 0}, or { X g = 0}, which arc clearly the same as { X g > 0}, { X g 3 0}, or

-

{Xg

= 0}, respectively, where

-

XT

X& -

~

BN

(provided that BN

> 0). This explains

why, in the discussion of the ‘presence’ or ‘absence’ of arbitrage on a (B,S)-market,

--

one can restrict oneself to (B,S)-markets with

B,

= 1 and

- = -. s , In other S,, B,

words, if we assume that B, > 0, then we can also assume without loss of generality that B,nE 1.

3 2b.

Martingale Criterion of the Absence of Arbitrage. First Fundamental Theorem

1. In our case of discrete time n = 0 . 1 , . . . , N we have the following rerriarkable result, which, tilie to its iniportance, is called the First f u n d a m e n t a l asset pricing theorem.

THEOREM A . Assume that a (B,S)-market on a filtered probability space (n,9, (.a,), P) is formed by a bank account B = (B7),), B, > 0, and finitel,y rnariy assets S = (Sl, . . . , S d ) : Si = (SA). Assume also that this market operates at the instants n = O , l , ..., N , {0,n},and 9~= 9. Then this -(B, 5’)-market is arbitrage-free if a n d only if there exists ( a t least one) measure P (a ‘martingale’ measure) equivalent to the measure P such that the d-dimensional discounted sequence

90=

-

is a P-martingale, i.e.,

for all i = 1:. . . , d and n = O , 1 , . . . , N and

for n = 1 , .. . , N . We split the proof of this result into several steps: we prove the sufficiency in 5 2c and the necessity in § a d . In §2e we present another proof, of a slightly more

Chapter V. Theory of Arbitrage. Discrete Time

414

general version of this result. Right now, we make several observations concerning the meaningfulness of the above criterion. We have already mentioned that the assumption of the absence of arbitrage has a clear economic meaning; this is a desirable property of a market to be ’rational’, ‘efficient’, ’fair’. The value of this theorem (which is due to J. M. Harrison and D. M. Kreps [214] and J. M. Harrison and S. R. Pliska [215] in the case of finite R and to R. C. Dalang, A . Morton, and W. Willinger [92] for arbitrary 0 ) is that it shows a way to analytic calculations relevant to transactions of financial assets in these ‘arbitrage-free’ markets. (This is why it is called the First f u n d a m e n t a l asset pricing theorem.) We have, in essence, already demonstrated this before, in our calculations of upper and lower prices (3 3 l b , c). We shall consistently use this criterion below; e.g., in our considerations of forward and futures prices or rational option prices (Chapter VI). This theorem is also very important conceptually, for it demonstrates that the fairly vague concept of e f i c i e n t , rational market (Chapter I, 3 2a), put forward as some justification of the postulate of the martingale property of prices, becomes rigorous in the disguise of the concept of arbitrage-free m a r k e t : a market is ‘rationally organized’ if the investors get no opportunities for riskless profits. 2. In operations with sequences X = (X,) that are martingales it is important to in terms of indicate not only the measure P, but also the flow of cr-algebras (9,) which we state the martingale properties: the X, are 9,-measurable,

EtXnI < m, E(X,+l

I 9,)= X ,

(P-a.s.).

To emphasize this, we say that the martingale in question is a P-martingale or a (P, (9,))-martingale and write X = (X,, gn,P). Note that if X is a (P, (9,))-martingale, then X is also a (P, (%,))-martingale with respect to each ‘smaller’ flow (%), (such that %, C: S,)?provided that the X, are %,-measurable. Indeed, by the ‘telescopic’ property of conditional expectations we see that E(Xn+1 I %)

E(E(Xn+1 I 9,) I %) = E(Xn I %n) = Xn

(P-a.s.),

which means the martingale property. Clearly, if X is a (P, (9,))-martingale, then there exists the ‘minimal’ flow (%), such that X is a (%,)-martingale: the ‘natural’ flow generated by X , i.e., gn = cr(XO,X1,.. . , X,). We can recall in this connection that we defined a ‘weakly efficient’ market (in was generated by Chapter I, 32a) as a market where the ‘information flow (9,)’ the past values of the prices off all the assets ‘traded’ in this market, so that (9,) is just the ‘minimal’ flow on such a market.

2. Arbitrage-Free Market

415

3. One might well wonder if the theorem still holds for d = m or N = CQ. The following example. which is due to W. Schachermayer ([424]), shows that if d = 'cc (and N = 1). then there exists an 'arbitrage-free' market wzthout a 'martingale' measure, so that the 'necessity' part of the above theorem fails in general for d = x).

EXAMPLE1. Let R = { 1 , 2 . . . . }, let 30= { 0 . n}, let 9 =

9 1 be the o--algebra

m

generated by all finite subsets of Q, and let P =

2-'6k,

i.e.. P { k } = 2 - k .

k= 1

We define the sequence of prices follows:

A S i ( u )=

S

=

i

(SA)for z

= 1 , 2 , . . . and n = 0 . 1 as

1, w = 2 ,

-1,

0

w =a+

1.

otherwise.

Clearly, the corresponding (B,S)-market with Bo = B1 = I is arbitrage-free. However. marginal measures d o not necessarily exist. In fact. the value X ; ( w ) of an arbitrary portfolio can be represented as the sum

i= 1

z= 1

x

where

*Y;

= co

+ C c,

(here we assume that

C lc,l

< x ) . If X$ = 0 (i.e.,

2=I

co

+

x

c, = 0 ) . then by the condition -K;

2 0 we obtain

z= 1

. l - T ( l ) = C l 3 0. .Y;(2)

= Cz

-

c1

3 0. . . . S ; ( k ) = ck - C k - 1 2 0.

Hence all the c, are equal to zero. so that .U; = 0 ( P - a s . ) . However, a martingale measure cannot exist. For assume that there m s t s a measure ? P such that S is a martingale with respect to it. Then for each 1 = 1 . 2 . . . . we have

-

-

-

i.e.. P(z} = P{z + 1) for z = I. 2 . . . . measure.

.

Clearly. this is impossible for a probability

The next counterexample relates to the question of whether the -sufficiency' part of the theorem holds for iV = x.It shows that the existence of a martingale measure does not ensure that there is no arbitrage: there can be opportunities for arbitrage described below. (Note that the prices S in this counterexample are not necessarily positive. which makes this example appear somewhat deficient.)

416

Chapter V. Theory of Arbitrage. Discrete Time

EXAMPLE 2. Let [ = ( [ l L ) n 2 0 be a sequence of independent identically distributed random variables on (n,9, P) such that P(ln = 1) = P(<, = -1) = $. We set SO= 0, S, = <1 . . . tn,Bn = 1, and let

+ +

where

As is well known, we can treat Xz as the gain of a gambler playing against a ‘symmetric’ adversary, when the outcomes are described by the variables (he gains if
i.e., he is a net loser. However, if he gains a t the next instant k becomes =

x; + 2k = -(2 k

+ 1, i.e., if [ k + l -

1)

=

1, then his gain

+ 2k = 1.

Hence if the concept of ‘strategy’ embraces (in addition t o a choice of a portfolio) also a (random) stopping instant 7 , then the gain of our gambler can be positive. For let 7

(h)‘,

= inf{k:

X; = I}.

Since P(T = k ) = it follows that P(T < m) = I , and therefore EX: = 1 because P(XF = 1) = 1 (although the starting capital X $ was zero). Thus, there exists an opportunity for arbitrage on our ( B ;S)-market with BI,= 1. Namely, there exists a portfolio 7r such that X l = 0, but the expectation EX: = 1 for some T . We note by the way that the strategy of doubling the stake after a loss used here implies that either the gambler is immensely rich or he can borrow indefinitely from somewhere; both variants are, of course, hardly probable. For this reason, in the considerations of various issues of arbitrage theory, one should impose some sound, ‘economically reasonable’ constraints on the classes of admissible strategies. (See Chapter VII, l a on this subject.)

2 . Arbitrage-Free Market

417

fj 2c. Martingale Criterion of the Absence of Arbitrage. Proof of Sufficiency

We claim that if there exists a martingale measure such that the seauence

equivalent to the measure P

is a (P, (3T,,))-martingale,then there can be no arbrtrage on the ( B ,S)-market. As already nirntioned (see the end of fj2a), the condition B, > 0, n 3 0 (which we assume to hold), allows one to set B, = 1. We use formula (2) in 9 2a, in accordance to which,

-

where S = (STL) is a P-martingale. To prove the required assertion we must show that if that X g = 0 and P(X& 3 0) = 1, i.e.,

7r

E S F is a strategy such

k=l

-

-

(P-a.s., or, equivalently, P-as.), then G; = 0 (P-as., or, equivalently, P-as.). We use the theorem and the lemma in Chapter 11, fj lc. The sequence ( G ; ) o l n G ~ is a martingale transform with respect to P and, therefore, by the theorem a local martingale. Since G& 3 0, it follows by the lemma that ( G ; ) 0 G n G ~is in fact a P-martingale and therefore, EpG& = GG = 0.

-

Hence X g = GL = 0 (P-as. and P-as.).

2d. Martingale Criterion of the Absence of Arbitrage. Proof of Necessity (by Means of the Esscher Conditional Transformation) 1. We niust now prove - that the absence of arbitrage nieans the existence of a probability measure P N P in ( Q 3 )such that the seqnence S = ( S T l ) O , < n gis~ a P-martingale. There exist several proofs of this result and its generalization to the continuoustime case (see, e.g., [92], [loo], [171], [215], [259], [443], and [455]). They all appeal in one or another way to concepts and results of functional analysis (the Hahn- Banach theorem, separation in finite-dimensional Euclidean spaces, Hilbert spaces, etc.).

Chapter V. Theory of Arbitrage. Discrete Time

418

At the same time, they all are ‘existence proofs’ and suggest no explicit constructions of martingale measures, to say nothing about an explicit description of all martingale measures P equivalent to P. It would be interesting for this reason to find a proof of the necessity part of the theorem where one explicitly constructs all martingale measures or a t least some subclass of them. This becomes important once we take into account that in the calculation of upper and lower prices we must find the upper and lower bounds over the class of all measures equivalent to the original measure P (see 3 l c ) . It is in this way of an explzcit construction of a martingale measure that we shall carry out the proof. We shall follow the ideas of L. C. G. Rogers [407] and use the construction of equivalent measures based on the Esscher conditional transformations. 2. To explain the main idea we consider first the single-step model ( N = l), where we assume for simplicity that d = 1, Bo = B1 = 1, and 90= { 0, n}. We also assume that P(S1 # So) > 0. Otherwise we shall obtain an uninteresting trivial market and we can take the original measure P as a martingale measure. Now, each portfolio 7r is a pair of numbers (p,7 ) . If X t = 0, then only the pairs such that /3 7So = 0 are admissible. The assumption of the absence of arbitrage means that the following two conditions must be met on such a (non-trivial) market:

+

P(AS1 > 0) Hence Fig. 54 in

> 0 and P(AS1 < 0) > 0.

3 2a takes now

W

the following form:

1

c

FIGURE 55. Typical arbitrage-free situation. Case N = 1

We must deduce from (1) that there exists a measure

N

P such that

1) EpIAS1I < 00, 2) EpASl = 0. It can be useful to formulate the corresponding result with no mention of ‘arbitrage’, in the following, purely probabilistic form.

2. Arbitrage-Free Market

419

LEMMA1. Let X be a real-valued random variable with probability distribution P on (R,.%(R)) such that P(X

> 0) > 0 and P ( X < 0) > 0.

-

(2)

I

Then there exists a measure P

P such that

< 00

(3)

Ei;lXl < W .

(4)

Eijeax

for each a E R; in particular, Moreover,

p has

the following property:

EpX

(5)

= 0.

Proof. Given the measure P, we construct first the probability measure Q ( d z )= ce-"*P(dx),

2

E R,

where c is a normalizing coefficient, i.e.,

for a E R and let

P and it follows by the construction of Q that p(a) < 00 for each Clearly, Q a E R that p(a) > 0. It is equally clear that Z a ( z )> 0 and EQZ,(Z) = 1. Hence for each a E R we can define the probability measure N

Pa ( d z ) = Z a

(.)a( d z )

(8)

such that Fa Q P. The function cp = cp(a) defined for a E R is strictly convex (downwards) since p/l(a) > 0. Let p* = inf{p(a) : a E R}. N

N

Then two cases are possible: 1) there exists a, such that cp(a,) = cp* or 2) there exists no such a,.

420

Chapter V. Theory of Arbitrage. Discrete T i m e

In the first case we clearly have cp'(a,) = 0 and

Thus. we can take P., as the required measure P in case 1). We now claim that assumption (2) rules out case 2). Indeed. let {a,} be a sequence such that I

This sequence must approach fx or --oo since otherwise we can choose a convergent subsequence and the minimum value is attained a t a finite point. which contradicts assumption 3 ) . a n and let u = limu, (= + I ) . Let u, = la, I By (2) we'obtain Q ( u X > 0) > 0. Hence there exists 6 > 0 such that

and we choose 6 that is a continuity point of Q . i.e..

Q ( u X = 6) = 0. Consequently,

so that for

R

sufficiently large we have

which contradicts (9). where p* 6 I

-

Remark. The above method of the construction of probability measures Pa, which

- 5d .a -) defined by(7) is a knownin the "UX

is based on the Esscher transformation z

actuarial practice since F. Esscher [lU](1932). As regards the applications of this transformation in financial mathematics. see. for instance, [177] and [178], and as regards its applications in actuarial mathematics. see the book [52].

2. Arbitrage-Free Market

421

3. It is easy to see from the above proof of the lemma how one can generalize it to a vector-valued case, when one considers in place of X a n ordered sequence (Xo, X I , . . . , X ), of 9,,-rneasurable random variables X,,, 0 n, 6 N , defined on a filtered probability space ( C 2 , 9 , ( P) with 90= {a,C2} and .BN = 9.

<

LEMMA2. Assume that P(X, > 0 I ,YT&-1) > 0 and for 1 < n

P(X,,

< 0 1 9 n ->~0)

(11)

-

<

N . Then there exists a probability measure P equivalent bo P in tho space (Q9) such that the sequence (Xo,X I , . . . , X,) is a (p; (9,,))-rnartin&e difference . Proof. If necessary, we can proceed first from P to a new measure Q such that N

Q(dw) = (:rip{ -

1X:(w)}

P(dw)

i=O

and the generating fiinction EQ exp{

,?aiXi) is finite.

z=o

We can construct the required measure P = P ( d w ) as follows (cf. (8)). Let p,,(a;w) = E(eaxn I . ? F n - ~ ) ( w ) .

(13)

For each fixed w these functions (in view of (11)) are strictly (downwards) convex in a. As in Lemma 1 we can show that there exists a unique (finite) value an, = a,,(w) such that the sniallest value inf p n ( n ; w) is attained at a,. U

Since here inf = i n f , where Q is the set of rationals, the function p T L ( w )= a

aEQ

inf cp,(a; w)is .9,_1-nicasurable, which means that a,(w) is also a 9:,-1-measurable a

function. Indeed, if [ A ,B] is a closed interval, then {LJ:a,,(w)

E [ A , B ] }=

r'l

(J

{ w : pn(a;w)

< pn(w) +

m a@n[A.B]

and therefore a , (w) is .?F,,-l-nieasurable. Next, we define recursively a sequence Zo,Zl(w), . . . , Z,(w) and

,> 1

E 9n-1,

by setting 20 = 1

for n 3 1. Clearly, the variables Zn(w)are 9,,-nieasurable and form a martingale:

EQ(&

I SrL-1) = Zn-l

(P-as.)

Chapter V. Theory of Arbitrage. Discrete Time

422

We define the required measure P by the formula

P(dw)= Z,(w)

P(dw).

As in Lemma 1, it easily follows from this definition that E,lX,l and 1 n N. Ei;(Xn I .Fn-l) = 0 ,

(15)

< 00,0 < n < N ,

< <

(16)

By Lemma 1, EpXo = 0. Hence the sequence ( X O X , I , . . . , X,) is a martingale difference with respect to

P,which proves the lemma. I

4. For d = 1, the necessity of the existence of a martingale measure P

P (in an arbitrage-free market) is a consequence of Lemma 2. For let X O = So, X i = ASl, . . . , X N = AS,. Since there can be no arbitrage, we can assume without loss of generality that N

for each n = 1,.. . , N . Indeed, if P(AS, = 0) = 1 for some n, then we can skip the instant n because no contribution to the value X c of an arbitrary self-financing portfolio 7r can be made at this time. On the other hand if there exists n such that P(AS, 3 0) = 1

<

or P(AS, 0) = 1, then P(AS, = 0) = 1 due to the absence of arbitrage. (Otherwise it is easy to construct a strategy 7r such that X c > 0 with positive probability.) Again, the corresponding contribution to X & is zero. Thus, we can assume that (17) holds for all n N , and the required necessity follows directly by Lemmas 1 and 2 as applied to X O = SO and X , = AS,, l
<

5 . We consider now the general case of d 3 1. Conceptually, the proof is the same as for d = 1; it can be carried out using the following generalization of Lemma 2 to the vector-valued case.

, . . , X,) be a sequence of 9,-measurabJe d-vectors LEMMA3 . Let ( X O Xi,.

O}, defined on a filtered probability space ( O , 9 , ( ~ , ) O G , < N , P) with 90= (0,

9, = 9.

423

2. Arbitrage-Free Market

Assume also that if

is a non-zero 9,-1-measurable vector-valued variable with bounded component ( j y i ( w ) j 6 c < 00, w E such that

a)

P((yn,X,)

> 0 I 9,-1) > 0 (P-a.s.1,

then also P((Yn, X n ) < 0 I 9,-1)

> 0 (P-a...),

where (m, X,) is the scalar product. Then there exists a probability measure is equivalent to P on (Q, 9)such that the-sequence ( X o ,X I , . . . , X,) is a d-dimensional martingale difference with respect to P: EplX,l < 00, EpXo = 0, and Ep(X, I 9,-1) = 0, 1 n N.

< <

Proof. If the topological support (i.e., the smallest closed carrier) of the regular conditional probability P(X, E . I 9,-1)(w) does not lie in a proper subspace of Bd, then, as for d = 1, the functions

are strictly convex and the smallest value inf 'p,(a; w ) is attained at a unique point a , = a,(w) E Bd; moreover, the functions a,(w) are 9,_1-measurable. The case of a regular conditional distribution P(X, E . I 9 , - 1 ) ( w ) concentrated at a proper subspace of Bd is slightly more delicate. As shown in [407], we can find in this case a unique 9,-1-measurable variable a , = a,(w) delivering the smallest value inf pn ( a ;w ) . The required measure is can be constructed as in (15) and (14) if we treat an(w)Xn(w) (in (14)) as the scalar product of the vectors a,(w) and X,(w). 6. The above construction of a martingale measure based on the Esscher conditional transformation gives us only one particular measure, although the class of martingale measures equivalent to the original one can be more rich. We devote the next section to several approaches based of the Girsanov transformation that can be used in the construction of a family of measures equivalent to or absolutely continuous with respect to P such that the sequence of the discounted prices is a martingale with respect to these measures. The Esscher transformation has long been in use in actuarial calculations (see, e.g. [52]). 'Esscher transformations' are less familiar under this name in financial mathematics (see nevertheless the already mentioned papers [177] and [178]), where several I. V. Girsanov's results on changes of measures known as 'Girsanov's theorem' play an important role.

Chapter V. Theory of Arbitrage. Discrete Time

424

In fact, these transformations have very much in common and we shall discuss this in detail in $ 3 of this chapter. Here we only mention in connection with Lemma 1 that if X is a normally distributed random variable ( X N ( 0 ,l)),then 1,2 p ( a ) = EpeaX = e 2 and (see (7))

-

A reader acquainted with Girsanov’s theorem will see a t once that the ‘Girsanov’ 1

exponential eax-Ta form (7).

2

involved in this theorem (see 53a) is just the Esscher trans-

2e. Extended Version of the First Fundamental Theorem 1. Let 9 ( P ) and 91,,(P) be the sets of all probability measures the discounted prices

is

-

P such that

are martingales and local martingales, respectively, relative to these measures. Let 9b(P) be the set of measures is in 9 ( P ) such that the Radon-Nikodyrn dis dis derivatives - are bounded above: -((w) 6 C(P) (P-a.s.) for some constant C(P). dP dP We can formulate Theorem A (the First fundamental theorem in 3 2b) as follows: the conditions (i) a ( B ,S)-market is arbitrage-free

and (ii) the set of martingale measures 9 ( P ) is not empty (9’(P) # 0)

are equivalent. Theorem A* below is a natural generalization of this version of the first fundamental theorem; it provides several equivalent characterizations of an arbitrage-free market and clears up the structure of the set of martingale measures. (We state and prove this theorem following [251].) First of all, we introduce our notation. Let Q = Q (dx) be a probability measure in (Rd, .“A(Rd)) and let

K ( Q ) be the topological support of Q (the smallest closed set carrying Q; [335; vol. 51); L(Q) be the closed convex hull of K ( Q ) ; H ( Q ) be the smallest a&e hyperplane containing K ( Q )(clearly, L ( Q ) C H ( Q ) ) ; L”(Q) be the wlative interior of L ( Q ) (in the topology of the hyperplane H ( Q ) ) .

2. Arbitrage-Free Market

425

For instance, if Q is concentrated a t a single point a, then H ( Q ) coincides with this point and L ( Q ) = L " ( Q ) = { u } . Otherwise H ( Q ) has dimension between 1 and d. If the dimension of H ( Q ) is 1, then L ( Q ) is a closed line segment and L " ( Q )is an open interval. Let Q,(w, . ) and a,(w, . ) be the regular conditional distributions P(AS,€

. l c q r L - l ) ( w ) and

P(A(&)

E . l.qn-1)(w),

<

l
We note that since B, > 0, 0 6 71 N , arid the B, are 9,_1-measurable, the sets K ( Q ) ,L ( Q ) ,and L " ( Q ) are the same for Q = Q , and Q = Hence we shall assume without loss of generality in the statements and proofs below that B, G 1, n 6 N.

on.

THEOREM A* (an extended version of the First fundamental theorem). Assume that the conditions of Theorem A are satisfied. Then the following assertions are equivalent: a) a ( B ,S)-market is arbitrage-free; a') a ( B ,S)-market is weakly arbitrage-free; a") a ( B ,S)-market is strongly arbitrage-free; b) %(P) # 0; c) W)# 0; d) %OC(Pk# 0; e) 0 E L o ( Q T r ( w.,) ) for each n E {I, 2 , . . . , N } and P-almost all w E R. First of all: we make several general observations concerning the above assertions. As regards the definitions of arbitrage-free and weakly or strongly arbitrage-free markets, see 5 2a. It follows by the lenima in Chapter 11, fi Ic that, in fact, 9 ( P ) = 910c(P), and therefore c) d). Further, if the properties a), a'), a", c), o r e ) hold with respect to some measure P equivalent to P, then they also hold with respect to P. If b) holds with respect

-

dP

,

to a measure P P (i.e., 9 b ( P ) # 0) and the derivative - is bounded, then b) dP holds also for the original measure P. We observe next that we can always find a measure P P such that all the N

-

dP

variables S,, n 6 N , are integrable with respect to it and the derivative dP bounded. For instance, it suffices to set ,

where C is a norirializing coefficient,.

d

N

,

IS

Chapter V. Theory of Arbitrage. Discrete Time

426

Hence we can assume throughout the proof of the theorem that EIS,]

n

< N , for the original measure P.

< 00,

Then, in view of the obvious implications a”) ==+ a)

=-=+ a’)

and b)

we must prove only that a’)

--r‘

c),

==+ e)

b)

and c) ===+ a”). 2. We introduce now several concepts that are necessary for the proof of these three implications and state two auxiliary results (Lemmas 1 and 2). Let Q = Q ( d z )be a measure in (Rd,B(Rd) such that

(we shall set Q to be equal to the regular conditional distributions Q,(w, d z ) with n < N in what follows, and (1) will hold (P-as.) in view of the assumption that E(S,( < m for n < N . ) Let x’ = ( x , w ) E E = EXd x ( 0 , ~ and ) let Q’ = Q’(dz’) be the measure on the Borel 0-algebra 6‘ of E associated with Q = Q ( d x ) in the following sense: Q is the ‘first marginal’ of Q’, that is, Q ( d z )= Q’(dz,(0, m)). Let Z(Q’) be the family of positive Borel functions z = z ( z ,w) in E such that

L-

IzIz(x,w ) Q’(dz;dw) < 00, z ( z ,w ) Q’(dz;dw) = 1,

and t,he functions u z ( x ,w ) are bounded. Let B ( u , E )be the closed ball in EXd with center at a and of radius the collection of all families 9=

(h(% E i , f f i ,c 4 ) i = l , . . . , k )

s u c h t h a t k E { 1 , 2, . . . } , a ~ E E X d , & ~ > O , c r ~ > O , a ~ > O . With each g E G we associate the positive function

(on E ) , where BC= Rd \ B.

E.

Let G be (4)

427

2 . Arbitrage-Free Market

If EQ/zg = 1, then (1) shows that zg E Z(Q’). Thus, we can define for such functions zg the vcctors 4 9 )=

q ( z ,u)Q’(dz,du)

(6)

(the barycenters). We set = { c p ( g ) : g E G , E Q / Z= ~ l}.

(7)

LEMMA1. We have the following relations:

If 0 # L”(Q),then there exists y in Rd such that Q(z:(y,z)3 0) = 1 and

Q ( z :(7.z) > 0)

> 0.

(10)

Proof. First we shall prove that L ( Q ) 5 Let y E K ( Q ) . We claim that there exists a sequence y, in @(Q’) such that y, + y. In other words, each point y in the topological support of Q is the limit of some sequence yn = cp(gn), where g, E G , n 3 1. Let A, = B ( y , be a ball of radius 1 / n with center at y. We set

A)

a,, = Q”(A,),

We choose 6, such that &a,

= Q”(A,“,)

+4 = 1. 7b -

Since y E K ( Q ) ,it follows that a , > 0. It is also clear that supn a; < m. A fortiori, 6, > 0 (for large n at any rate). Consequently, if g, = (1, (y,

for n large.

A, S,

A)),

then

Chapter V. Theory of Arbitrage. Discrete Time

428

We set yn = cp(yTL).Then y, = 6,6,

+ -.b71:L

where s u p 16F11 < m.

Since hnun t 1 and b, - yu,, t 0, it follows that yn lies in the closure 5(Q’) of the set

Now, note that

+ y,

arid therefore K ( Q )

is a convex set. Hence it follows from (13) that

is the barycenter of a probability On the other hand each point y E measure equivalent to Q (see (6). ( 7 ) ,and (11)),so that y lies in the closure of the convex hull L ( Q )of K ( Q ) . Hence 2 L ( Q ) ,and therefore .(Q’) C L ( Q ) ;together with (14) this shows that L ( Q )= We now use the fact that a convex set contains the interior of its closure. For the convex set (considered in the topology of H ( Q ) ) this means that L ” ( Q )C V ( Q ’ ) . Thus, we have proved (8) and (9). We procced now to the proof of the second assertion of the lemma. Let 0 @ L”(Q). Then there are three possible cases, which can be treated using the standard separation machinery of convex analysis; see, e.g., [406]. The first case: 0 $ H ( Q ) . In this case let y be a vector directed towards the affine hyperplane H ( Q ) and orthogonal to it. Then (7,z) > 0 for z E H ( Q ) ,which, of course, means that Q ( x : (-y,z) > 0) = 1. The second case: 0 E H ( Q ) , but 0 $ L ( Q ) . Then there clearly exists a vector y E H ( Q ) such that (y,z)> 0 for all z E L ( Q ) ,so that Q ( x :(y,x)> 0) = 1. The third case: 0 E H ( Q ) ,but 0 E L ( Q )\ L”(Q). Then both subsets L ( Q )and K ( Q ) of the plane H ( Q ) of some dimension q lie to one side of some ( q - 1)-diniensional hyperplane H’ of H ( Q ) containing 0. (If q = 1, then H’ is reduced t,o the point {O}.) By definition, H ( Q ) is the smallest affine plane containing K ( Q ) . Hence K ( Q )does not lie in H’ and y can be constructed as follows. We consider an arbitrary non-zero vector y in H ( Q ) that is orthogonal to H and satisfies the irieqiiality ( 7 , ~ 3 ) 0 for each z E L ( Q ) ,so that Q(z: (y,z)3 0) = 1. Next, we note that there exists z E K ( Q )such that (7,:c) > 0. Bearing in mind that K ( Q ) is the topological support of Q we see that Q ( z : ( 7 , ~>) 0) > 0. This completes the proof of Lernrna 1. 3. The next result that we shall require relates to the problem of the existence of a ‘measurable selection’.

2. Arbitrage-Free Market

429

LEMMA2 . Let (E,G, p ) be a probability space and let (G, %) be a Polish space with Bore1 a-algebra 9?. Let A be a 6 %measurable subset of E x G. Then there exists a G-valued &/%-measurable f h c t i o n ( a selector) Y = Y ( x ) ,x E E, such that (2, Y ( I I :E) )A for p-almost all II: in the E-projection

x ( A ) = { z : (z, y) E A for some y E G}. Remark 1. Here we mean by a Polish space a separable topological space that can be equipped with a metric consistent with the topology and making it a complete metric space. In the literature devoted to measurable selection (see, e.g., [ll])one can find various versions of results on the existence of measurable selectors under various assumptions about the measurable spaces (E,8)and ( G ?%). The most easy way for us is to refer, e.g., to [102; Chapter 111, Theorem 821 or [ll;Appendix 1: Theorem 11, where one can find the following result (in a slightly modified form).

PROPOSITION. Let (E,8)be an arbitrary measurable space and let (G, be a Polish space. If A E & ~3%, then there exists a universally measurable function ? = ? ( x ) , z E E , S U C ~that (z,?(z)) t A for all z E 7r(A). (We recall that a G-valued function ? = p(z)in ( E ,E ) is said to be universally measurable if it is ~/‘8-measurable,where E = Ep is the intersection of all the 0-algebras ELLthat are the completions of E with respect to all finite measures p in (E,G).) Thus, it follows from the above proposition that there exists a z/%measurable function p = ?(z), x E E , such that (z; ~ ( I I : ) E ) A for all z E 7r(A). Since 2 = 0, & ,, the function is, of course, Gp/%-measurable for each finite measure p . Using the fact that (G, %) is a Polish space and the 0.-algebra 8, is the completion of & with respect to p, it is easy to conclude (approximating Y by simple functions) that there exists a G/%-measurable function Y such that Y = p (p-as.). Hence Lemma 2 is a consequence of the above proposition. h

n,,

h

4. Proof a’) ==+ e). Assume that the market is arbitrage-free, but e) fails. Then there exist n E { 1 , 2 , . . . , N } and a .Fn-l-measurable set B with P(B) > 0 such that 0 lies outside L 0 ( Q r L ( w.,) ) for almost all w E B. The set

is Sn-1 L %(Rd)-measurablcand B is its projection r ( A ) by the last assertion in Lemma 1.

Chapter V. Theory of Arbitrage. Discrete Time

430

By Lemma 2 on measurable selection there exists a 9:,-1-measurable g = g(w) such that Q n ( w , { x : ( . ~ ( w ) l x )> 0))

Q n ( w , { x : ( g ( w ) , z ) 3 0)) = 1,

vector

>0

for P-almost all w E B. Since B is a ,F,_i-measurable set, the function i ( w ) = g(w)lB(w) is 9,-1-measurable and, in accordance with (15), (g,AS,) 3 0 ( P - a s ) and P ( ( i , AS,) > 0) > 0. We set yi = GI,=,, i = 1,.. . , N , and consider a self-financing strategy 7r = (p,y)

of value X T satisfying the equalities X t = 0 and AX? = (?i,ASi). (We determine the parameters Pi-e.g.,

Xq

=

in the case of

(yi,S i ) + & must be equal to

Bi

E

1-from

the requirements that

i

C (yj,A S j ) ) . j=1

Clearly, for this strategy 7r we have X t = 0.X p = 0 for i < n , and X,.i = Xz 3 0 for all i 3 n; moreover, P(X& > 0) > 0, which contradicts condit,ion a’). 5 . Proof e) b). We shall define the martingale measure formula dF = Z d P , where

E 9,(P)

by the

N n=l

for some 9:,-measurable functions Z,. We consider the regular conditional probability Qn(w,dz)= P(ASn t d z I Sn-l)(w),

which is also called the transitional probability (from (0,3FTL-1) to (R’ .)), )@I(% In what follows, we construct from Q,(w, d z ) , by means of a special procedure, transitional probabilities Q:,(w, d x , dw) (from (0,9T,-1) to ( E ,8 ) with E = Rd x (0, m)) such that 4; (w,d z , (0, m)) = Q n ( w ,d z ) . (17) We now return t o our discussion in subsection 2 and set Q’(dz,dv) there to be equal to the measure QL(w, drc, dv) ( n = 1,.. . , N , w E 0 ) . We also set (see (5)-(7))

Note that the above-introduced space G of elements g defined by (4) is a Polish be its Bore1 r-algebra. space (with product topology). Let , 1 8 ?!I-measurable, arid its projection 7r(A) is equal to R by The set A is 3 assumption e).

2. Arbitrage-Free Market

431

Hence, by Lemma 2 there exists a G-valued 9:,-1-measurable function gn=gn (w) such that (w,g,(w))E A for P-almost all w E R. We now consider the function

in

R x E.

It is

and

9n-1 @

&-measurable, and (P-as.) we have

IE

zz,(w,2,v ) Q’(w, dz, dv) = 0.

By (51,

sup v.,(w,

z, v) < co

V

P-almost surely. Using Lemma 2 again we see that there exists a 9:,-1-measurable tion Vn-l = V,-i(w) such that

positive func-

for all ( I C , ~ )and P-almost all w E R. We proceed now to a construction (by backward induction) of the sequence of measures QL = QL(w, drc, dv) and the corresponding sequence of functions Zn(W,5,v),12=N,N-1, . . . .

QL

To this end we set n = N and V,(w) = 1, w E 0. Let = Q&(w,dz,dw) be the regular 9n-1-measurable conditional probability of the vector (AS,, V N ) . Clearly, the ‘first marginal’ of this measure is precisely Q N = QN(w,dz). Let z~(w,z,w) be a function associated with Q& in accordance with the construction (18) and let V,-l(w) be defined by z ~ ( wIC,, v) in accordance with (21). Then we define Q h P l as the regular SN-z-measurable conditional distribution of the vector (ASN-1,VN-l). In general, given QL, we define zn(w,z,’u) in accordance with (18) and Vn-l(w) in accordance with (21). Next we set QLPl to be equal to the 9n-2-measurable conditional distribution of (AS,-l, Vn-l) and so on. We now set %(w) = ~ n ( W , A S n ( W ) , V n ( W ) ) (22) and

N

Z(w) =

Zn(W).

n=l

Chapter V. Theory of Arbitrage. Discrete T i m e

432

Then, in view of (21) and (22),

By (23), (24) and 1,earing in mind that Vr\i(w) = W,(w)

a4 <

= 1 we obtain

Vo(w), (25) the function Vo(w) is constant where Vo(w) < 00 (P-as.). Since 90= {0,C2}, (P-as.). Further, from the definition of QL and conditions (19) and (20) we see that (P-a.s.) E(%L I $,-1)(w) =1 (26) and E(AS,& I 9 n - 1 ) ( ~ ) = 0. (27) By (26) we obtain that, E Z ( w ) = 1, and therefore we can define a new probability nieasure P by setting (28) P(dw) = Z ( w ) P(dw). It is clear froni (25) that p P. I

I

-

Since and

-

EjASnl

=

EzjASnI

< r/oEIASnl < x

-

E(AS, 1 .!F,-l) = E(AS,Yn I9n-1) = 0 by Bayes's formula (see formula (4) in 33a) and (27), the sequence S = (S,) is a P-martingale. E Yb(P). This By the above construction of the measure p we obtain that proves the implication e) ==+ b).

s

-

6. Proofe) ===+ a"). Let p E g ( P ) # 0.Then = ( S T L ) ,
Remark 2. If we do not assume that B, conditional probabilities

= 1, n < N , then in place of the regular

Q,(u, d z ) = P(AS, E dx I Fn-i)(w).

one must consider directly the regular modifications of the conditional probabilities

The argunierit, in which one must bear in mind that the B,. n and 9,,-1-measurable, remains essentially the same.

< N , are positive

3. Construction of Martingale Measures by Means of an Absolutely Continuous Change of Measure

3a. Main Definitions. Density Process 1. Let (R, 9 > (,Fn)n>l,P) be a filtered probability space. Then we say that a measure i? in ( Q , 9is) ,absolutely continuous with respect to P (we write << P) if P(A) = 0 for each A E 9 such that P(A) = 0. We say that nieasures P and i? in the same measurable space (!2,9) are equivalent (we write F P) if F << P arid P << i?. In many cases the condition of absolute continuity or equivalence is too restrictive or simply redundant: one can do with a weaker condition of local absolute continuity that can be explained as follows. Let P, = P I 9,be the restriction of the probability measure P t o the a-algebra 5Fn C 9.(In other words, P, is a nieasure in ( 0 ,9,)such that

Pn(4 =P ( 4

-

for each A E gTL.) Then we say that a measure P is locally absolutelg continuous

- loc

with respect to P (we write P

<< P) if

p,,

for each n 1. Two measures, P arid

-P loc << P arid P 2 F. If, e.g.,

R

i?,

Pn

- loc P) if

are said to be locally equiiialent (we write P

N

is the space EXco, the coordinate space of sequences w = (21, xz,. . . ),

9,= o ( w : q ,. . . , z), is the c-algebra generated by the first 7~ coordinates functions, 9 = B(Rco), and P, i? are probability measures in ( n , 9 ) ,then the relation

-P loc << P means precisely that their corresponding finite-dimensional probability dis-

tributions are absolutely continuous.

Chapter V. Theory of Arbitrage. Discrete Time

434

<

Note that if we have n N < m, then local absolute continuity is the same as absolute continuity. Thus, the ‘local’ concept can be of interest only for models with time parameter n E N = { 1 , 2 , . . . }. Besides the restrictions P, = P I 9,of the measure P to the a-algebras we shall also consider the restrictions P, = P 1 9,of P to the 0-algebras 9,of sets A E 9 such that { ~ ( w ) n } n A E 9,for each n 3 1. We emphasize that, as usual, 9m= 9 (and gm-= V g n ) ;see [250; Chapter I, 5 la].

<

-

loc

2. Let P << P. Then P, << P, for each n E N and there exist (see, e.g., [4393) Radon-Nakodym derivataves denoted by

and defined as 9,-measurable functions Z, = Z n ( w ) such that

dP, . is defined only up to P,,-indistindP, guishability, i.e., if (1) holds for two functions, & ( w ) and Z h ( w ) , then

Remark. The Radon-Nikodym derivative

~

Both Z,(w) and Zk ( w ) are ‘representatives’ of the Radon-Nikodym derivative dF, in this case. Saying that ‘2,= is the Radon-Nikodym derivative’ we mean dP, that we have chosen o n e representative that we shall consider in what follows. It is easy to see that we can choose versions such that not only P(Z,(w) 3 0) = 1, but also Z n ( w ) 3 0 for all w E R and each n 3 1. It is for this reason that one usually includes the non-negativity in the definition of the Radon-Nikodym derivative of a probability measure. In what follows, we shall call the discrete-time process ~

the density process (of the measures F, with respect to the P,, n 3 1, or of the measure P with respect to the measure P). For the convenience of references we combine in the next theorem several simple but useful and important properties of the density process.

435

3 Construction of Martingale Measures

- loc

THEOREM. Assume that P << P. Then we have the following results. a) The density process Z = (Z7,) is a non-negative (P, (sn))-martingale with EZ,, = 1. Then the following conditions are b) Let 9 = 9:oo-, where 9,- = equivalent:

v9,.

(i) P << P; (ii) Z = (2,)is a iiniformly integrable (P, (S:,))-martingale; (iii) P sup Z, < = 1.

-(

->

n

c) Let r = inf{n: Z,, = 0} be the first instant when the densityprocess vanishes. Then it ‘remains at the origin indefinitely’, i.e.

o for some n 2 r ( w ) } = 0.

P { w : z,(w)

-

-

d) Let r be some stopping time. Then the restrictions P, = P I 9,and P, = P 19,to the a-algebra 9,(see Definition 2 in Chapter 11, 5 If) satisfy the relations P, << P, and dp, - dP,

I

z

e) We have the equality P infz, -( 71

f ) If P(Z,

> 0) = 1 for each

71

> 0)

3 1, then P

= 1.

loc

(3)

-

<< P and

p loc P. N

Proof. a) By ( l ) ,

for A E 9,, and therefore EIAZ, = E1~2,,+1.- Hence E(Z,+1 I 9,) = 2, (P-a.s.) for each n 3 1. It is also clear that EZ,, = Pn(R) = 1. Thus. Z is a (P, (9,))martingale. b) (i) ==+ (ii). We recall the following classical result of martingale theory ([log]; see also Chapter 111, 5 3b.4 in the continuous-time case).

Chapt,er V. Theory of Arbitrage. Discrete Time

436

DoOB’S CONVERGENCE THEOREM. Let X = (X,) be a supermartingale with respect to the measure P and the Aow ($,) such that there exists an integrable random variable Y such that X, 3 E(Y 1 9,)(P-as.) for each n 3 1. Then X, converg-es (P-as.) to a finite limit X,.

(The proof can be found in many handbooks, e.g., [439: Chapter VII, 5 41.) To prove the implication (i) ==+ (iii) it suffices to observe that since 2, 2 0, it follows by this Doob’s theorem that there exists with P-probability one a finite limit limZ,,. However, << P, therefore with P-probability one there also exists a 71

finite limit, so that P supZ,,

-(

< ..) = 1.

11

(iii) i (ii). The u n i f o ~ r nintegrability of a family of random variables means that l ~ s 111l P E ( I E n l ~ ( l E > ? , lN ) ) = 0 In our case (where

ET1 = Z T z )by,

(t,)

(iii) we obtain

which proves (ii). , (P-as.). Hence the uniform (ii) i (i). By Doob’s theorem, 2, i 2 integrability of the sequence (2,)ensures also the convergence in L1(f2, S ,P), i.e., EJZ, - Z,l t 0 as n + 00. For sets A E 9 , and for n 3 m we have

However. E(2,

-

Z,/

+ 00,and

for each A E gnL we obtain I

P(A) = EIAZ,. Hence, using the standard techniques of monotonic classes ([439; 5 2, Chapter 111) we conclude that the same equality holds in 9,and in 9 = g(u 9,)(z.!Ern =

u

V 9,). Thus, is << P and, moreover,

, = limZ,. where 2 c) To prove this property we require another classical result of martingale theory ([log]; see Chapter 111, rj3b.4 again for the continuous-time case).

437

3. Const,riiction of Martingale Measures

DOOB’SOPTIONAL that

STOPPING THEOREM.

Let X = ( X 7 %he ) a snpcrmartingale such

X n 3 E(Y 1 g n ) ,

n31

for some integrable random variable Y . Then the variables X u and X , are integrable for any two Markov times a and 7 , and E(X, I 9u) 6 X u (P-as-.) in the set {a 6

7}.

(See the proof, e.g., in [I091 or [439: Chapter VII, 921.)

Remark. On { w : T ( W ) = m} we set the value of X,(w) to be equal to X,(w), the limit of the X,(w), which exists by Doob’s convergence theorem. Besides 7 = inf{n 3 1: 2, = 0} we shall consider the times om = inf{n 3 1: 2, > l/m}. It is easy to see that 7 and om are stopping times, i.e., the sets ( 7 6 n } and {a, 6 7 1 ) belong to 9,for each n 3 1 and all m 3 1. (We recall that, as always, 7 ( w ) = 00 if Zn(w)> O for all n 3 1.) By Doob’s optional stopping theorem,

1 9,) 6 2, = 0

E(Z,

Hence Z,,Z(r means that

on the set

{w:

T(W)

< m}.

< m) = 0, m 3 1, so that nnL= m (P-a.s.), m 3 1, which precisely P{w:

371.

3 T ( W ) with & ( w ) # 0}

= 0.

d) Let A E 9,. Then E[IA . I{,<,}

.

-GI

=

1

E[IA

. I { , = ~ }‘2.1 =

ria1

=

C P(A n

-

1(7

C E [IA . I { , = ~ }. z,,] n>1

= n ) ) = P(A n { T

< w}),

n>l

which proves the required result. e) We set

m

which is equivalent to the required equality P inf 2, > 0) = 1.

-(

n

Chapter V. Theory of Arbitrage. Discrete Time

438

- loc

f ) If P << P and P(2, > 0) For A E $n we set

I

= 1. 71

3 1, then also P(2, > 0)

Qn(A)=

/

= 1.

Zllp(dw).

A

Then F,(dw) = Z,, Pn(dw), and therefore

Hence

22

so that P F. Thus, we have proved all assertions a)--f) of the theorem.

3. The following ‘technical’ result is useful in the considerations of conditional expectations with respect to different measures. We shall use it repeatedly in what follows and call it the ‘conversion lemma’. Forniula ( 4 ) is often called ‘(generalized) Bayes’ formula’ ([303; Chapter 71). LEMMA.Let Pn << P, and let Y be a bounded (or F-integrable) Sn-measurabJe random variable. Then for each ni n we have 1 E(Y 1 9m,) = -E(YZ, 19,) (P-a.s.). (4)

<

z 7 n

Proof. For a start we observe that P(Z, > 0) = 1 (see assertion e) in the above theorem). Further, we also have &(w) = 0 (P-a.s.) in the set {w:Z,(w) = 0) for n 3 m. Taking this into account we shall assume that the right-hand side of (4) vanishes in this set. By definition, E(Y 1 Sm)is a 3,-measurable random variable such that I

E [ I A . E(Y 1 9,,)] = E[IA . Y ]

(5)

for each A E Sm,so that we need only verify that for the 9m-measurable function on the right-hand side of (4) we have

1 -

1 gm) = E [ I A . Y ] In fact, this follows from the following chain of equalities:

3 . Constriiction of Martingale Measures

439

where ( a )holds by the definition of the conditional expectation E(YZ,, 1 .Fm) and (p) is a consequence of the 9m-measurability of IAY and the fact that Z = (Z,) is a martingale. fj 3b. Discrete Version of Girsanov’s Theorem.

Conditionally Gaussian Case 1. It is reasonable to start our discussion of the construction of the probability measures F, that are (locally) absolutely continuous or equivalent to the original basic measure P involved in the definition of the filtered space

with a discrete (with respect to time) version of a result established by I. V. Girsanov [183] for processes of dzffusion type, which became the prototype for a number of results for martingales, local martingales, local measures, semimartingales, and so on. (See, e.g., [250; Chapter 111.) Let n 3 1 be the time parameter and let E = ( q , ~ 2 ,. .. ) be a sequence of g:,-measurable random variables E~ with distribution Law(€,

1 Tn-l;P) = N ( 0 , l ) .

(1)

-

In particular, this means that E is a sequence of independent standard normally distributed random variables, E, J V ( 0, l ) . Assume that, besides F = ( ~ ~ ) ~ we ~ 2 are 1 , given predictable sequences p = (pn)n21 and e = (en),>l, i.e., sequences of 9,-1-measurable variables p, and en ( 9 0 = {0,0}). Moreover, we assume that 0, > 0, which is based on the interpretation of this parameter as ‘volatility’ and the fact that we can simply leave observations with en = 0 out of consideration. We set h = (hn),21r where h, = pn

+

(2)

On&,.

By (1) we obtain that the (regular) conditional distribution P(h, be defined as follows:

< . I 9,-1) can

or, symbolically, Law(& I 9,-1;

2

P) = . J y ( P n , e n ) ,

(4) which allows one to call h = (h,) a conditionally Gaussian sequence (with respect to P) with (conditional) expectation E(hn I s n - 1 ) = p n

(5)

440

Chapter V. Theory of Arbitrage. Discrete Time

n

n

n

we can say that the variables Hn can be represented as follows in the conditionally Gaussian case: Hn = An M n ,

+

where A = ( A n )is a predictable sequence and M = ( M n )is a conditionally Gaussian martingale with quadratic characteristic n

n

1~

k Awn , = E ~ A, H , = h,, and A = 1. Then we can k=l rewrite (2) in the difference form as follows:

We set Wn =

AH,

= pnA

+ anawn,

which one can regard as a discrete counterpart to the stochastic differential

dHt

= pt

dt

+ 0 t dWt

of some It6 process H = ( H i ) generated by a Wiener process W = (Wt)t2o,with local drift ( p t ) t 2 o and local volatility (ot)t>o (see, e.g., [303; Chapter 41 and Chapter 111, fj3d). In the present case of a conditionally Gaussian sequence ( 2 ) this discrete analog of Girsanov’s theorem (which, as already mentioned, was proved by I. V. Girsanov in the continuous-time case) has a relation to the question of the existence of a measure such that p i s absolutely continuous or equivalent t o the measure P and the s e q u e n ~ eh = (11,) i s a (local) martingale difference with respect t o P. It is worthwhile to point out in this connection that the right-hand side of (2) contains two terms: the ‘drift’ p n and the ‘discrete diffusion’ which is a martingale difference (with respect to P). The meaning of the above question lies in the existence of a measure p << P such that ( h n ) has no ‘drift’ component with respect to is << P and reduces to ‘discrete diffusion’, i.e., is a (local) martingale difference. I

3. Construction of Martingale Measures

441

-

2. Our construction of the measure P is based of the sequence of (positive) random variables

c al, -; 71

z , =~exp{

-

%k

k= 1

LEMMA.1) The sequence 2 = (Z,),,,

n 3 1. 2) Let 9 =

C(%j2}, k=l

n 3 1.

(7)

is a (P,(S,))-martingale with €2, = 1,

v 9,and assume that

(the ‘Novikov condition’) Then 2 = (.&),>I is a uniformly integrable martingale with limit ( P - a s ) 2, = IimZ, such that

Proof. 1) This is obvious since for each k 3 1 we obtain

Fk by the Sk-1-measurability of the and conditionally Gaussian property (1) (here $0

= (0,Ol).

“k

2 ) The proof that the family (Zn) is uniformly integrable if (8) holds is fairly complicated and can be found in the original paper [368]by A. A . Novikov as well as in many textbooks (see, e.g., [303; Chapter 71 or [402]). However, this proof becomes relatively elementary once one imposes a stronger condition: there exists E > 0 such that

Hence it would be reasonable to present this proof, which is what we do at the end of this section (see subsection 5).

442

Chapter V. Theory of Arbitrage. Discrete Time

3. We fix some N 3 1 and shall consider the sequence ( h n ) only for n 6 N . For simplicity we shall assume that 9 = 9 ~ so t, hat PN = P 19~ = P. Since ZN > 0 and EZN = 1, we can define a probability measure is = P(dw) in (R,9)by setting -

P(dw) = z N ( W ) P ( d w ) .

(13)

We - emphasize that we do not only have the relation is << P, but also P << p. Hence P P. We consider now the properties of the sequence ( h , ) , < ~ with respect to P. By Bayes’s formula (4) in §3a, for each X E R and n N we obtain N

<

-

(P-as.), where we use the equality ~ e ( z A o ~ - ~ ) & n - ; ( z A ~ n - ~=) z

and the fact that the 0; are 9,-1-measurable. We obtain the equality

-

A 2 2

E(eixhn 1 Tn-l) = e - 9

(P-a.s.),

(15)

which means that the sequence h = (h,) remains conditionally Gaussian with respect to this new measure p, but has now the trivial ‘drift’ component: 1

Law(h,

I 9,-1;

P) = “0,

cz),

(16)

so that we obtain the following analogs of (5) and (6) in this case:

One can say that the transitions from P t o the measure is eliminates (‘kills’) the drift p = ( p , ) , < ~ of the sequence h = ( h n ) n G but ~ , preserves t h e conditional variance.

3. Construction of Martingale Measures

443

We can also conclude from (16) that if ? = ( E l n ) n G ~ is a sequence of s n - m e a surable variables with distribution

(one can always construct such a sequence, although it may be necessary to enlarge our initial probability space), then

~ as a local martingale Hence it is clear that the sequence ( h T L ) , <'behaves' with respect t o P, while in terms of the original measure P a difference (o,E,),,~N property similar to (20) can be expressed as follows: Law(h,

-

pnr n

< N 1 P) = Law(cr,En,

n

< N 1 P).

(21)

We now shift our standpoint. We shall treat a measure P and a sequence h = (h,) satisfying (20) as original ones. Then the transition from r? to a measure P (in accordance with (13)) g'ives us property (21), which can be interpreted as drift appearing in the local martingale difference (crnEln),<~. This interpretation is more convenient when one is willing t o state the corresponding general result on the transformations of local martingales under absolutely continuous changes of measure (see 3d below). Before summarizing the results just obtained we point out the following. Assume that ~ ; ( w )is independent of w (= 0 ; ) . Using induction we obtain by (14) that I

-

Thus, with respect to the measure P the sequence ( h , ) , < ~ consists of independent norinally distributed random variables h, A ( 0 , cr;), with expectations zero. (We could arrive to the same conclusion on the basis of (17)-(20).) We now sum up the above results in the following theorem, which we could call the discrete analog of Girsanov's theorem.

-

THEOREM. Let h = ( h n ) n G N he a conditionally Gaussian sequence such that Law(h, 19,-1; P) = ."(FTL, Let 9~= 9 and let defined in (7).

2

0,))

12

6 N.

be the measure defined by formula (13) with density Z,

Chapter V. Theory of Arbitrage. Discrete Time

444

Then 1) the sequence h = ( h , , n ) n G is ~ conditionally Gaussian with respect to P :

I

Law(h,,

I 9,-1;

-

P) = A ( o , ~ ; ) n,


<

2) if the 02 = o;(w),n N , are independent on w , then h = ( h n ) r r G is ~a sequence of independent Gaussian variables with respect to P:

v

Law(hn 1 F) = J ( O ,

n


3) if 9 = gn and condition (8) is satisfied, then properties 1) and 2) hold with respect to the measure F(dw) = Z,(w) P ( d w ) for all n 3 1, where Z,(w) is as in (9). 4. We note that the sequences ( P n ) and (an)from the definition of hn (= pn+on&n)

are directly involved in our construction of the measure in accordance with (13) and (7). For this reason and also to establish connections with our special choice of the values u n ( w ) in the discussion of the Esscher transformations (see 5 2d), we consider now the sequence of processes Z @ )= ( Z(,6 )) l G n 6defined ~ by the equalities

where the bk = bk(w) are 9k-l-measurable. ( h ) ( w ) = 1, we have a well-defined probability measure Sirice EZN

-

( b )(w)P ( d w ) P ( h ) ( d w ) = ZN

(23)

in 9~= .9. The expectation with respect to this measure is

-

E(”)(h, 1 ,!Fn-i) = -b n

It is now clear why one sets b, =

-Pn,

n

Pn

- -

an

.

(24)

6 N , in Girsanov’s theorem: it is under

on this choice that the sequence h = ( h n ) n Gbecomes ~ a local martingale dzifference. Further, if X , = p n E ~ then ,

+

vn(a;w)E ~

1

( e I 9n-l) ~ ~ = n eya

2

p a ~ n .

Hence

w) inf vn(a;w ) = vn ( a n(w);

<

where a n ( w ) = P n , n N . We used just these ‘extremal’ variables a n ( w ) in $2d, in our construction (by means of the Esscher transformation) of a measure making the sequence (X,) a martingale difference. Thus, in the present case (of on = 1) both Girsanov and Esscher transformations bring us to the same measure P . I

3. Construction of Martingale Measures

445

5. We now claim that if (12) holds, then the family Z = (Zn)n21of the variables Z,,, defined in ( 7 ) is uniformly integrable. Let DTL= - k . Then t,he condition (12) takes the following form: there exists

b

>0

fT*

such that

El:XP(

(; i d ) F p ; } <

00.

k=l

In accordance with (7),

Assume that

E

> 0 arid p

1. We set

and

To prove the uniform integrability of ( Z n ) ~ > it 1 suffices to show that sup EZ;+€

< 00

n

for some

E

> 0 (see, e.g., [439; Chapter 11, 56, Lemma 31). Since (1) (2) = $ n $rl 1

it follows by Holder's inequality that

(l/p

+ l/q

= l ) , where we use the equality

E($A'))'

+

(see (11)). We set p = 1 6 and q = (1 holds. We now find E > 0 such that E(1+

E)

=1

+ 6 ) / 6 , and we choose 6 > 0 such that (25)

< (1 + 6 )6(21 + 26) .

(31)

Chapter V. Theory of Arbitrage. Discrete Time

446

Then

so that

by ( 2 5 ) , which proves the required uniform integrability of (2,)

3 3c.

Martingale Property of the Prices in the Case of a Conditionally Gaussian and Logarithmically Conditionally Gaussian Distributions

1. Let ( n , 9 , ( g n ) , P ) , n 3 0, be the original filtered probability space. In our discussions of martingale measures turning the sequence of discounted prices =

(&),

where B = (B,) and S = (S,),into a martingale we consider first a

somewhat idealzzed model of a ( B ,S)-market: we shall assume that B, S,=S0+Hn,

n31,

= 1, (1)

n

where H , =

C

hk

and SO = Const. We shall also assume that h = (h,) is

k=l

+

a conditionally Gaussian sequence: hn = b, cr,~,, where the p, and nT1are 9n-l-measurable, and E = ( E ~ is ~ )a sequence of independent, A ( 0 , 1)-distributed, gn-measurab1c variables cn, n 3 1 (see the preceding section for details). Thus, we allow the prices S, to take also negative values. (This is the ‘idealization’ we have mentioned.) Note, however, that L. Bachelier [I21 considered precisely a model of this kind; see Chapter I, fj 2a. Let p, = 0. Then

s,

SO

+

OkEk.

(2)

k(n

Since the are 9k-I-measurable and E ( E ~ l g k - 1 ) = 0, it follows by (2) that the sequence of prices S = (S,)is a martingale transformation and, therefore, a local martingale (see the theorem in Chapter 11, 3 Ic). Assuming additionally, that, e.g., E l c r k ~ k l< 00, k 3 1, we obtain that S = (S,) is a martingale with respect to the original measure P. (As regards general conditions ensuring that a local martingale is a martingale, see Chapter 11, 3 Ic).

3. Construction of Martingale Measures

447

<

Assume now that the p, do not vanish identically for all n N . In this case the discrete version of Girsanov’s theorem (3 3b) give us tools for a construction of measures (see formulas (8) and (6) in 53b) such that the sequences (S,),
2. We consider now a more down-to-earth situation where

<

in our ( B ,S)-market and B, = 1, n N . As mentioned in Chapter 11, 3 l a , the representation (3) of ‘compound interest’ kind is convenient for a statistical analysis but it is not perfectly convenient for the aims of stochastic analysis. The problem is that t o verify the martingale property of the sequence S = (S,) one would better have a result saying: “in order that a sequence S = (S,) defined by ( 3 ) be a martingale it is sufficient that the sequence H = (H,) be a martingale”. This is, however, not so in general, which explains why we turn to the representation

s, = s,qa),

(4)

(‘simple interest’), where (see Chapter 11, 3 l a )

G,=H,+x(eAHk

(5)

-AHk-1)

k
h

and &(H)= ( & ( f i ) , ) 7 1 2 0 is the stochastic exponential constructed from H = (H,) by the formulas

g(ii),

= eHn

IT

(1

+ Aiik)e-A’k,

n

2 1,

(6)

k
and 8 ( H ) o = 1. Of course, we could simplify the right-hand sides in (5) and (6) and write these equations as ii, = (,AH,, - 1) (7) A

c

k
and k
Still, it is useful to point out again (see Chapter 11, 3 l a and Chapter 111, 5 5c) that bearing in mind a similar representation in the continuous-time case, we should regard as ‘right’ formulas ones close to (5) and (6) and not t o (7) and (8). This has something to do with the problem of the convergence of the corresponding ‘slims’ CsGtand ‘products’ , which, generally speaking, have inzfinitely many terms and factors for each t > 0 in the continuous-time case. The advantage of (4) over ( 3 ) lies in the following result.

n,,,

448

Chapter V. Theory of Arbitrage. Discrete Time

PROPOSITION. In order that a sequence S = ( S n ) defined by (4) be a martingale it is sufficient that the sequence H = (Hn),21 be a local martingale with AHn -1 for n 3 1.

>

Indeed. from (6) or (8) we see that

A&(E?), = & ( i t ) n - l A f i n

(9)

for n 2 1. Assume that (2,) is a local martingale, and therefore is a martingale transformation (see the lemma in Chapter 11, 5 Ic), which admits a representation

with Bk-1-measurable ak and some martingale hl = ( M n ) . From (9) and (10) we see that

A G ( E? ) n = a, G ( E?) - 1A Mn , i.e.. & ( H ) is a martingale transformation and. therefore. a local martingale. If AH, 3 -1. then. surely, &(H), > 0. Hence, by the lemma in Chapter 11, 5 Ic the local martingale & ( g is ) actually a martingale. The condition AH, 2 -1 holds in our case because

3. We shall assume that the h, = A H , are conditionally Gaussian variables with h, = p n + C T ~ E , . Then it is natural to say that the sequence S = (S,) with S, = S"eHn is logarzthmzcally condztzonally Gaussian. as in the title of § 3c. First, we consider the question of the conditions ensuring that the sequence S = (S,) is a martingale with respect to the orzgznal measure P. We have already seen that it is sufficient to this end that the sequence 8 = (fin) with Afi, = eAHn - 1 be a local martingale. i.e.. E(lASn1 l 9 n - i ) < x and E ( L H , i ~ ~ - =1 0.) or. equivalently.

Since we assume that

= p,

which is equivalent to the relation

+ on€,.we can rewrite condition

j

11) as follows:

3. Construction of Martingale Measures 1

449

2

The left-hand side here is equal to e5"n. Thus, we arrive a t the condition 2

/An

+ gn2

=0

-

(P-a.s.),

n

1,

(14)

ensuring that the logarithmically conditionally Gaussian sequence

is a martingale with respect to P. Of course, this is what one could expect because the sequence

is a martingale, as already mentioned.

4. We now proceed to the case when (14) fails. Assume that n < N . We shall construct the required measure on 9~= 9 by means of the conditional Esscher transformation, in the following form:

-

P ( d w ) = Z,(w)

P(dw)

where we shall choose the 9k-l-measurable variables ak = ~ ( w(here ) $0 = (0, O}) such that the sequence ( S n ) 7 a is < a~ (p, (Tn))-martingale. In our case, when the prices are described by formulas ( 3 ) ,this means that, with necessity, E[e("n+l)hn I sn-l] = E[eanhn 1 9n-1 ] . (17) Bearing in mind that h, = p n

+ on€,, we see that equality (17) holds if 2

Pn

2 + grl 2 = -anan,

i.e., pn

%=---I T :

1 2.

-

If (14) holds for all n 6 N , then a , = 0 and Z N = I , i.e., P = P

Chapter V. Theory of Arbitrage. Discrete Time

450

Choosing a, in accordance with (19) we obtain

Thus.

and

N

2,

= exp{

-

c[ (g+

+ (:;

n=l

Hence the sequence

+

3)2]}.

(20)

S = (S,),
-

I

is a P-martingale with ES, = S O ,and the density ZN of the measure is with respect to P is described in (20). If (14) holds for n 6 N , then is = P and (S,),,, is a (P, (9,))-martingale, a martingale with respect to the original measure.

3 3d.

Discrete Version of Girsanov’s Theorem. General Case

1. We have already mentioned that the discrete version of Girsanov’s theorem for the conditionally Gaussian case had paved way to similar results on stochastic sequences H = (H,), where h, = AH, can have a more general structure than the one suggested by the representation h, = p n c,~,. To find generalizations of a ‘right’ form we shall analyze our proof of the implication

+

(established in the conditionally Gaussian case) once again. This proof in 3 3b was based to a considerable extent on the conversion formula for conditional expec-

-

loc

tations ((4) in 33a), which has the following form for P << P, Y = Hn (where EIH,I < M), and m = n - 1: 1

I

E(Hn I 9,-1)

=

~

2,-1

E(H,Z,

I 9n-1)

(p-a.~.),

n

2 1.

(1)

Here E is averaging with respect to p and the right-hand side is set equal to zero if Z,-l(w) = 0. We also set 90= {0,0} and Z O ( W3) 1 in what follows.

3. Construction of Martingale Measures

451

We claim that one can easily deduce from (1) the following equivalence:

- loc zf

P

<< P ,

then

HE

&(is)

H Z E &(PI,

(2)

where &(P) and &(is) are the sets of martingales with respect to the measures P and p (see Chapter 11, 5 l c ) . Indeed, if H E A ( P ) , then i ( H n 1 9 7 L --1 )= Hn-l (P-as.), and it follows from (1) that HTL-lZTL-l= E(HnZn 1 .Fn-1) ( P - a s ) . This equality holds not only p-a.s., but also P-as., for its both sides vanish on the set { & - I = 0) because we also have Z,, = 0 (P-as.) in this set. As regards the set {Zn,-l > 0}, the measures P and P are equivalent there (in the P({Zn-l > 0 } n A ) = 0 sense of the equivalence P({Zn-l > 0 ) n A ) = 0 for A E Sn-l), therefore the left-hand and the right-hand sides of this equality coincide also P-almost surely. Thus, we have proved the implication ===+ in (2). In a- similar way, if H Z- E A ( P ) , then Hn-lZn-l = E(HnZn1Fn-1) (P and P-a.s.). Since Zn-l > 0 P-a.s., it follows that

Hn-l

-

1 =

~

Zn-

1

E(HnZn I gn-l) (P-a.s.).

Hence it follows by (1) that H E &(is). It is worth noting that the Bayes’s formula (4) in § 3 a (and, therefore, also formula (1))is a consequence of the implication ==+ in (2). For let Y be a s n - m e a -

- loc

surable variable - such that ElYl < 00 and P << P. Then we consider the martingale ( H m ,gm,P ) m < n rwhere Hm = E(Y 1 9m). By (2) we obtain I

E(YZ,, I 9m) = HmZm

(P-a.s.)

-

In particular, E(YZ,j S7,-1)= Hn-lZn-l

( P and P-a.s.)

Since p(ZTL-,> 0) = 1, it therefore follows that

-

-

Together with the equality Hn-l = E(Y I .qn-1) (P-a.s.), this proves the required formula (4) in the ‘conversion lemma’ in 9 3a:

- loc

P << P

and

E(Y lST,-l) =

E ( Y Z , IT^-^) (F-a.s.).

~

zn-1

Hence the property (2) is a sort of a ‘martingale’ version of the conversion lemma for absolutely continuous changes of measures.

Chapter V. Theory of Arbitrage. Discrete Time

452

2. It can be useful to formulate (2), together with its 'local' version, as the following result (cf. [250; Chapter 111, $ 3bl).

- loc << P and let Z = (2,)be the density process:

LEMMA.Assrime that P

z -

dPn -- dP,

-

with P, = P I gT1 and P, = P 1 S T l . Let H = (H,, 9,)be a stochastic sequence. Then

a) H is a P-martingale ( H E &(P)) ifand only i f t h e sequcncc H Z = (H,z,, is a P-martingale ( H Z E A ( P ) ) : H E

A(P) +==+ H Z E

.F,)

A(P)

-

- loc

b) I f , in addition, P P, then H is a local P-martingale ( H only if H Z is a local P-martingale ( H Z E Aloc(P)):

A1oc(P)) if and

H E .AZ~~,-(P) a H Z E Aloc(P).

(3)

Proof. a ) We have already proved this using formula (1) (which is also of interest on its own right). However, this can also be proved directly, starting from the definition of a martingale. We choose 7ri n and A t .qm.Then E(IAH,) = E(IAH,,Z,), so that

<

-

E ( I A H ~ ,= ) E ( I A H ~ )U E(IAH,Zn)

=

E(IAH~Z,).

However, E(IAH,Z,) = E(IAH,Z,). Hence H E A(F) H Z E A(P). b) We claim that H Z E A l o c ( P ) ==+ H E dlloc(P) (even if we assume only that P Let

2; P). (7,)

be a localizing sequence for H Z and let

- loc

and P(7 < 00) = EZ71(7 < m ) = 0 (since P proved in $3"). Hence P(T = 00) = 1. We note that

7

= lim7,.

Then P(T = m) = 1

<< P and by property d) in the theorem

I

( H 7 n z ) I ,= H P Z k = ( H k z k ) 7 n + HTn(Zk - Z,,I(k

3 7,)).

Consequently, H7nZ is a P-martingale-and H T n E .A(P)by assertion a). However, P(lirri7, = m) = 1. Hence H E A1oc(P). I

-

- loc P, then H E .A1,,(P)

We claim now, t h a t , conversely, if P

-----I.

H Z t Aloc(P).

3. Construction of Martingale Measures

-

453 I

Let ( g r L )be a localizing sequence for H E ,Aloc(P). Then P(limm, = m) = 1 and Hun t A(F).Hence H'nZ E A ( P ) by a ) , and sirice

(cf. the above formiila for ( H T n Z ) k ) ,it follows that (HZ)En E A ( P ) . However, P

loc

-

<< P, so that

P(lim0, = 00) = 1. Hence H Z E Aioc(P). The proof is complete.

3. Properties (1)' (2), and ( 3 ) arc of fundamental importance for the problem of the verification of the martingale property with respect t o one or another measure p for the sequences ( H n ) , (&,), (S,), etc., because they reduce this task to the verification of the martingale property with respect to the original, basic measure P for the sequences (If,&), ( g , Z n ) , (&Z,). In fact, we have already used this in our proof of the discrete analog of Girsanov's theorem in the conditionally- Gaussian case, when Hn = hl . . . h, and h, = p, CJ~E,, n N , and the PN are the measures with densities

+

<

+

+

explicitly constructed from ( p k ) and (ak). However, the general case of the construction of the measures PN is considerably more complicated. The simplicity of the formulas for the 2 , in the conditionally Gaussian case is essentially the effect of the simple formulas for the Hn: A H , = h, = p, On&,. There exist various generalizations of Girsanov's theorem in the discrete-time case. For a more clear perception of the results below as some generalizations of this theorem it seems reasonable to reformulate its already established version (i.e., the theorem in 5 3b) in the conditionally Gaussian case. We set

+

Then

ri

and if M , =

a k & k , then k l

hf

=

(M,)

E

+Aloc(P)and it is easy to show that

Chapter V. Theory of Arbitrage. Discrete Time

454

Thus, leaving aside the questions of integrability for a while, in the conditionally Gaussian case we can state the theorem in 3 3b as follows:

-

In other words, the inclusion M E A l o c ( P ) means that the sequence M = (Gn),
k=l

-

is a local martingale with respect to the measure P(dw) = Z,(W) P(dw)

We emphasize an important point in the above analysis. In our discussion of A M , we were primarily interested in the sequence H = (H,) with A H , = p, the martingale component of H . In effect, we ‘traced’ the change of the martingale component under a continuous change of measure. As we see, if we consider the measure p, then (M,) is no longer a martingale: it has the representation

+

-

-

C;=,

where M = (M,) is a P-martingale and A = ( A , = E(akAMk 19,+1)) is some predictable drift. It is the appearance of this additional ‘drift’ term after an absolutely continuous change of measure that enables one to .kill’ the drift components of the original sequences H = (H,) by means of a transition to measures P that are absolutely continuous (or locally absolutely continuous) with respect to P. 4. This interpretation of the above discrete version of Girsanov’s theorem (in the conditionally Gaussian case) enables us to state the following general result on local n

martingales, where we do not speczfy that kf,=

C q&k. k=l

3. Construction of Martingale Measures

455

- loc dP, THEOREM 1. Let M E Al0,.(P) with A40 = 0. Assume that P << P, let 2, = __ dPn rL >, 1, be the corresponding densities, and let on = __ 2, I(Zn-l> O), where &-I

20= 1. Also, let E(lAMnlan I c9n-i)< 00

- Then the process

=

(P-a.s.),

n 2 1.

(10)

(Gn) defined in (8) belongs to Aloc(F) (i.e., is a local

P-martingale) .

Proof. Again (as in the proof for the conditionally Gaussian case in §3b), we use Bayes’s formula (4) in 0 3a:

Hence, by assumption (10): we obtain

-

< E(Ia,AMt(

E(IMn/ 1 S n - i )

1972-1)

+ IMn-11

<

(P and “Pa.s.) From (11) and (8) we immediately see that E(IMnl 1 Tn-l) < 00 and

_ -

--

E(Mn I %-1)

i.e., % is a local F-martingale

=

(% E ..“Il,,,(P))

Gn-I’

(12)

by the theorem in Chapter 11, 5 1c.l.

5 . Now, assume that the basic sequence H = (Hn)n21 has the representation

Hn = A ,

+ Mn,

(13)

where A = (A,),>l is a predictable sequence (the An are 9:,-1-measurable n 3 1, where 90= and A0 = 0) and M = ( M n ) n 2 E ~ J&I~~(P). Since E(lAM,I 1 9,-1) < 00 for a local martingale, it follows that

{@,a},

E(jAHn/ I Fn-1) so that the H,, n

2 1, have n

< IAAnI + E(laMnI I gn-1)

also the representations n.

< 00,

for

456

Chapter V. Theory of Arbitrage. Discrete Time

which we have called the generalized Doob decomposition (see Chapter 11, l b ) of the sequence H = (Hn)n21. As in the case of t,he usual Doob decomposition, the representation of the form (13) with predictable ( A n ) is unique, and therefore

and

Theorem 1 above has the following simple generalization.

THEOREM 2. Let H

=

(Hn),>1 be a sequence with generalized Doob decomposi-

tion (14) and assume that (10) holds. Let is be a measure such that the sequence H = (Hn)7L21has a representation

H, =

An + G,,

p2 ; P.

Then

(17)

or, equivalently,

k=l

k=l

(a generelized Doob decomposition), where

and the sequence &? of the variables

is a local P-martingale (&? E

A~~,-(F)).

Proof. This is an immediate consequence of Theorem 1 applied to the sequence = (Mn)n21 with M , = H, - An.

A4

3. Construction of Martingale Measures

457

6. Under the hypothesis of Theorem 2 we assume now that M and 2 are (locally)

square integrable martingales. Then they have a well-defined predictable quadratic covariance ( M ,2 ) = ( ( M ,2 ) n ) n 2where ~,

One sometimes calls ( M ,2 ) simply the angular brackets' of M and 2;as regards the corresponding definitions in the continuous-time case, see, e.g., [250; Chapter 11, § § 4a, b] or [439; Chapter VII, § 11 and Chapter 111, § 5b. We also recall that and Y = (Yn),>o is the the qiiadratic covariance of the sequences X = (X,),,, sequence [X,Y ]of variables n

C AXkAYk.

[X,YIn =

k=l

By (21) and (22) we obtain that, in the case of (locally) square integrable martingales, the difference [ M ,Z ] - ( M ,2 ) is a local martingale (see [250; Chapter I, 3 4e]). We shall now assi~niethat P

1Er P. Then 2, > 0 (p and P-a.s.) and

Note that if ( M ) , ( w ) = 0 where ( M ) (= ( M , M ) )is the quadratic characteristic, then also ( M , Z ) n ( u )= 0. Hence the left-hand side of (23) can be rewritten as follows (here ( M ) , = ( M , h & ) : "

z)n= -a,A(M),,

where

and where we set

A ( M ,Z)n to be equal, say, to 1 if A(M), A(W71

Thus, (19) can be written as follows: n

= AT,-

k=l

= 0.

Chapter V. Theory of Arbitrage. Discrete Time

458

Hence we can make an important conclusion as regards the structure of the original sequence H (with respect to P): if this sequence i s a local martingale with respect

- 1EC P (z.e., A- = 0 ) , t h e n

to a measure P

n

Hn

=

1a d ( M ) k + Mn,

71.3 1,

(26)

n 3 1.

(27)

k=l

or, in t e r m s of increments,

AH,

=u

~ A ( M+)A~M n ,

7. So far we have based our arguments on the mere existence of the measure P, without specifying its structure or the structure of the sequence a = (a,) involved in the definition of the Radon-Nikodym derivatives

By ( 2 3 ) and (24) we obtain

ann(M)n

1

E[(1- a n ) A M n I9n-11.

(29)

This relation can be regarded as an equation with respect to the ($,-measurable) variable a,, and one can see that it has the following ( n o t necessarzly unique) solution: an = 1 - unAMn. (30)

Of course, only those solutions a , satisfying the relation P ( a , suitable for our aims. If this holds, then

> 0) = 1, n 3 1, are

where 6‘ = ( 8 ( R ) nis ) the stochastic exponential (see Chapter 11, 3 1):

6‘(R)n= eRn

n

k
(1

+ A R k ) e P A R k = JJ(1+ ARk).

(32)

k(n I

Let be a probability measure such that its restrictions P, = P I Sn can be recovered by formulas (31). Our original sequence H = ( I f n ) ,which satisfies (27), becomes a local martingale with respect-to this measure because AA, = a n A ( M ) , n + E(anAMn 1 $,-I) = 0 for n 3 1 and A0 = 0. We have already pointed out that this probability measure P, a martingale measure, is not unique in general. However, it has certain advantages. First, it can be explicitly constructed from the coefficients an. Second, it has certain properties of ‘minimality’, which justify its name of m i n i m a l measure (see [429] and Chapter VI, 3d.6). I

3. Construction of Martingale Measures

459

3e. Integer-Valued Random Measures and Their Compensators. Transformation of Compensators under Absolutely Continuous Changes of Measures. 'Stochastic Integrals' 1. Let H = ( H r L ) n 2be ~ a stochastic sequence of random variables H , = H n ( w ) defined on a filtered probability space (n,9, (9,),20, P). We shall assume that HO = 0 and 90= {@, n}. There exist two ways to describe the probability distribution Law(H) of H : in terms of the unconditional distributions of the variables H I , H 2 , . . . , H,,

Law(H1, H 2 , . . . , H,)

(1)

(or, equivalently, Law(AH1, AH,, . . . , AH,)) or in terms of the (regular) conditional distributions of the variables AH,:

P(AH, E j 9,-l), '

n 3 1.

(2)

The second way has certain advantages because, given conditional distributions, one can, of course, find the unconditional ones. (In addition, the conditional distributions exhibit the dependence of the AH, on the 'past' in a more explicit form.) On the other hand, starting from unconditional distributions (1) one .I where 9fPl= can recover onlg the conditional distributions P(AH, a ( w : H I , . . . , H r L - l ) ,so that C FTL-l (this can also be a proper inclusion).

9F-l),

9FPl

2. We consider a &dimensional stochastic sequence

X = ( X n ,9:n)n>O

(3)

on a filtered probability space (0,9, (gn),20, P). Let X o = 0 and let 90= {@,R}. We associate with X the sequence p = (p,( .)),>I of integer-valued random measures defined as follows: p n ( A ; w )= I A ( A x n ( w ) ) ,

A E B(Rd),

i.e.,

CLn(A;w)=

{

1 if A X n ( w ) E A, 0 if AX,(w) @ A.

Further, let v = (vn( .)),>I be the sequence of regular conditional distributions vn( . ) of the variables AX, with respect to the algebras 9,-1, i.e., of the functions v n ( A ; w )(defined for A E % ( E L d ) and w E R) such that 1) v , , ~ (; .w ) is a probability measure on (ad, B(Rd)) for each w E R; 2 ) vn(A;w ) , regarded as a function of w for fixed A E % ( E L d ) , is some realization of the conditional probability P(AX, E A I 9 , - 1 ) ( w ) , i.e.,

vll(A;w= ) P(AX, E A I 9 , - 1 ) ( w )

(P-as.).

460

Chapter V. Theory of Arbitrage. Discrete T i m e

(The proof of the existence of such a realization of the conditional probability can be found, e.g., in [439; Chapter 11, 5 71.) For regular conditional distributions the conditional expectations

E [ f ( A X n )l9n-11(u) can be calculated for a non-negative or bounded function f by means of integration over the regular conditional distributions un( . ) for fixed w :

E [ f ( A X n )I %I]

(w) =

ld

f(5)v n ( d z ; w ) (P-a.s.1.

Thus, in the case under consideration we have

vn(A; . )

= E [ p n ( A ; w )I9n-1](. ),

so that for each A E B(Rd) the sequence

( P n ( A )- vn(A))7L>1 with p r L ( A )= p n ( A ; w )and v,(A) = vn(A;w)is a martingale difference with respect to the measure P and the flow We set

(sn).

k=l

k=l

Then, clearly, for each A E B ( R d ) the sequence (P(O,n](4w ) - V ( O , n ]( 4w ) ) T L d ]

is a martingale. This property explains why one calls the (random) measure V ( ~ , ~. )I the ( co,mpensator of the (random) measure p(O,nl(. ) and why one calls the sequence P - v = b ( O , n ] ( ' -) v ( O , n ] ( ' ) ) n g l a random martingale measure. Note that the representation p =v

+ (p

-

v)

with predictable measure v = ( V ( ~ , ~ I )is~ a> kind ~ of the measure p = of the Doob decomposition (Chapter 11, 3 l b ) into the sum of a predictable and a martingale components.

3. Construction of Martingale Measures

461

Remark. One can also associate with the sequence X = ( X n ,c F n ) r r bthe ~ integervalued randoni nieasures of jumps p x = (P$,~,(. ) ) l L a lwhere

and p f ( A ; w ) = I(AX,(U) E A, AX,(w)

# 0).

Clearly, if A E %(rW’\{O}), then p n ( A ; w )= p Z ( A ; w ) . All distinctions between these measures are concentrated a t the ‘no-jump’ events { w : A X n ( w ) = O}. If P { w : A X , ( w ) = 0} = 0, then the measures p and px are essentially the same. We note that these are the random jump measures px, rather than p , that play the central role in the continuous-time case, in the description of the properties of the jump components of stochastic processes in terms of integer-valued random measures. (See Chapter VII, 5 3a of this book and [250; Chapter 11, 1.161 for greater detail.)

3. In this subsection we consider the ‘stochastic integrals’

with respect to the just introduced random measures p , v , and p - v. Let w = ( w k ( w ,x ) ) k a l be a sequence of 9 @ B(IWd)-measurable functions. We denote by w * p the (dependent on w ) sequence of the sums of the Stieltjes integrals n

Our integer-valued random measures values, 0 and 1. Hence

-

pk

are very special in that they take only two

Consequently, we have in fact n

(w

* p ) n ( w )=

1

W k (w;A

Xk(4).

k=1

In a similar way we can define the ‘stochastic’ integrals w * u and w * ( p u ) with respect to the nieasures Y arid p - v in ternis of Stieltjes integrals. Note that ~

462

Chapter V. Theory of Arbitrage. Discrete Time

we must impose the following condition of integrability on the uik(w:rc) for the existence of these ‘integrals’:

for all (or almost all) w E R and k 3 1. It is easy to see in this case that w

* ( p - Y ) = w * p - w * u.

(We must warn the reader against extending this property automatically to the case of general integer-valued random measures, e.g., the jump measures of stochastic processes with continuous time: the integral w * ( p - Y ) may be well defined while w * p and w * Y are equal to +m, so that their difference has no definite value; see [250; Chapter 1111 for greater detail.) Assuming additionally that the functions w k ( w , z) are ,Yk-1-measurable for each z E Rd we obtain the predictable (i.e., .Yn-l-measurable) integrals (w* u),. If, moreover, E

Ld

I ~ & J , .)I

vrc(dz;w ) < 00

(4)

for each k 3 1, then it is easy to see that the sequence w * ( p - v ) = ( w * ( ~ - v ) , ) ~ ~ ~ I is a martingale. Replacing (4) by the conditions

where (T,) is some localizing sequence of Markov times obtain a sequence w * ( p - u ) that is a local martingale.

(T,

6

~ , + 1 , T,

t

m), we

4. We now consider the Doob deconiposition of the sequence H = (&),>I,

we assume that EJh,/

< 00 for h,

= AH,, n

where 3 1. We have (see Chapter 11, 3 l b )

where

and M T L

=

c

[hk

-

(7)

3. Construction of Martingale Measures

463

We can represent the variables A,, and M , as follows in terms of the measures p = (p1L)n21introduced above and their compensators v = (v,),al:

For brevity, one denotes the right-hand sides of (8) and (9) by (X

* u)n

and (z

* (P - u))n

respectively (see [250; Chapter 111). Thus,

H n = (z * v ) n + (z * ( P - v

) ) ~ ,

or, in the coordinate-free notation,

H

=z

*v +z*

( p - v).

Of course, H = z * p in our case, so that (13) is in fact the equality z * p=5

* v+z*

( p - v),

as obvious as is the Doob decomposition under the assumption Ejh,l < 00, n 3 1. In place of the condition that Elh,l < 00 for n 3 1 we shall now assume that (P-a.s.) E(IhnI I s n - 1 ) < 00, n 3 1. (14) Under this assumption, the sequences A = ( A n )and M = (M,) are certainly well defined (by formulas (6) and ( 7 ) ) ,and M is a local martingale since

Hence, if (14) holds, t h e n the sequence H = (H,) has a generalized Doob decomposition H=A+M (15) where A = ( A , ) and M = ( M n ) are as in (6) and (7). Moreover, A is a predictable sequence and M is a local martingale. The representation (15) can be rewritten in terms of p and v , as equality (13).

Chapter V. Theory of Arbitrage. Discrete Time

464

Remurk. We recall (see Chapter 11, 3 l b ) that E(h, 1 , F n - ~in) (6) and (7) is the generalized conditional expectation defined as the difference E(h:

19n-1)

-

E ( G ISn-1)

on the set { w : E(lh,,/ 1 9,-1) < co} and in an arbitrary manner (e.g., as zero) on { u : E(/h,,/ I .Fn-l) = w}. In the general case where (14) can fail, one can obtain an analog of (15) or (13) as follows (we already discussed that in Chapter 11, 5 l b ) . Let p = p(z) be a bounded truncation function, i.e., a function with compact support that is equal t o z in a neighborhood of the origin. A typical example is the 'standard truncation function' p(z) = zl(jz/6 1).

(16)

Then 71,

n

71

k=l n

n,

71

Using the notation of (12) and (13). we obtain the following representation: K

1

= (P(4

* V),?, + ( P ( 4 * ( P - 4), + ((.

- cp(x))*

dn7

(18)

or, in the coortliriate-free notation,

H

=p

* v + p * ( p - ).

+ .(

-

p) * p .

(19)

DEFINITION. We call (18) a r i d (19) the canonical representations of the sequence H = ( H , l ) T , > oHo , = 0, with triincat,ion function cp = cp(z). With an eye to the continuous-time case it is useful to compare this definition anti the canonicul representation of seminiurtingales in [250; Chapter 11, 5 2c] and in Chapter VII, 3 3a of the present hook.

3. Construction of Martingale Measures

5. Let H = (H,),>l

465

be a sequence with generalized Doob decomposition

H,

=A,

+ Mn

and assume that condition (10) in §3d is satisfied. Then by Theorem 2 in the same 3d we obtain the following representation with respect to an arbitrary mea-

- loc

sure P << P:

where EAlloc(~). We now write down the canonical representations of H with respect to the measures P and F:

H =p *u +p and

H

=

p

* ( p - u ) + (x - p) * p

* V + cp * ( p - i?) + (z - p) * p

(with respect to P)

(with respect to

is),

(21)

(22)

where p is the jump measure of the sequence H . It is important for the stochastic calculus based on the canonical representations (21) and (22) that one knows how to calculate the compensators i; for given compensators u and the characteristics of the density process 2 = (2,). In particular, we are interested in formulas describing the transformation of the ‘drzft’ terms p * u and p * i? under, a change of measure.

- loc

We shall discuss this issue more closely under the assumption that P << P. Let

and

be the regular modifications of the corresponding conditional probabilities. Bayes’s formula (4) in 3a assumes the following form for Y = 1 ~ ( h , ) , A E %(EX \ {O}), arid m = n - 1:

Chapter V. Theory of Arbitrage. Discrete Time

466

Hence the following conjecture looks plausible: for each w E R the conditional distributions Vn( . ;w ) are absolutely continuous with respect to vn( . ; w ) , i.e., there exists a B(B \ (0))-measurable (for each w E n) function Y, = Y n ( zw , ) such that

If it were so. then

i.e., the function Y n ( z w , ) would play the role of the density of one measure (more precisely, of one conditional distribution) with respect to the other.

- loc

Wc present now the proof of formula (24) (under the assumption P << P),which w),n 3 1. simultaneously delivers an ‘explicit formula’ for the density Yn = Y,(x, We consider the conditional expectation on the right-hand side of (23). By definition, for each B E Sn-l we have

Let M n ( d z ,dw) be the ‘skew product of measures’ pn(dz;w)(PISn-d(dw)

on %(R\{O})@9n-l and let EM,(. 1 B(B\{O})@9n-1) be the conditional expectation (with respect to the algebra 93(R\{0))@Sn-~and measure Mn = M n ( d z ,d w ) ) defined in a standard way (see, e.g., 1439; Chapter 11, §71), on the basis of the

3. Construction of Martingale Measures

Since B is an arbitrary subset in

E(1,4(hL)*2,-1 where

I

Sn-1, it

9,-1)(w)

467

follows that (P-a.s.)

= LY,(x,w)v,(dz;~),

(5 I B ( R \ (0)) @ %l)

Yn(,, w ) = EMn 2,-1

(27)

w).

(28)

sA-

Comparing ( 2 3 ) with E ( 1 ~ ( h , ,I )9,-1) = v,(dz; w ) and formulas (27) and (28) we see that V,(. ; w ) << v,(. ; w ) for each w E 0 and (25) holds. This formula provides a n answer to the above question: the 'drift' terms p * V and cp * v in (22) and (21) are connected by the relation p * V = 'P*v+cp(Y - l ) * v (at any rate, if (ip(z)(Y - 1)1* v),

(29)

< m , n 2 1)

$3f. 'Predictable' Criteria of Arbitrage-Free ( B ,S)-Markets 1. By the Fzrst fundammtal asset przczng theorem (3 2b) a ( B .S)-market formed by a bank account B = (B,) and d assets S = (S',. . . , S d ) ,Sz = (SL), 0 n N-, is arbitrage-free if and only if there exists a probability (martingale) measure P equivalent to P on the, initial filtered probability space (0,9, ( 9 n ) nP)asuch ~, that the d-dimensional sequence of discounted prices

< <

-

is a P-martingale. The description of the class 9 ( P ) of all such martingale measures P P is also very interesting because we have already seen in 5 l c that in looking for upper and lower prices one must consider the greatest and the least values over the class 9 ( P ) . Searching such measures it is reasonable to start -with a slightly more general problem of the construction of martingale measures P, that are locally absolutely continuous with respect to P, leaving aside the question whether a measure so obtained satisfies incidentally the relation P P.

-

-

I

2. In the previous sections we exposed all the material pertaining to 'absolutely continuous changes of measure' that is necessary for a discussion of the problem of

- loc

the construction of nieasiires P << P. We start with the case where d = 1, B, 3 1, S = (S,) is the only 'risk' asset, and = SoeH". (1)

s,

Chapter V. Theory of Arbitrage. Discrete Time

468

We assume that we have a filtered probability space (n,9, (S,),P) and that the variables H , are .q,-measurable. As before, we shall set h, = A H , and NO = 0. Throughout, we shall assume that 90= (0,n). Using the notation and the results of 5 3c; for n 3 1 we set

and

we also set HO = 0. Note that

A(@),

=

&(ii)ndlE,. -

(4)

loc

a martingale measure P << P such that S = (S,) is a P-martingale we can follow directly the above-described pattern: write down the canonical representation for S = (S,)(of type (13) in 5 3e) and then find a measure

- In our construction of -

loc

P << P 'killing' the drift term. We could proceed in that way; however, we have an additional property: prices are positive. This property enables us to consider the logarithms of the prices (i.e.>the sequence H = (H,)), which, as statistical analysis shows, have a more simple structure that the sequence S = (S,) of prices themselves. Let

p be a measure such that

- loc P << P.

dPn We set 2, = -, where P, = P I 9,and P, = P I 9,. In what follows we set dPn PO = PO, so that 20 = 1. Let X = (X,),,, be some sequence of 9,-measurable variables X,, and let X O = 0. By the lemma in 5 3d, I

if

P2 ; P, then

x E &(is) * X Z E A ( P ) , and

if

is 'z P,

then

(5)

469

3 Construction of Martingale Measures

Now let , Y E A1oc(P). By the theorem in Chapter 11. rjlc the sequence X is a martingale transform, and therefore AX, = a,AM,, - where the a, are Yn_1-measurable and JI is a martingale (with respect to P). Hence

and therefore & ( X )is also a martingale transform. so that (again. by the theorem in Chapter 11. $ l c ) G ( X ) E A I ~ , -Thus. (~).

Assuming that &(X)# 0 and considering

we can see in a similar way that

Hence, in view of (6). we obtain the following result.

LEMMA 1. Let

p

P and assume that &(X)# 0 ( P - a x ) . Then

€(S) E .J$lloc(P) e==+

x E AIoc(F)L xz E A l o c ( P ) .

h s u m e now that 6(.Y) 3 0 and (S(-Y) E A ( P ) Then, by the lemma in Chapter 11. $ l c we obtain that G ( X ) € A ( P ) Consequently. in view of ( 5 ) , we have the following result LEMM.2

2 Let

p $2 P and assume

&(.Y) E ,?qoc(P)

-

that & ( X )3 0

e=? G ( S ) E

A(P)

(P-LLs.).

Then

&(.Y)Z E A ( P )

LVe point out the following implications (which must be interpreted componentwise) holding by ( 3 ) :

and

A4Y > -1 '+=? G ( X ) > 0. A

Applying Lemmas 1 and 2 to the case of formula ( 2 ) , we obtain the following result.

X

=

I?. where H

is related to H by

Chapter V. Theory of Arbitrage. Discrete Time

470

LEMMA3 . Let S = ( S n ) n 2 ~where , S, = SoeHn,

Ho = 0,

A

and let AH, = eAHn - 1. Ho = 0. Then

s, = SO&(fi),

and if

p % P.

then

if

p 'Ec P,

then

-

3. These implications indicate the way in which one can seek measures P such that

s E A(P).

The sequence 2 = (2,)is a P-martingale. In accordance with (10) and ( l l ) ,we must describe the non-negative P-martingales Z such that EZ, 3 1 and, in addition,

- loc

either & ( f i ) ZE A ( P ) if we are looking for a measure P

<< P

orfiZ€&~,,jP)if we

require that p '2P. The corresponding class of martingales Z = (Zn) in the conditionally Gaussian case is formed by the martingales

(with 3,_1-measurable bk: see. for instance, formula ( 7 ) in 53b) satisfying the difference equations AZ, = Z,-,AN,, (13) where the variables - ebncn-$bz - 1

n -

(14)

make up a generalized martingale difference. i.e..

Hence a natural way to look for the density processes 2 = ( Z n )can be as follows.

3. Construction of Martingale Measures

471

We shall seek the required densities 2, (necessary for the construction of the measures P, by P,(dw) = & ( w ) P r L ( d w ) )in the form (13), i.e., we assume that

2, = € ( I q n ,

2, = 1,

(15)

where N are some (specified in what follows) local martingales with NO = 0, A N , 3 -1, and E&'(N), = 1.

- loc

The question on the volume of the class of so obtained measures P << P is far from simple. The point is that, first, it is not an easy problem to define whether (for a- family of consistent 'finite-dimensional' distributions {is,}) there exists a measure P such that 1 9,= P,, n 3 1. (See a counterexample in [439; Chapter 11, 3 31.) Second, the question of the structure of all the martingales (or local martingales) P) is not simple in principle. (See [250; Chapter 1111 on this on (n,8,(l, issue.) In what follows, we take a way that, while giving us no exhaustive answer to the

- loc

-hc

question of the structure of all measures is such that P << P or P P, is nevertheless fairly simple technically and brings one to a broad class of such measures. First of all, we make several observations. It is clear from the above that, given a measure P, all measures is such that p loc P are completely (as regards their finite-dimensional distributions {is,})

-

described by their densities 2 = (2,). It follows from the assumption P 'zcP that P(Z, > 0) = 1, so that we can construct from 2 = (Z,) a new sequence N = (N,) with No = 0 and

AN, =

Az, ~.

(16)

2,-1

Clearly, N E Aloc(P), and the sequences Z and N are in a one-to-one correspondence thanks to the equality 2, = &'(N),.

- lzc P from the densities

Hence we can turn i n s u r construction of measures P

dP, 2 = (&), where 2, = __ , to the appropriate sequence N = (N,), which must dP, satisfy the inequalities A N , > -1 to ensure that 2, > 0 (P-as.) for n 3 1. Thus, we shall assume that SO > 0 , S, = S Q ~ ( @for , n 3 1, and, in addition,

S, > 0 (which is equivalent to the relation AG, > -1). Also let Z = (Z,), where 2, = g(N),_with A N , > -1, so that 2, -> 0 (P-as.). Assume that there exists a measure P such that its restrictions P, = P 1 9, - loc P. satisfy the relations P, P, n 3 1, i.e., P Then, in view of ( l o ) , N

s EA(F)

-

+==+ & ' ( i i ) & ' ( N E)&(P).

(17)

Chapter V. Theory of Arbitrage. Discrete Time

472

4. The following Yor's formula (M. Yor; see, e.g.. [402]) can be immediately veri-

fied:

&(H)&(iV) = where

&(I? + N + [a,N ] ) .

(18)

n

[I?.N ] , = C AHkANk. k= 1

By assumptions

&(H)> 0: G ( N ) > 0, and equivalence (17) we obtain

s E A(P) +=+ &(a+ iv + [ H , N ] E) A ( P ) ii + N + [a.N] E A,,,(P). Hence i f N E .Al,,.(P).

s E AqP) +

- a [a.

A N > -1, 2, = & ( N ) n qand dP,

+

=

2, dP. then

-V] E A l o c ( P )

+

+

Since A(a [fi.A r ] )= AI?(l A N ) , the inclusion fi [ H , N ]E A l O c ( P )is equivalent t,o the condition that the sequence A Z ( 1 A N ) = ( A E n ( l + ANn)), is a local P-martingale difference, or. the same (see the lemma in Chapter 11, 5 Ic), that it is a generalized P-martingale difference and satisfies (P-as.) the relations

+

E [ 1 Afin( 1 i- A N n )I I gn-1 ] <

(19)

OG

for all 3 1. We note that conditions (19) and (20) are formulated in terms of the conditional expectation E( . 1 ,FTL-1) (i.e.. in .predictable' terms). Conditions (19) and (20) can be expressed in various forms. For instance. bearing in mind that AZ,, = Z,,-lANn, A N n > -1. and AH, = eAHn - 1. we see that (20) is equivalent to the following condition on 2:

-

- loc

Of course. we could derive this condition directly because if P

<< P. then

by Bayes's forniula - ((1)in $ 3 a or (1) in 3 3d). and therefore (20) is equivalent to the relation E(S,, 1 .3,L-l) = SrL-l ( P and P-as.). In a similar way. (19) tj I

3 . Construction of Martingale Measures

473

-

E(S, I 2Jn-1) < 00. Since the S, are non-negative, conditions (19) and- ( 2 0 ) ensure that the sequence S = (S,) is a martingale with respect t o the measure P constructed f r o m the sequence N = (N,). The above method can easily be extended to the multidimensional case, when S = (S’, . . . , S d ) and we consider the question of whether

S -

B

E

&(P)

- l%c

with respect to some measure P P. This is the question we discuss next. 5 . Our choice of the bank account B = (B,) as the normalizing factor has the advantage that the variables B,,are 9,-1-measurable, which, as already mentioned, brings certain technical simplifications. However, there are no serious obstacles to considering any other asset in this role. In this connection we discuss now the following situation. Let So = (St)and S1 = (Si) be two assets. We assume that

and AHA = eAHA where HA = 0, and let AZ, = Z,-,AN, Setting S; and S: to be constants we consider the ratio

and ask whether

S1

-

E

so-

such that p 1 9,= p, and 9 = SN.) Clearly

&(is).

-

1.

(To avoid the question of the existence of a measure p

-

where p,(&) = Z,(w)P,(dw), we can assume that n

s1

-2

so

s; s;

-20.

€(ill)

&-1(HO).

B(N)

(we use the coordinate-free notation). It is easy to verify directly that

where


Chapter V. Theory of Arbitrage. Discrete Time

474

S1

Hence -2 SO

is a product of three stochastic exponentials:

s; s,o

s 1

-2

= -20

so

'

&(@)

'

&(-@*)

'

&(N),

so that using Yor's formula (18) we successively obtain

8(ii')

'

&(-H*). &(N) S1

Hence one necessary and sufficient condition on N ensuring that - E SO is the following inclusion:

&(P),

or, equivalently,

Since for d assets S1,. . . , Sd the question on the martingale property of the vector of discounted prices

,$)

jS. = ( "

SO'"' can be answered by component-wise analysis, we obtain from (29) the following general result.

THEOREM 1. Let ( S o ,S1,.. . , S d ) be d + 1 assets defined on the filtered probability space (0,9, ( 9 n ) TP)Lwith <~ 9,= 9~such that S i = SieHk,

where

H i = 0, i

or, equivalently,

= 1,..., d ,

SA-~AH;, A

AS:

=

1

< n 6 N,

.

where h .

AgA

= eAHi

-

1,

Hi

= 0.

Assume also that the constants Sg are positive for all i = 0 , 1 , . . . , n.

3. Construction of Martingale Measures

475

Further, let Z = ( Z n ) o G n Gbe~ a sequence of random variables such that

AZ, = Zn-lANn,

20= 1,

(32)

where ANn > -1. Then the ratio - IS a d-dimensional martingale with respect to a measure

s .

such that

SO

PN

-

p N ( d w )= z N ( w )p ( h ) if and only if for all i and n. z = 1 , . . . ,d and 1 n N , we have ( P - a s , )

< <

(33)

and

COROLLARY. Let So = (S:) be a ‘risk-free’ asset in the sense of the Sn-1-measurability of the S: (for example. So = B can be a bank account with fixed interest rate ( A H : = r ) ) . Then (in view of the 9,-1-measurability of the 6:) conditions (34) and (35) can be written in the following form: E[lAHA(l+ AlY,)I I 9n-11 < 1>3

(36)

for i = 1 . . . . , d and n between 1 and N . In particular. if AH: z 0, then these conditions have the following form:

E[(AG;(~+ E \ A&(I

+

AN^)^ 1 9n-l] < cO,

(38)

AN^) 1 9 n - 1 1 = 0.

(39)

i.e.. are the same as earlier obtained conditions (19) and (20). If, in addition, A N n E 0. then the conditions in question reduce to

E[pH;l 1 . 9 4 < 3c and

E [ AHA 1 9 n - l ] = 0 , moreover. (41) is equivalent to the condition

E[eAHA I Fn-l]= 1 (cf. (11) in 5 3c). which is an obvious condition of the inclusion Si E d l ( P ) .

(40)

Chapter V. Theory of Arbitrage. Discrete Time

476

6. We consider now several examples illustrating the above criteria. To this end we observe first of all the following. Assume that a ( B ,S)-market is formed by two assets: a bank account B = (&) such that AB, = rnBn-l

with $n-l-measnrable

T,

and stock

S = (S,)

such that

with ,FT1-measurablep I l . Then the conditions (36) and (37) can be rewritten as follows:

On the other hand, setting

B,, = Bn-lern

and

S,

= Sn-lePn,

(44)

we can rewrite (36) arid (37) as

and

E [ (epn - 1)(1+ AN,)

1 9 7 L - 1 ] = ern - 1.

(46)

EXAMPLE 1. We consider a single-step niodel (44) with ri = 0 or 1, where we set .90 = ( 0 , Q )and Zo = 1. Then 1 + AN1 = 21 and by (46), EeP'Z1 = e r .

We shall assume that R = R, p l ( z ) = 5 , 21 = Z(z). and let F = F ( z ) be the probability distribution in 0 . Then the above condition is equivalent to the equality ezZl(z)d F ( z ) = er.

(47)

F

= F(z) equivalent to F = F ( z ) Thus it is clear that finding all distributions (in the sense of the equivalence of the corresponding Lebesgue-Stieltjes measures) is the same as describing all positive solutions 2 1 = Z,(z) of equation (47) satisfying the condition

Z1(z) d F ( z ) = 1.

(48)

3. Construction of Martingale Measures

For instance, if F form:

N

477

JY(rn,a2)with o2 > 0, then (45) and (46) take the following

1 where p m , r (x) r ~ = ___

&Gze

(z-m)2 2u2

is the density of the normal distribution.

From (49) we see that if we seek a martingale measure in the class of normal distribvtions N ( 6 ,“2) with a2 > 0 , i.e., if

then ‘admissible’ pairs (6) “2) must satisfy the condition

J-Ca

which is equivalent to 7n

+

-

2

= T.

In other words all the pairs (&,Z2) with a2 > 0 satisfying (51) are admissible. We have already encountered this condition with r = 0 in 5 3c (see (14)). Note that, besides the solutions Z,(x) of the form (50), the system (49) has other solutions, and their general form is probably unknown. This shows the complexity of the description problem for all martingale measures even in the case of the above simple ‘single-step’ scheme. Our next example can be characterized as ‘too simple’ in this respect, for we have there a unique, easily calculated ‘martingale’ measure.

EXAMPLE 2 . The CRR-model. Let

<

where n N and Bo and SO are positive constants. It is assumed in the framework of this model that (p,) is a sequence of independent identically distributed random variables taking two values, b and a , such that -1 < a < T < b

and PN(Pn = b) = P,

where 0

< p < 1 and p + q

= 1.

P N ( P n = u) = q,

(53)

Chapter V. Theory of Arbitrage. Discrete Time

478

Since the variables p,, ri 3 1, are the unique source of 'randomness' in the model, our space of elementary outcomes can be the space R = { a , b}N of sequences (21,. . . , X N ) with zi = a , b, and the functions pn = p n ( z ) , z = ( X I , . . . , XN), can be defined coordinate-wise: ,on(.) = 5 , . We can define in a standard way a probability measure PN = PN(X1,. . . , X N ) such that p1,. . . , PN are independent variables with respect to PN and (53) holds; namely, pN(Zl, . . . , z N ) = pvb("l,...,z.v) 4 N - - Y b ( 2 1 , . . . 7 N ) where vb(z1,. . . , Z N ) =

N

C Ib(xi) is the number of the zi equal to b. i=l I

-

We shall construct the measure P N - PN - in several steps: we set P, = PN I9,, where 9,= a(p1,. . . , p,), and define PI, P z , . . . , P N by the formulas I

where we shall (successively) find the Z, from condition (43), which, bearing in mind that 1

+ AN,

z,

= -, can be rewritten as Zn-1

For n = 1 (with 9 0 = {B, R}) we obtain by (54) that

which, together with the normalization condition (56)

delivers with necessity the equalities

r-a Zl(b) = - . b-u We set

-

1 -

p

r-a

p = b-a

Then

and

and

Zl(a) =

-

b-r q = -. b-a

b-r b-u

~

1 -. 9

(57)

3. Construction of Martingale Measures

479

I

To find P 2 (and, after that, P 3 , . . . , PN) we use (54) again. Based on the fact that p1 and p2 are independent with respect to P2, we obtain by (52) that

An additional condition on the values of &(b, b) and Z 2 ( b , u ) can be obtained from the martingale property

EP2[Z2(P1,P2)1P1 = b l = Zl(b), which brings us to the equality

Comparing (59) and (60) with (55) and (56) we see that

In a similar way,

Hence

It is now clear that the variables p1 distributed - and p2 are identically - and independent with respect to the measure Pa; moreover, P2(pi = b ) = j7 and P2(pi = u ) = q, 2 = 1,2. The next steps, the specification of P y , . . . , PN, proceed in a similar manner, which brings us to the following result.

THEOREM 2. A martingale measure PN in the CRR model defined in (52) and ( 5 3 ) is unique and can be defined b y the formula

PN(21, ‘ where

,...

- V b (ZI ,ZN) - N - Y b

1

ZN) =P

4

(ZI,. ..,ZN) 1

(61)

480

Chapter V. Theory of Arbitrage. Discrete Time

Remark. We note that it was not a priori obvious that the martingale measure PN was a direct product of ‘one-dimensional’ distributions:

N

i.e., that the variables p1, . . . , p ~which , are independent and identically distributed with respect to the initial measure P N , are also independent and identically distributed with respect to the martingale measure PN.

4. Complete and Perfect Arbitrage-Free Markets

5 4a.

Martingale Criterion of a Complete Market. Statement of the Second Fundamental Theorem. Proof of Necessity

1. In accordance with the definitions in 3 lb, a (B,S)-market defined on a filtered probability space (R,$, ($,), P), where 0 n 6 N , 90= (0, n}, and 9~= 9, is said to be complete (perfect) or N - c o m p l e t e ( N - p e r f e c t ) if each $N-measurable bounded (finite) pay-off function f~ = f ~ ( w is ) replicable: there exist a selffinancing portfolio T and an initial capital z such that X{ = II: and

<

X&

=

f~

(P-a.s.).

Let 9 ( P ) be the collection of all martingale measures p

N P, such that the disS counted prices - are martingales. It is assumed (see 3 l a , 2a) that B = ( B , ) 0 4 n c ~ B is a risk-free asset and S = ( S n ) o Q n Gwith ~ S, = (SA, . . . , S:), d < 00, is a multidimensional risk asset. The asset B is usually regarded as a bank account; the assets Siare called stock. In what follows we assume that B, > 0 for n > 0. Then we can set B, = 1, n 3 0, without loss of generality. The next result is so important that it may well be called the ‘Second fundamental asset pricing theorem’.

THEOREM B. An arbitrage-free financial ( B ,S)-market (with N < 00 and d < m) is complete if and only if the set 9 ( P ) of martingale measures contains a single el em en t . Thus, while the absence of arbitrage means that

Chapter V. Theory of Arbitrage. Discrete Time

482

the completeness of an arbitrage-free market can (provisory) be written as / 9 ( P ) i = 1. We now make several observations relating to the proof of this theorem. It is well known in stochastic calculus (see, e.g., [250; Chapter 1111) that the uniqiieness of a martingale measure is intimately connected with the issues of ‘representatability’ of local martingales in terms of certain basic martingales. The corresponding results (especially in the continuous-time case) are considered to be technically difficult, because one must essentially use ideas and tools of the stochastic analysis of semimartingales and random measures in their proofs. Incidentally, in the discrete-time case this range of issues pertaining to the ‘representability’ problem for local martingales and the completeness of a ( B ,S)-market can be discussed on a relatively elementary level. We start our discussion with the case of d = 1 ( 3 3 4a-4e). The general case d 3 1 is considered in 3 4f. 2. The idea of the proof of Theorem B in the case of d = 1 is to establish the following chain of implications (involving the concepts of ‘conditional two-pointedness’ and ‘S-representability’, which are introduced below, in 3 5 4b,e, and the equality gn = 9 : meaning that the u-algebra 5Fn coincides with the u-algebra

9 : = u(S1,.. . , S,) generated by the random variables S1,. . . , S, up to sets of P-measure zero):

I completeness I

I

‘conditional two-pointedness’

I 1-

‘S-represent ability

w1

4. Complete and Perfect Arbitrage-Free Markets

483

Implication {l},i.e., the necessity in Theorem B. has a relatively easy proof that proceeds as follows. . set f N = I A ( w In)view . of the assumed completeness there Let A E L q ~We exists a starting capital II: and a self-financing strategy 7r such that if X $ = ic, then Xl; = f N (P-a.s.). Since 7r is self-financing strategy, it follows that

Let Pi, i = 1 , 2 , be two martingale measures in the family 9 ( P ) . Then ( X z ) n G ~ is a martingale transform, and since X $ = I,, the sequence X“ = ( X z ) n Gis~ a martingale with respect to each martingale measure Pi, i = 1 , 2 , by the lemma in Chapter 11, 5 lc. Hence II: = X $ = Ep,(Xl; 190) = EP,IA = Pi(A) for i = 1 , 2 , so that P1(A) = Pz(A), A E 9 ~ . Thus, the measures P1 and P2 are in fact the same, which proves that the set 9 ( P ) (non-empty due to the absence of arbitrage in our (B,S)-market) has a t most one element ( ( 9 ( P ) (= 1). This proves the necessity in Theorem B (implication {I}). In the next section we consider the issues of ‘representability’, which are important for the discussion of implications (4) and ( 5 ) .

5 4b.

Representability of Local Martingales. ‘S-Representability’

From the standpoint of the ‘general theory of martingales and stochastic calculus’ (see [102], [103], [250], [304]) the assumption of ‘completeness’is in fact equivalent to the so-called property of ‘S-representability’ of local martingales ([250; Chapter III]).

DEFINITION.Let (n,9, ( g n )P), be a filtered probability space with P) a &dimensional (basic) martingale S = ( S n ,9,, and a (one-dimensional) local martingale X = ( X n ,9,, P). ( g n )P) , admits an ‘S-repreThen we say that the local martingale X on (Q9, sentation’ or a representation in terms of the P-martingale S if there exists a prewhere yn = ( r:, . . . ,-&), such that dictable sequence y = (m),

Chapt,er V. Theory of Arbitrage. Discretc Time

484

P-as. for each 71 3 1, i.e., X is a martingale transform obtained from the P-martingale S by 'integration' of the predictable sequence y;see Chapter 11, 5 lc. The next result relates to implication (5) in the chain of implication on the previous page.

LEMMA.Let ( B ,S) be an arbitrage-free market over a finite time horizon N , let B,, = 1 for n 6 N , and let 9 ( P ) be the family of measures equivalent to P where 9 = 9 ~ such ) ,that S = (ST,),2o is a P-martingale. (on (R, 9), Then this market is complete if and only if there exists a measure p E 9 ( P ) such that each bounded martingale x = (x,,sn,P) (with IX,(w)/ < C' ri 6 N , w E R) on (0,
Proof. (a) Assume that our (arbitrage-free) market - is complete. We take an arbiP ) , Q , Tbe a trary measure in 9 ( P ) as a required measure P. Let X = (X,, ,FTL, martingale with IX,(w)I 6 C , n N , w E R. w e set f N = X N . The completeness assumption means that there exists a self-financing portfolio T and an initial capital z such that (P- and p-a.s.) I

<

<

and X & = f N = X N . However, l f N l C , therefore X" = ( X : ) n ~is~a Pmartingale (the lemma in Chapter 11, 5 IC), so that the p-martingales X" and X , which have the same terminal pay-off function f N , actually coincide (P- and P-as.). Hence the martingale X admits a n 'S-representation'. (b) Now let f N = f ~ ( wbe ) a 9N-measurable bounded function (if" such that each bounded P-martingale has an 'S-representation'. We consider one such martingale

x$

Since Ifr\ii 6 C , X is a bounded (Lkvy) martingale and it has a representation (1) with some 9k-1-measurable variables $, j = 1 , .. . , d , k N . For these variables we construct a portfolio 7r* = (p*,y*) such that y* = y and

<

p;

=

x,

c rbs,j,. d

-

j=1

4. Complete and Perfect Arbitrage-Free Markets

By (1) we obt,ain t h a t the [J;

are ,Fn_1-nieasurable. Moreover,

d

=

d

c

+1 TAWl a

s;l-lny;l

-

J=1

J=1

Hence

7r*

485

(c7;s:) d

= 0.

J=1

is a self-financing portfolio arid

x;* = p:, + -

cy;sil d

=

x,;

3=1

in particular, Xl;* = X N = f~ (P- and P-a.s.), i.e., our which proves the lemma.

(B, S)-niarket is complete,

Remark. If we do riot assume that B, 1, n 6 N , then all our results are valid S for the P-martingale - = in place of the P-martingale S = ( S n ) 7 1 c ~ . I

I

5 4c.

Representability of Local Martingales (‘p-Representability’ and ‘(p-v)-Representability’)

1. The issue of ‘S-rcpresentability’ is, as shown in the preceding section, closely related to the ‘completeness’ of the corresponding market and the fact t h a t the evolution of the capital X” is described by (2). As shown in 54d, we have ‘S-representability’ in the CRR-model, so that the market is complete in this case. Generally speaking, completeness (and therefore also ‘S-representability’) is an exception rather than a rule. It is reasonable therefore to consider here another kind of representations of local martingales, which uses the concepts of random measures p and martingale random measures p-v; see s3e. It will be clear from what follows that ‘p-representations’ and ‘(~i~v)-representations’ are significantly more widespread than ‘S-representations’. Hence it often makes sense to find a ‘ p ’ or ‘(p--)-representation’ first and attempt to transform them into an S-representation after that. 2. Let S = (Sl, . . . , S d ) be a d-dimensional martingale on a filtered probability space (a, 9, (Sn), P) with 9 0 = (0, a} (with respect to the original measure P and the flow (.5FrL)). Let 9;= a(SL, k 6 n, j = 1 , .. . , d ) ’

n 3 1, be the a-algebra generated by the prices and let

x = ( X n ,9:,P) be a local martingale.

486

Chapter V. Theory of Arbitrage. Discrete Time

The increments AX, = X , - Xn-l are 9$measurable, a Bore1 function f , = f n ( z l , .. . , zn), x i E Wd such that

therefore there exists

(Since we have set 9 0 = {@,a},the vector SO is not random, and there exists a one-to-one correspondence between (S1,. . . , Sn)and (ASl, . . . , ASrl).) For each n 3 1 we now set

x),B(Wd)-measurable in z for each w E R This function is clearly measurable in (w, and 92-1-measurable for each z E Wd. Let p n ( A ; w )= I(AS,(w) E A ) , A E B ( W d ) , be the integer-valued randoni measure constructed from the increments AS,(w), n 3 1. Then

so that we obtain the so-called ‘ p r e p r e s e n t a t i o n ’ of X :

or, in a more compact form,

X

=

xo + w * p

(3)

(see 3 3e). We now note that since X is a local martingale by assumption, it follows that E(lAX,I I SZ-,) < 00 and E(AX, I 9Z-l) = 0, n 3 1. Hence if

then we can see that

Thus, besides (2) and ( 3 ) , we obtain the so-called ‘ ( p - u ) -representation’

4. Complete and Perfect Arbitrage-Free Markets

487

or, in a more compact form,

x = xo + W * ( p - v).

(6)

In niay seem odd that, alongside the natural representation ( 3 ) , we also consider the representation (6), which can be obtained from the first one in a trivial way, and honor it with a special name '(p-v)-representation', assigning to it certain importance by doing so. The reason is as follows. First, in some more general situations (continuous time, more general flows of a-algebras, etc.) similar representations of local martingales involve stochastic integrals just of the kind W * ( p - v ) (see, e.g., [250; Chapter 111, 5 4.23,4.24]). Second, the expressions of type W * ( p - v) (as compared with W * p ) have one advantage: in general, there is no unique way to define the function W in these representations, while in expressions of the type W * ( p - v) these functions can often be chosen pretty simple. We present the following example as an illustration. Let A E 9 ( R d ) and let X ( A ) = (XLA),Sz, P) be a martingale with Xi") = 0 and

On the other hand, setting

we obtain

so that X(A)=W

* p.

Clearly, the function W(A) is simpler as W . These arguments show, in particular, that the integral W * p does nor change after the replacement of the functions W,(w,z) by W,(W,Z) gA(z), where gh(z) satisfies the equality

+

Chapter V. Theory of Arbitrage. Discrete Time

488

In turn, the integral W W n ( w ,z) + .9E(w) because

L

* ( p - v ) does

g;(x)(p,(dX;w)

not change if we replace W n ( w , z ) by

-

v,(dz;w)) = 0

In the next section we show how one can easily deduce an ‘S-representation’ from a ‘(p-v)-representation’ in the CRR-model of Cox-Ross--Rubinstein. 4d. ‘S-Representability’ in the B i n o m i a l CRR-Model 1. As shown in §3f (Example 2 in subsection 5 ) , there exists a martingale measure in the CRR-model (so that the corresponding market is arbitrage-free), which (assuniing that discuss the coordinate probability space) is the unique martingale measure. By Theorem B, this is equivalent to the completeness of the corresponding market. It would be interesting for that reason to trace down how the uniqueness of the martingale measure in this particular model delivers the ‘S-representability’ and, therefore, the completeness of the market (following by the lemma in 3 4b). First we recall some notation. As explained in Chapter I, 3 le, the CRR-model of a ( B ,S ) market defined on a filtered probability space (n,S,(9,), P),ao is described by two sequences B = ( B , ) , ~ Oand S = (S,),>o such that

B, = &-1(1+ r n ) ,

(1)

s, = Sn-1(1+

(2)

Pn),

where the T, are Sn-1-nieasurable, the pTL are Sn-measurable, and Bo SO > 0 are constants. Since s, - sn-1 1 + P ,

B,

B,-1

> 0 and

(3)

l+rn’ I

it clearly follows that

(where !

I

is a martingale with respect to a measure P if, first,

is averaging with respect to

p) and, second,

In view of the 9r:,,-1-nieasurabilityof the variables r, the condition (4) reduces to the equality E(P, I Sn-1) = 7-n. (5)

4. Complete and Perfect Arbitrage-Free Markets

489

2. In the Binomial CRR-model one has T, E r , where r is a constant and ( p 7 L ) n 2 1 is a sequence of i n d e p e n d e n t identically distributed random variables taking two values, b and a , with positive probabilities

and

p = P(pTL= b )

+

p q = 1. (We also agree that a and let 90= {0,fl>.

-

(6)

< b.) Finally, let 9,= cr(p1, . . . , P,) for n 2 1 S

If we require that the sequence

-

q = P(p, = a ) 7

-

B

-

=

(3

be a martingale with respect

-

,>O

loc

to P,-then the quantities P, = P(pn = b ) and - a measure P such that P P(p, = a ) , in view of the equality Ep, = T , must satisfy the condition

-

bpn which, in view of the normalization

+ aq,

4, =

= r,

P,+ 4, = 1, shows that

In order that these values be positive one requires that a < T < b. We shall also assume that a > -1. Then S, > 0 for all n 2 1 because SO > 0. Let X = ( X , , 9 , , P ) n 2 0 be a martingale and let 90= ( 0 , Q )and 9, = a ( p 1 , . . . , p,) for n 2 1. For n 2 1 we set , ~ , ( A ; w= ) I(p,(w) E A ) and & ( A ) = Ep,(A;w). Since pn takes only two values, the measures p , ( . ; w)and V,(. ) are concentrated at a and b. Moreover, p n ( { u } ; u )= q P n ( 4 = a ) , G ( { a > )= 4 and p n ( { b } : w )= I ( P n ( 4 = b ) ,

GL({bj)

Let gn = g,(zl,. . . ,x,) be functions such that Xn(W)= S n ( P l ( 4 , ' . ' > P n ( 4 ) >

and therefore

'P.

Chapter V. Theory of Arbitrage. Discrete Time

490

or, equivalently,

In view of (7), this can be rewritten as follows:

We proceed now to 'p-representations'. In accordance with (1) in 5 4c,

where

Setting

W A ( w ,x) =

x) w n (w, x-r

1

we see from (10) that

Note that, in view of (9), the function WA(w,z)is independent of z. Hence, denoting the right-hand side (or, equivalently, the left-hand side) of (9) by y k ( w ) , we obtain

Thus, for X = ( X n ,Sn,is) we obtain the representation

4. Complete and Perfect Arbitrage-Free Markets

491

Since

it follows that

pn - r = (1

+r

) k A ( z ) , sn-1

and therefore

are 9k-1-rneasurable functions. The sequence

(s) BI,

-

is a martingale with respect t o the measure P. Hence

k.20

(16) is just the ‘S-represeGtation’of the P-martingale X with respect to the (basic)

P-martingale

(2)

k ~ o

Using the lemma n; 5 4b we see that the ( B ,S)-market described by the CRRmodel is complete for each finite time horizon N . 3. Remark. It is worth noting that it was essential in our proof of the uniqueness of the martingale measure in the CRR-model in §3f (Example 2 in subsection 5) that the original probability space was the coordinate one: R = {z = ( X I ,x2,.. . ) } , where xi = a or xi = b; 9n= ~ ( 5 1 ,. .. ,x,) for n 3 1, and 9 = V 9,.It can be seen from what follows (see 5 4f) that in an arbitrary filtered space ( R , 9 , (9n), P) the condition that a martingale measure be unique implies automatically that 9,= p(S1,. . . , S,) for ri 2 1 (up to sets of P-measure zero).

5 4e.

Martingale Criterion of a Complete Market. Proof of Necessity for d = 1

1. In accordance with the diagram of implications in 5 4a.2, to prove the necessity in Theorem B (i.e., the implication ‘ l 9 ( P ) i = 1’ ===+ ‘completeness’) we milst verify implications {2}, { 3 } , and (4) in this diagram. (We recall that we established implication (5) by the Iemma in 5 4b and d = 1 by assumption.) We start with the proof of implication (4)’ where we assume that B, = 1 (and therefore T , zz 0) for n 3 1, which brings no loss of generality, as already mentioned. To this end we point out that it was a key point of our proof of the ‘S-representability’ for the CRR-model that the probability distributions Law(p, 1 p), n 3 1, were concentrated a t two points ( a and b, a < b ) .

Chapter V. Theory of Arbitrage. Discrete Time

492

In other words, it was important that these were ‘two-point’ distributions. The corresponding arguments prove to be valid also for more - general models once 1: or, equivalently, the the (regular) conditional distributions Law(AS, I 9,-P)

-

conditional distributions Law(p,, I 9,-1; P) with pn

8, = a ( S 1 , .. .,S,,),

= - are 1

sn-

‘two-point’ and

71. 3 1. Formally, the ‘conditional two-pointedness’ means that there exist two predictable sequences a = (a,) and b = (&), of random variables a , = a,(w) and b = b,,(w), n 3 1, such that

-

-

P(P, = a , 19n-l)(W)

+ P ( P n = b, I S , - l ) ( W )

=1

(1)

<

and a,(w) 0, b n ( w ) 3 0 for all w E R, n 3 1. (If t,he values of a,(w) and b,(w) ‘merge’, then we clearly have a n ( w ) = b,(w) = 0; this is an uninteresting degenerate case corresponding to the situation when AS,(w) = 0. We can leave this case out of consideration from the very beginning and without loss - of generality.) Let & ( w ) = P ( p , = b, 1 2Fn-1)(w) and let & ( w ) = P(p, = a, 1 ~,-I)(LJ). - The martingale property of the sequence S = (S,, F,?p) delivers the equalities E(pn 1 9,-1) = 0, n 3 1, which mean that b,,(w)P,(w)

+ an(w)q;,(w) = 0 ,

n

3 1.

(2)

By (1) and (2) we obtain (cf. formula (7) in 34d)

i.)

(If U , ~ ( W ) = b,(w) = 0, then we agree to set ?T,(w) = & ( w ) = Let X = (X,, 92,P) be a local martingale and let y, = g T L ( z l ., . , .rn) be with formula (8) functions such that X,(w) = g,(pl(w), . . . , ~ ~ ~ ( wBy ) )analogy . in 5 4c we obtain I

S,(Pl(w),”.

1

Pn-I(4,b7&4)

-

Sn-l(PI(4,.

, Pn-1 (-4)

Trl(w)

- 971--1(Pl(W)..4 7 1 - - 1 ( w ) )- Sn(Pl(W),- 4,-1(w),a,(w)) 5

Further, following the pattern of (9)-(17) tation’:

1

(w)

(5 4c) we see that X

.

(4)

has an ‘S-represen-

n

x n = XO

+

‘Yk(d)Ask(w) k=l

with S:-l-measurable

functions T k (w),k 3 1.

(5)

4. Complete and Perfect Arbitrage-Free Markets

493

Thus, implication {4} is established. We consider now the proof of implication {a}, which says that the uniqueness of the niartingale measiire means the ‘conditional two-pointedness’ (for d = 1). Taking into account that one can operate the conditional probabilities I

P(ASTl E

I YTl-l)(U)

like ordinary ones (for each fixed w E a); the required ‘conditional two-pointedness’ is equivalent to the following result. I. Let Q = Q(dz) be a probability distribution on (R, .%(R)) such that

(’the martingale property’). Let 9 ( Q ) be the family of all measures Q = Q(dz) equivalent to Q = Q(dic) and having the properties

If the family 9 ( Q ) contains only the (original) measure Q, then, of necessity, this is a ‘two-point’ measure: there exist a 6 0 and b 3 0 such that Q({a>) + Q({b>) = 1, although these points can ‘merge’ a t the origin ( a = b = 0). We can put this assertion in the following equivalent form: 11. Let Z ( Q ) be the class of functions z = z ( z ) , ic E R,such that Q{ic:

O

< z(x)< m } = 1,

We assume that, for some measure Q, the class Z(Q) contains only functions that are Q-indistinguishable from one (Q{x: z(x) # l} = 0). Then, of necessity, Q is concentrated at two points a t most. Finally, the same assertion can be reformulated as follows. 111. Let = ((x) be a random variable on a coordinate space with distribution Q = Q(dz) on (R, !%(R)). Assume that Elti < ca, E< = 0, and let Q be a measure such that if Q N Q, El(] < 00,and EC = 0, then Q = Q. Then the support of Q consists of a t most two points, a 0 and b 2 0, that can stick t,ogether at the origin ( a = b = 0). I

I

<

Chapter V. Theory of Arbitrage. Discrete Time

494

To prove these (equivalent) assertions we observe that each probability distribution Q = Q ( d x ) on (R. B(JR))can be represented as the mixture

ciQi

+ c2Q2 + c3Q3

of three: a purely discrete one ( Q l ) , an absolutely continuous one ( Q 2 ) , and a singular one ( Q s ) , with non-negative constants c l , c2, and cg of unit sum. The idea of t,lie proof becomes already transparent in the ‘purely discrete’ case, when we assume that Q is concentrated at three points x-, zo. and z+, 2zo 6 z+, with non-zero mass p - , p o , or p+ at each point. The condition EE = 0 means that

<

+ zopo + x+p+ = 0. = 0, then (6) assumes the form z - p - + z+p+ = 0. z-p-

If

20

We set

P- o = -1+ - ,PO p + = P+ T, (7) 2 2 which corresponds to ‘pumping’ parts of the niasses p - and p+ at z- and x+ to the point zo = 0. It is clear from (7) that the measure Q = {F->&,F+} concentrated at the three points z-, xo, and z+, is a probability measure, Q Q , and EE = 0: moreover, Q # Q , which contradicts the uniqueness of Q. Thus, Q cannot be concentrated at three points including zo =-0. Now let zg # 0. Then the idea the construction of a measure Q Q by ‘mass pumping’ from x- and x+ to xo can be realized; e.g., as follows. We set

p-- = - ,P-

2

-

-

-

p- = p-

- E--,

Po = po + ( E - + &+),

p+

= p+

-E

-

For sufficiently small E- and E + , Q = {@-,&,p7} is a probability measure-and we must show that we can choose positive coefficients E- and E+ such that EE = 0, i.e., = (x-p-+zopo+z+p+)-(&-IC-+(&-

2-p-+xopo+x+p+

+

Since EE = x-pzopo satisfy the equality

+ x+p+

= 0, these positive coefficients &+ - ”0 -~ E-

We set X =

zo

-

z-

~

z+

-

zo

(>O).

+&+)xo-E+Z+) E-

and

E+

= 0.

must

z-

x+-zo

Then it is clear that choosing sufficiently small E-

first and setting E+ = XE- after that we can achieve the inequalities Pand p7 > 0.

> 0, Po > 0,

4. Complete and Perfect Arbitrage-Free Markets

495

-

I

Hence Q = {F-,Fo,p7} is a probability measure, Q Q, Q # Q, and EE = 0, which is again in contradiction with the uniqueness of the martingale measure Q, so that all the three masses p-, PO, and p+ of the distribution Q cannot be positive. This construction is easy to transfer to the case when a purely discrete martingale measure Q is concentrated at a countable set {xi,i = 0, fl, f2,. . . } , . . . < z - 2 < 2-1 < z o < z1 < x2 < . . . , with respective probabilities { p i , 2 = 0, f l ,f 2 , . . . }. If the set { x i , i = 0 , f1,f2,.. . } contains the origin, say, 20 = 0, then we must set N

and

Po = I

Then

CPi = 1 and EE = C ziPi i a -

=

1 - CifOPi

1 xipi = 0. i

-

-

The measure Q = {pi,i = 0, fl,5 2 , . . . } is a probability measure, Q Q, # Q, and EE = 0, which is incompatible with our assumption that the martingale measure is unique. Now let {xi,i = 0, Irtl, 1 2 , . . . } be a set of non-negative points zi # 0. We construct a new distribution Q = {Fi5i = 0, + 1 , + 2 , . . . }, where P, = pi for i = f2,f3,.. . and, as above,

a

I

-

P-1 = P-1

-

E-1,

P+1 = P+1- &+1,

Po = P o + (E-1+

&+1).

Then

-

EE = E< - E - ~ z - I +

(E-1

+ E + ~ ) Z OE + I Z + =~ -

E+(ZO

+

-~ + 1 )

E-~(ZO

- z-I),

and the same choice of ~ - and 1 E+I as in the above case of three points ( z - , zo, z+) brings us to a new martingale measure Q distinct from Q but equivalent to it, which contradicts the assumption of the uniqueness of the martingale measure Q. In a similar way we can consider cases where Q has absolutely continuous or singular components. 2. We now turn to the proof of implication (3). We claim that the uniqueness of the martingale measure means that the a-algebras 9nare generated by the prices S:

gn= 9 : = O(SOl.. ., Sn),

n

< N.

We shall proceed by induction. (Note that the 0-algebras 90and 9fare the same since, by assumption, 90= { 0 , 0 }and So is a non-random variable.) Let ( R , 9 , (Tn), P ) n G be ~ a filtered probability space and let S=(Sn,g n , P ) n ~ ~ be a sequence of (stock) prices with S , = (SA, . . . , S:). To avoid additional notation we shall assume that P is itself a martingale measure.

Chapter V. Theory of Arbitrage. Discrete Time

496

Assuming that

.’Yn-l = 9:-1 L=1

we consider a set A E 9,. Let

+ -21( I A

-

E ( I A 1 T Sn ) ) .

(8)

Clearly, 6 z 6 $ arid Ez = 1. Hence the measure P’ with P’(dw) = z(w) P(dw) is P. Let z; = E(z 1.9i). By Bayes’s formula (see (4) a probability measure and in 3 3 4 , N

:-1 it .follows from (8) that E(z I ,Fn-l) In view ofour assuniption that Fr2-1= 9 = 1. At the same time

z is a 9,-measurable function. Hence

zi

__ = 1 for i zi-1

so that 1 . 9 - 1 ) = 0 for all i # n. z,, Since __ = z , E(z/B,S-,) = 1, and AS, are $:-measurable, zn- 1 by (12) that

# n,

it follows

1 gn-l) = E(zAS,, 1 .F,-l) = E(zAS, I c 9 : - 1 ) = E(AS,E(Z

= E(E(zAS,

19:)I 9 i - 1 )

1 9;) 1 .F:-~)= E(AS,, I 9z-l) = 0,

where we also use the equality E(z 1 9 : )= 1 holding by (8). Hence the sequence of prices (S,, 9 n ) n is Ca ~ P’-martingale. Thus, the assumption that P is a unique martingale measure brings us to the equality z = 1 (P-as.). so that for each A E
I A = E ( 1 ~ 1 9 , : ) (P-as.) by (11). Hence, Sn = 9;up to sets of P-ineasure zero. Using induction on n we can verify these relations for all implication ( 3 ) .

71

6 N , which proves

3. Thus, the uniqueness of the martingale measure P ensures the conclusions of the implications { a } and { 3 ) . Combined, they mean ‘S-representability’, which, in turn, delivers t,he completeness of the market (by the Lemma in s4a). This done, we have established the sufficiency part of Theorem B (for d = 1).

Remark 1. It is worth noting that, as shows the above proof of Theorem B, a discrete-time complete arbitrage-free market (with N < 00 and d = 1) is in fact discrete also in the phase variable in the following sense: the a-algebra 9~is purely atomic (up to P-measure zero) and contains a t most 2N atoms. This is an immediate consequence of a ‘conditional two-pointedness’. (In the case of arbitrary d < 00 the number of atoms in 9~is a t most ( d l)N.)

+

Remark 2 . The fact that for N < 00 and d < the a-algebra 9p~corresponding to a complete arbitrage-free market has a t most (d l ) Nelements shows that completeness and perfection mean the same for such markets. % 1

+

4. Complete and Perfect Arbitrage-Free Markets

497

4f. Extended Version of the Second Fundamental Theorem 1. The above proof of Theorem B relates to the case of d = 1. (We have explicitly used this assumption in our proof of implications {a} and {4}.) In the general case of d 3 1, it seems appropriate to present an extended version of this theorem including the equivalence of the cornpleteness and the uniqueness of a martingale measure and several other equivalent characterizations. First, let us introduce our notation. We set

s, (the discounted 3, = -

prices), Bn Q ~ ( . ; w=) P(AS,r, t ’ / 9 ; n - l ) ( ~ ) Q.I L ( . ; w ) = P(ASn E ’ I ~ T L - ~ ) ( w ) . We recall that vectors a l , , a k wit11 ai E 2 6 k d 1, are said to , k } such that the k - 1 vectors be a f i n e l y i n d e p e n d e n t if the , k , j # i , are linearly i n d e p e n d e n t . If this property holds for then it also holds for each integer i = 1,.. . . k . Note that this dependence of d-dimensional vectors a1 , . . . , a k is equivalent to the fact that the minimal affine plane containing al,. . . , ak has dimension k - 1.

< +

THEOREM B* (an extended version of the Second f u n d a m e n t a l t h e o r e m ) . Assume that we have an arbitrage-free (B,S)-market with B = ( B , ) O G n Q ~ and S = (S,)O<~Q,T,S, = (Sk, . . . , S:), where the B, > 0 are %7,-~-measurable~the SR 3 0 are 9n-measurable, N < 00,and d < 00. Then the following conditions are equivalent. a) The market is complete. b) The market is perfect. c) The set of martingale measlire 9 ( P ) contains a unique measure. (1) The set of local martingale measures 9ploc(P) contains a unique measure. e) There exists a measure in .910c(P) such that each martingale M = ( M n ,.qn, P ’ ) T L G Nhas an ‘S-representation’ n

+

M , = MO Cyi~Si: n 6 N , with 9i-1-rneasurablc yi. i=l f ) 9,, = a(S1,.. . S,) u p to sets of P-measure zero, and there exist ( d + 1)predictable Rd-valued processes ( a ~ ., .~. ,~ad+l,n)r , 1 6 71 N , such that their values are affinely independent (for all n and w ) and the supports of the measures Q T L ( . ; w ) lie (P-as.) in the set { a ~ , ~ ( w. .). ,, a d + ~ , ~ ( w ) ) . g) s n = a(S1,.. . , STL)u p to sets of P-measure zero and there exist ( d + 1)predictable Rd-valued processes (Z1,7z,. . . , u d + ~ , ~1) , 7~ N , such that their values are affinely independent (for all n and w ) and the supports of the measurcs G7L( . ; w ) lie (P-a.s.) in the set { E I , ~ ( w ) , If these conditions hold, then the a-algebra 9~is purely atomic (with respect to P) and consists of a t most ( d l l N atoms.

.

<

< <

+

Chapter V. Theory of Arbitrage. Discrete Time

498

Proof. For d = 1 this was explained in the preceding sections. In the general case of d 3 1 the corresponding proof can be found in [251]. We refer the reader there for the technical detail related to the fact that the prices S = (S’, . . . , Sd) are vector-valued; we concentrate ourselves on outlining the proof and pointing out the differences between the cases d = 1 and d > 1. First, we note that the equivalence of f ) and g) is a simple consequence of the relation

between the ti;,, and the ai,,. Further, we clearly have the implication b)==+a) and by Theorem A* (§2e), d) -c). Herice to prove the theorem we must prove that a) =+

4,

c) ==+

dl

g) ===+ b ) , a) + g )

e)

=j

el,

==+ a ) .

The implication a)==+d) can be established in the same way as for d = 1 (see §4a.2), where, in place of the martingale measures P,, i = 1 , 2 , one must consider local martingale measures. As regards -the implication c)==+g), we already proved the equality 9,= a(S1,.. . , S,) in 5 4e.2, in our proof of implication ( 3 ) from 4a.2. (The corresponding proof is in fact valid for each d 3 1.) The most tedious part of the proof of the implication c)-g) is to describe . ; w ) . For d = 1 these supports the structure of the supports of the measures were ‘two-point’. In the general case of d 3 1 these supports consist of at most d 1 points (in !Itd).This part of the proof is exposed at length in [251] and we omit it here. (Conceptually, the proof is the same as for d = 1 and proceeds as ;w ) follows. Assume that P is itself a martingale measure. If the support of contains more than d 1 points, then, using the idea of ‘mass pumping’ again, we can construct a new measure by the formula P’(dw) = z ( w ) P(dw). For a suitable choice of the SN-measurable function z ( w ) , is a martingale measure, P, and # P. This contradicts, however, our assumption of the uniqueness of a martingale measure. In a similar way we can also prove that the Rd-valued , ~ affinely independent.) variables ti^,^, . . . ,u , j + ~ are We consider now the implication g)==+b). Let f N be a 9N-measurable random variable and assume that the original measure P is itself a martingale measure. Then it follows from g) that, in fact, f~ is a random variable with finitely many possible values.

a,(

+

+

N

a,(.

4. Complete and Perfect Arbitrage-Free Markets

499

w e claim that f N can be represented as follows:

x N

+

fN =

(1)

TkaSk.

k=l

The sequence

ynf x +

11

C ?,AS,, < N , is a P-martingale, therefore the relations 71

i=l

x = EfN and = E(fN I tqn) - E(fN 1 9 n - 1 ) (2) must be satisfied. Hence, to obtain the representation (1) we can set z = EfN and then show (as for a! = 1) that using condition g) we can construct 9,-1-nieasurable functions yn with required property (2). (See [251] for greater detail.) We now turn to the implication a) g) 3 e). By g) the a-algebra 9~is purely atomic. Hence all SN-measurable random variables can take only finitely many values and are therefore bounded. Let P’ E 910c(P) and let M = (Mr1,STL, P ’ ) n be ~ ~a martingale. By (a) there

+

N

exist

IC

E R and a predictable process y = ( y r L such ) that MN = x

The sequence M’ = (MA,971, P’),
n

+ C yiASi. i=l

+ C yiASi is a local i=l

P’-martingale and, therefore, a martingale because all its elements are bounded. Since MN = M h , the martingales M and M’ coincide (P’-a.s.), which gives us assertion e). Finally, to prove the implication e)==+a) it suffices t o make the following observation (cf. Lemma § 4a). Assume that each martingale M = ( M n ,gn,P’) with E 9ploc(P) admits an n

‘S-representation’ Mn = Mo

+ C -yiASi. i=l

Let f N be a FN-measurable bounded function. w e consider the martingale Mn = fN I g n ) n, N , where is averaging with respect to P’. By assumption,

<

N

i=l

Hence we can represent f N as a sum: fN = x

+

N

TiaSi

(P-a.s.),

i=l

where x = MO and y = ( y i ) i < ~is a predictable sequence, which means precisely that the market is complete. These arguments complete our discussion of the above-described implications required in Theorem B*.

Chapter V. Theory of Arbitrage. Discrete Time

500

2. We present now several examples illustrating both Theoreni B* and Theoreni A*.

EXAMPLE I (cl = 1). In the framework of the CRR-model (see 5 4d) with B, = 1, n < N , we assume that ( p , ) , < ~ is a sequence of independent identically distributed random variables taking two values, a and b, a < b. Since AS,,, = S,-lp,, it follows that AS, = S,,-la or AS,, = S,-lb. By Theorem A*, in order that the market be arbitrage-free the interval ( a ,b ) must contain the origin. Hence a < 0 < b. To ensure that the prices S are positive we must also set u, > -1. In this case AS,, = S,-lpn, so that AS, can take two values: S,-lb (‘upward price motion’) and Sn-la (‘downward motion’). Hence the supports of the conditional distributions ; w) consist of two points: S,-l(w)a and S,-l(w)b,while the ‘price tree’ (So,5’1,S2, . . . ) (see Fig. 56) has the ‘homogeneous Markov st,ructure’: if (So,ST].. . , Sn-l) is some realization of the price process, then the next t,rarisition brings S, = S,-1B with probability p = P(p,, = h) and S, = S,-1A with probability q = P ( p r l = a),where B = 1 + b and A = 1 + a.

a,(.

sn

t

I

F

1 2 FIGURE 56. ‘Price tree’ (So,S1,Sz,. . . ) in the CRR-model

If -1 < a < 0 < b: then there exists a unique martingale measure, so that the corresponding (I?,S)-market is arbitrage-free and complete. By Theorem B* we obtain that for d = 1 all complete arbitrage-free markets have fairly similar ‘dyadic’ branching structure of the prices. Namely, for fixed ‘history’ (So,S1,. . . , S1,-l) we have S,n = S,-l(l p,), where the variables pn = pn(SO,S1,...,Sn-l) can take only two values, a,n = aT1(S0,S1,. . . ,&-I) and b, = bn(Sg,S1,.. . , S n - l ) . In the above model of Cox-Ross-Rubinstein the variables a , and b, are constant ( a , = a and 6, = b). In general, they depend on the price history, but again. to obtain a complete arbitrage-free market with positive prices, we must set -1 < a,, < 0 < by,.

-+

4. Complete and Perfect Arbitrage-Free Markets

501

EXAMPLE 2 ( d = 2 , N = 1). Let Bo = B1 = 1 and let S = (S1,S2)be the prices of two kinds of stock, where S A = 5': = 2. We consider a single-step model ( N = 1); let AS1

=

(;$)

be the vector of price increments with AS; = Sf- Si = Sq - 2, i = 1 , 2 . In accordance with Theorem B*, to make the corresponding arbitrage-free market complete we need that the support of the measure P(AS1 E . ) consist of three

and the three corresponding vectors in R2must be affinely independent. As already mentioned, this is equivalent to the linear independence of the vectors

(

a2

-

ug

)'

and h2 - h3 For instance, assume that the probability of each of the vectors

i.

is equal to These vectors are affinely independent, and the measure assigning them probabilities and respectively, is martingale.

i, i,

i,

Chapter VI. Theory of Pricing in Stochastic Financial Models. Discrete T i m e .....

503

Risks and their reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main hedge pricing formula. Complete markets . . . . . . . . . . . . . Main hedge pricing formula. Incomplete markets . . . . . . . . . . . . . Hedge pricing on the basis of the mean square criterion . . . . . . . . Forward contracts and futures contracts . . . . . . . . . . . . . . . . . . .

503 505 512 518 521

2. A m e r i c a n hedge pricing on a r b i t r a g e - f r e e m a r k e t s . . . . . . . . . . . .

525

fj 2a. Optimal stopping problems. Supermartingale characterization . . . 3 2b. Complete and incomplete markets. Supermartingale characterization of hedging prices . . . . . . . . . . . 2c. Complete and incomplete markets. Main formulas for hedging prices . . . . . . . . . . . . . . . . . . . . . . . . 3 2d. Optional decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

525

1. E u r o p e a n hedge p r i c i n g o n a r b i t r a g e - f r e e markets

5 la. lb. fj lc. fj Id. fj l e .

535 538 546

3 . Scheme of series o f ‘ l a r g e ’ a r b i t r a g e - f r e e m a r k e t s and asymptotic arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

One model of ‘large’ financial markets . . . . . . . . . . . . . . . . . . . . fj 3b. Criteria of the absence of asymptotic arbitrage . . . . . . . . . . . . . . fj 3c. Asymptotic arbitrage and contiguity . . . . . . . . . . . . . . . . . . . . . . 3 3d. Some issues of approximation and convergence in the scheme of series of arbitrage-free markets . . . . . . . . . . . . . . . . . . . . . . . .

3 3a.

4. European options on a b i n o m i a l (B,S)-market . . . . .

553 555 559 575 588

fj4a. Problems of option pricing . . . . . . . . . . . . .

. .. .. . .... .... .. Rational pricing and hedging strategies. Pay-off function of the general form . . . . . . . . . . . . . . . . . . . . . . fj 4r. Rational pricing and hedging strategies. Markovian pay-off functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4d. Standard call and put options . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4e. Option-based strategies (combinations and spreads) . . . . . . . . . .

595 598 604

. ... . ..

608

. . . . . . . .. . . ........... ........... ...........

608 611 621 625

588

3 4b.

5 . A m e r i c a n options on a b i n o m i a l (B,S ) - m a r k e t fj 5a. fj 5b. fj 5c. fj 5d.

American option pricing . . . . . . . . . . . . . . . . . . . . Standard call option pricing . . . . . . . . . . . . . . . . . Standard put option pricing . . . . . . . . . . . . . . . . . Options with aftereffect. ‘Russian option’ pricing .

590

1. European Hedge Pricing on Arbitrage-Free Markets

fj la. Risks and Their Reduction

1. H. Markowitz’s theory ([332], 1952), as expressed in his mean-variance analysis (see Chapter I, SZb), suggests an approach to the pricing of investment risks and the reduction of their nonsystematic component that is based on the idea of diversification in the selection of an (optimal) investment portfolio. Several other optimization problems arising in financial theory can, in view of the ‘environmental uncertainties’, be (as in H. Markowitz’s case) ranked among the problems of stochastic opti7nization. In the first line, it should be mentioned that finance brings forward a series of nontraditional, nonstandard opt-imizat,ion problems relating to hedging (see Chapter V, l b about the notion of ‘hedging’). They are nonstandard in the following sense: optimal hedging as a control must deliver certain properties with probability one, rather than, e.g., on the average, as is usual in stochastic optimization. (As regards problems involving the m e a n square criterion, see 3 I d below.) In what follows we put an accent on the discussion of hedging as a method of dynamical control of an investment portfolio. It should be noted that this method is crucial for pricing such (derivative) financial instruments as, e.g., options (see subsections 4 and 5). We can say more: it is in connection with the pricing of option contracts that the importance of hedging as a protection instrument has been understood and its basic methods have been developed.

2. We recall that we already encountered hedging in 5 lb, Chapter 5, where in the simplest case of the single-step model we deduced formulas for both initial capital sufficient for the desired result and optimal (hedging) portfolio itself. Explicit formulas for hedges are of considerable interest also in several-step problems, in which an investor seeks capital levels at a certain (fixed in advance) instant N that, with probability 1 (or, more generally, with certain positive probability), are not lower than the values at these instants of some fixed (random, in general) performance functional.

Chapter VI. Theory of Pricing. Discrete Time

504

Problems of this kind have a direct relation to the pricing of European options; this connection is based on the following remarkably simple and seminal idea of F. Black and M. Scholes [44]and R. Mertori [345]: in complete arbitrage-free markets

the dynamics of option prices m u s t be replicated by the d y n a m i c s of the value of the optimal hedging strategy in t h e corresponding i n v e s t m e n t problem. In the case of A m e r i c a n options, besides the hedging of the writer, which realizes his ‘control’ strategy, we have one new ‘optimization element’. In fact, the buyer of a European option is i n e r t : he is not engaged in financial activity and waits for the maturity date N of the option. American-type options are another matter. Here the buyer is a n acting character in the trade: the contract allows him to choose the instant of exercising the option on his own (of course, within certain specified limits). In selecting the corresponding hedging portfolio the writer must keep in mind the buyer’s freedom to exercise the option a t any time. Clearly, the contradictin,g interests of the writer and the buyer give rise to optimization problems of the m i n i m a x nature. The present section ( § § l a - l d ) is devoted to European hedging. This name is inspired by an analogy with European options and means that we are discussing hedging against claims due at some moment that was fixed in advance. We consider American hedging in the next section (see, in particular, 3 2c for the corresponding definitions). 3. As already mentioned, the buyer’s control in American options reduces to the choice of the instant of exercising the contract: or, as it is often called, the stopping time. Here. if, e.g., f = ( f o , f l , . . . , f N ) is a system of pay-off functions ( f i = f i ( w ) , i = 0 , 1 , . . . , N ) and the buyer chooses a stopping time 7 = ~ ( w )then , his returns are ”f7 = f T ( u ) ( 4 We represent now f r as follows:

Note that the event { k follows:

<

T }

belongs to 9k-1.

Hence (1) can be rewritten as

N k=l

where ffk = I ( k

<

T)

is a 9k-1-nieasurable random variable.

1. European Hedge Pricing on Arbitrage-Free Markets

505

We can express this otherwise: the terms of an American option contract allow the buyer to exert only a predictable control cy = ( a k ) k < ~of the form ffk = I(k T). In principle, one could imagine contracts allowing other forms of (predictable) control cy = ( c y k ) k < ~from the buyer’s side. One possible example is a ‘Passport option’ (see, e.g., [ti]) with pay-off function

<

<

where I a k / 1 and S = ( S , L . ) ~is~the N stock price. It is clear in this case that the writer of the contract must select a (hedging) portfolio T = ~ ( c y ( w w ) , ) such that for each admissible buyer’s control cy = a ( w ) the value X & ( c y ( w ) , w )a t the terminal date N is a t least f N ( a ( w ) ) (P-a.s.). 4. Hedging (by an investor or the writer of an option) and the control exercised by

the option buyer, say, are the two main ‘optimization’ components that should be reckoned with in derivatives pricing on complete markets. For z,ncomplete markets, we must add also the third component determined by ‘natural factors’. This means the following. There can exist only one martingale measure in complete arbitrage-free markets. However, in incomplete arbitrage-free financial markets ‘Nature’ provides one with a whole range of martingale measures and, therefore, with drstznct versions of the formalization of the market. Which measures and formalizations are in fact operational remains unknown to the writer and the buyer of an option. Hence both writer and buyer, unless they have some additional arguments, must plan their strategy (of hedging, or the choice of the stopping time, etc.) bearing in mind the ‘most unfavorable combination of natural factors possible’. We shall express this formally by considering in many relations the least upper bounds over the class of all martingale measures that can describe potential ‘environment’.

3 lb.

Main Hedge Pricing Formula. Complete Markets

1. We shall consider a complete arbitrage-free (B,S)-market with N < 00 and d < m (see the general scheme in Chapter V, 5 2b). By assertion (f) of the extended version of the Second fundamental theorem (Chapter V, 3 2e), such a discrete-time market is also discrete with respect t o the phase variable, and all the SN-measurable random variables under consideration can take only finitely many values because the a-algebra 9~consists of a t most ( d + l)N atoms. Thus, there can be no problems with integration in this case, and the concepts of complete market and perfect market are equivalent.

Chapter VI. Theory of Pricing. Discrete Time

506

DEFINITION. T h e pnce of perfect European hedging of a n (9N-measurable contingent claim f N ) is the quantity @( f N ; P) = inf{z: 3~ with (cf. Chapter V,

x$ = z with xc = f N

(P-a.s.)}

(1)

3 lb).

Since the market in question is arbztrage-free and complete by assumption,

(anGN

1) there exzsts a martingale measure

p equivalent to P such that the sequence

is a martingale (the First fundamental theorem)

and

2) this measure is unzque. and each contingent claim f~ can be replicated, i.e.. there ezzsts a (‘perfect‘) hedge T such that = fN (the Second fundamental theorem). = 2 and = fN (P-a.s.), then Hence if K is a perfect (x,fN)-hedge, i.e.. (see formula (18) in Chapter V, S la)

xk

x$

and therefore -fN E-=---, x BN

BO

1.e..

We note that the right-hand side of (3) is zndependent of the structure of the in question. In other words. if T’ is another hedge, then the initial prices x and x’ are the same. Hence. we have the following result. ( 2 ,f,v)-hedge 7~

THEOREM 1 (Mazn formula of the pnce of perfect European hedgzng zn complete markets). The price @( f ~P):of perfect hedging in a complete arbitrage-free market is described by the formula

1. European Hedge Pricing on Arbitrage-Free Markets

507

2. In hedging one must know not only the price @ ( f ~ P), ; but also the composition of the portfolio bringing about a perfect hedge. A standard method here is as follows (cf. Chapter V, 54a).

-

-

We construct the martingale M = ( M n ,9 n , P ) n c ~ with Mn = E

(21 ~

t9n

.

Since our market is complete, it follows by the Second f u n d a m e n t a l theorem (or the

S B

lemma in Chapter V, 5 4b) that M has a n ‘--representation’

with 9k-l-measurable yk. We set

7r* =

Sn (p*,r*),where y* is equal to y = ( y n ) in ( 5 ) and p i = M n - yn -.

Bn It is easy to verify that this is a self;financing portfolio (see nevertheless the proof of the lemma in Chapter V, 3 4b). Further, by construction,

and

Hence, for all 0 6 n 6

N we have

and, in particular, I

XG* = fN Thus, the portfolio

T* constructed

(P- and P-a.s.).

S B

on the basis of the ‘--representation’ is a perfect

hedge (for f ~ ) . We can sum up the above results as follows.

THEOREM 2 (‘Main formulas for a perfect hedge and its value’). In a n arbitrary arbitrage-free complete market there exists a self-financing perfect hedge T * = (/I*, y*) with initial capital

508

Chapter VI. Theory of Pricing. Discrete Time

that replicates f N faithfully:

The dynamics of the capital

Xz-is described

by the formulas

,s

the components y* = (T;) are the same as in the . --representation’

B

and the components ,!?* =

(0;)can

he defined from the condition

3. We consider now the issue of hedge pricing in a somewhat more general framework. assuming that, in place of a single function f N , we have a sequence fo,f l . . . . . f N of pay-off functions and f i is 9i-measurable, 0 6 i N . Let 7 = T ( W ) be a fixed Markov time ranging in the set {0,1,. . . , N} and let f r be the terminal pay-off function constructed for this T and fo, f l , . . . , f N .

<

THEOREM 3 . If an arhitrage-free (B.5’)-market is N-complete, then it is also r-complete. that is, there exist a self-financing portfolio T and an initial capital z such that X $ = 3: and X : = f T (P-as). Proof. This is simple: we construct a new pay-off function f,$ = f,-,,N; then the perfect hedge T * for &. is simultaneously a perfect hedge for the initial pay-off function fT. The corresponding price @(fT;

P) = min{z: 3.ir with ,Y$ = x and

can be evaluated by the formula

-fT BT

@(f7; P) = Bo E-

X :

= f T (P-as.))

1. European Hedge Pricing on Arbitrage-Free Markets

509

4. One can put the following question in connection with ‘main formula’ (4).

Assume that we consider a complete and arbitrage-free

(B, S)-market and let

S P be a martingale measure for the discounted prices -. B

It will be useful to

I

,;( E z,. , c) (i.e., the process

formulate the last property in the following equivalent way: the vector process S1 . . i s a P-martingale.

B

B

We assume now that there exists another (positive) discounting process B = (E,),
-

B’

””)

B’””

is a P-martingale. One would expect the value of the price f N ; P) defined by (1) to be zndepen- _ dent of the choice of the corresponding pairs p) and (B,P ) . In fact, this is just the case: we have the equalities

e( (B,

moreover, even the ‘price processes’

are the same. We now - discuss the factors underlying this coincidence. To this end we assume BN that E= = 1 (which is not a serious constraint). Then we can introduce a new

BE

measure P (on 9 ~ by ) setting d p = r??N dj?,

The measure p is a probability measure, and by Bayes’s formula (see the lemma in Chapter V, S3a)

510

Chapter VI. Theory of Pricing. Discrete Time

is a martingale both with respect to

Hence the sequence

and with

respect to F. However, if the market in question is complete. then the martingale measure must be unzque, and therefore ^P = P, that is, 2, = 1 for n 6 N , which, by the defin'tion of the Z,, brings us to the equalities

-

A

showing that

This proves (9) and ( l o ) , so that. indeed, the value of the price @(fp,; P) in a complete market is independent of our choice of a discounting process. In Chapter VII. § l b we shall consider discounting in the continuous-time case (and in greater detail). At this point, however, we mention only that a successful choice of a discounting process can in many cases considerably reduce the analytic complexP) and the corresponding perfect hedges. See, for ity of finding the prices Cjf~; instance. the calculations for .Russian options' in 5 5d and Chapter VIII, 2c. 5 . Thus. formula ( 4 ) fully solves on a complete arbitrage-free market the pricing problem for perfect hedges in the case of the discounting process B. If is is the

corresponding martingale measure

(

i.e.,

->

5B is a martingale with respect to P

,

then a transition to another discounting process B changes also the martingale measure: the new measure p, in accordance with ( l l ) ,can be obtained by t h e formula -

BN -

dP = -dP.

(12)

BN

On the other hand. if we are in an incomplete market. then there exist several martingale measures and it is no longer easy to understand what we m_ust cuff the hedging price. If we have two distinct martingale measures, P i and p2, and therefore can give distinct definitions of what it means for a market to be arbitrageI

- E- T fN free. then the quantities Bo - and Bo E (see 5 lc below).

p1 BN

k, are. in general, also distinct

p2 E N

1 . European Hedge Pricing on Arbitrage-Free Markets

511

6. For a n illustration of our above discussion of various discounting processes and the calculation of conditional expectations with respect to different measures we consider the following example. Let f N be the price of some contingent claim (expressed in USD). If we consider a complete arbitrage-free (dollar) market. then the price of a perfect hedge is - -fN Bo ET-, where B = ( B n ) n ,
Next, we consider a market with prices in German marks (DEM). Expressed in marks, the value of the same contingent claim becomes f N s N (DEM). where DEM

SN

=

(EdN

is the cross rate a t time N . If B = ( B n ) n G is ~a DEM-bank account and the corresponding market is complete and arbitrage-free. then the price of a hedge against the contingent claim fivS,v (in DEM) is

Bo --SNfN E v ,

-

BN

which, converted into dollars. makes up

What are the conditions ensuring that. as one would anticipate. this dollar price is equal to

B-o -E-.f N BN

or. in

3.

more general form

We assume that the DEM-market is arbitrage-free: then for the cross rate

Hence

Chapter VI. Theory of Pricing. Discrete Time

512

-

is a P-martingale,

If the USD-market is also arbitrage-free, then ( g ) n < N

so that

By (15), (16),and our assumption that the USD-market is complete, and therefore the measure P is unique, we obtain I

-

_BN _ . - dP -=1 B N dP (cf. (12)), and for all n 6 N , -B, . _ _dp, = 1. Bn dP, -

-

dp,

Since the process Z = (Z,, Sn,P ) n c ~with Z , = __ is a martingale, equal-

-

dp,

is also a P-martingale. In fact one could foresee

ity (18) inearis that ($)n
that this property would ensure the coincidence of the (dollar) prices of hedging against f N in the USD and the DEM-market before any calculations: we could regard B = (Zn),<x as one of the basic securities on the dollar market with dollar bank account B = ( B n ) n ~ ~ . I

5 lc.

Main Hedge Pricing Formula. Incomplete Markets

1. As shown in the preceding section, the price @ . ( f N ;P) of perfect hedging on a complete arbitrage-free market has the following expression:

-

I

where E is averaging with respect t o the (unique) martingale measure P such that

2B is a F-martingale.

A similar question of hedging prices can be put, of course, also on a n incomplete market. However, there does not necessarily exist a perfect self-financing hedge on such a market, therefore we must modify our definition of the hedging price and consider a somewhat wider class than self,financing strategies to which we could stick in the case of a complete market. We recall that the value X" of a self-financing strategy 7r = (0, y) on a complete market can in fact be defined in two ways: we either write

1. European Hedge Pricing on Arbitrage-Free Markets

513

(see Chapter V, 3 l a for greater detail). Representation (3) is more convenient in a certain sense for it visualizes the dynamics of the growth of capital: X $ is the share of the initial capital in Xz, while AX: = &LIBn (4)

+

is the increment. In our case of hedging on an incomplete market it seems reasonable to consider, alongside the portfolio 7r = (0, y),also the consumption process C = ( C n ) n 2 which ~, is a non-negative non-decreasing process with g:,-measurable C, and CO= 0. In fact, we already discussed this situation in Chapter V, 3 l a , where we called it the case with ' c o n s u m p t i o n ' . We assumed there that, in place of (4), the dynamics of the capital X"ic corresponding to a portfolio 7r and consumption C can be described by the relations = PnABn

+ ynASn

-

AC,,

(5)

where AB, + y n A S n is the share of the total increment that can be ascribed to the composition of the portfolio and is due to the 'market-related' changes AB, and AS,, while AC, characterizes the expenses on consumption (e.g.: the expenditures on the changes in the portfolio). Thus, we shall now assume that the value X"ic of the strategy (7r,C) can (by analogy with ( 3 ) ) be calculated by the formulas

which is equivalent to

A ( g ) =T~A($)

Remark 1. Setting

/j't = p k

-

~

ABk

-= ncn ,

we see from (6) that

n31.

Chapter VI. Theory of Pricing. Discrete Time

514

This formula is very much similar to ( 3 ) . However, while measurable, is now only Bk-measurable.

pk

in (3) is 9 k - l -

Remark 2. We have already mentioned that, generally speaking, perfect hedging is unattainable in incomplete markets, i.e., there does not necessarily exist hedging T = (P,?)such that X g = f~ (P-as.) At the same time, this does not, rule out the possibility that modifying our definition of a n admissible strategy we can attain the level of terminal capital oflsetting (P-a.s.) the pay-off f ~As.will be clear from the proof of the theorem below, our calling upon 'consumption' enables us t o find a strategy ( T , C ) such that = f N (P-as.). This is one 'technical' argument in favor of considering consumption C alongside the portfolio T . O n the other hand, our introduction of strategies with 'consumption', which must satisfy constraints of the form AC, 3 0, has clear economic implications.

"2"

2. DEFINITION.The upper price of European hedging (against a 9N-measurable pay-off f ~ is)the quantity @ * ( f j ~ P) ;

= inf{z:

3 ( T , C )with

xi"

= 2 and

' ; x

3

fN

(P-a.s.)}.

(7)

Remark. Besides the upper price we can also consider the lower price of hedging (see the definition in 3 l b ) . In our subsequent discussions we shall consider only the upper price arid call it simply the price of hedging. Let 9 ( P ) be the set of all martingale measures P equivalent to P. We assume that P ( P ) # 0. The central result of pricing theory on incomplete arbitrage-free markets, which generalizes formula (I), is as follows.

THEOREM 1 (Main formula of the price of European hedging on incomplete markets). Let fN be a non-negative bounded 9N-measurable function. Then, on an P) ;can be calculated by the incomplete arbitrage-free market, the price @* ( f ~ formula @ * ( f N ;P) =

sup

1

B,E-f'V,

P€9(P)

BN

-

where EP is averaging with respect to measure P. We have already encountered a special case of this result (Theorem 1 in Chapter V, $ lc; see also [93])in the case of a single-step model. The crucial point in the proof of (8) is the so-called optional decomposition (see 5 2d below), which has a rather technical proof. The existence of this optional decomposition and formula (8) were established for the first time by N. El Karoui and M. Quenez [136] and D. 0. Kramkov [281];as regards generalizations and other proofs, see also [99], [163],and [164].

1. European Hedge Pricing on Arbitrage-Free Markets

515

3. Proof of the theorem. Let (l/r,C)be an (z'fN)-hedge, i.e., assume that = z and x;" 2 f N (P-a.s.1. Then (cf. formula (2) in 3 lb)

x;"

so that for each measure

because Ep

N

yk A k=l

E 9 ( P ) we have

(2)

=

0, which is a consequence of assertion 2) of the lemma

in Chapter 11, 5 l c and the inequality

N

Yk A k= 1

(2) -% 3

X

following by (9).

Hence SUP

P€kP(P)

Bo E---

fN BN

< C * ( ~P). N;

(11)

To prove the reverse inequality we set

Y, = _esssup EP€.9(P)

(fiv I 9,), BN

where, by definition, the essential supremum Y, is the g,-measurable random variable that, on the one hand, satisfies the inequality

for each measure p E P ( P ) , and, on the other hand, has the following property ('minimality'): if Y, is another variable dominating the right-hand side of (13), then Yn Fn (P-as.). As shown in §2b, the sequence Y = (Y,,~,),
<

516

Chapter VI. Theory of Pricing. Discrete Time

Recall that, as follows from the classical Doob decomposition (5 l b , Chapter II), for each particular measure Q we can find a martingale M Q = (M?, 2Jn,Q ) o G n c ~where , MF = 0, and a predictable non-decreasing process AQ = (A,, Q 9TL-1,Q ) ~ G ~ ~ G N , where A$ = 0, such that

+

Y,, = YO M: - A.: (15) It is remarkable that if Y = (Yn, $n) is a supermartingale with respect to each measure Q in the family P ( P ) . then Y has a uniwersal (i.e., independent of Q) decomposztion Y,=Yo+M,-C,, (16) where % = ( M n , S n ) is a martingale with respect to each measure Q E 9 ( P ) is a non-decreasing process with = 0. and = (En,Sn) We point out that while the process AQ in the Doob decomposition (15) is in (16) predictable (i.e., the A? are $;,-I-measurable), the process ??= (En,Sn) is only optional (i.e., the En are $,-measurable). This explains why (16) is called the optional decompositzon. In the special case of the supermartingale Y = (Yn, defined by (12) we can further specify the structure of the martingale = (a,,

co

sn) sn):

where 7 = ( T 7 a , 9 n - 1 )is a predictable process. (We emphasize that this is far from trivial and can be established simultaneously with the proof of the optional decomposition; see 3 2d.) For the processes 7, -and YOintroduced by (16) and (17) we shall now conand the consumption process 2; such that for the struct the portfolio Z = (/3,7) corresponding capital we have X,F>c = Bo

c,

P€P,(P)

BN

course, this means that

which, together with (ll),brings us to (8). The required portfolio ii: = 7)and the consumption process as follows:

(p,

cTl

2;can be defined

n

=

Bk-1

Ack,

k= 1

where 7 and

??are as in the optional decomposition of the supermartingale Y

(19)

1. European Hedge Pricing

For .ii and

0x1

Arbitrage-Free Markets

517

c so defined the starting capital is

In the scheme with consumption we assume (see Chapter V, ment in the capital can be described by the formula

5 la.4) that thc incre-

which, as already mentioned, yields

(cf. also (27) in Chapter V, By (16)- (19),

9 la).

~(g) = AY,,

xo".c

and sincc -= YO,it follows that

Bo

Thus, X;'

--

= f N , so that the proposed strategy

Xf,'

= BoYo = Bo

sup

(E,

c)with starting capital

E-- fN

P€9(P) pBN

gives one a perfect hedge: X?' Hence

=fN.

C*(f N ; P)

< Bo

SUP

P€9(P)

E- __ fN

.

BN

Together with (11) this proves required formula (8) (provided that we have the optional decomposition). Thus, we have proved Theorem 1 and, incidentally, established the following result (cf. Theorem 2 in 5 l b ) .

Chapter VI. Theory of Pricing. Discrete Time

518

THEOREM 2 ( M a i n formulas f o r a perfect hedge, its value, and consumption). On an arbitrage-free market i t is possible to find a self-financing hedge rr* = (p*,y*) and consumption C* such that the value of this hedge, XE* = BABn+yGSn, changes in accordance with the balance condition AX;* = ,B;AB, + y;ASn - ACG and satisfies the relations

and

x&*= fN

(P-a.s.).

The value Xz* of this hedge can be calculated by the formula

BN

the components y* = (7:) position

and C* = ((7;)

can be found from the optional decom-

and the components p’ = (p;) can be found from the condition X:’

5 Id.

= PiB,+yGS,.

Hedge Pricing on the Basis of the Mean Square Criterion

1. Let f N = f N ( W ) be a SN-measurable pay-off. In a complete arbitrage-free market there exist a starting capital 2 and a strategy 7r such that a trader can replicate f N faithfully (in the sense that x&(z) = fN with probability one, where X & ( z ) is the value X N of the strategy rr such that X [ = z). However, if a market (with or without arbitrage) is incomplete, then the situation becomes considerably more complex and one cannot hope to replicate f N faithfully. In 5 l c we considered the calculations of the hedging price @*(f N i P) on an arbitrage-free market when the hedging strategy rr was to ensure that

Xf;(cC*(f~; P)) 3 SN

(P-a.s.1.

In the present section we shall understand optimal hedging in a somewhat different sense, as the replication of f N with ‘maximum precision’. Our choice of the measure of accuracy is, in a sense, a matter of convention; it depends on one’s aims, the chances of finding a precise solution of the corresponding optimization problem, and so on.

519

1. European Hedge Pricing on Arbitrage-Free Markets

In what follows, we measure the quality of replication by the mean square deviation R N ( T ,). = E[X%(x)- f N 1 2 , (1)

which in many cases enables one t o find the ‘optimal’ components x* and ing about the minimum of E[Xg(x) - f ~ ] ~ : inf R N ( T z) ; = R N ( T *x*). ;

7r*

bring-

(2)

(TJ)

(a,

2. Let 9, ( 9 n ) nP) (~ be, a fixed filtered probability space, let 9 0 = (0, a}, and let 9~ = 9.Let S = (SA, . . . , S $ ) n , ~be a sequence of prices of some &dimensional asset and assume that E;f < 00. If the sequence of prices is a martingale and, moreover, a square integrable martingale with respect to the original measure P, then the optimization problem ( 2 ) is easy to analyze in the class of strategies 7r satisfying the inequality E(XG(X))~ < m. (We emphasize that we do not assume the uniqueness of the martingale measure P and, therefore, the completeness of the market.) For a strategy 7r = (r’,. . . , T ~ ) where , the yi = ( T ; ) ~ ~are N predictable variables, let n

n

d

x;(x)=r+C(Yk:aS,)=x+C(C?;asl)) k=l

k=l

(3)

i=l

be its value. Since the sequence ( X ~ ( X ) ) ~is
EX&(x) = z. Setting

(4)

< = x&(x)- f N we see in view of the obvious equality EE2 = (EE)2 + E(< E 0 2 -

that RN(T’C)= [ W N

--.)I2 + E [ ( X & ( 4 - ).

-

(fN - E f N ) 1 2 .

(5)

We shall show below that for each pair ( T , z) with E ( x f ; ( ~ ) ) < ~ 00 there exists a pair ( T * ,z) such that R ~ ( 7 rx); 3 R N ( T *x) ; and X&* (x)- z is independent of z. Hence it follows by (5) that the smallest lower bound inf inf R N ( Tx)] ; is attained for X* = EfN.

Fori=l,

...,d w e s e t

520

Chapter VI. Theory of Pricing. Discrete Time

(where we put 0/0 = 0) arid consider the martingale L* = ( L f i ) 7 L G with ~

Clearly, f~ has the following decomposition:

Using the definition (7) we can verify directly that

E(ALk. (7,) AS,) 1 Fn-l)= 0.

(10)

Note that this property is equivalent to the ‘orthogonality’ of the square integrable martingales (L;),
5( n , A S k ) )

(k1

n
in the sense that their product is

also a martingale. It is worth noting in this connection that (9) is called the Kunita--Watanabe decomposition in the ‘general theory of martingales’. By (9) arid (10) we obtain that for each pair ( ? r , x ) ,

= R N ( T *Z) ; 3 R N ( T *x*). :

(11)

Moreover, t,he first inequality turns to a n equality for y = y*. Thus, we have proved the following result.

THEOREM. Assume that the original measure P is a martingale measure. Then the optimal hedge 7r* = (y*l, . . . , Y * ~ )in the problem (2) (in the sense of the mean square criterion) is rlescribcd by forniulas ( 7 ) ,x* = EfN, and

1. European Hedge Pricing on Arbitrage-Free Markets

521

Remark. If P is not a martingale measure (with respect to the prices S ) , then the question on the existence of a n optrmnl pair (z*, T*) and its search become fairly complicated. See, e.g., the papers of H. Follmer, M. Schweizer, and D. Sondermann [167], [168], [425], and also [195], [194] in this connection. We note also that the concept of a mznzmal martingale measure that we mentioned in passing at the end of \ 3 d , Chapter V has been developed precisely in connection with the above problem of hedging on the basis of the mean square test.

5 le.

Forward Contracts and Futures Contracts

1. In the present section we shall show how t o price forward contracts and futures contracts, important investment instruments used on financial markets alongside options. In accordance with the definitions in Chapter I, 3 l c , forwards and futures are sale contracts for some asset that must be delivered a t a specified instant in the future at a specified price (t,he 'forward' or the 'futures' price, respectively). There exists a n cssential distinction between futures and forwards, although both are sale contracts. Forwards are in fact mere sale agreements between the two parties concerned, with no intermediaries. Futures are also sale agreements, but they are concluded at a n exchange and involve a clearing house through which the payments are made and which is the guarantor of the contract. 2. Assume that the market price of the asset in question can be described by a stochastic sequence S = ( S ~ ) ~ Q where , T , N is the maturity date of the contract, which we identify with the instant of delivery. Clearly, if the deal is struck at time N , when the market price of the asset is S N , then for any reasonable definition of the forward or the futures price it must be equal to SN.This beconies another matter, of course, if the contract is sold at time n < N ; the crucial question here is how we must understand the fair price of the contract (on an arbitrage-free market). For a formalization, let us consider the scheme of a ( B ,S)-market described in Chapter V, 3 la, where B = (B,) is a bank account and S = ( S n ) is the traded asset. (If the asset in question is in fact one component of a d-dimensional vector of risk assets, then, in the arbitrage-free case, all our conclusion remain valid). We shall consider the case with dividends discussed already in la.4, when the y) is described by the formula value X = ( X ~ ) , G Nof the buyer's strategy 7r = (0,

and its changes arc described as

Chapter VI. Theory of Pricing. Discrete Time

522

where Y~ is the number of ‘units’ of the asset bought and D = ( D n , 9 n ) n G ~ , Do = 0 , is the (not necessarily positive) process of overall dividends connected to the asset S . We describe now the structure of dividends in the above cases of a forward or a futures contract and find the ‘fair’ prices of these contracts. 3. Assume that a forward contract is sold at time n and that both parties agree (on the basis of the ‘information’ gn) that the (forward) delivery price of the asset will be F,(N). Then, by the very mechanics of forward contracts, the overall dividends (positive or negative) can be represented as follows:

Dk=O,

n
(3)

and

DN = SN - IFn(N). By (1) and (2) (see also (24) in Chapter V,

5 l a ) we obtain

<

Clearly, for a forward contract sold at time n we can set Y k = 0 for k n and >, n 1, where -yn+l is the number of units of the asset S requested in the contract. By (6) we obtain

-yk = -yn+l for all k

+

and we immediately arrive at the following conclusion. Assume that the (B, 5’)-market under consideration is arbitrage-free and com-

-

plete. Let P be the unique martingale measure such that

is a martingale

with respect to it. Assume also that the 9;,-measurable prices F,(N) satisfy the relation

i.e.. let

1. European Hedge Pricing on Arbitrage-Free Markets

523

so that the forward contract sold a t time n a t the price IFn(N) defined by (9) is arbitrage-free (i.e., if X z = 0 and P(X& 3 0) = 1, then P(X& = 0) = 1; see Definition 2 in Chapter V, 3 2a). Of course, the value of I F , ( N )(the forward price) can be regarded as the fair price of the forward contract. Note that our assumption about an arbitrage-free (B,S)-market gives us the eaualitv

Thus, we see from (9) that the arbitrage-free forward prices IFn(N) can be defined as follows:

4. We now proceed to futures contracts. Assume that we sell a contract of this kind at time n, a t an 9n-nieasurable price Q n ( N )(the futures price). Immediately, the mechanism of settling through the clearing house comes into play. This can be roughly (skipping the issue of the margin account, the amounts deposited into it, and so on) described as follows in terms of (positive or negative) dividends. If the market futures price a t time ri 1, Q n + l ( N ) ,turns out less than Q n ( N ) , then the buyer puts the amount Q n ( N )- Q n + l ( N ) into the seller's account. If QTL+1(N) > Q n ( N ) ,then. conversely, this is the seller who deposits the amount Q n + l ( N )- @ n ( N )into the buyer's account. We shall set 60 = @ o ( N )and

+

6, Also, let SO

= Q n ( N )- @,-l(N),

ri

21

D7,= 60 + 61 + . . . + 6nr

that AD, = 6,) n 3 1. By (6) we see that (cf. (7))

-

Hence. siniilarly to forward contracts, we conclude that if P is a unique martingale measure for the (B,S)-market in question, then the condition

524

Chapter VI. Theory of Pricing. Discrete Time

imposed on the prices @,(N), . . . , @,+l(N) ensures that the futures contract sold at time n is arbitrate-free. Assume that the sequence D = ( D n ) n Gis~ a martingale with respect to P. Then (14) holds for each 71 3 0. In fact, since the predictable variables BI, are positive, we also have the converse result. The condition that D = ( D r L ) n cbe~ a martingale means that

@ n ( N )= E ~ ( S N I cpn),

n

< N,

(16)

and comparing with (11) we obtain the well-known result that for a deterministic bank account B = ( B T t the ) forward and the fut,ures prices are the same.

2. American Hedge Pricing on Arbitrage-Free Markets

3 2a.

Optimal Stopping Problems. Supermartingale Characterization

1. The superniartingale characterization in 5 l c of the sequence Y = (Y,) with respect to each measure in the family 9 ( P ) is not that striking if one treats the operation of taking the essential supremum in (12)’ 5 l c as an optimization problem of finding the ‘best’ probability measure. For this interpretation, the above supermartingale property is a mere consequence of the well-known ‘optimality principle’ for the price process (the ‘Bellman function’) in a stochastic optimization problem. A special case of such a problem is the optimal stopping problem for a Stochastic , seems a reasonable starting point in our discussion sequence f = ( f 7 L ) n < ~ which of ‘supermartingale characterizations’ in optimization problems. The emphasis o n this case is also justified by our discussion of American options below (the buyer of such an option has a right t o choose the date of exercising, so that the latter can be regarded as an ‘optimization element’), and also because the necessity to have a ‘suficien,tlz/rich’ class of objects when considering ess sup is clearly visible there.

(a, $, ($,)o<,<~,

2. Let f = ( f T 1 , , ! X T L ) ~ C n be < ~ a stochastic sequence on where 9 0 = (0,O)and 9~= 5. We shall assume that

EIf,l

<

P), oc for all

n
V,”

= sup E f T r rEmn,N

(1)

where the supremum is taken over the class 9Jl: of all stopping times r such that n r 6 N , and 2) the optimal stopping t i m e (which is well defined in our case).

<

We do not formulate here the optimal stopping problem in its most general form (see subsection 4 below), when N can be infinite (and 9Jr l is the class of all finite

Chapter VI. Theory of Pricing. Discrete Time

526

stopping times 7 3 n ) ; we restrict ourselves to finrte 'time horizons'. The main reason is that this case is relatively easy to study, but on the other hand one must already use backward induction, which is one of the main tools in the search of both prices V," and the corresponding optimal stopping times.

3. We introduce a sequence 7 N =

(-yz)~~~~~~

by the following ( a d hoc) definition:

We also set 7,:

= min{n

< i < N : f i = 7;>

< <

for 0 n N . The following is one of the central results of the theory of optimal stopping problems for a finite time interval 0 n N (cf. [75; Chapter 31 and [441; Chapter 21).

< <

THEOREM 1. The sequence 7 N = (T;),
(4 7,"

E

m ;:

(b) E(f.,N 19,) = 7,;; (c) E ( f r 1 97L) E(fr21 .Fn) = 7,"for each

<

E 2Jlz; (d) 7; = esssup E ( f r I 9,) and, in particular, 70"= sup Ef, (e)

v

7

TEm:

= E7,

=

Efri;l;

TE9.X;

N '

Proof. We drop for simplicity the superscript N in this proof and throughout subsection 3 arid write yn: V,, Mn,and r, in place of r ;, V,", and 7,". Property (a) is a consequence of the definition of T ~ Properties ~ . (b) and (c) are obvious for n = N . Now, we shall proceed by induction. Assume that we have already established these properties for n = N , N-1, . . . , k . We claim that they also hold for n = k - 1. Let 7 E MI,-^ and let A E 9 k - 1 . We set 7 = max(7,k). Clearly, 7 E MI, and for A E 9 k - 1 , bearing in mind that { T 3 k } E 9 k - 1 , we obtain

Mr,

E[lAfr]

= E[lAn{~=k-l}fr] =

+ E [ 1 A n { ~, >I,}f7]

E[I~n{7-=1,-i}f1,-il

= E[lAn{~=k-l}fic-l] E[lAn{~=I,-l}fic-l]

-

E [IAAylC-l]

+ E[Im{r>lc}E(fr

I91,-1)]

+ E[lAn{~>k}E(E(fFI 9 k ) 19I,-l)] 1 91,-1)]

+ E[lAn{r>k}E('%

1

where the last inequality is a consequence of (2)

(3)

2. American Hedge Pricing on Arbitrage-Free Markets

527

Thus, E( f T 1 9 k - 1 ) ,< yk-1. We shall have proved required assertions (b) and ( c ) for n = k - 1 once we have shown that E(fTk-,

I %-1)

= 7%-1.

(4)

To this end we consider the chain of inequalities in ( 3 ) ;we claim t hat we actually have the equality signs everywhere in (3) for T = ~ k - ~ . Indeed, by the definition of ~ k - 1 we have 7 = ~k in the set { q - l 3 k } . Since E( f T k I 9 k ) = yk by the inductive hypothesis, it follows that in (3) we have

1

CIA f%-1 =

[IAn{ T~ - = k - 1) fk- 11

1

-

I

+ E [ l A n { ~ k - l t k } E ( E ( f ~ k 9 k ) 9k-1)] E I I A n { ~ k - l = k - l } f k - l ] -k E I I A n { ~ k - l > k } E ( Y k

= E[IAYk-l]

1 9k)]

3

wherethelast equalityfollowsfromthedefinitionofyk-1 a s m a x ( f k - 1 , E ( y k I gk-l)), which means that yk-1 = fk-1 on { q - 1 = k - l},while the inequality fk-1 < yk-1 in the set { q - 1 > k - 1) means that yk-1 = E(yk 1.9k-1) there. Thus, we have proved (b) and ( c ) ~and therefore also assertion (d). Finally, since for each 7 E Wk we have EfT 6 E f T k = Eyk by (c), it follows that V k = sup EfT = EfTk = E y k , i.e., assertion ( e ) holds. TEmk

Remark 1. We used in the above proof the fact that if T E W k - 1 , then the stopping time 7 = max(r,k) belongs to Wk. In our case the class MI, has this property simply by definition. This means that the Wk, k 6 N , are ‘sufficiently rich’ classes in a certain sense. (See the end of subsection 1 in this connection.)

COROLLARY 1. The sequence y = ( y n ) n Gis~ a supermartingale. Moreover, y is the smallest supermartingale majorizing the sequence f = ( f n ) n f N in the following sense: if 7 = ( T n ) n is ~ ~also a supermartingale and qn 3 f n for all n N , then yn yn (P-as.), n 6 N.

<

<

Indeed, the fact that y = ( y n ) n Gis~a supermartingale majorizing f = ( fn)nGN follows from recursive relations (2). Further, it is clear that 7~ 3 f N and, for n < N , Tn 3 max(fn1 E(Tn+l I s n ) ) .

Since YN = f N , it follows that TN-1

TN 3 f N

(5)

and

3 max(.fN-1, E(7N I 9 N - 1 ) )

3 max(.fN-1, E(yN I 9 N - 1 ) ) In a similar way we can show that

= YN-1.

yn 3 yn for each n < N

-

1.

Chapter VI. Theory of Pricing. Discrete Time

528

COROLLARY 2. Corollary 1 can be formulated otherwise: if y = solution of the recursive system of equations

< Tn for n < N

with YN = f N , then Y~~

Thl 3 f N and

and each sequence 7 = (

I

Y~ 3 rnax(fn. E(Tn+1

I

971))r

n

T n ) n such ~~

is a

that

< N.

(7)

We claim that among all such solutions 7 = ( + n ) n we ~ ~can find the smallest solution y = with Y N = f N satisfying the system of equations (6). We set 7~= f N , and let

r7,

<

for n < N . Clearly, f7& for all 71 N and 7 = ( T ~ ) ~
Consequently, Y N = 7~= f N and, in view of (9), for all n

< N we have

Thus, the smallest supermartingale y = ( Y ~ ) ~ < majorizing N, the sequence f (f n ) n 6 N satisfies the equations (6) with Y N = f N .

COROLLARY 3 . The variable

is an optimal stopping time in the class

i.e.,

=

2. American Hedge Pricing on Arbitrage-Free Markets

529

4. We now consider the possible generalizations of Theorem 1 to the case when

%Jlr

N = m. More precisely, we shall assume that %Jl:L 3 is the class of all fifinrte Markov times -r = T(W) such that ~ ( w >, ) nl w E R. We denote the class "0" by

m*. Further.

let

f

( Q 9 1 ( 9 n ) n > 0 1 P),

= ( f n , 9 n ) n y )be and let

Of course one would expect that 7;

a stochastic sequence defined

= lirn 7," N-im

on

(under certain conditions) and

that one can make the limit transition in ( 2 ) . which yields the following equations for y* = (7;): 7; = nlax(fn. E(7;,+l I %)). (13) In a similar way it would be natural if the class %Jli in the following sense:

7;-

defined in (12), were an optimal time in

and V,* = Ey,;. As shown in the general theory of optimal stopping times (exposed, e.g., in [75] and [441]), these results hold, indeed, under certain conditions (but not always!). Referring to the above-mentioned monographs [75] and [441] for details we present just one. fairly general, result in this direction.

THEOREM 2. Let f

=

(fit.

F n ) n 2 ~be a stochastic sequence with Esupfr;

Then we have the following results. a) The sequence y* = (y;)71>o, where

satisfies the recursive relatioris

and is therefore a supermartingale niajorizing the sequence f = ( f T L ) .

n

<

30.

Chapter VI. Theory of Pricing. Discrete Time

530

b) The sequence y* = (y;),>o

is the smallest supermartingale majorizing f =

(fn)n>O.

c) Let

T;

= inf{k 3 n : f k = yz}.

If Esuplf,l < cm and P ( r i < cm) = 1, then n.

ri is an optimal stopping time:

d) For each n 3 0, 7,"

(P-as.) as N

t 7:

(20)

--t 00.

5 . We have already mentioned that for a finite time horizon ( N < m) the optimal stopping problem can be solved by backward induction, by evaluating the variables ~N ~ N, y ~ . . , - ~recursively. , . This is possible since yfvN = fN and the 7," satisfy recursive relations (13). On the other hand, if the time horizon is infinite ( N = m), then the problem of finding the sequence y = ( Y ~ ) becomes ~ ~ o more delicate: in place of an explicit formula describing the situation at time N we must use additional characteristics and properties of the prices V,, n 3 0. For example, one can sometimes seek the required solution of the system (17) using the observation that it must be the smallest of all solutions. The techniques of solution of optimal stopping problems are better developed in the Markovian case. For a description we assume that we have a homogeneous Markov process X = (xn, P), with discrete time n = 0 , 1 , . . . , with phase space ( E ,2),and with probability measures P, on 9 = V s n corresponding to each initial state z E E (see [126] and [441] for greater detail). Let T be the one-step transition operator (i.e., Tf(z) = E,f(q) for a measurable function f = f(z1) such that E,lf(xl)j < co for z E E, where Ez is averaging with respect to the measure P, 3: E E ) . Also, let g = g ( z ) be some 2-measurable function of z E E . We set f n (see the above discussion) to be equal to g ( x n ) , n 3 0 , and assume that Ez[supg-(zn)] < co,z E E. We also set

#

sn,

a}

where 337* = { T : T ( W ) < 00, w E is the class of all finite Markov times. The optimal stopping problem for the Markov process X consists in finding the function s(z) and optimal stopping times T* (satisfying the equality s(z) = E,g(zT*), z E E ) or €-optimal times T,* (satisfying the relation s(3:) - E 6 E,g(Xr;), z E E ) , provided that such times exist.

2. American Hedge Pricing on Arbitrage-Free Markets

531

The advantages of the Markovian case are obvious: the above-considered condidepend on the 'history' cFnonly through the value tional expectations Ex(. I 9n) of the process a t time n , i.e., through xn. In particular, yn and 7," are functions of xn alone. We present now a Markovian version of Theorems 1 and 2 , which we shall use in what follows (in Chapter VI), e.g., in our analysis of American options.

THEOREM 3 . Let g = g(z) be a B(R)-measurable function with E,g-(xk) x E E,k < N , and let S N ( Z ) = sup E x d z r ) ,

<

00,

(22)

TED$

where s g ( x ) = g ( x ) ; (c) the Markov time is optimal in the class m t : = SN("),

z E

E;

(d) the sequence yN = (y,",9 m ) m with~7," ~= S gale for each N 0.

>

(27)

N - ~ ~ ( Z is ~a )supermartin-

Proof. This result and its generalizations can be found in Chapter 2 of the monograph [441] devoted to the Markovian approach t o optimal stopping problems. Of course, Theorem 3 is also a consequence of Theorem 1 above, with the only exception: one must in any case study the structure of the operators Qg(x) and their iterations Q"g(x). (See 3 2.2 in [441] for a fuller account.) It follows from this theorem that the optimal stopping times in the, class 3nf = ( 7 : 0 < 7 < N } have the following structure for fixed N < 00. Let 0 < TZ < N , D f = { z : S N - ~ ( X ) = g(Z)}, (28) and let

c,N = E \ 0 .:

Chapter VI. Theory of Pricing. Discrete Time

532

In accordance with Theorem 3 the optimal time r t can be expressed as follows:

In other words, DON, OF,. . . , D$ = E is a sequence of stopping d o m a i n s while = 0 is a sequence of d o m a i n s of co,ntinued observotio~ns. Note that D ~ . .C. ~ D , N = E

C t ,C y ,. . . , C i

DON c

and

c,N 2 c,N 2 . . . 2 c,N

= 0.

Hence no observations are carried out for zo E D: and 70” = 0 in that case: whereas for 20 E C t one performs an observation and if z1 E D Y ,then the observations are terminated. If z1 E C F , then the next observation is performed, and so on. All observatioris stop at the terminal instant N (i.e.? DE = E ) .

Reninrk 2. It follows from Theorem 1 that the qualitative picture changes little if, ) by (21), we consider the prices in place of the price s ~ ( z defined

with discount arid observation, f e e s (here 0 < /3 < 1 and c(z) 3 0 for z E E : if r then the expression in [ ’1 is set to be equal to g(z)). Formula (25) holds good with Qg(z) = rriax(g(x),PTg(x) - c(x)),

and recursive equation (26) takes the following form: s N ( z ) = r n a x ( , ~ ( z ) , P T s N - l ( z) c(z)).

where

SO(Z) = g ( x ) . See

EXAMPLE 1. Let

E

[441; Chapter 21 for greater detail.

= (€1,~ 2 ,. . ) be Bernoulli variables,

P(€Z = 1) = P(&i = -1) =

Let

2, =

z

1 2

-

+ ( ~ +1 . . . + E ~ ) where , z E E = { 0 , & 1 , .. . }, and let

=

0,

2. American Hedge Pricing on Arbitrage-Free Markets

533

TON

If /3 = 1, t h m Qg(z) = g ( x ) for g(z) = z, z E E , and we can take = 0 as an optimal stopping tirnc. On the other hand, if 0 < /3 < 1. then for g ( z ) = z we see that Q"g(z) = z for z = 0 , 1 , 2 , . . . and Q n g ( z )= /3"z for L = -1, - 2 , . . . . Hence s ~ ( x=) max(z, PNz) in this case. The optimal time is now

(If zn E (-1, - 2 , . . . } for all 0 6 n 6 N , then we set Note that for 0 < 4 < 1 we have s N ( z ) t z+ = max(0, z)

as

TON

N

to be equal to N . )

+ oc.

6. We consider now the optimal stopping problem for infinite horizon ( N = m ) .

We set s(z) = sup EZg(zr),

(31)

r€LJJl*

where TR* = { T : 0 < T ( W ) < co} is the class of all finite Markov times. To state the corresponding result on the structure of the price s = ~ ( z )5, E E , and the optimal (or &-optimal)stopping times we recall the following definition.

DEFINITION (see, e.g., [441]). We call a function f < a,z E E , arid f(.) 3 T f ( z )

=

f(z)such that E z 1 f ( q ) 1 (3'4

an excessive function for the honiogeneous Markov process X = ( z r lSn) , Prc)n20;

x E E. If, in addition, f(z)3 g(z), then we call f = f ( z ) an ezces.sive majorant of the function g = g(z). Clearly, if f = f(x) is an excessive majorant of g = g ( z ) , then

The following theorem reveals the role of excessive rnajorants in optimal stopping problems for hornogeneous Markov processes

THEOREM 4. Let g = g ( z ) be a function such that EZ[supg-(zn)] <

30,

n

Then (a) the price s = s(z) is the smallest excessive majorant of g = g(z);

z E E.

Chapter VI. Theory of Pricing. Discrete Time

534

(= $rirsn(z)) and s(z) satisfies the equation

(b) s(z) = lim QTLg(x) n+Oo

(cf ( 3 3 ) ) ; (c) ifE,[sup/g(z,)/] < 00,z E E , then the time n

(35) is €-optimal for each

E

> 0, i.e., (36)

(d) let L

UP n

lg(zTL)jJ<

00

and

r* = inf{n

< 00)

i.e., r* = 7;; if P,(T*

> 0: s ( z n ) = g(z,)}, = 1, z

E E , then

T*

is an optimal time:

S(Z) = E,g(z,*), z E E; (e) if E is a finite set, then r* is an optimal time.

(37)

As regards the proof of this theorem and its applications, see [441; Chapter 21. In subsection 5 we discuss its applications to pricing of American options.

Remark 3 . By analogy with (21’) we could consider also the optimal stopping problem in the case with discounting (0 < 6 1) and observation fee c(z) 3 0. We set 7-1

4.)

= SUP E,

[

(31’)

Pkc(x4

P7g(ZT) k=O

where we take the supremum over the class

{

%JITp,c, =

7-1 T

p k c ( z k )< 00, z E E

E M*:E, k=O

>

and assume that g(z) 0. Under these assumptions the price s(x) is the smallest (p,c)-excessive majorant of g(z) (see [441; Chapter 2 ] ) , i.e., the smallest function f ( z ) such that f ( z ) g(z) and f k ) 3 PTf(.) - 4.). (33’) Moreover, s(x) = max(g(z),PTs(z) - c(z)) (34’) and 4x1 = n+QD lim Q ( p , c ) ~ ( z ) ,

>

where

Q(p,cp(x)= max(g(x), P T g ( z ) - c(z)).

2 . American Hedge Pricing on Arbitrage-Free Markets

535

EXAMPLE^. L e t x , = x + ( & l + . . . + & , ) , w h e r e z E E={0,*1, . . . } a n d & = ( is a Bernoulli sequence from Example 1. For 2 E E we set

where we consider the supremum over all stopping times 7 such that E z r such r we have Ezx: = z2 Ez7,

< 00. For

+

so that E,((zT/ - c.)

Hence

2

=

s(x) = C 2 2

E ~ )

+ E z ( j ~ -1 c ( z T (2).

(39)

(40)

+ Sup Ezg(2,),

where g(z) = 1x1 - ~ 1 x 1and ~ the supremum is taken over r such that E,r < cm. 1 . 1 . Since g(z) attains its maximum for ic = i--, It follows in the case when - I S 2c 2c an integer that 1 s(x) ex2 - . 4c

<

Passing to the limit as N 1 see that E z r c 6 < ~

for 1x1

5 2b.

< 2c1

-

+

+ co (and using the monotonic convergence theorem) we 03.

Hence rc satisfies (39), which shows that if 1x1

. It follows by (41) that

7c is

an optimal stopping time (for 1x1 6

< 2c1

-

1 -). 2c

Complete and Incomplete Markets. Supermartingale Characterization of Hedging Prices

1. We now return to the proof of formula (8) for hedge prices on incomplete markets exposed in 3 lc. As mentioned there, this proof is based on the following two facts: the supermartingale property of the sequence Y = ( Y n ) , < with ~ respect to each measure in the family 9 ( P ) and the optional decomposition for Y = (Y,),
Chapter VI. Theory of Pricing. Discrete Time

536

In the present section we consider the supermartingale property not only for the sequence Y = ( Y n ) , < defined ~ by formulas (12) in lc, but also for a more general sequence defined by formula (1) below, which makes it possible to study American hedging (see the reinark in 5 la). The optional decomposition for Y = ( Y n ) n G is ~ discussed in 5 2d. 2. Let (a, 9, ( $ n ) n G ~P) , be our underlying probability space and let (B,S ) be a market formed by a bank account B = (B,),
<

< <

THEOREM. The sequence Y = ( Y n ) n Gis~a supermartingale with respect to each measure in .9(P). Proof. Basically, this can be carried out along the same lines as the proof (in the preceding section) of the fact that the sequence y = ( y n ) n Gis~ a supermartingale. We can proceed as follows. We choose some (‘basic’) measure in the set 9 ( P ) . To avoid extra notation, let P be this measure (P is therefore assumed t o be a martingale measure). We shall verify that Y is a supermartingale with respect to P. If lp E Y(P), then we set

-

-

dP

dp,

z, = $ p , zn = ~, For

71

=

dP7,

-

where

-

-

= Pj

P,

Sn, and

P, = P I gn.

0 we sct 20= 1.

Let

Since

p

-

P, it follows that

-

I

-

P(Z7,-1 > 0) = P(Z,-1

for each

71

< N.

Setting nl,, = IJn

-

-

1, Mn =

n

> 0) = 1

-

C i%k, and Mo = 0 we have IC=l - -

AZ, = Zn-l AM,.

(3)

2. American Hedge Pricing on Arbitrage-Free Markets

537

From (3) we can see that

z,, q M ) , =

n,

n,

(1

=

+ AMk) =

is,,

(4)

k=l

k=1

&(M)

where is the stochastic exponential (see Chapter 11, § l a ) . It follows from the above that having chosen - P to be the ‘basic’ measure we can completely characterize P and its restrictions P,, n 6 N , by one of the sequences (%), or By Bayes’s formula ((4) in Chapter V, §3a), for each stopping time T (with respect to (9,)) and 71 6 N we have I

(Ed, (&%I.

1

Ep(fr 1 Lqn) = -;-Ep(fTZT

zrl

I 9,)

= Ep(&+l . . ‘ 6

f T

1 9,)

= Ep ( P I . . .Pn . P n + l ’ . .PTf-i-

I9 n )

Ep(fTZT I s n ) ,

(5)

__

wherepl=...=p,=l,Pk=Pllc, k > n , and Z ~ , = p l . . . &. It is interesting to note that defining a measure p by the equality dp = ZNdP we obtain

-

Clearly, P P. In view of our notation we can rewrite the definition of (1) as follows:

Y, = esssup ~ p ( f , Z , j .F,),

< <

where ess sup is taken over the class 9Jlr of stopping times T such that n 7 N , and over P-martingales in the class 2,”of positive martingales = (Z,+)~
z

9Jlrc9Jlt-l and

2Zf

C

%El,

which play an important role in the proof of the supermartingale property of the sequence Y = (Y,, g T L ) , < ~ . By the definition of the essential supremum (see, e.g., [75; Chapter 11);there exist sequences of times ~ ( 2 )and martingales belonging to the classes 9Jlf and T ,; respectively, such that

z(2)

esssup rEmn,N,ZEF/

E(frZr 1 9 k ) = @ l ~ E ( f r ( ~ ) IZg ~k )) ,i ~ a

(7)

Chapter VI. Theory of Pricing. Discrete Time

538

where lirn? is the limit of an increasing sequence. i

Hence, by the monotone convergence theorem

6

EP (fT-ZT I S k - 1 )

ess SUP TEm;,ZEY;

<

esssup

EP(fTZ

r€rnF- 1,Z€YE-1

I %-1)

= Yk-1,

which is just the required supermartingale property. Remark. We can extend Theorem 1 to the control with stopping case, with T

replaced by a functional

a k ngk

+

fT,

fT

where g = (go,g1,. . . ,g N ) is a sequence

k=l

of gn-measurable functions gn and a = ( a l ,. . . , C U N )is a control belonging to a sufficiently rich class of predictable sequences. (See 5 la.2 in this connection.)

5 2c.

Complete and Incomplete Markets. Main Formulas for Hedging Prices

1. By the main formula of the price @* ( f ~P ); of European hedging on a n arbitragefree ( B ,s)-market ( § l c ) ,

We proceed now to a more complex financial instrument, A m e r i c a n hedging on a ( B ,S)-market. We shall assume here that the set of martingale measures 9 ( P ) is non-empty. We have mentioned on several occasions that we must often (e.g., in the study of American options) consider in place of a single pay-off function f N an entire system of functions f = ( f n ) n G ~interpreted as follows. If the buyer exercises the option at time n, then the corresponding amount (payable by the writer) is described by the 9t,-measurable function f n = f n ( w ) . Of course, the writer must choose only strategies T of value X” = ( X E ) n c ~ satisfying the hedging condition

X: 3 f T (P-as.), (2) which guarantees his ability to meet the contract terms for each stopping time T = T ( W ) that can be chosen by the buyer to exercise the contract.

2. American Hedge Pricing on Arbitrage-Free Markets

539

2. For more precise formulations of the corresponding problems we introduce now several definitions. Following 3 2a we set M,N =

If

K

=

(7-

= .(w): n

< .(w) < N , w E n}.

(p = (pn)n
+ YnSn,

n

< N,

(3)

then we shall understand its self-financing property in the sense of the following balance condition: AX: = PnABn - ACn (4) where C = (Cn)+o is a non-negative process, Co = 0, and Cn is Sn-measurable. (Cf. the 'consumption' case in Chapter V, 3 la). To point out the dependence of the capital X z on the 'consumption' we shall denote it by X,"'c (as in 3 l c ) . DEFINITION1. By the upper price of American hedging (against a system of 9n-measurable payment functions f n , n N) we mean the quantity

<

c(f~; P) inf{z: 3 ( x , C )with X;lc =

= IG with

X,Xic 3 f r (P-as.) V r E

Imr}. (5)

DEFINITION2. We say that a strategy ( x , C ) is perfect if X,"" n 6 N and X>' = f~ (P-as.).

3 fn for each

THEOREM 1 ( M a i n formula for American hedging price). Assume that 9 ( P ) # 0 and let f = ( fn),
fn

sup E - - < m , F € 9 ( P ) P Bn

n < N

Then the upper price of American hedging is

Proof. We have already made necessary preparatory work for the proof of this formula. As in the case of European hedging we prove first that

Chapter VI Theory of Pricing Discretp Time

540

e(f~;

If the set of hedges is empty, then P) = 30 [by Definition 1). and (8) is obvious. Now let 7r be a sclf-financing hedge (with consumption C ) such that X;" =

x < 00. By analogy with (9), wc see from

5 l c that

for each r E m t . In particular,

Hence we obtain by assertion 2) of the lenirna in Chapter 11, 3 l c that the sequence

is a martingale with respect to each measure p E P ( P ) . Applying Doob's stopping theorem (see Chapter V, 3 3a) we see from (9) that

and therefore (8) holds. The proof of the reverse inequality to (8) is more complicated. It is sufficient to -

(3,r)and 'consumption' 2;such that the capital X'.'

find a portfolio .ii = the 'balance' conditions

the starting capital is

and (P-as.)

satisfies

2. American Hedge Pricing on Arbitrage-Frre Markpts

-

541

-

To this end we consider a sequence Y = ( Y r l ) , <with ~

-

(Z?),
It follows by the theorem in S2b that Y = is a supermartingale with respect to each measure E 9 ( P ) . O n the other hand it follows from 3 2d that the - Y = ( Y n ) 7 Lhas c ~ the optro,nal decomposition (holding P-as. for superinartingale each measure P E 9'(P)) I

with some predictable - sequence 7 =- (T,)ncN and non-negative sequence C = (&),
The value of the s t r a k g y (Z,

c)is -

-

and 'balance' condition (11) is satisfied in view of (15). Since Xh'C = B n K L .the I

capital XX"

has the following representation by (14):

We conclude from this forniula that 1) the initial capital of the strategy (Z, 2;) is defined by (12); I

I

2) (Z,C) is a hedgzng strategy in the following sense: X,T'c 3 or, equivalently, (13) is satisfied:

3) (Z, 2;) has the following rephcatzon property:

fiz

for

11


--

x2c = fN (P-a.s.).

The proof is complete and, on the way, we have established also the following result (cf. Theorem 2 in 5 lc).

Chapter VI. Theory of Pricing. Discrete Time

542

THEOREM 2 ( M a i n formulas f o r hedgzng, consumption, and capital). Under the assumptions of Theorem 1 there exists a hedge .Ti = (Ply) and consumption -

-

I

such that the dynamics of the corresponding capital X,"lC = ,&Bn

+ ynSn satisfies I

'balance' conditions (11). Moreover, Xf" satisfies (12), the dynamics of Xz" is determined by (18), the components of? = (Tn) and consumption C = (Cn)can be found from the optional decomposition (15), and = can be found from (16).

p (pn)

4. In connection with the assumption ' 9 ( P ) # 0'in Theorems 1 and 2 of this section and the assumption of the absence of arbitrage in Theorem 1 and 2 of 5 l b we can make the following observation (taking the hedging of option contracts as an example). The standard definition of the absence of arbitrage (see Definition 2 in Chapter V, § 2a) relates to some particular instant N , e.g., the maturity date of a European option. On the other hand, in dealing with American options one must consider in place of a single instant N an entire class Mf of stopping times 7 . For that reason, in place of the assumption ' 9 ( P ) # 0'in Theorems 1 and 2, it would be more logical to assume that the market is arbztrage-free in t h e strong sense (see Definition 3 in Chapter V, 3 2a). By the extended version of the Fzrst fundamental theorem (Chapter V , 3 le) the 'arbitrage-free' state (whether in the strong sense or not) is equivalent in our case of discrete time ( n N < 30) and finitely many stocks ( d < 30) to the condition

<

WP)# 0.

The reasons why we have actually put this condition in the form 9 ( P ) # 0 in the statement of the theorem are as follows. First, this property can often be verified (even in the continuous-time case); second, the term .arbitrapfree in the strong sense' is not widely accepted and the question of the equivalence of different interpretations of the 'absence of arbitrage' and the condition 9 ( P ) # 0 is not always that simple, especially, in the continuous-time case. (See Chapter VII, 9 3 l a , c in this connection.) 5. As regards the writer of an American option, he must first of all choose a strategy

( r , C )enabling him to meet the terms of the contract. This imposes the following

-

I

constraint on the capital XTic: the 'hedging condition' X,"" 3 fr must be satisfied (P-a.s.) for each 7 E Now, there arises the natural question on the particular instant T = T ( W ) at which it would be reasonable for the buyer to exercise the contract. We shall consider the case of a complete arbitrage-free market, which in the present context of discrete time ( n N < cm)and finitely many stocks ( d < a) is equivalent to the existence of a unique martingale measure is (the Second f u n d a mental theorem). Before answering this question we reformulate and somewhat improve on Theorems 1 and 2 in the case under consideration.

mf.

<

2. American Hedge Pricing on Arbitrage-Free Markets

543

-

THEOREM 3 . Let P be a unique martingale measure (Ip(P)[ = 1) and let f

=

fn

(f n ) n G N be a sequence of non-negative pay-off functions with E- - < 00,n 6 N . P Bn Then we have the following results. 1) The upper price is

-

2) There - exists a self-financing strategy ( E , C) such that the corresponding capital Xi;,' satisfies the 'balance' condition

AX:>'

=

,&AB, + Tn A

-

Ac,:

The dynamics of X'>" is described by the formulas

(m)

-

-

3) The components 7 = and C -= (C,) - can he - determined from the Doob decomposition for the supermartingale Y = (Yn,gnT P),GN with

which has the following form:

I

-

The components D = ( , & ) n G ~are defined by the relations

4) In the optimal stopping problem of finding

Chapter VI. Theory of Pricing. Discrete Time

544

the stopping time

0

< n < N : Y, = -

is optimal, i.e.,

Moreover,

-

I

and the sequence (Y,),
Proof. Assertions 1)-3) follow from Theorems 1 and 2. It should be noted only that, since the martingale measure is unique, we need not refer to optional decornpositions; it suffices to use directly the Doob decompositrori

-

-

-

t',

-

-

= MTL -

-

c,

(30)

of the superrnartingale Y = (Y,, 9,, P ) n < ~(see Chapter 111. 5 1b). By the extended version of the Second fundamental theorem (Chapter V, 54f) - the martingale M = ( M n ,9,, p) can be represented as follows:

Together with (30), this brings us to the required representation (24). As regards assertion 4), this is a special case of Corollaries 1 and 3 to Theorem 1 in 52a. 6. We now proceed directly to the question whether (judging on the basis of the information contained in the flow (9,)) it would be 'reasonable' of the buyer to choose the exercise time 7. Both buyer and writer operate with the understanding that the option price f N ; P ) defined by (19) is mutually acceptable. (See Chapter V, - 5 lb.4.)

e(

We consider now all strategies ( F ,

c)with initial capital X2c = c(fry; P) that --

<

bring about hedging, i.e., strategies such that X,"" 3 f n for n N . We denote f N ; P)). the class of these strategies by This class contains a strategy ( Z , of the mznzmiim value, such that

n(e( c)

for each (TI??) -

Y,, =

xT7"

~

Bn

,n

n(e(fj~; P ) ) . For it follows from the 'balance' condit,ions that <

-

fn

N , is a P-supermartingale majorizing -, n Brl

<

N : while it

2. American Hedge Pricing on Arbitrage-Free Markets

-

follows from assertion 4) in Theorem 3 that the sequence Yn,n

-

fn

P-supermartingale majorizing -,

Bn

Hence

n

545

< N , is the smallest

< N.

yn < Y, for n < N . Together with the relation

this proves inequalities (32). These inequalities show that for each stopping time

Clearly, the buyer must chose

7 such

-

7

we have

c) n(e(f N ; P)) the

that for no strategy (3, E

writer would get profits XT" - f7 > 0 with positive probability. In other words, the buyer may consider only the stopping times T such that -

fr =

x?"

(P-a.s.),

v(F,C)= n ( @ . ( f N ;PI).

(34)

All this justifies the following definition. DEFINITION. Stopping times

7

satisfying (34) are called rational exercise times.

THEOREM 4. Each stopping time T that solves the optimal stopping problem (26) (i.e., each stopping time satisfying (28)) is a rational exercise time. I

Proof. Let @,@)

=

JJ(@.(f N ; P)). Then, in view of the P-supermartingale property

-

of the sequence

Y,,=

x;>"

~

Bn

,n

< N , we see that

Hence

-

-

which, coupled with the property XF' (P-as.), i.e., T is a rational time.

2

f,,

proves that, in fact, X,""

= fr

Remark. It may be useful t o repeat a t this point that a solution of the optimal stopping problem (26) provides one with both value of the rational price @.(f~;P) and rational exercise time. Usually, one cannot find C(f N ; P) or 7 separately; they can be found only in t a n d e m , by solving (26).

Chapter VI. Theory of Pricing. Discrete Time

546

5 2d.

Optional Decomposition

1. For a filtered probability space (Q, 9, ( 9 , ) ,
<

-

-

-

I

-

where M(') = (MLp),9,,p) is a martingale, C(') - = ( C i p ) , 9 , - l , P ) is a non-

-

x

decreasing predictable process, Mip)= 0, and CAP) = 0. The components A d p )

-

I

and dP) in (1) depend on our choice of the measure P, which we emphasize by our notation. In the theorem following next we describe another decomposition of X , the so-called optional decomposition. It is remarkable for its universal nature: its components (see (2)) are the same for all is E 9 ( P ) .

THEOREM. If a process X is a supermartingale with respect to each martingale measure is E 9 ( P ) , then it has an (optional) decomposition

with predictable Rd-valued process y = ( ~ C = ( C n ) n
k ) k
non-decreasing process

Before turning to the proof we note - that there exists an essential difference between (1) and (2): the variables Cip) in (1) are 9,-1-rneasurablel whereas C, in (2) is 9,-measurable. It is just for this reason that (2) is called an optional decomposition. is Remark. In the present, discrete-time, case one says that a process C = ( C n ) , < ~ optional with respect to ( 9 , ) , , ~ when it is merely consistent with (or adapted to) the filtration ( 9 , ) , ( ~ i.e., , when Cn is g,-measurable, n 3 0. See Chapter 111,s5a and [250; Chapter I, lc] for the concept of optional process.

s

2. American Hedge Pricing on Arbitrage-Free Markets

547

First versions of the above theorem were established in [136] and [a811 for the continuous-time case (as already mentioned in 5 l c ) . Before long, several other papers devoted to both discrete and continuous time were published ([99], [163]-[165]), in which, in particular, the assumptions of [136] and [281] were weakened and various versions of proofs were suggested. The proof below follows the scheme of H. Follmer and Yu. M. Kabanov [163], [164], which is based on the idea of treating the yk in (2) as Lagrange multipliers in certain problems of optimization with constraints. (We shall also use several results of Chapter V, 3 2e in the proof). 2. In accordance with [163] and [164], we shall have established the decomposition (2) once we have shown that for each n = 1 , .. . , N the variables AX, = X, - X,-1 can be represented as the differences

Ax, = (m, AS,)

-

c,

(3)

where -yn is a 9,-1-measurable Rd-valued variable and c, is a non-negative %,-measurable variable. To obtain such a representation of AX, it is in fact sufficient to show that (under the above assumptions about X and S ) there exists a 9,-1-measurable Rd-valued variable y7,,such that

In this case, we can take (y,, AS,) - AX, as the required variable c,. We note also that if p E P ( P ) , then

If P, and

pn

are the restrictions of P and

Bayes’s formula (Chapter V, 3 3a),

where z , =

p

to 9 and 2, = __ d P 7 L , then by dP,

-. 2, 2,-1

Hence it is easy to see that (4) is a consequence of the following general result (where we set E = AX, and 77 = AS,), which is also of independent, ‘purely probabilistic’, interest.

Chapter VI. Theory of Pricing. Discrete Time

548

LEMMA.Let ( R , 9 , P) be a probability space and Jet [ and 71 = (7i1,. . . , q d ) be a real-valued variable and an Rd-valued variable in this space. Let $9 be a a-subalgebra of 9 and let 2 be the set of all random variables z > 0 such that, P-as.,

and E(zr/ 1 %) = 0 .

If 2 #

0 and

E ( 4 I g)< 0

for all z E 2 , then there exists a %measurable Rd-valued vector A* such that [

+ (A*,

TI)


(P-a.s.).

(12)

Proof. a) The idea of the proof is already transparent when 3 ' ' is a trivial usubalgebra, i.e., Y? = { 0 , R } . In this case the required vector A* is nonrandom and can be treated (as shown below) as a Lagrange multiplier in a certain optimization problem. If (8 is a nontrivial a-subalgebra, then we can carry out the same arguments for each w and, again, obtain a vector A* (not uniquely defined, in general), which depends on w . After that, the entire problem is reduced to the proof that A* can be chosen $!?-m,easurable. We recall that we encountered the same measurability problem in Chapter V, 3 212,in our proof of the extended version of the First fundamental theorem (in the discussion of the implications a') ===+ e) and e) 3 b)). We referred there to certain general results on the existence of a measurable selection (Lemma 2 in Chapter V, § 2e). The same techniques are applicable in our context for the proof of the existence of a %'-measurable selection A*. Referring to [163] and [164] for detail we note that there is no such problem of measurability in the case of a discrete space 0.

b) Thus, we shall assume that % = (0, R}. Let Q = Q ( d z , d y ) be the measure in R x Rd generated by the variables [ and 7 = (ql, . . . , q d ) , i.e., Q(dz,d z ) = P(< E d z , q E d y ) . Without loss of generality we can assume that the family of random variables 71, . . . ,,id is (P-a.s.) a linearly independent system, i.e., if alr/' . . . adqd = 0

+

+

(P-a.s.) for some coefficients a l , . . . a d , then a' = . . . = ad = 0. Indeed, $,. . . 7d enter (12) in a linear manner. If they were linearly dependent, then the problem could be reduced to another, with vector 7 of lower dimension.

2. American Hedge Pricing on Arbitrage-Free Markets

549

As in Chapter V, $2e, let L " ( Q ) be the relative interior of the closed convex hull L ( Q )of the topological support, K ( Q )of the measure Q. Let x' = (x,y), II: E Rd,y E Rd+', and let Z ( Q )be the family of Bore1 functions L = ~ ( d >) 0 such that EQZ = 1 and E Q ~ x ' / z < 00. Let = { P(Z) : P(Z) = EQZ'Z, z E Z ( Q ) }

@(a)

be the family of barycenters (of the measures dQ' = z d Q ) . By Lemma 1 in Chapter V, 52e we obtain that L " ( Q ) C @(a)(in fact, L " ( Q )= @(a)) and if 0 $! L " ( Q ) ,then there exists y' E Rd+' such that Q { x ' : ( y ' , ~ ' )2 0} = 1 and

Q{d:( y ' , ~ ' )> 0} > 0.

(13)

To prove the existence of a vector A* with property (12) we consider separately the following two cases: (i) 0 $! L " ( Q ) , (ii) 0 E L"(Q).

Case (i). By (13), there exist nnrribers y and y', . . . , yd such that (P-a.s.) YE

+ (ylvl + ' . + y d v d ) 3 0

(14)

'

and, with positive P-probability,

+ (y'vl + ' . . + y d v d ) > 0. +

(15)

+

We claim that y # 0. For if y = 0, then ~-' 7 1 ' ydqd 3 0 (P-as.). By hypothesis, there exists a martingale measure P P such that t..

N

Ep(7'~i'

+ . . . + y d v d ) = 0,

therefore y1711 + . . .,+ydvd = 0 (both p- and P - a s ) . Since T I ' , . . . , are assumed to be linearly independent, this means that y' = . . . = yd = 0, which, however, contradicts (15). Hence y # 0 and we see from relation (14) and our assumptions that Ep< ,< 0, Epq' = 0 that y < 0. i . Setting X i --7 -, z = 1 , .. . , d , we obtain by (14) the following inequality: Y

[

+ (A'??' + ' . + Advd) < 0, '

which proves (12) with A* = (A', . . . , A d ) in case (i). Ca.se (ii) is slightly more complicated. I t is in this case that we shall use the idea of [163] and [164] about Lagrange multipliers.

Chapter VI. Theory of Pricing. Discrete Time

550

z E Zo(Q) ===+ cpc(z) 6 0.

(16)

If 0 E L ” ( Q ) ,then, as already observed, L ” ( Q )C @ ( Q ) ,and by Chapter V, §2e we obtain that there exists zo E Zo(Q) such that cpc(z0) = 0. Hence, in case (ii),

which we can interpret as follows: in this case the value o f f * in the optzmization problem “find f* = sup cpc(z)” (18) z€zo(Q)

is equal to zero. Following [163] and [164] we now reformulate (18) as an optimization problem with constraints:

“find f *

sup cpc(z) under the additional constraint p q ( z ) = 0”.

(19)

ztz(Q)

According to the principles of variational calculus, for some non-zero vector A* (the Lagrange multiplier) the problem (19) is equivalent t o the following optimization problem: “find f * E sup (cpc(z) A*cp,(z))”.

+

~EZ(Q)

(For simplicity, we denote here and below the scalar product ( a , b) of vectors a and b by ab.) We shall now prove that the problems (19) and (20) are equivalent (this is interesting on its own, although it would suffice for our aims to show that, under the assumption (ii), we have

a t any rate, for some non-zero vector A*).

2. American Hedge Pricing on Arbitrage-Free Markets

551

Let

A

= {(z,y) E

R x Rd: x < cp&)

and y = cpl?(z)for some z E Z ( Q ) } .

This set is nonempty and convex. By assumption (ii), (0,O) $ A , therefore we can separate the origin and A by a hyperplane, i.e., there exists a nonzero vector X = ( X i , A,) E Iw x Rd S U C A that X1x

+ x2y 6 0

(22)

f o r all (2, y) in the closure of A . (See, e.g., [241; 0.11; note that we have in fact employed the idea of 'separation' also in case (i)). Note that A contains all points (z, 0) with negative IC. Hence X1 3 0 in (22). Further, if we assume that A 1 = 0, then we see from (21) that EQ(X2Y). = X2EQYz = X2cP77(z)

<0

(23)

for all z E Z(Q). Further, z E Z ( Q ) ,so that X2y 6 0 (Q-a.s.), i.e., X 2 ~ 76 0 (P-a.s.). Since 9 ( P ) # 0,it follows that EpX2q = 0 for some measure p in 9 ( P ) .

<

Combined with the property X 2 q 0 (p-a.s.) this shows that we have linear dependence (A277 = 0), which is ruled out by the above assumption. Hence X2 = 0. Consequently, if A1 = 0, then also X2 = 0, which contradicts the assumption that the vector (XI, X2) in (21) is not zero. Thus, X i > 0. A2 We now set A* = -. Then we see from (21) and the definition of A that for all

z

A1

E

Z(Q) and

E

> 0 we have (P&)

Passing to the limit as

E

-E)

+ 0 we obtain

which is equivalent to the inequality

We now observe that, obviously,

for each X E R d .

+ X*cps(.)

6 0.

552

Chapter VI. Theory of Pricing. Discrete Time

If (ii) holds, then the right-hand side of (25) is equal to zero, while the left-hand side vanishes for X = A*. Hence, under assumption (ii) we obtain

which establishes the equivalence of (19) and (20) and can be interpreted as follows: the Lagrange multiplier 'lifts' the constraint pa(z) = 0 in the optimization problem (19). We now turn back t o (24) or, equivalently, to the inequality

which can be rewritten as follows: EQZ('G

+ X*y) < 0,

z E Z(Q).

The space Z(Q)is 'sufficiently rich', therefore z + X * y required property (12) in the case of $4' = ( 0 ,a}.

< 0 (4-a.s.), which proves

In the general case, as already mentioned, one n u s t corisider in place of Q ( d x ,dy) regular conditional distributions

and prove the existence of X* = X*(w)for each w.At the final stage one must show that we can choose a %-measurable version of the function X* = X*(w).w E R, by using general results on measurable selections. such as Lemma 2 in Chapter V, fj2e. See the details in 11631 and [164].

3. Scheme of Series of ‘Large’ Arbitrage-Free Markets

and Asymptotic Arbitrage

5 3a.

One Model of ‘Large’ Financial Markets

1. We encountered already ‘large’ markets and the concept of asymptotic arbitrage in Chapter I, 5 Zd,in the discussion of the basic principles of Arbitrage pricing theory (APT) pioneered by S. Ross [412]. As in H. Markowitz’s theory ([331]-[33317 see also Chapter I, SZb), based on the analysis of the m e a n value and ,variance of the capital corresponding to various investment portfolios, asymptotic arbitrage in Ross’s theory is also defined in terms of these parameters. In the present section we define asymptotic arbitrage in a somewhat different manner, which is more consistent with the concept of arbitrage considered in Chapter V, § 2a and more adequate to the martingale approach permeating our entire presentation. 2. Developing further our initial model of a (B,S)-market formed by a bank account B = ( B k ) k G n arid a d-dimensional stock = ( S k ) k ~(where ~ S k = ( s i , .. , both defined on some filtered probability space

si))

s

(Q 9, ( . F k ) k < n , p), we assume now that we have a s c h e m e of series of n-markets

(B7L1 S n ) = (B?. S?)k
S:

=

n,d(n)

(S;;””, . . . , S,

), and each niarket is defined on a probability space

p7,, 972, ( q 3 k < k ( n ) , p7,) ‘of its own’. Here wc assume that n 3 1, 9;=

d ( n ) < 00.

{O;Rn},S n = ,P with k ( n ) < k(n)

(1) 00

and

Chapter VI. Theory of Pricing. Discrete Time

554

n

We are mainly interested in the cases when k ( n ) + 00 and (or) d ( n ) -+ 00. It is in this sense that we shall speak about 'large' markets.

00

as

4

Remark. In our considerations of the scheme of series of probability spaces the index n is the number of a series, while k plays the role of a t i m e parameter.

3. Let X"(") = ( X i ( n ) ) k g k ( n )be a capital corresponding to some self-financing portfolio ~ ( nin) a (Bn, Sn)-market. Recall that we agree to assume that the variables l3; are positive and 9E-lmeasurable. We explained at the end of § 2 b in Chapter V that we can assume without loss of generality that BZ = 1, which corresponds to a transition to discounted prices. In this case,

1=1 471)

where (-f,

AS?) = C

Ty>iASy>i.

i=l

DEFINITION 1. We shall say that a sequence of strategies T = (7r(n)),>lrealizes an asymptotic arbitrage in a scheme of series (B, S ) = { ( B n ,S n ) , n 3 1) of n-markets ( B n ,S n ) if l inm X l ( n ) = 0,

x;::; where 0

< c ( n ) -1 0 as n +

00

2 -c(n)

(pn-a.s.1,

(2) n 2 1,

(3)

and

Using the above-introduced concepts and notation we can say (slightly broadening the arguments of Chapter V, s2a) that the asymptotic arbitrage in APT considered in Chapter I, 3 2d occurs if there exists a subsequence (d)C ( n )and a

sequence of strategies ( ~ ( n 'such ) ) that X;(n') = 0 and EX:lZ,Y -+ 00, DXZ{Z,Y -+ 0 as n f + 00. Definition (4) of asymptotic arbitrage is more convenient and preferable from the 'martingale' standpoint to that of APT, which can be explained as follows. First, we can consider (4) as a natural generalization of our earlier definition of opportunities for arbitrage (Chapter V, 3 2a), and the latter, as we know from the First fundamental theorem, relates arbitrage theory and the theory of martingales and stochastic calculus in a straightforward way. Second, taking the definition (4) or a similar one (see [260], [26l], [273]) we can find effective criteria of the absence of asymptotic arbitrage. They include criteria in terms of such well-known concepts of the theory of stochastic processes as Hellinger integrals and Hellinger processes (see [250; Chapter V]).

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

555

(B,8)-market that is a collection { (B", S"), n 3 l} of n-markets is locally arbztrage-free if the market (Bn, Sn)is arbitrage-free (Chapter V, S2a) for each n 3 1. 4. DEFINITION 2 . We say that a

The central question in what follows is to find conditions ensuring that there is no asymptotic arbitrage (in the sense of Definition 1 above) on a locally arbitrage8)-market. free (B, The existence of asymptotic arbitrage as n + 00 can have various reasons: the growth in the number of shares ( d ( n ) + oo),the increase of the time interval ( k ( n ) + 00; see Example 2 in Chapter V, §2b), breaking down of the asymptotic equivalence of measures. It can also be brought about by a combination of these factors. In connection with the above-mentioned asymptotic equivalence of measures and the related asymptotic analogues of absolute continuity and singularity of sequences of probability measures it should be noted that their precise definitions involve the concepts of contiguity and complete a s y m p t o t i c separability (see [250; Chapter V], where one can find also criteria of these properties formulated in terms of Hellinger integrals and processes). The importance of these concepts in the problem of asymptotic arbitrage on large markets has been pointed out for the first time by Yu. M. Kabanov and D. 0. Kramkov [26l] who have introduced the concepts of asymptotic arbitrage of the first and the second kinds. (In accordance with the nomenclature in [26l] the asymptotic arbitrage of Definition 1 is of the first k i ~ i d . ) Later on, the theory of asymptotic arbitrage has been significantly developed by I. Klein and W. Schachermayer [273] and by Yu. M. Kabanov and D. 0. Kramkov in [260].

5 3b.

Criteria of the Absence of Asymptotic Arbitrage

(Xl("))kGk(rL)

1. Let X"(") = be the capital corresponding to some self-financing portfolio ~ ( nin) a (B", Sn)-market with BE = 1:

1=1

If Q" is a measure on (On, 9") such that Q" << Pn, then we obtain by Bayes's formula ((4) in Chapter V, fj3a) that (Qn-as.)

Q; = Q" Ic!Ft, and PE = P" 19;.(Here we assume that the dP;' expectation on the left-hand side of (2) is well defined.) where Z n

-

dQ'

~

Chapter VI. Theory of Pricing. Discrete Time

556

We assume that each of the n-markets ( B n , S n ) ,71 3 1, is arbitrage-free, and therefore. in accordance with the First fundamental theorem (Chapter V, 5 3 2b, e), the faniily of martingale measures 9( Pn) is nonempty. Let P l L E 9 ( P 7 L )and let ~ ( nbe) a strategy such that

EP, (X:;j)-

< 00.

(3)

Then we obtain by the lemma in Chapter 11, fj lc, that the sequence X"(n) = is a P7'-niartingale. Consequently, EP, ( X " ( n ) < 00 and (XP)k
k(n)

X l c n ) = Epa (Xl:; fork

1

1st)

(4)

< k ( n ) . Clearly, E P ~ / X ~ { ~ =~ EpIXl{:'I Z ~ ( ~ ) I < 00, and, by (2) and (4),

-

X ~ ( 7 0= Z Ep, ~ ( X k" ( n )Zk(lL) n 19;) (P"- and Pn-as.). Hence assumption (3) ensures in our case of discrete time k

(5)

< k ( n ) < 00 that

are martingales. (Cf. the lemma in Chapter V, 3 Sd.) In particular,

Each of these equivalent relations can be used in the search of conditions ensuring that the strategy ~ ( nis )arbitrage-free and the sequence of strategies 7r = (7r(n))+l is asymptotically a~bitrage-free.(Note that, in fact, we used just these relations in our proof of the sufficiency in the First f u n d a m e n t a l theorem, in Chapter V, 5 2c.) Remark. It is not necessary for formulas (4)-(7) to hold that F" Pn. It suffices that F" << PIL. We must nevertheless assume that Pn << if we want to deduce from ( 7 ) ,say, the absence of arbitrage. For a n explanation assume that = 0; Xl:; 3 0 (P"-a.s.), and A = {X"(n) > 0 ) . Then it is clear from (7) that we k(ni can say nothing on the probability Pn(A) of a set A if ZF(n, = 0 on this set. This N

Fn

= 0) = 1 from the assumption explains why we can deduce the equality Pn(X7i(7z) k(")

P"(Xl:T:;

3 0)

= 1 and the relation 0 = Ep, %(") "(")Zn k ( n ) only if P"(ZZ(,,

(By assertion f ) of the theorem in Chapter V. 5 Sa, this means that Pn Thus, the condition (in the definition) that the martingale measure alent to the nieasure P" ensures, in particular, that

> 0)

= 1.

<< P".)

Pn be equiv-

0

< Z&) < 00 (Pn-a.s.).

(8)

3. Scheme of Srries of ’Large’ Arbitrage-Free Markets and Asymptotic Arbitrage

557

2. For simplicity, we start our search of criteria of the absence of asymptotic arbitrage from the s t a t i m n r y case, when we have a (B, S)-market, (B, 8)= { (Bn, S 7 L )n, 3 l}, of the following structure. There exist a probability space (0,.9, ( 9 k ) ~ oP), , 9= .i%k and a ( d + 1)dimensional process ( B ,S) = ( B k , S,+)k20 of ,Fk_1-measurable variables BI, and 9k-measurable variables Sk = ( S i , . . . , Sf) such that each n-market has the structure (Bn,,S”)= ( B k , S k ) k G k ( n ) with k(,n) = n. Clearly (using the language of the scheme of series), we can assume that the market (B”,Sn)is defined on its own probability space (0,Sn,( . F r ) k G n , P”), which has the properties 9”= 9,, 9;= 9 k , k n, and Pn = P 19*. We can express this otherwise: in the ‘stationary case’ the number d ( n ) of shares is independent of n ( d ( n )= d ) arid the market (Bn+’,Sn+l) is an ‘extension’ of the market - (B”, - S”)for each n. } the family of sequences (Pk)k>l of martingale measures - Let P = { ( P k ) ~ 1 be Pk that have the property of compatibility: Pk+l 19k = Pk, 5 3 1. Given such sequence of measures (Pk)k21 we can consider the associated se-

v

<

I

I

quence

Z = ( Z k ) k 2 ~of Radon Nikodyni derivatives Z k

Let

=

dPk

-,

dPk

k 3 1, 20= 1.

and let

Although we do not assume the existence of a measure P on (0,,F) such that P k = p I 9 k , note that the sequence (Zk, 9 k ) k 2 0 is nevertheless a (positive) P-martingale thanks to the property Pk+1 I 9 k = Pk. Hence, by Doob’s convergence theorem (Chapter V, S3a) there exists (P-a.s.) limZk (= Zx)%and moreover, O<EZ,
THEOREM 1 (the stationary case). If (B,S) = {(B7’,S”),71 3 l} is a locally arbi trag-e-free market , then the condition

is necessary and sufficient, while the condition

is sufficient for the absence of asymptotic arbitrage.

558

Chapter VI. Theory of Pricing. Discrete Time

Proof. The sufficiency of condition (9) is relatively easy to establish; we present its proof below. (The sufficiency of (10) is a consequence of the implication (10) =+ (9).) The proof of the necessity is more complicated. It is based on several results on contiguity of probability measures and is presented in the next section, 3 3c (subsection 9). Let (P,),>l E P,and let T = ( ~ ( n ) ) , > l be a sequence of strategies on a (B,S)market, (B,S) = { ( B n , S n ) n, 3 1}, satisfying condition (3) in g3a. Then we have (3) and therefore also ( 6 ) , which takes the following form in our 'stationary case': .rr(n) = EZnX:(n), (11) X0 Hence, choosing E > 0 we obtain

EZ,X:(7L) = E Z n X : ( n ) ( I ( - c ( n ) 6 X z ( 7 1 < ) 0)

+ I ( 0 < x;(7L) < &) + q x p 3 &)) 3 -c(n) + EZ,X,"'"'I(Z, 3 &)I(X;(,) 3 3

-c(n)

+ &2P(X,"(") >,

3

-C(.)

+ &2[P(Xz(n)3 &)

E,

E)

2, >, F ) -

P(Zn,&)I,

and therefore

x;(71) + c ( n )+ &2P(& < &) If conditions (2) and (3) in

3 2 P ( X p 3 &).

(12)

5 7a are satisfied, then we see from (12) that

-

I

(because (P,),>l is a n arbitrary sequence in P. Hence it is clear that if (9) and, of course, (10) hold, then the sequence of strategies 7r = ( 7 r ( 7 1 ) ) , ) 1 with properties (2)-(4) in 3 3a cannot realize asymptotic arbitrage.

-

COROLLARY. Let (P71)71>1be some the condition P(Z,

sequence in P and let

-

-d P ,

Z , = lirn -. dP, > 0) = 1 ensures the absence of asymptotic arbitrage.

Then

3. We now proceed to the general case of arbitrage-free n-markets ( B n ,S n ) , n 3 1, defined on filtered probability spaces

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

559

'of their own', where 92(,= )9,.

-

If )P ,(;

is a martingale measure, PtCn, 1

N

P(:),

and Zz(n, =

";cn,

, then in a

dP;(n)

similar way to (12) we obtain

+c(n) +

E2

P"(Z&)

< &) 2 E2 pyx;;;;

2 E).

We set

THEOREM 2 . Let (B,S) = { ( B n , S n ) ,n 2 l} be a 'large' locally arbitrage-free market. Then the condition

is necessary and sufficient for the absence of asymptotic arbitrage.

Proof. The sufficiency of (15) follows from (14) as in the 'stationary' case. The proof of the necessity see in 3 3c.9.

5 3c.

Asymptotic Arbitrage and Contiguity

1. It is clear from our previous discussion of arbitrage theory in this chapter that the issue of the absolute continuity of probability measures plays an important role there. As will be obvious from what we write below, a crucial role in the theory of asymptotic arbitrage is that of the concept of contiguity of probability measures, one of important concepts used in asymptotic problems of mathematical statistics. To introduce this concept in a most natural way we consider the stationary case first (see 3 3b.2 for the definition and notation). Let ( p n ) T l > ibe a sequence of (compatible) martingale measures in P. Assume that there exists also a measure p on 9)such that p 1 -9,= p,, n 2 1. We recall (see Chapter V, 53a) that two measures, P and P, are said to be

(a,

locally equivalent

(is 'E P) if P,

N

P, n 3 1. It should be pointed out here that the

relation 'Zc P does n o t m e a n in general that It is clear from Theorem 1 in §3b that if P(2,

where 2, = limZ,, 2

-

P << P, P << P, or p

-

P.

> 0) = 1,

dPn -, then there is no asymptotic arbitrage. dP,

(1)

Chapter- VI. Theory of Pricing. Discrete Time

560

By the theorem in Chapter V, $ 3 a condition (1) is equivalent to the relation

- lofc

P). Thus, we can regard the relation P << is (under our assumption that P P << F (or (1) if' one likes it better) as just an additional (to p 'ofc P) restriction on thc behavior of the probability measures associated with n-markets that prevents asymptotic arbitrage as n + 00. In the case when there exists n o measure P on the probability space (a, 9, (9,),>1) with 9 = 9, such that isis, = P, n 3 1, the following definition is helpful for finding a counterpart t>othe assertion

v

I

DEFINITION1. Let Q" and Qn,n 1, be probability measures on measurable spaces ( E n ,E n ) . Then we say that the sequence (Qn,),21 is contiguous with respect (our notation is a ( Q " ) ) if for all sequences of sets An E Gn such to (Q"),,l that Q n ( A T L-)+ 0 as n -t 00 we have Qn(A7') + 0 as n + 00.

(an)

Remark 1. In the case when the sets (ITn, Gn) and the-measures Q" and @, are a independent on n ( ( E n ,E n ) = ( E ,&), Q" = Q , QTLE Q ) contiguity becomes the standard property of the absolute continuity Q << Q of the measures Q and Q on ( E , & ) . I

(an) (an)

-

THEOREM 1 (the stationary case). Let ( P l l ) be a sequence of martingale measures in Then

F.

P ( Z ~ > 0) = 1 u (P,)

-

a (is,).

(3)

The condition of contiguity ( P T L ) a ( P T L )ensures the absence of asymptotic arbi trag'e. If (P,) is a unique martingale sequence, then the condition (PI') a (P,) is necessary and siifficient for the absence of asymptotic arbitrage.

Proof. This is an immediate consequence of Theorem 1 in the previous section and Lemma 1 beIow, which contains several useful contiguity criteria. For a formulation we require additional definitions and notation. 2. Let Q and Q be two probability measures on the measurable space ( E , 8 ) ,

d Q 3- = =, d 6 and Z = d-. (Here we choose representatives Q = '2( Q + Q ) . Wc set 3 = =, dQ dQ d 3 and j of the Radori-Nikodyrn derivatives such that 2 + j = 2 . ) We recall that, in accordance with - the Lebesgue decomposition (see, e.g., [439; Chapter 111, 5 9]), we can represent Q as follows: -

a

= Ql

to,;

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

where

-

Q l ( A ) = EQZIA and

561

Q s ( A )= Q ( An (2= K ) ) .

Since Q ( 2 < m) = 1, it follows that Ql << Q and Q 2 1 Q . Hence 2 is just tlie Radon -Nikodym derivative of the absolutely continuoils coniponerit of Q with respect to the measure Q.

It is in this sense that we shall

use the notation Z = -. 2 . Let DEFINITION

(Y

E (0.1) and let

p ( Q , Q)=

dl

~

H ( Q , Q).

(6)

We call the quantity H ( a ;Q , Q) the Hellanger integral of 0rde.r (Y (of the rrieasures Q and Q); we call the qiiantity H ( Q , Q) simply the Hellinger integral, and p ( Q l Q ) is the Hellanger distance between Q and Q. It can be provcd (see, e.g., [250; Chapter IV, 5 la]) that H ( n ;Q , Q) is in fact independent of the dominating measure 0. This explains the widespread synibolic notation

E X A M P L E . Let Q = Q1

Gaussian measures on

and

x Q2 x . . . , Q

(R, %(EX))

= Q,x Q2

with densities

x "., where QI, and

Qk

are the

562

Chapter VI. Theory of Pricing. Discrete Time

Then

and therefore

LEMMA1. Let ( E n ,8") be measurable spaces endowed with probability measures dQn Q" and @, n >, 1. Then the following conditions are equivalent dQ" 2" = ---, and dQn a)

1 a" = -(Q" + Qn) : 2 -)

(6") a (Q"); _I

b) lirnlimQ"(~" < E ) = 0; EL0

c) lim

"

l i m Q n ( ~> N)= 0;

NTCC n

d ) limlirnH(cu; Q", Q") = 1. "LO 11

Proof. This can be found in [250; Chapter V, Lemma 1.61. The proof of equivalence (3) in Theorem 1 is an immediate consequence of the equivalence of a) and c) in Lemma 1 in the case of Q" = is, and Qn = P,:

dPn which exists P-as. , is equal to lim --, where 2 dPn The absence of asymptotic arbitrage under the assumption (P,) a (p,) is a consequence of the corollary to Theorem 1 in 5 3b and property ( 3 ) just proved.

3. Scheme of Series of 'Largc' Arbitrage-Free Markets and Asymptotic Arbitrage

563

3. The extension of Theorem 1 to the general (nonstationary) case proceeds without complications. Wc shall stick to the pattern put forward in 5 3b.3.

THEOREM 2 . Let

(p2(n))n>l be

a 'chain' of martingale measures on

I

markets (here Pg(nl P z ( n , ) . Then the condition (P;(",) absence of asymptotic arbitrage. N

-

(B", Sn)-

a (Pz(",) ensures the

Proof. Since conditions a) and c) in Lemma 1 are equivalent, it follows that

(P(;),

The required absence of asymptotic arbitrage in the case of the contiguity a (p:(,,) is a consequence of (11) and Theorem 2 in 5 3b.

4. So far, we have formulated conditions of the absence of asymptotic arbitrage in

terms of the asymptotic properties of the likelihood ratios Z;(",(Theorems 1 and 2 in 3b) or in terms of contiguity (Theorems 1 and 2 in the present section). - Lemma 1 above suggests a necessary and sufficient condition of the contiguity (Q") a (4") in terms of the asymptotic properties of the Hellinger integrals of order a E ( 0 , l ) :

It is often easy to analyze this integral and to establish in that way the absence of asymptotic arbitrage. (See examples in subsection 5.) A simplest particular case here is that of the direct product of measures. Namely, assume that

Chapter VI. ThPory of Pricing. Discrete Time

564

and lini lini H ( n :Q", 4.0 rL

6")= 1 +=+

k(n)

lini

CYJO

I

(1 - H ( t r :QE3QE)) = 0.

(14)

k= 1

Thus, in the case of a direct product we have k(n)

5. We now present several examples, which on t,he one hand show the efficiency of the criteria of the absence of asymptotic arbitrage based on Hellirigcr integrals of order N and, on the other hand, can clarify the argunients and conclusions of APT. the theory described in Chapter I, 5 2d. Examples 1 and 2 are particular cases of examples in [26O].

EXAMPLE 1 ('Large' stationary market with d ( n ) = 1 and k ( n ) = n ) . We consider the probability space (Q.9, P), where bt = { -1, l}" is the set of binary sequences z = ( z 1 , 2 2 , . . . ) ( x i = +1) and P is a measure such that P{z: (21,. . . , x T I ) }= 2-". Let q ( z ) = x i , ,i = 1 . 2 , . . . . Then E = (q, ~ 2 ;. .. ) is a sequence of Bernoulli random variables with P ( E ~= 51) = $. We assume that each (B", S'L)-market defiricd on (0,27",P") with 9'' ='971 and Pn = PIS" has the properties Bt = 1 and S" = (5'1,. . . , SrL) , with SI, = Sk-l(l + p k ) arid SO = 1, (16)

+

where P k = pk Q & k , o k > 0. ant1 max(--ak, ok-1) < pk < ok (cf. condition (2) in Chapter V, 5 Id). We can rewrite (16) as follows:

S k = sk-l(1

+pk(&k

-

bk))

(17)

where bk = -(pk/ok). (Note that jbk) < 1.) It follows from (17) and Theorem 2 in Chapter - V, 5-3f that - there exists - a unique martingale measure, which is a direct product P" = P;" x PT x . . x PE and has the following properties: the variables ~ 1~ ,2 . .. . , E" are independent with respect to this measure arid

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

565

it follows by (12) and (15) that

Hence it is easy to conclude that 00

(PTL)a

(iSn)

b: < 00

+==+ L=l

Recalling that bk =

(z)2<

Pk

--

and using Theorem 1 we see that the condition

0 C l

ce is necessary and sufficient for the absence of asymptotic arbitrage

k=l

on our 'large' stationary market.

EXAMPLE 2 ('Large' market with k ( n ) = 1 and d ( n ) = n ) . Consider a single-step S i , . . . , SF"-') with k = 0 model of (Bn,Sn)-markets, where Bn = ( B Z ) ,Sn = (S,", or 1, BZ = BY = 1, and

si1 - sz& + p i ) , -

s;>o,

(19)

where P" = FLg P2 = Pi

+ aOEOI

+ 0i(CiEO + C Z E i ) , 0, ci > 0, c? + E:

i 2 1.

We also assume that ai > = 1, and E = ( E O , ~ ., . . ) is a sequence of iritlependent Bernoulli random variables taking the values f1 with probabilities With an eye on the theories CAP211 and APT, we reconinlend to interpret S i , i 3 1, as thc price at time k of some stock that is traded on a 'large', 'global' market and S! as some general index of this market (for example, the S&P500 Index of the market of the 500 stocks covered by it; see Chapter I, 3 lb.6). GLOi Let /3~= -, i 3 1, and let

i.

00

PO bo = -00

and

b, = POPZ

-

0zG

Pz

'

Chapter VI. Theory of Pricing. Discrete Time

566

Using this notation, we obtain by (19)-(21) that

s,o = S,O(l+r r O ( € O and

sf= s; (1 + OiCi(E0

-

bo)

- bo)),

+ Oi.i(Ei

(23)

-

bi))

(24)

for i 3 1. Sticking to the above-described scheme of series of (Bn,S7')-markets we can assume that each of the markets is defined on a probability space (a, 9", P"), where S n = ~ ( € 0q, ,. . . , &"-I), Pn = P I S",and SZ and P are the same as in the above example. It is easy to see from (23) and (24) that, in the framework of this scheme, for each n 3 1 there exists (at least one) martingale measure. For we can consider a measure F" (having again the structure of a direct - product) such that the variables E O , ~ 1 ,. ..,&,-I are independent with respect to P", namely,

It is straightforward that

H ( a ;F", P")

=

7L-1

JJ

[ (I +

bi)a

+ (1

-

bi)a

2

i=O

I.

As in the previous example, we conclude that oz)

(P")

1

a (P") -e=+ b: < 00, i=O

so that, by Theorem 2, the condition

(in addition to

151<

1 and

I

1

'' < 1, i

1) ensures the absence of the

asymptotic arbitrage. (Compare formula (26) with (4) in Chapter I, 3 2c and (19) in Chapter I, 5 2d).

Remark 2. We point out one crucial distinction between the above examples. In the first of them, where k < n, there exists a unrgue martingale measure which

Pn,

00

C

allows 11s to claim (on the basis of Theorem 1) that the condition

-

k=l ..

is necessary and sufficient for the absence of asymptotic arbitrage.

3. Scheme of Series of ‘Large’ Arbitrage-Free Markets and Asymptotic Arbitrage

567

On the other hand, in the second example, where n is the index of a series, the measure p n is not unique for n 3 1. This explains why (26) is only a s u f i c i e n t condition of the absence of asymptotic arbitrage.

(As shown in [260], the condition

that is both necessary and sufficient has the form l h [ m i n ( l + b i , 1 - b i ) ]

> 0.)

i

EXAMPLE 3. We consider a stationary logarithmically Gaussian (Chapter V, 3 3c) . . , S n ) , where market (B,S) = {(El,, Sn),n 3 1) with El? = 1 and Sn = (So,S1,. Sk = Soehl+-+hk ,

so > 0.

(27)

+

We assume that h k = ,uk qq,, k 3 1, where ( ~ 1~,2 ,. ..) is a sequence of independent, normally distributed (JV(0,l)) random variables defined on a probability space (0,9, P) and crk > 0, k 3 1. Let 9,= ~ ( q. , . , E. ~ and ) let P, = PIS,, n 3 1. It was shown in Chapter V, § 3 c that if

I

then the sequence of prices ( S k ) k < , is a martingale with respect to the measure P, such that dP, = 2, dP,; moreover, Law(hk I P), = JY(j&, ak),where

It is now easy to see from formula (10) that

By (12) we obtain

so that it follows from Theorem 1 that the condition

ensures the absence of asymptotic arbitrage.

Chapter VI. Theory o f Pricing. Discrete Time

568

pk ak R e m a r k 3 . If += 0, i.e., Ok 2

then t,he initial probability measure P is a martingale measure for S = (Sk)k>~. Note that condition (30) is also necessary and sufficient for the relation (is") a (P"). Hence this condition- is necessary and sufficient in order that the sequence of measures (P") and (P7') be mutually contiguous; we denote this property by (P") d D (is"). 6 . We now discuss briefly the concept of complete asymptotic separability, a natural 'asymptotic' counterpart of the concept of singularity.

DEFINITION 3. Let Q" and Q",n 3 1, be sequences of probability measures on rncasurable spaces ( E T 1 , 8 " ) .Then we say that (Qn),>l and (Q"),>l have the property of complete asymptotic separability (and we write (Q")A ((2")) if there exists a subsequence ( n k ) , 7 i k 1' 30 as k t 30, such that for each k t,here exist a subset A7'k E &"k such that Q ' n k ( A n k ) 4 1 and Q n k ( A n k ) + 0 as k t 00. LEMMA2. Let ( E n ,G71) be measurable spaces endowed with probability measures Q" and

a)

12

Q91,

(F")

3 1. Then the following-properties are equivalent

A (P"));

b) limQn(bn 3 E ) = 0 for all E 7L

-

<

> 0;

c) GQ"(2"N ) = 0 for all N 1

> 0;

Proof. This can be found in [250; Chapter V, Lemma 1.91.

7. Our analysis in Examples 1-3 of the cases when there is no asymptotic arbitrage demonstrates the efficiency of criteria formulated in terms of the asymptotic properties of Hellinger integrals of order Q > 0. For filtered probability spaces (as in Examples 1 and 3 ) it can be useful to consider also the so-called Hellinger process: we can also formulate criteria of absolute

3. Scheme of Series of 'Large' Arbitrage-Free Markets a n d Asymptotic Arbitrage

569

continuity, continuity, arid other properties of probability measures with respect to one another in terms of such a process. We now present what can be regarded as an introduction into the range of issues related to Hellinger processes by means of a dascrete-trme example. (See [250; Chapters 1V and V] for greater detail.) Let P and P be probability measures on a filtered measurable space (0,9, (97,)n>o), where 90 -= (0, -R} and 9 = V .9n. Let P, = PI ,%, and Pn = P 1 ,Fnbe their restrictions to Sn, n 3 1. let Q = $(P lp), and let Q, = Q I 9 , . -

+

-,dP,, an

dP, -,dn and Pn = (here we set O/O = 0; dQ, dQ, bn-1 dn-1 = 0). recall that an = 0 if d n - l = 0 and = 0 if Using this notation we can express the Helliriger integral H,,(a) = H ( a ; P, P,) of order a as follows: H,,(cu)= ~ Q a E j h - " . (32)

We set

an

=

I

= __ , [j,,,=

an

We consider now the process Y ( a )= (Y,(a)),>o of the variables

Y,,(a)= a:

a p .

(33)

Let f a ( u , . u ) = ua2i1-". This function is downwards convex (for u 3 0 and u 3 0), and therefore we have EQ(Y~(Q) 1 cFT,,) Ym(cl.1 (34)

<

<

( Q - a x ) for m n by Jensen's inequality. Hence the sequence Y ( a ) = ( Y n ( a ).F ,, Q) is a (bounded) supermartingale, which, in view of the Doob decomposition (Chapter 11, 5 lb), can be represented as follows: Y , I ( Q I ) = M n ( n )- & ( a ) ; (35) where M ( a ) = ( M n ( a ) .Fn, , Q) is a martingale and A ( @ )= ( A n ( a )9,-1, , Q) is a nondecreasing predictable process with A0 ( a )= 0. The specific structure of Y ( a )(see ( 3 3 ) ) enables us to give the following representation of the predictable process A ( a ) : n

where h(a)= (hk(n))k20,ho(a) = 0, is some nondecreasing predictable process. In general, there is no canonical way to define this process. For instance, both processes

k=l

Chapter VI. Theory of Pricing. Discrete Time

570

and

- "a, < 1, satisfy the above requirements. where p a ( u , u)= ~ u + ( ~ - - Q ) u - - u ~ v 0~ < This can be proved by a direct verification of the fact that the process M ( a ) = ( M ~ ( - Q9 )n ,, Q) with

c n

M,(a) = Yn(Ct.1 +

Yk-l(-Q) & ( a )

(39)

k= 1

is a martingale for both of them. (See also [250; Chapter IV. 3 le].)

DEFINITION 4. By a Hellinger process of order a E ( 0 , l ) we mean an arbitrary predictable process h ( a ) = (hk(a))k>o,ho(a) = 0 such that the process M ( a ) = ( M k ( a ) ,9 k ,Q)k20 defined by (39) is a martingale. Remark 4. Pn =

zn ~

- loc Assume that P << P, i.e., P, << P, for n 3 0. Let Z, I

-

dPn and let = dP, ~

. Then the two processes hn(a)defined by (37) and (38), respectively,

zn-1

have the following representations:

and

Remark 5. We consider now the 'direct product- scheme' - by - setting R =El xE2 X . . . , 9 = g1 8 g2cg... , P = Q1 x Q 2 x . . . , and P = Q1 x Q2 x " ' , where Qi and Qi are probability measures on ( E i ,gi). Inthiscaseweset Sn= @ 1 @ . . . @ g nPn , = QlX...XQ,, and Pn = Q 1 X . . . X Q n . I

I

- loc

-

Then the property P << P is equivalent to the relation Q, << Q,, n 3 1, and we set d& Pn = -, dQ, Since Eppn = 1, the right-hand sides of (36) and (37) are the same. The ) corresponding Hellinger process h ( a ) = ( h n ( a ) with n

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

571

k=l

pn), then in our 'direct product case' we have H ~ ( c Y=) H n - ~ ( a ) H ( aQn; ; Qn) -

If & ( a ) = N ( a ;P,,

= Hn-l(a)

[I - (1- H ( a :Qn, Q n ) ) ] .

Using the notation (42) we obtain

A H n ( a )= - H ~ - ~ ( cAY~) ~ ( c Y ) . We already encountered difference equations of this kind in Chapter I1 (see formula (11) in fj l a ) . We expressed their solution in terms of stochastic ezponentials:

where

This agrees completely with the expansion n

H ( a ;Q k ,

Hn(a)= H o ( Q )

6,)

k=l

holding in the present case. The next results (relating to the 'scheme of series') reveal the role of the stochasasymptotic separability tic exponential in the issues of the contiguity and complete of sequences of probability measures ( P n ) , > 1 and (P"),>l on filtered measurable = gn and 9;= {B, On}. spaces (Qn,9n,( g F ) k < k ( n ) ) , n 2 1, with k(n) By analogy with (39) (but modifying our notation in an obvious way so as to fall in the scheme of series) we shall denote by

sn

Wn)

h;cn)(.)

EQ" (1 - ( ~ ; ) a ( ~ ~ ) IlS-t -a1 )

=

(42')

k= 1

a Hellinger process of order

CY

E ( 0 , l ) corresponding to the measures Pn and

P"

572

Chapter VI Theory of Pricing Discrete Time

LEMMA3 . The following conditions are equivalent: a) b)

(Pn) u (P‘~)); (Pi!) u (P;) arid

for each

E

> 0.

COROLLARY. Consider a stationary case, when P and P areprobabilitymeasures on a filtered measurable space (0,.4”, (9k)k2”). Let PN = P 19~ - and PN = P 19~. Then, for each N 3 1 we have the absolute continuity PAT<< PN if and only if I

I

LEMMA4. If

lim Pyh;(n)(;) n for each N

2 0, then (Pn) A

>N)=1

(45)

(P”).

COROLLAY. In the statzo,nary case both conditions P o l P o and

P ( h N ( + ) = m) = 1 are sufficient for the relation

(46)

PNIPN.

p << P and P i P take a particularly simple form if loc we assume additionally that P << P (i.e., P,, << P,, n 3 0). Namely, Criteria for the properties

dP, dP,, The proofs of Lemmas 3 and 4 and the corollaries to them can be found in [250; Chapter V, 3 2c and Chapter IV, $ 2 ~ 1 .

where a,, =

~

3. Scheme of Series of 'Large' Arbitrage-Frce Markets arid Asymptotic Arbitrage

573

8. We proceed now to the proof of the necessity of conditions (9) and (15) in Theorems 1 and 2 of 5 3h. It can be recomniended to this end to use the following generalization of the concept of contiguity introduced in [260] in connection with asymptotic arbitrage on incomplete markets. For each n 3 0 let ( E " :&',) be a measurable space with probability measure Q" and let = {Q"} be a fnmiky of probability measures Q'l in this space. (In what follows Q7' will always be a family 9 ( P " ) of ,martingale measures.) We associate with a family Q" = {Q"} of measures Qrl their u p p e r envelope supQ", the function of sets A E G" such that

We shall also denote by conv Q" the convex hull of Q".

-

DEFINITION 5. We say that a sequence of measures (Qn)7r21is con,tiguous t o the sequence of upper envelopes ( S U ~ Q " (we ) ~ write ~ ~ ~ ( Q T L ) a (supQn)) if for each 1% 3 1 such that (supQn)(A") + 0 we also have sequence of sets A'" E &7', Q n ( A n )+ 0.

+ 4").

for Q E conv Q", where = (Q The following result of [260] is a straightforward generalization of Lemma 1.

LEMMA5. The following conditions arc equivalent: a)

(a")a (supQ");

b) l i r n G ELO

c) lini

Ntm

inf

Q"(b"(Q)

QEcorivQn

lirrl 71

inf QEconvQn

< E)

= 0;

Qn (z"(Q) > N ) = 0;

-

d) lirnl@ sup H ( a ; Q , Q n ) = 1. do QEconvQn 9 . Proceeding to the direct proof of the necessity of (9) and (15) in Theorems 1 and 2 in ji3b we shall set Q" = P" (zP" ) and, as already mentioned, we take k(n)

-

the family Y(P") of all martingale measures PgCn, as Q", n

3 1.

Clearly, 9 ( P 7 ' ) is also equal to convQ" in this case, therefore condition c) in Lemnia 5 takes the following form:

Chapter VI. Theory of Pricing. Discrete Time

574

-

which is equivalent to the following condition (since P;(TL)

-

PE(n)):

Since (48) is precisely the same as (15) in 5 3b, it follows by the equivalence of conditions a) and c) in Lemma 5 that the 'sufficiency' part of Theorem 2 in § 3 b can be reformulated as follows: t h e condition

means the absence of asymptotic arbitrage. Hence (again, by the equivalence of a) and c) in Lemma 5) to prove the necessity in this theorem we must show that the absence of asymptotic arbitrage m e a n s (49). We shall carry out a proof by contradiction. Picking a subsequence if necessary we can assume that the sets An E S n k(n) satisfy the relation (SUP p;(,))

( A n )+ 0,

(50)

,(; + Q > 0. but ,P We claim that we have asymptotic arbitrage in this case. For a proof we consider the process

X,"

esssup

=

p:(n,EF(p;(n,)

E-

p;cn,

( I A IS;), ~

k

< k(n).

(51)

By the theorem in §2b, X n = ( X , " ) is a supermartingale with respect to each E 9'(P;(.,) and by the theorem in 5 2d it has an optional decompomeasure Pn k(n) sitiori I

k

j=1

where C," = 0, the Cr are 9,"-measurable, and the 7; are 9z-l-measurable. Based on (52)' we define (for each n 3 1) a strategy 7rn = ( P n , y n ) with ,On= n

( @ ~ ) , Q o and yn = (r;)k20

+ C ($, AS;). j=1 + (yg,S;) = X,"

such that its value X;" is equal to X g

To this end it suffices to choose and 7; such that (for simplicity we always set El; = 1), to take the 7," with k 3 1 as in the decomposition (52), and t o define P;;" from the condition of self-financing.

3. Scheme of Series of ‘Large’ Arbitrage-Free Markets a n d Asymptotic Arbitrage

For such strategies 7rn we clearly have Xc” = XF and as n -+ 30 we have

x;“

=

-

E-

SUP

p;(n,Ep(pliL(n,)

p;cn,

I A= ~-

+ Cz > 0 for all k

SUP

6 k(n),

P;(,,(An)

-+

0

p;( n ) @ ( p ; ( n ) )

by assumption (50). Thus, conditions (2) and (3) in Definition 1 in §3a hold for the strategy

(T”)n>l. To complete the proof it remains to observe that the strategy satisfies also condition (4) in the same Definition 1 because limPn(Xt(n) n

575

> 1) > limPn(XZ(,) n

= 1) = limPn(An) = cy n

7r

?r

=

= (?rn)n>l

> 0.

This proves the necessity in both Theorems 2 and 1 COROLLARY. To emphasize once again the importance of the concept of contiguity in the problems of asymptotic arbitrage in models of ‘large’ financial markets, we put Theorem 2 in the following (equivalent) form: the condition (Pzcn,)a (sup PTcnj)is necessary and sufficient for the absence of asymptotic arbitrage on a ‘large’ locally arbitrage-free market (B, S) = { ( B n ,Sn), n 3 1). Clearly, for a complete market this is the ‘standard’ condition of the contiguity

(PT(.)) of the family (P;(n))n21

-

a (FT(n))

(53)

of original probability measures Pn

-

k(n)

(P;(n))n21 of martingale measures Pn

k(n)

to the family

(which are unique for each n 3 1).

Remark 6. In Theorems 1 and 2 we formulated sufficient conditions of the absence of asymptotic - arbitrage in terms of the contiguity ( 5 3 ) to some ‘chain’of martingale measures (P;(,,). It is worth noting that, as shown in [260], the converse result is in fact also true: if we have asymptotic arbitrage, then there exists a ‘chain’ of martingale measures )n21 satisfying condition of contiguity ( 5 3 ) .

3 3d.

Some Issues of Approximation and Convergence in the Scheme of Series of Arbitrage-Free Markets

1. In the models of ‘large’ financial markets discussed in § $ 3 a , b , c we always assume that there exists a scheme of series of n-markets ( B n ,S n ) , n 2 1, each of which is arbitrage-free; after that we consider the question of the absence of asymptotic arbitrage. It should be noted that we make there no assumptions about the existence of a ‘limit’ market ( B ,S ) .

Chapter VI. Theory of Pricing. Discrete Time

576

In the present section we discuss the case when, besides .prelimit' n-markets

( B n ,S") defined on the probability spaces ( O n , S n ,P'')), n 3 1, we also have a 'limit' market (B,S) defined on (0,,9, P) and the (weak) convergence Law(Bn, S" I P")

+ Law(B, S 1 P)

(1)

as ri + m. We are mainly interested in understanding (under the assumption (1)) the issue of the convergence Law(Bn, ST' I Pn) --t Law(B, S I P), (2)

-

I

where PTLand P are some or other martingale measures in- the cl 9 ( P ) , respectively, and in finding appropriate measures P" and P such that the convergence (2) can be ensured. It seems timely to recall in this connection that we have already come across various methods of the construction of martingale measures, based, for example, on the Girsanov and the Esscher transformations. We also recall that the concept of mininial martingale measure discussed in Chapter V, 3 3d, has been developed (in works of H. Follmer and M. Schweizer, e.g., [167] and [429])just in connection with the question on martingale measures in 9 ( P " ) 'eligible' to enter the construction of chains of measures (Pn),>l used for financial calculations. (It seems appropriate to point out here that, actually, one uses martingale measures Pn and P rather then the origiria1~---physical, as one puts it sometimes-measures P" and P in, say, pricing of hedges or rational option pricing; see, e.g., the main pricing formula for European-type hedges on incomplete markets ((8) in 3 lc) or formula (20) in 3 4b.) I

2. Proceeding to a discussion of the above questions we should recall the following two 'classical' models of ( B ,S)-markets distinguished by their simplicity and popularity in the finances literature: the Cox-Ross-Rubinstein model (or the binomial model; see Chapter 11, l e ) in the discrete-time case and the Black-Merton-Scholes model (or the standard diffusion model, based on geometric Brownian motion; see Chapter 111, § 4b) in the continuous-time case.

As is well known, the first of them is (for a small time step A > 0) a satisfactory approximation to the second, and pricing (of standard options, say) based on the first model yields results close to those obtained for the second model of a ( B ,S ) market, in which

where W = (Wt)t>o is a standard Wiener process.

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

577

In line with the invariance principle well known in the theory of limit theorems (see, e.g., [39] and [250]), a Wiener process can be a result of a limit transition in a great variety of random walk schemes. It is not surprising therefore that, for instance, in binomial models of (B", Sn)-markets (with discrete time step A = l / n ) defined on some probability spaces (On, P,P") we have convergence to the (B,S)-model of Black--Merton-Scholes in the sense of (1) as n + 00, where P is a probability measure such that W = (Wt)t>ois a Wiener process with respect to P. Note that the existence of the convergence (1) for these two 'classical' models as well as for other models of financial markets is, as we see, only a part of the general problem of the convergence of ( B n ,Sn)-markets to some 'limit' (B,S)-market. An equally important question is that of the convergence as n + 00 of the distributions

L ~ ~ ( B ~I Pn) , s "+ Law(B, s I is),

(4)

where Pn and P are martingale (risk-neutral) measures for ( B n , S n ) -and ( B , S ) models, respectively. Here one must bear in mind the following aspects related to the completeness or incompleteness of the (arbitrage-free) markets in question. If ( B n ,S")-markets are complete, then (at any rate, if the Second f u n d a m e n t a l theorem holds; see Chapter V, 3 4a) each collection 9 ( P " ) of martingale measures contains a unique element, the question as to whether (4) holds under condition (1) is connected directly to the contiguity of the families of measures (P") and (P"), and it can be given a fairly complete answer in the framework of the stochastic invariance principle, which has been studied in detail, e.g., in [250; Chapter X, 3 31. However, the situation is more complicated if the arbitrage-free ( B n ,Sn)-markets in question are incomplete. In this case the sets of martingale measures 9 ( P " ) contain in general more - than one element and there arises a tricky problem of the choice of a chain (P"),>l of the corresponding martingale measures ensuring the convergence (4). In accordance with the definition of weak convergence we can reformulate (4) as the limit relation Ei;,f(B", S n ) + Epf(B, S ) (4') for continuous bounded functionals in the space of (cadkg) trajectories of processes under consideration (which, in the above context, are usually assumed to be martingales). dPn dP Note that if Z'l = __ and Z = -, then dP7l dP

and

578

Chapter VI. Theory of Pricing. Discrete Time

and therefore, clearly, (4') is most closely connected with the convergence Law(Bn, S n , Z n 1 P")

-+ Law(B, S, Z 1 P),

(5)

relating to the 'functional convergence' issues of the theory of limit theorems for stochastic process. See [39] and [250] for greater detail. It is worth noting that the convergence (4') follows from (5) if the family of random variables { Z n f ( B n ,S n ) ; n 3 1 ) is uniformly integrable, i.e.,

3. As an example we consider the question whether properties (1) and (2) hold for the 'prelimit' models of Cox-Ross-Rubinstein and the 'limit' model of BlackMerton-Scholes, which are both arbitrage-free and complete. Let ( O n , g n ,P") be a probability space, and assume that we have a binomial (B", S")-market with piecewise-constant trajectories (Chapter IV, 3 2a) defined in this space in accordance with the 'simple return' pattern (Chapter 11, § l a ) : for O < t < l , k = l , . . . ,n , a n d n > l w e h a v e

where the bank interest rates are

and the market stock returns are n

P

p>o.

PI,=-+[;,

n

In the homogeneous Cox-Ross-Rubinstein independent and identically distributed, with

Pn([F=$)=p

and

P"

(9)

models the variables

([ f = - -3

where a , b, p , and q are some positive constants, p

+ q = 1.

=q,

[y,

. . . , [,"are

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

By (9) and (10) we obtain

Ep,

1 ( ~ 2=) -(pb2 ~ + qa2) + 0 n

and

For sufficiently large n we have 1

+ pz > 0 and

Assume that

p b - qa = 0. Then, setting cr2 = pb2

and bearing in mind that p q ( b

+ a)'

+ ga2

= o2 and

we obtain

as n + 00. The following Lindeberg con,dition is clearly satisfied in the above case:

579

580

Chapter VI. Theory of Pricing. Discrete Time

Hence it follows by the functional Central limrt theorem ([250; Chapter VII, Theorem 5.41) that, (as Sz + SO) Law(St7); t

< 11 P") + Law(&

t

< 11 P),

(21)

where

and W = (Wt)tclis a Wiener process (with rcspect to the measure P). Thus, if B t + Bo, then Law(B,", S,"; t 6 1 I P")

+ Law(&, St; t 6

11 P ) ,

(23)

i.e., we have convergence ( I ) . We turn now to an analogue of - property (23) in the case when P" and P are replaced by martingale iricasiires P" and p.

[T,.. . ,t: are independent with respect to Pn, I

for k = 1,.. . , n, and the variables then the martingale condition

brings us to the relations

which are already equivalent to the condition

Taking account of the equality 6"

p"

=

+ Qn = 1, we find that (I ~

a f b

+--, ,1h ur +- pb '

(Cf. Example 2 in Chapter V, §3f, where the miqueness of the martingale measure Fn and the independence of ty,. . . , t; with respect to this measure were also established.)

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

581

Hence

where Z2 = ab. Note that the conditions p q = 1 and p b In other words, o2 = 5 ' and therefore

+

by

-

qa = 0 mean that ab = pb2

+ qa2.

The Lindeberg condition (20) withstands the replacement of the measure Pn P". Hence using the functional Ccntral limit theorem (as Son t So)we obtain Law(sp; t

< 1 1Fn) t Law(St; t < 1 IF),

(26)

where

-

-

-

W = (Wt)tGl is a Wicner process with respect to P, which is a unique martingale measure existing by Girsanov's theorem (Chapter 111, 3 3e):

-

In addition, Wt = Wt + cL - I-t. m Thus, we have the following result. ~

THEOREM. If the parameters a > 0, b > 0 , p > 0, and q > 0 in the 'prelimit' Cox-Ross-Rubinstein models defined by (6)-(10) satisfy the conditions pb - qa = 0 and p q = 1, then one has the convergence (21) and (26) to the 'limit' BlackMerton-Scholes models.

+

582

Chapter VI. Theory of Pricing. Discrete Time

4. We now modify slightly the 'prelimit' Cox-Ross-Rubinstein models, dropping the restrictive condition p b - qa = 0 so as to retain the right to use the functional limit theorem. To this end we assume that the variables ~2 are still defined by formulas (9) for odd k = 1 , 3 , .. . , whereas for even k = 2 , 4 , . . . we have

P; =

P

n +v;,

-

P

> 0,

where

Then for k = 2 , 4 , . . . we obtain

Hence, in view of (11)-(13) (for odd k ) ,

where u2 = p q ( a

+b y .

Thus, in our case of the inhomogeneous Cox-Ross-Rubinstein the same result as in the homogeneous case; namely, Law(Sr; t

< 1 I P")

+Law(&; t

< 11 P),

model we obtain

(27)

+

where S = (St)tGl is a process defined by ( 2 2 ) with u2 = p q ( a b ) 2 . Now let pn be martingale measures such that the variables p?,. . . , p: are independent again and

3. Scheme of Series of 'Large' Arbitrage-Fkee Markets and Asymptotic Arbitrage

for odd k , where

5; = 6" and g;

583

= Qn and

for even Ic with 52 and $t; defined (due to the martingale condition) by the formulas p ; = p^" and = with

F;

cn

a - -1 r-p 5" = -

g"

a+b b =a f b

fia+b' 1 r-p f i a f b '

+--

Then for all k = 1,.. . , n we have

ab Ei;"(P;)2

=n

+0

and

Consequently,

where 82 = ab, and in view of the Lindeberg condition (which is still satisfied) we see that (as SF --t So) Law(sF; t

< 1 I P)-+ Law(St; t < 1 I P),

(28)

where

and

^w = ($t)tG1

h

is a Wiener process with respect to the measure P such that

584

Chapter VI. Theory of Pricing. Discrete Time

+

We point out that, in general, ab # pq(a b ) 2 , antl therefore Z 2 # a2. Hence if the 'limiting' model has volatility a2 and the parameters a > 0, b > 0 , p > 0, and q > 0 satisfy the equalities p q = 1 and o2 = p q ( a b ) 2 , then we have the functional convergence (27); however the 'limit' volatility Z 2 in (28) can happen to be distinct from u2 (if we drop the 'restrictive' condition p b - aq = 0). This example of an inhomogeneous Cox-Ross-Rubinstein model shows that in choosing such models as approximations to the Black-Merton-Scholes model with parameters ( p ,0 2 )one must be careful in the selection of the parameters ( p , q , a , b) of the 'prelimit' models, since even if there is convergence (27) with respect to the original probability measure, the corresponding convergence (to the Black-MertonScholes models with parameters ( p , a 2 ) )with respect to martingale measures may well fail. This, in turn, means that the rational (hedging) prices Cy in the 'prelimit' models d o ,not necessarily converge to the (anticipated) price C1 in the 'limiting' model. Clearly, a similar situation can arise in the framework of other approximation schemes.

+

+

5 . In connection with the above cases of the convergence of the processes

S"

= (SF)t1 is nevertheless tight (see the monograph [250; Chapter X, $ 3 ] ) ,and therefore, in general, can have several limit measures (corresponding to different subsequences). One standard trick ensuring the uniqueness of the limit probability measure is that, besides the condition of the weak convergence of the laws Law(Sn 1 P") and the contiguity (P") Q (P"), we assume the weak convergence of the joint distributions -

<

Law(S7',2" I P"), where 2" = (Z?)t
dp"

are the densities of the dPr Py with respect to the Py (here and Py are the restrictions of pn and Pn to (.9;t")t
3. Scheme of Series of 'Large' Arbitrage-Free Markets arid Asymptotic Arbitrage

585

6. For illustrations of these results we turn again to the Cox-Ross-Rubinstein model (6)-(7) considered above, set for simplicity r = 0 and B$ = 1, and put this model in the form used already in the construction of the minimal martingale measure in Chapter V, 5 3d.7 (see also [392]). k

Let H z =

1pa.

If p h

-

qa = 0, then the sequence MrL= ( M ; ) k G n , where

1=1

k

M; =

C [i',

is a square integrable martingale with quadratic characteristic

1=1

k

( M n ) = ( ( M T 1 ) k ) k G n where ,

DEY = 0 2 k . o2 = ph2

(M")k =

+ qa2 (= ah).

1=1

Setting a t p/a2 we can write the Doob decomposition ( Chapter 11, fj l b ) of H7L= (H;;L)kcn in the following form (in terms of the increments):

AH?! k -

k

A ( M " ) k + AM;.

Assunie that b p < n2. Then the (minimal) measure

n

(29)

pn

such that

n

d P =

(1 - a; AM,.) dPn

k=l

(cf. formula (31) in Chapter V, 33d) is a prohahility measure and, moreover, a (unique) martingale measure for the sequence H'n = ( H ; ) k G n r as follows from Chapter V, s3d.7 and can be verified directly. We set

,

and represent Sr as follows:

Let also M = ( M t ) t G 1be a square integrable martingale (on some stochastic basis (0,9, ( 9 t ) t G 1 , P)) with quadratic characteristic ( M ) = ( ( M ) t ) t ~ let l, a = ( a t ) t < l be a predictablc process with a 2 . ( M ) 1< cc i.e., with

loa: d ( M ) t< x), 1

Chapter VI. Theory of Pricing. Discrete Time

586

and let

zt =

.(

-

s,'

,

a, d M J t

st = So.(H)t.

(34)

The structure of relations (31)-(34) suggests the conditions that one must impose on these processes to obtain the weak convergence Law(Sp, 2,"; t 6 1 I P")

+ Law(&,

Zt; t 6 I 1 P).

(35)

Thus, if a" = ( a y ) t G l , M n = (Mp)tGl and ( M n ) = ( ( M " ) t ) t < l are piecewiseconstant processes constructed from ( a E ) k G n , ( M 2 ) k G n ,and ( ( M n ) k ) k G n , then it is clearly sufficient for (35) that the distributions

'

'

k=l

k=l

/

converge to

and St + SO. Various conditions ensuring such convergence of martingales and stochastic integrals can be found, e.g., in [250; Chapter 1x1 and [254]. In particular, it follows from theorems 2.6 and 2.11 in [254] that the required convergence occurs once Law(Mp, a?; t 6 1 I P")

+ Law(Mt, at; t 6 1 I P)

and the jumps of martingales satisfy the following condition of uniform smallness: sup Ep, [sup IAMFI] n

< 00.

t
This clearly holds for the model (6)-(7) and (as seen from subsection 3) we can l process with Mt = aWt and a2 = pb2 qa2, take as the limit M = ( M t ) t ~the where W = (Wt)t
+

St=soexp{ ( p - $ ) t + n U i ) . We also see from (34) that

+

+

3. Scheme of Series of 'Large' Arbitrage-Free Markets and Asymptotic Arbitrage

587

In the present case we have the contiguity (F") a (P"), which can be proved as in Example 1 in 3c.5. Hence it follows from the above-cited generalized version of LeCam's third lemma that

where dP = Z l d P . Since, by Girsanov's theorem ( Chapter 111, 33e), the process

-

with Wt = Wt

+ -t

P .

is Wiener with respect to the measure

7 l

@

= (6&)tcl

F, it follows that

Hence Law(St; t

02

< 11P) = Law(Soe-Tt+uwt; t < 1 I P);

which has been already proved in subsection 3 by a direct application of the functional Central limit theorem (see (26)).

4. European Options on a Binomial (B, S)-Market

4a. Problems of Option Pricing 1. Following a long-established tradition in finance and in accordance with the nomeiiclaturc in Chapter I, 5 l a we distinguish two kinds of financial instruments and, in particular, securities, basic (primary) and d e r i m t i v e (secondary). We discussed basic securities (stock, bonds, currency) a t length in the first chapters. We considered various models of their dynamics, and the results of statistical analysis revealing such pheriorriena in the behavior of firiancial d a t a as the cluster p,roperty, fractality, long memory, and some other. We have also paid much attention to the theory underlying derivatives pricing, which is based upon the concept of arbitrage-free, ‘fair’ financial rnarket. As pointed out in Chapter I, not only speculators may develop a n interest in derivatives. Importantly, they play the role of hedging instruments, which protect one from financial risks incurred by uncertainties of the price development. For instance, if the current price of a share in corporation A is SO= 100 and the investor anticipates its growth (Sl= 120), then he can buy a share (at time n = 0) and then (at time 71 = 1) sell it pocketing the profit of S1 - SO = 120 - 100 = 20. Of course, the price can also drop (So = 100 -1 S1 = 80), and then selling the share will bring losses: S1 - So = 80 - 100 = -20. Thus, we have two possible patterns of development:

so = 100

/ \

s1= I2O

S1 = 80

4. European Options on a Binomial

(B. S)-Market

589

and .large’ gains (= 20, i.e., 20% of the price So = 100) are accompanied by a ‘large’ risk of losses (= -20, i.e., 20% of So = 100). Besides the above strategy (of buying and selling basic securities), an investor can also look a t the derizratives market. For example, he can buy a call optmionwith maturity time n = 1, which gives him a right (see Chapter I, 3 l c ) to buy a share at time n = 1 a t a price of K = 100, say. In this case, if So = 100 and S1 = 120, then the investor buys the share a t the price K = 100 (fixed in advance) and immediately sells it (on the so-called spot marketa) a t the market price S1 = 120, pocketing the profit of (5’1- K ) + = 20. Of course, for buying this option that grants the right of a purchase a t a fixed price ( K = 100) one pays certain premium t o the writer. Assume that this premium is cC1 = 10. If the stock moves up (So = 100 1‘ S1 = l20), then the buyer’s net profit is 10. On the other hand, if the prices drop (So = 100 J S1 = 8 0 ) , then the buyer does not exercise his option (there is no sense in the purchase of a share a t the price K = 100 when one can buy it a t the lower market price S1 = 8 0 ) , and takes losses in the amount of the premium (= 10) paid for the option. Hence, the purchase of a call option (which is just one kind of derivatives) reduces the risks of an investor (he can now lose only 10 in place of 20 ‘units’), but, of course, his potential profits have also slimmed (10 ’units’ in place of 20). Thus, we can say that the strategy of a speculator anticipating a rise of price (of a ‘bull’, using the term of Chapter I, 5 lc) is better insured against possible losses if it is based on the purchase of a call option than when it is based on transactions involving stock directly. We consider now a ’bear’, a speculator who anticipates a drop of prices (the price of a share, in our case). In principle, it is not against the rules on many markets to sell stock one does not have a t the moment. Assume that our ‘bear’ undertakes to sell stock a t time n = 1 and the corresponding exercise price is 100 ‘units’. If the price S 1 drops to 80, in line with the ‘bear’s forecasts, then he will buy a share a t this (market) price S1 = 80 and take a profit of 20 ’units’. However, if S1 = 120, then his losses will be 20 ‘units’. Again, both huge profits and huge losses are possible. Similarly to the ‘bull’, the ‘bear’ can turn to the derivatives market. For instance, he can buy a put option (for 10 ‘units’, with K = l o o ) , which gives him the right to sell the share (which he does not necessarily have a t the moment) at the price K = 100. Later on, if S1 = 80, then he buys a share a t this market price and sells it for K = 100 ‘units’, as stays written in the contract. Allowing for the premium, his net profit is 20 - 10 = 10 ‘units’ if prices actually drop (So= 100 J S1 = 80). On the other hand, if they rise, then the ‘bear’ takes losses aThe following ternis are often used in the finances literature (see, e.g., [50]): deals (contracts) providing for a delivery or certain actions a t some moment in the future (options, futures, forwards, etc.) are usually said t o be forward, while the ones including instant delivery are said to be spot.

Chapter VI. Theory of Pricing. Discrete Time

590

of 10 ‘units’. Hence, the purchase of a put option reduces speculative risks, but reduces accordingly also the possible speculative gains. The above figures are fairly arbitrary; nevertheless, our illustration of the ‘speculative’ and ‘protective’ functions of derivatives is adequate on the whole. 2. One cardinal issue relating to options is the value of the fair, rational premium paid for the purchase of an option contract. This is important for the buyer and also for the writer, who sells the derivative and must use the premium to provide for his ability to meet the terms of the contract. Of course, the writer of options is also interested in the assessment of the overall profits or losses from their flotation on the market. Note that a pricing theory for some or other derivatives must be built upon concrete models describing basic derivatives and general assumptions about the structure and the mechanisms of securities markets. The simplest in this respect is the ( B ,S)-market described by the binomial CRR-model of Cox-Ross-Rubinstein (see Chapter 11, l e ) . For all its simplicity, in its analysis one can more easily understand general principles and find examples of pricing based on the concept of ‘absence of arbitrage’. Our discussion will be built around options, both for their own sake and because many problems related to the markets of other derivatives can either be reformulated in the language of options or benefit by the well-developed techniques of option pricing, which is based on the plain and fruitful idea of hedging.

5 4b.

Rational Pricing and Hedging Strategies. General Pay-Off Functions

1. In the CRR-model of a (B,S)-market formed by two assets, a bank account B = (B,) and a stock S = (S,), one assumes that

where (p,) is a sequence of independent random variables taking two values, a and b, a < b, and T is the interest rate, -1 < a < r < b. Moreover, we assume that the sequence p = (p,) of variables defined on the underlying filtered probability space ( R , 9 , (9,), P) has the property P(p, = b ) = p

and

P(p, = a ) = q ,

+

where p q = 1, 0 < p , q < 1, and the pn are g,-measurable for each n. All the randomness in this model is due to the variables p,, therefore, we can take as the space R of elementary outcomes either the space RN = { ~ , b of} ~ finite sequences x = ( z l , x 2 , .. . , z ~ such ) that x n = a or 2 , = b (if n N) or the space R, = { a , b}OO of infinite sequences z = (zl, 2 2 , . . . ) with z, = a , b (if n E { 1 , 2 , . . . }). Then p,(x) = 2 , and since both spaces RN and R, are discrete,

<

4. European Options on a Binomial

(B, S)-Market

591

probability measures PN and P on the corresponding systems of Bore1 sets are completely defined by their finite-dimensional distributions P, = P n ( q , . . . , z,), where n N or n < 00.

<

If vb(z1,. . . , z n ) =

i

< n, then, obviously,

n

C Ib(xi) is the number of the components zi equal to b for i=l

pn($., .

, ,

-

, n ) = pVb(21r...rZn)QR-Vb(~l ,...,z~)

(2)

Putting this another way we can say that P, is equal to Q 8 . . . @ Q, the direct 71 times

product of the measures Q such that Q ( { b } ) = p and Q ( { a } ) = q , where p > 0, q > 0, and p q = 1. As shown in Chapter V, 5 I d the CRR-model is arbitrage-free and complete, while, by the First and the Second- fundamental theorems, for each n 3 1 there exists a unique martingale measure P, P,, which has the following simple structure (cf. (2)):

+

N

p n ( z l , . . . ,z,)= pvb(z1 ,...,zn)-n--L/b(31,...,2,) 4 where

r-a p=-b-a' I

-

(3)

-

b-r q = -b - a

(4)

We - see from (3) that P, like P,, has the structure of a direct product: P,=Q@...@Q, whereQ({b})=pandQ({a})=c. n times

2. We shall now consider European options of maturity N < c23 with pay-off (contingent claim) fN depending, in general, on all the variables SO,&, . . . , or, Chapter I, l c for various concepts reequivalently, on SO and p1,. . . , p ~ (See . lated to options.) We have already mentioned that both writer (issuer) and buyer of an option contract come across the central problem of the correct definition of the 'fair' ('rational') price of this contract. In accordance with Chapter V, 3 l b , if a market is complete and arbitrage-free (as is our binomial ( B ,,")-market), then as a f a i r price, one can reasonably regard the following quantity (the price of perfect European hedging):

s ~ ,

@ ( f ~ P) ; = inf{z 3 0: 3n such that X [

= z and

X;

= f~ (P-a.s.)},

(5)

where X" = ( X Z ) o G n 6 ~ is the value of the self-financing strategy n = (P,r).(See Chapter V, 5 l b and 5 l b in the present chapter for greater detail.) Moreover, one can calculate @ ( f ~P) ; by forniula (4) in 5 l b ; namely,

Chapter VI. Theory of Pricing. Discrete Time

592

-

where E is averaging with respect to the niartingale measiire PN. For the model (1) we have BN = & ( l + T ) ~ Hence . we obtain there

and, in principle, this gives one a complete answer to the question on the rational price of an option contract with pay-off f ~ . Remarkably, the option writer in this model, on taking the premium C ( ~ N P); from the buyer, can build a portfolio ?? = (p, of value X' = ( X i ) n Greplzcatzng ~ faithfully the pay-off f~ at the instant N . As mentioned, e.g., in 5 Ib, one standard way of finding this portfolio ii = (P,;i) is as follows. We consider the martingale M = (A&, Fn,PN)~GN, where

r)

I

I

I

S

In view of the '--representataon', there exists a predictable sequence B

-

Setting @I, = Mk

-

7= ( 7 z ) r c ~

__ we obtain (see Chapter V, 5 4b and 5 lb in the present ""

Bk chapter) a self-financing hedge .7i = (p,7)of value

such that

x:

= @ ( f r \ i ; P)

and we have the property of perfect hedging:

x;==f N . I

Since

it follows by (8) that

(9)

4. European Options on a Binomial ( B ,S)-Market

593

where 6t)and r k are connected by the equality

(PI , 9n, -PN),
of the variables

is a martingale. As will be clear from what follows, it makes sense to consider alongside the sequence p = (p,) also the sequence 6 = (6,) of the variables

6,

Pn

-

a

=-

b-a

Clearly,

and 9,= a ( p 1 :, . . . Since

-

Pk-r

&-p=-

b-a

it follows that, in addition to (8) and (ll),we have also the representation n

M , = M o + X a l -,4 6 ) m k(4

1

k=l

where the sequence m(’) = ( m f ) 9, k ,PN) of variables n

mi6)=

C(S, -

1=1

is a martingale and

6(J)= ( b - a ) @ . We sum up the above results as follows.

Chapter VI. Theory of Pricing. Discrete T i m e

594

THEOREM 1. 1) In the framework of the CRR-model (l),for each N and each 9N-measurable pay-off f N the fair price @.(f N ; P) can be described by the formula

-

where E is averaging with respect to the martingale measure P N . 2) There exists a perfect self-financing hedge % = 7)of value X’ = ( X z ) n G ~ such that = C ( f N ; P), =fN

(p,

xz

xf

and

-

3) The components p = rela tion

-

( P n ) n c ~and 7 = ( 5 n ) n G ~of the hedge % satisfy the

m

P n = M n - - - - , Sn Bn I

yn,

S

n 6 N , can be determined from the ‘--representation’ B martingale A4 = ( M n ,Sn,P N ) ~ < Nwith

where

-

(8) for the

3. As seen from the statement of this theorem, finding a perfect hedge 77 = is intimately connected with the representation of the martingale M = ( M n ,Sn,F N ) ~ Q Nin one of the equivalent forms (8), ( l l ) ,or (17). The next result concerns one interesting case of such a representation.

(p,?)

THEOREM 2 . Assume that a pay-off function is as follows:

+ .+

where g = g ( A N ) is a function of AN = 61 . . 6 ~ . Then the coefficients in the representation (17) are

where A0 = 0 and

4. European Options on a Binomial

( B ,S)-Market

595

-

Proof. First of all, we note that Mn = E(MN1 Tn). where M N = -.f N Since A M n = Gi6)Amn (6) , the value of 6.i')

=

$)(S,,

BN

..., 6,-1) can be ex-

pressed as follows:

In the set { w : ATL-l = x, 6, = l} we have

and

Hence we obtain there

which, in view of (25), brings us to the required representation ( 2 3 ) .

5 4c.

Rational Pricing and Hedging Strategies. Markovian Pay-Off Functions

1. We shall now assume that the pay-off function f N has the 'Markovian' form fN = f (SN), where f = f (x)is a nonnegative function of z 3 0. Let

be the value of a perfect hedge Z at time n; in particular,

Chapter VI. Theory of Pricing. Discrete Time

596

We set

c n

E',,(z;p) =

+

+U)n-k)c:pk(l

f (z(l b)"l

-

P y .

k=O

We have

with A, = 61

p=

+ . . . + S7, for 61, = ____ 'lC ', b-a -

and therefore

T - U

-. b-a Taking finally into account the equality

with

we obtain by (5) the following result.

THEOREM 1. The value X' = ( X z ) n < of ~ a perfect hedge .Ti in the CRR-model with Markovian pay-off function f N = f ( S N ) can be described by the formulas

xz = (1+ T)-"-"'FN-n(s,;p).

(7)

In particular, the rational option price is

-

2. We consider now the question of the structure of the perfect hedge ?? = (P,?). We set

Then, by Theorem 2 in the preceding section, the coefficients in the representa-

k=l

4. European Options on a Binomial ( B ,S)-Market

for

are predictable functions:

where

Settirig here x = Ai-1 arid taking account of the equality

and notation ( 3 ) ,we obtain by (11) that

Now, note that, by formulas (12) and (16) in §4b,

By (12) and (13) we obtain

As in 5 4b, we set

-

The strategy iF = (p,7)is self-financing. therefore

597

Chapter VI. Theory of Pricing. Discrete Time

598

so that

Bearing in mind ( 7 ) and ( 1 4 ) we obtain

We now sum up the results so obtained.

-

-

THEOREM 2. The components /3 = ( P i ) i < ~and T = ( 7 i ) i of ~ a~ perfect hedge ?i = in the CRR-model with f N = f(S,) can be defined by formulas ( 1 4 ) and (15).

(p,?)

COROLLARY 1 . The predictable functions pi and Ti depend on the 'past' only through the variable Si-1:

COROLLARY 2 . Let f = f (x) be a nondecreasing nonnegative function. Then it is a perfect hedge, then Ti 3 0 for all follows from (3) and (14) that if F = i N.

<

(p,r)

Remark. One can interpret negative Ti as borrowing stock (short-selling). Then Corollary 2 means that if f ( z ) is non-decreasing, then no short-selling is necessary for perfect hedging. 34d. Standard Call and Put Options 1. For a standard call option,

where N is the maturity time and K is the strike price. Of course, formulas for the rational price and perfect hedge obtained in the preceding section look more simple in this case.

4. European Options on a Binomial

(B, S)-Market

599

By definition ( 3 ) in $ 4 ~ :

Let

KO= Ko ( a , b, N ;

$)

be the smallest integer such that s o ( l + a ) N ( ~ ) K f > l

For f N = ( S N - K)'

K.

we shall denote for brevity @ ( f ~ P) ; by @.N (or by

CN (K)

if we want to underline the dependence on K ) . If KO > N , then F N ( S O ;=~0, and therefore the rational price C N is equal to zero (see (8), 5 4 ~ )this ; is understandable since we surely have SN < K in this case, and the purchase of an option brings no profits. We shall assume for that reason that KO 6 N . Then @.N = ( 1

+ T)-NFN(SO;p) N k=Ko

N

(3) k=Ko

We set

N k=j

Using this notation, we can formulate the result so obtained (which is originally due to J. C. Cox, S. A. Ross, and M. Rubinstein [SZ]) as follows.

THEOREM. The fair (rational) price of a standard European option with pay-off f ( S N ) = ( S N - K ) + is CN = SOB(K0, N ; p * )- K(1 f r ) - N B ( K o ,N ; p ) , where

If Ko

K

>N,

then @N = 0.

l+a

(6) (7)

Chapter VI. Theory of Pricing. Discrete Time

600

2. Since

( K - sN)+ the rational (fair) price

= (sN

-

K ) + - SN

+K,

PN of a put option can be defined by the formula

P N = E(1

= CN

+r)-N(K -

-

-

SN)+

+

E(1+ T ) - ~ S N K(1+

T ) - ~ .

(8)

Here E ( ~ + T ) - ~= S SO. N Hence we have the following identity (the cull-put purity):

PN = (CN- so + K ( 1 + r ) - N . 3. Let f =

f(S,) be

a pay-off function and let

=

(9)

BOEf ( s N ) be the correBN

sponding rational (fair) price. The following observation (see, e.g., [121], [122]) shows how one can use the rational prices

(SN- K ) + . K 3 0, in the search

corresponding to the pay-off

of the values of Cl;f’ for options with other types of pay-off functions f . Assume that the derivative of the pay-off function f = f ( z ) , z 3 0. can be expressed as an integral: f’(x) = [ p ( d y ) , where p = p ( d y ) is a finite measure (not necessarily of constant sign) on (R+, 9?(R+)). (If f ( y ) has a second derivative in the usual sense, then p ( d y ) = f / / ( y ) dy.) Then it is straightforward that

f ( x ) = f ( 0 ) + zf’(0) + and therefore

f(s,)

=

f ( o )t s N ~ ’ ( o ) +

1

J,% K)+!4dW -

co

(sN -

K ) + , u ( d ~ ) (P-a.s.1.

We consider now the expectation with respect to the martingale measure and obtain

so that, by formula (6) in

Note that if f ( z ) =

PN

5 lb,

(2-

K*)+, K , > 0, then p ( d K ) is concentrated at the as one would expect.

and C$) = point K , (i.e., p * ( d K ) = d{~,)(dz))

@I;“.’,

4. European Options on a Binomial

(B, S)-Market

601

4. Formulas (6) and (9) answer the question on the rataonal przce of put and call options. It is also of considerable practical interest to the option writer to know how to find a perfect hedge 71. = this can be carried out on the basis of formulas (15) and (14) in the preceding section. We do not analyze these formulas thoroughly here; we content ourselves with one simple example, the idea of which is borrowed from [162]. (See also [443] and a similar illustrative example a t the beginning of this chapter .)

(Pi?);

EXAMPLE. Consider two currencies, A and B. Let S, be the price of 100 units of A expressed in the units of B for n = 0 and 1. Let So = 150 and assume that the price S1 a t time n = 1 is expected to be either 180 (the currency A rises) or 90 (the currency A falls). We write s 1 = SO(l+ Pl), (11) and see that p1 can take two values, b = and a = -$, which correspond t o a rise or a drop in the cross rate of A. Let Bo = 1 (in the units of B) and let r = 0. Thus, we assume (for simplicity) that funds put into a bank account bring no profits and no interest on loans is taken. Let N = 1 and let f(S1) = (Sl - K ) + , where K = 150(B), i.e., K = 150 (units of B). Thus, if the currency h rises, then a buyer of a call option obtains 180 - 150 = 30 (units of B),whereas if the exchange rate falls, then f(S1) = 0. So far, nothing has been said on the probabilities of the events p1 = b and p1 = a. Assuming that A can rise or fall with probability we obtain that Ef(S1) = 30.8 = 15. A classical view, dating back t o the times of J. Bernoulli and C. Huygens (see, e.g., [186; pp. 397-4021), is that Ef(S1) = 15 (units of B) could be a reasonable price of such an option. It should be emphasized, however, that this quantity depends essentially on our assumption on the values of the probabilities p = P(pl = b ) and 1 - p = P(p1 = u ) . If p = then, as we see, Ef(S1) = 15(B). However, if p # then we obtain another value of Ef(S1). Taking into account that, in real life, one usually has no concluszve evadence an favor of some or other values of p , one understands that the classical approach to the calculation of rational prices is far from satisfactory.

a

a,

a,

Rational pricing theory exposed above works under the assumption that p , 0 < p < 1, is arbztrary. The value that must enter the (classical scheme of) calculations is - r-u p = -. b-a In our example o+g-2 p=--I

;+;

3'

Chapter VI. Theory of Pricing. Discrete Time

602

If N = 1, then the corresponding value KO = KO( a ,b, 1;S o / K ) is 1 for a = - 52,

b=

i , and SO= K = 150, therefore

by (3). Hence the buyer of an option must pay a premium @1of 20 units of B, which can now be regarded as the starting capital X o = 20 (B) of the writer (issuer), who invests it on the market. Werepresent now X Oin thestandard form (for (B,S)-markets) X O= ,LIoL?o +yoSo. Setting Bo = 1 and SO= 150 we can express the capital X o = 20(B) as follows: 20 = 0 . 150. That is, Po = 0 and 70= which can be deciphered as follows: the issuer puts 0 units of B into the bank account, while yo . SO = . 150 = 20 units B can be converted into the currency A. Assume that the issuer can also borrow money (from the bank account B , in the currency B), which, of course, should be paid back in the future. Then we can represent the initial capital X O = 20 (B) as the sum X O = -30 . 150, which ) (-30, f ) meaning that the issuer borrows corresponds to the portfolio ( P o , ~ o= 30 units of B and can now exchange f . 150 = 50 units of B for 33.33 units of A. Assume that, as an investor on our (B,5’)-market, the issuer chooses (PI, 71) = (/30,70). What does his portfolio bring a t the instant N = l? By the assumption that B1 = Bo = 1, there will be P1B1 = -30 units of B in the bank account. If the currency A rises (180B = lOOA), then 33.33 units of A will be worth 60 units of B,of which 30 is the outstanding debt. On paying it back the issuer still has 60 - 30 = 30 units of B,which he will pay to the buyer of the option to meet the conditions of the contract. On the other hand, if A falls, then 33.33 units of A will be worth 30 units of B, which should be paid back to the bank. Nothing must be paid t o the buyer (who has lost), so that the issuer ‘comes clean’. Our choice of the portfolio (/?l,n) = (-30. f ) may appear ad hoc. However, these are just the values suggested by the above theory. In fact, by formula (14) in 4c the ‘optimal’ value 71 = yl(S0) that is a component of a perfect hedge can be calculated as follows:

+&

&,

&

+5

4. European Options on a Binomial

The value

p1

(B, S)-Market

603

= Po can be defined from the condition

xo = Po + roso. Since X O = 20, 70 = $, and So = 150, it follows that = ,Do = -30, just as chosen above. It is clear from these arguments that the buyer’s net profit V ( S 1 ) (as a function of S1 for fixed K ) can be described by the formula

V(S1) = (Sl- K ) + - c1. The graph of this function is as follows: V(S1) f

Of course, the question on the writer’s profits also seems appropriate in this context. It is easy to see that there are none in the above example, for both cases of raising and falling currency A. So, how can there be someone ready to float options and other kinds of derivatives on financial markets? In fact, the situation is more complex because, first of all, one must take into account overheads, the broker’s commission, taxes, and the like, which, of course, increases the size of the premium calculated above. For instance, the commission can be regarded as the writer’s profit. Moreover, one must bear in mind that the writer has control (not necessarily for long) over the collected premiums and can use these funds for gaining some money for himself. Some may also wonder at the variety of kinds of options and other derivatives traded in the market. One possible explanation is that there are always some who expect currencies, stock prices, and the like to rise or fall. Hence there must exist someone who derives profit from this. This is what issuers are doing by floating call options (designed for ‘bulls’), put options (designed for ‘bears’), or their combinations with derivatives of other types.

Chapter VI. Theory of Pricing. Discrete Time

604

5 4e.

Option-Based Strategies (Combinations and Spreads)

1. In practice one can encounter most diversified kinds of options and their combinations. We listed several kinds of options (‘with aftereffect’, ‘Asian’, etc.) in Chapter I, glc. Some of them, due to their peculiarity and intricacy, are called ‘exotic options’ (see, e.g., [414]). We now list (and characterize) several popular strategies based on different k i n d s of options. Usually, one classifies these strategies between c o m b i n a t i o n s and spreads. The distinction is that combinations are made up from options of different kinds, while spreads include options of o n e kind. (See, e.g., [50] for a more thorough description, details of corresponding calculations; and a list of books touching upon financial engineering, which considerably relies upon option-based strategies.) 2. Combinations

Straddle is a combination of call and put options for the same stock with the same strike price K arid of the same maturity N . The buyer’s gain-and-loss function V ( S N )( = f (S,) - CN))for such a conibination is as follows:

V(S,)

=

I S ,

-

KI

-

CN.

Its graph is as in the chart below.

Strangle is a combination of call and put options of the same maturity N , but of different strike prices K1 and K2. The typical graph of the buyer’s gain-and-loss function V ( S N )is as follows:

4. European Options on a Binomial

(B; S)-Market

605

Analytically, V ( S N ) has the following expression:

Strap is a combination of one put option and two call options of the same maturity N , but, in general, of different strike prices K1 and K2. If K1 = K2 = K , then V ( S N )= 21SN

-

KII(SN > K ) + ISN

-

K / I ( S N < K ) - CN.

The graph depicting the behavior of V ( S N )is now asymmetric:

Strip is a combination of one call option and two put options of the same maturity N , but, in general, of different strike prices K1 and K2. The gain-and loss function is

and its graph has the following form:

606

Chapter VI. Theory of Pricing. Discrete Time

3. Spreads B u l l spread is the strategy of buying a call option with strike price K1 and selling a call option with (higher) strike price K2 > K1. Here

and the graph of this function is as follows:

It is reasonable to buy a bull spread when an investor anticipates a rise in prices (of some stock, say), but wants to reduce potential losses. However, this combination restricts also the potential gains. B e a r spread is the strategy of selling a call option with strike price K1 and buying a call option with strike price K2 > K1. For this combination

The corresponding graph is as follows:

4. European Options on a Binomial

K1

( B ,S)-Market

607

\

Such a combination makes sense if the investor anticipates a drop in prices, but wants to cap losses due to possible rises of stock. As regards other kinds of spreads, see [50], 3 24. 4. On securities markets one comes across other combinations besides the abovementioned ones, involving standard (call and put) options. For instance, there exists a strategy of buying options ( a derivative security) and stock (the underlying security) a t the same time. Investors choose such strategies in an attempt to insure against a drop in stock prices below certain level. If this occurs, then the investor who has bought a put option can sell his stock a t the (higher) strike price, rather than at the (lower) spot price. (See [50], $ 2 2 . )

5. American Options on a Binomial ( B ,S)-Market

§ 5a. American Option Pricing 1. For American options the main pricing questions (both in the discrete and the continuous-time cases) take the following form:

(i) what is the rational (fair, mutually acceptable) price of option contracts with fixed collection of pay-off functions? (ii) what is the rational time for exercising an option? (iii) what optimal hedging strategy of an option writer ensures his ability to meet the conditions of the contract?

In the present section, concerned with American option pricing in the discretetime case, we pay most attention to the first two questions, (i) and (ii). In principle, questions of type (iii), on particular hedging strategies, are answered by Theorems 2 and 3 in 5 2c. 2. We shall stick to the CRR-model of a (B,S)-market described in fj4b. That is, we assume that CB, = rB,-1 and AS, = p,S,-1; here p = (p,) is a sequence of independent identically distributed random variables such that P(p, = b ) = p and P(p, = a) = q , where -1 < a < r < b, p q = 1, 0 < p < 1. An additional assumption enabling us to simplify considerably the analysis that follows is that b=X-1 and a = X - ' - l (1)

+

for some X > 1. Thus, in place of two parameters, a and b, determining the evolution of the prices S,, n 3 1, we must fix a single parameter X > 1, which defines a and b by formulas (1). Clearly, in this case we have

5. American Options on a Binomial ( B ,S)-Market

609

where P(E, = 1) = P(pi = b ) = p and P ( E ~= -1) = P ( p i = a ) = q (cf. Chapter 11, § le). Assuming also that SO belongs to the set E = { A k , k = 0, &13 . . . }, we see that for each n 3 1 the state S, also belongs to E . The sequence S = ( S n ) n >described ~ by relation (2) with So E E is usually called a geometric r a n d o m walk over the set of states E = { A k k = O , = t l , . . . } (cf. Chapter 11, le). Let z E E and let P, = Law((S,),>O I P, SO= z) be the probability distribution of the sequence ( S n ) n 2with ~ respect to P under the assumption that So = IC. In accordance with the standard nomenclature of the theory of stochastic processes, we can say that the sequence S = ( S n ) n >with ~ family of probabilities p,, x E E , makes up a homogeneous Markov r a n d o m walk, or a homogeneous Markov process (with discrete times). Let T be the one-step transition operator, i.e., for a function g = g ( x ) on E we set T g ( z )= E,g(S1), x E E, (3) where E, is averaging with respect to the measure P., In our case (2) we have

3. The ( B ,S)-market described by the CRR-model is both arbitrage-free and complete, and the unique martingale measure P has the following properties:

-

P(E2 =

-

1) = P(p2 = b ) =

-

b-r P(&Z = -1) = P(p2 = a ) = _ _ , b-a

r-a ~

b-a’

(See, e.g., Chapter V, Id.) The ‘arbitrage-free martingale’ ideology of Chapter V requires that - all probabilistic calculations proceed with respect to the martingale measure P, rather than the original measure P. To avoid additional notation we shall assume that P = P from the very beginning, so that

p=-

r-a b-a

Bearing in mind (1) we see that

where a = (1

+

and

6-r b-a

q= -

(5)

Chapter VI. Theory of Pricing. Discrete Time

610

Let f = ( f o , f l , . . . ) be a system of pay-off functions defined, as usual, on a filtered probability space ( R , 9 , ( 9 n ) n > o , P) with 90= {a,O}. In accordance with §’La, let CV:l be the class of stopping times T such that n T N . Let be the class of finite stopping times such that r 3 rt. The buyer of an American option chooses himself the instant T of exercising it and obtains the amount of f r . If the contract is made a t time n = 0 and its e q z r y date is n = N , then the buyer of an American option can choose any r in the class !?X[ as the time of exercising. Of course, the writer must allow for the most unfavorable buyer’s choice of T and (in incomplete markets) the ‘Nature’s choice’ of a martingale measure among the possible ones. Thus, in accordance with 5 l a , the writer of an American option must opt for a strategy bringing about American hedging. Our ( B ,S)-market is complete, and the upper price of American hedging (see (5) in fj 2c)

< <

CV rl

-

@ . ~ (P)f ;= inf{y: 37r with X g = y, X: 3 f r (P-a.s.), V r E

m:},

(7)

which can be reasonably taken as the price of the American option in question, can be calculated by the formula

(see (19) in $ 2 ~ ) . Recall that E here is averaging with respect to the (martingale) measure P. 4. We put the accent in the above discussion on the questions how and under what conditions the seller of an option can meet the terms of the contract. - According to the general theory of American hedges (subsection a), the premium @ ~ ( fP) ; for the option contract defined by (8) is the smallest price a t which the writer can meet these terms. The buyer is aware of this fact, and the price P) is in this sense acceptable for both parties. Now, in accordance with the general theory, the writer can select a hedging portfolio ?i such that its value X T is a t least f r for each r E We now discuss the question how the buyer, who agrees t o pay the premium C,(f; P) for the contract, can choose the time of exercising it in the most rational way. Clearly, if it is shown a t time cr when X g > fo, then the writer obtains the net profit X g - fo after paying f o to the buyer. Hence the buyer should choose instant cr such that X g = fo. Such an instant actually exists and, as follows from Theorem 4 in 5 2c, it is the instant obtained in the course of the solution of the optimal stopping problem of finding the upper bound

e~(f;

mf.

I

TO”

5 . American Options on a Binomial (B, S)-Market

611

5. We see from (8) that finding the price @ . ~ (P) f ;reduces to the solution of the optimal stopping problem for the stochastic sequence fol fl, . . . , fN. In $ 3 5b, c, following mainly [443], we shall consider the standard call and put options with pay-off functions f n = (S, - K ) + and f n = ( K - S,)+ (or slightly more general functions fn = /3"(Sn - K ) + and fn = p n ( K - S,)+),respectiveIy. Together with the Markov property of the sequence S = (Sn)">O,this special form of the functions fn enables us to solve the optimal stopping problems in question using the 'Markovian version' of the theory of optimal stopping rules described in 3 2a.5.

3 5b.

Standard Call Option Pricing

1. We consider a standard call option with pay-off function that has the following form a t time n: f T A ( 2 ) = P"(z - K ) + , z E E, (1)

w h e r e O < ~ < l , E = { z = X k : k = O , ~, .l . . } , a n d X > l . For 0 < n 6 N we set

where Sn+k = SnXEn+l+"'+En+k It is worth noting that

and S, = x.

V,"(z) = (aP)"vf-"(x)

(3)

and, in accordance with relation (8) in § 5 a (and under the assumption So = x), the price in question is I

Q1"f;

P) = V , N ( X ) .

(4)

By Theorem 3 in 5 2a and our remark upon it,

hi'(.)= Q N g ( z ) ,

(5)

Q g ( z ) = max(g(s),Q P T g ( z ) ) .

(6)

where g(z) = (z - K ) + and

The optimal time

TON exists in the class J !R r and can be found by the formula = min(0 < n < N : v,-"(s,)= g ( S n ) > .

(7 )

Setting 0: = {x E E : V/-"(x) = g(z)} = { z E E : V,"(x) = ( a / 3 ) n g ( z ) } ,

(8)

Chapter VI. Theory of Pricing. Discrete Time

612

we see that 70”

= min(0

< n 6 N : S, E D:}.

(9)

Thus, given a sequence of stopping domains

and a sequence of continuation domains

with C,” = E \ D Z , we can formulate the following rule for the option buyer concerning the time of exercising the contract. If SO E DON, then :7 = 0; i.e., the buyer must agree a t once to the pay-off

(So - K ) + . On the other hand, if SO E C[ = E \ D: (which is a typical situation), then the buyer must wait for the next value, S1 and take the decision of whether 71” = 1 or 71” > 1 depending on whether 5’1 E DY or S1 E and so on. In our case of a standard call option it is easy to describe, in qualitative terms, the geometry of the sets 0: and C,”, 0 < n N , and therefore also the buyer’s strategy of choosing the exercise time.

Cp,

<

2. We see from (4)-(7) that finding the function V ~ ( Iand G )the instant 70” reduces to finding recursively the functions Vc(z)= Q n g ( z ) for n = 1 , 2 , . . . , N . 1. We claim that the case of [j = 1 is elementary. By assumption, 0 < In fact, the sequence (cPS,),>o is a martingale with respect to each measure P,, IG E E ; therefore (cy”(Sn - K)),>o is a submartingale and, by Jensen’s inequality for the convex function IG + ,+, the sequence (an(& - K)+)n20 is also a submart ingale. Hence for each Markov time 7 , 0 7 6 N , we have

<

<

E , ~ ( S~ I)+

<E,~~(sN

-

I)+

(12)

by Doob’s stopping theorem (Chapter V, S3a). This immediately shows that we can take 70” = N as an optimal stopping time in the problem sup E,aT(Sr

-

K)’,

T€m: and therefore if

SO= z, then

e,(f; P) = V?(IG)= Ea,

N

(SN - K)+.

(13)

Translating this into more practical terms we obtain the following result of R. Merton [346]:

if the discount factor 0 i s equal t o 1, t h e n t h e standard A m e r i c a n and European call options are ‘the same’. In addition, the value of VON(z) can be found by formula (6) in 5 4d.

5. American Options on a Binomial

3. We consider now a more interesting case of 0

(B, S)-Market

613

< P < 1.

THEOREM 1. For each N 3 0 there exists a sequence of numbers :z 0 6 n 6 N , such that

E E U {0},

0: = { z E E : x E [x:, x)),

c,"= { z E E : z E ( O J : ) } and

and

The rational price @ . ~ (P) f ;is equal t o V T ( S 0 )

Proof. We set for simplicity K = 1, set consecutively N = 1 , 2 , . . . , and analyze Q"g(x) for n 6 N . Let N = 1 and let z = 1 be the initial point, i.e., z = A'. By formula (4) in 5 5a, we see for the function g(z) = (z - 1)+ (bearing in mind the inequality X > 1) that

Hence the use of the operator Q 'rises' the value of g = g(x) a t the point z = 1 to Qg(1) 1aPp(A - 1). In a similar way, Tg(A) = p(X2 - 1) and

Hence if /3 6

Qg(A) = (A

-

A-1 then Q does not change the value of g(X) = A - 1. However, A-a'

~

A-1 then Q 'rises' the value of g(X) to /3(X A-ff' Now let z = X k , k > 1. Then

ifp>-

( El)-

1)max 1,P-

-

a ) (> X

-

1).

Chapter VI. Theory of Pricing. Discrete Time

614

Since X"1-

p) 3 1 - ap

it follows that Q g ( X k ) = g(Ak)

* P+1(1- p) 3 1 ==+Qg(X"')

-

ap,

= g(X"'),

which can be interpreted as follows: if X k E DA, then the points A"', A"', ... belong to DA. By (16) we obtain that for sufficiently large k the quantity X k belongs t o Db. Let x; = min{z E E : Qg(z) = g(z)}. Then it follows from the above that [xh,cm) DA. Moreover, we can say that [z;, m) = DA. Indeed, let x = X k with k 6 -1. Then T g ( z ) = 0, Qg(z) = 0, so that both instantaneous stopping at these points and continuation of observations (by one step) bring one no gains. For that reason we can ascribe the points II: = X k with ,k 6 -1 to the continuation domain Ci. Of course, this domain also contains x = Xo = 1 and the points x = X k with k > 1 such that Ak < x;. Thus, if N = 1, then To'

=

io I

for SO E [z;,cm);

for

SO

E (0, X;).

The analysis for N = 2 , 3 , . . . proceeds in a similar way; the only difference is that while we considered the action of Q on g ( x ) on the first step we shall now study its action on the functions Q g ( x ) ,Q'g(z), . . . , each of them downwards convex (as is g = g(x); see Fig. 58 below) and coinciding with g ( x ) for large x. These properties mean that for each N there exist x," such that DC = [x,",m). Not plunging any deeper into the detail of this simple analysis we note also that it is immediately clear for N = 2 that Df = DA and, therefore, xf = Considering the set Do2 = { z : Q 2 g ( x ) = g(x)} = {x: g ( x ) 3 T Q g ( x ) }we see that Do2 = [ x i , m ) for some xi. In addition, 0 = z; 6 x; xi, so that 70" has the following structure:

%A.

<

=;. {

o

for SO E

xi,^),

1 for SO E ( 0 , ~ ; ) ; ~1 E [ x f , m ) , 2

for SO E (o,zi), ~1 E ( 0 , x f ) .

Fig. 57 and 58 below give one a clear notion of the structure of the stopping domains 0," and the continuation domains C," and also of the functions Vdy(x6)= QN.9(x).

5 . American Options on a Binomial ( B ,S)-Market

7

01

1

2

To"

N-1

N

FIGURE 57. Call option. Stopping domains DON = [xf,m),Df = [xf,m), . . . = [0,m), and continuation domains N N co = (O,Z0N ), c, =(0,N q ), . . . ,c,N = 0 . The trajectory (So,S1,S2,. . . ) leaves the continuation domains at time

TO"

FIGURE 58. Graphs of the functions g(z) = (x - 1)+ and V ~ ( Z )= Q N g ( z ) for a discounted call option with pay-offs fn = p"g(x), 0 < P < 1 , 0 < TI < N , X > 1

615

Chapter VI. Theory of Pricing. Discrete Time

616

e~(f;

4. As follows from the above, finding the rational price P) for So = z reduces to finding the functions V,"(x) = Q N g ( x ) ,which can be calculated recursively by the formula

Q"g(x) = max(Qn-lg(x), aPT&"-'s(z)) = max(g(x), cypTQn-'g(x)).

(See also [441; 2.2.11.) Clearly, V , ( z ) 6 V,""(x),

(17)

and therefore there exists

By Theorem 4 in Chapter V, fj6a the function V * = V*(z)is the smallest apexcesszue majorant of the function g = g ( x ) , i.e., V* = V*(z)is the smallest function U = U ( x ) such that U ( x ) 3 g ( x ) and U ( z ) 3 (a,B)TU(x).In addition, V* = V * ( x )satisfies the equation

V * ( x )= m a x ( g ( 4 , (aP)TV*(z)),

(19)

following from (17) and (18). By the same theorem V ( x )is just the solution of the optimal stopping problem in the class L J X r = { T = T ( w ) : 0 6 T ( W ) < 00,w E i.e.,

a},

The knowledge of V* = V * ( x )can be interesting also in the following respect: V*(So)is equal at the same time to the rational price

cm(f;P)

= inf{y: 37r with

X{

= y,

X: 3 P'g(S,),

VT E mr}

(21)

for the system f = (fn) of pay-off functions f n ( z )= ~ " ( z- IT)+, n

0,

(22)

in the case when the buyer can choose an arbztrary stopping time 7 in the set "0" to exercise the contract. (The corresponding proof is similar to the proof of the theorem in 3 lc.) The analysis of options with exercise times in the set m r in place of the class 9Rf with finite N can appear unappealing from the practical point of view. However, one should take into account that the discount factor 0, 0 < ,B < l , does not allow optimal stopping times to be 'excessively large'. Incidentally, the analytzc solution of problems of type (20) is much easier than the solution of problems of the type ( 2 ) with finite N ; and if N is sufficiently large, then by V*(z) we can make a (probably, crude) guess about the values of the function VoN(x).

5. American Options on a Binomial ( B ,S)-Market

617

V*(x), which, as we know, must satisfy (19). Note that DON 2 DON+’; and therefore xf xf”. Hence three exists the limit lim xf = x*, and by (19) the function V * ( x )must have the following form:

5 . We now proceed to the function

<

N+CC

V*(Z)=

x 3 x*, { g(x)l ( f f P ) T V * ( x ) ,x < x*.

We point out that both ‘boundary point’ x* and function V*(x)are unknown and must be found here. Problems of this sort are referred to as free boundary or Stefan problems (see, e.g., [441]). In general, a solution (x*,V*(x)) to ( 2 3 ) is not necessarily unique and we may need some additional conditions to select the ‘right’ solution. We discuss below the arguments behind these additional conditions. Let C* = (O,x*)and let D* = [x*,m).Then the function V*(Z) in the domain C* satisfies the equation 94x1 = a P T d z ) , (24) i.e. (in view of (4) in 3 5a),

In accordarice with the general theory of difference equations (see, e.g., [174]) we shall seek the solutions of this equations in the form y ( x ) = Then y must be a root of the equation 1 =p[ffpxY+ff(l -p)x-”’].

Recall that p = from the condition

T--a ~

b-a’

b=X

-

1, and a =

(26)

- 1. We have in fact found p

(see (4) in Chapter V, 3 4d), i.e., the relation f f x p + a(1 - p ) x - l Comparing (26) and (27) we see that if and another root 7 2 such that

Since

it follows that 7 2

P

I-p-ax-1 xp A-ff

< 0.

= 1.

(27)

= 1, then (26) has the root y1 = 1

1,

Chapter VI. Theory of Pricing. Discrete Time

618

Hence if

p = 1, then the general solution to cp(z) = c1x

(26) has the following form:

+ c2xy2,

where 7 2 < 0. By the nature of the problem the required function and nondecreasing. Hence c2 = 0, and therefore V*(X)

=

{

z-1, CIX,

(29)

V*(z)must be nonnegative

z>x*, x < x*,

where x* and cl are still to be defined. By the submartingale property of the sequence (an(& - l)+)nao with respect to each measure P, x E E , we obtain that x* = 00,because if p = 1 for each point x 3 1, then aTg(z)> dz), and therefore it would certainly be 'more advantageous' to make a t least one observation than to stop immediately. Further, c1 3 1 in (30), for if cl < 1, then z* < 00. On the other hand, cl cannot be larger than 1 in view of the (additional) property that V * ( x ) is the smallest a-excessive majorant of g ( x ) , and the smallest function c l z with c1 3 1 clearly has the coefficient cl = 1. Thus, for ,6' = 1 and g(z) = (x - 1)+ we have V*(z) = sup Ezcu7g(Sr) = z, TEDIF

and there exists no optimal stopping time (in the class m?). However, for each E > 0 and each x E E we can find a finite stopping time rz,Esuch that Eza'"~Eg(S,,,)

3 V * ( z )- E .

(See [441; Chapter 31 for greater detail.) 6. Assume now that 0 < p < 1. Then equation (26) has two roots, 7 1 > 1 and < 0, such that the quantities y 1 = Arl and y2 = A r z that are the solutions of the quadratic equation ? 4 = P [ a m 2 + a ( 1 -dl (31)

72

can be expressed as follows:

where A = (cypp)-l and B = (1 - p ) p - l . Thus, if 0 < ,O < 1, then the general solution p ( x ) of (25) can be represented as the sum: p(z) = C l X Y l czz7Z. (33)

+

5. American Options o n a Binomial

(B, S)-Market

619

For the same reasons as in the case p = 1, the coefficient c2 must here be equal to zero and it, follows from ( 2 3 ) that the required function is V*(X)

=

{

x-1,

x>x*, c*xy1, x < x*,

(34)

where x* and c* are constants to be defined (see (40)-(43) below.) To find x* and c* we use the observation that V*(z) must be the smallest cup-excessive majorant of the function g ( x ) = (z - l ) + ,x E E . The following reasoning shows how, in the class of functions

with x,zE E . F > 0, and y1 > 1, one can find the smallest majorant of the function g ( x ) = (x - l ) + . (Then, of course, we must verify that the function so obtained is cyp -excessive.) To this end we point out that for sufficiently large C the function cpc(x) = cxy1 is knowingly larger than g ( x ) for all x t E . Hence it is clear from (35) how one can find the smallest majorant g ( x ) among the functions vc(z;Z) We now choose C sufficiently large so that pc(x) > g ( x ) for all z t E , and then make c smaller until, for some value of the function c p (z) ~ ~‘meets’ g ( x ) at some point 51. The functions cpc(x),ic E E , are convex, therefore, in principle, there can exist another point, T2 t E , such that Z2 > T I and c p ( Z ~ 2~) = g ( Z 2 ) . In our case the phase space E = { x = Ic = 0 , 51,.. . } is discrete. However, if we assume that X = 1 A, where A > 0 is small, then the distance between and T2 is also small and, moreover, these points ‘merge’ into one point, i?, as A J, 0. Clearly, i? is precisely the point in the interval (0, m ) where the graph of cpc(z) = cxY1 for some value of C touches the graph of g ( x ) = (z - l ) + ,x E (0, co). Obviously, C and E can be defined from the system of two equations

+

cpm m> =

which yields

Moreover, it is clear that the function

(36)

Chapter VI. Theory of Pricing. Discrete Time

620

is a good approximation to the least functions of the form (35) if A small. (Cf. formula (37) in Chapter VIII, fj 2a).

> 0 is sufficiently

Remark. We must point out condition (37) of ‘ s m o o t h fittzng’, which enters our discussion in a fairly natiiral way. This condition often plays the role of an addztzonal requirement in optimal stopping problems and enables one to single out the ‘right’ solution of a problem (see [441] and Chapter VIII in the present book.) The above-described qualztatrve method of finding the smallest majorant of g(z) brings one after more minute analysis (see [443]) to the following ‘optimal’ values z* and c* of 5 and C ensuring that the corresponding function V*(z) = (x;z*) is not only the smallest majorant of g(z), but also an a,B-excesszwe majorant:

vc*

(40)

c* = min(c;, c;),

.;

where

4, 1*2. = ( A l b g A q - 1 1 - 71[lOgx4-Y1, =p g x4

-

1

71 [l%A

(41) (42)

and

(here [y] is the integer part of y and E is defined in (38)). It is obvious that the function V*(z)so obtained is crp-excessive for z < z* because, by construction, a,BTV*(z)= V*(z)for such z. On the other hand, if z z*, then we can verify directly that aPTV*(z)6 V * ( z )once we take into account (40)-(43) and the fact that V*(rc)= 5 - 1 for such x.

7. By Theorem 4 in $2, our function V * ( x )is precisely equal to the supremum sup E,(ap)’g(S,),

and the instant

TErnJlom T*

= inf{n:

V*(S,)= g(S,)}

= inf{n:

is an optimal stopping time, provided that PZ(7* CIear 1y, PJT*

> N ) = P,

S, 2 x * }

< cm)= 1, z E E .

)

and since P ( E ~= 1) = p and P(E$= -1) = q , the probability on the right-hand side converges to zero as N + 00 for p 3 q .

5. American Options on a Binomial

(B; S)-Market

621

By (5) the inequality p 3 q is equivalent to the relation

a+b 2 Bearing in mirid that b = X - 1 and a = X-' each x < x*, provided that r>-.

(45) -

1 we see that P,(T*

< m) = 1 for

On the other hand, if x 3 x*, then P,(T* = 0) = 1 without regard to (46). Summing up we arrive at the following result.

THEOREM 2. Assume that 0 < /3 < 1 and that (46) holds. Then the rational price P) of an American call option with pay-offs f , = P"(S, - l ) + ,n 3 0, is described by the forrniila E d f ; P) = V * ( S o ) , where

ew(f;

V*(SO) =

{ c*s;', so -

l'

so < x*, so

and the constants c* and x* can be found by (40)-(43). The optimal time for exercising the option is r* = inf{n: S, 3 x*}. In addition,

V*(So)= E~,(cYP)~*(S,*- 1)'.

5 5c.

Standard Put Option Pricing

1. The pay-off functions for a standard p u t option are as follows:

fn(Y) = P"(K - Y)+, x E E, where 0 < fl 6 1, E = { y = A k : k = O , + l , . . . } , X > 1. By analogy with the preceding section, we set

(1)

and V*[y) =

SUP

Ey(a/3)T(K - ST)+

T€DtF

(3)

These values are of interest because V,N(Y) = @."f:

PI,

Y = so,

(4)

so,

(5)

and V*(Y)

=I:

E d f ; PI,

c ~ ( f ; e,(f;

Y =

where the prices P) and P) for the system f = ( f n ) n 2 of ~ functions fn = fn(y) defined by (1) are as in formula ( 7 ) in 5 5a and formula (21) in 5 5b, respectively.

Chapter VI. Theory of Pricing. Discrete Time

622

THEOREM 1. For each N 3 0 there exists a sequence y,", in E U {+m} such that

0 6n

< N; with values

and

and

The rational price E N (j; P) is equal to V , (SO).

Proof. This is similar to the case of call options discussed in 5 5b; the proof is based on the analysis of the subset of points y E E at which the operators Q" increase the value of the function g(y). It is worth noting that, of course, the operator Q raises the value of g(y) at the point y = K (we have assumed above for simplicity that K = I) and Qg(y) = g(y) = 0 for y > K . Hence these values of y t E can be ascribed both to the stopping domains and to the continuation domains. As seen from (6) and ( 7 ) , we have actually put these points in the continuation domains. 2. We consider now the question of finding the function V*(y)

(= N-02 lini

Vt(y)),

the quantity y* = lim y f , and the optimal time r* such that N+x,

V*(y) = E y ( ~ P ) T( *K - ST*)+

(10)

(for simplicity we set, K = 1 again). Let C* = (v*,oo)and let D* = (O,y*]. As in §5b, we see that the function V*(y) in the domain C* is a solution of the equation

d Y ) = 4 [ P j ^ ( W + (1 Its general solution is clyyl in 5 5b).

+ c2yY2

where y1

-d.(y >

1 and

72

< 0 (see (31) and ( 3 2 )

5. American Options on a Binomial ( B ,S)-Market

01

1

2

N-l

To”

623

N

FIGURE 59. Put option. Stopping domains N N N DF = (o,Yf],. . . , D N - ~= (O,YN-~]>DN= (o,CO), and continuation domains N N C,”=(y/,”,CO) , . . . , C”-1=(YN-l,CO),CN =la. The trajectory ( S OS1, , Sz, . . . ) leaves the continuation domains at time #

0 .

,)-z

01

>

,

x-1 1

,

x

x2

x3

y=x

k

FIGURE60. Graphs of the functions g(y) = (1 - y)+ and V t ( y ) = QNg(y) for a put option with payoffs fn = Dng(y), where 0 < ,O 1 , 0 6 n 6 N ,and A > 1

<

Chapter VI. Theory of Pricing. Discrete Time

624

Since V*(y) of functions

< 1, it follows that c 1 = 0, so that we must seek V*(y) in the class

where the ‘optirnal’ values c* and y * of C and 5 are to be determined on the basis of the above-mentioned (3 5b.6) additional conditions that V*(y) = (y; y*) must be the smallest ap-excessive majorant of g(y) = (1 - y)+. Following the scheme (exposed in 55b) of finding, for small A = 1 - X > 0 , approximations C and y” to the parameters c* and y*, we see that they can be determined from the system of equations

vc*

Pdid

=g(9,

Solving it we obtain

Once we know the values and Ecorresponding t o the ‘limiting’ case (A J, 1) we can find (see 14431) the values of y * and c* in the initial, ‘prelimit’ scheme (with X > 1) by the formulas

and

(17)

The fact that the so-obtained smallest majorant V*(y) of g(y) excessive can be established by a n immediate verification. We finally observe that the condition a f b r < 2

(1 - y)+ is

cwp-

XfA-l

- ~-

2

<

(cf. (45) in 5 5b) ensures that PY(7*< a)= 1 for y E E and T * = inf{n: S, y*}. (If y y*, then PY(7* = 0) = 1.) Thus, if condition (14) holds, then T * is optimal in the following sense: property (10) holds for all y € E .

<

5. American Options on a Binomial

(B, S)-Market

625

<

THEOREM 2. Assume that 0 < p 1 and condition (18) holds. Then the rational price P) of an American put option with pay-oEs f n = p"(1 - Sn)+,71 3 0, is defined by the formula

em(f;

?co(f;

where

V*(So)=

{

P) = V*(SO)l

(19)

-so, so 6 Y*, so > y*,

1 c * q ,

and the parameters c* and y * can be foiind by (14)-(17). The optimal time of exercising the option is r* = inf{,n: S, y * } . Moreover,

<

V*(So)= E s , ( ~ t ~ ) ~ -* (Sl T * ) + .

3 5d.

Options with Aftereffect. 'Russian Option' Pricing

1. The pay-offs f,?,for the put and call options considered above have the Markovian, structure: fn = pn(Sn,- K ) +

and

fn, = p n ( K - STL)',

(1)

respectively. It is of interest both for theory and financial engineering to consider also options with aftereffect. Examples here can be options with the following pay-off functions:

or with pay-offs

where 0

< p 6 1, a 3 0.

Chapter VI. Theory of Pricing. Discrete Time

626

Options with pay-offs (4) and (5) are called A s i a n (call and put) options. Call and put options with pay-offs (2) and ( 3 ) have been considered for a = 0 in [434] and [435], where they are called ' R u s s i a n options' (see also Ill81 and [283]). In what follows we stick to [283]. 2. We consider the CRR-model in which p, can take two values, X - 1 and X - l - 1, where X > 1. In addition, for definiteness, we shall consider an American p u t option with pay-off function ( 3 ) , where p, 0 < p < 1, is the discount factor. In accordance with the general theory (see subsection a), the rational price of such an option can be found by the formula

e

+

where a = (1 r)-l and E is averaging with respect to the martingale measure P such that p and q are described by formula (6) in 3 5a. Since sup E ( ~ B ) T (max S, (7)

6=rE3nflom

and S , = S O X ~ ~ + . ' the . + ~quantity ~,

O
&)+

e is definitely apX

< 1.

We set Y, = max s k . Clearly, k(n

Moreover, (Sn,Yn),>o is a Markov sequence and, in principle, one can solve the optimal stopping problem (7) on the basis of general results on optimal stopping rules for two-dimensional Markov chains. (see [441] and 5 2a). However-and this is remarkable-our two-dimensional Markov problem can be reduced to some one-dimensional Markov problem if one uses the idea of change of measures and chooses an appropriate discounting asset (numkraire). (See also Chapter VII, 3 l b on this subject.) We recall that B, = Boa-, with a = (1 r)-', so that Let T E

2Xr.

+

sn/so. Then 2, > 0, and the sequence 2 = (Z,, gn,P ) , ~ ois a Bn/BO P-martingale with EZ, = 1. We set 2, =

~

5. American Options on a Binomial

(B, S)-Market

627

For A E 9,we set

Pn,(A) = E ( Z ~ I A ) . A

Clearly, the measures in the collection (P,)n>O are compatible (i.e., Pn+l I 9,= P, n 3 0) and by Ionescu Tulcea's theorem on the extension of a measure (see, e.g., [439; Chapter 11, 5 91) there exists a measure ^P (in the space R = {-1, l}cc) such that P I 9,= P, n 3 0. Hence

We now set

and observe that Xn+l = m a x

(

A&"+' X,rL , I)

(13)

~

and, moreover, all X , range in the set E = { 1,A, X2, . . . }. With respect to the new measure P the sequence E = (&,),>I is also a sequence of i n d e p e n d e n t identically distributed (i.i.d.) random variables with p = P(E, = 1) = E I ~ , n , l ) ~ X E 1 = aXp

and A

(14)

Q

q = P ( E T L = -1) = -(1 - p ) .

x

We shall consider the sequence (X,),>O defined by recursive relations (13) under the assumption that XO = z E ,!?. Let ^Pz be the probability distribution of this sequence. Then X = with z E and 9, = c(Xo,X1,. . . , X,), n 3 0, is a Markov sequence and, therefore, to find the price one must consider the optimal stopping problem h

(X,,gn,Pz)

P(x)=

sup EzPT(XT- a)+,

z E

I?,

(16)

T€@?

<

f67

where is the class of finite stopping times T = T ( W ) such that { w : T ( W ) 7 1 ) E gn,n 2 0. The price in question is connected with the solution c(1)of this problem by the formula i? = SoP(1). (17) R e m a r k 1. Strictly speaking, the supremum in (7) is taken over the class and therefore formula (17) is valid if, in the definition (16) of P(z), we consider the supremum over the wider class in place of However, these two suprema are the same, which follows from the general theory of optimal stopping rules for Markov sequences (see [441]) and is in effect proved below (see Remark 2).

"nr

"0"

"nr.

Chapter VI. Theory of Pricing. Discrete Time

628

3. Let g ( x ) = (z- a)+, z E

55:

where Fig. 61).

E, and let

h

is the class of stopping times

7

in W? siicli that

T(W)

< N , w E (2 (see

J

A

FIGURE 61. Graphs of functions g(z) = (z QoN(z)= ljNg(:l;.) for o < a < I

-

a ) + and

We also set

and

0 . m = m a x ( f ( 4 , P%)) By Theorem 3 in 5 2a and our remark upon it,

?t(i.) =QNg(z) and the optimal stopping time:?

(20)

E% : can be described as follows (cf. (9) in 5 5b):

where

5:

c

c

O

= {z €

h

E : v,"-7yz,

E

= y(z)}.

Clearly, 5: @' . . . E g = G (1, A, x2, . . . >. In the same way as in $5b, considering successively the functions Gg(z), . . . , Q N g ( z) and comparing them with g(z) we see that the stopping domains 6: are as follows:

5. American Options o n a Binomial ( B ,5')-Market

629

5:

The qualitative picture of the stoppirig domains and the continuation domains C[ = E\EN is as in Fig. 57 in 3 5b (with the self-evident change of notation nSi 4 X i arid E + E , . . . , z{ = 0 + 2; = 1). h

Remark 2. If

Vf(.)

= sup ~ z P ' g ( X , ) ,

'Ern:

then it follows by Theorem 3 in 52a that V t ( z ) = Q N g ( z ) . Comparing this with (20) we see that pf(z) = Vf(x),z E E , and that the instant 7: defined by (21) is optirnal riot only in the class f m f , but also in the broader class f m f . A

4. Since g(z) 3 0, it follows by Theorem 4 in 52b that p ( x ) = lini v/(z). N-iW

now set

7 = inf{n

> 0 : C(X,) = g ( ~ , ) } = inf{n > 0: X ,

We

E D},

A

where D = {z E E : x E [x,m)} and 2 = lirn 2:. N-tW

-

By the same theorem, ? is an optimal stopping time in the problem (16), provided that Pz(? < m) = 1, x E g. Leaving aside for the moment this property of ? we proceed to 3 and $(x). The function p(z) satisfies the equation

P(z)= max(g(z),P?P(z)), therefore, in the continuation domain

or, in more explicitly,

In particular, for z = 1 we have

while for x 3 A,

=

\ 6 it

z

E

E,

(25)

is a solution of the equation

Chapter VI. Theory of Pricing. Discrete Time

630

It would be natural to seek a solution of (29) in the form of a power function zY (cf. 5 5b.5). Then we obtain for y the equation

1 -

P

=j7A-r

+ (1

- v^)XY,

(30)

with two solutions, y1 < 0 and 7 2 > 1, such that the quantities y1 = y2 = A?* are defined by the formulas

with

and

A

A=

1

(1 - F ) P

B = -. P I-P

and

(32)

For z 3 X the general solution p(z) of (29) can be represented as c&(z); where ?,bb(”) = bZY1 (1- b)zY2. Since $ b ( l ) = 1, it follows that c = ~ ( 1 ) . Substituting p(X) = p(l)‘&(X)in (28) and bearing in mind that p(1) # 0 due

+

to the nature of the problem, we obtain the following equation for the unknown b: 1=/3{j7+(1-j7)[bArl

+(I-h)AY”}.

(33)

It has the solution

Since 71 and 7 2 can be determined from (30), it easily follows that 0 < < 1. Let Vc0(z) = cg$-(x) for z < q,where co and xo are some constants to be h

b

determined. Clearly, the required function G ( x ) belongs to the family of functions

Here the ‘optimal’ values 2 and P of co and zo can be found by means of the following considerations: the required function = V ( x ;P) must be the smallest p-excessive majorant of g(z), i.e., the smallest function satisfying simultaneously the two inequalities

e(z)

Q ( 4 3 dz), V(z) 3 pT^G(z) for each z E

= { 1, A,

. . .}

h

(36)

5. American Options on a Binomial

(B, 5’)-Market

631

The solubility of this problem can be proved and the exact values of 2 and 2 can be found in precisely the same manner as for a standard call option (see 5 5b.6 and [283]).Namely, if A = X - 1 is close to zero, then we can consider approximations C and E of E and 2 obtained as follows (cf. a similar procedure in 5 5 5b, c.) We shall assume that the functions @(x), p c 0 ( x ) ,g ( z ) , and ~ & , ( x ; x oon )

,?? = (1, X, X 2 , .

. . } are defined by the same formulas on [l,m). Then the approximations C and II: can be found from the additional conditions

&(E) = g(Z),

Bearing in mind that A

V,(X) = E$J-(X) b

g(z) = (z - a ) + , and, for sure,

=

c[%1 + (1 - ; ) X y q ,

2 > a , we see that C and 2 are the solutions of the

system of equations

In particular, for a = 0 we have

I

X

I

C =

yl;II:71 + - y 2 ( 1

-;p .

(40)

The above considerations show that for positive, sufficiently small A the quantity

p~(1) is close to c ( 1 ) . Hence, in view of (17) and the equality pz(l) = C, we obtain M SO . C for small A that Chapter VIII, 5 2d.)

> 0.

(See [283] for a more detailed analysis; cf. also

Chapter VII . Theory of Arbitrage in Stochastic Financial Models. Continuous Time 1. I n v e s t m e n t p o r t f o l i o in s e m i m a r t i n g a l e m o d e l s . . . . . . . . . . . . . . . . l a . Admissible strategies . Self-financing. Stochastic vector integral . . l b . Discounting processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . €j l c . Admissible strategies . Some special classes . . . . . . . . . . . . . . . . . 2 . Semimartingale m o d e l s without o p p o r t u n i t i e s f o r a r b i t r a g e . Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fj 2a . Concept of absence of arbitrage and it5 modifications . . . . . . . . . fj 2b . Martingale criteria of the absence of arbitrage . Sufficient conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fj 2c . Martingale criteria of the absence of arbitrage . Necessary and sufficient conditions (a list of results) . . . . . . . . . . €j 2d Completeness in semimartingale models . . . . . . . . . . . . . . . . . . .

3 . Semimartingale and m a r t i n g a l e measures ...................... 5 3a . Canonical representation of semimartingales. Random measures . Triplets of predictable characteristics . . . . . . 5 3b. Construction of marginal measures in diffusion models . Girsanov's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3c . Construction of martingale measures for L6vy processes . Esscher transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . li 3d . Predictable criteria of the martingale property of prices . I . . . . . . fj 3e . Predictable criteria of the martingale property of prices . I1 . . . . . , 3 3f. Representability of local martingales (' ( H C p-vj-representability') § 3g . Girsanov's theorem for semimartingales. Structure of the densities of probabilistic measures . . . . . . . . . . .

4 . Arbitrage. c o m p l e t e n e s s . and hedge p r i c i n g i n d i f f u s i o n modelsofstock ...................................................... $4, . Arbitrage and conditions of its absence . Completeness . . . . . . . . fj 4b . Price of hedging in complete markets . . . . . . . . . . . . . . . . . . . . . €j 4c . Fundamental partial differential equation of hedge pricing . . . . . .

5. Arbitrage, c o m p l e t e n e s s . a n d hedge p r i c i n g in d i f f u s i o n models of bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fj 5a . Models without opportunities for arbitrage . . . . . . . . . . . . . . . . . §5b. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fj 5c . Fundamental partial differentai equation of the term structure ofbonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

633 633 643 646 649 649 651 655 660 662 662

672 683 691 694 698

701 704 704 709 712 717 717 728 730

1. Investment Portfolio in Semimartingale Models

fj la. Admissible Strategies. Self-Financing. Stochastic Vector Integral In this section we shall consider models of two securities markets with continuous time: a ( B ,S) m a r k e t f o r m e d b y a bank account B a n d stock m a n y kinds,

S = (S’, . . . ,Sd) offinitely

and a ( B , P )m a r k e t f o r m e d also by a bank account B a n d , in general, a continual f a m i l y of bonds P = { P ( t , T ) ;0 < t < T , T > O}. In 3 5 1-4 we are concerned with ( B ,S)-markets. A ( B ,P)-market has some peculiarities and we defer its discussion to fj 5.

+

1. We consider a financial market of d 1 assets X = ( X o ,X 1 , .. . , X d ) that operates in uncertain conditions of the probabilistic character described by a filtered probability space (stochastic basis) (a, 9, (Bt)t>o, P), where ( 9 ; t ) t 2 ois the flow of incoming ‘information’. Our main assumption about the assets X i = (Xj)t>o, i = 0 , 1 , . . . , d , is that they are positive s e m i m a r t i n g a l e s (see Chapter 111, 5 5a). By analogy with the discrete-time case we call an arbitrary predictable (see Chapter 111, fj 5a) ( d 1)-dimensional process 7r = ( T O , T I ,. . . , n d ) , where ri = ( ~ f ) t > o an , i,nvestment portfolio, and we shall say that T describes the strategy (of an investor, trader, . . . ) in the above market. The process X T = ( X ; ) Q O , where

+

d

x; = X T f X f , i=O

Chapter VII. Theory of Arbitrage. Continuous Time

634

or, in the vector notation,

-v ( W X t ) , =

(2)

is called the value (or the value process) of the portfolio 7 r . The value z = X $ is the initial capital, which one emphasizes sometimes by the notation X" == X " ( x ) . 2. In Chapter V, § l a , in the discussion of the discrete-time case we introduced the concept of selj-financing strategy 7r and explained the role of these strategies, in which all changes of the value X K must be the results of changes in the market value (price) of the assets X iand no in- or outflows of capital are possible. The definition of self-financing becomes slightly more delicate in the continuoustime case. In the final analysis this is related to the problem of the description of the class of integrable functions with respect to the semimartingales in question. We recall that in the discrete-time case (see Chapter V, § l a ) we say that a portfolio n = (no,T I , . . . , 7 r d ) is self-financing (nE S F ) if for each n >, 1 we have n

Xz

=

X:

+ C ( n k ; AX,),

(3)

k= 1

or, in a more expanded form, n

d

,=l i=o

In the same way, a reasonable definition of a self-financing protfolio or a selffinancing strategy n (we shall write 7r E SE) in the continuous-time case could be the equality (5) for each t

> 0. Equivalently,

which we can symbolically express as follows:

dXT = (nt, d X t ) . Of course, we must first define the 'stochastic vector integrals' in (5). One way is simply to set

1. Investment Porfolio in Semimartingale Models

635

by definition, i.e., to treat a ‘stochastic vector integral’ as the sum of usual ‘stochastic integrals’. This is quite sensible (and we shall use this definition) in the case of ‘simple’ functions; in fact, this is the most natural (if not the unique) construction suggested by the term ‘integration’. It turns out, however, that the definition (7) does not cower all cases when a ‘stochastic vector integral’

/ t ( x S , d X s )can be defined, e.g., as a limit of some 0

integrals of ‘simple’ processes x(n)= (7rs(n)),>o, n 2 I , approximating 7r = (7rs),>0 in some suitable sense. The point here is as follows. t . First, even the usual (scalar) stochastic integrals xi.X j = xi d X j can be defined for a broader class of predictable processes xi than the locally bounded ones discussed in our exposition in Chapter 111, 5a. (What makes locally bounded processes xi attractive is the following feature: if X iE J Z Z ~ then ~ ~ , the stochastic integral 7ri . X iis also in the class A l l o c ; see property (c) in Chapter 111, 5 5a.7.) Second, the ‘component-wise’ definition (7) does not take into account the possible ‘interference’ of the semirnartingales involved; in principle, this interference can extend the class of vector-valued processes x = ( T O , xl,. . . , 7 r d ) that can be approximated by ‘simple’ processes 7r(n),n 2 1. 3. We explain now the main ideas and results of ‘stochastic vector integration’ that takes these points into consideration; we refer to special literature for detail (see, e.g., [74], [172], [248; Chapter 111, [249], [250], [303],[304], or [347]). Let X = ( X ’ , . . . , X d ) be a d-dimensional semimartingale admitting a decomposition

X

= Xo

+A

+M,

(8)

where A = ( A 1 , . . . , A d )is a process of bounded variation and M = (M’, . . . , M d ) is a local martingale ( A E -Y and M E A l l o c ) . Clearly, we can find a nondecreasing adapted (to the flow F = ( . 9 t ) t 2 0 )process C = (Ct)t,o, CO= 0, and adapted processes cz = ( c f ) and czj = (c?), i, j = 1,.. . , d , such that t .

Af =

cidC,,

t > 0,

(9)

while the quadratic variations satisfy the equality

(As regards the definition of the processes [ M i ,M j ] and their property [ M i ,Mj]’12 E d‘~,,,see $5b, Chapter 111.) Let x = ( T ’ , . . . ,)‘x be a predictable process.

Chapter VII. Theory of Arbitrage. Continuous Time

636

We shall say that 7T

E L"adA),

if (for each w E Q)

We also write

* E L:,(")

i.e., if there exists a sequence of Markov times that

T~~approaching 00

as

71

+x

such

+ +

T

If there a z s t s a representation X = X o A M such that a predictable process belongs to the class Lvar(A) n Lyo,(M), then we write 7r

t LQ(X)

(15)

(or also 7r E L q ( X ;P, IF), if we must emphasize the role of the underlying measure P and the flow IF = ( 9 t ) t 3 0 ) . Importantly, the fact that T belongs to L q ( X ) is independent of the choice of the dominating process C = (Ct)t>o(see, e.g., [249]). Both in the scalar and in the vector cases one standard definition of the stot chastic integral (7rs, dX,), t > 0. for T E L ' J ( X )consists in setting 0

rt

rt

where

is the sum of (trajectory-wise) Lebesgue--Stieltjes integrals and rt

is the stochastic integral with respect to the local martingale M = (Ad':. . . , M d ) .

1. Investment Porfolio in Seniirriartirigalc Models

637

The definition of Lebesgue- Stieltjes integrals with respect to a process of bounded variation (for 7r E L,,,(A) and arbitrary w E encounters no difficulties. The main prolilern here is

a)

a) to give a definition of the (vector) integrals (18) with respect to a local martingale M (for 7r E L : ~ , ( M ) ) and

b) to prove the consistency of the definition (16) or, in other words, to show

/t

that the values of the resulting integrals ( r s dX,) , are independent of a 0 particular semimartingale decomposition (8)

Rem.ark 1. There are several pitfalls in the ‘natural’ definition (16). First, one does not automatically get, say, the property of h e a r i t y

It is not a priori clear whether the integrability withstands the replacement of the measure P by some equivalent nieasure F, i.e., whether L q ( X ;P: F) = L * ( X ;is, F) and whether the values of the corresponding integrals are the same (if only P - a s . ) . Neither is it clear whether this definition is invariant under a reduction of the flow of o-algebras F = ( 9 t ) t 2 0 . Namely, assume t h a t X is a F-semimartingale such that the X t are %-measurable, where G = (%t)t20 is a flow of a-algebras satisfying general conditions (Chapter 111, 55a) and %t 5 9 t , t 3 0. It is well known (see, e.g., [249] and [250]) that X is also a 6-semimartingale in that case. Hence one would anticipate that if a process 7r is 6-adapted, then 7r

E L 4 ( X ; P,F)

*

7r

E LQ(XP : ,G)

and the value of the stochastic integral is independent of the particular stochastic basis, (Q9, F) or (Q9, G ) , underlying the processes T and X . It is shown in [74], [248], and [249] that all these properties hold nicely (for each 9 3 1). 4. We describe now the construction of the integrals (18) with respect t o a local

martingale M for 7r E L:,, ( M ) . If r E L ~ ( M )i.e., ,

t

then the stochastic integral (7r . M ) t 3 (T,, d M 3 ) can be defined as in the scalar case for square integrable martingales (see Chapter 111, 3 5a.4).

Chapter VII. Theory of Arbitrage. Continuous Time

638

Namely, we find first a sequence of simple predictable vector processes ~ ( n=) . 3 1, such t h a t

(7r1(n),. . ,7rd(n)), n

117r -

~ ( n ) l l ~ --t z (0 ~ as )

n

+ 30.

(20)

For these processes ~ ( 7 1 the ) integrals ( 7 r ( 7 ~ ) . M ) t can be defined component-wise by formula (7). By the Burkholder-Gundy-Davis inequality (see, e.g., [248;2.341 or [304; Chapter 1, 3 9 1 ~

with some universal constant (22. We conclude from (20) and (21) that

Since the L2-variables make up a complete space, there exist for each t 3 0 a random variable, denoted by ( T . M ) t or ht(7rs, d M s ) and called the stochastic vector integral of

T

E

L 2 ( M )with respect to the local martingale M , such that

E s u p l ~ u j i i g ( n d) ,M s ) v
d

I

~

2

+~ 0

)as

71

+ oo

It is easy to see (cf. the proof of Theorem 4.40 in [250; Chapter I] that we can choose the variables

t 3 0, such that the process

ht(7rs, dM~3),

(fi(7rs,

dMs))

t>O

is

adapted to the flow F = (.!Ft)t>o and has right-continuous trajectories with limits from the left for each t > 0. Using the standard localization trick we can extend these definitions for T E L 2 ( M ) to the class of predictable processes 7r E L f o c ( M ) ,i.e., to processes such that (14) holds with q = 2. It is much more complicated to construct stochastic integrals with respect to local martingales M for processes 7r in the class L t o c ( M ) ,i.e., for processes such that

Even in the scalar case ( d = l ) ,where this condition has the simple form

1. Investment Porfolio in Semimartingale Models

639

the construction of the stochastic integral T . M involves some refined techniques based on some properties of local martingales that are far from trivial (see [304; Chapter 2, $21).

R e m a r k 2. It is worth noting in connection with the condition ( n 2. [ M ,M])lI2E that it definitely holds for locally bounded processes T since, as observed earlier, each local martingale M has the property [ M ,MI1l2 E &lo,. As regards different definitions of stochastic vector integrals (T . M)t = dloc

ht(7rS,dMs)with respect to local martingales A4 and for T or of stochastic integrals

(T

. X)t

t

= J0 ( T ~dX,) ,

E

L l o c ( M )(= L&,(M))

with respect to semimartingales

X for T E L(X) (= L1(X)),see [74], [249], and [250], where the authors establish the following, well-anticipated properties (here X and Y are semimartingales): a) if p is a predictable bounded process and n E L ( X ) , then p~ E L(X), p E L(T. X ) and ( p ~. X ) = p . (T . X); b) if X E A l l o c , then Lloc(X)C L(X); c) if X E Y , then Lvar(X) L(X); d) L ( X ) n L ( Y ) C L(X + Y ) , and if T E L(X) n L ( Y ) ,then T . X T Y = T .(X Y); e) L(X) is avector space and T’.X+T’’.X = (.~“+T’’).X.for T’,T’’ E L(X). The question whether L(X)is the maximal class of processes T where a)-e) hold is discussed in [249] and [250]. Another argument in favor of its ‘maximality’ is the following result of J. Mkmin [343]: the space { T . X : T E L(X)} is closed in the space of semimartingales with respect to the E m e r y topology ([74], [138]).

+

+

5 . If the process

T

=

(T’,

. . . , r d )is locally bounded and M E Aloe, then the sto-

chastic vector integral process ( L t ( x s d, M s ) )

t>o

is a local martzngale ([74], [249]).

In the scalar case we mentioned this property in Chapter 111, 3 5a.7 (property (c)). It is worth noting that if there is n o local boundedness, then the stochastic t

integral xs dMs with respect to a local martingale M zs not, in general, a local martingale even in the scalar case, as shown by the following example.

EXAMPLE (M. Emery [137]). We consider two independent stopping times a and T on a probability space (a, 9, P) that have exponential distribution with parameter one. We set 0, t < min(a, T ) , Mt= 1, t min(a, T ) = a, (22) -1, t 2 min(a, T ) = T .

{

>

If 9 t = a(M,, s < t ) , t 3 0 , then it is easy to see that A4 = (Mt, 9t, P) is a martingale. We consider a deterministic (and therefore, predictable) process T = (7rt)t20 with T O = 0 and Tt = l / t for t > 0.

Chapter VII. Theory of Arbitrage. Continuous T i m e

640

The niartirigale M is a process of bounded variation and 7r is integrable with respect to M (in the Lebesgue-Stieltjes sense) because the random variable min(o-,7 ) is strictly positive with probability one.

/t

By property b) in subsection 4, the integral rS dMs with respect to M coin0 cides with the Lebesgue-Stieltjes integral. Let yt = hf7rs d M s , t 3 0. By (22) we obtain

t < min(a, 7 ) ' 1 rnin(a, T ) '

I-

1

t

> min(a,

T)

= o-,

>

t min(o-,7 ) = T . min(a, T ) ' The process Y = (Yt)t>Ois not a martingale because EIKl = 00,t > 0. Neither is this process a local martingale because EIYT~= 30 for each stopping time T = T ( w ) , not identically equal to zero (see [137] for greater detail). 6. In connection with M. Emery's example there arises a natural question on conditions ensuring that. a stochastic vector integral ( L t ( r s , d X , ) )

t>O

is a local mar-

tingale if so is the process X . We have already mentioned one such condition: the local boundedness of T . The following result of J.-P. Ansel and C. Stricker [9; Corollaire 3.51) g'ives one conditions in terms of the values of the stochastic integral itself, rather then in terms of T (this is convenient in discussions of arbitrage, as we shall see below).

THEOREM ([Y]). Let X = ( X ' , . . . , X d ) be a P-local martingale and let T = ( T I , . . . , ? r d ) be a predictable process such that the stochastic integral T . X is well defined and bounded below by a constant ( T . X t 3 C , t > 0). Then T . X is a local martinga-ale.

7. We now return to the issue of self-financing strategies.

+

DEFINITION 1. Let X = ( X O X , I , . . . , X d ) be a ( d 1)-dimensional nonnegative semimartingale describing the prices of d + 1 kinds of assets. We say that a strategy 7r = ( T O , T ~ ,. .. , T " ) is admissible (relative to X ) if 7r E L ( X ) . DEFINITION 2 . An admissible strategy T E L ( X ) is said to be self-financing if its value X K = (XT)t>o defined by (1) has a representation ( 5 ) . We denote the class of self-financing strategies by SF or S F ( X ) ;cf. Chapter V,

3 la. For discrete time, self-financing (see (3) or (4)) is equivalent to the relation d i=O

(formula (13) in Chapter V,

5 la).

1. Investment Porfolio in Semimartingale Models

64 1

We consider now the issue of possible analogs of this relation in the continuoustime case, under the assumption (7). To this end we distinguish the strategies 7r = ( n o T; I , . . . , 7 r d ) whose components are (predictable) processes of bounded variation (7rz 6 'Y, i = 0 , 1 , . . . , d ) . One example of such strategies are simple functions that we chose to be the starting point in our construction of stochastic integrals (Chapter 111, 5 5a). Under the above-mentioned assumption ( 7 r i E "Y) we obtain

(see property b) in Chapter 111, 3 5b.4), so that self-financing condition ( 5 ) is equivalent to the relation

or; symbolically,

( X t - : d r t ) = 0. In a more general case where 7r = ( T O , 7r1, . . . , 7 r d ) is a predicrtable sernimartzngale we obtain (using It6's formula in Chapter 111, 3 5c)

7r

Thus, the condition of self-financing of a semimartingale (predictable) strategy assumes the following form:

i=O

In particular, if

.

7r

E

.

Y , then (Xzc,7rzc)= 0 and we derive (25) from (27).

8. The main reason why one mostly restricts oneself to the class of semimartingales in the consideration of continuous-time models of financial mathematics lies in the fact that (as we see) it is in this class that we can define stochastic (vector) integrals (which is instrumental in the description of the evolution of capital) and self-financing strategies. (This factor has been explicitly pointed out by J. Harrison, D. Kreps, and S. Pliska [214], [215], who were the first to focus on the role of semimartingales and their stochastic calculus in asset pricing.) This does not mean at all that semimartingales are 'the utmost point'. We can define stochastic integrals for many processes that are not semimartingales, e.g.,

Chapter VII. Theory of Arbitrage. Continuous Time

642

for a fractional Brownian motion and, more generally, for a broad class of Gaussian processes. Of course, the problem of functions that can be under the integral sign should be considered separately in these cases. We recall again that in the case of (scalar) semimartingales stochastic integrals are defined for all locally bonunded predictable processes (Chapter 111, $ 5a). It is important here (in particular, for financial calculations) that the stochastic integrals of such functions are also local martingales. The situation becomes much more complicated, however. if we consider locally unbounded functions. M. Emery’s example above shows that if 7r is not locally bounded, then the integral process h ( 7 r s , d M s )(even with respect to a martingale M ) is not a local martingale in general. This is in sharp contrast with the discrete-time case where each ‘martingale transform’ (rk,A M k ) is a local martingale. (See the theorem in Chapter 11, 5 lc).

1

k<.

The following definitions seem appropriate in this context.

DEFINITION 3 . Let X = ( X t ,9 t , P)t>o be a semimartingale. Then X is called a martengale t r a n s f o r m of order d ( X E A T d ) if there exists a martingale A4 = ( A l l , . . . , M n ) and a predictable process 7r = ( d , . . . , 7 r d ) E Ll,,(M) such that

xt = xo +

Lt

(7rs,dMs),

t30

(28)

(cf. Definition 7 in Chapter 11, 5 lc).

DEFINITION 4. Let X = ( X t ,9 t , P)t>o be a semimartingale. Then X is called a local martingale t r a n s f o r m of order d ( X E AT:,) if there exists a local martingale M = (M1,.. . , M d ) and a predictable process 7r = (7r1,. . . 7 r d ) E J ~ ; ~ , ( M such ) that the representation (28) holds. ~

For discrete time the classes Aloe, A T d , and A T & are t h e s a m e for each d 3 1 (see the theorem in Chapter 11, $ 1 ~ )This . is no longer so in the general continuous-time case, as shows M. Emery’s example. 9. The above-introduced concept of self-financing, which characterizes markets without in- or outflows of capital, is one possible form of financial constraints on portfolio and transactions in securities markets. In Chapter V, $ l a we considered other kinds of constraints in the discrete-time case. Such balance conditions can be almost mechanically transferred to the continuous-time case. For instance, if X o = B is a bank account and X1 = S is a stock paying dividends, then the balance conditions can be, by analogy with Chapter V, 3 la.4, written as follows: d X T = Pt dBt + Yt (dSt + a), (29)

1.

Investment Porfolio in Semimartingale Models

643

where Dt is the total dividend yield (of one share) on the time interval [ O , t ] . In that case

The conditions in the cases involving ‘consumption’ and ‘operating expense’ can also be appropriately reformulated (see formulas (25)-(35) and (36)-(40) in Chapter V, 3 la).

lb. Discounting Processes 1. Comparing the values (prices) of different assets one usually distinguishes a ‘standard’, ‘basis’ asset and valuates other assets in its terms. For instance, the discussion of the S&P500 market of 500 different stocks (see Chapter I, l b or, e.g., [310] for greater detail) it is natural to take the S&P500 Index, a (weighted) average of these 500 assets, for such a basis. ) exposed briefly a popular CAPM pricing model, in In Chapter I (see $ 2 ~ we which one usually takes a bank account (a riskless asset) for a basis, and the ‘quality’ and ‘riskiness’ of various assets A are measured in terms of their ’betas’ P(A). In our considerations of d 1 assets X o , X 1 , . . . , X d we shall agree to choose one of them, say, X o , as the basis asset. It is usually the asset that has the ‘most simple’ structure. It should be pointed out, however, that, in principle, we could choose an arbitrary process Y = ( K , .9t)t>o t o play that role, as long as it is strictly positive. There exist also purely ‘analytic’ criteria for our choice of Y ; namely, if such a discounting process (or ‘nunibraire’ as it is often called; see, e.g., [175]) is.siiitably X” . chosen, then the process - is sometimes more easy t o manage than X” itself; see

+

Y

the remark a t the end of Chapter V, 5 2a. 2. If Y = ( Y t , 9 t ) t ) t 2 0 is a positive process defined on the stochastic basis (@9, ( 9 t ) t > o ,P) in addition to X = ( X o ,X 1 , . . . , X d ) , then for i = 0 , 1 , . . . , d we set.

If 7r is a self-financing portfolio (relative to X ) , then one would like to know whether = . . ,F d ) . it is self-financing with respect to the discounted portfolio 1 . To this end we assunie that we have property ( 7 ) in $ l a and Y-l = - 1s a

x (xo,xl,. Y

predictable process of bounded variation (Y-l E ”Y). Then d z = U,-l dXd

+ Xj-

dI’-’

(2)

Chapter VII. Theory of Arbitrage. Continuous Time

644

Hence we see from condition of self-financing (6) in

l a that

Assume that (P-as.) for t > 0 we have

Then

and by (4) we obtain

because dXT =

d

1.i,id X f , and AX? i=O

=

Cd

. 7ri

AX; by the properties of stochastic

i=O

integrals (see property f ) in Chapter 111, 5 5a.7). By (3) and (6) we obtain d

i=O

which means that the discounted portfolio financing property.

x = (X , X

--O -1

, . . . ,-d X ) has the self-

Remark. We assume in the above proof that Y is a positive predictable process, Y-' E "Y, and (5) holds. As regards other possible conditions ensuring the preservation of self-financing after discounting, see, for instance, [175].

1. Investment Porfolio in Seniimartingale Models

645

A classical example of a discounting process is a bank account B = ( B ~ ) Qwith o

where T = ( ~ ( t ) ) t is~ osome, stochastic in general, interest rate that is usually assumed to be a positive process. A bank account is a convenient 'gauge' for the assessment of the 'quality' of other assets, e.g., stock or bonds. 3. Let Y 1 and Y2 be two discounting assets and let t E [0, TI. We assume that the

x.

is a ( ( d + 1)-dimensional) martingale with respect t o some Y measure P' on (n,gT). We now find out when there exists a measure P2 P1 such that the discounted x. process - is a martingale with respect t o P2 (cf. our discussion in Chapter V, 3 4).

discounted process

Y2

To this end we assume that the process

< Y

is a (positive) martingale with

respect to P'. For A E .!FT we set

Clearly, P2 is a probability measure in ( O , . ~ T and ) P2 N PI. By Bayes's formula (see (4) in Chapter V , 3 3a),

x .

Hence the discounted process - is a martingale with respect t o the measure P2 Y2

defined by (9), It should be taken into account that if f~ is a .qT-measurabk nonnegative random variable, then it follows from (9) and (10) (provided that 9 ; o = {0,!2}) that

and (P2-, P1-a.s.)

Chapter VII. Theory of Arbitrage. Continuous Time

646

3 lc.

Admissible Strategies. Some Special Classes

1. In accordance with Definition 2 in 5 l a , the value X " = ( X F ) t Gof~ an admissible strategy 7r E SF(X)can be represented as follows:

where h'(7rs,dX,) is a stochastic vector integral with respect to the (nonnegative)

semimartingale x = (x',xl,. . . , xd). We shall assume in what follows that XO is positive

(Xp> 0, t 6 2') and take it

as the discounting process. To avoid 'fractional' expressions 1, i.e., the original semimartingale X = we assume from the outset that Xp (1,X1,. . . , X d ) is a ( d 1)-dimensional dascounted asset.

+

2. We shall now introduce several special classes of admissible strategies; their role will be completely revealed by our discussions of 'martingale criteria' of the absence of the opportunities for arbitrage (see $ 5 4 and 5 below).

DEFINITION 1. For each a 3 0 we set rI,(X)

= (7r

t SF(X):

xt"3 -a,

t E [O;T]}.

(2)

The meaning of a-admissibility ''X? 3 - a , t E [O,T]" is perfectly clear: the quantity a 3 0 is a bound (resulting from economic considerations) on the possible losses in the process of the implementation of the strategy T . If a > 0, then the value X " can take negative values, which we interpret as borrowing (either borrowing from the bank account or short selling, say, stock) d

If a = 0, then the full value XT

=

C 7r;X:

must remain nonnegative for all

i=O

O
1. Investment Porfolio in Semimartingale Models

647

3. The classes II,(X), a 3 0,are by far not the only ‘natural’ classes of admissible strategies . The following definition is consistently used by C. A. Sin [447].

DEFINITION 2. Let g = ( g o ,g l , . . . , g d ) be a (d

+ I)-dimensional vector with nond

negative components and let g(Xt) = (g,Xt)

( = i=OC g i X j )

We set

rIT,(X)=

(7r

E S F ( X ) :XT

3 -g(Xt), t E [O,T]).

(3)

As in the case of Definition 1, condition XT 3 - g ( X t ) , t E [O, TI, is transparent: at each instant t the quantity g ( X t ) imposes a bound on the maximum possible losses or debt levels considered admissible in an ‘economy’ formed by funds go in a bank account and gi shares of each of d assets, i = I, . . . , d. Clearly, if go 3 a , then & ( X ) C IT,(X). 4. For a discussion of various issues of arbitrage in semirnartingale models it can be useful to introduce several classes of 9T-measurable ‘test’ pay-off functions

+ = $(w) strategy

T

that can in principle be majorized by the returns f ( . r ; , , d X , ) in one of the classes of admissible strategies introduced above.

DEFINITION 3. For a

of a

3 0 let

and

where I I + ( X ) =

u II,(X).

ad0

DEFINITION 4. For g = (go, gl, . . . , g d ) with g i > 0, i = 0,I , . . . , d , we set

where L,(R, 9 ~ P) , is the set of 9T-measurable random variables 141 S ( X T ) .

<

$J

such that

648

Chapter YII. Theory of Arbitrage. Continuous Time

5 . As usual, we introduce the following norm in the space (n,9 ~ P) of , random variables li, (although it would be more accurate to speak about equivalence classes of randorn variables; see, e.g., 1439: Chapter 11, 5 lo]):

This makes L , a complete (and therefore, Banach) space. We shall denote the closures of the sets q a ( X ) ,a 3 0, and q + ( X ) with respect to this norm 11 . ,1 by q a ( X ) and T + ( X ) ,respectively. The space q g ( X )is endowed with the norm /I . defined by the formula

We denote the corresponding closure of \ l r g ( X )by q g ( X ) .

2. Semimartingale Models without Opportunities for Arbitrage. Completeness

5 2a.

Concept of Absence of Arbitrage and Its Modifications

<

1. In the case of discrete time ( n N < m) and finitely many kinds of assets ( d < m) the extended version of the Fzrst fundamental theorem (Chapter V, 3 2e) states that for ( B ,S)-markets

ELMM

E M M e N.4.

(1)

Here N A is the No-Arbitrage property in the sense of Definition 2 in Chapter V, § 2a. The properties E M M and E L M M mean the existence of a n Equivalent Martingale Measure and an Equivalent Local Martingale Measure, respectively. Thus, if there is no arbitrage in the market in question, then the implicaP), which tion N A =j E M M means the existence of a martingale measure (p (as shown in the previoiis chapter) enables one to use the well-developed machinery of martingak theory in calculations. On the other hand, if there exists a t least one martingak measure in our niotlel of a ( B ,S)-market, then the implication EMM N A means that we have a ‘fair’ market (in the sense of the absence of opportunities for arbitrage). The first two implications jand +== in ( I ) are also or principal importance: they show that martingale and locally martingale measures are in fact th,e same in our case. Clearly, one would also like t o have results of type (1) for the continuous-time case (if only for seniiniartingale models). However, the situation there is more complicated, although in essense, there is ’absence of arbitrage’ (in the sense of an appropriate definition) if and only if there exists a n equivalent measure with certain ‘martingale’ properties (which we specify below).

-

*

Chapter VII. Theory of Arbitrage. Continuous Time

650

It will be clear from what follows that for a satisfactory answer to the question of the validity of (1) we require different versions of the ‘absence of arbitrage’, depending, in the long run, on the classes of admissible strategies. We recall in this connection that one requires essentially no constraints on the strategies n- = (/3,r) to prove (1) in the discrete-time case (apart from the standard assumptions of predictability and self-financing) . Continuous time is different: for a mere definition of self-financing we must t use there stochastic vector integrals ( r S , d X s ) , and their existence requires the 0 constraint of admissibility 7r E L ( X ) on (predictable) strategies n-. Of course, if we let into consideration only ‘simple’ strategies, finite linear combinations of ‘elementary’ ones (Chapter 111, 3 5a), then there are no ‘technical’ complications due to the definition of vector integrals (see 3 l a ) . Unfortunately, in the continuous-time case we cannot in general deduce the existence of martingale measures of measures with some or other ‘martingale’ properties from the absence of arbitrage in the class of ‘simple’ strategies. (The class of ‘simple’ strategies is too ‘thin’!)

[

2. We proceed now to main definitions related to the absence of arbitrage in semimartingale models X = (1,X1,.. . , X d ) , Xi = (X;)~
(2)

DEFINITION 2 . We say that the properties NA, and NA+ hold if

Q,(X) n L Z ( R ,Fr,P) = (0)

(3)

or

*+(XI n L L ( R , 3 T , p) = (01, respectively, where q , ( X ) and \ k + ( X ) are as defined in 5 l c and the subset of nonnegative random variables in L,(R, ST, P).

(4)

Lk((R,S T ,P) is

It is easy to show that (4) is equivalent to the condition @‘o+(x) n L&,(fl,S T , p) = yo>,

(5)

where

1

for some strategy n- f rI+(X) . DEFINITION 3 . We say that the property

m+holds if

(6)

2. Semimartingale Models without Opportunities for Arbitrage. Completeness

651

m+,

The property which is a refinement of N A + , is consistently used by F. Delbaen and W. Schachermayer (see, e.g., [loo], [loll), who call it the N F L V R property: No Free Lunch wrth Vanishing Risk. This name can be explained as follows. In the discussion of the absence of arbitrage in its NA+-version we take for ‘test’ functions $ only non,negative functions t h a t are smaller or equal to the ‘returns’ hT(7rs, dX,) from the strategy 7r E II+(X). However, in our considerations of the m + - v e r s i o n of the absence of arbitrage P), , e.g., we can take (also nonnegative) ‘test’ functions $ E G + ( X )n L L ( O , 9 ~ of some sequences of elements ones that are the limits (in the norm I / . /lo ) ?,bk ( k 3 1) of q + ( X ) , which can take, generally speaking, negative values (in particular, these can be some integrals

LT(7r$, dX,)).

Since ll?,bk-$llm + 0 as k + 00,we can assume that ?,bk 3 - l / n (for all w E a), which can be interpreted as vanishing risk. Remarkably, the absence of arbitrage in the m + - v e r s i o n has a transparent (‘martingale’) criterion established in [loll (see Theorem 2 in 3 2c). 3. We present now versions of the concept of absence of arbitrage based on the use of strategies in the class n , ( X ) .

DEFINITION 4. Let g = (go,g l , . . . , g d ) , where g i > 0, i = O , l , . . . , d. We say that are satisfied if the properties NA, and

m,

or

-

S , ( X ) n L&(R, 9 T , P)

=

(0)’

respectively. In [447], the property %%, is called the NFFLVR-property: N o Feasible Free Lunch with Vanishing Risk.

2b. Martingale Criteria of the Absence of Arbitrage. Sufficient Conditions

+

1. We shall assume that our financial market is formed by d 1 assets = (1,X1,. . . ,X d ) , where X i= ( X f ) , , , are nonnegative semimartingales on a filtered probability space ( R , 9 , ( 9 t ) t G T ,P) with 9l= 9, 90= {@, O}. P and X is a martingale If p is a probability measure on ( 0 , s ~ such ) that with respect to this measure, i.e.,

X

-

652

Chapter VII. Theory of Arbitrage. Continuous Time

or it is a local ,martingale, i.e.,

then we say that we have the property

EMM or

ELMM, respectively. The next theorem, which describes some s u f i c i e n t conditions of the absence of arbitrage, is probably the most useful result of the theory of arbitrage in sernirnartingale models from the standpoint of financial mathematics and engineering.

THEOREM 1. In the semimartingale model X = (1,X I , . . . , X')), for each a 3 0 and g = (gO,gl,. . . ,.qd) with gz 3 0, i = 0 ; 1 ) . . , d, we fiave ELMM

NA,, EMM ===+ NA,.

Proof. Let X" be the value of a strategy

7r

E n,(X):

(7rs,dXs),

t


(3)

I

Assume that P is a rnartingale rrieasure equivalent to P. As meritioiied in Remark 1 in $ la, the integrability of T with respect to X withstands the replacement of P by the equivalent measure p. Hence if 7r E n , ( X ) , then the stochastic vector integral in (3) is well-defined also with respect to is. The proof of the absence of arbitrage for the strategy 7r E II,(X) with X$ = 0 (in the sense of property NA,) is based on the denionstration of the supermartingale property of X" with respect to P. Indeed, if the supermartingale property holds, then

EFX?

< EpX;

= 0,

(4)

so that we immediately obtain the required relation X $ = 0 (P- and F-as.) from the condition X ; 3 0 (P and P-as.). Thus, let us establish the P-supermartingale property of X". If 7r E ng(X),then (P- arid P-as.) I

rt

2 . Semimartingale Models without Opportunities for Arbitrage. Completeness

653

By the linearity of stochastic vector integrals from ( 5 ) we obtain .i”(Ts

+ s,d X s ) 3 -x; - ( 9 ,X o ) .

(6)

The process X is, by clefinition, a martingale with respect to P, and since the stochastic vector integral in (6) is uniformly bounded below, it is a local martingale by a result of J.-P. Ansel and C. Stricker (see 31a.6), and therefore (by Fatou’s lemma), a supermartingale. Consequently,

xt”= x; +

Lt

(Ts

+ 9,dXS)

-

( 9 ,xt

-

XO),

(7)

I

where the stochastic integral is a P-supermartingale and ( g , X t - X O ) ~ < Tis a P-martingale. Hence for T E rII,(X) the process X is a supermartingale with respect to the measure p, which together with (4) proves the required assertion (2). To prove (1) it suffices to observe that, as follows by (6) with g = ( a ,0 , . . . , O ) ? a stochastic integral (with respect to a local martingale) is itself a local martingale. Since X o = 1, it follows that(g, X t - X O ) = 0 for g = ( a ,0 , . . . . O ) . Hence the right-hand side of ( 7 ) is a p-local martingale, and the proof of (1) can be completed in a similar way to (2). COROLLARY. I t follows from (1) that

ELMM

==+

NA+.

(8)

2. Assertions ( l ) ,(a), anti (8) can be improved in the following way.

THEOREM 2 . In the semimartingale model X = (1: X I , . . . , X d ) we have

E L M M ==+ NA+, and i f g = (go, g l , . . . , g d ) with gz > 0, i = 0 , 1 , . . . , d , then E M M ==+ Proof. Let $ E %,(X) with $ functions in Q g ( X )such that

m,.

(10)

3 0. Then there exists a sequence

Without loss of generality we can assume that for all w E

_ _1

(9)

‘

$(w)

-$“4 < %1 >

g(xT(w))

R we

have

($‘IC)k21

of

Chapter VII. Theory of Arbitrage. Continuous Time

654

Since

g k E Q g ( X ) ,there

exists a strategy n k E II,(X) such that

Together with (12) this brings us to the inequality

which shows that, for the sequence of strategies ( r k ) k 2 1 , the negative part of the returns described by stochastic integrals (the ‘risks’) approaches zero as k increases (‘vanishing risk’). Inequality (14) is clearly equivalent to the relation

Since

< g ( X T ) / k ,we obtain in view of (13), (15), and Fatou’s lemma, that

I

where the last inequality is a consequence of the P-supermartingale property of the stochastic integrals (r: k:9 , d X , ) , t 6 T .

ht

+

I

Hence P($ = 0) = P($ = 0) = 1, which proves the implication (10). To prove (9) we assume that $ E $+(X) and $ 3 0. Then there exists a sequence of functions ( G k ) k 2 1 in Q + ( X )such that -

$k~/co

= esssup l$(w) w

1

-

+‘(w)I


-

+ 0,

(17)

2. Semimartingale Models without Opportunities for Arbitrage. Completeness

655

By (17) arid (18) we obtain

Further, as in (16), we see that

-

where the last inequality is again a consequence of the P-supermartingale property (T:,d X , ) , t T . This completes the proof. of the stochastic integrals

<

3 2c.

Martingale Criteria of the Absence of Arbitrage. Necessary and Sufficient Conditions (a List of Results)

1. In the present section we state several results concerning necessary and suficient conditions of the absence of arbitrage (in one or another sense). We recall again the implications E M M u NA in the discrete-time case, which has become a prototype for various result about general semimartingale models. Moreover, if the implication E M M ==+ N A is easy to prove, then the proof of the reverse implication E M M + NA, where one must either suggest an explicit construction or prove the existence of a martingale measure, is based on designs that are far from simple (even in the seemingly simple case of discrete time!; see fj 2 in Chapter V). Little wonder, therefore, that the proof of the corresponding results in the continuous-time case (due mainly to F. Delbaen and W. Schachermayer [loo], [loll) is fairly complicated and we restrict ourselves to a list of several interesting results, referring the reader to the indicated special literature for detail.

THEOREM 1 ([loo]). a) Let X = (1,X1,. . . , X d ) be a semimartingale with bounded components. Then (1)

b) Let X = (1,X1, . . . , X d ) be a semimartingale with locally bounded components. Then (2)

656

Chapter VII. Theory of Arbitrage. Continuous Time

2. To state the result concerning necessary and sufficient conditions of property ___

NA+ in general semimaryingale models let us introduce, following [ l o l l , the concept of a-martingale and u-martingale measure. To this end we recall that for discrete t i m e each local martingale X is incidentally a martingale transform, i.e., X = X O y .M (see the theorem in Chapter 11, l c ) , where y is a predictable sequence and M is a martingale. Analyzing the proof of this theorem in Chapter 11, 5 l c one observes that each local martingale X can be represented as a martingale transform X = X O y . M , where y is a positive predictable sequence. Bearing that in mind and following [loll we shall call a martingale transform with positive elements of y a a-martingale. Further, if there exists a measure p P such that X is a a-martingale with respect to it: then we shall talk about the EaMM property. Using these concepts, in the discrete-time case we can now put the First fwndamental theorem (Chapter V, 5 5 2b, c; see also (1) in 5 2a) in the following form:

+

+

-

IEMM

ELMM

E c M M INA.1

(3)

This form is useful since it suggests routes for generalizations of the First f u n d a mental theorem to continuous tz,me. It was a substantial success of the authors of [loll that they have developed a clear perception of the following approach: to find necessary and sufficient conditions for the absence of arbitrage in general semimartingale models in its version one must turn to a-martingales and a-martingale measures. We now give the corresponding definitions.

m+-

DEFINITION 1. We call a semimartingale X = ( X ’ , . . . , X d ) a c-martingale if there ~ an M-integrable predictable positive exist a IWd-valued martingale M = ( M t ) t
-

+

DEFINITION 2. If there exists a measure P P, such that a semimartingale X is a a-martingale with respect to this measure, then we say that P is a semimartingale measure and the Ecr MM-property holds. Remark 1. As already mentioned, t,he term ‘a-martingale’ has been introduced in [loll. Before that such processes had been called (see, e.g., [73] or [137]) semirriarti,ngales of the class (&). We point out that a-martingales are special cases of martingale transforms (see Definition 3 in 3 l a ) . The following results can be seen as the culmination of the process of search of necessary and sufficient conditions for the absence of opportunities to arbitrage in its NA+-version.

THEOREM 2 ([loll). I n general semimartingale models

2. Semimartingale Models without Opportunities for Arbitrage. Completeness

657

R e m a r k 2. M. Emery’s example (5 la.5) shows that a a-martingale is not necessarily a local martingale. For a clearer insight into the connections between Theorems 1 and 2 and the corresponding results in the discrete-time case (see (3)) we reformulate them as follows. COROLLARY. In general semimartingale models X = ( l , X z ) l < i < d where , X2 = ( X j ) t < ~i = , 1,.. . , d , T < 00, we have IEMM ==+ ELMM

jEaMM

===+

m+l

(5)

For locally bounded semimartingales X = (l,XZ)l
i = 1,.. . , d , T

EMMI

ELMMI

/ EMM

EaMM-NA+

I

(7)

mgI

(8)

We have already established the implication *. The proof of the reverse implication is based on the following considerations. Let X = (1,X 1 , . . . , X d ) be a semimartingale. Then, as shown in [447], X satisX fies condition if anti only if the dzscounted prices satisfy condition NA+. g(X) Since - is a bounded semimartingale, it follows by (7) that there exists an

m,

~

dX)

equivalent measure, which proves (8). To complete our list of results we dwell on two counter-examples.

+

EXAMPLE 1 (EMM N A ) . We consider a ( B ,S)-market with Bt = 1and St = Wt, where W = (Wt)t>ois a standard Wiener process (the linear Bachelier model; see Chapter VIII, l a ) . For a self-financing strategy 7r = (0, y) its value is t

X;=pt+ytSt

= X $ + l yu.dSu.

We set T = inf{t: St = l} and yu = I(” X c = 0, then X: = 1 (P-a.s.).

<

T).

Then XF = X $

+ S,,

so that if

Chapter VII. Theory of Arbitrage. Continuous Time

658

Clearly, there exists in this case a martingale measure (namely, the Wiener measure); however, our choice of the self-financing strategy 7r = (P,y) with yu.= I(. < T ) shows that we have opportunities for arbitrage.

EXAMPLE 2 ( E L M M + NA, E L M M +

m+,but + NAg). Let XF = 1, t E [0,1],

and let

where

yt = exp(wt and W = (Wt)tGl is a Wiener process.

-

/

1

ysdXg = 1 for ys = 1, it follows The process X1 is a local martingale. Since 0 that we have arbitrage in its classical interpretation. Simultaneously, it follows from (2) that we have property As regards property NA,, it breaks down here: let us set (as in [447]) 7rF = 1 and 7ri = -1. Then X: = 0, X T = 1 - X ; 3 - g ( X t ) with g ( X t ) = 1 X ; and X r = 1.

m+.

+

4. All our previous discussions of arbitrage related to the case where the processes

describing the evolution of assets were semimartingales. It would be natural to consider now the issues of existence or nonexistence of opportunities for arbitrage also in the nonsemimartingale case. In the following two examples we consider (B, S)-markets in which the process S = (St)t>ois constructed from a fractional Brownian motion BW = (BF)t20 with < H < 1, which (as already mentioned in Chapter 111, 52c) is not parameter a semimartingale, and therefore lacks a (local) martingale measure. This feature is an indirect indication (cf. Chapter I, 52f.4) that there may be opportunities for arbitrage in the corresponding 'fractal' models; and indeed, they occur in the examples below.

EXAMPLE 3 (NELMM

+ A ) . Consider

a (B,S)-market of the following linear

structure: &El,

St=l+B!,

t30,

where BW= (Bp)t20 is a fractional Brownian motion with linear Bachelier model in Chapter VIII, 3 l a . ) We consider the (Markov) strategy 7r = (p,y) with

Then its value

(9)

<W

< 1.

(Cf. the

2 . Semimartingale Models without Opportunities for Arbitrage. Completeness

659

Using It6's formula (22) in Chapter VIII, 5c we obtain

dXT c d ( B y ) 2= 2Bi4dB,W. From (9) and (11) we see that

dXT = Y t dSt. This meails that the strategy 7r in question is self-financing. Since X $ = 0 and XT = > 0 for t > 0. it follows that there occurs arbitrage in this ( B , S ) market (in the class of 0-admissible strategies) a t each instant t > 0. From the financial point of view this is a fairly artificial example in which the prices St, t > 0 , can take negative values. Our next example does not have this deficiency.

EXAMPLE 4 ( N E L M M =+ A ) . Consider a ( B ,S)-market such that dBt = T Bd ~ t,

Bo = 1,

+

dSt = S t ( ~ d t adBF),

(12)

So

= 1,

(13)

where BW = (B?)t20 is again a fractional Brownian motion with view of formula (33) in Chapter III, 5 5c we obtain

< IHI 6 1. in

Bt = ert, st = ert+uBF We consider now the strategy

7r

pt

=

( p ,y) with

= 1 - e'UBtW,

Yt = 2 ( P t " - 1).

For this strategy we have

XT

= ,&Bt

+ YtSt = ert(euBF - 1)'.

Using It6's formula (32) in Chapter 111, § 5c we obtain dXT = Terf(euB? -

dt

+ 2g,Tt+uB?(,uB?

-

1) dBF,

and it is easy to see that the expression on the right-hand side is just the expression for ,& dBt Yt CtSt that can be obtained taking account of (14)-(17). Thus, dXT = Pt dBt yt dSt ,

+

+

which means that the strategy 7r defined by (16) arid (17) is self-financing. Since for this strategy we also have X $ = 0 and XT > 0 for t > 0, this model (as also the one in Example 3) leaves space for arbitrage (in the class of 0-admissible strategies) for each t > 0.

Chapter VII. Theory of Arbitrage. Continuous Time

660

5 2d.

Completeness in Semimartingale Models

1. By analogy with the terminology used in the discrete-time case (see Definition 4 in Chapter V, 5 l b ) we say that a semimartingale model X = (Xo,X1, complete (or T-complete) if each non-negative bounded 9T-measurable contingent claini f~ can be replicated, i.e., there exists an admissible self-financing portfolio T such that X ; = f~ (P-a.s.). Of course, tjliis property of replicability depends on the class of self-financing strategies under consideration. Recall that by the Second f u n d a m e n t a l theorem (Chapter V, 5 $ 4a, f) an arbitrage-free model with discrete tinie n N < oc and finitely many assets ( d < co) is complete if and only if the set of martingale measures consists of a single element, a nieasiire P equivalent to P.

<

-

2. We present below one s u f i c i e n t condition of completeness in general semimartingale niodels such that the corresponding class 9 ( P ) of equivalent martingale measures is non-empty. Under t,his assumption we have the following result.

THEOREM. A ~ S U I Ithat M the set of martingale measures contains a unique measiire P. T h there exists a strategy T in the class S F ( X ) such that X $ = f~ (P-;t.s.) for each pay-off function f~ with Eplff < tm.

Proof. This can be established in accordance with the following pattern (cf. the diagram in Chapter V, $ 4a): IS(P)I = 1

(1)

==+

21 ‘X-representability’ (====+ completeness.

Here the ‘X-representability’ with respect to a martingale measure p E 9 ( P ) means that each martingale M = ( M t ,.Ft, p)tGT defined on the same filtered prob( 9 : t ) t G T P) , as the p-martingale X has a representation ability space (0,.9,

Mt

= Mo

+

io”

(Ys,as),

t

< T,

where y E L ( X ) (cf. ‘S-representability’ in Chapter V, $ 4b). The implication (1) follows from general results due to J. Jacod (see [248; Chapter 111);for the purposes of arbitrage theory it has been stated for the first tinie by J. Harrison and S. Pliska [215]. The implication (2) can be proved as in the discrete-time case (see the proof of the lerrirria in Chapter V, 3 4b).

As regards particular exampIes of complete markets, see 3 $ 4 and 5 below. 3. Remark 1. The issues of the ‘X-representability’ of (local) on incomplete arbitrage-free markets have been considered, e.g., in [9].

2. Sernirnai tirigale Models without Opportunities for Arbitrage. Completeness

661

Rem,ark 2. We now dwell on the question of the relations between the concepts of arbitrage, (locally) martingale m m u r e , and completeness the reader conies across throughout the wholc book. It should he emphasized that each of these concepts can be introduced in,&pendently of the others. Both arbitrage arid conipleteness were defined in tcrms of the initial (‘physical’) probability measure P, and there was no word about a martingale measure. In the case of discrete times and finitely many assets ( N < cx;; d < w) it turned out (the First fundamental theorem) that the absence of arbitrage has a simple equivalent characterization: the existence of a martingale measure (9( P) # o ) , while the completeness in such arbitrage-free models turned out (the Second f u n damental theorem) to be equivalent to the uniqueness of the martingale measure

(IW? = 1).

However, if we take into consideration also more general models, then the absence of arbitrage does not require the existence of martingale measures, as shown in Example 1 in Chapter VI, 3 2b. In a similar way, the reader should not think (in connection with the Second fundamental theorem) that one can discuss completeness only when there are no opportunities for arbitrage or there exist martingale measures. It is formally quite possible that, e.g., a) there can be completeness both in arbitrage-free models and in models with arbitrage; b) there can be completeness in the conditions where a ‘classical’ (i.e., nonnegative) martingale measure exists, but is not unique; c) there can be completeness in the conditions where there exists no ‘classical’ martingale measure, but there exists a unique ‘nonclassical’ (i.e., of variable sign) martingale measure.

3. Semimartingale and Martingale Measures

5 3a.

Canonical Representation of Semimartingales. R a n d o m Measures. Triplets of P r e d i c t a b l e C h a r a c t e r i s t i c s

1. In the discrete-time case we discuss the canonical representation in Chapter 11, 3 l b and Chapter V, 3e. We recall the essential features of this representation. Let H = ( H n ,9n)n>0 be a stochastic sequence, let /in = A H , (= H , - H,-l) for n 3 1, and let g = g(z) be a bounded truncation function, i.e., a function with compact support equal to z near the origin (one often uses the function g ( x ) = zI(Iz1 1)). Then

<

n

Hn

==

n

n

HO t

and therefore, since the functions g ( h k ) are bounded, it follows from (I) by the Doob deconiposition that

k=l

k= 1

Taking into consideration the j u m p measures p k ( A ) = I ~ ( h k )A, E 9 ( R \ {0}), = P(hk E A 1 Sk-l),

I; 2 1, and their compensators Q ( A ) = E ( I ~ ( ( h l c )1 g k - 1 )

3. Semimartingales and Martingale Measures

663

we see from (2) that

In a similar way to Chapter V, 3e relation (3) can be rewritten as follows:

H =g

* v + g * ( p - v ) + (z - 9 ) * p.

(4)

We call (4) the canonical representation of the sequence H = ( H n ,9 n ) n b ~ . It is worth noting that for Elhkj < 00,k 3 1, we still have the representation (4) if we take g(z) = x. In this case the Doob decomposition of the sequence H = ( H n ,$:n)n>O has the following form:

H =z * v +z

* ( p - v),

(5)

2. We proceed now to a discussion of the canonical decomposition for semimartingales H = ( H t ,9 t ) t > o in the case of continuous time. Let g = g ( z ) be a truncation function. We set

Note that A H , - g ( A H , ) # 0, provided that lAHsI > b for some b > 0. Since for semimartingales we have C (AH,)' < 00 (P-a.s.) for each t > 0 (see (24) and (25) s
in Chapter 111, 3 5b), the sums in (6) contain actually only finitely many terms and V

H ( g ) is well defined as a process of bounded variation. The process H ( g ) = ff - &(9) (7) has bounded jumps (lAH(g)l b ) and therefore it is a special semimartingale (see Chapter 111, 5 5b), i.e., it has a canonical decomposition

<

H ( g ) = Ha

+ M ( g )+

a),

(8)

where B(g) = ( B t ( g ) , 9 t ) t > o is a predictable process of bounded variation with Bo(g)= 0 and M ( g ) = ( M t ( g ) 9t)taO , is a local martingale with Mo(g) = 0. From ( 7 ) and (8) we see that

H = Ho

+ M ( g ) + B ( g )+

[AH, - g(AHs)]

'

(9)

SQ.

This is a continuous analogue of (2). Now, to deduce an analog of (4) from (9) we shall need the concepts of random measure and its compensator.

664

Chapter VII. Theory of Arbitrage. Continuous Time

3. Let ( E ,6') be a measurable space.

DEFINITION 1. A r a n d o m measure in R+ x E is a family p = { p ( d t ,d x ; w ) ; w E

R}

of non-negative measures (R+ x E , g ( R + ) @ &) ~ ( ( 0 x) E ; w ) = 0 for each w E R.

satisfying

the

condition

EXAMPLE 1. A classical example of a random (and in addition, integer-valued) measure p is the P o i s s o n measure defined as follows. For A E %(R+) 8 8 let m ( A )= Ep(A;w), where m ( A ) is a 0-finite (positive) measure. We assume that m ( A ) < 00 for each A E B ( R + ) 8 & such that A c ( t ,m ) x E , where t E R+ is arbitrary, and the random variable p ( A , . ) is independent of the particular a-algebra 9:t. If the measure m (of intensivity) has the property m ( { t } x E ) = 0 for each t E R + , then we call p a Poisson measure. If, in addition, rn(dt, d z ) = dt F ( d x ) , where F is a positive a-finite measure, then we call p a homogeneous P o i s s o n measure. The attribute 'Poisson,' can be explained by the following property of this nieasure. Let (Ai)i21 be a sequence of pairwise disjoint measurable subsets of R+ x E such that m ( A ; )< m . Then the random variables p ( A i ) ,i 3 1, are independent and p ( A i ) has Poisson distribution with expectation 7 r ~ ( A i )i.e., ,

(See [250; Chapter 11. 3 lc].) In Chapter 111, $ 5 a , we introduced two a-algebras, B and 9, of subsets of R+ x R,which we called there the a-algebras of optional and predictable subsets. An important role in the applications random measures in - of integer-valued -

R+ x E is assigned to the a-algebras 6 = B 8 & and 9 = 9 8 8 , which are also called the optional and the predictable a-algebras of subsets of R+ x R x E . If W = W ( t ,w , z) is an optional function on R+ x R x E and p is a random measure? then we shall define by W * p = ( ( W* p)t(w),9:t)t2o the random process

where the intcgral is interpreted as the Lebesgue-Stieltjes integral for each w E and it is assumed that

R

3. Semimartingales and Martingale Measures

665

DEFINITION 2 . We say that a random measure p is optional (predictable) if the process W * p is optional (predictable) for each optional (predictable) function W = W(t,w,.). DEFINITION 3. An optional measure p is said to be @-a-finzte if there exists a .!%"-measurablepartitioning (An)n21of the set R+ x R x E such that each variable ( I A , * p)m is integrable. The next theorem is a direct generalization of Corollary 2 (Chapter 11, 5 5b) to the Doob-Meyer decomposition.

THEOREM 1. Let p be an optional @-+finite random measure. Then there exists a unique ( u p to P-indistinguishability) predictable random measure Y , which is called the compensator of p, that satisfies each of the following two equivalent conditions: (a) E(W * v ) =~E(W * p)m for each non-negative 9-measurable function W on B+ x R x E ; (b) for each $-measurable function W on R+ x R x E such that IWI * p is a locally integrable process, the process IWI * Y is also locally integrable and W * p - W * v is a local martingale. The proof, together with various properties of random measures and their compensators] can be found in [250; Chapter 111 or [304; Chapter 31.

Remark. One obvious property of compensator measures v is as follows. Let A E 8. Then the process X = ( X t ) t 2 0 with X O = 0 and

is a local martingale. The measure p martingale measure.

-

v is called for that reason a (random)

EXAMPLE 2 . For a Poisson random measure p introduced in Example 1 its compensator v is the intensity measure m.

* (p-Y) of a @-measurable function W = W ( t ,w , x) with respect to the martingale measure

4. We discuss now an important concept of stochastic integral W p - u.

If IWI * p E dzc,then IWI reasonable to set by definition

*v

E

dLCby Theorem 1 and therefore it seems

It is easy to show that the so defined process W has the following two properties:

* ( p - v) = ( W * ( p - ~

a) it is a purely discontinuous local martingale (see Chapter 111, 5b);

) t ) ~ > ~

Chapter VII. Theory of Arbitrage. Continuous Time

666

b) it makes the ‘jumps’

where

wt

= J W ( t , w , z ) p ( { t }x d z ; w ) -

s

W ( t , w , z ) v ( { tx} d z ; w ) .

This properties suggests the following definition (cf. [250; Chapter 11: 3 Id] and [304; Chapter 3, 51). I

DEFINITION 4. The stochastic integral W * ( p - v ) of a 9-measurable function W = W ( t ,w , z) with respect to the martingale measure p - v is a purely discontinuous local martingale X = (Xt)t>O such that the processes A X = (AXt)tgo and W = ( w t ) t g o are indistinguishable. We have already seen that if the compensator v of a measure p has the property IWl * v E (or, equivalently, lWl * p E then we can take the process W * p - W * v as a purely discontinuous local martingale X . However, the condition IWI * v E dZc,which ensures the existence of such a process X with A X = W can be loosened. Here we restrict ourselves to the introduction of the necessary notation and the statements of the corresponding results. We refer t o the already mentioned books [250] and [304] for detail. Let

dLc),

dLc

-

We assume that for each finite Markov time ~ ( w we ) have

3. Semimartingales arid Martingale Measures

THEOREM 2. Let W

=

667

W ( t ,w , z) he a @-measurable function such that

Then there exists a unique ( u p to stochastic indistinguishability) purely discontinuous local martingale W * ( p - u ) such that

a(w * ( p COROLLARY.

Let

@ = 0.

-

u)) =

-

w

If

then the stochastic integral W * ( p - u ) with respect to the martingale measure p - u is well defined as a purely discontinuous local martingale such that

(

q w * ( P 4 ) = pw w ,). -

Remark. Let

@ = 0.

F ( W x d;.

w)

).

Then

which is a consequence of the observation that

It clearly follows from the above equality (16) that for the predictable quadratic variation we have (W * ( p - u ) , * ( p - v)) = w2* u.

w

5 . Special cases of random (moreover, integer-valued) measures are presented by the jump measures pH of processes H = ( H t ,9 : t ) t 2 o with right-continuous trajectories having limits from the left (in particular, of semimartingales):

where A E B ( R \ (0)). Such a random measure p H in R+ x E (with E = R \ (0)) is @-a-finite, and therefore, by the above theorem, the compensator uH of p H is well defined. We consider the canonical decomposition (9) again.

668

Chapter VII. Theory of Arbitrage. Continuous Time

Using the random jump measure p H the last term on the right-hand side of (9) can be written as follows:

c

[A&

~

g(AHs)] = .(

~

gb))

* pH.

S<'

As mentioned in Chapter 111, 5 5b.6, each local martingale can be represented (and moreover, uniquely) as the sum of a continuous and a purely discontinuous local martingales. Hence the local martingale M ( g ) in (9) can be represented as follows: M ( g ) t = M b ) o + J - f ( g ) Z + M(dZ1, (17) where M(g)' is the contrnuous and M ( g ) d is the purely drscontznuous component. The continuous local martingale M(g)' is in fact rndependent of g, and, as mentioned in Chapter 111, 5 5b.6, is usually denoted by H C . As regards the purely discontinuous component which is a purely discontinuous local martingale, it can be represented as follows: XR

d.1

4PH

-

vH).

(18)

dLc

To prove this we must verify that, first, G ( g ) E and, second, the jumps of the local martingales on the right-hand and the left-hand sides of (18) are the same. It is proved in full in [250; Chapter 11, $ 2 ~ and 1 [304; Chapter 3 , 551. Here we consider one special case. Assume that v H ( { t }x E ; w ) = 0; then * U

H

and the local integrability of this process is a consequence of the fact that g = g(z) is a truncation function and of the inclusion (z2A 1) * uH E dzcholding for each semimartingale H . Hence the integral in (18) is well-defined. Further, A M ( g ) d= A M ( g ) = g ( A H ) - A B ( g ) ,where

([250; Chapter 11, 2.141). Hence, assuming that v H ( { t }x E ; w ) = 0 we see that, AB(g)t = 0, and therefore A M ( g ) d = g ( A H ) . However, Ag * ( p H - v H ) is also equal to g ( A H ) , and the jumps of the purely discontinuous local martingales on the right-hand and the left-hand sides of (18) are the same. Thus, from (9), in view of (17)-(18), we obtain the following representation:

H = Ho

+ B ( g )+ H C + g

* ( p H - U H ) + .(

g(z)) * p H ,

(20) which is called the canonical representation of the semimart,ingale H . Comparing (20) and (4) in the discrete-time case we see that their main difference is the presence of the continuous component H C in (20). -

3. Semimartingales and Martingale Measures

669

6. There exist two predictable terms in (20): B ( g ) and v H . The third important characteristic of H is its ‘cingular brackets’ ( H C ) which , is the (predictable) compensator of the continuous locally square integrable martingale H C .

DEFINITION 5. Let H = ( H t , g ; t ) t > obe a semimartingale arid let g = g(z) be a truncation function. We set B = B ( g ) :C = ( H e ) ,and v = v H . Then we call the collection

a triplet of predictable characteristics of H .

It must be noted that the components C and L/ of T are independent of our choice of the truncation function g = g ( x ) . On the other hand B depends on 9 . Moreover, if g and g’ are two distinct truncation functions, then B ( g )-

= (Y - 9’)

* v.

7. We now discuss several properties of semimartingales that can be stated in terms of predictable characteristics B , C , and v. a) If H is a semimartingale, then (2’

i.e., the process

(.l(o,t,

XR

(2’

A

A 1)dv)

1) * v E

t>o

&lo,,

is locally integrable. In other words, there

exists a sequence of Markov times r,, r,

t 00

(P-a.s.) such that

b) The semimartingale H is special (in particular, it is a local martingale) if and only if (x2 A )1.1 * v E dloc. c) The semimartingale H is locally square integrable if and only if

x2 * v

€

,.e,.

The proof of a) is based on the inequality

C (AH,)’ < 00 (P-a&), t > 0, which

sst

holds for semimartingales; see Remark 3 in Chapter 111, 3 5b. As regards the proofs of properties b) and c), see [250; Chapter 11, 3 2b]. d) If H = Ho+N+A is a canonical decomposition of a special semimartingale H , then H = Ho H e X * ( p - V ) + A . (22)

+

+

Chapter VII. Theory of Arbitrage. Continuous Time

670

In other words, for special martingales H we can take g(z) = z in the canonical decomposition (20). e) We associate (for each 8 E R) the triplet T = ( B ,C , v) of predictable characteristics of a semimartingale H with the following predictable process of bounded variation: Q(0)t =

i0Bt

-

82 -ct + 2

pQZ

-

1 - i B g ( z ) ) v ( ( 0 ,t] x dz;w),

(23)

which is called the cumulant (of H ) ; cf. Chapter 111, 5 l b Let G(Q)=

qQ(Q)),

(24)

where E ( Q ( 8 ) ) = ( 8 ( Q ( Q ) ) ) tis> the o stochastic exponential constructed from @ ( 8 ) (see Chapter 111, 5 5c, Example 1): 8(Q(B)),= eIk(0)t

n

(1 + AQ(8),)e-a13'(Q)s.

(25)

O<s
THEOREM 3 . Let AQ(B)t# -1, t > 0. Then the following properties are equivalent: 1) H is a semimartingale with characteristics ( B ,C , v); 2) for each 0 E R the process

is a local martingale.

(As regards the proof based on It8's formula for semimartingales see [250; Chapter 11, § 2dl.) f ) The most simple processes H = ( H t ,9 ; t ) t 2 0 in the class of semimartingales are ones with independent increments. Their characteristic feature is that the corresponding triplet T = ( B , C , v ) is non-random. In other words, B = (Bt)t>o, C = (Ct)tao,and the compensator measure v = v ( d t ,d s ) are zndependent of w (see [250; Chapter 11, 3 4 ~ 1 ) . Hence the cumulant q ( 8 ) of such processes is independent of w and if A s ( 0 ) # -1, which corresponds to the case of a process H = ( H t ) continuous in probability, then for Ho = 0 we obtain from (22) the Le'vy-Khintchine formula

1

1 - iQg(z))v ( ( O , t ]x d s ) . (27)

For Lkvy processes we have

3. Semimartingales and Martingale Measures

671

where

(The function $(0) is called the cumulant along with Q(0)t.) The measure Y = v(dx) satisfies the conditions

~((0)= ) 0,

(x2A 1) * v < 00

(29)

and is called the Le'vy measure. (Cf. Chapter 111, 3 lb.) g) We consider now a special case of L6vy processes: 'a Brownian motion with drift and Poisson jumps'. More precisely, let

Ht

= mt

+ CWt +

Nt

(k,

(30)

k= 1

where W = (Wt)t>ois a Wiener process (Brownian motion), [I, &, . . . are independent identically distributed random variables with distribution function F ( z ) = P(& x),and N = (Nt)t>o is the standard Poisson process with parameter X > 0 (ENt = A t ) . We assume that W , N , and ([I, (2,. . . ) are jointly independent. The following chain of relation brings us easily to the canonical representation, giving us the triplet of predictable characteristics:

<

Hence

Chapter VII. Theory of Arbitrage. Continuous Time

672

h) Random sequences H =

(Hn,9,),~o with discrete time can be fitted in a

natural way into continuous-time schemes (see Chapter 11, 3 If). If H , = Ho+

n

11k k=l

with h k = AH,, then the triplet T = ( B ( g )C , , u ) has the following structure:

with A E B ( R \ (0)).

5 3b.

Construction of Marginal Measures in Diffusion Models. Girsanov’s Theorem

1. The results of 5 3 2b, c on necessary and sufficient conditions of the absence of arbitrage demonstrate that it is important for arbitrage theory to find martingale or locally martingale measures equivalent to the original probability measure One, fairly common, method of the construction of martingale measures is based on Girsanov’s theorem and its various generalizations. Another method, long known in actuarial studies, is based on the Esscher transformation (see 3 3c below). We presented Girsanov’s theorem in its original formulation from [183]in Chapter 111, 5 3e. In the present section we shall prove it, discuss some generalizations, and suggest criteria for the local continuity and the equivalence of probability measures corresponding to diffusion and It6 processes.

2. We consider a process X = ( X t ,9 t ) t > o , defined on a filtered probability space (0,9, (g:t)t>o, P), which is an It6 process (see Chapter 111, 3d) with differential dXt = a t ( w ) d t

+dBt,

”0 = 0 ,

where a = ( a t ( w ) ,9 t ) t 2 0 is a process satisfying the condition

and B = ( B t ,9 t ) t > o is a standard Brownian motion. Assume now that ii = ( & ( w ) , 9 t ) t 2 0 is another process and

(1)

3. Semimartingales and Martingale Measures

673

Then there exists a well-defined process 2 = ( Z t ,9 t ) t 2 o (cf. (21) in Chapter 111, 5 3d) with

which is a nonnegative local martingale, e.g., for the localizing times

By Fatou’s lemma this process is a (nonnegative) supermartingale, and therefore by Doob’s convergence theorem (see Chapter 111, 5 3b) there exists with probability one a finite limit lim Zt (= Zoo). t+oo

Let EZ, = 1. (5) (This is equivalent to the uniform integrability of the family {Zt, t 3 O}.) Then we can define another probability measure F on (Q, 9) by setting

dlp

=

Zoo d P .

(6)

THEOREM 1 (I. V. Girsanov, [183]). The process B = ( B t ,9tt)t>O with I

Bt = Bt

-

bt

(as -

a,) ds

(7)

-

is a standard Brownian motion with respect to the measure P and dXt = & ( w ) dt

-

+ dBt.

(8)

We present the proof in subsection 3, while here we discuss several consequences and observations.

COROLLARY 1. Let

rt

X t = Bt

where X E

-

X I o - a, d s ,

(9)

R,and let

Assume that EZA = 1 and set d p X = Z& d P . Then the process X = ( X t ) t 2 0 is a standard Brownian motion with respect to the measure PA. If EZ; = 1 for some finite T , then X = ( X t ) t > ois a standard Brownian motion on the interval [0,T] with respect to the measure p$ such that d F i = Z$ d P T , where PT = P I 9T.

Chapter VII. Theory of Arbitrage. Continuous Time

674

COROLARY 2. Let

T

= T ( W ) be a finite Markov time and Jet

(

Eexp XB,

3 2

- -7-

(

We set dPx = 2; dP, where 2; = exp XB,--7with

X t = Bt

-

X . (t A

= 1.

. Then the process X = ( X t ) t 2 0

7)

is a standard Brownian motion with respect to the measure

PA

Remark 1. We formulated Girsanov’s theorem in Chapter 111, § 3 e under the assumptions that 0 < t T and EZT = 1. This is in fact a special case of the current setting, where 0 t < 00, for we can set at = 0 and Zit = 0 for t > T .

<

<

Remark 2. We know of some conditions ensuring property (11). These are (see, e.g., [288], [303], or [402]):

Eei7

< cx, (‘the Nowikou condition’)

(12)

and sup EeiBTAt< 00 t>o

Since sup EeiBTht t>O

(‘the Kazamaki condition’).

(13)

< ( E e i 7 ) l l 2 , condition (13) is looser than (12).

If we set, e.g., r = inf{t: Bt = l}, then E f i = 00, and therefore Eei7 = 00. Thus, (12) fails in this case. Nevertheless, condition (13) holds for such times, too: 1

E ~ ~ ~ -( zT) B , = 1. If the stopping times T are Markov with respect to the flow ( 9 F ) ~ generated o by a Brownian motion, then we can relax (12) and (13).

THEOREM 2 ([282]). Let cp = y ( t ) be a nonnegative measurable function such that

lim (Bt - y ( t ) )= +cc

t-tm

(P-a.s.),

(14)

and let r be a Markov time with respect to the Aow ( 9 F ) t > o . Then each condition.

lim

1 sup Eexp{ - ( r A o)- y ( r A o)} <

N+oo uEm?

2

00

3. Semimartingales and Martingale Measures

675

or

where !3Xf is the class of Markov times 0 (with respect to ( 9 ? ) t > o ) such that P(0 u N ) = 1, is sufficient for the equality

< <

Remark 3. For 2’ = (2:)tao defined by (10) the corresponding ‘Nouikov and Kazamaki conditions’ can be stated as follows:

As regards the proofs and extensions to processes 2 = ( 2 t ) t 2 0 with

zt

= exp{lt

1

-

i(L,L)t},

(20)

(

where L = (Lt)t>ois a continuous local martingale e.g., Lt = see [288], [303],or [402].

1t a,(w) dB,, t 3 0 ) , 0

3. Proof of Girsanov’s theorem. As in the discrete-time case (see Chapter V, $3b), it suffices to verify that (P-a.s.)

E-P [,i@(&-Es) for 0 6 s 6 t and 6 E R. To this end we set a , = ?is

-

a,,

02 1 9,] = e- T ( t - s )

(21)

& = Bt - 10 a , ds, and t

(see (7) and (10)). By Bayes’s formula (Chapter V, 5 3a),

02

and we claim that the right-hand side of (22) is equal t o e - y ( t - - s ) .

Chapter VII. Theory of Arbitrage. Continuous Time

676

For simplicity we shall consider the case of s = 0. Let Ut = eisBt. Then by It6’s formula (Chapter 111, 5 5c) we see that I

i.e.,

ZsUs(a, + ie) dB,

~

Hence, using similar argunlerits to the proof of Lkvy’s theorern (Chapter 111, 5 5a), we obtain the following equation for EpZtUt:

from which we conclude that

Formula (21) can be verified in a similar way, which proves that the process Relation (8) is a consequence of (1) arid (7). This complete the proof of the theorem. = ( & ) t a o defined in ( 7 ) is a standard Brownian motion.

+

4. Hence if the process X has the differential d X t = a t ( u ) d t dBt with respect to the measure P, then its differential with respect to the measure is defined in (6) is d X t = & ( w ) d t dBt, where B = (Bt)t20is a Brownian motion with respect to P. It should be noted that if all our considerations proceed for a time interval [O, TI,

+

-

-

-

I

where T can be also a Markov time, then we can replace the condition EZ, by a weaker one, EZT = 1.

Remark 4. Assume in Girsanov’s theorem that Zt

-

Bt = Bt

+

1

3

t

a, d s ,

t

= 1

0, i.e.,


and

If EZT = 1, then the process

xt = Bt +

Lt

a,ds

coincides with & and is a Brownian motion with respect to the measure PT such that dF, = Z, dP (cf. Chapter V, 5 3b), therefore it is a martingale measure. I

3. Semimartingales and Martingale Measures

677

5. Let X be an It6 process on the (filtered) space (C,%,(%‘t)t>O) of continuous functions w = ( w t ) t > o with differential (1) and let px = Law(X 1 P) be the probability distribution of this process. Girsanov’s theorem is a convenient tool in answering the question when the restrictions pf = px 1 %t of the measures px are absolutely continuous, equivalent, or singular with respect to the measures p? = p B I ‘&. The measure p B is just the Wiener measure (Chapter 111, 5 3a), so that we are discussing the properties of the measure px corresponding to the process X in its relations to the Wiener measure p B . If p: << ,I? or pf << p:, then one would dP? also like to have ‘explicit’ expressions for the Radon-Nikodym derivatives __ dPB dPt” and

dP?

These issues are studied in much detail in [303; Chapter 71 for the case of It6 processes arid in [250; Chapters 111-V] for semimartingales (through the use of the Hellinger distance and Hellinger processes). For that reason we restrict ourselves to several results. We consider some time interval [0, TI. Assume that

1 Lt

P ( i T a ; ( w ) d s < 00 = 1.

(23)

Then there exists a well-defined process Z = (Zt)t
(- 1 t

2, = exp

THEOREM 3 . I f EZT

a,(w)d B , - 2

a&)

ds)

.

= 1, then

d

(25)

and

where 9 : = a(w:

x,(w),

s


Proof. We define a measure r?, by setting d r ? ~= Z T ~ P T(cf. ( 6 ) ) , where -PT = P 1 9 ~ .The process X = ( X t ) t < T is a Brownian motion with respect to PT by Girsanov’s theorem, and therefore p $ ( A ) = P,(X E A ) =

b

w:

X(w)EA}

ZT(W) P(dw)

Chapter VII. Theory of Arbitrage. Continuous Time

678

Let @ ~ ( zbe) a %+measurable functional such that E[ZT I 9 $ ] ( w ) = @ ~ ( x ( w ) ) . Then from (27), using the formula of the change of variables in a Lebesgue integral (see, e.g., [439; Chapter 11, § 61) we deduce the equality

and therefore pf! << p g . In addition,

and

) 0) = 1, We claimnow that p$ << p g . To prove this we observe that P T ( Z T ( ~> and therefore PT << (cf. Chapter V, 5 3a) and

PT

Hence

6. We consider now the special case when an It6 process X is a dzffuszon-type process, i.e., let u t ( W ) = A ( t , X ( w ) )in (l),where A ( t , z )is a nonanticipating functional ( A ( t x) , is measurable in ( t ,x) and for each fixed t the functional A ( t ,z) is %&measurable in z). In this case, for dXt = A ( t ,X ) d t dBt, (33)

+

we can deduce from Theorem 3 the following result.

3. Semimartingales and Martingale Measures

6 79

THEOREM 4. Let T

P(k

A 2 ( t , X ) d t < m) = 1

(34)

and let Eexp(-lTA(t,X)dBt-

(35)

or, equivalently,

Then

-

,LL$

pg,

and

Remark 5 . If we give up the condition EZT = 1 in Theorem 3, then we can prove (see [303; Theorem 7.41) that

and

7. Comparing Theorems 3 and 4 we see that while we have ‘explicit’ formulae (37) and (38) for diffusion-type processes, the corresponding formulas for It6 processes (see (26)) require the calculation of the conditional expectation E(. Is$).The following result, which is also of independent interest, is useful in the search of ‘explicit’ formulas for the Radon-Nikodyrn derivatives also in the case of It6 processes since it allows one to ‘transfer’ the conditional expectation in (26) under the integral sign.

Chapter VII. Theory of Arbitrage. Continuous Time

680

THEOREM 5 . Let X be an It6 process with differential (1). where

Let A ( t ,x) be a nonanticipating functional such that

Then the process

3 = (&)t
with

Bt = X t -

1"

A ( s ,X ( W ) )ds

is a Brownian motion (with respect to the Aow (.F:)tg) I f , in addition to (41),

(46) T

P(Jd

A2(t,B)dt <

-)

=1

(47)

and

-

Proof. The fact that the process B = (Bt)tGT, which in the case of 9 : = 9x t ( t 6 T ) is called an innovation process, is a Brownian motion, can be easily proved

3. Semimartingales and Martingale Measures

681

on the of Itb's formula for semimartingales (Chapter 111, 3 3d) as applied to - basis e2X(Bt-B,) for t 3 s and X E R.Jn fact.

and

t

- -

$>:I

= E [ L e"(Hu-B5)E(a,(w) - A ( u , X ( w ) )

]:.I

du

= 0,

we see from (50) that (P-a.s.)

and therefore (P-a.s.)

so that B is a Brownian motion. (The trajectories of (Bt)tGT are continuous by (431.) To prove the remaining assertions it suffices to observe that

and to use Theorems 3 and 4.

Chapter VII. Theory of Arbitrage. Continuous Time

682

8 . Theorems 1-5 can be generalized (see [303; Chapter 71 for greater detail) to multivariate processes X or to the case where (in place of diffusion coefficient 1) we allow the diffusion coefficients in (1) and ( 3 3 ) to depend on X and t . We present the following result obtained in this direction. Let X = ( X t ) t < T be a diffusion-type process with

+

d X t = a ( t ,X ) d t P ( t , X ) d B t , (52) where a ( t ,x) and p(t,z) > 0 are nonanticipating functionals, the stochastic integral

Jut

P(s, X ) d B s

/ T Ia(s,X)I d s <

is well defined for t 6 T , and

(P-a.s.).

00

0

Let G ( t ,z) be another nonanticipating functional with ST1G(s, X ) l ds O a.s.) such that

We set q s , X ) - a ( s ,X )

zt = ex&){

P(s,X

)

f l(

< 00 (P-

) - a ( s ,X ) )2ds}. @(S,X)

G(S,X

dBS

-

(54)

If EZT = 1 and is the measure such that d p T = ZT d P T , then X is a diffusionand its differential is type process with respect to

PT

dXt where

= G ( t ,X

)dt

+ @(t,X ) d&,

is a Brownian motion (with respect to

EXAMPLE. Let X t

i.e., a ( t , X ) =

(

,111

(55)

PT).

= emt+aBt. Then

3

+-

X t and p ( t , X )= a X t . We set G ( t ,X ) = 0. By (54),

-

I

and if a measure has the differential ~ P = T ZT d P , then the process X = ( X t ) t 6 ~ has the stochastic differential

-

dXt = UXt d B t with respect to PT. In other words, the process X = (Xt)t
xt = exp{ a&

-

$}.

3 . Semimartingales and Martingale Measures

683

fj 3c. Construction of Martingale Measures for L&y Processes. Esscher Transformation 1. If X = ( X t ) t Gis~a process of diffusion type with respect to the original measure PT, with local characteristics a ( t ,X ) and p(t,X) (see formula (52) in 3 2c), - then Girsanov’s theorem suggests an explicit construction of another measure PT such that X has local characteristics G ( t , X ) and B ( t , X ) with - respect to PT. - If now CY(t,X ) = 0, then X is a local martingale with respect to PT; one calls PT a locally martingale measure for that reason. The construction of this measure proceeds by the formula

-

dP, =

z,dPT,

(1)

where Z, is defined by equality (54) in the preceding section. The Esscher transformation suggests another construction of a new measure, which is essentially based on the same idea. Namely, assume that the initial process X = ( X t ) t G has ~ independent i n c r e m e n t s (e.g., is representable as the sum of independent random variables). 2. Recall that we encountered already the Esscher transformation and its generalization (‘the conditional Esscher transformation’) in Chapter V, § 2d (see, in particular, the remark in subsection 2 ) . It should also be noted that, as a the method of the construction of ‘risk-neutral’ probability measures assigning ‘larger weights to adverse events’ and ‘smaller weights to beneficial events’ the Esscher transformation is known in actuarial practices since 1932, when F. Esscher’s paper [144] was published. For instance, insurance companies are based in their calculations of life-insurance premiums not on the (quite precisely known) distribution PT of life expectancy (‘mortality tables of the second kind’), but on another, different, distribution PT (‘mortality tables of the first kind’) that has the above property of shifting the balance between beneficiary and adverse events.

3. Before a discussion of the Esscher transformation in the general case we consider the following simple example (cf. Chapter V, fj 2d), which is a good illustration of the true meaning of this transformation. Let X be a real-valued random variable with Laplace transform @(A) = EeXX < a, X E R,and let P = P ( d z ) be its probability distribution on (R, B ( R ) ) . We consider a f a m i l y of probability measures P(.), a E R,defined by means of the Esscher transfornation:

Setting

Chapter VII. Theory of Arbitrage. Continuous Time

684

we see that

,da)(s)> 0, EZ(a)(X)= 1, the measure P(.)

is equivalent to P, and

P(U)( d z ) = d a (z) ) P(dz).

(4)

It is also clear that

and therefore

Al=i

Ep(a)X =

@"u)

.(a)

'

(6)

We showed in Chapter V, §2d that if a random variable X has the properties P(X > 0) > 0 and P(X < 0) > 0, then the function @ ( a )attains its maximum a t some point iZ, where, obviously, @(E) -= 0. E,,g,X = 0 with respect t o the meaHence we obtain the expectation EX sure P = P('), which one sometimes expresses by saying that P is a 'risk-neutral' probability measure. We can consider the property EX = 0 also as a 'single-step' version of the martingale property. which explains why one also calls P a martzngale measure. I

4. Now let X = (Xt)t
function (see (27) in $ 3 a here and Chapter 111, 3 l b j &Xt

(7)

= ,t@(@),

where the cur/r,dantis

and g ( z ) is a truncation function (e.g., g ( z ) = z l ( / z l 6 1)). From (7) and (8), setting formally Q = - i X we see that

where p(A) = X b

+ -cA22 + l ( e A z - 1

-

X g ( z ) ) v(dz).

(10)

The easiest way to a rigorous proof of the representation (9)-(10) for the Laplace transform is based on the following observation: the process

3. Semimartingales and Martingale Measures

with

zp = exp{ x x t

-

@(A)}

685

(11)

is a martingale. (This is an immediate consequence of It6's formula for sernimartingales. Of course, one must also assume that the integral in (10) is finite.) By analogy with (2) we introduce now for each a E R the probability nieasiire

Pg) defined by means of the Esscher transformation: dP$) =

zpdPT.

(12)

THEOREM 1. The process X = (&)t
where p'"'(X) = p(a

+ A)

-

p(a).

(14)

Proof. This is a consequence of Bayes's formula (Chapter V, $3d), which states that (Pg)-a.s.)

Hence if the measure Pg' is defined by the Esscher transformation ( l a ) , then X = (Xt)tcris a L6vy process also with respect to this measure and its Laplace transform is defined by (13) and (14). (Cf. Girsanov's theorem in 4 3b.)

THEOREM 2 . The local characteristics

( d a )c("), , ~ ( ~ of1 the ) process X

= (Xt)tGT

with respect to the measure P g ) , a E R,can be determined from the local characteristics ( b , c, u ) by the following formulas (where g ( x ) is a truncation function):

686

Chapter VII. Theory of Arbitrage. Continuous Time

Proof. By Theorem 1 the process X is a L6vy process with respect to P$) and

+

Bearing in mind that p(")(X) = p(a A) - p(X), X E 'transition formulas' (15)-(17) by (18) and (10).

R,we immediately obtain

5 . Let X = ( X t ) t be ~ a~ LCvy process (with respect to the measure PT). Since it is a semimartingale, this process has a (non-unique, in general) canonical decomposition X = M + A , where M is a local martingale and A is a process of bounded variation. We consider now the following question: what conditions on the local characteristics ( b , c, V ) make X a local martingale (with respect t o PT). We can express this otherwise: we are interested in the conditions making PT a martingale measure for the process X. First we must point out that if X is a local martingale, then it is a special semimartingale, and therefore, with necessity,

(see (20) in 5 3a). Let X be a special semimartingale, let

be its canonical decomposition (where N is a local martingale and A is a predictable process of bounded variation), and let

X

=B

+ XC+ g * ( p - V ) + (z - g ) *

,LL

(21)

be its canonical representation, which (in view of (19)) can be rewritten as follows:

Comparing (20) and (22) we see that

and therefore we can say that a special semimartingale X with triplet of predictable characteristzcs ( B ,C, V ) is a local martingale if and only if

B+

(Z -

g)

* v == 0.

(24)

3. Semimartingales and Martingale Measures

687

All this shows that a L6vy process X with triplet of local characteristics (6, c, v ) is a special semimartingale if and only if

J(2A 1x1) v(dz) < co

(25)

(see (20) in 3 3a), and it is, in addition, a local martzngale if and only if b

+

L!

(26)

(z - g(z)) v(d5) = 0.

If g ( x ) = z I ( l z (6 l), then condition (26) assumes the following form: 5 +

v(d5) = 0.

L,>1

Of course, it may happen that (26) breaks d o w n with respect to the initial measure P. Then one could consider the measures P g ) constructed by means of the Esscher transformation. Since the 'new' local characteristics (dU), c("), ~ ( ~ can 1 ) be defined by (15)-(17), it follows by (25) and (27) that a L6vy process is a local martingale with respect to the measure P g ) if and only if

and

+ a c + Jz,>lz v(dx) + ;I,II:(eaz I>v(dz) = 0. (29) EXAMPLE 1. Let X i = mt + aBt + k N t , where B is a standard Brownian motion b

-

and N is a standard Poisson process with parameter v > 0 (ENt = v t ) . We now find a value of a such that X = (Xt)t
~g).

measure We represent X t as follows:

Xt = (m

+ k v ) t + aBt C k(Nt

-

vt),

(30)

and see that it is certainly a martingale with respect to the original measure, provided that m k u = 0. (31)

+

+

Assume now that m k v # 0. Then the process ( k N t ) t a o makes jumps of amplitude k , the Lkvy measure is v(dz) = v . I ( k } ( d z ) ,

688

Chapter VII. Theory of Arbitrage. Continuous Time

and the local characteristics b and c of X (for the truncation function g ( x ) =

zI(lxl 6 k ) ) have the following form:

b =m

+ kv,

2

c = o .

By (29) we obtain the following equations for a (in view of our choice of the truncation function g(z) = zI(1zl 6 k ) , integration over the set {z: 1x1 > l} is replaced by integration over {x: IC > k } ) :

( m + kv) + ao2 + v q e a ’ c - 1) = 0 .

(33)

If o2 # 0, then this equation has a root a, and therefore X becomes a martingale with respect to the measure P g ) . On the other hand, if o2 = 0, then the root a can be found from the equation

which is soluble if m is not zero arid of distinct sign from k . 6. We assume now that, the price process S = (St)t
St = ex,

(35)

Below we consider that following question, which is important for the probleni of the absence of arbitrage: is the process S a martingale with respect to the initial measure PT or some measure Pg) constructed by means of the Esscher transformation? As in subsection 3, we start from a simple example. Let X be a real-valued random variable and let @ ( a )= Eeax. (We assume here that @,(a)< 00, a E R.) Clearly, if Q(1)= 1, then the random variable S = ex has the ‘martingale’ property ES = 1 (with respect to the original measure). If cP(1)# 1, then we can look for Z such that this ‘martingale’property E (Q)S = 1 (Plholds with respect to the measure defined by the formula (2). Since

the value of iZ must be a root of the equation

@(a

+ 1) - q u ) = 0.

(37)

3. Semimartingales and Martingale Measures

689

For instance, if X is a normally distributed random variable with parameters m and g 2 , then (no)'

@.(a)= EeuX = e u m + T ,

(38)

and we find from (37) that

We consider now the process S = (St)tGT defined in ( 3 5 ) . The first question to ask here is about the conditions ensuring that this process is a martingale with respect t,o the original measure PT.

THEOREM 3. In order that the process S it is sufficient (and also necessary) that

and

1 b + -c 2

+ (eZ

-

= e x be a martingale

1 - g(z))

*v

with respect to PT

= 0.

Proof Condition (40) together with the inequality ( x 2 A 1) * v the integral (eZ - 1 - g(z)) * v is finite. Clearly, the expectation

(41)

<

30

ensures that

where p(1) is as in ( l o ) , is precisely the expression of the left-hand side of (41). Hence ~ ( e 1 c~ ~ ts=) e x s ,

which proves that S = ex is a martingale. (The necessity of (40) arid (41) is shown in [250; Chapter X, 52.1.)

7. Assume that (41) fails. Then by Theorem 1,

Hence it is clear that if U is a root of the equation

then the process S = ( S t ) t Gis~ a martingale with respect t o the measure P,(4 By (43), in view of (18), we obtain the following result.

Chapter VII. Theory of Arbitrage. Continuous Time

690

THEOREM 4. Assume that iT satisfies the inequality leaz(ez - 1) - g ( z ) l * L/

and b

+ (i; + i ) c + (eaz(e"

-

< oo

1) - g(z))

* v = 0.

(44)

Then the process S = ( S t ) t c is ~ a martingale with respect to the measure P,(Z) .

EXAMPLE 2. Let St = e x t , where X = (Xt)t>o is the process introduced in Example 1. Then equation (44) takes the following form:

If o = 0 and k # 0, then (45) becomes the equation m eak = n ( e k - 1) ' which certainly has a solution if m # 0 and k and m have distinct signs. If k = 0, then St = emttnBt

(47) is a standard geometric Brownian motion (Chapter 111, 5 4b). By It6's formula

l m From (45) or (39) we find that iZ = - - - - and 2 o2

Since

it follows that

-)

Law(Xt I ~ $ 9 = )N ( - -cr2t , o2t 2 2 This means that the process ( X t ) t < has ~ the same distribution with respect t o 02

the measure P g ) as (crWt - - t ) where W = (Wt)tcT is a standard Wiener 2 t
Remark. As regards the Esscher transformation and its applications to option pricing, see H. U. Gerber and E. S. W. Shiu [178], [179].

3. Semiinartingales and Martingale Measures

3 3d.

691

Predictable Criteria of the Martingale Property of Prices. I

1. We devote this and the next sections to predictable criteria (i.e., criteria in terms of the triplets of predictable characteristics of a process in question) ensuring that the prices, the semimartingales S = (St)t>o,are martingales or local martingales (with respect to the initial measure P or some measure P << P; cf. Chapter V, 53f for discrete time). We start with the observation that different representat,ions of prices can be convenient in different problems. If S = (St. 9 t ) t a o is a semimartingale, then, by definition, I

st = So f a t + 7 n t ,

(1)

where a = ( a t , 9 t ) t > 0 is a process of bounded variation and rn = (rnt,S:t)t>ois a local martingale. This decomposition is not uniquely defined. For instance, if S = (St)t>ois a standard Poisson process (So = 0 and ESt = A t ) , then we can set either at = St; ‘rnt = 0 or at = A t , mt = St - At in (1). Expansion (1) is ‘additive’. However, if S = (St)t>o is a special positive semimartingale and (1) is its decomposition with positive process a = (at)t>o, then, provided that StAat # 0, S has the multiplicative decomposition

+

where O<s
is the stochastic exponential and the processes 2 = (&) and % = (&) defined by the formulas

can be

Formula (2) can be easily established on the basis of ItA’s formula; see the details in [304; Chapter 2. $51. The product of two stochastic exponentials in ( 2 ) can be, by Yor’s formula (see (18) in Chapter 111, 5 3f), rewritten as a single exponential: (@)t&(&)t = & ( H ) t ,

and

(5)

692

Chapter VII. Theory of Arbitrage. Continuous Time

Hence we obtain the following representation for (St)t>o:

which is very useful in the analysis of this process 'for the martingale property', because the stochastic exponential 8(H)is a local martingale if and only if @ is a local martingale. We have already mentioned (Chapter 11, 5 l a ) that, from the standpoint of statistical analysis, in place of (8) one could more conveniently use a representation of 'compound interest' type

St =

(9)

with some semirnartingale H = (Ht)t>o. The representation (9) is usually chosen in financial mathematics as the starting point, while the transition from (9) to (8) proceeds by the formula

which can be written in the .difference-differential' form as follows:

To prove (10) we observe that, by ItA's formula for f ( H ) = e H we obtain

On the other hand, by (8)and the properties of stochastic exponential,

Comparing (12) and (13) we arrive to formula (10).

Remark 1. The infinite sum in (10) is absolutely convergent (P-a.s.) since for each seminiartingale H there exists (P-as.) only finitely many instants s 6 t such that

lAH,I >

1

( A H , ) 2 < 00

and

(P-as.);

O<s
see Chapter 11, 55b. For the same reason the infinite product in the definition of the stochastic tlxponential (3) is also absolutely convergent.

3 . Scminiartingalrs and Martingale Measures

693

2. Let H be a serriirnartingale arid let

be its canonical representation (with respect to some truncation function g = g(z); here p = ,uH is the jump measure of H and v = v H is its compensator; see fj 3a). By (10) and (14) we obtain the following representation for H : A

ii = H + 21( H C )+ (e" -

= Ho

/wI

-

1 - ).

* ,u

1 + B + H C + -(If") +9 * (p 2

-

v)

+ (x

-

g(z)) * p

+ (ez

-

1 - z)

* p. (15)

To transforrri the right-hand side of (15) we use the fact that IWl *v E aritl, moreover.

.dLc

* ,u E d&

if

(see 5 3a). Hence we see from (15) that if

then

where H C

i? = K + Ho + H C + (e"

+ (e"

-

1) * ( p

K

~

-

1) * ( p - v).

v ) E J&~(,~(P) and

=B

+ 21( H ' ) + (e" -

- 1 - g ( x ) ) * v.

Thus, the following result is a conseqiierice of (18).

THEOREM. Assume that condition (17) is satisfied. Then (P) if and only if

E

Aloc(P)

and S E

dlloc

Kt = 0

(P-a.s.),

t > 0.

In that case the local martingale H has the representation

(19)

Chapter VII. Theory of Arbitrage. Continuous Time

694

EXAMPLE. Let H be a L6vy process with triplet ( B ,C , u ) of the following form: Bt = bt, Ct = c2t, ~ ( d td, ~ =) d t F ( d z ) , where F = F ( d z ) is a measure such that F ( ( 0 ) ) = 0 and

(21)

/ ( 2A 1)F ( d z ) < m.

(22)

We can assume also a stronger version of (22): /(lzl~(lzl

< 1) + e " l ( / z l > 1))~ ( d z<) m.

(23)

Under this assumption the price process St = SOeHt is a martingale (with respect to the initial measure P) if ( b , c2,F ) satisfies the following relation: (24)

IF Bt = Boert is a bank account, then the discounted price process

S = B

-

is a martingale with respect to P if b

+ O22 + ~

I

(2- 1 - g(z)) F ( d z ) = T .

(25)

Remark 2. In accordance with notation (10) in the preceding section the left-hand side of (24) and (25) is the value of the 'cumulant' function p ( X ) for X = 1. Hence formula (25) can be put in the following form: p(1) = r.

(26)

<

Remark 3 . As Theorem 3 in 3c shows, the condition / ~ z ~ I ( 1) ~ Fz( d~z ) < m is redundant for LCvy processes. (This condition is a result of the 'regrouping' in (15) based on formula (16) .)

5 3e.

Predictable Criteria of the Martingale Property of Prices. I1

1. Without assumption (19) (see the theorem in 3 3d) the price process S = SoeH is not a local martingale with respect to the initial measure P. However, it is sufficient in many cases (e.g., in the problem of the absence of

- loc

-

- loc

arbitrage; see 5 2b) that there exists some - measure such that P << P or P P and S is a local martingale with respect to P. We have thoroughly discussed the question of the existence of such measures in models with discrete time (Chapter V, 5 5 3a-3f). Below, we consider this question in the continuous-time case for semimartingale models.

3. Semimartingales and Martingale Measures

695

(n,9, (.9:t)t201P) be a stochastic basis and let Pt = P 1 s t be the restriction - loc of P to 9 t . Assume also that P is a probability measure on 9 such that P << P, i.e., P, << Pt for all t 3 0. We also assume that - 90= {a,0) and PO= PO. 2. Let

Our analysis of the existence of measures P making one or another process a local martingale starts with Girsanov's theorem for local martingales, which demonstrates what happens to local martingales undergoing an absolutely continuous change of measure.

- loc

THEOREM 1. Assume that P

<<

P, let M E AiOc(P) with MO = 0, and let

dPt

Z = (Zt)tao, where 2, = -.

Assume that the quadratic covariance [ M , Z ] dPt has a P-locally integrable variation and let ( M ,2) be the predictable quadratic covariance (the compensator of [ M ,21). Then the process 1 M =M - -. (M,Z), (1)

2-

- -

-

is a local P-martingale and the P-characteristic ( M ' , M C )is the same (P-as.) as the P-characteristic (MC,M').

Proof. In accordance with the lemma in Chapter V. 5 3d,

XZ€

,A%

x E A(F).

(2)

(We have stated and proved this lemma in the discrete-time case; this can be trivially extended to continuous time.) From (2) we can easily derive the following local versions of this equivalence (see [250; Chapter 111, 5 3b] for the details):

xz € J4o,(P) ( X Z P E -4OC(P)

x E -4oc(P); ==+ x E ~ l o c ( P ) , ===+

(3)

(4)

where ( X Z ) T n = (XtAT,ZtAT,)t20 and Tn = inf(t: 2,< l / n ) . Thus, to prove that %? E &loc(P), it is sufficient to verify that (%?Z)TnE ~ l O C P ) n, 3 1. Let 1 A = -. (M,Z). (5) 2Then, by ItB's formula,

( M - A)Z = M Z - A2 = ( M - . Z + 2- . M + [ M ,2 1)- ( A . Z + 2- . A ) = (ML. . Z + 2- . M ( [ M Z] , - ( M ,2))) ( M , Z )- A . 2 - ( M , Z ) = M- . Z + 2- . M + ( [ M , Z ] ( M , Z ) )- A . 2 . (6)

+

+

Chapter VII. Theory of Arbitrage. Continuous Time

696

The first three terms on the right-hand side of (6) are local P-martingales. The same can be said for each 71 3 1 about the process (A.Z)*n. Hence, by assertion (4) the process G belongs to J & ~ ~ ~ ( F ) . Thus, M is a semimartingale with respect to the measure P, and it has the canonical decomposit,ion

M=G+A.

(7)

--

Hence by the definition of quadratic variations [ M ,M ] and [ M ,MI and consider-ing the limits as n + x of the Riemann sequences S ( " ) ( M , M )and S ( 7 2 ) ( M , M ) (see formula (10) in Chapter 111, 55b) we obtain that [ M , M ] = [ M , M ]up to P-indistinguishability. Finally, in view of formula (22) in the same section; 5 5b, of Chapter , C )and - - 111, we conclude that the predictable quadratic variations ( M C M (M', M c ) coincide ( a s . with respect to the measure P). x

-

I

3. Let St

=

S@t, where H

dpt and let Zt = -. dPt Assume that the process N = (Nt)t>o:

- loc

= ( H t ) t 2 0 is a semimartingale, assume that P

<< P,

Z = ( 2 t ) t 2 0 is generated by some P-local martingale dZt = Zt- dNt.

(8)

That is, let 2 = G ( N ) . We represent S as the prodnct S = SoF(I?), where I? can be found from H by the formula (10) in the preceding section. Let H be a special semimartirigale with canonical decomposition h

ii = HIJ + A^ + G,

(9)

where G E AlOc(P)and A^ is a predictable process of locally bounded variation. w e represent H as follows:

I? = Ho + A^ + G = Ho + A^ + (G,N ) + (G- (MI,N ) ) , and observe that

1 -'

2-

(10)

(G,2 ) = (G,N ) .

Then, by Theorem 1, the process G- (%,N ) is a local martingale with respect to the nieasiire p with dpt = 2,d P t , t 3 0. Hence we obtain the following result.

THEOREM 2 . If H is a special scmimartingale witl-i canonical decomposition (9),

-P loc << P, and the process Z = (Zt)t>ohas the representation

A^+ (GJ) = 0 ==+ H E Jzl1,,(P).

(8),then (12)

3. Semimartingales and Martingale Measures

697

4. Assume that condition (17) in the preceding section is satisfied and the process N has the following representation: N = p . H C+ (Y - 1) * ( p

v)

~

(13)

-

where = ( & ( W ) ) t > O is a predictable process and Y = Y ( t ,w , x). is a 9-measurable function o f t 3 0, w E R,arid 5 E R. (Here p = p H , v = v H , and we assume that the corresponding integrals with respect t o H" and p-v are well defined.) We use the representation (18) in 3 3d for 2:

If v ( { t }x d x ; w ) = 0, then we see that

(2, N ) = p . ( H e )+ (Y - l ) ( e Z

-

1) * v

(15)

(see the observation at the end of 53a.4). By ( l a ) , (14), (15), and also relation (19) in S 3d we obtain the following result.

THEOREM 3. Assume that the conditions (17) in 3 Sd. v ( { t }x drc; w ) = 0, and

are satisfied. If, in addition,

B

+ (; + /3 -

1

( H " ) + (eZ

-

1 - ~ ( x *) v)

+ (eZ

-

1)(Y - 1) * I/

= 0,

(17)

then the -processes H and - S = SoQ(fi) are local martingales with respect to the measure P such that dPt = Zt d P t , t 3 0.

EXAMPLE. Let H be the Lkvy process considered in the example of 33d. Let and let Y = Y ( z ) .Assume also that /3,(w) F b+

(5 +

/+2

+

-

Then the processes H and

1 - g ( 5 ) ) F(drc)

+ .i'(eZ

~

1 ) ( Y - 1)F ( d 5 ) = 0.

-

S = S o & ( f i )are P-local martingales

Note that condition (18) can be also written as follows: b+

(;

++2+

-

l)Y

-

y ( x ) ) F ( d z ) = 0.

For p = 0 a i d Y = 1 this is the same as condition (24) in 3 3d.

(18)

Chapter VII. Theory of Arbitrage. Continuous Time

698

We recall the representation for the cumulant function p ( X ) in 5 3c:

Setting 0 = of the equation

-

X and Y ( x ) = eAz, we find from (20) that (19) holds if X is a root cp(X + 1) - p(X) = 0. (21)

S form a local martingale with respect B to the measure p if dPt = Zt dPt and dZt = Zt- dNt, where If Bt

and

= BOert, then the discounted prices

x is the root of the equation p(X

5 3f.

-

+ 1) - p ( X ) =

T.

R e p r e s e n t a b i l i t y of Local M a r t i n g a l e s ( ‘ ( H e ,p-v)-Representability’)

1. We assumed in the previous section that the density process Z = (Zt)t20 with dpt 2,= (which is a P-local martingale) has a representation Z = 8 ( N ) . where dPt the P-local martingale N is a sum of two integrals, with respect t o He and p - I/ (see (13)). Comparing this with the ‘(p-v)-representability’ in the discrete-time case (Chapter V, S 4c) we see that the term ‘ ( H e p-v)-representability’ , fits very well in this context, so that we use it the title of this section. The issue of the representability of local martingales is considered in full generality in [250; Chapter 111, 3 4c]. Hence we discuss here only several general results related directly to arbitrage, completeness, and the construction of probability measures that are locally absolutely continuous with respect to the original measure. ~

2. We note first of all that to answer in a satisfactory way the question on the representation of local martingales in terms of the local martingale H e and the martingale measure p - v we must impose certain additional restrictions on the structure of the space R of elementary outcomes w . Namely, we shall assume in what follows that 2 ! is the canonical space of all right-continuous functions w = ( w t ) t 2 0 that have limits from the left. (See also [250; Chapter 111, 2.131 on this subject.) We shall assume all the processes X = ( X t ( w ) ) t > o below and, in particular, semimartingaies to be canonical (i.e., X t ( w ) E ~ t ) .

3. Semimartingales and Martingale Measures

699

We shall take for a filt,ration ( 9 t ) t > o the family of p-algebras

v

<

where 9 : = a ( w : w,, u s ) . We also set 9 = Fi. Let P be a probability measure on (0,9), Pt = P ] 9 t , t 3 0, and let H = ( H t ,9 t ) t 2 o be a semimartingale with triplet of predictable characteristics (B.C, v). For simplicity we assume that HO = Const (P-a.s.). The question whether the triplet (B,C, v ) defines the measure P unambzgously is of interest in many respects. This is not the case in general, as can be seen in the most simple, ‘deterministic’ examples. For instance, let H = (Ht)t>Obe a solution of an (ordinary) differential equation

(with non-Lipschitz right-hand side). Obviously, this equation has two solutions, z 0 and Eft(’) = t 2 . They are both semimartingales with respect to the measures and P(2),where the first is concentrated a t the trajectory wt z 0 and the second at ut = t2. At the same time, their corresponding triplets ( B ,C, u ) are the same: C = 0 , v = 0, and &(w) =

1 t

0

21ws11/2 d s .

3. The role played by triplets and the uniqueness of probability measure in the problem of ‘(H‘, p-v)-representability’ is revealed by the following result.

THEOREM 1. Let H = ( H t ,9 t ) t 2 0 , HO = Const, be a semimartingale with triplet (B,C, v ) on a filtered probability space (0,9, ( 9 t ) t > o ,P) and assume that the measure P is unique in the following sense: if P’ is another measure such that H loc

has the same triplet with respect to it, P’ << P, and Pb = PO,then P’ = P. Then each local martingale N = ( N t ,9 t ) has a representation

where f is a predictable process with f 2 . ( H C )E dZc and W is a F-predictable process with G ( W ) E dZc (3 3a). The proof of this result and its generalization (‘the F u n d a m e n t a l representation theorem’) can be found in [250; Chapter 111, 5 4d]. One can deduce from this theorem the following results concerning ‘ ( H Cp, - v)representability’. (They are useful, in particular, for complete arbitrage-free models)

700

Chapter VII Thpory of Arbitrage Continuous Time

THEOREM 2. Let (0,9, (.Yt)t>o,P) be the canonical filtered probahility space. a) If H = ( H t ,9 t ) t 2 0 is a Brownian motion. then each local martingale N ( N t ,9 t ) t > o has the following form: N

= HO

+ f .H ,

=

(2)

.dLc.

where f 2 . ( H ) E b) If a semimartingale H = ( H t . 9 t ) has independent increments, then each local martingale N = ( N t ,9 t ) t a o has a representation (1). The proof is a n immediate consequence of Theorem 1 in view of the uniqueness of the Wiener measure and the following fact: processes with independent increments have deterministic triplets, which uniquely define t2heprobability distributions (by the L6vy-Khintchine formula). Note that we have already encountered assertion a) (in Chapter 111, 4 3c). 4. Apart from ‘classical’ cases a) and b) of Theorem 2, we shall now discuss briefly

another case of the ‘(H‘, p-7))-representability‘ of local martingales. We consider the stochastic differential equation

dHt = b ( t , H t ) d t

+ g ( t ,H t ) d B t

+ g ( 6 ( t , H t , z)) ( p ( d t , dz;w )

-

v(dt,d z ; w ) )

+ g ’ ( 6 ( t , h t , z ) ) p ( d t ,d z ; w)

(3)

(cf. Chapter 111, 8 3c), where b, CT,and 6 arc Bore1 functions, g = g(z) is a truncation function. = z - g(z), B is a Brownian motion, and p is homogeneous Poisson measure with compensator v ( d t , d z ) = d t F ( d z ) (33a). It is well known (see, e.g., [250; Chapt,er 111, 2c]), that for (locally) Lipschitz coefficients satisfying the condition of linear growth stochastic differential equation ( 3 ) (with initial condition HO = Const) has a unique strong solution (Chapter 111, 3e). Moreover, whatever the initial probability space on which both Brownian motion and Poisson measure are defined, the probability distribution of the solution process H on the canonical space (0,s) is uniquely defined. The process H is a semimartingale with triplet (B,C , v)> where

v(dt,d z ; w)= d t K t ( w t , dZ) and K t ( w t : A )= /IA,{O] ( W , w t , z ) )F ( d z ) . Hence, if the coefficients satisfy the above-mentioned conditions (the local Lipschitz condition arid the condition of linear growth), then each local martingale N = ( N t ,. F t ) t > o admits a n ‘(H‘, p-v)-representation’. (See [250; Chapter 111, 3 2a] for greater detail.)

3. Semimartingales and Martingale Measures

3 3g.

70 1

Girsanov's Theorem for Semimartingales. Structure of the Densities of Probabilistic Measures

-

loc

1. If M = ( M t ,9 t ) t > o is a local martingale on (0,B, ( 9 t ) t > O ; P) and P << P, then M is a semimartingale with respect to this measure (see (7) in 3 3e). Remarkably, each semimartingale is transformed into a semimartingale again by such a change. That is, the class of semimartingales is stable under locally continuous changes of measure. (This is an easy consequence of It6's formula for seminiartingales; see Chapter 111, 3 5c.) From the standpoint of arbitrage the following question is of particular interest

- lzc

for financial mathematics: what can be said of the measures p such that P P - loc or P << P and the semimartingale X in question (describing,-e.g., the dynaniics of prices) is a local martingale or a martingale with respect to P? 2. One possible approach to this problem is to describe how the canonical representation (with respect to P)

of a semimartingale X with triplet (B,C , v ) transforms under a locally continuous

- loc

change of measure P

<< P into the canonical

representation

x = xo + E + xc+ g * ( p - G) + .(

-

g(2))* p

(with respect to p) of the same semimartingale with new triplet dPt Let Zt = --, t 3 0. We set dPr

Y = Ep ( - IZ( Z @

z-

> 0)

1 P),

(2)

(g, 2",5).

(4)

where EF is averaging with respect to t,he measure ME on (0 x R+ x E , 9 @ B(R+) @ 6') defined by the formula W * M p = E(W * p ) for all nonnegative measurable functions W = W ( w ,t , z). (Cf. the definition of Y,(z, w ) and M n ( d z jdw) in Chapter V, 3 3e). The processes [I arid Y are crucial in the issue of the transformations that triplets undergo under changes of measure. The following result is often called Girsanov's theorem for semimartangales.

Chapter VII. Theory of Arbitrage. Continuous Time

702

- loc

dPt

THEOREM 1. Assume that P << P, 2,= -, t 3 0, and let the processes /3 and dPt Y be defined in terms of Z = (Zt)t2o by formulas (3) and (4). Then B, and V . defined by the equalities

c,

E = B + p . c + g ( z ) ( Y- 1)* v, 2;= c, I

u=Y.v,

make u p a triplet of the semimartingale X with respect to the measure

p.

The proof of this result (not exclusively in the above case of one-dimensional semimartingales, but also for several dimensions) is presented in [250; Chapter 111, 5 3d] and is fairly complicated technically. Referring to this monograph for detail we comment now on the meaning of this theorem. Note first of all that the corresponding result in the discrete-time case was proved in Chapter V, 3 3e, where we explained the meaning of the discrete (relative to time) analogs of the measure M L and the variable Y . Assertion (5) displays the transformation of the 'drift' component B in the triplet (B,C , v). Assertion (6) says that the quadratic characteristics of the continuous martingale component X " do not change in fact under an absolutely continuous change of measure (up to P-stochastic equivalence). Assertion (7) means that Y is just the Radon-Nikodym derivative of i; with respect to v.

3. If X is a special semimartingale, then we can set g ( x ) = representation (2), so that

x = xo + B + xc+ ic * ( p

-

v).

ic

in the canonical

(8)

Hence we see that X is a local martingale if B = 0. Taken together with Theorem 1 this observation brings us to the following result.

-

loc

THEOREM 2 . Let P << P and, moreover, (ic2 A 1x1) * V E d&. Then a special semimartingale X is a local martingale with respect to the measure P if

B + @ .c +ic(Y - 1) * v

= 0.

(9)

4. Formulas (3) and (4) show the way for finding /3 and Y once the process

2 = (Zt)t>o is known. The converse question comes naturally: how, knowing

fl and Y ,can one find the corresponding process Z?

3. Semimartingales and Martingale Measures

703

A solution of this problem opens a way to the construction of the measure p such that X is a local martingale with respect to it. For if p and Y satisfy (9) and we reconstruct the corresponding process Z , then, of course, taking the measure p with dPT = Z , ~ P Twe see that X = ( X t ) t g T is a local martingale on [0, TI. Let X be a semimartingale defined on the canonical space (a, 9,( 9 t ) t > O , P ) . Assume that each P-martingale M admits an ‘(XC, p-v)-representation’: = Mo

+ f . X C + w * (p

-

v ).

(10)

(See formula (1) in 8 3f.)

-

loc

THEOREM 3 . Let P << P, let Z = (Zt)t>o be the density process, Jet v ( { t } x E ; w ) = 0 for t > 0, and Jet ,L3 and Y be defined by ( 3 ) and (4). Then (given the property of ‘ ( X “ ,C”-v)-representability’) the process Z satisfies the relation

2 = 20 + (Zq?) . XC+ z-(Y- 1) * ( p - v).

If p2

for all t

‘

+

* Vt < 00

(12)

1) * ( p - v)t

(13)

( X C ) t (1 - d F ) 2

> 0, then the process N

=

Nt = p .

(11)

(Nt)t>owith

x; + (Y

-

is a P-local martingale. The process Z = (Zt)t>ois a solution of Dol6ans’s equation

and can be represented in the following form:

where

8 ( j q t = eNt-+(P2C)t

(1

+~ l N , ) e - ~ ~ ~ .

(16)

o<s
In this statement we assume that v ( { t } x E ; w ) = 0. This means that the process X is qiiasi-left-continuous, i.e., for each predictable stopping time r we have AX, = 0 on the set (7 < m}. In the general case this theorem is stated and proved in [250; Chapter 111, 5 5a].

Remark. As regards direct applications of Theorems 2 and 3 to diffusion models, see the next section, 5 4a.

4. Arbitrage, Completeness, and Hedge Pricing

in Diffusion Models of Stock

5 4a.

Arbitrage and Conditions of Its Absence. Completeness

1. We have already discussed at length the issue of the construction of probability measures making price processes martingales or local martingales (both in the discrete and the continuous-time case). Our interest in this issue is mainly a consequence of the fact that the existence of equivalent martingale measures allows one to say about the absence of opportunities for arbitrage in a fairly general context (see 5 fj2b, c). In addition, the knowledge of all such measures enables one, for instance, to find the fair (rational) prices or hedging strategies, and so on, using the machinery of martingales. In the present section we consider the issue of the absence of arbitrage in the case when prices are It6 processes (Chapter 111, 3 3d).

2. Let (0.9, P) be a probability space with a Brownian motion B = ( B t ) t y j .We shall denote by ( 9 t ) t > o the Brownian ( W i e n e r ) filtration, i.e., the flow of a-algebras .!Ft = a(9: u N ) , where 9 := u ( B s ,s ,< t ) and Jy = { A E 9:P(A) = O}. (See Chapter 111, 5 3a for detail). In addition, we assume that 9 = V 9 t ( 2 a ( U Ft)). The filtered probability space (0,9, ( 9 t ) t > O ,P) satisfies the usual conditions (Chapter 111, 13a) and we shall regard it as the stocha.stic basis describing the probabilistic uncertainty and the structure of the flow of the incoming information. Let St = S0eHt (So > 0) be the price process for an asset, (some stock, say) with (1) where p = ( p t , 9 t ) and c = (at,9 t ) are two stochastic processes satisfying (P-a.s.) the conditions t

u2ds

< 30,

t > 0.

(2)

4. Arbitrage, Completeness, and Hedge Pricing in Diffusion Models of Stock

705

By It6‘s formula we obtain

where

i.e.. S has the stochastic differential

a # 0 , then we obtain the standard dzffusion model of If pt = p and at Samuelson [420] describing the dynamics of stock prices by means of a geometric Brownzan motzon (Chapter 111, 5 4b):

We set z t = exp(-1Bt LT

- -

(q 2 t )

(7)

2 a

Then EZt = 1 and by Gzrsanov’s theorem (see Chapter 111, 5 3b or 5 3e), for each T > 0 the process S = ( S t ,9 t ) t G T is a martingale with respect to the measure PT with I

-

d p =~

(8)

dPT,

where PT = P 1 .FT; its differential is

-

-

-

where B = ( B t , 9 t ) t Q T is a standard Brownian motion with respect to PT Thus, if EZT = 1 then the measure on (Q&) is equivalent - to PT and the process S = ( S t ,9 : t ) t G T becomes a martingale with respect to PT. It is worth noting that this measure is, is unique in the following sense: if QT is another measure such that QT PT and S = ( S t , 9 t ) t G T is a local martingale with respect to QT,then QT = s.i, This is intimately connected with the result on the representation of local martingales in terms of the Brownian filtration (see Chapter 111, 3 3c); we shall prove it below, in subsection 5.

-

Chapter VII. Theory of Arbitrage. Continuous T i m e

706

3. We consider now the case when the price process S has the differential ( 5 ) . Assume that the following conditions are satisfied: P-a.s. we have at

for t

>0

(10)

> 0 and jf(z)2d$

< 00.

(11)

Then there exists a well-defined process 2 = (Zt)t>owith

that is a positive local martingale, and the corresponding localizing sequence (rn)n21 can be taken in the form

-

-

-

If EZT = 1, then 2 = (Zt)tcT is -a martingale. the measure PT such that ~ P = T Z T ~ P Tis a probabzlaty measure, PT PT, and the process S = ( S t , 9 t ) t c T is a local martzngale with respect to this measure. To prove the last assertion we use Theorem 2 in Since Pt dZt = - Zt - dBt , ut if follows that (see formula ( 3 ) in

3 3g. (14)

5 3g)

In the triplet (B,C, v ) of the semimartingale S with respect to the measure P we havc v = 0.

However.

1

suffu

f f u .s2a2

du=0.

Hence, by Theorem 2 in 5 3g the process S = ( S t ,9 t ) t G T is a local martingale with respect to PT. Moreover (assuming that EZT = l ) , the measure PT is unique in the same sense as in the case of p t = p and at = a (see subsection 5 below). I

4. Arbitrage, Completeness, and Hedge Pricing in Diffusion Models of Stock

707

4. Now, for a market model consisting of a bank account B(0) = (Bt(O))t,o with Bt(0) 1 and stock S = (St)t>owith dynamics described by (5). we state condi-

=

tions ensuring the absence of arbitrage (in its m+-version; see 5 2a). Let 7r = (p,y ) be a strategy and let X K = (X?)t>o be its value,

x : If

T

= Pt

+rtst.

is a self-financing strategy, then

which, of course, implies that the stochastic integral in (17) must be well defined. As is clear from 5 l a , the integral in (17) is well defined for y E L ( S ) . In the current model (5) it would be reasonable to state conditions of the integrability of y with respect to s directly, in terms of t,he processes ( p t ) t < T and (crt)t
By the last condition and since S = (St)t>o is continuous, it follows that LtS;y;o; d u

<

00

(P-as.), and therefore, the stochastic integral

& Suy,cru dB, t

with respect to a Brownian motion is well defined (see Chapter 111, 5 3c and the present chapter). Further.

3 l a in

1

t

Hence it follows from (11) and (18) that the integrals y,p, d u and LtS,y,pu du, 0 t > 0, are well-defined and finite (P-as.). Thus, conditions (a), ( l o ) , (ll),and (18) ensure the existence of the stochastic integral in (17). The next result, which is an immediate consequence of the implication (9) in Theorem 2 in 2b is the best known assertion concerning the absence of arbitrage in diffusion models.

THEOREM. Assume that stock prices S = (St)t>o have the differential (5) and conditions (a), ( l o ) , (ll),and (18) hold f o r t 6 T. Let EZT = 1. Then the property holds and, in particular, there exist no opportunities for arbitrage in the class of a-admissible self-financing strategies for any a 3 0.

m+

Chapter VII. Theory of Arbitrage, Continuous Time

70%

5 . We turn now to a result already mentioned in subsection 2: the nieasiire PT such that d P T = Z T ~ P T where , 2, is defined in ( 7 ) ,is unique in the following sense: this is a unique measure equivalent-t o PT such t h a t the process S = ( S t ,9 ; t ) t c T is a local martingale with respect to PT. We shall consider a more general case, where we assume that S is as in (5) and the process 2 is defined by (12). Let QT be a measure equivalent t o PT such that S = ( S t ,.9t)tGT is a local martingale with respect to Q T . Then we construct the martingale

It is positive, therefore there exists a process

'p

= ( ' p t , .Ft)t
with hTp2 ds < oc (P-as.), arid E N T = 1 (see (20) in Chapter 111, 5 3c). We now use Theorem 1 in 53g, which describes the transformation of the triplet of predictable characteristics of semirnartingales under absolutely continuous changes of measure. Let (BP,CP,uP)be the triplet of S with respect to P. It follows from ( 5 ) that (21)

By Theorem 1 in 5 3g the triplet ( B Q C , Q ;v Q ) (with respect t o Q) is as follows:

where

Hence, from (21) and (22) we obtain

4. Arbitrage, Completeness, and Hedge Pricing in Diffusion Models of Stock

Since

S

= ( S t , . F t ) t < ~is a local martingale with respect t o

709

Qr, it follows that

BF = 0 (P-a.s.) for t 6 T. Hence we see from (24) that

(A x P ~ ) - a . s on . [O, T ] x 62: where X is Lebesguc measure. ~ stochastically indisHence the processes Z = (Zt)t
6. We discuss now the issue of T-completeness (see the definition in fj2d). A’. slime that the assumptions of the above theorem are satisfied. Assume also that for our (B(O),S)-market we have & ( O ) f 1 and the process S = (St)t
m+ m,-

fj4b. Price of Hedging in Complete Markets 1. We explained the riot,iori of hedging and methods of ‘hedge pricing’ in coniplete and iriconiplete markets for discrete-time case in Chapter VI. In general sernirnartingale models the discussion can proceed along a parallel route, except, maybe, that we m i s t describe concisely the classes of admissible strategies. We shall stick to the diffusion model of a (B(O),S)-market described in fj4a.4 and use the notation from there. Modifying slightly the definition of T-completeness given above (fj 2d) we shall say that a nonnegative 9T-measurable pay-off function fT with E Z T ~ T< cx can be replicated if there exists a strategy ?r E II+(S)such that XG = f~ (P-as.). Clearly, if f~ is bounded, then the condition E Z T ~ T< co is satisfied.

If a pay-off function f~ can be replicated, then we mean by the price of (perfect) European hedging (cf. the nomenclature in Chapter IV, 4 l b ) , or simply the hedging price, the quantity DEFINITION.

@ ( f ~ P) ; = inf{z 3 0: 3?r E II+(S)with X t = z, X $ = f ~ } .

(1)

Chapter VII. Theory of Arbitrage. Continuous Time

710

2. THEOREM. Let

Proof. If

7r

=

PT

be a unique martingale measure. Then

(p,y) is an a-admissible self-financing strategy, then

I

and (by the Ansel-Stricker theorem; see s l a . 6 ) X T = ( X F ) t G T is a PT-supermartingale, so that E-PT X G < X $ . (4) Hence if X ; = fT, then E- f T PT

< xt and

We claim - now that there exists a 0-admissible - self-financing strategy Z of initial value X{ = EZT~Tthat replicates f ~i.e.,, X $ = f~ (P-as.). We consider the process

Clearly, X = ( X t ,9 t ) t G T is a martingale with respect to the ‘Brownian filtration’ and by the representation theorem (see Chapter 111, 5 3c) there exists a process $ = (&, 9 t ) t C T with

T

$: ds < 30 such that

Note that the process ( Z c l X t ) t C T replicates f ~ :

-

We now claim that there exists a 0-admissible self-financing portfolio Z = (p,7) such that

xtz = z,-?xt. Since

dZt = -Zt- Pt d B t , “t

(9)

4. Arbitrage, Completeness, and Hedge Pricing in Diffusion Models of Stock

711

it follows by It6’s formula (Chapter 111, 5 3d) that

and

We set

Then we see that

d(Z,-lXt) =

rt dSt,

and moreover, ~t~~~~

du < 00 (P-a.s.1,

t 6 T.

(Cf. condition (18) in 5 4a.) Thus, for t < T ,

Setting

Pt = Z T 1 X t - r t s t ,

we see from (15) that the strategy E (Xt’)tGT such that

=

(ply)

is self-financing and of value X’

xi = E(ZTfT)

=

(19)

From (5), comparing (19) and (20), we derive required assertion (2). Note that the strategy Z so constructed is 0-admissible because X’t 3 0, t T .

<

712

5 4c.

Chapter VII. Theory of Arbitrage Continuous Time

Fundamental Partial Differential Equation of Hedge Pricing

1. We consider the model of a market formed by a riskless asset-a bank account with zero interest rate ( B t ( 0 )= 1)-and a risk asset S = (St,.5Ft)tao whose dynamics is described by relation (5) in 9 4a. As follows by the theorem in § 4 b , the hedging price C ( ~ T P);and the corresponding hedging portfolio ;ii = (0.7) can be found on the basis of the properties of the process Y = (Yt,. 5 F : t ) t c ~where , I

In addition, the quantity = E(ZTfT)

(2)

YT = f T >

(3)

is precisely the price C(fT; P),

<

and yt = Xt', i.e., yt is the value of the hedging portfolio at time t T . (These properties justify the name 'hedging-price process' for Y . As already nientioned on several occasions; our method of finding C ( ~ T P);is usually called the martingale method.) In many cases we can explicitly find E ( Z T ~ Tand, ) therefore, the price ( c ( f ~P). ; In particular, this can be done in the Blaclc--Merton-Scholes model, where ,ut and ut are constants. (See Chapter VIII below). 2. F. Black and M. Scholes [44], and R. Merton [346] (1973) proposed another method for finding the price @.(f~; P) and the hedging strategy. It is based on the so-called f u n d a m e n t a l equation that they have obtained. This method, which is now widely used in financial mathematics (see, in particular, 3 5c below), is essentially as follows. We consider the process yt defined in (I). Since

) ; the and S = ( S t . . F t ) t ~is a Markov process, we see that if f~ = ~ ( T , S T then process Y = ( K , .Tt)t2.This enables one to use ItA's formula, which brings

4. Arbitrage, Completeness, and Hedge Pricing in Diffusion Models of Stock

713

one to the following stochastzc partzal differential equatzon (where for simplicity we drop the arguments of functions):

We consider now another representation for

Y:

In view of self-financing,

Using the representations (5) and (7) of the special semimartingale Y = Y ( t .S t ) arid the uniqueness of the representation of special semimartingales (see Chapter 111, 55b) we see that the coefficients of dB (and of d t ) in (5) arid (7) must be the same. Since St > 0 for t > 0, it follows therefore that (P-a.s.)

moreover, the processes ( r t ) t G T and

G

are stochastically indistin-

-( t ,St)

guishable. Comparing the terms with d t in (5) and (7) and taking account of (8) and the

-

aY

equality ,& = Y ( t ,S t ) - St-(t,

3s

St),we obtain the following relation (that holds

(A x PI-as., where X is Lebesgue measure in [0, TI):

This, in turn, must hold if Y = Y ( t ,S) satisfies the following Fundamental partzal diflerentaal equatzon for 0 < t < T and 0 < S < m:

i3Y -(t, at

a2Y

S)+ 2" ( t ,S t ) S -(t, 1

2

2

as2

S)=0

with boundary-value condition

Y ( T ,S)= f(T, S),

s> 0.

(Cf. the backward Kolmogorov equation ( 6 ) in Chapter 111, 3 3f).

(9)

Chapter VII. Theory of Arbitrage. Continuous Time

714

We shall discuss the solution of this equation (which reduces-in any case, for a(t,s) a = Const-to the solution of the standard Feynman-Kac equation; see (19) in Chapter 111, 53f) in connection with the Black-Scholes formula in the case of f(T, S)= (S- K ) + , in Chapter VIII. Here we point out the following features of this equation. Assume that there exists a unique solution to (9)-(10). We find rt by (8) and define

& by setting

I

Pt = Y ( t ,St) - rtst.

(11)

(p,

Clearly, the value X‘t of the portfolio 5 = 7)is precisely Y ( t ,S t ) . It is not a priori clear if this portfolio ;ii constructed from Y ( t ,S) is self-financing, i.e., whether d Y ( t ,S t ) = rt dSt. (12) However, this is an immediate consequence of (5) and (9):

Thus, assume that the problem (9)-(10) has a unique solution Y ( t , S ) . Since X’t = Y ( t , S t ) ,it follows that X g = f ( T , s ~ while ) , X i = Y ( 0 , S o ) is the initial price of the portfolio ?r = (P, 7). The following heuristic arguments show that the price X t = Y ( 0 ,So) (obtained in the solution of (9)-(10)) has the properties of ‘rationality’, ‘fairness’, while the portfolio 5 = (p,7 ) is an ‘optimal’ hedge. In fact, let us interpret our problem as a search of a hedging strategy for a seller of a European call option such that the value of this strategy replicates faithfully the pay-off function f ( T :S T ) . Our solution of (9)-(10) shows that selling this option a t the price C = Y ( 0 ,So) the seller can find a strategy 5,such that X g becomes precisely equal to f ( T ,S T ) . Assume now that the price C asked for this option contract is higher than Y(O,So) and the buyer has accepted this price. Then, clearly, arbitrage is possible: the seller can obtain the net profit of C - Y ( 0 , So)meeting simultaneously the terms of the contract because there exists a hedging portfolio of initial price Y ( 0 ,So) that replicates faithfully the pay-off function. On the other hand, if C < Y ( 0 ,So), then due to the uniqueness of the solution of (9)-(10) the terms of the contract will not necessarily be fulfilled (at any rate, if one must choose strategies in the Markov class). There are several weak points in this method based on the solution of the fundamental equation, namely, the a priori assumptions of the ‘Markovian structure’ of the value of 5 (i.e., the representation XT = Y ( t ,S t ) ) and of the C1>2-regularity of Y ( t ,S ) (enabling the use of It6’s formula). I

x

4. Arbitrage, Completeness, and Hedge Pricing in Diffusion Models of Stock

715

Fortunately, there exist other methods for the construction of hedging strategies and finding the ’rational’ price C ( ~ TP); (e.g., the ‘martingale’ method exposed in 3 4b), which show, in particular, that a hedging portfolio does exist and its value has the form Y ( t ,S t ) with a sufficiently smooth function Y ,so that equation (9) actually holds. We present a more thorough analysis of the case of a standard European call option with f ( T ,S T ) = (ST - K)’ in Chapter VIII, 3 lb, where we discuss arid use both ‘martingale’ approach and approach based on the above fundamental equation. 3. In the above discussion we assumed that the riskless asset (a bank account) B(0) = (Bt(O))t,o has the form Bt(0) = 1. In effect, this means that we deal with discounted prices. However, in some cases one must consider the ‘absolute’ values of the prices rather than the ‘relative’, discounted ones. Here we present the corresponding modifications in the case when the bank account B ( r ) = (Bt(r))t>O (in some ‘absolute’ units) has the following form:

(here ( T t ) t 2 0 is a deterministic nonnegative function-the interest rate), and the risk asset (stock) is S = ( S t ( p ,c))t,o, So(p, 0)= So > 0, where

(Our assuniptioris about p t , ct, and the Brownian motion B = ( B t ,9 t ) t 2 0 are the same as in 5 4a.) Let ?? = 7)be a self-financing portfolio and let

(p,

xtF= PtBt(.)

+ 7tSt(P,c).

(16)

We assume that XtF is as follows:

xtF = Y ( t ,S t ) , where St = St(p,cr) and Y ( t ,S) E C1>2. Then for Y = Y ( t , S )we obtain the same equation ( 5 ) . On the other hand, since

it follows by the property of self-financing that

Chapter VII. Theory of Arbitrage. Continuous Time

716

Comparing tlie terms containing dB in (5) and (18) we obtain again that

aY dS By ( 1 0

qt = 7 ( t 3 S t ) .

and, as in our tlerivation of (9), we see that the coefficients of dt in (5) and (18) are the same if Y ( t ,S ) satisfies the following Fundamental equataon for S E pP+ arid

O
dY 1 d2Y + rs-aY as2 -ry, at as + -u2s22 with boundary condition Y ( T ,S) = f ( T , S ) ,S E R+. It should be noted that p = p . ( t , S ) does n o t enter the equation or boundary condition, and therefore Y ( 0 ,So) is independent of p . This might appear puzzling -

at first glance. (On the other hand we can consider this property to be desirable because investors can have different ideas about the values of 11 and 0 , and therefore, different notions of the actual dynamics of the price process S = (St)tao.) Probably, the best explanation can be given from the standpoint of tlie ‘martingale’ approach, in which the price satisfies the relations

(see forniula (2) in 54b) and the process S = (St)t
5 . Arbitrage, Completeness, and Hedge Pricing

in Diffusion Models of Bonds

3 5a.

Models without Opportunities for Arbitrage

1. In Chapter 111, 5 4c we considered several models of the term structure of prices of bond families. In particular, we observed that there existed two approaches t o the description of the dynamics of bond prices P ( t , T ) : the indirect approach (when we take some ‘interest rate’ process r = (r(t))t>Oas a ‘basis’ and assume that P ( t , T ) = F ( t ,r ( t ) ,T ) ) .and the direct one (when P ( t , T ) is defined in a straightforward way, as a solution to some stochastic differential equations). These approaches bring forward distinct models. In the framework of our notion of a ’fair’ market as a market without opportunities for arbitrage it would be natural to find out first of all what conditions ensure the absence of arbitrage in these models and how one can find ’explicit’ expressions for the prices P(t, T ) in these models. 2. Taking the indirect approach we assume that the interest rate process r = ( T ( t ) ) t > O is the solution of the stochastic differential equation (cf. (5) in Chapter 111, 3 4c) d r ( t ) = ~ ( rt (,t ) )d t b ( t , T ( t ) ) dWt, (1)

+

generated by a Wiener process W = (Wt,9t)t>o. (As regards the Brownian (Wiener) filtration (3Ft)tao, see 4a.) We also assume that the coefficients a = a ( t ,r ) and b = b ( t , T ) are chosen so that equation (1) has a unique strong solution (Chapter 111, 3 3e). In a natural way, one can associate with the interest rate T = (r(t))t>Oa bank accomt B ( T ) = (Bt(r))t>o with

Chapter VII. Theory of Arbitrage. Continuous Time

718

which, like in the case of stock or other assets, plays the role of a ‘gauge’ in the t valuation of various bonds. (We assume throughout that Ir(s)l ds < K (P-a.s.),

/

0

t

> 0.)

Let P ( t , T ) be the price of some T-bond (see Chapter 111. 5 4c), which is assumed to be 9t-measurable for each 0 6 t < T , and P ( T , T ) = 1. Below we assume also that the processes ( P ( t , T ) ) t > o are optional for each T > 0. Then, in particular, P ( t , T ) is 9t-measurable for each T > 0. By the meaning of P ( t , T ) as the price of bonds with P ( T , T ) = 1 we must also assume that 0 6 P ( t . T ) 6 1. Now let us introduce the discounted price

Bearing in mind the result of the Fzrst fundamental theorem about the absence of arbitrage (Chapter V, 52b) and based on the conviction that the ’existence of a martingale measure ensures (or almost ensures) the absence of arbitrage’ we assume that there exzsts a martingale (or risk-neutral) measure on 9~such that PT PT ( = P I3+)and (P(t, T ) ,.9t)tCT is a PT-martingale. Then we conclude directly from ( 3 ) that I

N

so that we havc the following result. 1

THEOREM 1. If there exists a martingale measure PT counted process (P(t, T ) ,9 t ) t G T is a PT-martingale, then

N

PT such that the dis-

This is an immediate consequence of (4) and the condition P ( T , T ) = 1, for we have

which delivers the representation (5). We see froni (5) that if T = (r(t))t?o is a Markov process with respect to P, then the price P(t, T ) can be written as follows:

I

P(t, T ) = F ( t , r ( t ) ,T ) . The ‘absence of arbitrage’ imposes automatically certain restrictions on the function

F ( t ,T , T ) (see 5 5c below).

5. Arbitrage, Completeness and Hedge Pricing in Diffusion Models of Bonds

719

Remark 1. We point out that the price P ( t , T ) of bonds cannot be unambigoulsy evaluated on the basis of the state of the bank account B ( r ) and the condition of the absence of arbitrage (more precisely, of the existence of a martingale measure). In fact there is no reason for to be unique, which means that P ( t , T ) can - be realized (in the form (5)) in various ways depending on a particular measure PT. We note also that if, instead of P(T, T ) = 1, we require that P(T, T ) be equal to fT, where fT is ST-measurable and EPT

1

~

fT

BT ( r )

~

< 00,then by (4) we obtain

3. We proceed now from one fixed T-bond to a famaly of T-bonds:

P = {P(t,T); 0 6 t ,< T , T > O}. DEFINITION 1. Let P be a probability measure on (0,9, (St)tao) with 9 =

We say that a measure

-

p ’2 P (i.e., Pt

for the family P , if for each T

N

v9t.

Pt for t 3 0) is a local martingale measure

> 0 the discounted prices P ( t , T ) = P ( t l T ) , t ~

Bt ( r )

I

< T,

are local PT-martingales.

To define an arbitrage-free (B,P)-market formed by a bank account B and a family of bonds P we must, first of all, discuss the notion of portfolio (strategy) in this case. DEFINITION 2 ([38]). A strategy 7r = (P,y) in a (B,P)-market is a pair of a predictable process p = (pt),,, and a family of finite (real-valued) Bore1 measures y = (yt( . ) ) t a such ~ that for all t and w the set function yt = y t ( d T ) is a measure in (R+, B(R+)) with support concentrated on [t, 00) and for each A E B (R + ) the process (yt(A))t>o is predictable. The meaning of /3 and y is transparent: /3t is the number of ‘unit’ bank accounts (at time t ) and y t ( d T ) is the ‘number’ of bonds with maturity date in the interval [T,T dTj.

+

DEFINITION 3 . The value of a strategy with

X? = PtBt

+

7r

is the (random) process X” = ( X T ) t 2 0

LW

P(t, T )rt(dT).

(7)

(We assunie here that the Lebesgue-Stieltjes integrals in (7) are defined for all t and w . )

Chapter VII. Theory of Arbitrage. Continuous Time

720

7r = (0, y) in a (B,P)-market. To this end, taking the direct approach (Chapter 111, $ 4 ~we ) assume that the dynamics of the prices P(t,T)can be described by the HJM-model; namely, for 0 t < T and T > 0 we have

4. We define now a self-financing portfolio

<

dP(t,T ) = P(t,T ) ( A ( tT, ) d t

+ B ( t ,T)d W t ) ,

(8)

where W = (Wt)t>ois a standard Wiener process, which plays the role of the source of randomness. We niust add to equations (8) the boundary conditions P(T,T ) = 1, T > 0. (For a discussion of the condition of the measurability of A ( t ,T ) and B ( t ,T ) and the solubility of (8),see Chapter 111, 4c). In view of the equation d B t ( r ) = ~ ( t ) B t (drt )

(9)

we obtain by It6’s forniula t,hat the process

has the differential (with respect to t for each T )

rlP(t,T)= F ( ~ , T ) ( [ A ( ~ , T~) ( t ) ] d t + ~ ( t . ~ ) d w ~ ) .(11) We have said that a strategy 7r = (p,y) of value X? = PtBt +rtSt in a diffusion ( B ,S)-market is self-financing if d X T = Bt dBt 1.e..

Xt“ = X $

+

+ Y t dSt,

t

1

(12)

t

Dud&

+

yZ1dSZI.

(13)

the present case of a diffusion (B,P)-market it is reasonable to say that a strategy 7r = ( 0 ,of~value ) X ; = a B t ( r ) + ~ ” P ( t . T ) y t ( c l iis) self-financzng ([38]) if (in the symbolic notation) 111

d X T = pt dBt (7.)

+

1

co

dP ( 4 T )Yt

which (in view of (8))should he interpreted as follows:

1

(14)

5. Arbitrage, Completeness and Hedge Pricing in Diffusion Models of Bonds

If a strategy

T

721

= ([j, y) is self-financing, then its discounted value

satisfies the relation d

z=

d P ( t , T )y t ( d T ) .

(17)

As in (14), relation (17) is symbolic and (in view of (11)) it means that

5 . To state conditions for the absence of arbitrage in (B,P)-models we discuss first of all the existence of martingale measures. To this end we define the functions A ( t ,T ) and B ( t ,T ) also for t > T by setting A(t,T ) = T ( t ) and B ( t ,T ) = B ( T ,T ) . Then we immediately see from (11) that in order that the sequence of prices (P(t,T))tsT be a local martingale for each T > 0 with respect to the original measure P it is necessary that

A ( t ,T ) = r ( t ) .

(19)

By (8) we obtain that in this case d P ( t , T ) = P(t, T )( r ( t )d t -k B ( t ,T) dWt)

and dP(t, T ) = F ( t , T ) B ( t T j )dWt.

~ the equality Taking into account relations (14) and (15) in Chapter 111, $ 4 and

a A ( t ’ T ) - 0 holding under the assumption (19) we obtain the following relation dT for f ( t ,T ) : df(t,T ) = a ( t ,T )dt b ( t , T )d W t , ~-

+

where

T

~(T t ,) = b ( t , T )

b ( t , S ) ds.

On the other hand, if (19) fails, then it would be natural (by analogy with the case of stock) to refer to the ideas underlying Girsanov’s theorem.

Chapter VII. Theory of Arbitrage. Continuous Time

722

Assume that besides the original measure P on the space

9 = V 9 t there exists a probability measure p such that

(n,9, ( 9 t t ) t > O ) with

-P loc N

-

P, i.e., Pt

N

Pt,

t 3 0. We set Zt =

dF,

-.

Since ( 9 t ) t > o is the Brownian (Wiener) filtration, it follows dPt by the theorem on the representation of positive local martingales (see formula (22) in Chapter 111, 3 3c) that

t

where the p(s) are $:,-measurable, p2(s)ds < 00 (P-a.s.), and EZt = 1 for each t > 0. By Girsanov's theorem (Chapter 111, 3e) the process = (@t)t>o with

w

is Wiener with respect to

with respect to

p. Hence

is and

+

d F ( t , T ) = P ( t ,T )[ ( A ( tT, ) p ( t ) B ( t , T )- T ( t ) ) d t (cf. (8) and (11)). - Hence (cf. (11))the processes P for all T > 0 if and only if

+ B ( t , T )@t]

(24)

( p ( t ,T ) ) t < T are local martingales with respect to

A ( t ,T ) + cp(t)B(t,T ) - ~ ( t=)0.

(25)

By (23) we obtain in this case that

+

dP(t, T ) = P(t, T )( ~ ( dtt ) B ( t ,T ) d w t ) ,

-

-

(26)

I

where W = (Wt)t20 is a Wiener process with respect to P 6. The definition of the property of a strategy 7r = (PIy) in a ( B ,")-market to be arbatrage-free (say, in the NA+-version) at an instant T is as in 5 l c . We say that a (B,'P)-market is arbitrage-free if it has this quality for all T > 0.

5. Arbitrage, Completeness and Hedge Pricing in Diffusion Models of Bonds

723

-

- loc

THEOREM 2. Let P P be a measure such that the density process Z = (Zt)t20 has the form (21) and condition (25) holds. Then there are no opportunities for arbitrage for any a 3 0 in the class of a-admissible strategies T > -a, t > 0) such that

(x:

[lm 2

t > 0.

B ( s ,T )-yS(dT)] ds < 00,

(27)

-

x"

Proof. If (25) and (27) are satisfied, then the process = (X;)t>o is a P-local martingale by relation (18), In view of the a-admissibility 3 -a, t > 0) this process is also a supermartingale. Hence if = 0, then EFT: 0 for each T > 0. However,

(x:

xi

-

P(x: 3 0) = P ( X ; completes the proof.

> 0) = 1. Hence X ;

<-

= 0 (P- and P-a.s.) for

T > 0, which

Remark 2. Assume that the function

is independent of T and

-

- loc

P one for each t > 0. In this case, looking for a measure with the property P could proceed as follows. T , introduce the process We denote the function in (28) by cp = cp(t), t Z = ( Z t ) t y ~defined by (21), and assume that EZt = 1, t > 0. Then for each t > 0 Pt. the measure is, with clPt = Zt dPt is a probability measure such that The family of measures { p t , t O} is compatible (in the sense that PS = Pt 19,

<

-

>

<

I

-

(a,

for s t ) and if there exists a probability measure P on 9)such that is '?!I P, then it is the required one. If the maturity dates of T-bonds in our (B,P)-market satisfy the inequality T TO,where TO< 00,then we can take PT,, as the required measure is. It is equally clear that if 2, = lim 2, and we have EZ, = 1 and P(Z, > 0) = 1,

<

tioo

then the measure erty is % ' ' P.

P with d P = Z,

dP is the required martingale measure with prop-

Chapter VII. Theory of Arbitrage. Continuous T i m e

724

7. We present now an example of an arbitrage-free ( B ,P)-model. Following [36] arid [219], we start from the forward interest rate f ( t . T ) with stochastic differential (with respect to t for fixed T )

d f ( t , T ) = a ( t ,T )d t

+ b ( t , T )dWt,

(30)

where

b ( t , T )= a > 0,

a ( t , T ) = 0 2 ( T- t ) , t < T Then (30) takes the following form: d f ( t ,T ) = 2 ( T - t ) d t

+

0

dWt,

(33)

and therefore

where f ( 0 , T ) is the instantaneous forward interest rate of 7'-bonds on the ( B ,P)market (at time t = 0). By (34) arid the definition r ( t ) = f ( t ,t ) we obtain r ( t )= f(0, t )

Hence the interest rate

T

=

+ -22t 2 + owt.

(35)

( ~ ( t ) ) ~ satisfies >o the equation

dr(t) =

(y + 1

0 2 t dt

+ adWt.

(Cf. the Ho Lee model (12) in Chapter 111, 14c.) The coefficients A ( t , T ) and B ( t ,T ) in (8) can be calculated on the basis of a ( t ,T ) = a 2 ( T- t ) and b ( t , T ) s 0 in (33) as follows:

.I T

A ( t ,T ) = ~ ( t-)

a ( t ,S) ds

+ A2

(AT

2

b(t, S) d ~ )= r ( t ) ,

B ( t , T )= -a(T - t ) .

(37)

(38)

Hence condition (19) holds in our ( B ,7')-model, and therefore the original measure P is a martingale measure and no arbitrage is possible. The prices P ( t , T ) themselves can be found from the equation d P ( t , T ) = P ( t , T )[ ~ ( dt t)- u(T - t ) dWt],

t < T,

5. Arbitrage, Completeness and Hedge Pricing in Diffusion Models of Bonds

which must be solved for each T We can also use the equality

725

> 0 under the condition P(T,T ) = 1.

P(t,T) = exp(

-

.6’f(t,s) ds)

,


t

(39)

holding by (2) in Chapter 111, 54c. By (34) 1

4’

f ( t ,). ds =

=

lT

[f(O, s)

i’

+

f ( 0 , S) ds

0%

(s

-

2 + -W(T 2

;)I -

ds

+ cr(T

-

t)Wt

+

t ) a(T - t)Wt.

Consequently,

P(t,T)= exp{

1

T

-

f ( 0 , s ) ds

02

-

-tT(T 2

-

t ) + a(T - t)Wt

Hence we obtain from ( 3 5 ) the following representation for P(t, T ) in terms of the interest rate r ( t ) :

2

(T - t)f(O,T) - -t(T 2

-

t ) 2- (T - t ) r ( t ) } .

(41)

(Cf. affine models in Chapter 111, $ 4 and ~ in $ 5 below.) ~ 8. In the above diffusion models we have assumed that the interest rates r = ( r ( t ) ) , the forward interest rates f = ( f ( t ,T ) ) ,and the bond prices

P = { P ( t , T ) ;0 < t

< T , T < m}

themselves have a sangle source of randomness: a Wiener process W = (Wt)t>o. In the vast literature on the dynamics of bond prices the authors discuss also some other models, where one Wiener process W = (Wt)t>ois replaced by a multzwanate Wiener process W = (W’, . . . ,W n ) .To take into account also jumps in the prices P(t, T ) one invokes other ‘sources of randomness’: point processes, marked point processes, LCvy, and some other processes. Referring the reader to the special Iiterature (see, e.g., [36], [38], [128], and the bibliography therein), we present here only a few models equipped with such ‘sources of randomness’.

Chapter VII. Theory of Arbitrage. Continuous Time

726

In [36] and [38]the authors generalize models of type (1) by introducing models of 'diffusion with jumps' kind: d

where p = p ( d t , d z ) is an integer-valued random measure in R+ x R x E and (W', . . . , W d )are independent Wiener processes. One must also make the corresponding modification in the description of the dynamics of P(t, T ) and f ( t , T ) : d

d P ( t , T ) = P ( t , T ) A ( t ,T ) dt

+ 1B i ( t ,T )dWj i=l

9. Following [la81 we consider now models with Le'wy processes as 'sources of randomness'. To this end we consider first equation (20), which we rewrite as follows: d P ( t , T ) = P ( t , T )d H ( t , T ) , where

1 t

g(t,T)=

[ ~ ( sd s)

+ B ( s ,T )dWs].

(45)

(46)

We also set

H ( t , T )=

i"( 0

Then (see (9)-(13) in

[ r ( s )-

T]+ ds

B ( s , T )d W s

3 3d) we have the representations P ( t , T ) = P(0, T ) 8 ( i l (' , T ) ) t

(47)

5. Arbitrage, Conipleteness and Hedge Pricing in Diffusion Models of Bonds

727

In view of (47),

If, for instance, the function B ( s , T ) ,s ,< T , is bounded, then we see that the expression on the right-hand side of (50) is a martingale. Now, we replace the Wiener process W = (Wt)t>owith a Levy process L = (Lt)t20 (see Chapter 111, 5 l b ) . What can be the form of the processes g ( t , T ) and H ( t , T ) if, in place of the integrals

s, B ( s ,T )dW,, we consider now the int

tegrals h t B ( s ,T )dL, interpreted as stochastic integrals over a semimartingale L = ( L s ) s Gwith ~ bounded deterministic functions B ( s ,T ) ? If the functions B ( s ,T ) are sufficiently smooth in s, then we can use N. Wiener's

(See Chapter 111, $ 3 on ~ this subject and see [128] in connection with L6vy processes.) Let

be the cumulant function (see 5 3c) of the LCvy process L = (Lt)t>o. We assume that the integral in (51) is well defined and bounded for all X such that 1x1 6 c, where c = sup IB(s,T)I. s
By the meaning of the cumulant function EeXLt = ,tY'(X).

(52)

Let X r = h t B ( s , T )dL,, t 6 T . The process X T = ( X r ) t < has ~ independent increments and its triplet ( B x T ,C X T ,u X T )of predictable characteristics can be found from the triplet (B", C L ,v") of L (see Chapter IX, 3 5a in [la81 and [250]). Using ItB's formula we can see (see detail in [128]) that

The process (exp(X&-hTp(XB(s, T ) ) d s ) )

t
is amartingale (cf. (11)in 5 3c).

Hence if we want that ( P ( t , T ) ) t G Tbe a martingale, then it seems reasonable to generalize (50) by setting

Chapter VII. Theory of Arbitrage. Continuous Time

728

Returning from with

P(t,T ) to the process P ( t , T ) we obtain that

the ( B ,P)-market

(543

has the following property: the discounted prices ( P ( t ,T ) ) ~

H ( t , T )z H ( t , T ) + $ ~ t R 2 ( s , T ) d s +

( e B ( s , T ) A L s - l - B ( s , T ) AL,) (56)

O<s
and

d P ( t , T ) = P(t-, T )d j l ( t , T ) .

(57) R e m a r k 3. Starting from equations for P ( t , T ) E. Eberlein and S.Raible [128]have analyzed the structure of forward rates and interest rates f ( t , T ) and r ( t ) and also considered in detail hyperbolic Le‘vy processes, i.e., L6vy processes such that the random variable L1 has hyperbolic distribution (see Chapter 111, 3 Id).

3 5b.

Completeness

1. Proceeding to the issue of completeness in (B,P)-models it is worth recalling that for dzscrete time n 6 N < co and finitely many kinds of stock the completeness of an arbitrage-free market is equivalent (by the Second f u n d a m e n t a l theorem) to the uniqueness of the martingale measure and to the existence of ‘5’-representation’ for martingales (with respect to some martingale measure). For a diffusion ( B ,P)-market generated by an rn-dimensional Wiener process W = (W’, . . .,W m )we have a multidimensional analogue of Theorem 2 in Chapter 111, 3c: each local martingale M = ( M i ,9 i ) admits a representation

s

with gS-measurable functions &(s) such that

As for ( B ,S)-markets (see 5 4b), this representation plays a key role in the study of completeness in (B,P)-models.

5. Arbitrage, Completeness and Hedge Pricing in Diffusion Models of Bonds

729

Let TO be a fixed instant of time and let f~~ be a ,~TO-measurablepay-off function. We assume that f ~ is, bounded ( l f ~ , I 6 C) and we shall say that this pay-off function can be replzcated if there exists a self-financing portfolio 7r = (PIy) such that XGo = fTo (P-as.). (2)

If this property holds for each TO and each bounded 9To-measurable pay-off function f ~then ~ we, say that our (B,P)-model is complete. Assume that the prices P ( t , T) of T-bonds satisfy the relations d P ( t , T ) = P ( t , T ) r ( t )d t

(

+

1

m

Bi(t,T ) d W j . i=l

(3)

We also assume that 0 < B,(t, T ) 6 C = Const. Then the original measure P is martingale in the following sense: the family of prices ( p ( t ,T ) ) t G Tis a local martingale and m

I

=Xb+C i =~ l [ ~ Y B i ( s , T ) P ( s , T ) y l ( ddW,. T)

(4)

We set

Then M = ( M t , 9 : t )admits the representation (1) and comparing with (4) we see that to replicate the variables M i , t 6 TO,by the value z;, t 6 TO,of some self-financing portfolio 7r it is necessary and sufficient that ( [ 3 8 ] )

<

((dPxdt)-a.s.) for i = 1,. . . , m and t TO. If there exists a soIution ( $ ( d T ) , t 6 TO,T

0;= Mt we see that

7r*

-

< T O }then , setting

iTOp(t,Th;(dT),

(7)

= (p*,y*) is a self=financing portfolio such that -7r

X,

-7r*

In particular, XTo =

TO-co,mpleteness.

~

BTo

(> .

= Mt,

t 6 To.

and therefore X;:

=

f~~ (P-a.s.), i.e., we have

730

Chapter VII. Theory of Arbitrage. Continuous Time

2. EXAMPLE ([36], [ 3 8 ] ) . Assume that there are finitely many, specifically, d bonds in a (B,P)-market having maturity times T I , .. . , T d . (Hence the supports of the measures y t ( d T ) are concentrated a t the points {TI},. . . , { T d } . ) There are ‘IIL ‘sources of randomness’ and it seems plausible that we need sufficiently many kinds of bonds to replicate the pay-off function f ~d must ~ :probably be not less than m. Let d = m.Then the system (6) takes the following form:

where i = 1,.. . , d. It is clear from (8) that for each t TOthis system has a solution if and only if the matrix llBi(t,TJ)llis invertible. If m = d = 1, then the systeni (8) turns t o the single relation

<

$l(t) = Bl(t,Tl)P(t.Tl)Yt({Tl)),

5 5c.

(9)

Fundamental Partial Differential Equation of the Term Structure of Bonds

1. By contrast to the direct approach to the description of the dynamics of bond prices P(t, T ) by stochastic differential equations (see 5 5a), taking the indirect approach we assume that the prices P(t,T ) have the following form: P ( t , T )= F ( t , r ( t ) , T ) ,

(1)

where r ( t ) is some ‘interest rate’ taking, as a rule, only nonnegative values. The indirect approach (1) was historically among the first few. Later on it has been overshadowed (first of all, in theoretical studies) by the direct approach. However, as regards obtaining simple analytic formulas, the approach (1) has retained its importance and is still popular.

2. It should be noted from the outset that this method works only under the assumption that the interest rate process ( r ( t ) ) t > ois a Markov process satisfying the stochastic differential equation

d r ( t ) = a ( t . r ( t ) )d t

+ b ( t , r ( t ) )dWt

(2)

or an equation of ‘diffusion-with-jumps’ kind (see equation (6) in Chapter 111, 5 4a).

5. Arbitragc, Completeness and Hedge Pricing in Diffusion Models of Bonds

We shall assume that for each T > 0 the function FT C1,2(in t and r ) . Then

731

= F ( t ,r , T ) is in the class

Assuming that FT > 0, we now rewrite this equation as follows (cf. equation (8) in 55a):

dt+BT(t,r(t))dWt).

(4)

where

dFT

- + u p

A T ( t , r )=

at

dFT 1 2a2FT +-b dr 2 dr2 FT ~

(5)

and

In finding additional conditions on the functions FT (besides the obvious condition F T ( T ,r ( T ) )= F ( T ,r ( T ) ,T ) = P(T,T ) = 1)we shall be based on the condition that the ( B ,P)-market in question must be arbitrage-free. Then, comparing (4) and formula (8) in fj 5a and taking account of relation (25) in 3 5a we see that in order that the market be arbitrage-free there must exist a function p ( t ) such that

A T ( t ,r ) - r B T ( t ,r )

=

-dt)

(7)

<

for all t and T , t T . (For each function cp = p(t) we can construct the corresponding ‘martingale’ measure by formula (21) in fj 5a.) In view of (5) and (6), we see from (7) that if the functions FT = F ( t , r , T ) , T > 0 , satisfy the fundamental equation

dF

-

at

+ ( u + cpb) ddFr + -b21 2 ddr2 F = T F , -

t

< T:

with boundary condition F ( T , r , T ) = 1, T > 0 , r 3 0, then a ( B ,P ) - m a r k e t with P ( t , T ) = F ( t ;r ( t ) ,T ) i s arbitrage-free. Equation ( 8 ) is very similar to the Fundamental equation of hedge pricing for stock (see (19) in 5 4c). However, there is a crucial difference between these cases: the function cp = cp(t) in ( 8 ) cannot be defined in a unique way on the basis of our assumptions and must be set a priori. As pointed out above, the martingale measure is is defined in terms of this function. Hence the choice of the latter is equivalent to a choice of a ‘risk-neutral’ measure operative, in investors’ opinion, in our ( B ,P)-market.

732

Chapter VII. Theory of Arbitrage. Continuous Time

3. Following notation (11) of Chapter 111, fj 3f, where we discussed the forward and the backward Kolniogorov equations and the probability representation of solutions to partial differential equations we set now

The operator L ( s ,T ) is the backward operator of the diffusion Markov process T

= ( T ( t ) ) t 2 0 satisfying the stochastic differential equation

Rewriting (8) as

dF

-_ =

L ( s ,T ) F -

i)S

we observe that this equation belongs (see Chapter III,S 3 f ) to the class of FeynmanKac equations (for the diffusion process T = ( T ( ~ ) ) Q O ) . The probabilistic solution of this equation with boundary condition F ( T ,r , T ) = 1 can be represented (cf. (19’) in Chapter 111, S3f and see [l23], [170], [288]for detail) as follows:

F ( s , r , T ) = E,;,{exp(-

p l d u ) } ,

(12)

where Es,, is the expectation with respect to the probability distribution of the process ( T ( u ) ) , ~such ~ ~ ~ that T ~ ( s=) T . Note that the formula (12), which we have derived under the assumption of the absence of arbitrage; is in perfect accord with the earlier obtained representation (5) in fj 5a, because in the Markov case we have

4. It is worth noting at this point that all the models of the dynamics of stochastic interest rates discussed in Chapter 111, fj4a (see (7)-(21)) count among dzffuszon Markov mod& of t,ype (10). Their variety is primarily a result of their authors’ desire to find analytically treatable models producing results compatible with actually observable data. As noted in Chapter 111, fj 4c one important subclass of such analytically treatable models is formed by the (affine) models ([36], [38], [117], [119]) having the representation T ( t ) , T ) = exp{a(t, T ) - T ( t ) P ( t , T I } (13)

qt,

with deterministic functions a ( t ,7‘) and

p(t,T ) .

5. Arbitrage, Completeness a n d Hedge Pricing in Diffusion Models of Bonds

733

The model that we shall now discuss (borrowed from the above-mentioned papers) can be obtained as follows. Assume that in (10) we have

4 4 7 )+ c p ( t ) b ( t ,.) and

= a l ( t ) i7-a2(t)

+

b(t,r ) = Jbl ( t ) rb2(t) Then (8) takes the following form:

Seeking the solution of this equation in the form (13) with F ( T ,T , T ) = 1 we see that a ( t , T )and P ( t , T ) are defined by a l ( t ) , a 2 ( t ) , b l ( t ) , and b z ( t ) in accordance with the following relations:

BP

-

at

+ a2P

-

1.

and (16) Relation (15) is the Riccati equation. On finding its solution p(t,T ) we can find a ( t ,T ) from (16), which gives us the affine niodel (13) with these functions a ( t ,T ) and P ( t , 2').

EXAMPLE. We consider the VasiCek model (see ( 8 ) in Chapter 111, 5 4a) d r ( t ) = (ii - b r ( t ) )d t where ii, 5, and i? are constants. Then we see from (15) and (16) that

and

Consequently,

and

+ZdWt,

Chapter VIII . Theory of Pricing in Stochastic Financial Models. Continuous Time 1. European options in diffusion (B. S)-stockmarkets . . . . . . . . . . . . .

3 l a.

Bachelier’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . formula . Martingale inference . . . . . . . . . . . . . . . . 5 l c . Black-Scholes formula . Inference based on the solution of the fundamental equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Id . Black-Scholes formula . Case with dividends . . . . . . . . . . . . . . . .

5 l b . Black-Scholes

2 . American o p t i o n s i n d i f f u s i o n (B,S)-stockmarkets. Case of a n i n f i n i t e t i m e horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 2a .

Standard call option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2b . Standard put option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2c . Combinations of put and call options . . . . . . . . . . . . . . . . . . . . 3 2d . Russian option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . American options in d i f f u s i o n ( B ,S ) - s t o c k m a r k e t s . Finite t i m e h o r i z o n s . . . . . . . . . . . . . . . . . . .

735 735 739 745 748 751 75 1 763 765 767

....

778

53a . Special features of calculations on finite time intervals . . . . . . . . 3 3b . Optimal stopping problems and Stephan problems . . . . . . . . . . 3 3c . Stephan problem for standard call and put options . . . . . . . . . . 3 3d . Relations between the prices of European and American options

778 782 784 788

4 . European and A m e r i c a n o p t i o n s i n a d i f f u s i o n (B,? ) - b o n d m a r k e t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 4a . Option pricing in a bondmarket . . . . . . . . . . . . . . . . . . . . . . . . 5 4b . European option pricing in single-factor Gaussian models . . . . . 4c . American option pricing in single-factor Gaussian models . . . . .

792 792 795 799

1. European Options in Diffusion (B, 5’)-Stockmarkets

3 la.

Bachelier’s Formula

1. The content of this chapter relates to continuous time, but, conceptually, it is in direct connection with our discussion of the discrete-time case in the sixth chapter. Here we shall be mostly interested in options. We take them as examples where one can clearly see the role of arbitrage theory and stochastic calculus and the opportunities they give one in calculations related to continuous-time financial models. 2. As mentioned before (Chapter I, 52a) L. Bachelier was by all means the first person to describe the dynamics of stock prices using models based on ‘random walks and their limit cases’ (see [la]), i.e., Brownian motions in the contemporary language. Assuming that the fluctuations of stock prices are similar to a Brownian motion, Bachelier carried out several calculations for the (rational) prices of some options traded in France at his time and compared their results with the actual market prices. Formula ( 5 ) below is an updated version of several Bachelier’s results on options [12], which is why we call it Bachelier’s formula. In the linear Bachelier model one considers a ( B ,S)-market such that the state ~ the same (Bt l ) ,while the share price of the bank account B = ( B t ) t Gremains S = ( S t ) t
where W = ( W t ) ~iso a standard Wiener process (a Brownian motion) on some probability space (R, 9, P). Prices in this model can also take negative values, therefore it cannot adequately reflect real life. Nevertheless, its discussion could be of interest from different points of view: historically, it is the first di;giLsion model, on the other hand this model is both arbitrage-free and complete (see Chapter VII).

Chapter VIII. Theory of Pricing. Continuous Time

736

(2) and let 9 t , t 6 T, be the u-algebra generated by the values of a Wiener process {Ws,s 6 t } and conipleted by the addition of the sets of P-probability zero. We define the new measure P, on (n,9 ~ by ) setting

-

dPT = ZTdP T

(3)

(cf. (8) in Chapter VII, 3 4a), where PT = P 19~. Note that is a unique martingale measure in this model (see Chapter VII, 4a.5), i.e., the only measure such that PT PT and the process S = ( S t ) t is~ a~ P-martingale. Moreover, by Girsanov's theorem (Chapter 111, (i 3e or Chapter VII, 5 3b), ~ a (SO w pt u ~; t 6 t T I IPT) = Law ( so u ~; t t T 1 PT). (4) I

N

+ +

+

<

THEOREM (Bachelier's formula). The rational price CT = @ ( f ~ P) ; of the standard European call option with pay-off function fT = (ST- K ) + in the model (1) is defined by the formula

where cp(X)

1 -2 fie

= l;P(Y)dY.

3

= __

In particular, for SO = K we have

Proof. Analyzing the proofs of the theorems in (i(i4a,b we can see that the results obtained for the model ( 5 ) (with positive prices) in 54a remain valid for the present model (1). (The 'key' relation (14) in § 4 b assumes now the form

with Zt as in (2).) Thus, the ( B ,S)-market now is arbitragefree, T-complete, and the rational price is rt = 2;' ( $ + X t -

c72 'I>

@.T

E(ZTfT) =

(fT).

By (4) and the self-similarity of Wiener processes, E-

PT

(ST- K)'

(So+ p T

+ UWT - K)' = EpT (SO K + ~ W T ) ' = E(S0 K + uJ?;Wi)+. = E-

PT

-

-

(7)

1. European Options in Diffusion ( B ,S)-Stockmarkets

Note that if then

E

737

is a random variable with standard normal distribution N ( 0 ,l),

for a E R and b 3 0. Setting

a = So

-

K,

b = u&,

we obtain required formula ( 5 ) from ( 7 ) , ( 8 ) , and (9). 3. Now let Z = ( P l y ) be a strategy (in the class of self-financing portfolios) of initial value X: = CT that replicates the pay-off function f ~ i.e., , let X g = f~ (P-a.s.). By Chapter VII, § 4 b the value X' = (Xtz)tGT of this strategy satisfies the relation xtz = E P T ( f T 1%). (10) Since f~ = (ST - K ) + and S = (St)t
where a = St K and b = u r n . For 0 6 t 6 T and S > 0 we set ~

C ( t ,S ) = (S- K ) @

S-K

Then we see from (11) that Xtz = C ( t ,St).Simultaneously,

By It6's formula for C ( t ,St) we obtain

S-K

Chapter VIII. Theory of Pricing. Continuous Time

738

Comparing (14) and (13) and applying Corollary 1 to the Doob-Meyer decomposition (Chapter 111, 5b) we conclude that

Differentiating the right-hand side of ( l a ) , after simple transformations we obtain

The suitable value of

In other words,

,& can be found from the observation that

Pt == C ( t ,S t ) - rtst.

In view of (12) and (16) we see that

-

The following peculiarities of the behavior of r t and Pt as t t T are worth noting. Assume that close to the terminal instant T the stock prices St are higher than K . Then we see from (16) and (19) that

as t

t T . On the other hand, if St < K , then

as t t T . Both relations appear to be quite matter-of-course. For if St < K close to time T , then the pay-off function f~ vanishes and, clearly, the capital X g equal to zero at time T is sufficient for the option writer, which is just the case if (21) holds. On the other hand, if St > K close to time T , then f~ = S, - K , and the seller needs the capital X T = S, - K . Since XtF = pt r t S t , the required amount will be available if (20) holds, since XtF = TtSt + S, - K .

,& +

+

1. European Options in Diffusion ( B ,S)-Stockmarkets

3 lb.

739

Black-Scholes Formula. Martingale Inference

1. As already mentioned, the main deficiency of the linear Bachelier model

st = so + pt + awt

(1)

is that the prices St can assume negative values. A more realistic model is that of a geometric ( e c o n o m i c as some say, see [420]) Brownian m o t i o n , in which the prices can be expressed by the formula

st = So&, where

(2)

"

In other words,

St = Soe (,'-$)t+owt

(4)

Using It6's forniula (Chapter 111, 5 3d) we see that

dSt

=

St(pdt+adWt).

This is often expressed syrnbolically as

'St St

--

~

pdt

+ adWt,

which emphasizes analogy with the formula

which we used above, (e.g., in the Cox-Ross-Rubinstein model in the discrete-time case; see Chapter 11, 3 l e ) . The model of a geometric Brownian motion (2) was suggested by P. Samuelson [420] in 1965; it underlies the Black-Merton-Scholes model and the famous Black-Scholes formula for the rational price of a standard European call option with pay-off function f~ = (S, - K ) + discovered by F. Black and M. Scholes [44] and R. Merton [346] in 1973. 2. Thus, we shall consider the Black-Mertori-Scholes ( B ,S)-model and assume that the bank account B = (Bt)t20evolves in accordance with the formula

dBt = rBt d t , whereas the stock prices S = (St)t20 are governed by a Brownian motion:

(5)

dSt = S t ( p d t +adWt).

(6)

Bt = BOeTt,

(7)

Thus, let

Chapter VIII. Theory of Pricing. Continuous Time

740

THEOREM (Black-Scholes formula). The rational price (CT = @(f T ; P) of a standard European call option with pay-off function fT = ( S , - K ) + in the model (5)-(6) is described by the formula

In particnlar, for So = K and r = 0 we have

and CT

-

K n E

as

T

+ 0 (cf.formula (6) in

la).

We present the proof of this formula, as borrowed from [44] and [346], in the next section. Here we suggest what one would call a 'martingale' proof; it is based upon our discussion in Chapter VII. Using notation similar to that in the preceding section we set (11)

and let PT be a measure on ( n , 9 ~such ) that dP* = ZT ~ P T . By Girsanov's theorem (Chapter VII, 53b) the process = (@f)t
w

wt +

~

(T

Hence Law(St; t

< T IFT) = Law (~ o e ( p - + ) t + c w t ; t < T I isT)

< T I PT).

= Law ( ~ o e ( ~ - + ) t + o w t ;

(12)

From the theorem in Chapter VII, §4a we see that, taking the class of O-ad-

hT

missible strategies 7r = (0, y) with 7:s; du < 00 (P-as.) we can describe the rational price @.T = @( f T ; P) by the following formula: fT C T = B O E - -. PT

BT

(13)

1. European Options in Diffusion (B, 5')-Stockmarkets

741

Since f T = (&-K)+ in this case, it follows in view of (12) and the self-similarity of the Wiener process (Law(WT) = L a w ( O W 1 ) ) that

< - .N(O,1).

where

a = SoerT,

b=c f i ,

It is an easy calculation (similar to (9) in

3 l a ) that

Thus, it follows from (14)-(16) that

Setting here a = S0erT and b = aJ?; we arrive at the Black-Scholes formula (9), which completes the proof.

Remark 1. Setting

Y& =

In $$

+~

$1

( fr

9

c-JT

we can write (9) in a more compact form:

Let PT be the rational price of the standard European put option with pay-off function f~ = ( K - ST)+. Then, since

742

Chapter VIII. Theory of Pricing. Continuous Time

(cf. 'call-put parity' identity (9) in Chapter Vl, 3 4d), it follows that

or IPT = -So@(-y+)

+ Ke-rT@(-y-).

(19) 3. The model in question is T-complete (see Definition 1 in Chapter VII, fj2d), and there exists a 0-admissible strategy E = of value X' = (Xt')tGT such that Xf = CT and XF replicates f~ faithfully:

(B,?)

-

X$

= fT

(P-a.s.).

By the theorem in Chapter VlI, 5 4b,

where

a = Ster(T-t), b = a

m

,

6

-

N ( 0 ,l ) ,

and the variables St and [ = WT - Wt are independent with respect to the initial measure P. Taking (16) into account, we see from (20) that the price C ( t ,S t ) = X,' has the following expression:

1. European Options in Diffusion

(I?, S)-Stockmarkets

743

As in 3 la (see subsection 3) we can show that, for an optimal hedging portfolio

By (21),after simple transformations we obtain In $

rt = @

(cf. formula (16)in

+ (T

3 l a ) , and since &Bt

It is easy to see that 0

<

Tt

-

t)(T

+ $1

+ TtSt = C ( t ,St), it follows that

< 1 and

,& is always

borrowing from the bank account under the constraint

negative, which indicates K -

--

< /It.

BO As with the Bachelier model, we have properties (20) and (21) in fjla; namely, if t T T and St > K close to time T then YtSt

+ St

and

&Bt

+ -K;

and af

t T T and St < K close to time T then YtSt

+0

and

PtBt

+ 0.

Remark 2. The above price C ( t ,S t ) depends, of course, also on the parameters T and a specifying the particular model. To indicate this dependence we shall write C = C ( t ,s, r , CT) (with St = s). It is often important in practice to have a knowledge of the ‘sensitivity’ of C ( t ,s , T , a ) to variations of the parameters t , s, T , and a. The following functions are standard measures of this ‘sensitivity’ (see, e.g., [36]and [415]):

dC ( 9 -

at

dC a=as

dC dr

p=-,

(Here ‘V’ is pronounced ‘vega’.) For the Black-Scholes model, from (21)we see that

v = -dC . aa

Chapter VIII. Theory of Pricing. Continuous Time

744

where

4. Our calculations of @T in (14)-(16) could be carried out in a somewhat different way, on the basis of an appropriate choice of a dzscountzng process ('num6raire'), as discussed in Chapter VII, lb. To this end we rewrite (13) with f~ = (ST - K)' as follows:

By (12), the calculation of E-

PT

I ( S T > K ) encounters no complications:

To calculate Bo E F ST ~ ~ I ( >S KT) we consider the process

z= (Ztt)tGTwith

z z~

It is important that is a posztaue martzngale (with respect to the 'martingale' measure PT) with E= 1. Hence we can introduce another measure, PT, by PT setting I

dFT = -Z, ~ P T .

(The measure PT is called in [434] the dual-to PT-martingale BY (7) and ( 8 ) , Zt = e U W t + ( p - - T - g - ) t =

-

where Wt = Wt

+ "-Tt U

(28) measure.)

(t < T ) is a Wiener process with respect to

PT.

Using Girsanov's theorem (Chapter 111, 3e or Chapter VII, 3 3b) it is easy to verify that

-

-

W t = Wt - at

(= Wt + (5 a ) t ), t < T, -

(29)

1. European Options in Diffusion

is a Wiener process with respect to

PT.

( B ,S)-Stockmarkets

745

In view of the above notation,

therefore it follows from (25) that

By analogy with (12),

In particular, i f f

N

"0,

l ) ,then

Hence

E- I(ST > K ) =

In

$ + T ( r + $)

PT

A combination of (30), (as),and (32) proves the Black-Scholes formula (9) for @r in a different way, as promised a t the beginning of the subsection.

5 lc.

Black-Scholes Formula. Inference Based on the Solution of the Fundamental Equation

1. We now present the original proof of the Black-Scholes f o m v l a for the ratzonal przce of option contracts, suggested independently by F. Black and M. Scholes in 1441 and R. Merton in [346] (1973). Of course, the first question before the authors was about the definition of the rutzonal przce. Their (remarkable in simplicity and efficiency) idea was that this must be just the minimum level of capital allowing the option writer to build a hedging portfolio. More formally, this can be explained as follows. Consider a European option contract with maturity date T and pay-off function f ~ Then . the rutzonul (fazr) pmce Yt of this contract at a time t , 0 t < T ,

<

Chapter VIII. Theory of Pricing. Continuous Time

746

is (by the definition of F. Black, M. Scholes, and R. Merton) the price of a perfect European hedge Qllt,T]= inf{z: 37r

with XT = z and XG = f~ (P-as.)}.

(1)

(Cf. the appropriate definitions in Chapter VI, 5 l b and Chapter VII, 5 4b; we also denoted before C[O,T] by C)., In general, it is not a priori clear whether there exist perfect hedges. The results of Chapter VII, 5 5 4a, b show that such hedges do exist in our model of a (B,S)-market and, moreover, yt = elt,,] is equal to the quantity Bt EpT

($1

9 1 1 , where

is the martingale measure. This is why we called

our scheme in 5 l b a ‘martingale’ inference. Papers [44], [346] were written before the development of the ‘martingale’ approach and used another method for the calculation of Yt = C [ t , T ~which , we describe now. Since both process S = (St)t>o and pay-off function f~ = (S, - K ) + are Markov, it is natural to assume that the 9t-measurable variable yt depends on the ‘past’ only through St: Yt = Y ( t ,S t ) . Assuming that the function Y = Y ( t ,S) on [O,T)x (0, m ) is, in addition, suffi~ ) authors , of [44] and [346] obtain the ciently smooth (more precisely, Y E C ~ T the following fundamental equation:

dY

-

dt

1 d2Y + rS-dasY + -n2s22 dS2

=ry

with boundary condition

Y ( T ,S)= (S- K ) +

(3)

(One can find a derivation of (2) in Chapter VII, 5 4c; see equation (9) there.) The next step to the Black-Scholes f o r m u l a (that is, to the expression for Y ( 0 ,So)) is to find a solution of ( 2 ) - ( 3 ) . Equation (2) is of Feynman-Kac kind (see (19) in Chapter 111, 33f) and it can be solved using the standard techniques developed for such equations. We consider the new variables

Z=InS+ and set

(

3

r--

(T-t)

(5)

1. European Options in Diffusion

( B ,S)-Stockmarkets

747

In these variables ( 2 ) - - ( 3 )is equivalent to the following problem:

Relation (7) is the heat equation, and by formula (17') in Chapter 111, §3f the solution to (7)-(8) can be expressed as follows:

V ( 0 , Z )= E(eWO+'

-

K

I+,

(9)

where W = (Wo) is a standard Wiener process. We set a = e '+:, I!=&, and [ - N ( O , l ) Then

Using formula (16) in

3 I b we see that

Finally, using notation (4) and (5), by (6) and (11) we obtain the formula

Y ( t ,S ) = e-'(T-t)V(B, 2)

(Cf. (21) in $ lb.) Setting here t = 0 and S = So, we obtain the required Black-Scholes formula (formula (9) in lb). As shown in EJ I b , the portfolio Z = ( & , T t ) t G ~with rt = dY -(t, S t ) , and pt = Y ( t ,S t ) - StTt is a hedge of value X g replicating faithfully the dS pay-off function fT = (ST - K)+.

Chapter VIII. Theory of Pricing. Continuous Time

748

2. In conclusion we make several observations concerning the above two derivations of the Black-Scholes formulu. The ‘martingale’inference in fj l b is based on the existence of a unique martingale measure in the model of a (B,S)-market in question. This means the absence of arbitrage and enables one to calculate the rational price @T by the formula CT = Bo E P , f~T, which (for fT = (ST- K ) + ) gives one just the Black-Scholes formula. The approach based on the solution of the ’fundamental equatzon’ brings one to the same formula. It is worth noting that the lack of arbitrage and the existence of perfect hedges are reflected there by the fact that, due to the unique solubility of (2)-(3), the resulting price Y ( 0 ,So) is automatically ‘arbitrage-free’, ‘fair’: if the price asked for the option is lower than Y ( 0 ,So) then the seller cannot in general fulfill hi5 obligations, while if it is higher than Y ( 0 ,So) then the seller will for sure cash a net profit (‘have a free lunch’}. See Chapter V, 3 l b for detail. fj Id. Black-Scholes Formula. Case with Dividends 1. We assume again that a ( B ,5’)-market can be described by relations (5) and (6)

in

5 Ib, but

Bt

Here 6 3 0 is a parameter characterizing the rate of dividend payments. If = 1, then it follows from (I) that

the stock also brings dividends (cf. Chapter V, la.6). More precisely, this means the following. If S = (St)t>o is the stock market prrce then wrth dzvrdends taken into account the capital 3 = ( s t ) t > o of the stockholder is assumed to evoIve (after discounting) by the formula

dSt = dSt

+ SSt d t ,

(2)

so that the increase over time dt in the capital of the stockholder is the sum of the increase dSt in its market price and the dividends SSt dt proportional to St. Since dSt = St ( p dt u d W t ) and

+

it follows by (1)that

1. European Options in Diffusion ( B ,S)-Stockmarkets

We set

-

wt = wt +

p--T+6 CT

749

t

and 1 p-r+6

wT--( 2 Then, defining the measure

PT by

)2T).

the formula -

dPT = ZT d P T ,

w = ( W t ) t < T is a Wiener

we see by Girsanov’s theorem (see Chapter 111, 3 3e) that process with respect to PT. Hence Law(pt+aWt:

~
and Law(St; t Let XT = PtBt

< T / P T ) = Law(soe(T-6-g)t+gWt; -

+ ytSt, t

t

< T 1 PT).

(7)

6 T, be the value of a self-financing strategy x

(p,7 ) . Since the discounted capital

.

is a martingale with respect to P T ,

x belongs t o the class of 0-admissible strategies with follows that

=

x,.

/ 7:s: T

0

du

< 00

(P-a.s.), it

X? PTBT Bo Hence (cf. (13) in $ l b ) we obtain that the rational price @ ~ ( dr ;) of a call option is expressed by the formula E-

750

Chapter VIII. Theory of Pricing. Continuous Time

Now let P T ( ~T ); be the corresponding price of a put option in the case of dividend payments. It is easy to see that Cc~(6; r ) and P T ( ~T ); are connected by the following identity of ‘call-put parity’ (cf. (9) in Chapter VI, 5 4d):

Comparing (9) and formula (9) in i l b for @,(O;T) account we arrive at the following conclusion.

(SE

CT)and taking (10) into

THEOREM. The rational prices c ~ ( 6 T ) ;and p ~ ( 6T ;) of call and put options in the case of dividend payments are described by the formulas

I C*(6;

T)

= e-%T(O;

T -

6)

I

where CT(O;T - 6) and PT(O;T - 6) can be defined by the right-hand sides of (9) and formula (18) in 5 I b (‘the case without dividends’) with T replaced by T - 6.

2. American Options in Diffusion ( B ,S)-Stockmarkets. Case of an Infinite Time Horizon

2a. Standard Call Option 1. In the Considerations of options and other derivative financial instruments one must sharply distinguish between two cases: of the time paramet,er t ranging over a finite interval [0, TI, and o f t in the infinite interval [ O , o o ) . Of course, the second case smacks of idealization, but it is much easier to study than the first case, when the decisions taken at time t depend significantly on the time T - t remaining till the expiration of the contract. This explains why we start with the discussion of the second case. We consider finite intervals [0, TI in 5 3 . 2. We assume that we have a standard Wiener process W = (Wt)tao on a filtered 9, ($t)tao, P) and that our diffusion ( B ,S)-market has the probability space (a, following structure:

dBt = r B t d t , dSt = S t ( p d t

Bo > 0 ,

(1) (2)

+ g d W t ) , so > 0.

For a standard discounted call option the pay-off function has, by definition, the following structure: g(St), where g(z) = (x - K)+, z E E = (0, cm),X 3 0. By analogy with the discrete-time case we set

(3)

ft z

where the supremum is taken over the class of finite stopping times

rnr = { T = .(w):

06

T(W)

< La,

w E

o},

Chapter VIII. Theory of Pricing. Continuous Time

752

-

and E, is the expectation with respect to the martingale measure P, such that the process S = (St)t>O has a stochastic differential SO=Z

dSt = S t ( r d t + a d W t ) ,

(6)

with respect to this measure. To simplify the notation we assume from the very beginning-that ,u-= T . Making this assumption we can drop the sign in our notation for P, and Ex. Thus, let ' ^ l '

V*(z)= sup EZe-(X+T)r(ST - K ) + .

(7)

T€mr

It is reasonable in many problems to consider. alongside !lJZn,X, also the class

330" = { 7- = .(w) : 0 < .(w) 6 00,w E a} of Markov times that can also assume the value

+CO

and to set

Finding V*(z) and v*(z)is in direct relation to the calculations for standard American call options because the values of V*(z)and v*(z) are just the rational prices, provided that the buyer can choose the exercise time in the class or % ; and SO= z. (The case of 7 = 00 corresponds to ducking the exercise of the option.) The proof of this assertion in the discrete-time case can be carried out in the same way as the proof of Theorem 1 in Chapter VI, 3 2c. The changes in the continuoustime case are not very essential: see, for instance, [33], [265], or [a811 for greater detail. Moreover, if T* and ?* are optimal times delivering the solutions to (7) and (8),respectively, then they are also the optimal strike times (in the classes m r and

!lRr

%fir).

3. Embarking on the discussion of optimal stopping problems (7) and (8) we single out the (noninteresting) case of X = 0 first. In that case

which shows that the process (eCTt(St- K)+)t>ois a submartingale and therefore if 7 E !JJtr,i.e., T ( W ) 6 T for w E 0, then

EzeCTT(ST- K ) + < E Z e - T T ( S ~- K ) + 6 5 . By the Black-Scholes formula (see (9) in

E x C T T ( S-~ K)' for each

T

3 0.

(9)

5 lb), --t z

as T

+ CO,

(10)

2. American Options. Case of an Infinite Time Horizon

753

Since V*(x) = lim V$(x) in our case, where T+@2

(see 1441; Chapter 31 and cf. Chapter VI, §5b), it follows by (9) and (10) that if X = 0 and r 3 0, then ‘the observations must be continued as long as possible’. More precisely, for each z > 0 and each E > 0 there exists a deterministic instant Tx,E such that E,

and Tx,E + m as

E

-rTx

I

E

(STZ,& -

w+3 x -

E

-+ 0.

4. We formulate now the main results obtained for the optimal stopping problems (7) and (8) in the case X > 0.

THEOREM. If X > 0, then for each x E (0, CQ) we have

-*

V*(x) = v (x)=

x < x*, K , x 3 x*,

c*xy1,

x

-

where

There exists an optimal stopping time in the class T*

= inf{t

mr, namely, the time

3 0 : St 3 x”}.

(16)

Moreover, 02

if? 3 - o r x 3 x*, 2

PZ(T* < m) =

ifr

02

(17)

< - and x < x*. 2

We present two proofs of these results below. The first is based on the ’Markovian’ approach to optimal stopping problems and is conceptually similar to the proof in the discrete-time case (see Chapter VI, 35b). The second is based on some ‘martingale’ ideas used in [32] and on the transition to the ‘dual’ probability measure (see 3 lb.4).

Chapter VIII. Theory of Pricing. Continuous Time

754

5 . The first proof. We consider optimal stopping problems that are slightly more general than (7) and (8). Let

-* V (x)= sup E,e-PTg(ST)l(~< m) TEEr

be the prices in the optimal stopping problem for the Markov process S=(St, 9 t , Pz), z E E = ( O , c o ) , where P, is the probability distribution for the process S with SO= z, p > 0, and g = g ( x ) is some Bore1 function. If g = g(x) is nonnegative and continuous, then the general theory of optimal stopping rules for Markov processes (see [441; Chapter 31 and cf. Theorem 4 in Chapter VI, S2a) says that: (a) V*(z)= v*(z), z E E; (20) (b) V*(z) is the smallest /3-excessive majorant of g ( x ) ,i.e.. the smallest function V ( x )such that V ( x )3 g(z) and V ( z )3 e-@TtV(z), (21) where T t V ( z )= E,V(St); (c) V*(z) = lim limQ$g(z), n N where Q n g ( z ) = max(g(z),e-0'2"T2-ng(x)); (d) if E, [sup e-fltg(St)] t

(22) (23)

< 00,then for each E > 0 the instant

is an &-optimalstopping time in the class !Blr,i.e., P,(T, v*(z)- E E,~-P~;~(S,,); (e) if 7 0 = inf{t: v*(st) e - f l t g ( s t ) }

<

< 00) = 1, x

E

E , and

<

is an optimal stopping time ( P z ( q < cm) = 1, x E E ) , then it is optimal in the class tmr: V*(z) = E,e-PTog(STo), z E E;

<

moreover, if 71 is another optimal stopping time, then P,(Q 71) = 1, x E E , i.e., is the smallest optimal stopping time. Let G* = {z E E : V*(z) > g(z)} and let D* = {z E E : V*(z) = g(x)}. It is easy to see from (22) and (23) (cf. Chapter VI, 5b) that the structure of V* = V * ( z )is rather simple: this is a downwards convex function on E = (0,cm)

7-0

2. American Options. Case of an Infinite Time Horizon

755

majorizing g = y ( ~ ) In . addition, there exists z* such that C* = ( 5 : z < z*} and D* = {z: z 3 z*}. Hence the solution of the problem (7) and (8) is reduced t o finding z* and, of course, the function V*(z) ( = V*(Z)). Analyzing the arguments used in the corresponding discrete-time problem in Chapter VI, s5b.6, one easily understands that the required value of z* and the function V*(z), the smallest (A + r)-excessive majorant of g(z). must be solutions of the following Stephan, or free-boundary, problem (see [441; 3.81):

L V ( Z ) = (A

+ r)V(z),

V ( z )= g(z),

where

L

z

ic

< E,

3 i?,

a ff2& ax + -2z a x 2

= rx -

is the infinitesimal operator (see [126]) of the process S = (St)t>owith stochastic differential dSt = S t ( r d t (5 dWt).

+

We shall seek a solution of (25) (in the so far unknown domain ( 0 , E ) ) in the following form: V(z) = CZY. (29) Then we obtain the following equation for y:

y2 - (1 -y);

-

2(X + r ) 7 = 0.

To simplify the notation we assume that g 2 = 1. 7' A . make the change T + -. X + - in the answers.

If

g2

# 1, then one must

(52

02

Equation (30) with n2 = 1 has two roots, 71 =

(i

- T )

+

and 72

=

($-

T ) -

/r)2+20

Jm.

Chapter VIII. Theory of Pricing. Continuous Time

756

Since X > 0, it follows that 71 > 1. (If X = 0, then y1 = 1.) The root Hence the general solution of (25) is as follows: I

V(x) = C ] Z Y l

72

+ c2x72.

is negative.

(33)

As in the discrete-time case (Chapter VI, § 5b), we see from (33) that c2 = 0 since otherwise V(x) i 00 as x 4 0, which is impossible in our context (we must have V * ( x )3 0 and V * ( x ) x). Thus, V(x) = c1x71 for x < Z, where c1 and the 'free' boundary Z are the unknowns. To find them we use condition (26) and condition (27) of kmooth pasting'. Condition (26) shows that I

C ] P

=x

-

K.

(34)

Condition (27) takes the form

We see from these two relations that

Thus, the solution v ( x ) to (25)-(27) can be represented as follows:

-

V ( x )=

ClXYl,

x < 2,

X-K,

x ~ Z ,

(37)

where 2 and c1 are defined by (36)

Remark. If K = 1, then ?(x) is just the function v ( x ) defined by (39) in Chapter VI, 5 5b, which is not that surprising if we take into account (22) and our method of finding v ( x ) in Chapter VI. We shall have proved the theorem once we shall have shown that the function c ( a ) so obtained is the price V * ( x )(see (7)) and the time ;T = inf{t

2 0 : St 3 Z }

2Jlr

(and in the class if Px(? < 00) = 1). is optimal in the class To this end it is obviously suffices to use the following test: for x E E = ( 0 ,m) we must-have (A) V ( x )= E,e-('+')'(S.; - K)+1(? < OC) and (B) V ( x )3 E,e-('+')'(S, - K ) + ~ (< T m) for each T E

wr.

2. American Options. Case of an Infinite Time Horizon

757

Further, since ( S , - K ) + I ( 7 < m) = V ( S , ) I ( 7 < W ) and v(z) 3 (x - K ) + , for (A) and (B) we must verify the following conditions: (A’) p(z)= Eze-(’+‘)‘v(S,)1(7 < m)

Usually, the verification of (A’) and (B’) is based on It6’s formula for V(z) (more precisely, we use its generalization, the It6-Meyer formula). It proceeds as follows. Let V = V(z) be a function in the class C 2 , i.e., a function with continuous second derivative. Then the ‘classical’ It6’s formula (Chapter 111, 55c) for the function F ( t , z ) = e-(X+T)tV(z)and the process S = (St)t>obrings one to the following representation:

Considering now the function v(z) defined in (37) we can observe that it is ) one poznt x = E. so that one can in the class C2 for z E E = ( 0 , ~ outside anticipate (38) also for V ( x )= V ( z ) if one chooses a suitable interpretation of the derivative at x 12. In our case V ( z ) is (downwards) convex and its first derivative ?(x) is well defined and continuous for all z E E = ( 0 , ~ ) its ; second derivative V”(x) is defined for z E E = (0, m) distinct from 2, where we have the well-defined limits I

?!(5)

= lirri?’(z) XT.

and

vi(2)= limV”(z). XcJ.

There exists a generalization of It6’s formula in stochastic calculus obtained by P.-A. Meyer for a function V(z) that is a daflerence of two conuex functions. (See, e.g. [248; (5.52)] or the Zt6-Meyer formula in 1395: IV].) Our function V(z) is (downwards) convex and for F ( t , z ) = e - ( X + T ) t V ( x )and S = (St)t>o we have the ItG-Meyer formula, which looks the same as (38), but where v ” ( 2 )is replaced by, say, @‘(2). Having agreed about this we obtain

where

Chapter VIII. Theory of Pricing. Continuous Time

758

It is worth noting that

L V ( 2 ) - (A

+ r ) V ( z )= 0

(41)

for z < E (by (25)), and it is a straightforward calculation that this equality holds also for z = Z, while for z > 5 we have

L V ( z ) - (A

+ .)V(s) < 0.

(42)

By (39), (41), and (42) we obtain (for SO = x)

As is clear from (40), the process M = (Mt)t>ois a local martingale. Let be some localizing sequence and let T E Then by (43) we obtain

wr.

and, by Fatou's lemma,

which proves (B'). We now claim (A'). If x E 5 = {z: z 3 Z}, then P,(7 = 0) = 1, and property (A') is obvious. Now let Z E = {x: z < Z}. Then by (41) and (39) we obtain

so that

(m)

2. American Options. Case of an Infinite Time Horizon

759

Since

and

Esup[e-xt eUWt-G't t>o

I-

(45)

(see Corollary 2 to Lemma 1 below), it follows from Lebesgue's dominated convergence theorem that lirn E,e-(X+T)(7"A')V(ST,Ar;)I(r < 00) = E,e-(X+')7"V(S~)I(7< m). n

Further, v(ST,)

(46)

< v(Z) < 00 on the set { w : F = oo},therefore

v

lim EZe-(X+r)Tn (STn)1(7= 03) = 0. n

(47)

Required property (A') is a consequence of (44), (46), and (47). To complete the proof of the theorem we must verify property (46) and prove (17) (with T * = 7). We shall establish the following result to this end. LEMMAI . For z 3 0, p E Iw, and n

> 0 we have

xnGt)

< x ) = @ (_ _ where @(z)=

e

~

2

-

e y @ (

y), (48)

dy.

Proof. For simplicity, let a2 = 1. By Girsanov's theorem (Chapter 111, § 3 e or Chapter VII, 5 3b)

(

+ W,) > z, pt + Wt < 2 = E I ( ~ g ( p+ s W s )> pt + Wt 6 x)

1

P max p s s
(

2,

(

= Eexp DWt

-

/'2L ) L t

-t

I maxW, > x, W, < z).

(49)

Chapter VIII. Theory of Pricing. Continuous Time

760

We set Tx = inf{t

> 0 : Wt = z}. Then, by D. Andrk's reflection principle wt = WtI(t 6 Tx) + (2%- Wt)I(t> T,)

(50)

is also a Wiener process (see Chapter 111, 0 3b, and also [124], [266], and [439]). By (49) and (50) we obtain

=C

(

9

( Y ) -e2pxEexp p W t - -t

I(Wt

> x)

The proof is complete. 1. If p COROLLARY

< 0, then

If p 3 0, then P(sup(pt t2O

+ OWt) 6 z)

= 0.

(51')

( + 3in

COROLLARY 2 (to the proof of (45)). Setting p = - X

Hence if X > 0, then (45) holds.

-

(50) we obtain

2. American Options. Case of an Infinite Time Horizon

76 1

COROLLARY 3 (to the proof of (17)). Let St = zert . e u w t - g t and assume that

x* > x. Then one sees from (51) with p

< x* and p = r

a2 -

2

< 0 that

(72

< - and x < x*. This formula is obvious for x 3 x*, while

which proves (17) for r for x

=r -

-

-2 ' o - 3 0 formula (17) is a consequence of (51'). 2

All this completes the first proof of the theorem.

p

6. The second proof. Let and let SO= 1. Setting

=X

+ r , assume that X > 0, let y1 be as in

zt = ,-@s-f1 t ,

(31),

(54)

we obtain

Hence Z = ( Z t ) is a P-martingale and

e-qst

-

K ) + = S,--f'(St - K)+&.

Setting, in addition,

G ( x )= X - ~ ' ( X

-

K)+,

we see that

The process S = (St)tLO under consideration is generated by a Wiener process W = (Wt)t>o;without loss of generality we can assume that (n,9, ( 9 t ) t 2 0 , P) is a coordinate Wiener filtered space, i.e., R = C[O,m) is the space of continuous functions w = ( w ( t ) ) t > o , 9 t = a ( w : w ( s ) , s < t ) , 9 = V.Fi, and P is the Wiener measure. - Let is be a measure in (0,s)such that the process W = (Wt)t>o with Wt = Wt - (y1a)t is a Wiener process with respect to P.

-

I

Chapter VIII. Theory of Pricing. Continuous Time

762

If Pt = P 1st and Pt = p 1st are the restrictions of P and and the Radon-Nikodym derivative is

is to 9 t , then

N

Pt,

where Z t is defined by (55). (See, for instance, Theorem 2 in Chapter 111, 3 3e.) Hence, if A E 9 t , then EIA = EIAZt! where

c is averaging with respect to p, and if A E 9,-, then I

E I A I ( T
(58)

(cf. formula (2) in Chapter V, 33a). Hence we obtain that i f f = f ( ~is)a nonnegative g:,-measurable function, then

EU(T
(59)

Taken together with (56), this shows that

In other words, the optimal stopping problem (8) is equivalent (for II: = 1) to another problem, (60), which can be easily solved by the following arguments. We consider the function G(z) = z-TI(z - K ) + . This function attains its maximum on E = (0, m) at the point z* = K- Y1 (cf. (15)), and Y1 - 1 maxG(z) = c* xEE

(= G ( z * ) ) ,

(61)

where c* is defined by (14). Hence by (60) we obtain

-*

v

(1)

< c*

sup EI(7

< w) < c*

TEE,-

Let 7-* = inf{t 3 0: St 3 z*} and let the initial value SO be equal to 1 6 z*. Since X > 0 by assumption, it follows that z* < 00.

LEMMA2. 1) For X > 0 we have

-

P(T* < co) = 1.

2

2) If X > 0 and, in addition, r 2 -, 2

(63)

then

P(7* < m) = 1.

(64)

2. American Options. Case of an Infinite Time Horizon

Proof. With respect to Girsanov's theorem

763

p, the process wt = Wt - ( y 1 r ) t ( t 3 0) is Wiener, and by

where the last equality follows from (51) and the relation

Thus, we have proved (63). Property (64) was established by Corollary 3 . This completes the proof of Lemma 2.

We return to (62). Since that

F(T* < m ) = 1 and

G ( S r * )= G(rc*)= c * , it follows

-

EG(S,*) = C*

and by (62) we obtain -

V * ( l ) = EG(S,*) = EG(S,*)I(7* < - Ee-(%T)T*

(S,.

-

W)

K ) I ( r *< m) = c*.

This gives us the second proof of formula (12) (for z = 1) and shows that r* is 02

an optimal stopping time. If r 3 -, then P(T* 2 this case an optimal stopping time in the class

< m ) = 1, and therefore

T*

is in

"0".

$2b. Standard P u t Option 1. We can consider put options with pay-off functions ft = ePxtg(St), where g(x) = ( K - II:)+,II: E E = (0, oo), in the same way as we have considered call options. For this reason we restrict ourselves to the statements of results and the main points of their proofs. We shall consider a diffusion ( B ,S)-market described by the representations (1) and (2) in 52a. Let

Chapter VIII. Theory of Pricing. Continuous Time

764

THEOREM. Assume that X 3 0. Then

where 2(A

Y2=(;-;)-/(;-;)+-

+r)

02

'

(4)

--oo

There exists an optimal stopping time in the class M, , namely,

Moreover,

This result is even slightly more simple than the theorem in 5 2a: the function g(z) = ( K - x)+ is now bounded.

By analogy with the corresponding discrete-time problem (Chapter VI, 3 5c) it would be natural to assume that the domains of continued observations C, and the stopping domain D, have the following form:

C, = {x E E : z > z * > = { Z E E :V*(Z) > g(z)} and

D , = {x E E : z 6 x*}

= (z

E E : V*(Z) =g(x)},

-

where z* arid U*(Z) are equal to the solutions ( Z and V ( x ) )of the Stephan problem

2. American Options. Case of an Infinite Time Horizon

765

In our case the bounded solutions of (9) have the form C ( x )= Ex72 for II: > 6, where 7 2 is the negative root of the square equation (30) in § 2 a given by (4). Using (10) and ( l l ) ,we can find the values of C and 6,which are expressed by the right-hand sides of ( 5 ) and (6). We can prove the equality U,(IC)= U ( x )and the optimality of T, by testing the conditions (A) and (B), using arguments similar to the ones presented in 3 2a. The proof of (8) is based on Lemma 1 in 5 2a and the corollaries to it.

Remark. The 'martingale' (i.e., the second) proof of the above theorem is based on the observation that in our case the process 2 = (Zt)t>owith

zt = e-PtSY2t ,

P=X+r,

is a martingale and

Hence

e-Ot(K

-

St)+ = SF"(K

-

St)+&

and we can complete as in the case of call options

3 2c.

< c,Zt,

(5 2a).

Combinations of P u t and Call Options

1. In practice, as mentioned in Chapter VI, 3 4e, alongside various kinds of options, one often encounters their combinations. One example here can be a strangle option, a combination of call and put options with different strike prices. In this section we present a calculation for an American strangle option, making again the assumption that it can be exercised at arbitrary time on [0,cm)and the structure of the ( B ,S)-market is as described by relations (l)-(2) in 5 2a. In other words, we assume that

and

where W = (Wt)t>ois a standard Wiener process and ,LL = T . The original measure P is martingale in this case. For a discounted strangle option the pay-off function is as follows (cf. (3) in 5 2a):

Chapter VIII. Theory of Pricing. Continuous Time

766

In accordance with the general theory (Chapter VII, $ 4 and Chapter VI, § 2 ) , the price is

fr

V * ( z )= sup &E,T€ZJZ?

9JTr

Br

= sup

(4)

r€mF

<

where = { T = ~ ( w: ) 0 T ( W ) < 00, w E a} is the class of finite stopping times and E, is averaging under the assumption that So = 3: E E = (0, m). 2. To determine the price V*(rc)and the corresponding optinial stopping time we use the ’martingale’ trick from [ 3 2 ] , which we have already used in the preceding sections (e.g., in the ‘second proof’ in 52a.6). We shall assume here that SO = 1 and the strike prices K1 and K2 satisfy the inequality K1 < 1 < K2. Let

(f-3)+ JW, + (; 3) / + 2(A

71 =

-

72 =

2(X

-

be the roots of the square equation (30) in $ 2a.

$ 5 2a, b,

As shown in

t 3 0, with /3

the processes Mjl) = e-DtSz1 and M i 2 ) = e-PtSp, are P-martingales. Hence the nonnegative process (1 - p ) M j 2 ) is a P-martingale for each 0 p 1, and

= X

Mt(p)= pMjl) +

+r

< <

V * ( l ) = sup Ele-(X+‘)r~(S,) ?-Em?

Proceeding as in

5 2a.6 we consider measures

P ( p ) such that

Then we conclude from (7) that

where E-

P(P)

is averaging with respect to

is(,).

The next step is to choose a suitable value of p in [0,1] (which we denote by p* in what follows) such that we can solve the corresponding optimal stopping problem (9).

2. American Options. Case of an Infinite Time Horizon

767

As shown in [32], the following system of equations for ( p , sl,5-2): s2--2

-

+ (1 - p ) s T

ps;'

--32

-

s2 - K2 --31

~1

where p E [O. 11, s2 Let c* =

K1- s1 ps?' + (1 - p ) s Y

mls;' + (1 - P)Y2S?

+ (1 p ) s T PYls:' + (1 P)Y2S:2 p ~ ? ' + (I -p)s? ps;1

-

- K1

> K2, and

-

s1

-

'

-

I

< K1, has a unique solution ( p * , s;, s;).

si; - K2 p*(s;)W (1 -p*)(s;)rZ

(=

+

p*(S;)%

K2 - SE + (1 - P*)($)YZ

A simple analysis shows that = sup "s? p*sY' +g(s) (1 p*)sY2

s < l p*sy1

-

In addition. the function G ( s ) = points S; < K1 and s; > K2. Hence, by (9) we obtain

+g(1( s )

- p*)sY2

ds)

p*sY'

+ (1 - p*)s72

(= c*).

takes its maximum at some

V * ( l )3 c*. We set

r* = inf{t: St = E-

(13)

ST or St = si;}.

By the properties of a Brownian motion with drift, P(T* G(S,*) = c*, and therefore

<

00)

= 1. Hence

YP')

V * ( l ) = C* and r* is the optimal stopping time.

Remark. If K1 = K2, then the strangle option becomes a straddle option (see Chapter VI, 4e.2).

5 2d.

Russian Option

1. We consider a diffusion ( B ,S)-market with

dBt = r B t d t , and dSt = S t ( r d t

Bo > 0,

+ gdWt),

So > 0 ,

Chapter VIII. Theory of Pricing. Continuous Time

768

or, equivalently,

Since

the process

st --S o Bt Bo

S -

em+

,

(3)

is a martingale with respect to the original measure P.

=

Let

t 3 0,

f t = e-%(s>,

where gt(S) =

(Inn;su

-

USt

>+.

u

(4)

3 0.

(5)

American options with pay-off functions (4) (which are called R u s s i a n options [434], [435]) belong to the class of put options with aftereflect and discounting. (Cf. Chapter VI, 5 5d.) Using the same notation (Ez, m r , . . . ) as in § 2a, b we set

a:,

and

By contrast to 'one-dimensional' optimal stopping problems for a Markov process S = (St)t>o considered in § § 2 a , b , the problems (6) and (7) are 'two-dimensional' in the following sense: the functionals f t ( S ) depend on a two-dimensional Markov process ( S t ,max S u ) . U
It is remarkable, however, that using the methods of 'change of measure' one can reduce these 'two-dimensional' problems to 'one-dimensional' ones, which makes and optimal stopping it possible to find explicit expressions for U*(z)(= U*(z)) times. 2. We have already explained rather explicitly the idea of this reduction of 'twodimensional' problenis to 'one-dimensional' in Chapter VI, $ 5 d ,for the discrete-time case. Here we proceed as in 5 2a.5 and assume that (R,$, ( $ t ) , P) is the coordinate Wiener space.

2. American Options. Case of an Infinite Time Horizon

Let p be a measure in each t and

(n,9)such that

its restriction

pt

769

is equivalent to Pt for

where

@ = (@t)t>O

with

is Wiener with respect to

^P by

The process

Girsanov's theorem, and for

7

E% :

we have

We consider now the process ($t)t>o of the variables

where $0 3 I . Clearly, if $0 = I , then $t =

max,Gt S,

st

'

and therefore h

[

E C X T maxu't ST

su

+ -

u]

I(' < m) = kxT[$T - a]+ I ( r < m).

(14)

Let p, be the probability distribution of the process ($Jt)t>ounder the assumption $0 = $ 3 I; we consider the following optimal stopping problems:

Chapter VIII. Theory of Pricing. Continuous Time

770

and

which can be regarded as pricing problems for discounted American call option with process of stock prices ($t)t2o and Bt = 1. (We point out that our initial problem relates to p u t options!) By (6), ( 7 ) ,and (15), (16) we obtain (in view of (11) and (14)) the equality

3. Before we state the main results on the solution of the optimal stopping problems (15) and (16) we discuss some properties of the process ($t)tbo.

LEMMA.1) The process ($t)t20 is a diffusion Markov process on the phase space E = [ l , m ) with respect to the measure ^P, with instantaneous reflection a t the point (1). 2) The process ($t)tbo has the stochastic differential

where ( p t ) t 2 0 is a nondecreasing process increasing on the set { ( w , t ) : &(W) = l}, and W = (Wt)t20 is a Wiener process (with respect to P). 3) If q = q ( $ ) is a function on E = [l,m) such that q E C 2 on (1, m) and there exists q’(l+) = limq’($), then A

-

1111

and q’(l+) = 0

Proof. By ( l a ) ,

771

2. American Options. Case of an Infinite Time Horizon

Note that for t

< u < t + A we have

.-

Hence, taking into account that W is a Wiener process with respect to the following ‘Markovian’ property:

p, we obtain

Law($t+A 1 9 t j p ) =Law($t+A I$t,^p).

To deduce (18) we set

Clearly, N = ( N t ) t 2 0 is a nondecreasing process of bounded variation. BY (2) and ( l o ) , dSt = St [ ( T n 2 )d t ~7d@t]

+

+

(23)

and

Hence, by It6‘s formula

or, in the integral form,

We now set

: and note that dN,(w) = 0 on the set { ( w , ~ )$,(w)

>

l} (in the sense of the

which shows even more clearly that the process (vt)t>ochanges the value only when the process ($t)t>,o arrives at the boundary point (1).

Chapter VIII. Theory of Pricing. Continuous Time

772

.I”

We claim that

I(&

= 1)du = 0

(k3.S.)

for each t > 0. By Fubini’s theorem

A

because P(& = 1) = 0, which is a consequence of the fact that the distribution of the pair (ma,@,, s
FU)has a density.

Thus, the process ($t)t>o stays zero time (^P-a.s.) a t {l},so that this point is an instaneously reflectzng boundary ([239; Chapter IV, 5 71). 4. THEOREM.Assume that A

> 0, a 3 0, and $ 3 1. Then

where

are the roots of the square equation

y2 - A y - B = O with

the ‘threshold’

6is the solution of the transcendental equation

in the domain $

> a. If a = 0, then

The stopping time A

?=inf{t>O:$t>$}

2. American Options. Case of a n Infinite Time Horizon

has the property P+(? and in %??.

773

< m) = 1, 11, 3 1, and i t is optimal both in the class 9 Xr

As in 3 3 2a, b we present two proofs, one (‘Markovian’) based on the solution of a Stephan problem and another based on ‘martingale’ considerations. T h e first proof. The same arguments as in 3 5 2a, b, based on the representation (22) in §2a, show that the domain of continued observations ??i and the stopping domain fi in (15) must have the following form:

e =(11, 3 1: 11, < $} and

=

( $ 3 1: 6($) >g(11,)}

6 = ( $ 3 1: $ 3 4}= {$ 3 1: G(11,) =g($)},

where g ( $ ) = (11, - a)+. As in § § 2 a , b , 6($) and the unknown threshold Stephan problem

$ make

up a solution of the

where the operator L is defined by (19). We seek a solution to (34) in the form 6($) = $7. Then we obtain square equation (32) for y,which has the roots y1 < 0 and 7 2 > 1 described by (31). has the following general Thus, equation (34) in the domain {$: 1 < $ < solution: = Cl11,Yl c2$’y2, (38)

G}

G(11,)

+

where c1 and c2 are some constants. To find and the constants c1 and c2 we can use three additional conditions, (35), (36), and ( 37) ,which, in view of (38), take the following form:

6

774

Chapter VIII. Theory of Pricing. Continuous Time

By (36') and (37'),

BY ( 3 5 7 , 72 c1 = --c2, 71

which yields equation ( 3 3 ) for

4.If a = 0, then it follows from this equation that

Finally, from (38) and (39), in view of ( 3 3 ) , we see that

in the domain ?i = { 1c, : $J < &}. We now claim that p ~ ( ? < m ) = 1 for 11, 3 1. To this end it suffices to show that

6

for each > 1. (Property (42) is obvious for We have

+ = 1.)

where yt=sup ugt

rf-

We consider the sequence of stopping times 00

3 I

[.(wu-wt)+ - - (

(q.)k20

(U-t)

such that

= 0,

~1 = inf{t

> 1: yt = 0},

.. ~ + l = i n f ( t > a k + l :y t = o } , . . . . Then for y^ = In

& 3 0 we have

.

2. American Options. Case of an Infinite Time Horizon

775

e}

yt(w) 3 are independent for distinct k and their sup gk
The events { w :

-{

c}

t<m

Borel-Cantelli lemma, so that p ~ ( < ? cm)= 1. It remains to show that ? = inf{t 3 0 : $t 3 $} is the optimal time in the problems (15) and (16). One way to prove this is to test properties (A) and (B) in §2a, which can be done in the same way as for call and put options. (See [444] for the detail.) We now present another proof, based on ‘martingale’ considerations (cf. 5 2a.5 and [32]). h

The seco,nd proof. We assume for simplicity that a = 0. (As regards the general case of a 3 0, see [32].) We set Mt = ePxt$th($t) (43)

and find a function h = h ( $ ) , $ 3 1 , such that M = (Mt)t>o is a local martingale with respect to the measure p. Using It6’s formula for e-xt$th($t) we see that

where

By (44), to make M = (Mt)t>o a local martingale we can take h = h ( $ ) , $ 3 1, equal to the solution of the problem

*

1 2 2 1 1 -0

2

h (+)

+ (2

-

T)$h’(*)

with boundary condition

h’(l+) We rewrite (47) as

+

?+h2h”(*) 2 (1

-

-

(A

*>

+ r ) h ( $ ) = 0,

1,

(47)

+ h ( l + ) = 0.

.>

$/I1($)

-

2

($)

=0

(49)

and seek its solution in the form h ( $ ) = $”. Then we obtain the following square equation for x: x 2 + 2 ( 1 - 2r) - 2(A r ) = 0. (50)

+

Chapter VIII. Theory of Pricing. Continuous Time

776

Comparing it with (32):

y2 - y ( 1 +

- 2x = 0,

2T)

+

we see that (51) turns into (50) after the change y = z 1, so that the roots zi and yi of these equations are related by the formulas yi = zi 1, i = 1 , 2 . The general solution of (49) (for cr2 = 1) is as follows:

+ d2$'12,

h($) = dl$'ll

$

+

> 1.

We consider h = h($) such that (48) holds. Then

dl =

1+22 ~

z2

-

-

21

&---.1 + z 1

-

72

,

7 2 -71 71

71 - 7 2

21 - 2 2

Consequently,

-

I

We have y1 equation

< 0, y2 > 1, and h'($)

= 0 for $ = $, where $ can be found from the

&72-71y1(y2 - 1) = 1, ^12(Y1

-

(53)

1)

4

&

Comparing (53) and (41) we observe that the quantity is the same as in (41). Moreover, the function h ( $ ) takes its minimum at the point $. We see from (43)-(48) that for the function h = h($) so chosen the process

is a nonnegative local martingale, and therefore a supermartingale. Hence for each 7 E iJXTn,- and $0 = 1 we have A

Ele-X7$/r = E1h-'(&)M7 = -

h-1($)E1M7 72

-

< Elh-'($)MT < h-l(&)EiMo

71

h

y2*71-1

If $0 = 1, then ? = inf{t 3 0: above (PI(? < ca) = l ) ,and

-

$.

- yl$72-1

$t

= h-'($) 72

-

71

-

71$ 7 2

A

7 2 $71

3 $} is finite with probability one as shown

A

Ele-"$?

.

= Elh-'($?)MF

= h-'($)

(=

2. American Options. Case of an Infinite Time Horizon

777

5JZr

for this stopping time, which means precisely the optimality o f ? in the class for $0 = 1. (Similar arguments hold for each $0 6 We proceed now to our original problems (6) and (7). By (ll),setting a = 0 we obtain E,e-(Xfr)TgT(S)I(i-< m) = zEe-XT$T1(i-< m). (54)

4.)

h

Here $0 = 1 and, as shown above, ? = inf{t: $t 3 $} is the optimal stopping time in the following sense:

and 7

Hence 3 is the optimal stopping time in the problem (7). Arguments used above, in the proof of the property F$(? < 00)= 1, are suitable also for the analysis of the process (&)t>o, where they show that ? is finite almost everywhere also with respect to P. Hence ? is an optimal stopping time in our original problems (6) and (7).

3. American Options in Diffusion ( B ,S)-Stockmarkets. Finite Time Horizons

5 3a.

Special F e a t u r e s of Calculations on F i n i t e T i m e Intervals

1. If the time horizon is infinite, i.e., the exercise times take values in the set [0, m), then one often manages to describe completely the price structure of American options and the corresponding domains of continued and stopped observations. For instance, in all the cases considered in $ 2 we found both the price V*(x)and the boundary point x* in the phase space E = {x:x > 0} between the domain of continued observations and the stopping domain. We note that this is feasible because a geometric Brownian motion S = ( S ~ ) Q O is a homogeneous Markov process arid there are no constraints on exercise times, so that the resulting problem has elliptic type. The situation becomes much more complicated if the time parameter t ranges over a bounded interval [0,TI. The corresponding optimal stopping problem is inhomogeneous in that case and we must consider problems that have parabolic type from the analytic standpoint. As a result, in place of a boundary point x* we encounter in the corresponding problems an znterface function z* = x * ( t ) ,0 t T , separating the domain of continued observations and the stopping domain in the phase space [O,T)x E = { ( t , ~0) :6 t < T , x > O}. (Cf. Figs. 57 and 59 in Chapter VI, 5 5 5b, c.) We also point out that, although the theory of optimal stopping rules in the continuous-time case (see, for instance, the monograph [441]) proposes general methods of the search of optimal stopping times, we do not know of many concrete problems (e.g., concerning options) where one can find precise analytic formulas describing the boundary functions x* = x * ( t ) ,0 t < T , the prices, and so on.

< <

<

3. American Options. Finite Time Horizons

779

In practice-e.g., in calculations for American options in some market-one usually resorts to quantization (with respect to the time and/or the phase variables) and finds approximate values of, say, the boundary function and prices by backward induction (see Chapter VI, 5 2a). Of course, this does not eliminate interest to exact (or 'almost' exact) solutions; in this direction we must first of all discuss several relevant questions of the theory of optimal stopping problems on finite time intervals and, in particular, one very common method based on the reduction of such problems to S t e p h a n problems (often called also fme-boundary problems) for partial differential equations. 2. For definiteness, we consider a ( B ,S)-market described by relations (1) and (2) in 5 2a, where we assume that the time parameter t belongs to [0,TI, p = T , and the pay-off functions have the form f t = e C x t g ( S t ) , where X 3 0, the Bore1 function g(z) is nonnegative, and z E E = (0,m). Let V ( T , z )= B o E Z - f T (1) BT and

be the rational prices of European and American options, respectively. In relations (1) and (2) the symbol E , means averaging with respect to the original measure (which is martingale since p = T ) under the assumption So = z.

R e m a r k . We proved formula (1) for V ( T ,z) in Chapter VII, 5 4b. The proof of (2) is based on the optional decomposition and its idea is the same as for discrete time (see Chapter VI, 5 5 2c, 5a). See, for instance, [281] for the detail specific to continuous time.

3. For t 3 0 and z E E = (0,m) we set

V ( t , Z )= E,e-Otg(St)

(3)

and

+

where ,B = X T and x = SO. In the discussion of the case where t E [O,T]it is also useful to introduce the functions Y ( t ,z) = V ( T - t , z) (5)

and Y * ( t , X ) = V*(T-t,2), where T - t is the remaining time.

Chapter VIII. Theory of Pricing. Continuous Time

780

Clear 1y,

Y ( t , x )= Et,ze-P(T-t)g(S~)

(7)

and

Y * ( t z) ,

Et,ze-P(T-t’g(ST),

= sup

(8)

T€DtF

where Et.z is averaging with respect to the original (martingale) measure under the assumption St = x , and 2Jl; is the class of optimal stopping times T = T ( W ) such that t 6 r 6 T . We considered the functions V = V ( t ,Z) (in the case of a Brownian motion and for p = 0) in Chapter 111, 3f, in connection with the probabilistic representation of the solution of a Cauchy problem. The same arguments (see Chapter 111, 3f.5 for greater detail) show that the function V = V ( t ,x ) (provided that it belongs to the class C1i2)satisfies for t > 0 and x E E the equation

dV -

at

+ pv = LV,

where

(9)

av + -1r 2 x 2 -a2v

L V ( t ,x) = T Z -

ax

2

3x2’

with initial condition V(0,

=dx).

By ( 5 ) , (9), and (11) we obtain that the function Y = Y ( t , z )satisfies for t the equation -d Y

-+,BY at


= LY

and the boundary condition

Y ( T ,). = g(z). Recall that we met already fundamental equatzon (12) (which is in fact a Feynman-Kac equation-see Chapter 111, 5 3f) in 5 l c , in our discussion of the methods used by F. Black, M. Scholes [44], and R. Merton [346] to calculate the rational price ( V ( T ,z) = Y(O,z)) of a standard European call option with g(z) = (Z - K ) + and X = 0. 4. We proceed now to the question of the rational price V * ( Tx) , = Y*(0.z).

We set T:

= inf(0

6 t 6 T : ~ * ( St) t , =g(St)}

(14)

E : Y * ( t , Z )= g ( x ) } ,

(15)

E : Y * ( t , x )> g(z)}.

(16)

and

0 :

= {Z E

CT =

{Z

E

3. American Options. Finite Time Horizons

781

F o r s G t w e h a v e Y * ( s , z ) = V * ( T - s , z )> V * ( T - t , x ) = Y * ( t , z ) .Hencefor

0

< s 6 t < T we obtain

DOT

C DT C DF

and

C,T 2

c:

2 CF.

For t = T we clearly have DF = E arid CF = 0. The domains DT = { ( t ,Z) : t E [O,T ) , x E D T } and

CT = {(t,rc):t

€

[O,T),5 E CF}

in the phase space [O,T)x E are called the stopping domain and the domains of continued observations, respectively. This is related to the fact that in 'typical' optimal stopping problems for Markov processes the stopping time is optimal (see, e.g., Theorem 6(3) in [441], Chapter 111, 3 4):

TOT

E,epo'"Tg(ST~) Since T:

= inf{O

it is clear why we call DT =

6t

= V * ( T5 , ).

< T : st E OF},

(17)

(18)

u ( { t } x 0;) the stopping d o m a i n : if ( t , S t )E D T , t
then the observations terminate. (It seems reasonable to include in this domain also the 'terminal' set T x D? = T x E.) The domains DT arid CT can have very complicated structure depending on the properties of the functions g = g(z) and, of course, of the process S = ( S t ) t G ~ . For instance, as subsets of [O,T)x E these domains can be multiply connected or consist of several stopping 'islands', and so on. On the other hand, for standard call and put options with g(z) = (x - K ) + and g ( x ) = ( K - z)+, respectively, the domains DT and CT are simply connected (see 3 3c below). For these options the boundary aDT of the stopping d o m a i n can be represented as follows: dDT = { ( t , Z ) : t [O,T),z = z * ( t ) } , where we have

x * ( t ) = inf{z E E : Y*(t,x) = (x -IT)+}, for a call option and

x * ( t )= sup{z E E : Y * ( ~ , = z )( K - z)'} for a put option.

782

Chapter VIII. Theory of Pricing. Continuous Time

3b. Optimal Stopping Problems and Stephan Problems 1. It follows from our discussion that for a description of the structure of the optimal stopping time and the doniains of continued and stopped observations one must find the function V* = V * ( t , x )or. equivalently, the function Y * ( t , x )= V * ( T- t , x ) . There exist various characterizations of these function in the general theory of optimal stopping rules. For instance, it is known (see [441])that the function Y * = Y * ( t ,x) is the smnllest p-excessive majorant of the (nonnegative Borel) function 9 = g ( x ) . In other words, the function Y * = Y * ( t , x )is the smallest among all functions F = F ( t , x ) such that e - p A T A F ( t , x ) 6 F ( t ,z), x E E, (1)

T?

for 0 6 t 6 t

+ a 6 T , where T A F ( t ,x) = Et,,F(t + A, st+A),x = St, and g ( x ) 6 F(t,cc),

x E E,

0 6 t 6 T.

(2)

It follows, in particular, that max{g(z), e - p A ~ A y * ( t , x )6) Y * ( t , z ) . For small A > 0 and t Y * ( t , x )to be 'close' to

=

(3)

0 . A . . . . , [T/A]A one would expect the function

where m r ( A ) are stopping times 7 such that 7 = kA,k = 0 , 1 , . . . , [ T / A ]t, 6 7 6 T , and {~:76Ica}~~:lc~(A),~:,,(a)=a{w: S A , S ~.A . . , S:lc~}. As shows the theory of optimal stopping rules, this conjecture on the 'closeness' of these functions for small A > 0 has a rigorous formulation (see [441; Chapter 111, § 21). Further, for YA(t,x) with t = 0, A , . . . , [ T / A ] Awe have the recursive relations Y A ( t ,i.) = maX{g(i.), C p A E t , Z Y A ( t st+A)} (5)

+ a,

(see the discrete-time case in Chapter VI, 5 2a and [441],Chapter 11, 3 4), therefore, assuming that Y * ( t , x )is sufficiently smooth and using Taylor's formula we see from (5) that

where

L Y * ( t ,i.) = TIC

aY*(t,x) 1 +-0 ax 2

2 ,a2Y*(t:Z) ic

3x2

'

3. American Options. Finite Time Horizons

783

By (6) we obtain that in the domain where Y * ( t z) , > g(z) (i.e., in the domain of continued observations) the function Y* = Y * ( tx, ) satisfies the equation

dY* -__ at + pY* = LY*.

(8)

Remark 1. Equation ( 8 ) is similar in form to (12) in 3a; this is not that surprising for the following reasons. Assume that there exists optimal stopping time T? in the class 9XT. Then

Since

Y ( t ,z)

= Et,re-P(T-t)g(S~),

Cr,

it is intuitively clear that if ( t ,x ) E then the (backward) equations with respect to t and 2 for the functions Y * ( t , z )and Y ( t , z )must be the same since their coefficients are determined by the local characteristics of the s a m e two-dimensional process (u, Su)tGuGTin a neighborhood of the inztzal point ( t ,z = S t ) .

Remark 2. There exists an extensive literature devoted to the derivation of equation (8) for Y * ( t , z )in the domain of continued observations; e.g., the monographs [266], [287], [441], [478] and the papers [33], [66], [134], [135], [179], [247], [265], [272], [340], [363], [467]. 2. We considered already the connection between optimal stopping problems and Stephan problems in Chapter IV, $ 5 , in our discussion of American options in a binomial ( B ,5’)-market. In the continuous-time case this connection has apparently been revealed for the first time in statistical seguentzal analysis, in testing problems for statistical hypotheses about the drift of a Wiener process ([67], [300], [349], [440]; see also the historico-bibliographical passages in [116] and [441]). One of the first papers in the finances literature considering Stephan (freeboundary) problems was that of H. McKean [340], devoted to the rational price of American warrants.

3. In mathematical physics, the S t e p h a n problems arise in the study of phase transitions ([413], [463]). One simple example of such a two-phase S t e p h a n problem is, for instance, as follows. Assume that our ‘time-state’ space R+ X E = { ( t ,z) : t 3 0, z > 0) is partitioned between two phases, = { ( t ,z): t 3 0, 0

and

d 2=) { ( t ,z) : t 3 0,

< z < x(t)}

z ( t ) < z < oo},

784

Chapter VIII. Theory of Pricing. Continuous Time

where z = z ( t ) , t 3 0, is the interface (e.g., between ice and liquid in freezing water). For each phase di)the temperature u ( t , z ) a t z at time t , i = 1 , 2 , is assumed to satisfy 'its own' heat equation

where (using physical terminology) the ci are the specific heats, the pi are the densities of the phases, and the ki are the thermal conductivities (see, e.g., [335; vol. 5 , p. 3241). Equations (9) are considered for the boundary condition ~ ( 0t ), = Const, the initial condition u(z,0 ) = Const and, e.g., for the following condition at the interface surface:

for t > 0, with additional assumption z(0) = 0. Under these assumptions the Stephan problem consists in finding the interface z = z ( t ) ,t 3 0, and the function u = u.(t,z) describing the temperature schedule in the phases. 4. We have presented an example of a Stephan problem borrowed from m a t h e m a t -

ical physics to emphasize its common and distinct traits in comparison with the Stephan problems arising in the search of optimal stopping rules and, in particular, in connection with American options. As already mentioned, for standard put and call options we also encounter a two-phase situation: looking for optimal stopping rules we can restrict ourselves to the consideration of only two simply connected phases, the donlain CT of continued observations, where Y * ( t z) , satisfies equation ( 8 ) , and the domain D T , where Y * = Y * ( t z) , coincides with the function g = g ( z ) . In the next section we formulate precisely the corresponding Stephan problems for these two options and describe the qualitative features of the corresponding solutions Y* = Y * ( t , z )and X * = z * ( t ) . fj3c. Stephan Problem for Standard Call and Put Options 1. A call option. We assume that a (B,S)-market can be described by relations (1) and (2) in § 2 a with p = T , 0 t T , and the pay-off function f t = e - x t g ( S t ) , where X 3 0 and g ( z ) = (z - IT)+,z E E = ( 0 , ~ ) The . main results about this option are as follows.

< <

3. American Options. Finite Time Horizons

785

1) The rational price V * ( Tz), , z = SO,of such a (discounted) option is defined, as mentioned in 3 3a, by the formula

+

where ,B = X r and E, is averaging with respect to the original (martingale) measure under the assumption SO= z. 2) For t E [O,T]and z E E let

where Et,z is averaging with respect to the (martingale) measure under the assumption 3: = St. The function Y * = Y * ( t x) , is the smallest ,B-excessive majorant of the f u n c t i o n g(z) (see 53b.l). 3) The rational price is

V*(Tz , )= Y*(O,z),

(3)

and the rational t i m e for stopping the buyer's observations and exercising the option is = inf{O 6 t T : Y * ( t St) , = g(St)} (4)

<

(we denoted this time by 7;. =

7-F in

3 3a) or, equivalently,

inf(0 6 t 6 T : ( t , S t )E D~

u { ( ~ , x )z:E E ) } .

(5)

4) The stopping domain DT and the d o m a i n of continued observations CT are simply connected and have the following structure: DT =

u { ( t , ~ Y) :* ( t , z ) u >

= g(z)},

(6)

{ ( t , x ) :Y * ( t , z ) g(z)}.

(7)

O
CT =

O
5 ) The function Y * = Y * ( t , z )on [0, T ) x E belongs to the class C1>2. For each fixed x E E the function Y * ( . , x )is nonincreasing in t ; for each fixed t E [O,T)the function Y * ( t ,. ) is nondecreasing and convex (downwards) in x. 6) The interface function x* = x * ( t ) is nonincreasing on [0,T ) ,and the sets CF and DT have the following form for t < T :

CT = {X E E : St < x * ( t ) } , DT =

{Z E

E : St 3 x * ( t ) } .

Chapter VIII. Theory of Pricing. Continuous Time

786

For t = T we have CF = 0 and DF = E . If X = 0, then z * ( t )= 30 for t < T , which corresponds to the equalities

for each t < T. In other words, for t < T observations should go on whatever the prices St can be, which is a consequence of the fact that the process (ePTt(St- K)+)t>ois a submartingale, so that by Doob’s optimal stopping theorem, E z e - T T ( S T - K)’ EZeCTT(ST- K ) +

<

for each 7 E LJI.;t A similar result is known in the discrete-time case (R. Merton, [346]), and we have interpreted it as follows (see Chapter VI, 55b): the standard A m e r i c a n and European call options ‘coincide 7) The function Y * = Y * ( t , z ) t, E [O,T],z E E , and the interface function z* = z * ( t ) ,0 t < T , are the solutions of the following ‘two-phase’ Stephan (or free- boundary) problem:

<

-

aY*(t,x) at

in the domain CT = { ( t , z ) z :

+ PY*(t,z) = L Y * ( t ,z)

< z * ( t ) ,t

E

Y*(t,

dx)

=

[O,T)}; (9)

in the domain DT u { ( T , z ) :z E E } ; while at the interface surface z* = z * ( t ) , 0 t < T , we have the Dirichlet condition (P. G. L. Dirichlet)

<

and the Neurnann-type condition (K. Neumann)

(called above the condition of smooth pasting) We shall now make several observations concern ng- the above results; for detailed proofs of these results see the papers mentioned at the end of 53b.l. We discussed already the validity of (1) at the end of 53a. The fact that T+ is optimal follows from the general theory of optimal stopping rules (see, e.g., [441; Chapter 111, 5 31). As regards the smoothness of Y * ( t z) , and the derivation of (8), see, e.g., [247], [363], and [467].

3. American Options. Finite Time Horizons

787

While condition (10) is fairly natural. condition of smooth pasting (11) is not that obvious. In [200] and [441; Chapter 111, $81 we have shown that (11) must hold at the boundary of the stopping domain under rather general assumptions. We also recall that we have already encountered conditions of smooth pasting in our considerations of approximations in discrete-time problems (5 5 in Chapter VI) and in our discussions of American options in the case of a n infinite time horizon (5 2 in this chapter). It is worth noting that while in the standard Stephan problem of mathematical physics that we considered in the previous section each phase satisfies an equation 'of its own', in the optimal stopping problems we have a differential equation for Y * ( t , x )only in one phase (in the domain of continued observations), while in the other (the stopping domain) Y * ( t , x )must coincide with the fixed function g ( X ) . We note also that

dPI

~

xJx* (t )

= 1 in our case of g ( x ) =

(X -

K ) + because

x * ( t ) > K , 0 < t < T . (It is easy to deduce the last inequality from the fact that Y * = Y*(t,rc)is a P-excessive niajorant of the function g = g(z).) As concerns the solubility of the Stephan problem (8)-(11) and the properties of the interface function x* = x * ( t ) see [467] anti [363] (see also the comments to the second paper). 2. P u t options. In that case g ( x ) = ( K - x)+. Properties 1)-4) remain valid and the function Y * = Y * ( t , s )is in the class C1i2 again. For each fixed x E E the function Y * (. , z) is nonincreasing in t ; for each fixed t E [0, T ) the function Y * ( t ,. ) is nonincreasing and (downwards) convex in x. For each X 3 0 the sets CF and DF have the following form for t < T :

c? = {x: s, > x * ( t ) } , DT = { x : St < x * ( t ) ) . For t = T we have C$ = 0 and DF = E . The interface function x* = X * ( t ) is nondecreasing in t ; if X = 0: then limx*(t) = K ttT

The Stephan problem for Y * ( t , x ) and z * ( t ) can be formulated in a similar manner. Here conditions ( 8 ) , (9), and (10) are the same, while (11) takes the following form for 0 t < T :

<

where

because g ( x ) = ( K - x)+ and z * ( t ) < K .

Chapter VIII. Theory of Pricing. Continuous Time

788

One can find additional information on the properties of the functions Y * ( t x) , and x * ( t ) in the paper [363] devoted to standard American put options and containing an extensive bibliography on other types of options.

3 3d.

Relations between the Prices of European and American Options

1. We have previously mentioned that American options are in fact more commonly traded than European ones. However, while for the latter we have such remarkable results as, e.g., the Black-Scholes formulas, calculations for American options in problenis with finite time horizon encounter great analytic difficulties. On the final count, this is due to difficulties with the solution of the corresponding Stephan problems. It is clear from (1) and (2) in 3 3a that V *( T ,z) 3 V ( T ,z); this is, of course, fairly natural since, by condition, American options encourage the buyer not merely to wait for the execution but to choose its timing. In this section we present several results on the relations between the prices of standard call arid put options with pay-off functions g(z) = (z - K ) + and g(z) = ( K - z)+, respectively. We shall assume that X = 0. Then formulae (1) and ( 2 ) in §3a can be written as follows: V ( T ,z)= E z e - ' T g ( S ~ ) (1)

and

V * ( T x, )

= sup

E,e-TT.9(S,),

TE!TRT

where z = So. 2. The question of the relation between the prices V ( T , z ) and V * ( T , z )for (z- K ) + is very easy to answer. In that case

g(z) =

V ( T ,x ) = V * ( Tx), , and

7;

= T is an optinial stopping time in the class

(3)

9JtT (see 3 3c).

Remark. We emphasize that if X > 0, t,hen (3) does not hold any longer because the process (e-(X+T)t(St- K)+)t>ois no longer a submartingale (cf. 5 3c). We proceed now to the question of the size of the 'deficiency'

a;(z) = V * ( T z) , - V ( T ,x )

(4)

for a standard call option (g(z) = ( K - z)'). We set X = 0 and denote by Z* = z * ( t ) ,0 6 t < T , the function defining the interface between the stopping domain and the domain of continued observations for optimal stopping time 7;.

789

3. American Options. Finite Time Horizons

THEOREM. For a standard call option,

COROLLARY 1. Let PT and P$ be the rational prices oE European and American call options (PT = V ( T ,S o ) , P$ = V * ( TS, o ) ) . Then

5 Ib)

where (cf. the notation in

and

Proof. Let

Y ( t ,X) = E t . z e - " T - t ' g ( S ~ )

(8)

and let

Y * ( t Z) , =

SUP Et,ze-T(T-t)g(ST), TEm?

(9)

where g ( x ) = ( K - z)+.Then for

we have

e-TtaTt

(

,) = E t , z { e - ' T T g ( s T ~ )- e P T T g ( S T ) } ,

r

%

(11)

where T? is the optirrial stopping time in the problem (9). By ItA's formula (Chaptcr 111, 3 5c)

d(e-',(K

-

S,)')

= e-',

d(K - S ,)'

- T ~ - ' ~ (-K S,)'

du,

(12)

Chapter VIII. Theory of Pricing. Continuous Time

790

and by the Itb--Meyer formula for convex functions (see Chapter VII, Chapter IV]; cf. also Tanaka’s forrnula (17) in Chapter 111, 5 5c) d ( K - ST,)+ = - I ( & where

L,(K)

5 4a;

1

< K ) dS,, + Z L , ( K ) ,

S”

= Iim €LO 2 E 0

I(1St - KI

(13)

< E ) dt

(14)

is the local time of process S = (St)taO (on [O: u ] )a t the level K . By (11)-(13)we obtain

i,, e?”{-I(S, T

= -Et,x

< K ) dSu + i d L , , ( K ) ~

T

= EtJ

e P ’ U {

-dL,,(K) x [TS,

+I(&

d7L

T(K

T;

LT(5)= Et,z[AT - At].

We represent now At as follows:

At where

= A:

SU)I(SU< K ) d u , )

+ as,dW, + (TK

= T , it follows from (15) that e-Tt

~


For t 6 T we set

Then, since

[395;

+ A:,

-

TS?‘)du]}

3. American Options. Finite Time Horizons

Since x*(u)< K for all u

791

< T , it follows that

The process A' = ( A i ) t is ~ ~a predictable submartingale and, by Corollary 2 in Chapter 111, 5 5b, it is the compensator of itself. A slightly rnore refined analysis (see [134], [135],arid [363])shows that the compensator of the process A2 = vanishes. Hence

so that (5) holds for AT(x). Finally, formula (6) in Corollary 1 is a consequence of (5) and relation (18) in l b for PT. This completes the proof of the theorem and Corollary 1.

COROLLARY 2 . The functions Y * ( t , z )and z * ( t )are connected by the relation

Y*(t,z*(t= ) )K

~

z*(t),

t

< T,

(20)

which can be regarded as an integral relation for the interface function x* = x * ( t ) ,

t 6 T , with x*(T) E lim x*(t). t+T

It should be pointed out that the function Y * ( t , z )is in fact also unknown. In place of this function one considers in practice approximations YL(t,x) calculated , by Y;i(t,z) in (19) by nieans of backward induction (see 5 3b.l). Replacing Y * ( tz) we obtain a function = x ; ( t ) , t 6 T , which is taken for an approximation to Z* = x * ( t ) ,t

< T.

x6

4. European and American Options in a Diffusion (B, p)-Bondmarket

4a. Option Pricing in a Bondmarket 1. So far we have considered only options in (B,S)-stockmarkets, whereas one encounters most various kinds of them in practice: options on eurodollars, futures, currency, etc., even options on options. Besides standard put and call options their various combinations are also traded. Many financial instruments in the option family have an elaborate structure as regards the corresponding pay-off functions and the securities underlying option contracts. The diversity of options, some of which are said t o be ‘exotic’, can be judged by their names: up-and-out put, up-and-in put, down-and-out call, down-and-in call, barrier option, Bermuda option, Rainbow option, Russian option; knock-out option, digital option, all-or-nothing, one-touch all-or-nothing, supershares, . . . (see [232], [414], or [415]). In our discussions of pricing for these options and other derivative instruments we should point out that the general methods remain the same as in the models considered by F. Black, M. Scholes, and R. Merton ([44], [346]) in the case of a ( B ,S)-stockmarket. Again, two approaches are possible: the martingale approach and the approach based on direct considerations of the ‘fundame,ntal equation’ (cf. 3 3 l b , c). 2. The discussion that follows relates to calculations for standard European and American options in the case where in place of a (B,S)-market we consider a (B,P)-market consisting of a bank account B = ( B t ) t
4. European and American Options in a Diffusion ( B ,?)-Bondmarket

793

the state of the bank accoiiiit (Bt)t
Bt = Bo exp

(l

r ( s )d s )

(1)

with some stochastic process of interest rate r = ( r ( t ) ) t < T . As regards the dynainics of the bond price process P = (P(t, T ) ) t G Twe assume that the discounted prices (2)

make up a martmyole with respect to the znrtral measure on ( 0 , 9 ( 9~ t), tGT). By Theorem 1 iii Chapter VII, 3 5a we obtain P ( t , T ) = E (exp

(- i i ’ r ( s )

ds)

~

T t )>

(3)

and by Theorein 2 in the same section the (B, P)-market in question is arbztrage-free (e.g., in the NA+-version of this concept). 3. From (1) and (3) we see that the dynamics of the processes ( B t ) t g and ( P ( t , T ) ) t in~ our ~ (B,P)-market depends considerably on the structure of the process T = ( r ( t ) ) t < ~ . Our main assumption about this process is that it is a drffusron Gauss-Mnrkov process described by the stochastic differential equation d r ( t ) = ( 4 t ) - a ( t ) r ( t ) dt )

+ y(t) dWt,

(4)

with Wiener procms (Wt)t
(5) Under these assumptions equation (4) has a unique (strong) solution

is the fundamental solrition of the equation

Chapter VIII. Theory of Pricing. Continuous Time

794

Remark 1. Recalling our discussion in Chapter 111, §4a we see that (4) is just the Hull--White model, of which the Merton, the VasiEek, and the Ho-Lee models are particular cases (see formulas (14), ( 7 ) ,(8), and (12) in Chapter 111, S4a). 4. Since

T

= ( r ( t ) ) t
We set I ( t , T ) =

T

r ( s )ds. Then we see easily from (6) that

Hence it is easy to obtain by ( 3 ) the following representation for

and

B ( t ,T ) =

1'

9( t )

du.

(14)

Remark 2. Using the terminology of Chapter 111, $ 4 we ~ call the models with prices P(t, T ) representable as in (12) single-factor a f i n e models. Our additional assumption that T = ( r ( t ) ) t < Tis a Gauss-Markow process enables one to carry out fairly complete calculations for European and American options in ( B ,P)-markets for such models (often called also single-factor Gaussian models). We devote 3 4b, c below to these problems.

4. European and American Options in a Diffusion ( B ,P)-Bondmarket

795

R e m a r k 3 . As regards the agreement between various models describing the dynamics of stock prices and empirical data, see, for instance, [257].

3 4b.

European Option Pricing in Single-Factor Gaussian Models

1. We shall consider a model of a ( B :?)-market consisting of a bank account and a bond arid completely determined by a single factor, the interest rate r = ( T ( ~ ) ) ~ Q that is a Gauss-Markov process satisfying stochastic differential equation (4) in 5 4a with (nonrandom) initial condition r ( 0 ) = ro. Let T o be some time (To < T ) treated as the t i m e of exercising a European ( P ( T o , T )- K ) + for a call option and f ~ =, option with pay-off function fTo ( K - P(T', T I ) +for a put option.

THEOREM. The rational price C o ( T o T) . of a standard call option in the singlefactor Gaussian model of a ( B ,")-market in question is described by the formula

I c0 (T0 ,T ) = P(0, T ) @ ( d + )

-

KP(0,TO)@(d-)

1

(1)

where In d& =

p(o, 5 L " T 0 , T)B2(TO,T) KP(0,TO) 2 2 o ( T o ,T ) B ( T " T , )

The rational price Po(TO, T ) o f a standard put option is described by the formula

I PO(T0,T ) = K P ( 0 ,T O ) @ ( - d - )

-

P(0, T ) @-d+) (

1

(6)

Before proceeding to the proof of (1) and (6), we point out that these relations are very similar to the formula for the rational prices @ ( T )and P(T) in the case of stock (see (9) and (18) in l b ) .

Chapter VIII. Theory of Pricing. Continuous Time

796

This is not that surprising given that, similarly to the prices St in the BlackMerton--Scholes model, the prices P(t, T ) in the present model have a loprzthmzcally normal structure: In P( t, T ) = A ( t ,T ) - ~ ( t ) B (Tt ), , where ( r ( t ) ) t
and ( W t ) t Gis~a Wiener (and therefore also Gaussian) process. It is maybe more surprising that it took so long between the publication of the Black-Scholes formula in 1973 and 1989, when F. Jamshidian published the paper [256] containing formulas (1) and (6) for the VasiEek model ( a ( t )= a , p(t) E p, y(t) =: y;see (4) in 3 4a and (8) in Chapter 111, 3 4a). We follow mainly [a571 in our proof below. 2. By pricing theory for complete arbitrage-free markets (see 3 5 in Chapter VII), assuming that the initial probability measure on (a, 9, ( 9 t ) t ~ ~9~ ) , = 9, is a martingale measure and setting

we obtain

Clearly, we have the coincidence of the events

where

r* =

1 n K - A ( T o ,T ) -B(TO, T )

and A ( t ,T ) and B ( t ,T ) are defined by (13) and (14) in 3 4a.

4. European and American Options in a Diffusion ( B ,P)-Bondmarket

797

Let

Then by (8) arid (9) we obtain

For a further simplification of this formula we bear on the following result, which can be verified by a direct calculation (see [257; Lemma 4.21).

LEMMA.Let ( X ,Y )be a Gaussian pair of random variables with vector of mean values ( p x ,p y ) and covariance matrix

and

EI(X

< x ) X e x p ( - Y ) = exp(-o$ 1 2

-

PY)

'

{ (Px

-

PxY)@(%

-

f l X U ( 2 ) ) . (13)

where

Taking formulas (6). ( l o ) , and (11) in $ 4 a into account it is easy to calculate that

Chapter VIII. Theory of Pricing. Continuous Time

798

and

By (11) arid (12) we obtain

Substituting here the above values of p t , p., pc, at, g7, a<, p ~ ? ,and pee, after some algebraic transformations (see [257; Appendix 4b]) we obtain required formula (1). Relation ( G ) is seen to be an immediate consequence of (1) once one takes into account thc equality

( K - P(T0, T ) ) += (P(T0,T ) - K ) + - P(T0, T )

+K

(Cf., e.g., the derivation of (9) in Chapter VI, 4d.) This completes the proof.

4. European and Aincricari Options in a Diffusion (B,P)-Bondmarket

799

3. It follows from (6) that the rational price Po(To,T ) can be determined from the 'initial' prices P(0, T o ) ,P(0, T ) ,the constant K , and the quantity a ( T o ,T ) B ( T oT, ) . which, in its turn is defined by the coefficients P ( s ) and y(s) for T o s < T . In the case of the VasiFek model we have P ( s ) E s, y(s) E y,anti it is easy to

<

see that

The initial prices P(0. T o ) and P(0, T ) can be found from the following formula (see (12) in 34a): P(0, t ) = exp{ A(0,t ) - roB(0,t ) } , where

B(0,t ) =

1

-

P

[l - e-Ot]

fj 4c. American Option Pricing in Single-Factor Gaussian Models 1. We continue our discussion of single-factor Gaussian (B,?)-models. We found formillas for @.'(To,T ) and Po(To,T ) in 3 4b; now let @ * ( T oT, ) and P*(To,T ) be the corresponding rational prices of American (call and put) options. Here we assume that the exercise times belong to the class

rn;O

=

{.

=

.(w): 0 6

.(W)

< TO, w E ( I } .

Tlie ( B ,?)-market, in question is both arbitrage-free and complete, and in accordance with the general theory of pricing in such models (see Chapter VI, 5 2 and Chapter VII, 5 5),

equalit,ies (1) arid (2) are related to standard optimal stopping problems of finding the suprema

for Markov processes T = ( ~ ( t ) )anti t ~nonnegative functions G ( r ,T ;~ ( 7 ) )There . exists a well-developed theory for such problems (see, for instance, [441]).

Chapter VIII. Theory of Pricing. Continuous T h e

800

2. The question of the value of C* ( T o T , ) is very simple because, as it is also for standard call options in ( B ,S)-markets, the process

is a submartingale arid C * ( T o T , ) = C o ( T o , T )by Doob's stopping theorem (we interpreted this equality in the following way in Chapter VI, § 5 b and in §3b of the present chapter: an American call option coincides in effect with a European option). Proceeding to the prices P*( T o ,T ) ,we consider the variables

where Et,r is averaging under the assumption r ( t ) = T , times 7 = T ( W ) such that t T ( W ) 6 T o , and

<

!lTlr

0 .

is the class of stopping

4

G ( t ,T ;~ ( t = ) )( K - P ( t , T ) ) += ( K - exp(A(t, T ) - r ( t ) B ( t , T ) ) ) with A(t, T ) and B ( t ,T ) as in formulas (13) and (14) in 5 4a. We set = { ( t ,7.1: ~ * ( rt ) ,> ~ ( T t ;,T ) . o t < T, r > o}

cT

<

DT = { ( t ,T ) : Y * ( t , r )= G ( t , T ; r ) ,0

< t < T , r > 0).

and

On the basis of the characterization of the prices Y * ( t , r )as the smallest excessive majorants of the functions G ( t ,T ;T ) (see [340], [363], [441; Chapter 1111, [467], [478]), we can show that there exists a continuous interface function T* = ~ * ( t ) , t < T o ,such that the domains CT and DT (of the continuation of observations and their stopping, respectively) have the following form:

CT

=

{ ( t , ~ ~) :( t<)~ * ( t 0) ,< t < T ,

T

>0)

DT

=

{ ( t , ~ ~) :( t3) r * ( t ) , 0 6 t < T ,

T

> 0}

and Here the function Y * = Y * ( t T, ) and the interface function of the following Stephnn problem:

aY*(t,7.) at

+ L Y * ( t , r )- T Y * ( t , T )

= 0,

T*

=~

(t,T) €

* ( tare ) solutions

CT,

4. European and American Options in a Diffusion

( B ,P)-Bondmarket

801

where

L Y * ( t ,T )

= ( N ( t ) - p(t).)

dY*(t;r) i)r

@Y*(t,r)

1

+ 5y2(t)

ar2

;

Y * ( t T, ) = G ( t ,T ;T )

(6)

in the domain D T , and we have the condition of smooth pasting on d D T :

ay2lrTr*(t)

1

-

3G(t,

(7)

rlr* ( t )

We know of no precise analytic solution to this problem (as, incidentally, also in the case of ( B ,S)-models; see 3 3c). At the same time, in view of the abundance of American options in real life one would like to have a n idea on the difference between the price P*(To,T ) of a n American option and the price Po(TO, T ) of a European option, and on the behavior of the interface T * = ~ * ( t t) < , To. Arguments similar to the ones we used in 4 3d, while looking for relations between the prices of American and European options in a (B,S)-market, can be used for (U,'P)-niarkets in our case and bring us to the following result (cf. (19) in 33d): for 0 t < T" we have

<

Y * ( t T, ) = YO@,7.)

+ K l T o E t , r { exp(

-

L'T(~)

1

du)r.(s)i(r.(s) 2 T * ( s ) ) d s , ( 8 )

where

Y o ( t T, )

= E t , r exp

(-lT:(,u) (K du)

-

= Et,r exp( -ifT':.(u) d u )

P ( T 0 ,T ) ) +

( K - exp(A(To, T ) - T ( T ' ) B ( T7'))). ~,

(9)

Using formulas (12) and (13) in 0 4b and the expressions there for ,q,p q , . . . , after simple transforniations we obtain

Y * ( t T, )

= YO@, 7.)

+K for

o < t < TO,

1

.TO

P(t, s ) {

(-u* ( t ,s ) )f ( t .s )

+ a ( t ,s)cp(u*( t ,s ) )} ds

where

a 2 ( t ,S )

= D(T(s)

I~

( t=)T ) ,

d f ( t ,s ) = -- In P ( t , Q ) ,

8.5

(See a detailed derivation of (8) and (10) in [257].)

(10)

Chapter VIII. Theory of Pricing. Continuous Time

802

In particular, sirice

P*(To,T)= Y * ( O , T ~and )

Po(To,T)= Y 0 ( O , r o ) ,

it follows that TO

P*(T”, T ) = PO@, T ) + K

P(0, S ) { @ ( - - Y * ( O , s))f(O,s)+a(O,S)’P(”*(O, s ) ) } ds.

In conclusion we point out that (10) enables one t o find (by nieans of backward induction), at, least approximately, also the values of the price P*(To,T ) and the interface function T * = r * ( t ) ,t < T O . As regards various methods of American option pricing (including the ‘case with dividends’), see, for instance, [as],[29], [56], [57], [179], [257], [376], [478], or [479].

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Index

APT-tlicory 56, 553 ( B ,S)-lIlaIkct 286, 383. 467, 588 ( B ,S)-stockmarket 778 CAPM line 53 FX-rnarke t 318 ( H I , IL - v)-r~,prcsentability698 R/S-analysis 367, 376 S-represerit,atbility 482, 483 &distribution 197, 199 X-representability 660 a-admissibility 646 a-stability 190 r-distribution 197, 199 X2-test 331 prepresentability 485 ( p - v)-rci)rescntability 485 a-algebra of predictable sets 297 r-martingale 656 0-time 358 Actuary 71 Admissible strategics 633 Anrnial discount rate 8 interest rate 8

Arbitrage 411, 647 Arithmetic random walk 111 Attractor 224 Autoregressive conditional lieteroskedasticity 63, 106, 153. 159. 288 Bachelier formula 19. 735 Bank account 7, 383 Bayes formula 438, 496 Best linear estimator 143 Black noise 234 Black-Scholes formula 20, 739, 740, 746 Brackets angle 305 square 305 Brownian bridge 238 Brownian motion 39 fractional (fractal) 221, 227, 228 geometric (economic) 39, 237, 284, 739 linear with drift 735 multifractional 230

826

Canonical representation of sernirnartingale 668 space 698 Cantor set 225 Chaos 221 Chaotic whit,e noise 181 Class of Dirichlct 301 Cluster 364, 588 Combination of options 604, 765 Compensator 100, 305, 461 Cornplcte asyniptotic separabilit,y 568 market 399, 505, 535, 661, 704, 728 Compound interest 7, 447 Concept of efficient market 60; 65, 414 Condition Dirichlet 786 Kazarnaki 674 Lipschitz local 265 Ncuriianri 786 Novikov 269, 441, 674 of linear growth 265 ‘smooth fit t ing’, ‘smooth pasting‘ 620, 756, 786, 787, 80 1 usual 243, 295 Conditional expection generalized 93 two-pointedness 482, 492 Conditionally Gaussian case 440, 451 Conjecture martingale 39 random walk 39 Contiguity of probability measures 559 Corrclation dirncnsion 183

Index

Coupon yield 9 Cross rate 329 Cumulant 202, 670 Current yield 11 Decomposition Doob 89, 301 Doob generalized 94, 456 Doob-Mcyer 301 Kunita-Watanabe 520 Lebesgue 560 multiplicative 691 optional 514, 516, 546 Differential variance 238 Diffusion 238 Diffusion with jumps 279 Dinamical systems nonlinear 231 Discrete version of Girsanov’s theorem 439, 451 Distribution binomial 197, 198 Cauchy 193, 197, 199 Gaussian\\inverse Gaussian 197 199, 214 generalized hyperbolic 214 geometric 197, 198 hyperbolic 197, 199, 214 infinitely divisible 194 invariant 181 Lkvy-Smirnov 197 lognormal 197, 199 logistic 197, 199 negative birioniial 197, 198 normal 193, 197, 199 of Pareto type 325 one-side stable 193 Pareto 192, 197. 199 Poisson 197, 198 stable 189

Index

Student 197, 199 uniform 198, 199 Diversification 46, 503 Dividend 13 DolCans exponential 83, 244, 261, 308 Domain of continued observations 532 Domain of stopping observations 532 Drift component 197 Effect of asymmetry 163 Equation Cameron-Martin 275 DolCans 308 Feynnian-Kac 275 heat 274, 275 Kolmogorov backward 272 Kolmogorov-Chapman 272 Kolmogorov forward 271 Langevin 239 Yule Walker 133 Estimator of maximum likelihood 133, 136 Event catastrophic 77, 79 extremal 79 normal 79 Excessive rnajorant 533 Extended version of the First fundamental theorem 425 of the Second fundamental theorem 497 Face value 9, 290 Filter of Kalmari-Bucy 173 Filtered probability space 82, 294, 324

First fundamental theorem 413 Financial engineering 5, 69 instruments 5 turbulence 234 Formula Bachelier 19, 735 Bayes 438, 496 Black-Scholes 20, 739, 740, 746 It6-Meyer 757 It6 257, 258, 307 Kolmogorov-It6 263 LCvy-Khintchine 195, 670 Tanaka 267, 311 Yor 472, 691 Forward (contract) 22, 521 Fractal 224 Fractal geometry 221, 224 Fractional noise 232, 347 Function elementary 252, 295 exessive 533 predictable 297 simple 252, 295 truncation 196, 464 Functional Brownian 256 measurable 270 progressively measurable 270 Functions adapted 252 Fundament a1 partial differential equation 712, 713, 716 solution 45, 793 Futures (contract) 21. 521 Gasket of Sierpihski 224, 225 Gaussian fractional noise 232, 371 Geographic zones 319

827

828

Geometric random walk 111 Heavy tails 334 Hedge 477 upper, lower, perfect 477 Hedging of American type 504, 538 of European type 504, 506 Helices of Wiener 229 Hellinger distance 561, 677 integral 555, 561 process 555, 677 Identities of Wdld 244 Index Dow (DJIA) 15, 376 S&P500 (Standard & Poor’s 500) 15, 376 stability 336 tail 336 Inequalities Doob 250 Kolmogorov-Doob 250 Infinite time horizon 751 Interday analysis 315 Interest rate 7, 278, 279, 291 Interniediarits 5 Kolmogorov’s axioniatcs 81 Kurtosis 88 Law of 213 234 of large riunibers 109 of large numbers strong 134 of the iterated logarithm 247 Lenirna conversion 438 Leverage effect 163 Leptokurtosis 329

Index

Linearly independent system 548 Local absolute continuity 433 drift 238 law of the iterated logarithm 246 Logistic map 177 Long memory 558 position 24, 28 Main formula of hedge pricing 505 Majorant smallest crp-excessive 616 smallest ,,%excessive 754, 782 srnallest (p,c)-excessive 534 smallest excessive 533 Margin 24 Market N-perfect, perfect 399 arbitrage-free weak, strong sense 412 arbitrage-free 411 complete 399, 505, 535, 661. 704, 728 currency exchange (FXmarket) 318 imperfect 399 large 553 semi-strongly efficient 41 strongly efficient 41 weakly efficient 41 Markov property 242 Markov time 114, 324 Martingale 41, 42. 89, 95 difference 42, 96. 157 difference, generalized 97 generalized 97 local 96

Index

local purely discontinuous 306 squaw intcgrahlr 92, 296, 298 transformation 98 uniformly integrable 96 Maturity date 9 Maximal ineqiialitirs 250 Maxirriuni likelihood method 133 Mean square criterion 520 Measurable selection 428 Measure 3-c-finite 665 absolutely continuous 433 dual 744 equivalerit 433 honiogcneous Poisson 664 Lbvy 195, 202 206, 671 local rriartingale 683, 719 locally eqiiivalcnt 433 martingale 413, 459, 666 rrmimal niizrtingalc 459, 585 optional 665 Poisson 664 random 664 Wiener 236 Mixture of Gaussian dis tribri tions 21 7 Modcl affincl 292 A R 125, 150, 288 ARCH 63, 106, 15.3, 160, 288 A RDM (Autoregressive Conditional Duration Model) 325 ARIhfA 118, 138, 141 ARMA 105, 138, 140, 151, 288 Bachclier linear 284, 735 Black Derman-Toy 280 Black Karasinski 280

Black-Merton-Scholes 287, 712, 739 chaotic 176 Chen 280 conditionally Gaussian 103, 153 Cox-Ingersoll-Ross 279 Cox-Ross-Rubinstein (CRR) 110, 400. 408. 478, 590, 606 Dothan 279 dynamical chaos 176 EGARCH 163 GARCH 63, 107, 153, 288 Gaussian single-factor 794 HARCH 166 H J M 292 Ho-Lee 279 HuIl-White 280 linear 117 M A 105, 119, 148. 288 hlA(cx;) 125 Merton 279 non-Gaussian 189 nonlinear stochastic 152 Sarnuelson 238, 705 Sandmann-Sondermann 280 Schmidt 283 semimartingale completeness 660 single-factor 292 stochastic volatility 108, 168 Taylor 108 TGAR CH 165 VasiEek 279 with discrete intervention of chance 112 with dividends 748 Modulus of continuity 246

829

830

Negative correlation 49 Noise black 234 pink 234 white 119, 232 One-sided nioving averages 142 Operational time 116, 358 Opportunity for arbitrage 411 Optimal stopping time 525, 528 Option 22,26 Anierican type 27, 504, 608, 779, 788, 792 Asian type 626 call 27, 30 European type 27, 504, 588, 735, 779, 788, 792 exotic 604 put 2 7,31 put, arithmetic Asian 31 put with aftereffect 31 Russian 625, 767 with aftereffect 625 Paranieter Hurst 208 location ( p ) 191 scale (0)191 skewness (0)191 Pay-off 514, 660, 709 Phenorrienon absense of correlation 50 ‘cluster’ 364 Markowitz 49 negative correlation 49, 355 Point process marked 323 multivariate 323 Port folio investment 32, 46, 385, 633

Index

self-financing 386 Position long, short 22, 28 Predictability 89, 278 Prediction 118 Price American hedging 539, 543 forward 523 futures 523 perfect European hedging 506 rational (fair, mutually appropriate) 31, 398 strike 31 tree 500 upper, lower 395 Principle reflection 248, 760 Problem Cauchy 274 Dirichlet 276 free-boundary 755 Stephan 755, 764, 773, 783 Process adapted 294 Bessel 240 Brownian motion with drift and Poisson jumps 671 c&dl&g294 counting (point) 115, 323 density 434 diffusion-type 678 discounting 644, 744 Process Hellinger 555, 677 innovation 680 interest rate 717 It6 257 L6vy 200 L6vy a-stable 207, 208 L6vy purely jump 203 multivariate point 115

Index

Orristein-Uhlenbeck 239 Poisson 203 Poisson compound 204 predictable 297 quasi-left-contiriuous 703 stable 207 stochastically indistinguishable 265 three-dimensional Bessel 102 Wiener 201 with discrete intervention of chance 112, 114, 322 with iriterniittency (antipersistence, relaxation) 232 zero-energy 350 Property E L M M 649. 652 E M M 649, 652 EcTMM 656 NA+, N A , 651, 652 NA,, N A , 652 N F F L V R ( ~ , ) 651 N F L V R ( ~ + 651 ) no-arbitrage 649, 650 Pure uncertainity 72

m+,

Quadratic covariance 303, 310 predictable variation 92 variation 92, 247, 303 Quantile method 331 Radon-Nikodym derivative 434, 561 Randoni process self-similar 226 vector stable 200 vector strictly stable 200 walk geometric 609 Range 222

Rank tests 331 Rational price 31, 399, 545, 591, 595, 736, 740, 750 time 545 Reinsurance 75 Representation canonical 662, 668 Returns, logarithmic returns 83 Risk market 69 systematic 50, 54 unsystematic 50, 54 Scheme of series of n-markets 553 Second fundamental theorem 481 Securities 5 Selector 429 Self-financing 386, 633, 640 Self-similarity 208, 221 Semimartingale 292 locally square integrable 669 special 302, 669, 686 Sequence completely deterministic 145 completely nondeterministic 145 innovation 145 logistic 184 predictable 89 regular 144 singular 144 stationary in the strict sense 127 stationary in the wide sense 121, 127 Shares (stock) 13,383 Short position 22, 28, 394 Simple interest 7 Smile effect 286 Solar activity 374

831

832

Solution probabilist,ic 275 strong 265, 266 Specification direct 12, 292 indirect 12, 292 Spectral represeritat,ion 146 Spread 322, 604, 606 Stability exponent 191 Stable 189 Stanclard Brownian niotion 201 Statistical sequential analysis 783 Statistics of 'ticks' 315 Stochastic basis 82, 294 differential eyiiat,ion 264 differential 440 exponent,ial (Doleans) 83, 244, 261. 308 integral 237, 254, 295. 298 partial diffaential equation 713, 746 process predictable 297 St,opping time 114 Straddle 604 Strangle 604 Strap 605 Strategy adniissible 640 in a ( B ,P)-iriarket 719 perfect 539 self-financing 386, 640 Strip 605 Subrnartirigale 95 local 96 Subordination 21 1 SiiI,"ri~iartingale 95 local 96 srna11est> 528

Index

Theorerri Doob (convergence) 244, 435 Doob (optional stopping) 437 Girsanov 439, 451, 672 Girsanov for sernirnartingales 701 Lhvy 243, 309 Lundberg- Cram&- 77 on normal correlation 86 Time local (L6vy) 267 operational 213, 358 physical 213 Transformat ion Bernoulli 181 Esscher 420, 672, 683, 684 Esscher conditional 417, 423 Girsanov 423 Transition operator 530, 609 Triplet ( B , C , v ) 195, 669 Triplet of predictable characteris of a sernirnartingale 669 Turbulence 234

Uniform integrability 96, 301. 518 Upper price of hedging 395, 539 Value of a strategy (investment portfolio) 385, 633, 719 Vector affinely independent 497 linearly independent 497 stochastic integral 634 Volatility 62, 238. 322, 345. 346 implied 287 White noise 119, 232 Wiener process 38, 201 Yield to the maturity date 291

Index of Symbols

dploc

307, 367, 638, 686 799 795 27, 31 749, 750 19, 740, 741 746 506 709 781 780 123 86, 123 18, 316, 329 227 78 1 780 664 83, 84 649 649 40 40, 81 40, 82 24 1

241 242 114 114 255 299, 639 290, 291 29 1 98 666 396 208, 227, 353 663 663 298 254, 298 298

331 298 88

834

Index of Symbols

98 642 642 96 97 92 92 649, 650 649, 650 651 664 664 799, 802 795 29, 600 741, 789 750 789 40, 81 10, 292 9 297, 664 665 367 367 291 290, 291 386, 640 331 193 192 368 316, 317 317 322 328 763 763 743 779 779

752 753 299 304 349 349 665 666 305 303 305, 306 200 95 200 663 663 780 780 191 52, 643 743 349 350 647 646 647 743 11

743 333