Stochastic Implied Volatility: A Factor-Based Model

Lecture Notes in Economics and Mathematical Systems Founding Editors:

M. Beckmann H. P. Kiinzi Managing Editors: Prof. Dr. G. Fandel

Fachbereich Wirtschaftswissenschaften Femuniversitat Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut far Mathematische Wirtschaftsforschung (IMW)

Universitat Bielefeld Universitatsstr. 25, 33615 Bielefeld, Germany Editorial Board:

A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Karsten, U. Schittko

545

Reinhold Hafner

Stochastic Implied Volatility A Factor-Based Model

4) Springer

Author Dr. Reinhold Hafner risklab germany GmbH Nymphenburger StraBe 112-116 80636 München Germany

Library of Congress Control Number: 2004109369 ISSN 0075-8442 ISBN 3-540-22183-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by author Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

42/3130Di 5 4 3 2 1 0

Ftir meine Eltern

Preface

This monograph is based on my Ph.D. thesis, which was accepted in January 2004 by the faculty of economics at the University of Augsburg. It is a great pleasure to thank my supervisor, Prof. Dr. Manfred Steiner, for his scientific guidance and support throughout my Ph.D. studies. I would also like to express my thanks to Prof. Dr. Glinter Bamberg for his comments and suggestions. To my colleagues at the department of Finance and Banking at the University of Augsburg, I express my thanks for their kind support and their helpful comments over the past years. In particular, I would like to thank Dr. Bernhard Brunner for many interesting discussions and also for the careful revision of this manuscript. At risklab germany GmbH, Munich, I would first of all like to thank Dr. Gerhard Scheuenstuhl and Prof. Dr. Rudi Zagst for creating an ideal environment for research. I would also like to express my thanks to my colleagues. It has been most enjoyable to work with them. In particular, I would like to thank Dr. Bernd Schmid. Our joint projects on stochastic implied volatility models greatly influenced this work. I am also indebted to Anja Fischer for valuable contributions during her internship and Didier Vermeiren (from Octanti Associates) for carefully reading the manuscript. Further, I am extremely grateful to Prof. Dr. Martin Wallmeier for his continuous support and advice, his thorough revision of the manuscript, as well as for many fruitful discussions. The results of our joint projects on the estimation and explanation of implied volatility structures entered this work. Most of all, I want to thank my girlfriend Heike for endless patience, encouragement, and support, and also my mother Lieselotte and my brother Jurgen for being there all the times.

Mering, May 2004

Reinhold Hafner

Contents

1

Introduction 1.1 Motivation and Objectives 1.2 Structure of the Work

1

1 5

2

Continuous-time Financial Markets 2.1 The Financial Market 2.1.1 Assets and Trading Strategies 2.1.2 Absence of Arbitrage and Martingale Measures 2.2 Risk-Neutral Pricing of Contingent Claims 2.2.1 Contingent Claims 2.2.2 Risk-Neutral Valuation Formula 2.2.3 Attainability and Market Completeness

9 10 10 13 15 15 18 20

3

Implied Volatility 3.1 The Black-Scholes Model 3.1.1 The Financial Market Model 3.1.2 Pricing and Hedging of Contingent Claims 3.1.3 The Black-Scholes Option Pricing Formula 3.1.4 The Greeks 3.2 The Concept of Implied Volatility 3.2.1 Definition 3.2.2 Calculation 3.2.3 Interpretation 3.3 Features of Implied Volatility 3.3.1 Volatility Smiles 3.3.2 Volatility Term Structures 3.3.3 Volatility Surfaces 3.4 Modelling Implied Volatility 3.4.1 Overview 3.4.2 Implied Volatility as an Endogenous Variable 3.4.3 Implied Volatility as an Exogenous Variable

23 24 24 25 27 29 32 32 34 35 38 38 39 41 43 43 45 51

X

Contents

3.4.4

Comparison of Approaches

56 59 60 60 61 63 63 68 68 70

4

The General Stochastic Implied Volatility Model 4.1 The Financial Market Model 4.1.1 Model Specification 4.1.2 Movements of the Volatility Surface 4.2 Risk-Neutral Implied Volatility Dynamics 4.2.1 Change of Measure and Drift Restriction 4.2.2 Interpretation of Terms in the Risk-Neutral Drift 4.2.3 Existence and Uniqueness of the Risk-Neutral Measure 4.3 Pricing and Hedging of Contingent Claims

5

Properties of DAX Implied Volatilities 73 73 5.1 The DAX Option 73 5.1.1 Contract Specifications 5.1.2 Previous Studies 75 76 5.2 Data 5.2.1 Raw Data and Data Preparation 76 78 5.2.2 Correcting for Taxes and Dividends 82 5.2.3 Liquidity Aspects 5.3 Structure of DAX Implied Volatilities 83 5.3.1 Estimation of the DAX Volatility Surface 83 92 5.3.2 Empirical Results 5.3.3 Identification and Selection of Volatility Risk Factors 99 102 5.4 Dynamics of DAX Implied Volatilities 5.4.1 Time-Series Properties of DAX Volatility Risk Factors . 102 5.4.2 Relating Volatility Risk Factors to Index Returns and other Market Variables 109 113 5.5 Summary of Empirical Observations

6

A Four-Factor Model for DAX Implied Volatilities 6.1 The Model under the Objective Measure 6.1.1 Model Specification 6.1.2 Model Estimation 6.1.3 Model Testing 6.2 The Model under the Risk-Neutral Measure 6.2.1 Risk-Neutral Stock Price and Volatility Dynamics 6.2.2 The Market Price of Risk Process 6.2.3 Pricing and Hedging of Contingent Claims 6.2.4 Model Calibration 6.3 Model Review and Conclusion

115 115 115 118 124 131 131 133 137 140 144

Contents

XI

7

Model Applications 7.1 Pricing and Hedging of Exotic Derivatives 7.1.1 Product Overview 7.1.2 Exotic Equity Index Derivatives 7.1.3 Volatility Derivatives Value at Risk for Option Portfolios 7.2 7.2.1 The Value at Risk Concept 7.2.2 Computing VaR for Option Portfolios 7.2.3 A Case Study 7.2.4 Beyond VaR: Expected Shortfall Trading Volatility 7.3 7.3.1 Definition and Motivation 7.3.2 Volatility Trade Design 7.3.3 Profitability of DAX Volatility Trading Strategies

145 145 145 147 153 158 158 160 162 167 170 170 171 178

8

Summary and Conclusion

187

Some Mathematical Preliminaries

A.1 Probability Theory A.2 Continuous-time Stochastic Processes

193 193 194

Pricing of a Variance Swap via Static Replication

201

A

B

List of Abbreviations

205

List of Symbols

207

References

215

Index

225

1 Introduction

1.1 Motivation and Objectives Financial derivatives or contingent claims are specialized contracts whose intention is to transfer risk from those who are exposed to risk to those who are willing to bear risk for a price. Derivatives are heavily used by different groups of market participants, including financial institutions, fund managers (most notably hedge funds), and corporations. While speculators intend to benefit from the derivative's leverage to make large profits, hedgers want to insure their positions against adverse price movements in the derivative's underlying asset, and arbitrageurs are willing to exploit price inefficiencies between the derivative and the underlying asset. During the last two decades the market for financial derivatives has experienced rapid growth. From 2000 to 2002 alone, global exchange-traded derivatives volume nearly doubled, to reach almost 6 billion contracts traded in 2002. With a market share of approximately 50%, equity index derivatives are thereby the most actively traded contracts.' Huge volumes of derivatives are also traded over the counter (OTC). In addition to standard products, the OTC market offers a wide variety of different contracts, including so-called exotic derivatives. Exotic derivatives were developed as advancements to standard derivative products with specific characteristics tailored to particular investors' needs. The latest development in this area are volatility derivatives. These contracts, written on realized or implied volatility, provide direct exposure to volatility without inducing additional exposure to the underlying asset. The increasing use and complexity of derivatives raises the need for a framework that enables for the accurate and consistent pricing and hedging, risk management, and trading of a wide range of derivative products, including all kinds of exotic derivatives. The first important attempt in this direction was the Black-Scholes option pricing model, developed by Black/Scholes (1973), formalized and extended in the same year by Merton (1973). It builds a corSee FIA (2003).

2

1 Introduction

nerstone in the theory of modern finance, and has led to many insights into the valuation of derivative securities. In 1997, the importance of the model was recognized when Myron Scholes and Robert Merton received the Nobel Price for Economics. The Black-Scholes model provides a unique "fair price" for a (European) option that is traded on a frictionless market and whose underlying asset exhibits lognormally distributed prices. Under the model's assumptions, an option's return stream can be perfectly replicated by continuously rebalancing a self-financing portfolio involving stocks2 and risk-free bonds. In the absence of arbitrage, the price of an option equals the initial value of the portfolio that exactly matches the option's payoff. The Black-Scholes model is often applied as a starting point for valuing options. However, the empirical investigation of the Black-Scholes model revealed statistically significant and economically relevant deviations between market prices and model prices. A convenient way of illustrating these discrepancies is to express the option price in terms of its implied volatility, i.e. as a number that, when plugged into the Black-Scholes formula for the volatility parameter, results in a model price equal to the market price. If the Black-Scholes model holds exactly, then all options on the same underlying asset should provide the same implied volatility. Yet, as is well known, on many markets, Black-Scholes implied volatilities tend to differ across exercise prices and times to maturity. The relationship between implied volatilities and exercise prices is commonly referred to as the "volatility smile" and the relationship between implied volatilities and times to maturity as the "volatility term structure". Volatility surfaces combine the volatility smile with the term structure of volatility. The existence of volatility surfaces implies that the implied volatility of an option is not necessarily equal to the expected volatility of the underlying asset's rate of return. It rather also reflects determinants of the option's value that are neglected in the Black-Scholes formula. The obvious shortcomings of the Black-Scholes model have led to the development of a considerable literature on alternative option pricing models, which attempt to identify and model the financial mechanisms that give rise to volatility surfaces, in particular to smiles. One strand of the literature concentrates on the nature of the underlying asset price process which was assumed to be a geometric Brownian motion with constant volatility in the BlackScholes framework. Here the main focus is on models which assume that the volatility of the underlying asset varies over time, either deterministically or stochastically. Derman/Kani (1994b), Derman/Kani (1994a), Dupire (1994), and Rubinstein (1994) were the first to model volatility as a deterministic function of time and stock price, known as local volatility. The unknown volatility function can be fitted to observed option prices to obtain an implied price process for the underlying asset. In an empirical study Dumas et al. (1998) 2 We use the term "stock" as a general expression for the underlying asset of a derivative security, although it could as well be an equity index, an exchange rate, or the price of a commodity.

1.1 Motivation and Objectives

3

conclude that, as fax as S&P 500 options are concerned, local volatility models are unreliable and not really useful for valuation and risk management. The stochastic volatility approach was motivated by empirical studies on the time series behavior of (realized) volatilities. They suggest that volatility should be viewed as a random process. Specifications for a stochastic volatility process have been proposed by a number of authors, including Hull/White (1987), Wiggins (1987), Scott (1987), Stein/Stein (1991), and Heston (1993). A problem of stochastic volatility models is that unrealistically high parameters are required in order to generate volatility smiles that are consistent with those observed in option prices with short times to maturity. 3 A third explanation for implied volatility patterns that is related to the asset price process are jumps. 4 When jumps occur, the price process is no longer continuous. Jumps have proved to be particularly useful for modelling the crash risk, which has attained considerable attention since the stock market crash of October 1987. In the attempt to correctly reproduce the empirically observed implied volatility patterns, neither (one-factor) stochastic volatility models nor simple jump-diffusion models are successful. Furthermore, both types of models are incomplete. Consequently, the requirement of no-arbitrage is no longer sufficient to determine a unique preference-free price of the contingent claim. Another problem of models based on the underlying asset to describe the dynamic behavior of option prices is that infinitesimal quantities such as the local or stochastic volatility or the jump intensity, are not directly observable but have to be filtered out either from pricing data on the underlying asset or "calibrated" to options data. In the first case, the quantity obtained is model-dependent and in the second case it is the solution to a non-trivial optimization problem. A second strand of the literature identifies market frictions as another possible explanation for the smile pattern. Transaction costs, illiquidity, and other trading restrictions imply that a single arbitrage-free option price no longer exists. Longstaff (1995) and Figlewski (1989a) examined the effects of transaction costs and found that they could be a major element in the divergences of implied volatilities across strike prices. Yet, Constantinides (1996) points out that transaction costs cannot fully explain the extent of the volatility smile. McMillan (1996) argues that the crash of 1987 lessened the supply of put option sellers, whereas at the same time fund managers showed a higher demand for out-of-the-money puts. Because hedging the risk exposure of written out-of-the-money puts turned out to be expensive, higher prices for out-of-the-money puts were charged. This could also partly explain the observed strike pattern of implied volatility. It is generally acknowledged that the above influences are interrelated, and no single explanation completely captures all empirical biases in implied volatilities. The increasing liquidity in the market for standard options, especially in the area of equity index options considered here, has had two major conse3 See,

e.g., Andersen et al. (1999), p. 3, and Das/Sundaram (1999), p. 5.

4 See,

e.g., Bates (1996a), Trautmann/Beinert (1999).

4

1 Introduction

quences: 5 First, there is no more need to theoretically price standard options. The market's liquidity ensures fair prices. Second, hedging of standard options becomes less important as positions can be unwound quickly. These developments and the inability of the models described above to accurately reflect the dynamic behavior of option prices or their implied volatilities have brought up a second modelling approach. In directly taking as primitive the implied volatility (surface), this approach is usually referred to as a "market-based" approach. 6 Market-based models have the advantage to be automatically fitted to market option prices. In difference to fundamental quantities such as an (unobservable) instantaneous volatility or a jump intensity, implied volatilities are highly regarded and continuously monitored by market participants. A market scenario described in terms of implied volatilities is therefore easy to understand for a practitioner. Due to the noticeable standard deviation found in time series of implied volatilities, deterministic implied volatility models do not seem to be appropriate. A natural step of generalization is to let implied volatilities move stochastically. In contrast to (traditional) stochastic volatility models, where the instantaneous volatility of the stock return is modelled, stochastic implied volatility models focus on the (stochastic) dynamics of either a single implied volatility (e.g., Lyons (1997)), the term structure of volatility (e.g., Schônbucher (1999)), or the whole volatility surface (e.g., Albanese et al. (1998) and Ledoit et al. (2002)). A major advantage of stochastic implied volatility models is their completeness. While the approaches of Schtinbucher (1999) and Ledoit et al. (2002) dealt with the problem of stochastic implied volatility from a theoretical perspective, Rosenberg (2000), Cont/Fonseca (2002), Goncalves/Guidolin (2003), among others, focus on the empirical aspects of the problem. For example, Cont/Fonseca (2002), using S&P 500 and FTSE 100 option data, suggest a factor-based stochastic implied volatility model where the abstract risk factors driving the volatility surface are obtained from a Karhunen Loève decomposition. A natural application of this model is the simulation of implied volatility surfaces under the real-world measure, for the purpose of risk management. However, the models are not intended to determine the consistent volatility drifts needed for risk-neutral pricing of exotic derivatives. "How best to introduce the ideas from these models into a no-arbitrage theory remains an open question" . 7 It has mainly been this question that motivated this work. The overall goal of this work is to provide a stochastic implied volatility model that allows for the integrated and consistent pricing and hedging, risk management, and trading of equity index derivatives as well as derivatives on the index volatility. As we assume that the evolution of the volatility surface See SchtInbucher (1999). approach is similar to the approach of Heath-Jarrow-Morton (HJM) in the field of interest rates. See Heath et al. (1992). 5

6 This 7 See

Lee (2002), p. 25.

1.2 Structure of the Work

5

is driven by a small number of (fundamental or economic) risk factors, the model is termed "factor-based". Specifically, in the first, theoretical part of this work, we aim at developing a unifying theory for the analysis of contingent claims under both the real-world measure and the risk-neutral measure in a stochastic implied volatility environment. Based on the theory developed, the objective of the second, empirical part is to specify, estimate, and test a factor-based stochastic implied volatility model for DAX implied volatilities. 8 In the final part of this work, we will present potential applications of the model.

1.2 Structure of the Work This work is organized as follows (see Figure 1.1). In Chapter 2, we discuss the principles of continuous-time financial markets in a rather general framework, which will also serve as a reference in the later chapters. Special emphasis is put on the valuation of contingent claims. We comment on the class of results — referred to as a fundamental theorem of asset pricing — which says, roughly, that the absence of arbitrage opportunities is equivalent to the existence of an equivalent martingale measure. The main result of this chapter is the risk-neutral valuation formula, which states that the arbitrage price of any attainable contingent claim is the expectation of the discounted claim under the equivalent martingale measure. Chapter 3, devoted to implied volatility, starts with a description of the Black-Scholes model. We present the model's assumptions, derive the BlackScholes formula for European-style stock options, and state the main option's sensitivities, better known as the Greeks. We then introduce the implied volatility concept and describe some well-known patterns in the behavior of implied volatility as the strike price and the maturity date of the option change, namely the volatility smile, the term structure of volatility, and the volatility surface as the combination of the two. In the remainder of this chapter, we discuss the various approaches to value options in the presence of implied volatility structures. We highlight their individual strengths and weaknesses and explain the difficulties induced by them. In Chapter 4, we develop a general mathematical model of a financial market in continuous time where in addition to the usual underlying securities stock and risk-free bond, a collection of standard European options is traded. The prices of the standard options are given in terms of their implied Black-Scholes volatilities, whose joint evolution is driven by a small number of risk factors. The chapter starts with a description of the financial market model under the real-world or objective probability measure. Then, we derive necessary and sufficient conditions that have to be imposed on the drift 8 The DAX option is one of the most heavily traded equity index options in the world.

6

1 Introduction Continuous-time financial markets (Chapter 2) special case

Black-Scholes model (3.1) leads to Implied volatility surface (3.2 & 3.3) is

endogenous (3.4.2) 21 Market frictions Stock price is not a geometric Brownian motion

exogenous (3.4.3) volatility surface is

stochastic

deterministic

General factor-based stochastic implied volatility model (Chapter 4)

Properties of DAX implied volatilities (Chapter 5) examine

Structure (5.3)

Dynamics (5.4)

Four-factor model for DAX implied volatilities (Chapter 6)

11,

Objective measure (6.1)

Value at risk (7.2)

Risk-neutral measure (6.2)

Volatility trading (7.3)

Figure 1.1.

1.

Pricing and hedging of exotic derivatives (7.1)

Structure of the work

coefficients of the options' implied volatilities in order to ensure discounted call prices to be martingales under the risk-neutral measure. We also discuss existence and uniqueness of the risk-neutral measure. Finally, we show how to price and hedge a general stock price dependent contingent claim. The goal of Chapters 5 and 6 is the specification, estimation and testing of a factor-based stochastic implied volatility model for DAX implied volatilities.

1.2 Structure of the Work

7

Chapter 5 concentrates on the identification of the main properties of DAX implied volatilities both in a cross-sectional ("structure") and a time-series ("dynamics") setting. Our database contains all reported transactions of options and futures on the DAX, traded on the DTB/Eurex over the sample period from January 1995 to December 2002. To the best of our knowledge, this database is one of the largest databases, at least in Europe, that has ever been used in such a study. In preparing the data, we have carefully accounted for potential biases such as tax effects or non-simultaneous options and underlying prices. Based on the empirical results of Chapter 5, Chapter 6 proposes a fourfactor model for the stochastic evolution of the DAX volatility surface. We begin with the specification of the model under the real-world measure, we then show how to estimate the model parameters from historical data, and finally we perform various in- and out-of-sample tests to assess the quality of the model. In the second part of this chapter, we derive the risk-neutral dynamics of the DAX index, the DAX volatility surface, and the instantaneous DAX volatility. We discuss the issues of existence and uniqueness of a martingale measure and show how to price and hedge contingent claims. Finally, we calibrate the model to market data. Chapter 7 presents applications of the factor-based stochastic implied volatility model in the fields of pricing and hedging, risk management, and trading. In particular, we consider the pricing and hedging of selected exotic derivatives, including derivatives on index volatility, then apply the model to calculate the value at risk and expected shortfall for an option portfolio, and finally discuss volatility trading. Here, we describe several ways of how to trade volatility, discuss the advantages and disadvantages of each strategy, and empirically test some of these strategies on their ability to generate abnormal trading profits. The work concludes with a short summary of the main results and suggestions for further research (Chapter 8). For readers who are not familiar with stochastic analysis, we give a short introduction to this subject in Appendix A. Finally, Appendix B gives a proof on the pricing of a variance swap via the method of static replication.

2

Continuous-time Financial Markets

Understanding a theory means to solve a certain problem.

(.

.)

understanding it as an attempt

Sir Karl Popper

This chapter discusses the principles of continuous-time financial markets in a rather general framework, which will also serve as a reference in the later chapters. Special emphasis is put on the valuation of contingent claims. Following the path-breaking work of Harrison/Kreps (1979) and Harrison/Pliska (1981), we start in Section 1 by developing a rigorous mathematical model of a financial market in continuous time. In distinction to Harrison/Pliska (1981), who model the evolution of asset prices by a possibly discontinuous, semimartingale process, we restrict ourselves to continuous processes of the Ito type. The agents' activities in the market are modelled by trading strategies. A particularly important class of trading strategies in the context of contingent claim valuation is the class of self-financing trading strategies. It is described in some detail. In Section 2, we first introduce the concept of an arbitrage opportunity and then comment on the class of results — referred to as a fundamental theorem of asset pricing — which says, roughly, that the absence of arbitrage opportunities is equivalent to the existence of an equivalent martingale measure. The last section focuses on the arbitrage pricing of contingent claims. We present the risk-neutral valuation formula, which states that the arbitrage price of an attainable contingent claim is the expectation of the discounted claim under the equivalent martingale measure, discuss the attainability of contingent claims and introduce the notion of a complete market.

10

2 Continuous-time Financial Markets

2.1 The Financial Market 2.1.1 Assets and Trading Strategies We start with a frictionless security market, where investors are allowed to trade continuously up to some fixed terminal time horizon T'> 0. 1 A security market is called frictionless, if there are no transaction costs or taxes, no bidask spreads, no margin requirements, no restrictions on short sales, and all assets are perfectly divisible. 2 The uncertainty in the financial market is characterized by the complete probability space (S2, F, P) where f2 is the state space, .7. is the a-algebra representing measurable events, and IP is the objective or real-world probability measure. 3 Information evolves over the trading interval [0, TX] according to the Brownian filtration F = {,Ft : t E [0, , generated by a (p + 1)-dimensional standard Brownian motion W =- {Wt : t E [0, T*]} , Wt = (Wo,t, • • . , W,4'. 4 The a-algebra .Ft represents the information available at time t. Throughout this work, we assume that F satisfies the usual conditions: IF is complete, i.e. To contains all P-null sets of F and IF is right-continuous. 3 Moreover, we assume that the a-field ..7-0 is trivial, i.e. F. = {0, n} , and that FT' The market consists of d + 1 (stochastic) primary traded assets (stocks, bonds, options, etc.), whose (spot) price processes are given by stochastic processes Zo, Zd. We assume that Z = {Z t : t E [0, T 3 ]} , Zt =follows a positive (d + 1)-dimensional Itô process 6 with re-Z4', spect to W on the filtered probability space (St, F, P, F). 7 The movement of the security prices relative to each other will be important to study, so it is convenient to normalize the price vector Z. We set Zt Dt

Zr = — = with Zr t = Zi, t /Dt, the numéraire.

j

,

,

Vt E [0, TX]

,

(2.1)

= 0, ... d. The (one-dimensional) process D is called

'This section follows closely the presentations given in Bingham/Kiesel (1998), Chapter 6, Korn/Korn (1999), Chapter 6, Musiela/Rutkowski (1997), Chapter 10, and Hasrison/Pliska (1981). 2 See, e.g., Bingham/Kiesel (1998), p. 7. 3 Appendix A provides a brief account to fundamental concepts of probability theory and the theory of continuous-time stochastic processes, so far as they are used in this text. For more information, the reader is advised to consult the references cited there. 4 x' denotes the transpose of the vector x. 'By definition, the natural filtration of a Brownian motion a (W, : s < t) is right- and left-continuous but not complete. However, if we extend -Fr by the a-algebra containing all IF-null sets of .7., we obtain a filtration with the desired property. See Lamberton/Lapeyre (1996), p. 30. 6 Note that this specification does not allow for jumps in the price processes. 7 We shall henceforth denote a probability space (SI, F , P) endowed with a filtration F a filtered probability space (0, T,P,F) .

yr, =

2.1 The Financial Market

11

Definition 2.1 (Numéraire). A numéraire D = {Dt : t E [0, T1 } is a price process which is P-a.s. positive for each t E [0, T*] . Mostly, the money market account or a zero-coupon bond is used as

numéraire. This explains why Z* is usually called the discounted price process. From now on, we assume that Zo is the numéraire, i.e. D = Zo• The market participants' activities over time are described by trading strategies or portfolio strategies. Definition 2.2 (Trading Strategy). Let us fix a time horizon T < T* . Then, a trading strategy or portfolio strategy over the time interval [0, 7 ] is an R d+1 -valued progressively measurable process 0 = {g5t : t E [0, T1 } , ,ffit = (0o,t , 0147 • • • 1 Old,t) such that the stochastic integrals fip t

08dZ8

fo 08 dZ:

and

exist. The portfolio holding 0i ,t denotes the number of units of asset i held at time t.8 Since we have assumed that the market is frictionless, 0i,t may be any positive or negative value. The value and the gains process associated with a trading strategy are introduced next. Definition 2.3 (Value and Gains Process). Let 0 be a trading strategy over the time interval [0, 7 ] . 1. The value of the portfolio 0 at time t is given by d

Vt (0) = Ot • Zt

E i=o

Vt E [O, T] ,

(2.2)

The process V(0) is called the value process or wealth process of the trading strategy 0 with initial value or wealth Vo (çb). 2. The gains process G(0) is defined by d

Gt (0) =

t

f .9 c1Zt s , °

f 0 8C1Z8 = E

Vt E [0, .

(2.3)

8 1n a more general framework, where asset prices follow a continuous-time semimartingale, the trading strategy has to be predictable. Intuitively, this means that the number of assets held at time t are determined on the basis of information available before time t but not t itself. In our setting, however, it can be shown that it is actually enough to require that 4 is progressively measurable. See also Musiela/Rutkowski (1997), p. 230.

12

2 Continuous-time Financial Markets

The value G(0) represents the gains or losses accumulated up to and including time t. Thereby, we implicitly assume that the securities do not generate any cash payments such as dividends. With Z0 as numéraire, we define the discounted value process Vt*(0) as

Vt (4))

Vt* (0) =

d

= ,t

E ot,tzzt,

Vt E [0, 7],

(2.4)

.

(2.5)

and the discounted gains process as d

G; (0)

=

E i=1 0

t

Vt

E [0,7]

Note that G;(0) does not depend on the numéraire. A trading strategy, where all changes in the value of the portfolio are due to capital gains, as opposed to withdrawals of cash or injections of new funds, is called self-financing. Definition 2.4 (Self-financing Trading Strategy). A trading strategy 0 is called self financing over the time interval [0, T] if the value process V(0) satisfies -

Vt E [0,

,

(2.6)

or equivalently: d

dVt(0)

=

E f ,=0 0

t Vt E [0, 7 ]

.

(2.7)

Our goal is to be very flexible with respect to the chosen numéraire. The next result underscores this. Theorem 2.5 (Numéraire Invariance Theorem). Self financing trading strategies remain self financing after a numéraire change. -

-

Proof. This can easily be shown using It6's product rule. For a formal proof, see Bingham/Kiesel (1998), p. 173. Using the Numéraire Invariance Theorem we can restate the self-financing condition (2.6) in terms of the discounted processes: A trading strategy 0 is self-financing if and only if

vt*(sb) = 11 (0) +

(

(2.8)

0),

and, of course, 14 (0) > 0 if and only if Vt*(0) > 0 for all t E [0, 7 ] . This result shows that a self-financing trading strategy is completely characterized by its initial value Vo* (q5) and the components 0 1 , ,ç5 Therefore, any trading strategy can be uniquely extended to a self-financing strategy q5 with initial value V0*(0) = y by setting .

2.1 The Financial Market d

00,t =

Ef i _1

t

o

13

d

— i=1

0, t z:, t ,

vt c [0,

.

(2.9)

In real life finance, there is usually a limit of how much loss an investor is willing to tolerate, or in other words, there is a lower boundary on the portfolio value. The class of tame trading strategies incorporates this restriction as the following definition shows. Definition 2.6 (Tame Strategy). A self-financing trading strategy 0 over the time interval [0,2 ] is called tame if 9 V * () > 0,

IF'-a.s., Vt E [0,T] .

The class of tame strategies over [0, T] is denoted by TT. Correspondingly, T = UT
2.1.2 Absence of Arbitrage and Martingale Measures Definition 2.7 (Arbitrage). A trading strategy çb E TT is called an arbitrage opportunity for time T if the value process V(0) satisfies the following set of conditions:

Vo(0) = 0 IP-a.s.,

VT(0) 0 1?-a.s.,

and P (VT(0) > 0) > O.

(2.10)

We say that a market is T-arbitrage-free (arbitrage-free) if there are no arbitrage opportunities in T T (T). An arbitrage opportunity would allow investors to make limitless profits without being exposed to the risk of incurring a loss. In order for the continuous-time market model to be reasonable from an economic point of view, it should be free of arbitrage opportunities. Unfortunately, it is very difficult to check directly if a model has any arbitrage opportunities. However, there is an important necessary and sufficient condition for the model to be consistent with the absence of arbitrage. This condition involves the concept of equivalent martingale measures. Definition 2.8 (Equivalent Martingale Measure). We say that a probability measure Q defined on the measurable space (SI, .F) is a (strong) equivalent martingale measure or a risk-neutral measure if:

/. Q is equivalent to IP, 2. the discounted price process Z* is a Q-martingale.

9 This condition can be weakened: In fact, it is enough to demand that Vt (0) is P- as. lower bounded. See, e.g., Musiela/Rutkowski (1997), p. 235.

14

2 Continuous-time Financial Markets

The set of equivalent martingale measures is denoted by P. A useful criterion in practical applications to test whether a given equivalent measure belongs to P is the observation, that the drift rates relative to the numéraire of all given primary securities under the measure in question must be the same. In particular, if the numéraire asset Z0 is the money market account B with price process Bt = ert, , Vt E [0, where r is the constant risk-free interest rate, the discounted asset prices Z* have an expected instantaneous rate of return of r under the measure Q. This is exactly the return a risk-neutral investor demands from an investment. For this reason, an equivalent martingale measure is alternatively called a "riskneutral measure". In the following we shall use the two terms interchangeably. In our model setting, all equivalent martingale measures QEP can be obtained using Girsanov's theorem (or the Cameron-Martin-Girsanov theorem). Theorem 2.9 (Girsanov's Theorem). Given the standard Brownian mo, tkp,t r, tion W defined on ( ft, P, F), let 71) = : t E [0, T*]1 , Ibt = OP

be a (p +1)-dimensional progressively measurable process satisfying the condition fot tivls
20

is a martingale. Then, under the equivalent probability measure Q defined on = ET., the process W* defined by (Il, .F) with Radon-Nikod#m derivative

g

Wt* = Wt

0.ds,

0

Vt E [0,21,

is a standard p-dimensional Brownian motion. The process W is called the market price of risk. Proof. See Karatzas/Shreve (1988), pp. 193-196. A sufficient condition for E to be a martingale is the Novikov condition: Ep

[exp

1

r

0

Os' tivis }


(2.11)

The existence of an equivalent martingale measure is now sufficient for the market to be free of arbitrage: Theorem 2.10. If there exists an equivalent martingale measure, i.e. P 0,

then the market model contains no arbitrage opportunities in T. Proof. See Korn/Korn (1999), pp. 155-156.

D

2.2 Risk-Neutral Pricing of Contingent Claims

15

A natural question to ask is now whether the converse is also true: does the absence of arbitrage opportunities imply the existence of an equivalent martingale measure. This would yield a fundamental theorem of asset pricing. In a discrete-time model it is in fact possible to prove a fundamental theorem of asset pricing, but the arguments rely heavily on the discrete time setting» In the present continuous-time context the restrictions on trading strategies introduced by the no-arbitrage condition turn out to be not strict enough, and further requirements have to be imposed on trading strategies to establish a continuous-time version of a fundamental theorem of asset pricing. Of course, these requirements should still be meaningful from an economic point of view. Here, we state - leaving aside the details - the well-known result of Delba.en/Schachermayer (1994): Theorem 2.11 (Fundamental Theorem of Asset Pricing). Them exists an equivalent martingale measure for the financial market model if and only if the market satisfies the NFLVR ("no free lunch with vanishing risk") condition. In fact, if there exists an equivalent martingale measure, the model does not contain any arbitrage opportunities in T, and it also satisfies the weaker condition NFLVR. Conversely, if the market satisfies the NFLVR condition, then there exists an equivalent martingale measure. To simplify matters, we shall henceforth say: the existence of an equivalent martingale measure is equivalent to an arbitrage-free market.

2.2 Risk-Neutral Pricing of Contingent Claims 2.2.1 Contingent Claims In practitioners' terms a contingent claim or derivative security is a financial instrument whose value depends on the values of other, more basic underlying variables (referred to as primary securities) - very often the prices of traded assets such as stocks or bonds. More formally, we define:" Definition 2.12 (Contingent Claim or Derivative). A (European) contingent claim (in short: claim) or (European) derivative security (in short: derivative) H with maturity date (or expiration date) T < T* is any nonnegative .FT -measurable random variable representing a payoff such that H <00,

EQ [

1° The first proof of a fundamental theorem of asset pricing in discrete time stems from Harrison/Kreps (1979) and Harrison/Plislca (1981). For an easy to read exposition, see Plislca (2000), pp. 94-95. "See, e.g., BjCirk (1998), p. 78.

16

2 Continuous-time Financial Markets

for all equivalent martingale measures QEP . If H is of the form H = ... , Zr), where 1, is a sufficiently integrable function (I> : Rd -- R, then H is called path-independent, otherwise H is called path-dependent. The function (I) is called the payoff or contract function. 12 One can think of a contingent claim as part of a contract that a buyer and seller make at current time t E [0, T*], in which the seller promises to pay the buyer the amount H at future time T> t. Note that this definition does not cover American-style derivatives (i.e. contingent claims which can be exercised at any time t up to the maturity date T) because the main focus of this work is on equity index derivatives and almost all exchange-traded equity index derivatives are of European-type. Derivative securities might be classified into three different groups: Forwards and futures, options and swaps. During this text we will mainly focus on options, although the techniques developed may be readily applied to forwards, futures and swaps as well. We shall now describe the basic features of these types of derivatives, as far as they are needed for the further understanding. 13 Therefore, let us consider the case d = 1 with the asset Zi being a non-dividend paying stock S." The stock price at the current date t (expiration date T) is denoted by St (ST). A forward contract is a particularly simple derivative. It is an agreement to buy or sell the underlying stock S at a certain future date T for a certain price K. A forward contract is traded in the over-the-counter (OTC) market — usually between two large financial agents (banks, institutional investors, etc.). The agent who agrees to buy the stock is said to have a long position, the other agent assumes a short position. The maturity date T is sometimes called delivery date and the prespecified price K is referred to as delivery price. The forward price F, (T) is the delivery price which would make the contract have zero value at time t, t < T. It is important to distinguish between the forward price and the delivery price. The two are the same when the contract is first entered into but are likely to be different at later times. Since a forward contract is settled at maturity and a party in a long position is obliged to buy an asset worth ST at maturity for K, it is clear that the payoff from the long position (from the short position, respectively) corresponds to the time T contingent claim H (—H, respectively), where H = (I) (ST) = ST — K.

(2.12)

"In the literature, as, e.g., in Pliska (2000), P. 112, it is sometimes distinguished between contingent claims and derivative securities. The term "contingent claim" is then reserved for a European-style derivative, whereas the notion "derivative security" also includes American-style derivatives. "See Hull (2000), pp. 1-10 and Bingham/Kiesel (1998), pp. 2-4. 14 We will often use the term "stock" as a general expression for the underlying asset of a derivative security.

2.2 Risk-Neutral Pricing of Contingent Claims

17

A futures contract is conceptually equal to a forward contract. However, unlike forward contracts, futures contracts are normally traded on an exchange. To make trading possible, the exchange specifies certain standardized features of the contract. The given price is now called the futures price and is paid via a sequence of installments over the contract's life. These payments reset the value of the futures contract after each trading interval - usually a day; the contract is marked to market. In general, the futures price is different from the forward price. However, it can be shown that if Zo is a deterministic process, e.g., in the case that Zo is the money market account and interest rates are a deterministic function of time, then the futures price equals the forward price. 15 Since this is the only relevant case in this work, we no longer distinguish between forward and futures prices. An option is a financial instrument giving the right but not the obligation to make a specified transaction at (or by) a specified future date T at a specified price K. Call options give the right to buy, and put options the right to sell the underlying asset S. The price K in the contract is known as strike price or exercise price and the date T as the option's maturity or expiration. The simplest call and put options are called standard or plain vanilla options. A standard European call option is formally equivalent to the claim H whose payoff at time T is contingent on the stock price ST, and equals H = (I) (ST) = max {ST — K; 0} .

(2.13)

Similarly, a standard European put option is a contingent claim of the form H

= '(ST) = max {K

—

ST;0} .

(2.14)

At a given instant t before or at expiry, we say that a standard call option is in - the - money (ITM) and out- of- the money (OTM) if St > K and St < K, respectively. Conversely, a put option is in-the-money (out-of-the-money) if St < K (S t > K). An option is said to be at- the - money (ATM) if St = K. In addition, the terms near-the-money for options close to being at-the-money and deep out-of-the-money (in-the-money) for options that are far off being at-the-money are frequently used. Instead of the spot price St , the forward or futures price Ft (T) is sometimes used in the above definitions. In addition to standard options, modern options markets offer a wide variety of different contracts, including so-called exotic options. Types include: Asian options, whose payoff depends on the average price over a period, lookback options whose payoff depends on the minimum or maximum price over a specified period, and barrier options, whose payoff depends on some price level being attained or not. Finally, a swap is an agreement whereby two parties undertake to exchange, at known dates in the future, various financial assets (or cash flows) according to a prearranged formula. A swap can always be decomposed into a basket of 15 See,

Hull (2000), pp. 60-62 and 85-86, and Musiela/Rutkowski (1997), p. 86.

18

2 Continuous-time Financial Markets

forwards and/or options. Therefore, to value a swap it suffices to be able to value the individual components.

2.2.2 Risk- Neutral Valuation Formula Assuming that there exists at least one equivalent martingale measure for the market model, we now approach the problem of pricing derivatives. The problem of interest is to determine the time t value of the payoff H. In other words: what is the fair price of claim H at time t that the buyer should pay the seller in order to satisfy both parties. One might suppose that the value of a contingent claim would depend on the risk preferences and utility functions of the buyer and seller, but in a many cases this is not so. It turns out that by the arguments of arbitrage pricing theory there is often a unique, "correct", value of the claim at time t, a value that does not depend on investors' risk preferences. In the remainder of this section, we are going to derive this value. Let us therefore fix an arbitrary equivalent martingale measure Q*EP and an arbitrary maturity date T < T*. Definition 2.13 (Admissible and Attainable Claims). 1. A tame trading strategy 0 E TT is called Qs-admissible for time T if V*(0) is a (2*-martingale. We denote the class of Q* -admissible trading strategies by TT(Q * )• 2. A contingent claim H with maturity date T is called (Q* )-attainable if it admits at least one trading strategy 0 E TT(Q* ) such that

VT(0) = H,

P- a.s.

We call such a trading strategy 0 a replicating strategy for H.16 Now we are ready to present the central idea of arbitrage pricing theory: If a T-contingent claim H is (Q* )-attainable, H can be replicated by a trading strategy 0 E TT(Q*). This means that holding the portfolio and holding the contingent claim are equivalent from an economic point of view. In the absence of arbitrage opportunities, the value process V(0) of 0 replicating H and the arbitrage price processII(H) = Mt (H) : t E [0, T] } of H must therefore satisfy

H(H) = Vt (0).

(2.15)

16 Remark: Strictly speaking, it is not necessary to demand a Q*-admissible strategy to be tame. See Bingham/Kiesel (1998), p. 177. Moreover, we want to emphasize that the definition of an attainable claim depends on the class of trading strategies, but not on the numéraire. It can easily be shown that an attainable claim in one numéraire remains attainable in any other numéraire and the replicating strategies are the same.

2.2 Risk-Neutral Pricing of Contingent Claims

19

Otherwise, an astute investor would buy (sell) the replicating strategy and sell (buy) the contingent claim to make a riskless profit of Illt (H) — Vt(0)1 > O. Naturally, the questions arise what will happen if there exists more than one replicating strategy for H, and what the relation of the arbitrage price process to the equivalent martingale measure is. The following central theorem is the key to answering these questions. Theorem 2.14 (Risk- Neutral Valuation Formula). In the standard market model the arbitrage price process II(H) = fli t (H) : t E [0, 7 ] } of any Q* attainable claim with maturity date T is given by the risk-neutral valuation formula

rit(H) =

H ZO,t 15()* [

4,0,T

Ft]

/

Vt e [0, 7] .

(2.16)

Proof. Since H is (Q*)-attainable, there exists a trading strategy 0 e TT (Qs) with value process V(0) which replicates H, i.e. VT (0) = H. As the discounted value process V*(0) is a Qs-martingale we can write

Vt* (0) = EQ. [ Tr; (95)1 Ft ] , and by the definition of the discounted value process

Vt(0) _ E r VT(0) Ft] Zo,t Q. L _0,T z

.

Using lit (H) = Vt (0), which holds by the absence of arbitrage, the assertion follows. 0 Taking a different perspective, the risk-neutral valuation formula (2.16) was first derived by Ross (1976) and Cox/Ross (1976). Based on the insight of Black/Scholes (1973) that the fair price of an option does not depend on the risk preferences of the individual investors, they assumed investors to be risk-neutral. Under this assumption, (2.16) follows directly. 17 The first mathematically rigorous proof of the risk-neutral valuation formula stems from

Harrison/Pliska (1981). Coming back to the question of uniqueness of the replicating strategy, it is apparent from the risk-neutral valuation formula (2.16) that if the replicating strategy would not be unique, it would be possible to construct an arbitrage strategy. Hence, the replicating strategy is unique, up to indistinguishability of stochastic processes. An apparent drawback of Definition 2.13 is the dependence of the class of admissible strategies on the choice of the martingale measure.' To circumvent 17 Thus the valuation is preference free in the sense that it is valid independent of the specific form of the agents' preferences, as long as they prefer more wealth to less wealth. 18 Note, that the martingale property of the discounted value process is in general not invariant to an equivalent change of a martingale measure. See Musiela/Rutkowski (1997), p. 235, footnote 5. -

2 Continuous-time Financial Markets

20

this problem we generalize our definition of an admissible trading strategy and an attainable claim to: Definition 2.15. A tame trading strategy 0 E TT is called admissible for time T if it is Q-admissible for some Q EP, and H is called attainable, if there exists an admissible strategy 0 replicating it. The next result shows that the arbitrage price process is unique. Theorem 2.16. For admissible trading strategies 01 E TT(Q1) and 02 E TT(Q2) replicating H, we have

H

lit(H) = Zo,tEQ1[ zo,T .Ft] = Zo,t 1E4:22

H

Vt E [0, 7].

Ft]

Proof. See Musiela/Rutkowski (1997), pp. 235-236. 2.2.3 Attainability and Market Completeness

If we wish to price a contingent claim H it is sufficient to find an equivalent martingale measure, but for hedging purposes we are more interested in the replicating trading strategy. The following useful result shows that the existence of a replicating strategy for H is equivalent to the existence of a stochastic integral representation of the claim. Theorem 2.17. Let Q EP be any equivalent martingale measure and H a contingent claim with maturity date T. If the Q-martingale defined by

r

Ht(H) (H) = Li ,t

H I

FAQ

1

Vt E [0, 7 ] ,

admits an integral representation of the form d

t

11; (H) = v +E f

Vt E [0,7'],

(2.17)

i=1 °

with 0i,t progressively measurable and such that fct, 0i,t d4"3 exists for i = 1, . , d, then H is attainable. Proof. See Bingham/Kiesel (1998), p. 180. From relation (2.17) in the above theorem, we see that attainability is closely related to martingale representation theorems. These exist in various degrees of generality. In our model setting, the most important version states that a square-integrable Brownian martingale, i.e. a martingale with respect to a Brownian filtration, can be represented in terms of an It8 integra1. 19 19 See,

e.g., Lamberton/Lapeyre (1996), P. 67.

2.2 Risk-Neutral Pricing of Contingent Claims

21

Provided that an equivalent martingale measure exists, we can use the risk-neutral valuation principle to derive a unique price for every attainable contingent claim. The problem, of course, is that the claim might not be attainable. In this case, it is not clear what the "correct" price should be. In particular there is no reason why it should equal its arbitrage price. We therefore need a convenient method for checking if a contingent claim is indeed attainable. One method was illustrated in theorem 2.17. But there exists an alternative, even more powerful, method. It builds upon the definition of a complete market: Definition 2.18 (Complete Market). A financial market model is called complete if every contingent claim H with arbitrary maturity date T < T* is attainable. Otherwise, the model is said to be incomplete.

The method establishes a connection between market completeness and the number of existing equivalent martingale measures, namely: Theorem 2.19. A financial market model is complete if and only if P consists of exactly one equivalent martingale measure.

Proof. See Harrison/Plislca (1981), p. 241 ff. This extremely powerful theorem tells us that we only have to prove the uniqueness of the equivalent martingale measure to infer the attainability of any contingent claim. To summarize, if the model is complete we know how to price all the contingent claims. On the other hand, if the model is incomplete, we know how to price some of the contingent claims, namely, all the attainable ones. But we do not know how to value the claims that are not attainable. It turns out, however, that we can identify an interval within which a fair, reasonable value for the claim must fall. To derive a unique value, we have to make assumptions on the risk preferences of the agents in the economy. This implies that the valuation of contingent claims is no longer preference-free. Of course this is not desirable from a modelling point of view, explaining partly the popularity of complete-market models, where the price of the contingent claim is unaffected by the agents' preferences. 2° Before we close, let us state an extremely useful rule of thumb, which helps us to judge whether a given model is respectively arbitrage-free or not and/or complete or not. Without proof, we formulate what Bji5rk (1998), p. 106, calls a "meta-theorem": Theorem 2.20 (Meta theorem). Let d +1 be the number of stochastic primary traded assets in a financial market model, and let p+1 denote the number of random sources. In general, we then have the following relations: -

20 See Pliska (2000). For a comprehensive treatment of questions on pricing and hedging in incomplete markets see, e.g., Carr et al. (2001).

22

2 Continuous-time Financial Markets

1. The model is arbitrage-free if and only if d < p. 2. The model is complete if and only if d > p. 3. The model is arbitrage-free and complete if and only if d = p.

3 Implied Volatility

A smiley implied volatility is the wrong number to put in the wrong formula to obtain the right price. Riccardo Rebonato

As a special case of the standard market model, the Black-Scholes option pricing model builds a cornerstone in the theory of modern finance and was the starting point of a whole branch of research concerning the pricing and hedging of contingent claims. In this text, the Black-Scholes model functions as a reference model to all other models. In Section 1, we describe the Black-Scholes assumptions about the financial market, derive the BlackScholes formula for European-style stock options, and state the main option's sensitivities, commonly referred to as the Greeks. Given the market price of an option, the volatility implied by this price can be determined by inverting the Black-Scholes option pricing formula. In Section 2, we give a formal definition of this so-called implied volatility, show how to compute it and discuss possible interpretations under different sets of assumptions. If the Black-Scholes model holds exactly, all options on the same underlying asset should provide the same implied volatility. Yet, empirical implied volatilities differ systematically across strike prices and maturity dates such that the misspecified model produces the correct market prices. Section 3 describes the most important of these implied volatility patterns: the volatility smile, the volatility term structure, and the volatility surface as the combination of the two. In the last section, we discuss the various approaches to price options in the presence of volatility surfaces and assess their individual strengths and weaknesses.

24

3 Implied Volatility

3.1 The Black-Scholes Model 3.1.1 The Financial Market Model Black/Scholes (1973) assume a frictionless security market with continuous trading up to some fixed terminal time horizon T* > 0 (Assumption 1). The uncertainty in the financial market under the objective probability measure is characterized by the complete filtered probability space (SI, F I?, 1F). The filtration is generated by a one-dimensional (i.e. p = 0) standard Brownian motion W = {Wt : t E [0, T* ] 1 which satisfies the usual conditions. The primary traded securities are the money market account and a non-dividend paying stock (i.e. d = 0). ,

Assumption 2. The price process of the money market account is given by the ordinary differential equation (ODE): dB t= rBt dt,

Vt E [0, 71,

(3.1)

where B o = 1 and the continuously compounded interest rate r is supposed to be constant and nonnegative. The solution of (3.1) is Bt = ers ,

Vt E [0, T*I .

(3.2)

In the Black-Scholes financial market the money market account is taken as the numéraire. Assumption 3. The stock price follows a geometric Brownian motion (GBM). Specifically, the dynamics of the stock price process S= {St : t E [0,71} is described by the linear stochastic differential equation (SDE):

dSt = PStdt + vSt dWt ,

Vt E [0,

,

(3.3)

where A E R, y > 0 are deterministic constants and So > 0 is the initial stock price. The coefficient is a constant appreciation rate of the stock price and the coefficient y, referred to as the (stock price) volatility, is interpreted as a measure of uncertainty about future stock price movements.' As volatility increases, the chance that the stock will perform very well or very poor increases. Related definitions stem from Taleb (1997), p. 88 and Natenberg (1994), p. 51. The former states: „Volatility is best defined as the amount of variability in the returns of a particular asset" and the latter describes volatility as a "measure for the speed of the market". More formally, the volatility of a 'See, e.g., Hull (2000), p. 241.

3.1 The Black-Scholes Model

25

stock is often defined as the annualized standard deviation of its continuously compounded returns . 2 Using formula, it is elementary to check that the stock price process given by S St = So exp

–

1

t + yWt ,

Vt E [0, T*I ,

(3.4)

is indeed a solution of (3.3), starting from So at time O. Since '1St and vSt are Lipschitz continuous for all t E [0,11, the solution S is unique, according to a general result of It45.3 It has the following properties: S is Yradapted, S has continuous trajectories, and S is a Markov process. It is apparent from

(3.4) that the continuously compounded (t – u)- period stock returns ln (P-) are normally distributed with mean (.1 – 10) (t – u) and variance y2 (t – u) under the objective measure P for any dates u
ln Su (

N ((iz – –1 v 2 ) (t – u) , v2 (t — U)) . 2

(3.5)

Since the logarithmic stock prices are normally distributed, the stock prices themselves are lognormally distributed. 3.1.2 Pricing and Hedging of Contingent Claims Under the objective probability measure P the discounted stock price process Et* = St IBt = St e', follows the SDE

dEt" =

r) *dt + vSi'dWt ,

Vt E [0,T*].

(3.6)

For S* to be a martingale, the drift term in (3.6) has to vanish. If we let tpt = = to be a process with constant value for all t E [0, 71, then Girsanov's theorem says, that there exists a measure Q equivalent to P under which the process W*, defined as

Wt* = Wt f Ouclu = Wt

V

Vt E [0, T*] ,

(3.7)

is a standard Q-Brownian motion. Clearly, the process Ot satisfies the boundedness condition ,f(; 02„du < oc and also the Novikov condition, since Ot is constant. Under the measure Q, the SDE for the discounted stock price S* becomes

dS:

– r)Si'dt + vS:(dWt*

dt),

= vS;dWt*. 2 See, 3 See

e.g., Roth (1999), p. 7, and the references cited there. Olcsendal (1998), Chapter 5.

(3.8)

3 Implied Volatility

26

This shows that S* is a Q-martingale. In fact, Q is the unique equivalent martingale measure. 4 Using It6's product rule, we find the Q-dynamics of S as dSt = rSt dt + vStdiVt * • (3.9) We see that the appreciation rate tt is replaced by the interest rate r, which justifies the alternative terminology risk-neutral measure. Because Q is the unique equivalent martingale measure, it follows from Theorem 2.10 and Theorem 2.19, that the Black-Scholes market is arbitragefree and complete. This can also be seen from the meta-theorem, since d = p = 0. The completeness of the Black-Scholes market accounts at least for part of the popularity of the model, as was pointed out by Harrison/Pliska (1981), p. 221: "It can be argued that the important and interesting feature of the model ... is its completeness ...". The arbitrage price process of any contingent claim in the Black-Scholes model is obtained from the risk-neutral valuation formula: Theorem 3.1. The arbitrage price process II(H) = {H(H) : t E [0, ID in the -

Black-Scholes model of any contingent claim H with maturity date T < T* is given by the HA-neutral valuation formula

Ht (H) =

H Ft = e -r(T-t) F44:2 [ H I .7 BtEQ [__ ]

-e

]

BT

Vt E [0,T] .

(3.10)

Proof. Since the model is complete any contingent claim is attainable. The rest follows, with Zo,t = Bt = e directly from Theorem 2.14. Consider, for instance, the case of a forward contract on the stock with delivery date T and delivery price K. According to Theorem 3.1, the arbitrage price of this contract H = ST — K at time t, t < T, is:

rit (H) = e—r(T—t)E Q [b

Ki .Ft] -= St — e_(T—t)K.

(3.11)

The strike price K for which the contract value H t (H) is zero at inception is called the forward price Ft (T). Theorem 3.2. The forward price Ft (T) at time t E [0, 7] of a stock S for the

delivery date T in the Black-Scholes market equals Ft (T) = St er(T—t) ,

Vt E [0,T] .

(3.12)

Proof. By definition, Ht (H) = 0. Solving (3.11) for K = Ft(T) proves the claim. It can easily be shown that formula (3.12) is not restricted to the BlackScholes model, but holds in any arbitrage-free model where interest rates are constant. 4 The

proof is a direct consequence of Girsanov's theorem.

3.1 The Black-Scholes Model

27

To derive the explicit form of the replicating strategy for a contingent claim H with maturity T in the Black-Scholes model, we apply the martingale representation theorem. We get

t 14(H) = v + f 01 uciSi *L, , o

Vt E [0 , T] ,

(3.13)

or, in differential notation di-4(H) = (151 ,t dS;' ,

Vt E [0,7] .

(3.14)

We see that we have to hold 01 t units of the stock at time t. Using equation (2.9), we get the number of units of the money market account to hold in t as

t - 0i ,t 57 Sbo,t = v ± f , o = 1-4(H) - 0i,v57 = e -rt (llt(H) - Shi,tSt) ,

(3.15) Vt E [0,7] .

A bit of rearrangement yields VT

(0) = (1)0,TBT + OLTST = (n(H) - 01,TSn = H(H)BT

BT ± OLTST

=H. For 0 = (00 ,01 ) to be a replicating strategy for H we have to prove that 0 is self-financing. With 14 (0) = 1-4(H)Bt we get successively:

dVt (0) = dril (H)Bt ± dBtili'(H) = (bi,tdS;Bt + dBtlIt"(H) = (151,tdSi " B t + dB t (0o,t + 01,tS;')

(3.16)

= çbo ,t dBt + 0 14 (dS;" B t + SciBt )

= 00,tdBt + sbi,tdSt,

Vt E [0,7] .

This shows that 0 is, indeed, self-financing. Moreover, 0 is tame, and such that 14*(0) is a Q-martingale. Hence, we conclude that 0 is the unique replicating strategy for H. 3.1.3 The Black Scholes Option Pricing Formula -

In their original paper, Black/Scholes (1973) focused on the valuation of standard European call options. Recall that the payoff function of this option is given by 4) (s) = max {s - K; 0 } , Vs E R. Applying the risk-neutral valuation formula (Theorem 3.1) to H = (I) (ST) , the famous Black-Scholes formula for European call options on non-dividend paying stocks follows:

3 Implied Volatility

28

Theorem 3.3 (Black-Scholes Formula). The arbitrage price Ct at time t E [0,7 of a European call option with strike price K and maturity date T < T* in the Black-Scholes market is given by the formula ]

Ct = CBS (t,

Vt G [0, Th

Se),

(3.17)

where CBS : [0, T] x R+ -- R is the Black-Scholes call option pricing formula CBS (t, S) --= SN (di (t, 8)) — K e -r(T-t) N (d2 (t, 8)),

(3.18)

with

di(t, s) =

In ( L) + (r -I- 1 v2 ) (T - t) -

K

-‘,/, d2(t, S) --.-- di(t, 8) — 217

7

ii7 V .

(3.19) and

z n(x)dx,

N(z) = f

.2 1 n(x) = —e --r,

Vx, z E R.

N/T7T

-CO

Here N(z) stands for the cumulative distribution function and n(x) for the probability density function of a standard normal random variable. Proof. According to the risk-neutral valuation formula, the Black-Scholes price process is given by Ct = e -r(T-t)EQ [ max { ,97, _ K; 0 } 1 Fe]. Evaluating the expectation using

1

St = So exp { (r - y 2 ) t + ylVt*} ,

Vt E [0,T] ,

(3.20)

yields, after some algebra, the Black-Scholes formula. For a full description of the calculations see, e.g., Bjork (1998), pp. 88-89. 0 Occasionally, it will be convenient to explicitly account for the dependence of the option's price on some or all of the parameters K,T,r and v. For example, to stress the dependence of the Black-Scholes price on the volatility y, we write CBs(y). The arbitrage price of a standard Europeanput option with payoff function (I) (s) = max {K - s; 0} for all s E R+ can easily be obtained using the put-call parity. This is a general, model-independent relationship between European call and put option prices that follows from simple no-arbitrage arguments. Theorem 3.4 (Put Call Parity). Provided that the money market account is modelled by (3.1), the arbitrage prices Cit and Pt of standard European call and put options on a non-dividend paying stock with the same expiry date T and strike price K satisfy the put-call parity relationship -

Ct

for every t E [0,7].

—

Pt = St —

(3.21)

3.1 The Black-Scholes Model

29

Proof. See Musiela/Rutkowski (1997), P. 123, and Hull (2000), pp. 174-175.

From Ct = CBS (t, St) and the put-call parity relationship (3.21), the Black-Scholes arbitrage price Pt of a European put option with strike price K and maturity T at time t E [0, T] follows: Pt = Pas (t, Se)

where PBS

Vt E [0, Tb

(3.22)

[0,7] x R + —> R is the Black-Scholes put option pricing formula: PBS (t7

S)

= Ke — r(T—t) N(—d2(t,$))— sN(— di (t,$)).

(3.23)

The functions di (t, s) and d2(t, s) are defined as in (3.19). Since in typical situations it is not difficult to find a proper form of the put-call parity, we shall usually restrict ourselves to the case of a call option. Originally, the Black-Scholes formula was derived by introducing a continuously rebalanced risk-free portfolio containing an option and underlying stocks. In the absence of arbitrage, the instantaneous return from such a portfolio needs to be equal to the risk-free rate. 5 This property leads to a partial differential equation, which is then solved for the price process of the option. More specifically, if H = (I) (ST) is a path-independent contingent claim, then the arbitrage price process is also given by H t (H) = f (t, St ), where f solves the (parabolic) Black-Scholes partial differential equation (PDE):6

af(t,$)

Of(t,$) 2 2 (92 f(t, rs as + 2 v s 0.92

S)

r f (t, s) = 0, V(t, s) E (0, T) x (0, oo) (3.24)

with terminal condition

f (T, s) = (1)(s).

(3.25)

In particular, solving (3.24)-(3.25) for (Ks) = max {s — K; 0} yields the BlackScholes call option pricing formula CBs(t, s). Theoretically, the connection between the risk-neutral valuation formula (3.10) and the Black-Scholes PDE for claims of the form H = (I) (ST), i.e. for path-independent claims, is established by the Feynman -Ka e stochastic representation formula, which basically expresses the solution of a parabolic PDE as the expected value of a certain functional of a Brownian motion. 7 3.1.4 The Greeks To assess the risk of an option's position, we will now examine the impact of the option's underlying (risk) factors on its price. The Black-Scholes option 6 1t appears, however, that the risk-free portfolio does not meet the formal definition of a self-financing trading strategy. See Musiela/Rutkowski (1997), p. 109. 6 The Black-Scholes PDE is a so-called Cauchy problem. For further details, see Musiela/Rutkowski (1997), pp. 124-129. 7 See Bj6rk (1998), pp. 58-60.

30

3 Implied Volatility

values of standard call and put options depend on the current time, the underlying stock price, the volatility, the interest rate, the maturity date, and the strike price. The sensitivities of the option price with respect to changes in the first four factors are commonly referred to as the Greeks. Each Greek measures a different dimension of the risk in an option position. At a first glance, the sensitivity of the option price to fluctuations in the model parameters volatility and interest rate seems self-contradictory, since a model parameter is by definition a given constant, and thus cannot change within a given model. In fact, these Greeks measure the sensitivity of the option price with respect to misspecifications of the model parameters. Denoting by Cgs = CBs(t,S, K,T,r,v) and PBS = PBS (t, K,T, r, v) the Black-Scholes pricing function of a standard European call option and a standard European put option, respectively, we can determine the Greeks by taking partial derivatives. The most common Greeks (or more precisely Greek functions) for call options are OCns

as

N(d 1 ) > 0

("Delta"),

(3.26)

82 CBS = 0 ("Gamma"), 052 t acBs s > 0 ("Vega"), Ov sn(di)v OCBs r K e - r(T-t) N (d2) < O ("Theta"), at 21,/7 t OCBs = K(T - t)e - r(T-t) N (d2) < 0 ("Rho"), Or where d1 = d1 (t, s, K, T, r, y) -

ln(*)

+ (r + ly2 ) y

(T - t)

t

d2 = d2(t,s,K,T,r, y) =d1 - v and t E [0, 7]. Similarly, in the case of a put option we get: OPBS

Os 492 PB s

= N(C11) — 1 G

aPBs

("Delta"),

n(di)_ 02 C

8.52 sv OP.6.3 =

av

0

s-Vn(di 7 )

sn(cli)v

0 sa2c

(3.27) ("Gamma"),

> 0 ( "Vega" ),

rice_ r(T-t) iv,„.

(_d2)

("Theta"),

- 2 °Pas = K(T - t)e - T (T-t) (N(d2 ) - 1) < 0. ("Rho"). Or Ot

3.1 The Black-Scholes Model

31

The delta of an option is defined as the rate of change of the option price with respect to the price of the underlying stock, when all else remains the same. At any time t, it gives the number of shares 01, , in the replicating portfolio for the option. Therefore, the option's delta is also called "hedge ratio". 'When the stock price changes, the delta changes, too. This effect is captured by the option's gamma, the second partial derivative of the option price function with respect to the stock price. The vega (also known as lambda) of an option measures the rate of change of the option price compared with the change in the underlying's volatility. Similar statements hold for the Greeks theta and rho. Theta is also sometimes referred to as the (deterministic) time decay of the option. In contrast to the theta of a standard call option which is always positive, the theta of a put option may become negative. Yet, in practice, such a case hardly ever occurs. Compared with the other Greeks, rho is small in magnitude, and is therefore often neglected. When we relax the assumption of a constant volatility in the later chapters, two further Greeks will become important. Following the nomenclature of Taleb (1997), p. 200, we define:

92 CBS

at,2

= .97 Nr'tn(di)

I92 CBS

d2

°say

y

di d2

=—n(c/i) —

a2 PBS at,2

a2 PBS

("DVegaDVol" ),

( "DDeltaDVol" ).

asav

DVegaDVol (DDeltaDVol) corresponds to the change in vega (delta) resulting from a change in volatility. For convenience, we introduce the following standard notation for the Black-Scholes call option Greeks: Definition 3.5. We define OCBs

&CBS

03

OCBs

acBS

as2

ABS

OV

OBS =

at ,

and VBS

a2 CBS

a2 CBS

(91) 2

080V

The Black-Scholes PDE (3.24)-(3.25) can be used to obtain the relation between the Greeks. In the case of a standard European call option, we have: 1 2

et ± rSt 6e + _ v 2 s2,-,t t

rCt

(3.28)

where we have assumed a priori that the arbitrage price of the call option at time t equals Ci = CBs(t, St). The Greeks at time t are given by (it = 8Bs(t,St), et = OB s(t,St ), and rt = rBB(t, St ). This relation also holds for a portfolio, when the portfolio value can be expressed as a function of time and stock price only. The portfolio can thus consist of a position in the

32

3 Implied Volatility

underlying stock itself, as well as positions in various forwards, futures and (path-independent) options written on the underlying stock. The portfolio Greeks are then obtained as the weighted sums of the Greeks of the portfolio components. In general, a portfolio which is insensitive with respect to small changes in one of the risk factors is said to be neutral or hedged with respect to this factor. Formally, this means that the corresponding Greek equals zero. For example, a portfolio is called delta-neutral or delta-hedged if the portfolio delta is zero. If we consider a stock with delta 1 and a call option on this stock with delta 0.5, delta neutrality can be achieved, among other things, by buying two call options and selling one stock. A portfolio which has a positive (negative) sensitivity with respect to small changes in one of the risk factors is said to be long (short) with respect to this factor. For example, a long call is said to be delta long or long delta.

3.2 The Concept of Implied Volatility 3.2.1 Definition The Black-Scholes formula relates the price of an option to the current time, the underlying stock price, the volatility of the stock, the interest rate, the maturity date, and the strike price. All parameters other than the stock's volatility can be observed directly in the market. Given that these parameters are known, the pricing formula relates the option price to the volatility of the underlying stock. If one may observe the market price of the option, then the volatility implied by the market price can be determined by inverting the option pricing formula. This volatility is known as the implied volatility. 8 Definition 3.6 (Implied Volatility). Let Ct (K,T) be the market price of a standard European call option with strike price K > 0 and maturity date T at time t E [0, T). The (Black-Scholes) implied volatility crt (K,T) is then defined as the value of the volatility parameter which equates the market price of the option with the price given by the Black-Scholes formula (3.18):9

Ct (K,T)= CBs(t,S t ,K,T,r,cr t (K,T)).

(3.29)

In fact, this definition can be somewhat misleading these days, since option traders often quote implied volatility directly, and then calculate the option's market price implied by this volatility quote. 1 ° 8 See Mayhew (1995), p. 8. 8 For ease of notation, we will use the symbol at(K,T) for both, the implied volatility of an option with fixed strike K and fixed maturity T and for the implied volatility function with respect to strike price and maturity. 1° In the OTC options markets it is very common to quote option prices in terms of their implied volatilities. On the other hand, for exchange-traded options typically the option's price is quoted.

3.2 The Concept of Implied Volatility

33

Discussions involving implied volatilities will typically also incorporate two other notions of volatility with very different meanings — instantaneous and realized volatility. Whereas implied volatility is derived from the market price of an option, the concepts of instantaneous and realized volatility are based on the price process of the underlying stock. The term instantaneous or actual volatility refers precisely to the volatility that appears in the SDE describing the evolution of the underlying asset. In general, it cannot be observed directly. Consider, for example, the BlackScholes model with stock price dynamics: dSt = p,St dt + vStdWt• Here, the constant y is the instantaneous volatility of the stock price. In a more general setting, the instantaneous volatility might also vary over time, either deterministically or stochastically. If there is no chance of confusion, we will simply refer to the instantaneous volatility as the "volatility". Historical data may also be used to estimate the volatility parameter, which can then be used to compute the theoretical option values. The most natural approach uses an estimate of the standard deviation based upon the ex-post continuously compounded stock returns measured over a specific sample period in the past. This estimate is usually called realized or historical volatility: 11 Definition 3.7 (Realized Volatility). Let us assume that we can observe the stock price process S under the objective measure P at N* + 1 equidistant points in time to, t 1 , , tN, where At denotes the length of the sampling t,,-1, n = 1, . . . , N* . The stock price at time tt, is interval, i.e. At = tn denoted by S. Then the realized volatility (or historical volatility) I', (N) of the stock at current time tti for the last N periods, N < n < N*, is defined as the annualized sample standard deviation of the continuously compounded st stock returns Rtn = ln ( —n— stn _ i :

\I At (N —1)

(Rt

i — Rtn(N))2 , — +1

(3.30)

where R tn (N) is the N-period sample mean at time tn Rin (N)

= — E Rtn-i+1 • i=1

Under the assumptions of the Black-Scholes model, realized volatility (and the sample mean) is independent of n, i.e. Ytn (N) = Y for all n, N < n < N*. b ee e.g., Hull (2000), p. 242. For an in-depth discussion of realized volatility, "See, see Figlewski (1997).

34

3 Implied Volatility

Statistically, 'Y it is a consistent estimator of the constant volatility parameter y. 12 Its standard error can be shown to be approximately i'')/./V. A word on terminology: Let us assume that the current date is to. Then the implied volatility at (K,T) at time t is more precisely called the spot implied volatility, if t = to, the past or historical implied volatility, if t < to, and the future implied volatility, if t > to . The same terminology applies to all other types of volatility used throughout, except for realized (or historical) volatility, which is, by definition, related to the past. 3.2.2 Calculation

Since the Black-Scholes call pricing formula CBS (y) = CBS(t, St, K ,T, r, y) is a continuous - indeed, differentiable - increasing function of y, with boundaries" liMCBS(V) = v-40

- Ke - r(T- t)

t

o

if St > K e- r(T-t)

if st < Ke -r(T-t)

(3.31)

liM CBS(V) = St)

V—.00

the inverse function exists. 14 Therefore, a unique positive implied volatility a-t (K,T) can always be found. Unfortunately, there is no closed-form solution for implied volatility, although the Black-Scholes formula is given in analytic form. Instead one has to use numerical methods to find the implied volatility. Perhaps the most commonly used method in this context is the Newton-Raphson iteration procedure. If the transcendental equation

Ct(K,T) = CBS (V) is to be solved for y, and vk is the current approximation, then the next approximation V k+1 is given by the Newton-Raphson formula CBS (Vk) — Ct(K,T) Vki-1 = Vk

8C25(v) v lv—vk

(3.32)

of the Black-Scholes formula with respect to where the derivative acBs(v) av the volatility parameter - the option's Black-Scholes vega ABs(v) - is always positive. The iteration step (3.32) is repeated until convergence to the desired 12 Notice that realized volatility is a biased estimator of true volatility y, but the bias disappears asymptotically. On the other hand, realized variance 1D2 is an unbiased estimator of true variance y 2 . For a definition of consistency and unbiasedness, see Kmenta (1997), pp. 156-169. "See Cox/Rubinstein (1985), p. 216. 14 See KOnigsberger (1995), p. 30.

3.2 The Concept of Implied Volatility

35

accuracy is achieved (typically only a couple of iterations). 15 Alternatively, implied volatility can be computed using the simple approximation formula of Corrado/Miller (1996). It provides the implied volatility of an option in closed-form across a wide range of underlying prices. In the absence of arbitrage and market frictions, the implied volatility of a European put option and a European call option with the same strike price and the same maturity date must coincide if the underlying stock pays no dividends. 16 To see this, suppose that, for a particular value of y, Cgs (t, St, K,T,r,v) and PR s(t, St, K,T,r,v) are the time t Black-Scholes values of European call and put options with strike price K and maturity date T. Because put-call parity is satisfied for the Black-Scholes model we must have Cgs (t, St, K,T,r,v)

—

PB s(t, St , K,T,r,v) = St

—

Ke — r(T—t) .

(3.33)

Since it also holds for the market prices Ct (K,T) and Pt (K,T):

Ct (K,T)

—

Pt (K,T)= St

—

Ke — r(T—t) ,

(3.34)

we get by subtracting (3.34) from (3.33)

CBs(t,S t ,K,T, r, y)

—

Ct (K,T)= PBs(t,St ,K,T,r,v)— P t (K,T).

(3.35)

Equation (3.35) shows that the absolute pricing error when the BlackScholes model is used to price a European call option with strike price K and maturity date T must be equal to the absolute pricing error when it is used to price the corresponding European put option. Let us now suppose that the implied volatility of the call option is at (K,T). This means that the left-hand side of equation (3.35) becomes zero when y = at (K,T) in the Black-Scholes formula. From equation (3.35), it then follows that Ps(K, T) = PBS(t, St, K,T,r,at (K,T)). This argument shows that in the case of a standard European option it should not matter whether one uses a call price or a put price to back out implied volatility. 3.2.3 Interpretation The interpretation of implied volatility depends on whether the Black-Scholes model provides a good description of reality or not. Let us first consider the case where the Black Scholes model holds. If it holds exactly, the volatility implied in an option's market price is widely regarded as the (subjective) -

15 Another frequently used method to derive the implied volatility is the bisection method. For background information on the Newton-Raphson and the bisection method, see Press et al. (1992), Sections 9.1 and 9.4. "See Carr (2000), p. 9 and Hull (2000), pp. 435-436.

3 Implied Volatility

36

market's expectation of the future constant volatility y of the stock's continuously compounded returns. 17 Relaxing the assumption of a constant volatility and assuming instead that volatility is time-dependent, i.e. the stock's actual volatility is a deterministic function y : [0, T1 R + of time only, it can be shown that the Black-Scholes formula is still valid when using the average volatility D(t,T) over the remaining life of the option T — t, i.e.

I

D(t,T) — \ I T 1 t ) T V 2 (u du

(3.36)

in place of y. 1 8 In that case, implied volatility is interpreted to be the market's expectation of the future average volatility £3(t, T).' 9 Now, if capital markets are informationally efficient 20 , implied volatility should be an efficient forecast of future (average) volatility. This means that implied volatility should subsume all information in the market information set, especially the historical price record of the stock, that can be used to predict the future volatility of the underlying stock. 21 On the other hand, if the Black-Scholes model does not hold, the logical foundation for the belief that implied volatility is the efficient market's forecast of the future (average) volatility may be rather weak. For example, if the actual volatility is a deterministic function of time and stock price or if it is stochastic, implied volatility can generally not be interpreted as a time-averaged volatility — not even as a true volatility measure. 22 Although implied volatility may, in fact, have little to do with the stock's actual volatility, it is nevertheless a highly regarded number by option traders and other market participants, but "See Christensen/Prabhala (1998), Figlewski (1997), and Mayhew (1995), among others. result was first shown by Merton (1973). "Note that the term "average volatility" can be somewhat misleading, since the quantity defined in (3.36) does not in general coincide with the value fT v(u)du, as one would normally expect from the name. However, this term is so common that we will use is here, too. "Financial economists have a strong belief that capital markets are infor?nationally efficient, at least in their weak form. In an efficient market, prices fully and instantaneously reflect all available relevant information. The concept of market efficiency was mainly developed by Fama (1970). For further details, see Copeland/Weston (1988), pp. 330-356. 21 More formally, implied volatility is called an efficient forecast of future volatility, if the forecast error is a white noise process that is uncorrelated with any variable in the market's information set. See also Christensen/Prabhala (1998). 22 0ne exception is the stochastic volatility model of Hull/White (1987). Assuming that actual volatility follows a GBM and volatility risk is unpriced in the market, Hull/White (1987) have shown that the option price is the Black-Scholes price integrated over the distribution of the average volatility. For a thorough discussion of the interpretation of implied volatility as a future average volatility under different models for the stock price, see Lee (2002). 18 This

3.2 The Concept of Implied Volatility

37

for quite different reasons. 23 For them, implied volatility is a measure of an option's price or as Lee (2002) points out "... a language in which to express an option price", one that controls for option-specific characteristics such as the strike price, the maturity, etc. The translation of option prices into implied volatilities brings the advantage of eliminating a substantial amount of nonlinearity. 24 Implied volatility is also interpreted as an implicit parameter which embodies all deviations from the Black-Scholes assumptions. If the BlackScholes assumptions are wrong and implied volatility cannot be interpreted as a volatility forecast, it is worth stressing that there is nothing special about using the Black-Scholes formula in the definition of implied volatility. The Black-Scholes formula is then just a convenient and well-known mapping from option prices to implied volatilities. Other functions with similar properties as the Black-Scholes formula will work just as wel1. 25 The question whether implied volatility efficiently predicts future volatility has been the subject of a vast number of empirical studies. 26 In these studies, typically a weighted average of implied volatilities, rather than the implied volatility of a single option, is used as a point estimate for the future volatility. 27 Early papers, e.g., Latane/Rendleman (1976), examine the informational content of implied volatilities in a cross-sectional setting. These papers essentially document that stocks with higher implied volatilities also have higher ex-post realized volatility. The time series literature on the predictive power of implied volatilities has produced mixed results. On the one hand, Canina/Figlewski (1993), for example, find for S&P 100 index options that implied volatility has little or no correlation at all with future return volatility. On the other hand, Christensen/Prabhala (1998), using the same data set, come to the conclusion that implied volatility is a useful source of information to predict future volatility, although it fails to be efficient. In particular, they find that historical volatility has no incremental explanatory power over implied volatility in some of their specifications. Although there is still a lot of discussion, the general conclusion to be drawn from this large body of research is that the informational content of implied volatility goes beyond that of past return data. 28 However, implied volatility fails to be an efficient forecast of future volatility. Moreover, it tends to be biased. this context, the term "implied volatility" is somewhat misleading. Rosenberg (2000), pp. 51-52. 25 In principle, every bijective function of the option price with respect to volatility is suitable. See also Ledoit et al. (2002). 26 For a systematic overview on tests of the Black-Scholes model, see Bates (1996b). See also Mayhew (1995). 27 The reasoning behind this approach is as follows: If the Black-Scholes model is correct, market microstructure effects such as price discreteness and nonsynchronous trading, causing implied volatilities to differ across options, represent noise, and noise can be reduced by using more observations. For an in-depth discussion of the different weighting schemes, see Bolek (1999), pp. 134-143. 28 See Mayhew (1995), p. 13. 23 In

24 See

38

3 Implied Volatility

If the Black-Scholes model holds exactly all options on the same underlying asset should provide the same implied volatility. Yet, as is well-known, empirical implied volatilities differ systematically across strike prices and across maturity dates such that the misspecified model produces the correct market prices29 or as Rebonato (1999), p. 78, puts it: "...implied volatility is the wrong number to put in the wrong formula to obtain the right price." This finding delivers the most striking evidence against the Black-Scholes model and the interpretation of implied volatility as an efficient forecast of future volatility. 30

3.3 Features of Implied Volatility In this section we describe some well-known patterns in the behavior of implied volatility as the strike price and the maturity date of the option change. 31 explanations for the existence of these patterns are given in the nextPosible section. 3.3.1 Volatility Smiles The most often quoted phenomenon testifying to the limitations of the BlackScholes model is the smile effect: that implied volatilities vary with the strike price of the option contract. Formally, we define the volatility smile as follows: Definition 3.8 (Volatility Smile). For any fixed maturity date T, T < T*, the function crt(K,-) of implied volatility against strike price K, K > 0, is called the (implied) volatility smile or just smile (for maturity T) at date t E [0, T). Henceforth, we shall use the term "volatility smile" to denote both the implied volatility function at (K, -) with respect to K and its graphical representation. Before the 1987 crash, implied volatilities in equity options markets were, in general, nearly symmetric around the prevailing underlying price, with ITM and OTM options having higher implied volatilities than ATM options. This corresponds with a "U-shaped" form - a smile shape - in the plot of implied volatility against strike (left graph of Figure 3.1). A skew or sneer pattern, however, is more indicative of the pattern since the crash - at least for longer term index options - with implied volatilities decreasing monotonically as the strike price rises (right graph of Figure 3.1). The skew curve tells us that there is a premium charged for OTM puts and ITM calls above their "See, e.g., Mayhew (1995), p. 14. 30 The

Black-Scholes model has been tested in a number of empirical studies. For a summary of the main results, see Hull (2000), pp. 448-450. 31 In the following, see Alexander (2001a), Chapter 2, Dumas et al. (1998), and Rebonato (1999), pp. 83-87.

3.3 Features of Implied Volatility

39

Black-Scholes price computed with the ATM implied volatility. Conversely, for OTM calls and ITM puts there is a premium received. In this work, if not stated otherwise, the term "smile" is used as a general expression for the shape of the implied volatility pattern across exercise prices. It covers a literal smile as well as skew.

// /

OTM puts ITM calls

......____

ATM

Implied volatility

Implied volatility

ATM

OTM puts FTM calls

OTM calls fl-M puts

OTM calls ITM puts

Strike price

Strike price

Figure 3.1. Ideal types of volatility smiles. Left graph: literal smile; right graph: skew

Other qualitative features of volatility smiles for equity index options are that smiles are not constant but vary over time and that smiles are more pronounced for short-dated options when expressed in the strike price K ("flattening-out effect") •32 3.3.2 Volatility Term Structures Next, we consider the relationship between implied volatility and maturity date for a fixed strike option. Definition 3.9 (Volatility Term Structure). For any fixed strike price K, K > 0, the function ci t e ,T) of implied volatility against maturity T, T < Ta, is called the term structure of (implied) volatility or (implied) volatility term structure (for strike K) at date t E [0, T). Again, the notion "term structure of volatility" stands for the function at (., T) as well as for its plot. The strike price K is usually chosen to be the ATM strike. Unless otherwise stated, we will follow this convention. In analogy to the terminology used in interest rate markets, the term structure of volatility is called normal, if implied volatilities for options with longer maturities are higher than those for options with shorter maturities. 3 2 See, e.g., Dumas et al. (1998). For a detailed analysis of the empirical properties of the strike profile of DAX option implied volatilities, see Section 5.3.

3 Implied Volatility

40

Implied volatility

Implied volatility

Conversely, we speak of an inverse shape, if short-dated options have higher implied volatilities than longer-dated options. The term structure is called flat, if the plot of implied volatility against maturity is a horizontal line. The three basic shapes are illustrated in Figure 3.2.

Maturity date

Maturity date

Maturity date

Figure 3.2. Basic shapes of the volatility term structure. Upper left graph: normal shape; upper right graph: inverse shape; lower graph: flat shape The concept of forward implied volatility is introduced next: 33 Definition 3.10 (Forward Implied Volatility). Given the term structura of volatility at(., T) (for some K) at time t, where cr?(.,T2)(T2 — t) > 01(.,To (Ti - t), the forward implied volatility Grr (T1 , T2) between dates T1 and T2 t
ar(71,7,2 ) = 101(. ,T2) (T2 — t) — 01(.,Ti) (T1 — t) T2

(3.37)

If the underlying volatility is at most time-dependent, then the forward implied volatility cri(Ti, T2) is equal to the average volatility U(T1, T2) over that period. The notion of forward implied volatility crr(T1 , T2) should not be confused with the concept of future implied volatility 0 T1( . , T2). Whereas at time t the forward implied volatility is known, the future implied volatility is not. In 33 See

Wilmott (1998), p. 290.

3.3 Features of Implied Volatility

41

fact, cr2 (-,T2 ) is an .FT,- measurable random variable. The collection of forward implied volatilities for different dates T1 and/or 7'2 is called the forward -1

(implied) volatility curve. 3.3.3 Volatility Surfaces

Volatility surfaces combine volatility smiles with the term structure of volatility to tabulate the implied volatilities appropriate for market consistent pricing of an option with any strike price and any maturity. Definition 3.11 (Volatility Surface). For any time t E [0, T*], the function crt : (0, oo) x (t, T*] —+ R± , which assigns each strike price and maturity date tuple (K, T) its implied volatility a t (K,T) is referred to as the (implied) volatility surface (or volatility matrix). A typical illustration of a volatility surface is given in Figure 3.3 in the case of options on the German stock index DAX.

-I

ena

e•W•41.

Figure 3.3. Volatility surface for DAX options as of August 1, 2000. DAX price:

7145 index points

42

3 Implied Volatility

The volatility surface o-t (K,T) will, in general, be a stochastic quantity with three variables, t, K and T, and, for each outcome in the underlying sample space, the dependence upon these variables will be different: •

•

For a fixed time t, at (K,T) is a function of K and T providing the implied volatilities or equivalently, the market prices, at the fixed time t for options of all possible strike prices and maturity dates. For a fixed strike K and fixed maturity T, cr t (K,T) (as a function of t) will be a scalar stochastic process. This process gives the implied volatilities or equivalently, the market prices, of the option with fixed strike K and fixed maturity T.

Empirically it is often advantageous to reexpress the volatility surface in terms of moneyness and time to maturity, defined as r = T - t. Definition 3.12 (Moneyness). Let m(t,s,K,T,r) be a function of time, underlying price, strike price, maturity date, and interest rate. Then the moneyness Mt at time t E [O, T* ] is generally defined as

A = m(t,S t ,K,T,

r).

(3.38)

The function m is referred to as the moneyness function. It is required to be increasing in K. For convenience, we suppress some or all of the arguments of the function m. If there is no chance of confusion, we will also suppress the subscript t and simply write M for the moneyness A at time t. The choice of the adequate moneyness metric is mainly an empirical issue and depends strongly on the application under consideration. The most simple measure of moneyness is the fixed strike moneyness M = K. When quoting option prices, traders often use the following moneyness measure: -

K M = —. St

Other measures of moneyness will be discussed in Section 5.3.1. When expressed in terms of M and T, we denote the volatility surface by at (M, r), where the implied volatility function is defined as at : R x (0, T* t] --> R+ . In contrast to the absolute volatility surface at (K,T), the function -cit (M, T) is sometimes called the relative volatility surface. Note that there is a one-to-one correspondence between crt (K,T) and at (M, T) of the form

r) = at (m - i(m),t

+ r),

(3.39)

with m -1 (M) being the inverse function of m with respect to M. Henceforth, we shall no longer strictly distinguish between the volatility smile at (K, .), the term structure of volatility a t (-, T) and the volatility surface crt(K , T). Instead, we will use the symbol a- t (K , T) to represent all three. The same applies to Frt (M, r).

3.4 Modelling Implied Volatility

43

3.4 Modelling Implied Volatility 3.4.1 Overview The Black-Scholes model has been very popular among practitioners over decades, although the existence of volatility surfaces, is in clear conflict with the model's assumptions. In reality, traders have always been very well aware of these shortcomings of the model, but have been so loath to relinquish its simplicity, robustness and intuitional appeal that they have preferred to account for all its imperfections and inadequacies by means of a skilful, often ad-hoc, "adjusting" of the volatility parameter. 34 Although this practice might be acceptable for some problems, such as the pricing of standard options, it is completely inadequate for others as, e.g., the hedging of standard options or the pricing of exotic options. For example, while a European option of arbitrary strike and maturity could clearly be valued using a volatility interpolated from observed implied volatilities, it is far from clear which volatility is appropriate to plug into the Black-Scholes model for pricing a barrier option: the implied volatility at the exercise price, barrier, or at some other value. 36 The search for an option pricing model which is theoretically consistent with the observable implied volatility patterns has brought on two different modelling approaches (see Figure 3.4). The first approach attempts to identify and model the financial mechanisms that give rise to volatility surfaces, in particular to volatility smiles (left branch of Figure 3.4). Within this approach, implied volatility is an endogenous variable. In general, smile patterns may be either due to market frictions (violation of Black-Scholes assumption 1) or to deviations of the underlying stock's price process from a GBM (violation of Black-Scholes assumption 3). 36 The assumption that the stock price process follows a GBM, or equivalently, that logarithmic stock returns are normally distributed is in conflict with empirical research which has accumulated convincing evidence for "fat-tailed" return distributions. 37 Moreover, the underlying asset distribution often turns out to be asymmetric. If it is positively skewed, there is more probability mass on the right side of the distribution than on the left, and vice versa. Three possible explanations for these empirical properties have been offered: 38

34 See Rebonato (1999), p. 73. In this context Taleb (1997), p. 109, also calls the Black-Scholes model "an almost nonparametric pricing system". 35 See Overhaus (1999), p. 8. 36 See Mayhew (1995), p. 14 and Section 3.1.1. 37 By a "fat-tailed" distribution we mean a distribution where extremely low or high returns have greater probability than assigned by the normal distribution. All moments are assumed to exist. For an alternative discussion of fattailed distributions with important implications for capital market theory, see Bamberg/Dorfleitner (2002). 38 See Gemmill (1993), p. 113.

44

3 Implied Volatility

1. The volatility of the underlying stock varies over time, either deterministically or stochastically. 2. The price process of the stock exhibits jumps. 3. The price process is continuous, but the innovation term shows a fat-tailed distribution. Time-varying volatilities present a violation of the parametric assumption of the Black-Scholes model of a constant volatility; the occurrence of jumps shows a violation of the Itt3 process assumption; finally, if prices move smoothly, but not according to a GBM, the distributional assumption of the Black-Scholes model is violated.

Identify and model financial mechanisms that give rise to volatility surfaces

Take implied volatility as primitive and model it directly

Implied volatility is

Potential are

CaLI1108

Market frictions (BS assumption 1)

Stock price does not follow a GBM (BS assumption 3)

Distributional Process assumption assumption is

Static

P8111MefriC

assumption

is violated

violated

is violated

Alternative distributions for the innovation term

Jumps

Deterministic or stochastic volatility

Risk-neutral density approach

Dynamic

Deterministic

Stochastic

Figure 3.4. Overview over volatility smile models

Over the last years the liquidity in the market for standard options, especially in the area of equity index options, increased rapidly. This has had

3.4 Modelling Implied Volatility

45

two major consequences: 39 First, there is no more need to theoretically price standard options. The market's liquidity ensures fair prices. Second, hedging of standard options becomes less important as positions can be unwound quickly. These developments have given rise to a second modelling approach (right branch of Figure 3.4). In directly taking as primitive the implied volatility (surface), this approach is usually referred to as a "market-based" approach." Within this approach, implied volatility is an exogenous variable. Depending on the assumptions made, we can, at the top level, distinguish between static and dynamic models of implied volatility. Static models only use the information that can be obtained from the current cross-section of standard option prices. They make, however, no assumption on the dynamics of implied volatility. On the other hand, dynamic implied volatility models explicitly model the evolution of implied volatilities, either in a deterministic or in a stochastic way. The only representative of the class of static implied volatility models considered here is the risk-neutral density method. In the following discussion, we will elaborate more on these models — sometimes called volatility smile models — that have been proposed to account for the observable volatility patterns.

3.4.2 Implied Volatility as an Endogenous Variable Deterministic Volatility

Derman/Kani (1994b), Derman/Kani (1994a), Dupire (1994), and Rubinstein (1994) were the first to model volatility as a deterministic function v : T*1 x R+ R+ of time and stock price, which is usually referred to as the local volatility. 41 Their work has subsequently been extended by Chriss (1996), Jackwerth (1997), Andersen/Brotherton-Ratcliffe (1998), among others. Within the local volatility framework, the dynamics of the stock price under the martingale measure Q are given by: dSt St

= r(t)dt v (t, St) diVt,

(3.40)

where the interest rate r : [0, 71 R+ is a deterministic function of time. At the current time, say to, the local volatility v(t, St ) at future time t > to is unknown since the future realization of the stock price St is not known. Contingent on the value of St at time t, however, the local volatility is uniquely

See Schônbucher (1999). approach is similar to the approach of HJM in the field of interest rates. See Heath et al. (1992). 41 The graph of the local volatility function v(t, St) and sometimes the function itself — is commonly termed the local volatility surface. See, e.g., Taleb (1997), p. 39

40 This

—

164.

46

3 Implied Volatility

determined. In this sense, y is a deterministic function of a stochastic quantity, the stock price.' Calibration of this type of models requires determining the local volatility such that model prices agree with observable option prices. This can be achieved by optimizing a parametric form for the local volatility function. One such form, with interesting properties, was proposed by Cox (1975): y(t,$)

= vs,

Vs E R+ ,

where y > 0 and 0 < < 1 are constants. Since k, (t,st) • stio,st) = the Cox model is frequently referred to as the constant elasticity of variance diffusion model (the CEV, model for short). As Rubinstein (1985) shows in an empirical study for the US options market, the CEV model is completely unable to describe the volatility term structure. This presents a severe restriction for its practical use. As an alternative to the fitting of a parametric form, Dupire (1994), Rubinstein (1994), Derman/Kani (1994b), and Derman et al. (1996), provided tree-based algorithms to nonparametrically extract the unknown local volatility function from today's option prices. Their approach is known as the implied tree approach. A procedure conceptually similar to the implied tree approach uses implied diffusion theory to directly extract the local volatilities. In particular, it can be shown that given (market) call prices Ct (K,T) at all strikes K and maturities T up to some horizon T*, the local volatility function can be retrieved. It is given by 8C0(K,T)

v (T, K) — (2

aco (K,T)

+ rK ax aT K282Co(K,T)

(3.41)

where v (T, K) is the local volatility that will prevail at time T if the future stock price is equal to K. 43 Yet, in practice, option contracts are only available for a discrete set of strike prices and maturities. Therefore numerical methods have to be employed to compute (3.41). According to Rebonato (1999), p. 130, estimating the local volatility surface directly from (3.41) tends to dominate the implied tree based models in terms of computational speed and numerical stability. Since the number of random sources equals the number of stochastic traded assets (i.e. p = d = 0), local volatility models are complete and are therefore independent of investors' risk preferences. Consequently, they always offer a consistent pricing and hedging scheme. Whilst exactly reproducing market option prices, they have the drawback that they do not allow for idiosyncratic stochastic dynamics in the option prices. This is in conflict with empirical 42 This is the reason why Rebonato (1999) calls this type of models "RestrictedStochastic Volatility Models" in contrast to "Fully-Stochastic Volatility Models" where the volatility of the underlying asset follows a stochastic process of its own. 43 For a derivation of this result, see, e.g., Dupire (1994).

3.4 Modelling Implied Volatility

47

observation." The poor results in an empirical test on their hedging performance and their predictive power performed by Dumas et al. (1998) are probably also due to this drawback. Stochastic Volatility

The stochastic volatility approach was motivated by empirical studies on the time series behavior of (realized) volatilities. They suggest that volatility should be viewed as a random process exhibiting mean reversion. 45 Moreover, volatility seems to be correlated with stock returns. Specifications for a stochastic volatility process {v t : t E [0, T*1 } have been proposed by a number of authors, including Hull/White (1987), Wiggins (1987), Scott (1987), Stein/Stein (1991), and Heston (1993). These processes can be divided into two groups: lognormal processes and processes exhibiting mean-reversion. Among all the stochastic volatility models that have been proposed so far, the model of Heston (1993) is the most popular. In this model, the volatility follows a mean-reverting square-root process (also used by Cox et al. (1985) in the area of interest rates) which is correlated with the stock price process. With this description, the dynamics of the stock price under the objective measure IP is

dSt

= tzdt + vtdWo,t, St dv?= x (v2c0 — dt

(3.42) tvvt dWi, t ,

(3.43)

2 , lc, and tv are the long-run varifor all t E [0, T*], where the constants v 00 ance, the speed of mean reversion, and the "volatility of volatility". The two standard Wiener processes Wo and W1 show constant correlation p, i.e.

dWo,tdWi,t =-- pdt.

(3.44)

Since volatility is not a tradable asset in the model (i.e. p> d), the model is not complete. This requires the specification of the market price of volatility risk. Heston (1993) assumes that this is proportional to the variance of the stock price. Call prices can then be readily obtained in closed-form using Fourier transformation techniques. This is an appealing feature of the model, which explains part of its popularity. Different correlation coefficients p will result in different probability distributions and smile patterns. If volatility is uncorrelated with the stock price, i.e. p = 0, a true smile occurs whose degree depends on the other parameters of the model, especially the volatility of volatility tv. A negative correlation 44

Empirical analyses tend to support stochastic implied volatilities and therefore stochastic option prices. See, Skiadopoulos et al. (1999), Hafner/Wallmeier (2001) and Cont/Fonseca (2002), among others. 45 See Schwert (1990b), Engle (1982), and Ebens (1999), among many others.

48

3 Implied Volatility

coefficient spreads the left tail of the distribution and thus produces a skew pattern. One possible reason for the increase in volatility when the stock price falls refers to the leverage effect. A lower stock price brings about a higher leverage ratio producing an increase in stock return volatility, and vice versa. Yet, in a study on the S&P 100 index, Figlewski/Wang (2000) find a strong leverage effect associated only with falling stock prices. They conclude that the variations of volatility have little direct connection to firm leverage. On the other hand, a positive correlation between stock price and volatility increases the probability of high returns and thus results in a positive skew pattern. In a market with stochastic volatility, the existence of smiles can also be explained by simple (model-free) no-arbitrage arguments. Let us consider an investor who buys an option with positive DVegaDVol (any OTM option, either high- or low strike) 46 If the investor now hedges the outright vega by selling an ATM option (which has roughly zero DVegaDVol), he ends up with a position which gains whatever direction volatility will move. To prevent arbitrage opportunities, thus, the OTM option must have an implied volatility which is above the ATM implied volatility. This explains the (literal) smile profile. The explanation of the skew pattern involves the Greek "DDeltaDVol". Yet, the argumentation is quite similar. A problem of stochastic volatility models is that unrealistically high parameters are required in order to generate volatility smiles that are consistent with those observed in option prices with short times to maturity. 47 This is not the case for long times to expiration. A further problem with stochastic volatility models results from their incompleteness. Since one has to specify and calibrate the market price of risk (process), the model depends on the risk preferences of the investors. This gives rise to specification error, which may, for instance, result in wrong hedging numbers. Moreover, there exists no hedging strategy eliminating all market risk related with volatility.

Jumps A further explanation for implied volatility patterns refers to jumps in the asset price process. 48 When jumps occur, the price process is no longer continuous. Jumps have proved to be particularly useful for modelling the crash risk, which has attained considerable attention since the stock market crash of October 1987. It is often argued that the increased sensitivity of market participants to the crash risk - sometimes called "crash-o-phobia" - has contributed to the skew pattern in S&P options prevailing since 1987. In this context, Bates (1991) has interpreted the high implied volatility of OTM puts prevailing in the forefront of the 1987 crash as an insurance premium against jump risk. 46 In the constant volatility Black-Scholes market DVegaDVol would be zero, by definition. 47 See, e.g., Andersen et al. (1999), p. 3 and Das/Sundaram (1999), p. 5. 48 see, e.g., Bates (1996a), Trautmann/Beinert (1999).

3.4 Modelling Implied Volatility

49

Merton (1976) was the first to study the impact of jumps in the stock price process on the pricing of options. The model of Merton combines a continuous diffusion process with a discontinuous jump component. The evolution of the stock price under the objective measure P is given by the following SDE

dS t

= (II. — Am) dt + vdWt + (I — 1) dIVi t (A),

(3.45)

where the symbols p and y have their usual meaning, g(A) denotes a Poisson process" with intensity A, / is the time-independent random amplitude of the jump, and m = EP [I — 1] is the average jump size. It is assumed that there is no correlation between the Brownian motion W and the Poisson process N(A), and no correlation between the size of the jump I and the occurrence of the jump, represented by N(A). Since the jump component cannot be hedged, there exists no unique replication strategy for contingent claims in this model. The model is incomplete. To avoid the specification of the market price of jump risk, Merton (1976) assumes that the jump component of the stock's return represents nonsystematic risk (i.e. risk not priced in the economy). This means that holding a portfolio consisting of a long position in a call option and a short position in S stocks must earn the riskless rate r over a time step dt. This argument leads to a PDE, which can then be solved for the call price. For the special case that the logarithm of / is normally distributed, there exists a closed-form solution for the call price. 50 In all other cases, numerical procedures have to be applied. The jump-diffusion model gives rise to fatter left and right tails than BlackScholes and can therefore explain the smile effect. Whereas the effect of stochastic volatility increases with longer time to maturity, the impact of jumps diminishes. This is due to the fact that in long time periods positive and negative jumps compensate each other. Therefore, jumps seem especially suitable for modelling the steep implied volatility smile for short maturities. On the other hand, stochastic volatility models are particularly qualified for the modelling of the relatively flat implied volatility smile for long maturities. It seems therefore natural to combine the two approaches. Two examples for stochastic volatility models with jumps are Chernov et al. (1999) and Jiang (1999). Although adding jumps to the stock price process undoubtedly captures a real phenomenon that is missing from the Black-Scholes model, jump-diffusion models are rarely used in practice. Basically, there are three main reasons for this: first, it is hard to find a solution for the pricing PDE, since the governing equation is no longer a diffusion equation, but a difference-diffusion equation; 49 For a discussion of the Poisson process, see, e.g., Brzezniak/Zastawniak (1998), Section 6.2. 50 For an easy to understand derivation of the call price formula, see Wilmott (1998), pp. 329-330.

50

3 Implied Volatility

second, the model parameters are difficult to estimate; and finally risk-free hedging is not possible, since the model is not complete. 51 Alternative Distributions for the Innovation Term A further possible explanation for the smile pattern is that prices move continuously but not according to a geometric Brownian motion. The true underlying distribution may thus be characterized by fat tails and skewness, even if the volatility is constant and jumps do not occur. For example, Eberlein et al. (1998) propose to describe the terminal stock price using a hyperbolic distribution. Analyzing five German stocks, the authors find that the hyperbolic model accurately fits the empirically observed returns and is also able to reproduce the observable smile pattern. Another interesting approach comesfrom Madan et al. (1998). They propose a three parameter generalization of Brownian motion as a model for the dynamics of the logarithm of the stock price. The new process, termed the variance gamma process, is obtained by evaluating Brownian motion at a random time change given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time. They provide control over the skewness and kurtosis of the return. Within an empirical study for S&P 500 options they demonstrate that their model can reproduce the observable implied volatility patterns. A recent development, with promising first results, concerns the use of a so-called fractional Brownian motion in the SDE of the stock price instead of the standard Brownian motion. For more details on the use of fractional Brownian motion in finance and especially in option pricing, the reader is referred to Sottinen (2001). Market Frictions Market frictions are another possible explanation for the smile pattern. Transaction costs, illiquidity and other trading restrictions imply that a single arbitrage-free option price no longer exists. Instead, there is a band of feasible prices. Since arbitrage is no longer sufficient to derive a definite option price, Longstaff (1995) proposes an "unrestricted Black-Scholes model", which does not impose the martingale restriction. In his study of S&P 100 index options the suggested specification is able to neutralize the pricing bias with respect to the strike profile. Longstaff concludes "that transaction costs and liquidity effects play a major role in the valuation of index options" (p. 1093). Similarly, Figlewslci (1989a) (and also Figlewski (1989b)) examined the effects of transaction costs by simulating a large number of price paths and found that they could be a major element in the divergences of implied volatilities across strike prices. Yet, Constantinides (1996) points out that transaction costs cannot fully explain the extent of the volatility smile. 51

See Wilmott (1998).

3.4 Modelling Implied Volatility

51

McMillan (1996) argues that the crash of 1987 lessened the supply of put option sellers, whereas at the same time fund managers showed a higher demand for out-of-the-money puts. Because hedging the risk exposure of written out-of-the-money puts turned out to be expensive, higher prices for out-ofthe-money puts were charged. This could partly explain the observed skew pattern. 52 Tax effects and the capital requirements associated with holding out-ofmoney options in options books are two other cases that could cause a smile in implied volatility. 3.4.3 Implied Volatility as an Exogenous Variable Option Pricing using the Risk-Neutral Density Implied risk-neutral density functions derived from cross-sections of observed standard option prices, or equivalently, the volatility smile, have gained considerable attention during the last years. 53 To introduce the concept of a risk-neutral density, let us consider an arbitrage-free and frictionless market where the money market account evolves according to Bt = exp (rt) for all t E [0, 71. Due to the risk-neutral valuation formula (Theorem 2.14), the time t price of a European path-independent T-contingent claim H = T E (t, T 4], on the stock S is given by

H(H) t(H) = e -r(T-t) EQ [HI .Ft1= e -r(T-t) f

oo

(s) qsT (s)ds,

(3.46)

with qs,,,(s) denoting the risk-neutral density (RND) of ST conditional on the information set .Ft at time t. Setting (I) (Sr) = max {ST — K; 0} yields the price Ct of a standard European call option with strike price K and maturity

T: t = e-r(T-t)

oc

max {.5 — K; qs,,,(s)ds.

(3.47)

Given the market prices Ct (K, T) of a continuum of European call options on the stock S with the same time to maturity T and strike prices K ranging from zero to infinity, we can apply the fundamental result of Breeden/Litzenberger (1978) to fully recover the RND in an easy and unique way.54 They have shown that the discounted RND is equal to the second See also Cochrane/Sag-Requejo (1996). the following, see Brunner/Hafner (2003). 34 Since Breeden/Litzenberger (1978) build their work upon the state-preference theory of Arrow (1964) and Debreu (1959), they speak of a "state-price density". We refrain from using this expression for two reasons: First, it is ambiguous. The term state-price density is also referred to the Radon-Nikodym derivative c/Q/cilP (see, e.g., Plislca (2000), p. 28). Second, the term RND is more popular in the recent literature. 52

53 In

3 Implied Volatility

52

derivative of the European call price function (3.47) with respect to the strike price K. qs, (8)

=

a2c,(K, T) aK2

(3.48) K=s

For any fixed maturity T, T> t, the relation between the RND and the volatility smile is obtained by successive application of (3.48) and the definition of implied volatility (3.29): (K2, T) (s ) .= er(T—t) &cat K

(3.49) K=s

= e r(T t) a2 CBS(t, St K,T,r,cr t (K,T))1 —

I K=s After applying the chain rule for derivatives, we get: 1 gsT (s ) = n(d2(s)) [ scr t (s,T)

2 2d1(s) acrt(K,T)I + c• t (s,T) alf I K=s

—t acrt(IC' T) 1 ) 2 T + scli(s)da(s)V 0,(s, arc K=s

s

(3.50)

maatK (K2" K=sj

for all s > 0, where ln (49t ) -

di(s)

(r

(cr t (s , T)) 2 ) (T — t)

at(s,T) ■/77--t-

, d2 (s)= cli (s)—cr t (s,T).

(3.51) and n(x) is the standard normal density function. For equation (3.50) to be properly defined, the implied volatility function crt (K,T) is required to be twice differentiable in K. If o-t (K, T) is given in closed form, so is the RND qs,.(s). As the above discussion has shown, for a given maturity T three functions contain essentially the same information: the RND qsT (s), the call price function Ct(K,T) with respect to strike, and the volatility smile o-t (K,T). This is also illustrated in Figure 3.5. In practice, option contracts are only available for a discrete set of strike prices within a relatively small range around the at-the-money (ATM) strike price. Therefore, all of the methods for estimating RND functions boil down to the completion of the call price function (or implied volatility function or RND function, respectively) by interpolating between available strike prices and extrapolating outside their range. RNDs have found various applications in finance: Central banks, among others, use the RND to assess the market participants' expectations about underlying asset prices in the future. 55 Ait-Sahalia/Lo (2000) and Jacicwerth 55

ers.

See Bahra (1997), Cooper/Talbot (1999), and Levin/Watt (1998), among oth-

3.4 Modelling Implied Volatility

53

(t, St, K,T,r,Ct(K,T))

CBS(t, St ,

K ,T, r, t (K ,T))

Figure 3.5. Relationship between call price function, volatility smile, and RND

(2000) compare the RND with the objective density to retrieve the investors'

risk preferences. Bates (1996a) uses the RND to estimate the parameters of the underlying stochastic process which generates this RND. In the field of option pricing, the implied RND allows to price illiquid path-independent exotic derivatives in a consistent way. Given qs, (s), we simply have to evaluate (3.46) to obtain the price of a derivative H ---- CST)• The main advantage of the RND approach is that we do not have to specify the stock price process. The continuum of option prices is sufficient to uniquely derive the RND qs, (s) which allows for the pricing of all path-independent options as, e.g., digital or power options. On the other hand, because we do not explicitly model the stock price evolution, we cannot price options where this information is needed. In particular, we are not able to value pathdependent options such as barrier or lookback options. For the same reason, the RND concept cannot be employed for hedging purposes and the analysis of (option) trading strategies. For applications of the RND concept in the area of risk management, the RND has to be transformed into the corresponding objective density. This affords additional assumptions on the risk preferences of the agents in the economy. Deterministic Implied Volatility Models The empirical observation that the implied volatility surface evolves with time and stock price has led practitioners to develop simple rules to estimate its evolution. Following the study of Derman (1999) on S&P 500 options, we distinguish between three different rules, where, in turn, each rule is associated

54

3 Implied Volatility

with a different market regime. 56 The first rule, the so-called sticky-strike rule , postulates that, when the underlying index moves, the implied volatility of an option with a particular strike and a particular maturity remains unchanged from time t to t + At, where At is a small time interval (e.g., a day) - hence the term "sticky-strike". Mathematically, the sticky-strike rule can be expressed as: V(K,T). (3.52) at+At (K) T) = at (K,T), If the market behaves according to the sticky-strike rule, the absolute implied volatility surface has no dependence on the index level. This implies that the ATM implied volatility decreases as the index increases, and vice versa. The sticky-strike rule is typical for situations where future underlying moves are likely to be constrained to a trading range, without a significant change in current volatility. In a trending market, Derman (1999) suggests to use the sticky-moneyness rule (also called sticky-delta rule). This rule stipulates that, when viewed in (M, r) remains constant from relative terms (M, r), the volatility surface time t to t ± At. Mathematically:

a,

at-I-At

(MI T) = at(m, 7),

v(m, T),

(3.53)

with moneyness defined as M = tF,‘-. According to this rule, the ATM (i.e. M = 1) implied volatility is constant over time. However, for a fixed strike option, the implied volatility increases as the index level increases." In a third market environment, characterized by jumpy markets, implied volatilities are supposed to follow the sticky-implied-tree model. As the name suggests, this rule is related to the implied tree approach, presented in Section 3.4.2. It says that the implied volatility of a fixed strike option will decrease when the index goes up, and increase when the index falls. The ATM implied volatility decreases roughly twice as rapidly as the index level increases. For more details on these rules, the reader is referred to Derman (1999) and Alexander (2001a). The above rules are in fact (simple) deterministic laws of motion for the volatility surface: given today's prices and given the current market regime, there is no uncertainty about the implied volatility surface tomorrow. These rules can be extended to models where a t (K, T) or (M, r) evolve in a more general deterministic way. In a detailed study, Balland (2002) shows that in fact the only arbitrage-free models in which at (K, T) is deterministic are Black-Scholes models with at most time-dependent volatility. As Cont/Fonseca (2002) point out, these simple rules are not verified in practice. They show that the implied volatility surface of S&P 500 options has "a noticeable standard deviation that cannot be neglected when considering either hedging or risk management of portfolios of options". Even the 56 See

also Cont/Fonseca (2002). that this is only true in the case of a volatility skew.

57 Note

3.4 Modelling Implied Volatility

55

ATM implied volatility, assumed to be constant or slowly fluctuating in the sticky-moneyness model, exhibits significant variation. These results are also supported by other empirical studies. 88 Stochastic Implied Volatility Models Due to the apparent shortcomings of deterministic implied volatility models, a

natural step of generalization is to let implied volatilities move stochastically. In contrast to (traditional) stochastic volatility models, where the instantaneous volatility of the stock is modelled, stochastic implied volatility models focus on the (stochastic) dynamics of either a single implied volatility, the volatility smile, the term structure of volatility or the whole volatility surface. Lyons (1997) was the first to model the stochastic evolution of a single implied volatility. Amerio et al. (2001) also models a single implied volatility to price derivatives on implied volatility. Following the path-breaking work of Heath, Jarrow, and Morton (HJM) 59 in the area of interest rates, SchtSnbucher (1999) models the evolution of the term structure of volatility in a continuoustime setting. He derives a no-arbitrage condition on the drifts of the options' implied volatilities which is similar to the drift condition derived by HJM for the instantaneous forward rates and furthermore he analyzes the restrictions that have to be imposed to ensure regularity of the option price at expiry. Albanese et al. (1998), extending the work of Scht5nbucher (1999), consider the arbitrage-free evolution of the volatility surface at (K,T) in a generalized 10 setup. Ledoit et al. (2002) independently arrive at a similar model as Albanese et al. (1998), however, their focus is on the modelling of the relative volatility surface Eit (M, 7-) with moneyness defined as M = St/K. They show that in a stochastic implied volatility model the instantaneous volatility v t of the stock can no longer evolve independently. In particular, they prove that the implied volatility of at-the-money options converges to the instantaneous volatility of the underlying asset as the time to maturity approaches zero. Because the number of random sources is smaller than or equal to the number of risky assets, stochastic implied volatility models are in general completemarket models. While the above approaches dealt with the problem of stochastic implied volatility from a theoretical perspective, Rosenberg (2000), Cont (2001), and Cont/Fonseca (2002), among others, focus on the empirical aspects of the problem. Rosenberg (2000) proposes a "dynamic implied volatility function model" to describe the (discrete) evolution of the implied volatility surface of S&P 500 futures options. Thereby he separately estimates the time-invariant implied volatility function with respect to moneyness and time to maturity and the stochastic process of the only state variable, the ATM volatility, that 58 See Alexander (2001b) for FTSE 100 options and Hafner/Wallmeier (2001) for DAX options. "See Heath et al. (1992).

3 Implied Volatility

56

together drive changes in the individual implied volatilities. Based on an empirical study of time series of implied volatilities of S&P 500 and FTSE 100 index options, Cont/Fonseca (2002) suggest a factor-based stochastic implied volatility model. The abstract risk factors driving the volatility surface are obtained from a Karhunen Lc:16\re decomposition. This model extends the stickymoneyness model used by practitioners, while matching some salient features of volatility surfaces. A natural application of the statistical models of Rosenberg (2000) and Cont/Fonseca (2002) is the simulation of implied volatility surfaces under the real-world measure, for the purpose of risk management. However, the models are not intended to determine the consistent volatility drifts needed for risk-neutral pricing of exotic derivatives. 60

3.4.4 Comparison of Approaches In the attempt to correctly reproduce the current volatility surface, neither (one-factor) stochastic volatility models nor simple jump-diffusion models are successful. 61 Local volatility models, on the other hand, are able to exactly reproduce the current volatility surface; however, they perform poorly in grasping its future evolution. Implied volatility models match the current market prices of standard options, by definition. Whether they also match the future option prices depends on the chosen model. Models based on the underlying asset to describe the dynamic behavior of option prices or their implied volatilities, except of local volatility models, are in general incomplete. Consequently, the requirement of no-arbitrage is no longer sufficient to determine a unique price of the contingent claim. Instead, we have several risk-neutral measures, and several market prices of risk. To derive a unique value, we have to make assumptions on the risk preferences of the agents in the economy. This implies that the valuation of contingent claims is no longer preference-free. In contrast, stochastic implied volatility models are complete-market models and thus independent of investors' risk preferences. This implies that there exists a unique replicating strategy for each contingent claim and thus a unique arbitrage price. The fact that we can invoke the risk-neutral valuation principle for valuation purposes in models where implied volatility is endogenous does not imply that any modelling approach that produces the same final price distributions and is consistent with risk-neutrality will lead to equivalent results. For example, hedging numbers may in fact be quite different. It is therefore essential, to identify the financial mechanism, which causes the smile. Since it is generally acknowledged that the financial mechanisms that are responsible for volatility smiles are interrelated, and no single explanation completely captures all empirical biases in implied volatilities, this can be very difficult. 62 60 See

Lee (2002). the following, see also Cont/Fonseca (2002). See Rebonato (1999), p. 93 and Hafner/Wallmeier (2001).

61 In 62

3.4 Modelling Implied Volatility

57

Implied volatility models, on the other hand, do not have these problems as they do not attempt to explain the volatility surface but use it as an input. Another advantage of market-based modelling of implied volatilities is that implied volatilities are directly observable and independent of any modelling assumptions on the processes involved. By contrast, quantities such as the local or stochastic volatility or the jump intensity, are not directly observable and have to be filtered out either from pricing data on the underlying asset using an econometric model or "calibrated" to options data. 63 In the first case, the quantity obtained is model-dependent and in the second case it is the solution to a non-trivial optimization problem. On the other hand, implied volatility models are automatically calibrated to market option prices. In contrast to fundamental quantities such as an (unobservable) instantaneous volatility or a jump intensity, implied volatilities are highly regarded and continuously monitored by market participants. A market scenario described in terms of implied volatilities is therefore easier to understand for a practitioner than the same scenario (re)expressed in terms of these fundamental factors. These arguments motivate the direct modelling of the implied volatility surface. Due to the inability of deterministic implied volatility models to capture the empirically observable fluctuations of the implied volatility surface, stochastic implied volatility models seem favorable. The main disadvantage of these models is their complexity, because, in addition to the stock price process, the (stochastic) joint dynamics of the implied volatilities of all strikes and all maturities have to be modelled in such a way that they are consistent with no-arbitrage. Fortunately, shifts in the level of implied volatility exhibit high correlation across strikes and maturities. This suggests that their joint evolution can be accurately described by a small number of risk factors.

63 Calibration usually means determining the parameters of a financial model such that model prices best possibly fit observed market prices. This is achieved by minimizing a prespecified error function, for example, the sum of squared errors.

4

The General Stochastic Implied Volatility Model

There is nothing more practical than a good theory.

Leonid Ilich Brezhnev

In this chapter we develop a rigorous mathematical model of a financial market in continuous time where in addition to the usual underlying securities stock and money market account, a collection of standard European options is traded. The prices of the standard options are given in terms of their implied volatilities. These, in turn, are described by risk factors, which are stochastic. To meet the objective of providing a framework that is applicable to the pricing and hedging, the risk managing, and the trading of contingent claims, consistent modelling under both the objective measure and the risk-neutral measure is required. In this respect, the model presented here differs from other models proposed in the literature which apply either the objective measure (e.g., Cont/Fonseca (2002) and Rosenberg (2000)) or the risk-neutral measure (e.g., Schtinbucher (1999) and Albanese et al. (1998)), but do not consider both simultaneously. The chapter is organized as follows. In Section 1, we describe the financial market model under the objective probability measure II). In Section 2, we derive necessary and sufficient conditions that have to be imposed on the drift coefficients of the options' implied volatilities in order to ensure discounted call prices to be martingales under the risk-neutral measure Q. In this context, we also discuss existence and uniqueness of a risk-neutral measure. Finally, we show in Section 3 how to price and hedge a general stock price dependent contingent claim.

4 The General Stochastic Implied Volatility Model

60

4.1 The Financial Market Model 4.1.1 Model Specification We consider a frictionless security marketi where investors are allowed to trade continuously up to some fixed finite planning horizon T* (Assumption 1). 2 The uncertainty in the financial market is characterized by the complete probability space (SI, .7',P) where SI is the state space, Y. is the a-algebra representing measurable events, and P is the objective probability measure. Information evolves over the trading interval [0, 71 according to the augmented, right continuous, complete filtration IF = {Ft : t E [0, T*11 generated by a p 1-dimensional standard Brownian motion W = {Wt : t E [0, T*]} , Wi,t, • • • 7 Wp,t) i ) initialized at zero. We assume that the or-field Wt=(o, contains all the P-null sets of .7 -, and that Y .T. = F. trivial and Yo is The primary traded securities are a non-dividend paying stock3 , a money market account and a continuum of standard European call options on the stock. Under the objective probability measure P, we make the following assumptions on the evolution of the money market account and the stock: Assumption 2: The price process of the money market account is given by the SDE: dB t= rBt dt, Vt E [0, 71, (4.1) where Bo =1 and the interest rate r is supposed to be constanti and nonneg-

ative. Assumption 3: The evolution of the stock price is governed by

dSt Stiltdt + StvtdWo,t,

Vt

[0, T *

]

7

(4.2)

with initial non-random stock price So > O. The drift process {A t : t E [0, T*1} is real-valued, progressively measurable and satisfies fot ISuA u l du < co P-a.s. for all t E [0, Tt] . The volatility process {v t : t E [O, T1 } is supposed to be nonnegative, progressively measurable with ft:t) Siiv 2u du < oo P-a.s. for all t E [0, At time t the continuum of European call option prices Ct (K, T) with strikes K > 0 and maturities T> t can be represented by the volatility surface 1

Note, however, that some market frictions are already reflected by market option

prices.

the following, see also Hafner/Schmid (2003). results can be easily extended to the case of a (even stochastic) dividend paying stock. 4 Our approach can be generalized to allow for stochastic interest rates by simply attaching a HJM-type model. 2 In

3 All

4.1 The Financial Market Model

61

t (I( T). Because the volatility surface can be more easily parameterized and estimated as a function in moneyness M and time to maturity r = T t, we rather consider the continuum of call prices dt * (M, T) which is represented by the relative volatility surface at (M, 7). Recall from Section 3.3.3 that there is a one-to-one correspondence between at (K,T) and ai m 7-) of the form —

at (m,T) = cit (m-i(m),t + T) ,

(4.3)

with m 1 (M) being the inverse function of m with respect to K. We demand the moneyness function and the corresponding moneyness to be valid according to the following definition: Definition 4.1 (Valid Moneyness). We call M. a valid moneyness function and M defined as M = Mt = m(t, St , K ,T,r)

a valid moneyness for our financial market model if m has the following properties: 1. m(t, s, K ,T,r) E C 2 ([0,T*] x R++ x R++ x (t, T*] x R+ ) , r)
,y,) E C 2 ([0, T*1 x R x (0, T* — t] x RP) The volatility surface at (m,r) at time t E [0, T* ] is completely described by p abstract risk factors Yi,t1 172,t, • • • Yp,t such that: • • •

t(M, T. ) = g (t, M, r, Yl,t, 12,t, • • • , YP,t) •

(4.4)

The dynamics of the i-th risk factor is modelled os cti , t dt +

E

i = 1, . , p, Vt E [0, T* ] ,

(4.5)

i=o

with initial value YO. For any i = 1,... 43, j = 0,...,p, the processes : t E [0, T*]} are progressively measurable, and : t E [0, T*]} and {ry i < 00 P-a.s. for all t E [0, T* 1. satisfy fot I Cti, u I du < oc P-a.s., fot 4.1.2 Movements of the Volatility Surface The implied volatilities are driven by the p+1 dimensional standard Brownian motion W, where Wo is the same Brownian motion that is driving the stock price. This is used to model the correlation between the implied volatilities and -

62

4 The General Stochastic Implied Volatility Model

the stock price. The dynamics of an implied volatility of an option with moneyness M and time to maturity T is determined by the dynamics of Y1, • • • , Yp• If we set g (t, Yi, , yp ) = g (t, M,T , yp ) , then by application of It45's rule to equation (4.4) we get

dg (t, Y1,t, • • • Yp,t)

(4.6)

= 2(t,y1,t,••., Yp,t) dt

ag

P

+E i=1 ay,

— (t,y, t, • • • , Yp,t)

P

1 x-■ -

I-

P

'7,2

a 9 ayiayk 1_, i=1 k=1

L.,

V`

with quadratic covariation d (Yi, Yk) t = for dYi , t (i = 1, ... , p) in (4.6) yields

dg (t,Yi, t , . .. ,Yp,t) = {

(t, Yi,t, • • • ,

=0 ryid,t ryk,i,t dt. Substituting

(t,,t, li • • • , Yp,t) +

E cr t ,t

P P P

2

Yp,t) d (Yi,Yk)t ,

a2 g ay,ayk

k=1 3=0

(4.5)

(t, Yi,t, • • • ,

( t,

,

+/ 14, • • • 7 Yp,t)

dt

P P

ag (t,Y1,t, • • • , I rp,t) dWi,t• +,=1E.JE=0 -Yit — ay,

(4.7)

Using (4.4), the dynamics of the implied volatilities of each fixed time to maturity T and moneyness M can be expressed as

(4.8)

dat(M, r) = iit (M,r)dt +i9t (M,r)diV t ,

where Fit = Fit (M, r) is given by {P

ag f kt, at

„

Fit =

Yp,t) i 1,t, • • •+,

ag i , Eat,t, kt, Yi,, • • • , i=1 uYi ayiayk (t, Y1,t 7 • • • , Yp,t)

i=1 k=1 3=0 and

(4.9)

j.

= 1-9t (M , 7) is the (p + 1)-dimensional row vector ;79-t =

• • • , -/-9'72,t)

(4.10)

7

with ;9- j,t =

ag E'Yi,i,t—

ay,

kt,

•••

7

Yp,t)

j = 0, • • , P.

(4.11)

4.2 Risk-Neutral Implied Volatility Dynamics

63

4.2 Risk-Neutral Implied Volatility Dynamics 4.2.1 Change of Measure and Drift Restriction If we define Wt* --= (n,t, • • , n t r for all t E [0, T*] , with dynamics given by

dWt* = dWi + O tdt,

(4.12)

where tPt = (00» , Op,t r is a p + 1-dimensional progressively measurable process satisfying

7"'

0

b tdt < CO

P-a.s. (4.13)

as well as Novikov's condition, i.e. 1

B [exp Bp

IT . 0 tit Ot dt}i
(4.14)

then, by the virtue of Girsanov's theorem, under the measure Q with RadonNikodSrm derivative Et –

dQ

dP

—

= exp (–

f

t ,

– 1 f t li;87,b,ds) , 2 0

(4.15)

the process Wt* is a multi-dimensional standard Brownian motion. The process is interpreted as the market price of risk (process) associated with the ,p). random factor Wt (i = 0, Choosing

lkt

Pt

r

vt

Vt E [0,71 ,

(4.16)

it is easy to prove that under the measure Q the discounted stock price St* = St/Be is a martingale:

1 dS;` = –rS;dt + —dSt = ((lit – rB)dt t + vtdWo,t) = S(Ol t – r – vt00,t ) dt + vt dig) = SvtdW. As in our model option contracts are primary traded assets as well, we have to show that under the measure Q the discounted option price processes are martingales, too, i.e. we need to look at the dynamics of the option price with fixed maturity date T and with fixed strike price K. Therefore, we have to determine the dynamics of the implied volatility o-t (K, T) of this option under the measure P, given the dynamics of at (m, r). Setting g (t, M, r) = g (t, M, r, , yp ), for convenience, we proceed in three steps:

4 The General Stochastic Implied Volatility Model

64

1. For fixed T we make the second argument of the implied volatility process t. Then, by apZft(M, T) change deterministically with time, r(t) = T plication of It6's lemma, —

09 (M,T - t) - — (t, M,T - t)) dt

dart (M,T - t) =

aT

+13t (M,T

-

(4.17)

t)dWt .

2. Using It6's lemma, we get the dynamics of the moneyness with respect to the stock price as

102m am am dMt(St) = — (t, St) dt + — (t, St) dSt + — (t, St) d (S, S) t

as

at

208

am am 1 a2m = (— (t, St) + Sat t — — (t, St )) dt as (t, St) +2 - v2t 0s2 at

am

+stutw-s- (t, St) dWo,t.

(4.18)

3. Finally, we use a generalization of the It6-Venttsel formula 5 to the processes given by equations (4.17) and (4.18) to get t (K , T) = dat (Mt ,T - t)

(4.19)

=

+ 1- (Stvt- na(t,st)) 2 a2g - (t, Mt,

as

2

na

+( at

a2m

- t)

+ sttl, as(t, na st) + _1 s,2v#2 2 ., 192m as2 (t,

, am,(t, , au kt, Mt , T - t)) dt +stut--kt,st,—

am as am (t, St) ag ±Stth

Os = t (K , dt + V t (K,T)dW t , where

5

See

Appendix A and the references cited there.

am (t, Aff T

–

4.2 Risk-Neutral Implied Volatility Dynamics

nt (K ,T) =

65

(4.20) (t, Mt , T - t)

(Ms, T - t) -

- t)

(St v tw 0m (t, St)) 2 421,94- ( t Mt, -1-(-Ft am

82 a-s 2. (t, St))g (t, St) + S?v?m ° (t, Mt, T - t)

+

am O

+Stvt; (t, St)

Ou

(t, Mt,

-

am as

- t) + Stvt — (t, st) — (t, Aft,T - t) ,

19 o,t(K ,T) =

OM

(4.21)

and ,T) = kt(Mt,T

-

t) ,

i = 1,

(4.22)

and u is the deterministic function corresponding to ;51 o,t• Evaluating the Black-Scholes Greek functions at y = at (K ,T) we define (5t = 6Bs (t, St , K,T, r,at(K,T))

(4.23)

rt =_ rB s (t, St, K, T, r, t(K, T)) At = ABS (t, St, K, T, r, c(-K ,T)) et = eBs (t, St , K , T, r, t (K , T)) , Vt = VBS (t, St, K crt(K,T)) 'Pt = 41 Bs (t, St , K,T,r,at(K,T)) and by applying lies lemma to the Black-Scholes call option pricing formula (3.18), we find the dynamics of the call prices Ct(K, T) under Il» to be 1 dCt (K ,T) = O t dt + 5tdSt + Atdat + Ftd (S, S), -

(4.24)

2

1 + - Vtd (a, a)t + xlit d (S, a) t 2

=

(et + 6tstpt + Atnt(K, T) + .1Ft SM 1 +-Vtt,t(K ,T)19't (K,T) + tStv090,t(K ,T)) dt 2 +6tStvtdWo,t + At/9t(K,T)dWt,

where crt = t (K ,T), for simplification. The random terms in this formula consist of a term that is proportional to the Black-Scholes delta and a term proportional to the Black-Scholes vega. Note, that the formula can be used to create a delta-vega hedge by using a

4 The General Stochastic Implied Volatility Model

66

portfolio consisting of the stock and p standard options. Let us finally pool the random terms to end up with the following formula: dCt (K,T) = (et + btSttit + AtrIt(K,T) + li rt s?v?

( 4.25)

+ 1 Vtt9t(K, T)tit (K ,T) + Stv090,t(K ,T)) dt -149t (K ,T) dW t , where

(4.26)

1)0,t(K ,T) = StStvi + Ati90,t(K ,T) , and ai,t(K,T) = Atikt(K ,T) , We choose 0, i = 1, 19t(If

(4.27)

j = 1, • • ,13.

, p, such that the drift or no-arbitrage condition

(4.28)

=

+ +

1 did2

2 1

1

2 crt (K ,T) (T — t) ( d2

c4(K , T))

v t 190, t (K,T) ,

crt (K ,T)

with6 d1 = d1 (t, St , K,T,r, t(K ,T)) d2 = d2 (t, St, K ,T,r,at(K ,T)) holds for all t E [O, T*] and all (K ,T). 7 Then, under this measure, the dynamics of Ci (K ,T) are given by dCt (K ,T) =

(et +6tstiit +Atilt (K ,T) + .1.'tStv? 1 +- Vti9t(K ,T) li t (K ,T) 2

tStvtt9c,,t(K ,T) —

(4.29) t (KT)0) dt

(K, T) dW t* Discounting and using equation (4.28) shows that the relative call prices (K ,T) =- Ct (K ,T) B t are indeed martingales: 6 For

a definition of di(.) and d20, see equation (3.19).

'A similar condition that restricts the drift coefficients of the instantaneous forward rates was derived by Heath et al. (1992).

4.2 Risk-Neutral Implied Volatility Dynamics

dC7 (K ,T) -rCt* (K ,T) dt + --B1 dCt (K ,T) =

(-149 + ( Str + qt (K,T) Bt At At 1 rt 2 Vt --A7t 19t(if,

+T-A-Tt Sblt

t (K

67

(4.30)

,T)

'6K ,T) - LCi(K ,T) - 15 1t (K , T) 1,b t ) dt --I I Stvt04( At At

=

Bi At

,T)

1

did2

( Te. P\ 19; (K ,T)

2 crt (K,T) -t `- '

1

1

+ 2 crt (K ,T) (T - t) (v?- cr?(K , T)) d2

crt (K ,T) 1 ,T) dW t* Bi

v t1904 (K ,T) -

dt

1 = —i9 t (K,T)dW t*.

Bt

Note that equation (4.30) holds because the Black-Scholes PDE

et

rt 2 2 6t + A-t St at (K ,T) + 71 St r -t-

-

T) = 0.

(4.31)

is satisfied for Ct (K, T) when the stock price volatility vt equals the option's implied volatility crt (K ,T). Finally, the dynamics of co. t (K ,T) and at (m, r) under Q are:

dcrt (K ,T) = =

(K ,T) - 9(K, T)

dt +19t (K, T) dWt*

(4.32)

where

(K ,T)

=

1 did 2 19t(K ,T)19't (K ,T) 2 at (K ,T)

1 1 2 a t (K ,T) (T - t) d2

t(K,T) VT

(4.33)

(4 _ cr?(K, T)) 05'04 (K , T))

and

c& t (M, T) = (Flt (M,T) - 13"t(M, T)

t ) dt + -ji(M,r) dWt*

(4.34)

68

4 The General Stochastic Implied Volatility Model

4.2.2 Interpretation of Terms in the Risk-Neutral Drift

In the following we want to examine the source of each of the three terms in the risk-neutral drift (K, T) of crt (K,T) given in (4.33).8 In the BlackScholes model, all implied volatilities are constant over time and equal to the stock's actual volatility. Thus, a naive implementation of a stochastic implied volatility model might simply set the risk-neutral drift 77; (K, T) of all implied volatilities equal to zero. Since the expected drift of a call option is nonzero, this naive model specification would create three sources of arbitrage profits: 1. The nonlinear relationship between the Black-Scholes model value of a call option and its underlying's volatility combined with randomness in the implied volatility would lead to a source of arbitrage. To correct for this source of arbitrage, we need to include the term 1 di d2 19t(K,T)Vt(K,T). 2 crt (K,T) 2. The divergence between the actual stock volatility and the implied volatility of an option would cause the gamma trading profit, which is due to the actual stock volatility, to not offset the option's time decay, which is due to the option's implied volatility. To remove this source of arbitrage, the term 1 1 (v 2 2 crt (K,T) (T t) —

—

t

is included. 3. The nonlinear relationship between the Black-Scholes model value of a call option and the underlying's price and volatility and the covariance between the underlying price and the implied volatility would lead to a new source of arbitrage if the implied volatility had no drift. This source of arbitrage is removed by including the term d2 crt (K,T)

7,./

v0%4 (K,T)

in the risk neutral drift. -

4.2.3 Existence and Uniqueness of the Risk Neutral Measure -

In Section 4.2.1, equation (4.28), we have implicitly assumed that we can always find a market price of risk process such that equation (4.28) holds for all traded options at all times. However, if there is a continuum of traded options, equation (4.28) can only be satisfied if we can define = 1, independently of K and T. In general, this might be impossible. Even if we allow for more options than there are sources of randomness we end 'In the argumentation we basically follow Albanese et al. (1998).

4.2 Risk-Neutral Implied Volatility Dynamics

69

up with an over-specified model. If we consider exactly p traded options we can expect the implicit function to be uniquely defined by equation (4.28). (Ks , T3 ) for all i Fix (K1, , , (Kp , Tp ) such that (K, j, and < Tp < T* . There exists an equivalent probability measure 0 < T1 < denoted by Q(KI,T1),. ,(1C,,Tp) such that C7(1(1,7'0 , • • • (Kp,Tp) are martingales if and only if equation (4.28) holds for (K1, , , (Kp, Tp ) and for all t E [0, T1 ). Depending on the specific choice of the processes 19, ?I, and y the right-hand side of equation (4.28) might explode P-a.s. and t9t <00 P-a.s. if t tends to T1. 9 In this case it is impossible to find a market price of risk process tb such that jo t , tbu tbu du < 00 and thus an equivalent martingale measure does not exist. 10 Nevertheless, if we can find an equivalent martingale measure for the call options Ct(Ki, TO , • • - , Ct (Kp , Tp ) , the measure is unique if and only if /9 1,t(Ki,T1) - - • 19p,t(Ki, Ti) (

--• :

(4.35)

191,t(Kp, Tp) - • • Vp,t(Kp,Tp)

is nonsingular P-a.s. Both the market prices of risk and the martingale measure, however, depend on the particular call options chosen. To guarantee that there exists a unique equivalent martingale measure Q simultaneously making all relative call option prices martingales, the market prices of risk must be independent of the vector of call options (Ct(Ki, T1) , • • • , Ct(Kp, Tp)) chosen. Formally, the following conditions are equivalent: 11 •

•

•

Q defined by Q = (Puri ,Ti),•• •,(1c,,Tp) for any (Ki , Ti) , • • • , (Kp, Tp) E R++ x (0, T*] , (Ki,Ti) 0 (K 3 ,T3) Vi 0 j, 0< Ti <...
unique equivalent probability measure such that q (K, T) = Ct (K , T) /B is a martingale for all T E (0, T*I , K E R++ , t E [0, TO . 1,bi,t ((Ki , TO , . . . , ( Kp , Tp )) = ?Pt t ((Kp+ 1, Tp+i ) , ... , (K2p ,T2p )) for i = 1, ... ,p and for all , • - • , Iç p, Tp) , (Kp+i,Tp+i) , . . . , (K 2 p , T2p ) E R++ x (0, Tl, (K i, TO 0 (K 3 ,T3 ) Vi 0 j, i,j E {1, ... ,p} , (Kt , TO 0 (1‘3 , Ti ) V i 0 j, i, j E fp + 1, ... , 2p1 , t E [0, T*] such that 0 < t < Ti < ... < Tp < T* and 0 < t < Tp+ 1 < ... < T2p < T* . For all T E (0, T*I and t E [0, T)

° Similar problems occur in the HJM framework. See Filipovic (2000), for details.

1° A

similar problem also occurs if one specifies the implied volatility dynamics already under the risk neutral measure. See SchOnbucher (1999). "These conditions are very similar to those in the HJM model framework. See in particular, Proposition 3, p. 86. For this reason, we omit the proof.

70

4 The General Stochastic Implied Volatility Model 19t (K,T)1,b t =

?it

+ +

(K,T) 1 did2

2 a t(K ,T) 1

t9t(K ,T) V;(K ,T)

1

2 at (K ,T)(T — t)(v - al(K,T)) d2

where 004 = Tp ties

and

(t, T*1

crt(K ,T).\/7

vt'19° 't(1( ' T)

=

, , (K,, T,)) for any maturi-

and times t E [0, TO.

In the following, we assume that there exists a unique equivalent martingale measure Q.

4.3 Pricing and Hedging of Contingent Claims This section demonstrates how to value and hedge contingent claims in the economy decribed above. For the valuation of a general stock price dependent claim the following result is central: 12 Theorem 4.2. The (Black-Scholes) ATM implied volatility converges to the

stock price volatility when the time to maturity goes to zero, i.e. for K = St er(T — t ) the following is true: vt

(M(t, St, Ste r(T-t) 7

r), T — t) =

t (m(t, St , St , t, r), 0) . (4.36)

Proof. Let us consider a standard European call option with strike price equal to the forward price, i.e. K = St er(T - t). Then, the terms d1 and d2 from the Black-Scholes option pricing formula are given by (t, st , K) = dl (t, st,

=

1

sterml st, st er(T-t)

r) , T

and d2 (t,St ,K) = d2(t,St ,St er(T—t) ) 1 =

(m(t, st, ster (T - t) , T,r),T —

Since for small T t the cumulative standard normal distribution function can be approximated by a first-order Taylor approximation -

' 2 A similar result was derived by Ledoit et al. (2002).

4.3 Pricing and Hedging of Contingent Claims

N (di (t, St, Ste r(T-t) ))

± 127r di

:,.4

1

2 N el 2 (t , St, Ster (T—t) ))

(t, St , Ste r(T-t) )

at (m(t,

+-

st , Si er( T — t) ,T,r),T - t)

VT:t

21,/rn c- it (To, St, Ste(T-t) , T , r), T - t) V7 -27 2 ■,/rn-

1 2

71

,

the ATM forward call option price, i.e. CB s (t, St, K) for K = Ster(T-t) , comes to CBS

(

t st, Ste r(T-t) ) ,

= St (N (di (t, St , St er(T-t) )) - N (d2(t, St , Ster(T-t) ))) 2 St

(m(t, st, sterm - t) , T,

r), T - t)

(

- St

,

+

(1

at (m(t, St, Ste r(T-t) ,T, r), T - t) 1./7)

2

21/Tir

at (m(t, St, St er(T-t), T, r), T

-

t) ■/T

Nr2-77

Therefore at(m(t, St, st, t,

co

at (70, st , st er(T- t), T, r), T -

= liM

r CB s (t, St, Ste r(T—t) ) St

Now it suffices to show that the last line of this equation is equal to Vt. This can be done on the basis of the payoff function for the ATM forward call option using the distribution properties of the Brownian motion. The price of the call option can be described as the expected discounted payoff under the Black-Scholes martingale measure Q CBS

(t, St, Ste r(T—t) ) = EQ [e

(T-t) max { ST_ ster(T-t) ; 0 } 1 ;

Here, the option is considered close to maturity so that the stock price ST can be approximated by ST

st er (T- t

St vt

Wt*

for small T t; the last term is the Brownian increment which is normally distributed with mean zero and variance T t. Consequently -

-

72

4 The General Stochastic Implied Volatility Model EQ [e -r(T-t) max {s7, _ ster(T-t) , 0 } ] EQ [e

(Tt) max {Stvt(Wi.' - Wt* ); 0}] •

V—'1-7 gives Taking the limit of the product of the right-hand side and 1/7 firn

1

EQ [e -r(T-t) max t

fstvt (w2!_ w • • t* ); vil = T

27r-_ ote -r(T-t) vt,

where the property IE[max {z;

= for a normal random variable z with mean zero and variance y is used. From this, the proposition follows immediately. Note that this result holds under both the objective measure P and the equivalent martingale measure Q. Using (4.34), (4.36), and the general riskneutral valuation formula (2.14), the arbitrage price process of a contingent claim in the (general) factor-based stochastic implied volatility model is given by the following theorem: Theorem 4.3. The arbitrage price Il(H) at time t [0,7] of any Qattainable claim H with maturity date T in the (general) factor-based stochastic implied volatility model is given by the formula

Ht(H)

_ EQ [ e - (T-t)H1 1

Ft ]

,

Vt E [O, T] ,

(4.37)

where the expectation is taken with respect to the joint diffusion of the stock price and its volatility under the measure Q:

dSt = Strdt + Stvtdnt ,

Vt E [0,7] ,

(4.38)

and dyt = (Tit (m(t, St , St , t, r), 0) - T9 t (m(t, St , St , t, r), 0) W t ) dt

(4.39)

+;-9 t (m(t, St , St , t, r), 0) dWt* , for every t E [0,7 ] . The processes iit (M,r) and (4.10) and (4.11).

r) are defined by

(4.9),

If an analytic solution is not available, standard Monte Carlo simulation techniques can be applied to approximate the expectation in (4.37). For hedging standard European call and put options we can use the BlackScholes Greeks as defined in equation (4.23). For example, our model allows a simple and straightforward approach to the modelling and hedging of volatility risk, defined in terms familiar to practitioners in the options market, namely that of vega risk defined via Black-Scholes vegas. Since volatility is stochastic in our model, the higher order Greeks DVegaDVol and DDeltaDVol are of crucial importance. In any hedging scheme, they have to be considered in addition to the classical Black-Scholes Greeks. In the case of exotic derivatives one typically has to calculate the Greeks numerically, e.g., by employing a finite-difference approximation.

5 Properties of DAX Implied Volatilities

A theory must be tempered with reality. Jawaharlal Nehru (1889 - 1964)

Based on the results of the previous chapter, the goal of this chapter and the next is the development of a factor-based stochastic implied volatility model for DAX implied volatilities. This chapter aims at identifying the main properties of DAX implied volatilities both in a cross-sectional ("structure") and a time-series ("dynamics") setting. The next chapter is then devoted to the specification, estimation, and testing of the model. The contract specifications of the DAX option and previous studies on DAX implied volatilities are presented in Section 1. The data are described in Air 7 7 Y1) Y27 • • • 7 Yp) Section 2. In Section 3, we aim at finding a function '4(t, and (abstract) risk factors Y1, Y2, ,Y such that the difference between the observable volatility surface Fit (M, r) and the estimated volatility surface at (M, r) becomes minimal, subject to some optimization criterion. Given such a parametrization, we generate a new estimate of the DAX volatility surface for each day in the sample. This results in a multivariate time series of (estimated) risk factors. Section 4 performs a statistical analysis of this time series and determines its main features. The chapter closes with a summary of the main empirical observations.

5.1 The DAX Option 5.1.1 Contract Specifications

The option's underlying, the Deutsche Aktienindex DAX, was launched on June 23, 1988 at the Frankfurt Stock Exchange and the index level was set

5 Properties of DAX Implied Volatilities

74

to 1000 points on the base date December 30, 1987.' The DAX reflects the German blue chip segment and comprises the 30 largest and most actively traded German companies that are listed at the Frankfurt Stock Exchange. Its component issues have recently been admitted to the new Prime Standard Segment of Deutsche Btirse. The DAX is a capital-weighted performance index, i.e. dividends are reinvested. It is conceived according to the Laspeyres formula. Index calculation is performed continuously every 15 seconds2 on every exchange trading day in Frankfurt, using prices traded on Deutsche Btirse's electronic trading system Xetra. 3 As long as opening prices for individual shares are missing, the respective closing price of the previous day is taken instead for computing the index. As soon as a daily price is available for at least one index component issue the DAX is distributed. The "official" closing index level is calculated using the respective closing prices (or last prices). Trading of DAX options started in August 1991 on the official German derivatives market, the Deutsche Terminbt5rse (DTB). With the merger of DTB, now Eurex Deutschland, with the Swiss Options and futures exchange SOFFEX in 1998, Eurex was created. From the very beginning the market experienced rapid growth and is now the leading derivatives exchange worldwide. With more than 44 million traded contracts during 2002, the DAX option is the most liquid index option on Eurex and ranks among the top index options contracts in the world. 4 DAX options are cash-settled European-style options which expire on the third Friday of the contract month. The option premium is payable on the day following the trade and the minimum price change is 0.10 index points. Given a contract value of 5 EUR per index point, this represents 0.50 EUR. Before the conversion to Euro took place in 1999, the contract value of the DAX option was 10 DM per index point and the minimum price change amounted to 1 DM. At any point in time eight option maturities with lifetimes of up to two years are available: the three nearest calendar months, the three following months of the cycle March-June-September-December and the two following months of the cycle June-December. For each option maturity, at least five different strike prices are offered: one ATM strike, two OTM strikes, and two ITM strikes The strike price interval depends on the time to maturity of the option and amounts to 50, 100 and 200 index points for times to maturity of up to 6, 12 and 24 months. The DAX future, launched in 1990, is clearly associated with the option contract, nevertheless, some differences can be noted: the contract value is 25 EUR per index point and the minimum price movement amounts to half of an index point, representing a value of 12.50 EUR; the expiry months are only I

In the following, see Deutsche Borse (2003), the Deutsche BOrse website

www.deutsche boerse.corn, and especially the Eurex website www.eureschange.com . -

'Before July 11, 1997 the time interval was 60 seconds. 'Before July 1999, index calculation was based on prices traded on the Frankfurt Stock Exchange. 4 See FIA (2003).

5.1 The DAX Option

75

the three nearest months within the cycle March-June-September-December; and the contract is marked to market at the end of each trading day. Trading hours changed several times from 1991 to now, but DAX option and DAX future were traded at least from 9:30 a.m. to 4:00 p.m. Presently, trading takes place from 8:50 a.m. to 8:00 p.m. 5.1.2 Previous Studies At the German market Ripper/Gtinzel (1997) analyze the standardized implied volatility surface of DAX options using settlement prices over the years 1995 and 1996. As a means of standardization each implied volatility is divided by the ATM implied volatility. In their study, Ripper/Gtinzel (1997) estimate only one surface for the complete sample period and thus implicitly assume that the smile profile and the term structure of implied volatilities are stable throughout the two years under study. For short-lived options they report a U-shaped profile across exercise prices, i.e. a true smile, whereas options with a longer time to maturity of up to three months show an almost linear decrease of implied volatilities when the strike price rises, i.e. a skew. Similar to findings in the US, the rate of decrease is negatively related to the time to expiration. In the same way as Ripper/Gtinzel (1997), Tompkins (2000) also analyzes standardized implied volatilities on the basis of closing prices of DAX options. 5 The sample period ranges from 1992 to 1996. In a first step, Tompkins (2000) estimates the smile profile of DAX options for 18 different times to maturity, using a simple regression model. His findings support, in principle, the results of Ripper/Gtinzel (1997). In a second step, Tompkins (2000) tries to explain the pattern of implied volatilities by fitting a regression model. The set of explanatory variables includes: the strike price, the time to maturity, the futures price, as well as dummy variables marking data records that occurred before or after an extraordinary event. Combinations of these variables are also incorporated into the model. The assumption of Tompkins (2000) that the regression parameters do not change over the sample period is critical, because there is strong empirical support for time-varying implied volatility structures. 6 Moreover, due to the high collinearity of the independent variables, the interpretation of the estimated regression parameters is difficult. For instance, the time to maturity goes into seven explanatory variables, which have to be considered together to assess the term structure of volatility. Hermann/Narr (1997) use a nonparametric technique to investigate the valuation of the DAX option. They use a neuronal network, with training 6 Another study of the structure of DAX implied volatilities was performed by Bolek (1999), p. 122-123 on the basis of data for the second half of 1995. For the period 1994-1996, Dartsch (1999) investigates the time series properties of DAX ATM implied volatilities, represented by the German volatility index VDAX. 6 See Dumas et al. (1998) for the US, Pena et al. (1999) for Spain, and Hafner/Wallmeier (2001) for Germany.

76

5 Properties of DAX Implied Volatilities

data from 1995, to determine the empirical valuation function. Their method allows them to compute the partial derivatives of the pricing function which are then compared with the respective Black-Scholes values. Implied risk-neutral density functions (RNDs) derived from cross-sections of observed DAX option prices are analyzed by Neumann/Schlag (1995) Using data for the first half year of 1994, they find systematic discrepancies between Black-Scholes prices and market prices, but more astonishing, the RNDs they derived independently for call and put options did not coincide. In the absence of arbitrage, this could only be due to market frictions or estimation errors. Hermann (1999), calculating implied volatilities for DAX options from 1992 to 1997, also reports substantial violations of put-call parity. He explains this phenomenon with transaction costs that have to be incurred when short selling the DAX portfolio. As, in practice, typically DAX futures, exhibiting significantly lower transaction costs, are involved in this type of arbitrage, the argumentation of Hermann (1999) is only partially cogent. In contrast to the aforementioned studies, Fengler et al. (2000) model daily changes in the structure of implied volatilities, using closing prices of DAX options for the year 1999. 7 From the nonparametrically estimated daily implied volatility surface, they construct times series of implied volatilities for different times to maturity and degrees of moneyness by evaluating the implied volatility function at the respective grid points. On the basis of these volatility series, Fengler et al. (2000) then perform a common principal component analysis. The results indicate that three factors explain about 95% of the daily variations in implied DAX volatilities. The three factors are respectively interpreted as shift, slope and twist.

5.2

Data

5.2.1 Raw

Data and Data Preparation

Our database contains all reported transactions of options and futures on the DAX, traded on the DTB/Eurex over the sample period from January 1995 to December 2002. 8 In total, the sample period comprises 2010 trading days. There are no restrictions concerning the time to maturity and the strike price. Each data record contains the following information: the trade time, accurate to the second, the transaction price, the volume traded, the maturity date of the contract, and additionally, in the case of the DAX option, the strike price. Information on bid and ask prices is not available. Our database also comprises daily series of the following interest rates: the overnight rate, the money market rates for 1, 3, 6, and 12 months, and the 'The study of Fengler et al. (2000) for DAX options is similar to the analyses of Skiadopoulos et al. (1999) for S&P 500 options and Cont (2001) for S&P 500 and the FTSE 100 options. 8 We are grateful to Eurex Deutschland for providing us with these data.

5.2 Data

77

two year swap rate. To be, in fact, a risk-free rate, the two year swap rate is converted into an equivalent zero rate, using the bootstrapping method. 9 For the years 1995 through 1998, the rates are denominated in DM, and after that they are denominated in Euro. All interest rates are converted to continuously compounded rates and expressed in the daycount convention Actual/Actual. For an arbitrary time period T, the r-period risk-free interest rate r at time t is obtained by linear interpolation between the available rates enclosing 7-.10 In a next step, we compute the implied volatility accurate to 10 -4 for each options trade, using the standard Black-Scholes (1973) option pricing formula and a combination of the bisection and Newton-Raphson method. The Black (1976) model, which could be regarded as an alternative, presumes the existence of a liquid futures contract for each option's maturity. This condition is not met at the German futures market. Apart from the option price and the strike price, three parameters are required to compute the implied volatilities: the time to expiration, the riskfree rate and the level of the underlying index. Let n E {0, , N}, N = 2009, denote the trading day and 4, the corresponding trading date. 11 The option's expiration date is symbolized by To. Whether the time to expiration (To - tr,) should be measured in trading days or in calendar days is debatable. If the volatility of the stock price is caused solely by the random arrival of new information about the future stock returns, then the use of calendar days might be advisable. On the other hand, if volatility is caused largely by trading, then time to expiration may be better measured in trading days. In any case, the difference between calendar and trading days, expressed as a fraction of one year, is small except for very short-term options. 12 In the following calculations the time to maturity is measured as a proportion of 365 (calendar) days per year. Let 1 (1 = 0, , L) be the trading minute of an options transaction." The underlying index S,,,i on day n at minute 1 is derived from the current price Fn ,1 of the futures contract most actively traded on that day." The maturity of this contract, which is normally the nearest available, is denoted by TF. The value F,1 (TF) corresponds to the average transaction price observed in the TF-futures contract in minute 1 on day n. To obtain the corresponding 9 For a detailed description of the bootstrapping method, see Miron/Swannell (1991), pp. 91-117. The riskless rate r is dependent on the current time t and the investment horizon T. For ease of exposition, we suppress these indices. More information on the interpolation of interest rates can be found in Cremers/Schwarz (1996). 11 For our convenience, we call N the sample size, though N is in fact the sample size minus one. 12 See Hull (2000), p. 256. "Since trading hours changed through time, L is time-dependent. To keep notation simple, we suppress the time index. 14 We use S, 1 as a shortcut for Si . The same notational principle applies to forward prices, implied volatilities, etc., in discrete time.

°

,

78

5 Properties of DAX Implied Volatilities

index level we solve the theoretical futures pricing model (see Theorem 3.2 in Chapter 3) 1-4,1 (TF) = Sn ,ie r(TF -t,. ) (5.1) for Sn ,1. 15 If no future is traded at minute 1, we exclude all options transactions that took place in this minute from our database. This procedure ensures simultaneous options and underlying prices, i.e. their respective time stamps diverge by not more than one minute. 5.2.2 Correcting for Taxes and Dividends

Put-call-parity requires that the implied call volatilities do not systematically deviate from the implied put volatilities with the same degree of moneyness. However, on a number of trading days a scatterplot as shown in Figure 5.1 was observed. Call and put implied volatilities are symbolized by circles and triangles, respectively.The systematic differences apparent from Figure 5.1 can be traced back to a biased index level caused by dividend payments. The DAX index calculation rests on the assumption that cash dividends are reinvested after deduction of the corporate income tax for distributed gains from the gross dividend DIV. If the marginal investor's tax rate km is smaller than the corporate income tax rate for distributed gains kd, he receives an extra dividend of ADIV = (kd - km )DIV. In the following, this is referred to as the "difference dividend". Since the value of the dividend payment to the investors right before distribution is higher than the reinvestment amount after fictitious taxes, the continuously updated DAX falls by an amount of ADIV on an ex-dividend day. If k m > kd , the opposite holds. As a consequence, the difference dividend has the same effect as an ordinary dividend in the case of unprotected options and futures. In the following we assume that dividends are sure payments. Letting AD/Vt ,2- denote the time T terminal value of the difference dividend incurred between dates t and T, we get a modified version of our futures pricing formula (5.1): F„,i(TF)= S„,ief(TF-t ^ ) - AD/Vt„,TF, (5.2) or equivalently = Fn,i(TF)e-

r(TF—t.)

LID/Vt„,TF

r(TF-4‘ ) .

(5.3)

The difference dividend does not only have an impact on the valuation of DAX futures, but also influences the valuation of DAX options and the form of the put-call-parity. The modified put-call-parity is given by the equation: (K, To) -

To) = Sn,1 - AD/Vt„,T o e - r(70-tn ) - Ke - r(lb -t n ) , (5.4)

I5 Using the futures-based implied index level rather than the reported index level as the underlying price has also been suggested in a study for the S&P 500 options market by Jackwerth/Rubinstein (1996), p. 1616.

5.2 Data

-0.05

-0.10

0.00 0.05 h (strike pricenuttres price)

0.10

79

0.15

Figure 5.1. Implied call and put volatilities for different degrees of moneyness

(defined as the log ratio of strike price to futures price) on March 27, 1995 (time to maturity: 25 days)

or, combining equations (5.3) and (5.4): Cn,i(K,To)-Pn,i(K,To)=Fn,t(TF)e -r(TF-40 ±ADIVt„,T F,To -Ke -r(7b-t" ) ,

(5.5) where Cn ,l(K,To) denotes the price of a DAX call with strike price K and maturity date To on day n at minute 1, Pt,n (K,To) is the price of the corresponding put and AD/Vt n ,TTo is defined by AD/Vt,‘,TF,To = AD/Vt„,T F e-r(TF-t)

- LIDIVt , To e- r(To -t. ) .

(5,6)

Only if the underlying price in the implied volatility calculation is set to = Sn,1 AD/1/2„,To e-r(70-t " ) , put-call-parity implies that the implied volatility of a call equals the implied volatility of a put. This corresponds to the usual treatment of dividends within the Black-Scholes model. Using (5.3) and (5.6) the adjusted underlying price ,§71,/ can also be written as: §72,1 =

,

Fo(TF)

e—r(TF —t n

Al-irlirli,Tp,To

•

(5.7)

Apparently, our original futures pricing model (5.1) yields the correct underlying price :Sçn ,i if the option's and future's expiration coincide. In all other

5 Properties of DAX Implied Volatilities

80

cases, the calculated underlying price, which was hitherto assumed to be equal to Fn ,i(TF)e — r(TF —t - ) , has to be adjusted according to equation (5.7). In the derivation of the difference dividend we have (implicitly) assumed that the DAX futures price is determined by private investors. Alternatively, we could also assume that the DAX futures arbitrage that leads to this price is mainly carried out by institutional investors. Under this assumption, Wallmeier (2003) derives a formula that is similar to (5.6), but additionally involves the tax rate for accumulated gains as a further key determinant of the difference dividend. 16 A further remark concerns the tax system. Formula (5.6) was derived on the basis of the so-called "Kbrperschaftsteuer-Anrechnungsverfahren" This tax system was in force until 2001. In 2001, it has been replaced by the "Halbeinkiinfteverfahren" . Under the "Halbeinkiinfteverfa.hren", it can be shown that the difference dividend is the same, no matter if the futures arbitrage is carried out by private investors, institutional investors or both. Furthermore, the difference dividend is supposed to be much smaller than it was before under the " Mirperschaftsteuer-Anrechnungsverfahren". The adjustment amount AD./14„,T,,T, is identical for all trades on day As the dividend information is publicly available, AD/Vt,„TF,To can, in n. principle, be estimated. However, we propose an implied method to estimate ADIV,T,,,To . This method, being more easy and not so error-prone, relies on the assumption that put-call-parity holds. In fact, this a very weak assumption as put-call-parity does not depend on a particular stock price model, but only postulates the absence of arbitrage opportunities. A major advantage of the implicit estimation method is that it can be applied without modification no matter if the futures price is determined by private or by institutional investors and regardless of which tax system is currently under use. If To = TF, the term AD/Vt n ,TF ,To is set to zero. Otherwise, we preselect all options with a log ratio of strike price to futures price ln(K/Fn ,i) between —0.1 and 0.1 and identify pairs of puts and calls with the same strike price and the same expiry provided that they are traded in the same 5-minute interval. For each pair, an estimate AD/V t„,T,,T, for ADIVt„,T F ,To is computed from (5.5). If the paired call and put options are traded in different minutes 11 and 1 2 within the 5-minute interval, the futures price in equation (5.5) is set equal to the average of F,11 (TF) and F,12 (TF). To avoid biases due to outliers, we do not consider ADIV t,TF,To values greater than 15 points. 18 If the number of matched pairs on day n exceeds one, we take the average of the individual . 17

Wallmeier (2003), pp. 173-176. distributed gains are only subjected to the personal income tax of the owners. This is achieved by the German "K6rperschaftsteuerAnrechnungsverfahren". For a thorough analysis of the influence of cash dividends on the pricing of DAX futures see Ri5der (1994), p. 86 ff. 18 A thorough analysis of those months where AD/1/4„TF,To is supposed to be high in absolute values reveals that 15 index points can be regarded as an upper bound to LID/3/4„,TF,T0• 16 See

17 Ultimately,

5.2 Data

81

adjustment amounts as final estimate of ADIV. The same procedure is repeated for all option's maturities traded on day n. Using these estimates of AD/V t„,TF ,To we compute the modified underlying index level according to (5.7) and newly calculate all implied volatilities. The adjustments are highest in April and on trading days after the third Friday in March. Here the nearest option's maturities are April and May, whereas the next future expires in June. Since most DAX corporations pay dividends in May, these fall into the period between the expiration dates of option and future. An inspection of all scatterplots reveals that this implied estimation of the relevant underlying index level solves the problem of the difference dividend. For example, after increasing the unadjusted DAX index by approximately 8 index points, Figure 5.1 turns into Figure 5.2, in which call and put implied volatilities no longer systematically deviate from each other. .

0.25

0.20 -

0.15 o

3a.

0.10

-

-

0.05 -

0.00 -010

0.00

0.05

0.15

In (strike price/futures price)

Figure 5.2. Implied volatilities on March 27, 1995 after increasing the underlying

index level by approximately 8 points (time to maturity: 25 days)

In a final step, we eliminate all options that have implied volatilities higher than 150%.

82

5 Properties of DAX Implied Volatilities

5.2.3 Liquidity Aspects

As Figure 5.3 shows, at the DAX options market, liquidity is very much concentrated on short-term options and declines exponentially with increasing time to expiration. Of the total number of 6,171,700 options (transactions), 87.44% expire within the next 90 calendar days and 95.19% within the next 180 calendar days.

ii 0 - 29

30- 59

60 - 89

90 - 179

180 - 359

360- 730

Time to maturity in days

Figure 5.3. Number of transactions for different classes of time to expiration over the sample period 1995-2002

The call trades distribution across degrees of moneyness, here defined as the log ratio of strike price to futures price, is clearly skewed to the left whereas the put trades distribution is skewed to the right (see Figure 5.4). This means that OTM options are traded far more frequently than ITM options. Since the estimation of the volatility surface in next section requires a sufficient variety of strike prices, we include both calls and puts in our empirical study. As can also be seen from the graph, the strike prices of most of the option contracts traded lie within a relatively small range around the ATM point. For instance,

5.3 Structure of DAX Implied Volatilities

83

61% of all traded contracts have a strike price that is not more than 5% away from the prevailing ATM strike.

Calls Puts 1,500,000

1,250,000

1,000,000

750,000

500,000

250,000

43 irl' ,A ._,, A- 6,,A. ...,- • n .). M.e. ij A." A ." A. N

■

N

K

IV '

11<-1 .

6 C% •

•

...\ Nci■\ c‘qj g...k)' Nr• p-...• ,,,,.ç .

>\

1,1:

116

,..../Zrfr 1,..1 ..

,,,. . rp..0 Y

l.43

ZI'fr m

..."-e

In(strike price/futures price)

Figure 5.4. Distribution of call and put trades over the sample period 1995-2002 for times to maturity below 180 days

5.3 Structure of DAX Implied Volatilities 5.3.1 Estimation of the DAX Volatility Surface Overall Estimation Strategy g(t, M) Ty Y17 Y27 • • • , yp ) and For any fixed time t, we aim at finding a function :Y3., such that the estimated implied volatil(abstract) risk factors Y2 • • • Y ities are as close as possible to the observed implied volatilities. Because the relationship between the implied volatility o-t (M, r), the moneyness M and the time to maturity r = T — t is supposed to vary through time, we include

84

5 Properties of DAX Implied Volatilities

only one trading day's data in any cross-sectional analysis and estimate the DAX volatility surface (M, T) separately for each day n E {0, , N}. One possibility to analyze the volatility surface from the daily database is to assign all records to different classes according to their degree of moneyness and their time to maturity." The volatility surface is then represented by the relationship between the moneyness classes, the time to maturity classes, and the groups' average implied volatilities. Yet, the determination of the class boundaries turns out to be difficult. If there are many classes, some of them will hardly be occupied. With a small number of classes, on the other hand, options with markedly different degrees of moneyness would be combined. A further problem occurs from the use of grouped data in the subsequent estimation of the relation between implied volatility, moneyness and time to maturity. Statistical inference becomes more complicated, since standard test statistics are often not applicable or difficult to interpret. As, on the other hand, the grouping procedure offers no apparent advantages over the direct estimation of the volatility surface from the original data, we refrain from using it. Ait-Sahalia/Lo (1998), Cont/Fonseca (2002), and Fengler et al. (2000), among others, use nonparametric methods (e.g., kernel regressions) to estimate the daily volatility surface. 20 However, these methods tend to be very data intensive. Also, they tend to fill in missing data in a non-intuitive way. For instance, in a downward sloping implied volatility smile where a range of strike prices has no observations, we would find question-mark shaped segments. This could create artificial arbitrage opportunities. 21 For these difficulties, we employ a parametric approach and model the DAX volatility surface for an arbitrary but fixed day n E {0, , NI across degrees of moneyness M and times to maturity T by the regression model 4(Mn,3

flj = -

,j3

f3 n)

En,j7

j = 1, • • Jrt

(5.8)

where 0 is a regression function, fi n is a vector of regression coefficients, e, is a random disturbance, and j indexes the J sample observations on day n. 22 The estimation procedure involves the following four steps: 1. 2. 3. 4.

Specify the moneyness measure M. Determine the functional form of -4(-). Specify the regression model and choose an appropriate estimation method. Test on violations of general no-arbitrage relations.

"In the case of smiles, Ane/Geman (1999), Pena et al. (1999) and Ripper/Gtinzel (1997) apply this method. 20 See Ait-Sahalia/Lo (1998), Cont/Fonseca (2002), and Fengler et al. (2000). 21 See Jackwerth/Rubinstein (1996). 22 Note that Fr„,, = an ,i, by definition.

5.3 Structure of DAX Implied Volatilities

85

Choice of Moneyness Measure The moneyness of an option should reflect the degree to which an option is ITM or OTM. This means that not the strike price itself but its relation to the

underlying price is crucial. In addition to the strike price and the underlying price, moneyness should also depend on time to maturity as the following argument shows: a call option with a strike price of 100 would be classified as far OTM if the current underlying price is 95 and the time to maturity is 1 day. Yet, the same call having now a time to maturity of, let's say 1 year, could be reasonably well rated ATM, because the probability that the underlying price reaches or exceeds the strike price is distinctly higher than it is in the first case. In the literature, different moneyness measures have been proposed that explicitly account for time to maturity. First of all, Natenberg (1994) has suggested the measure:

ln() (5.9) where Ft (T) = St er (T-0 denotes the forward price of the stock. When expressed with respect to this measure, standardized volatility smiles for different times to maturity look very much alike. 23 Dividing (5.9) by the ATM implied volatility and replacing the forward price Ft (T) by the spot price St leads to the moneyness definition of Tompkins (1994). This is a measure of moneyness in standard deviation units, sometimes referred to as standardized moneyness. Compared to the Natenberg moneyness, it has the drawback that the ATM implied volatility cannot be observed directly but has to be estimated. Finally, the option's Black-Scholes delta is sometimes used as a measure of moneyness. 24 With respect to this measure, a call option is termed ATM if its delta is 0.5, and ITM (OTM) if its delta is above (below) 0.5. The delta measure has the advantage to be bounded by 0 and 1 in the case of a call option, and -1 and 0 in the case of a put option. Although the dependence on time to maturity is in general a desirable property of each moneyness measure, in the context of our stochastic implied volatility model it causes a problem. To see this, let us consider, for instance, the Natenberg (1994) moneyness. Apparently, the limits of ln ( ster/&._,) ) m(t, St , K ,T, r) — and

Natenberg (1994), p. 411. moneyness in terms of the option's delta is common in the foreign exchange options markets. See, e.g., Malz (1997). 23 See

24 Expressing

86

5 Properties of DAX Implied Volatilities i n (k )

am ,

—(t St , K, T, r) —

2(T — t)1/17-

2 Or

t

do not exist when t approaches T. Consequently, the Natenberg (1994) moneyness (function) is not a valid moneyness (function) in the sense of Definition 4.1. Similar arguments hold for the Tompkins (1994) moneyness and the option's delta as a measure of moneyness. Considering further alternatives, one of the most common moneyness measures is the simple moneyness. 25 It is defined as the ratio of strike price to stock price, i.e. K/S t , or the inverse of it. Instead of the spot price St , it is also common to use the forward price Ft (T) = Ster(T-t) . Whether one should choose one or the other, depends on the assessment of what "ATM" means Traders tend to think of an ATM option as one whose exercise price is approximately equal to the spot price. However, according to the Black-Scholes model an option would be termed ATM if it has a delta of 0.5. This is exactly the case if the spot price in the Black-Scholes formula is replaced by the forward price. The second alternative is theoretically more correct and is thus preferred. 26 A slight modification of the simple moneyness is the log simple moneyness, defined by

M = Mt = ln

K Ster(T -0 )

K )

Vt E [0,71

(5.10)

r > 0, and all K > 0. As can easily be verified, the for all St > 0, T E simple moneyness (function) as well as the log simple moneyness (function) meet the requirements of a valid moneyness (function). When expressed in terms of log simple moneyness, an option is said to be ATM, if M = 0. A call (put) is said to be ITM (OTM) for M < 0 and OTM (ITM) for M> 0. This allows for an easy interpretation of the moneyness measure as the amount a put option is ITM or a call option is OTM as a proportion of the forward price. 22 In all further calculations, we use the log simple moneyness, which we shall often just refer to as "the moneyness". Given our sample data, we assign each options trade its log simple moneyness. Specifically, for an option expiring in To, we compute Aln,1

= in

()

The simple moneyness is used, e.g., by Cont/Fonseca (2002) and Ledoit et al. (2002). 'For a theoretical and empirical justification to define moneyness with respect to the forward price rather than the spot price, see Natenberg (1994), pp. 106110 and Roth (1997). It should be noted, however, that under normal interest rate conditions, there is not much difference between the "spot moneyness" and the "forward moneyness", at least not for short-term options. 27 This interpretation results from a first order Taylor series expansion of the log moneyness: ln(ICIFt (T)).:1K Ft (T) — 1. 25

5.3 Structure of DAX Implied Volatilities

87

where Fn,1 (To) is the theoretical To-futures price computed with (5.1) for Sn,1 replaced by the adjusted index level Functional Form of

4(.)

For a fixed time to maturity, the strike profile of implied volatilities is often modelled by a quadratic function

:4(m, 0 ) = 0 1 + 02m + 03m2

(5.11)

where /31 , (32 , and 03 are parameters." In principle, this function is able to capture a true smile as well as a skew pattern. An inspection of our data, though, reveals that this approach is slightly biased in that it sometimes underestimates the implied volatility of options with M > 0. Including the variable moneyness cubed as in Tompkins (2000) does not eliminate the bias. To account for the asymmetry of the strike pattern of implied volatilities Hafner/Wallmeier (2001) propose a quadratic spline function with the two segments M < 0 and M> 0:

•, /3) = fl +132M + 03M2 + /34 (D M2) ,

(5.12)

where the dummy variable D takes the value 0 for M < 0, and the value 1, otherwise. Obviously, (5.12) is only once differentiable in M at the threshold M = 0. However, this presents a violation of Assumption 4 of the general factor-based stochastic implied volatility model, which requires the function g 0 to be twice differentiable in M.29 Furthermore, the inclusion of the variable D M2 increases the correlations between the explanatory variables and the higher the correlations become, the less precise our parameter estimates will be.39 For these reasons, we refrain from using (5.12), but model the volatility smile by the quadratic function(5.11). Graphical analyses of the term structure of volatility "4(•, T, /3) reveal for most days in the sample period a pattern that can be very well approximated by a square root function or a logarithmic function. This is also supported by simple regression analyses. Because the first derivative of the square root function Irr evaluated at r = 0 is zero, the modelling of the volatility term structure by a square root function would lead to problems with regard to the factor-based stochastic implied volatility model. In fact, one can show that the implied volatility processes (expressed in terms of K and T) do not exist when the volatility term structure is represented by a square root function. On the other hand, if we model the volatility term structure by the function "See Shimko (1993) and Ripper/Ganzel (1997). problem can be solved by using a polynomial of higher degree. See, e.g., Brunner/Hafner (2003). "This problem is known as multicollinearity. See Greene (1993), pp. 266-273, for details. 29 This

88

5 Properties of DAX Implied Volatilities

ln(1 + r), no problem occurs, while at the same time, roughly the same fit to the data is achieved. Combining the findings from the analysis of the volatility smile and the volatility term structure, we posit three alternative functional forms for the volatility surface: 31 Model 1: Model Model

-4(M, r 0 ) = 01 ± 02M 133M 2 04 111(1 T), 2: (M T 0) = + 02M ± 133M2 ± 04 ln(1 + r) + 05 M ln(1 + 7), (M,T, 0) = 01 ± 132M ± 03M 2 + 04 ln(1 + r) + 05 M ln(1 + 7) 3: ,

-

+ )36M2 1n(1 + In Model 1, the volatility smile and the volatility term structure are supposed to be independent. Model 2 allows the slope of the volatility smile to vary with time to maturity, and Model 3 additionally accounts for a varying curvature of the volatility smile across different times to maturity. A preliminary statistical analysis reveals that Model 1 and Model 2 are not flexible enough to capture the typical shape of the DAX volatility surface. There is unexplained structure left in the residuals. In particular, the first two models cannot reproduce the "flattening-out" effect that is commonly observed. This effect refers to the fact that the volatility smile becomes flatter when the time to maturity increases. 32 Model 3, on the other hand, is expected to capture most of the variations in implied volatilities attributable to variations in the degree of moneyness and variations in time to maturity. Regression Model and Estimation Method The full specification of Model 3 is given by an,3 = /3 i,n + 02,nMn,3 + /33,n4.2y,3 + /34, n ln(1 +

(5.13)

+05,nAln,i In(1 + Tn,i) + 06,n ML ln(1 rn,i) + en,i, where n E {0,...,N and j = 1, , J. Although the model is expected to capture most of the cross-sectional variations in implied volatilities, it might yet not be well specified as the variables M and M ln(1 +r), on the one hand, and M2 and M2 ln(l+r), on the other hand, and hence also [32 and 06, and 03 and 06, are highly correlated. When we hypothesize that these relationships are stable over time, the model can be simplified by imposing the following restrictions on the regression coefficients: }

05,n = g102,n)

n= 0,...,N,

(5.14)

and 31 Similar parametrizations have been suggested by Dumas et al. (1998) and Ane/Geman (1999). 3 2 See also Das/Sundararn (1999).

5.3 Structure of DAX Implied Volatilities 06,n =

22i33,n,

n = 0,

, N,

89

(5.15)

where 21 and 22 are known constants. Substituting (5.14) and (5.15) into (5.13) leads to the following unrestricted regression model: an,j = /1,n I32iti ,nnj ( 1 + el ln

(1 + rn,i ))

(5.16)

+03, n721,i 11/ ( 1 + 22 in (1 + rn,j)) + )34,nln ( + Tn,j) En,j • In practice, 21 and 22 are not known but have to be estimated. Consequently, it is not possible to apply the least squares method directly to (5.16). Instead, we propose a two step estimation procedure.33 In the first step, we estimate the original regression model (5.13) for all days n = 0, , N. From the obtained time series of regression coefficients ;3i ,n (i = 2, 3, 5, 6), we then estimate 21 and 22 by fitting the auxiliary regression models -

g:42,n +

En",

n -= 0,

, N,

(5.17)

&,n = 02;33,n +

En

n = 0 , • • • , N,

(5.18)

and with €( 1 ) and e(2) being random disturbances. The final estimates of the regression coefficients (31 , , 04 are obtained in the second step by applying the least squares method to model (5.16) where the unknown constants 21 and 22 . , i.e. to: are replaced by their estimates "e,i and ".ô2 ln (1 + r)) = 01,n + 02,n Mn,3 (1 04,n ln (1 + r,) j +03,7114,3( 1 i321n (1 +

(5.19) Enj.

The implied volatility of deep ITM calls and puts is very sensitive to changes in the index level. Since small errors in determining the appropriate index level are unavoidable, the disturbance variance of regression models (5.16) and (5.19) is supposed to increase as options go deeper ITM. Residual scatterplots support this presumption. Using the White-test, the null hypothesis of homoskedasticity was rejected in about 70% of all regressions. To account for the heteroskedasticity of the disturbances we apply a weighted least squares estimation (WLS) assuming that the disturbance variance is proportional to the positive ratio of the option's delta and vega. 34 This ratio indicates how an increase in the index level by one (marginal) point affects the implied volatility of an option, if its price does not change 33 This estimation procedure is similar to the two-step Cochrane-Orcutt method that is sometimes used to estimate regression models where the error terms are autocorrelated. For a detailed description of the Cochrane-Orcutt method, see Kmenta (1997), pp. 314-315. delta and vega are computed using the implied volatility of the corresponding option. The delta of puts is multiplied by —1 to obtain a positive ratio. "

The

90

5 Properties of DAX Implied Volatilities

In view of the large number of intraday transactions it is not astonishing that some extreme deviations occur representing "off-market" implied volatilities. They can, for example, be due to a faulty and unintentional input by a market participant. In this case, the trade can be annulled if certain conditions are fulfilled. To exclude such unusual events we discard all observations corresponding to large errors of more than four standard deviations of the regression residuals where the standard deviation is computed as the square root of the weighted average squared residuals. We then repeat the estimation on the basis of the reduced sample until no further observations are discarded. This procedure is known as applying the "4-sigma-rule" or "trimmed regression".35 We examined the impact of this exclusion of outliers and found it to be negligible in an but very few cases. On the one hand, the precondition of stationarity is best achieved by selecting data from a short time window. On the other hand, however, if the environment does not change dramatically, a larger database may improve the precision of the regression estimates. Our analysis of this trade-off argues in favor of the second view. When selecting a two-hour interval of 2:00 to 4:00 p.m., for example, the average coefficient of determination drastically decreases compared to regressions based on all transactions of one day. In addition, imposing this restriction often strongly reduces the range of strikes and maturities for which call and put prices are available. Therefore, we do not restrict the time window. Certainly, new pieces of information and large intraday variations in the underlying index level may alter the shape of the volatility surface. But scatterplots of moneyness, time to maturity and implied volatilities suggest that intraday the volatility surface is roughly constant. On some very few days, however, the volatility surface experiences a parallel shift within the day. 36 A large percentage of all traded DAX options in the period from 1995 to 2002 features a degree of moneyness between —0.25 and 0.20 (see Figures 5.3 and 5.4) and a time to maturity below 180 days. We discard all observations outside this range in order to eliminate potential problems with extreme degrees of moneyness or time to maturity. As options with fewer than 5 days to maturity have relatively little or no time premium and hence the estimation of volatility is extremely sensitive to measurement errors, we also exclude them. To always ensure a good fit to the data, we eliminate all days from the sample where the adjusted coefficient of determination is lower than 60%. In total, these are 29. In general, plotting the residuals did not reveal any remaining violations of the assumptions of the chosen regression model.

Kmenta (1997), p. 219 and Sachs (1972), p. 265. "This happened, for instance, at the four most extreme market decreases caused by the Asian and Russian crisis and the September 11th Terrorists Attacks (October 28, 1997, August 21, 1998, October 1, 1998, September 11, 2001). 35 See

5.3 Structure of DAX Implied Volatilities

91

General No-Arbitrage Relations Let us consider the market for standard options at an arbitrary time t. If there are no arbitrage opportunities, then standard options satisfy four general arbitrage relations: 37 1. Hedge relation: for any maturity T, T > t, the value of a call is never greater than the stock price and never less than its intrinsic value: St > Ct (K,T)> max {St

—

K e —r(T—t) ;0} ,

VK > 0.

(5.20)

2. Bull spread relation: for any maturity T, T > t, the value of a vertical bull call spread is nonpositive or, respectively, the call price function with respect to strike is monotonically decreasing. The slope of the call price function is never less than -1:

—1 < -

OCt (K,T)

at(

<0

VK > 0.

(5.21)

-

3. Butterfly spread relation: for any maturity T, T > t, the value of a butterfly spread is nonnegative or, respectively, the call price function with respect to strike is convex: 82 Ct (K, T)

?°'

VK > 0.

(5.22)

4. Calendar spread relation: for any strike price K > 0, the value of a calendar spread is nonnegative or, respectively, the call price function with respect to time to maturity is monotonically increasing:

OCt (K,T) tn,

0,

VT > 0.

(5.23)

To check these conditions, we consider for each day n E {0, , NI an equally spaced grid of 200 options exhibiting degrees of moneyness between —0.15 and 0.10 and times to maturity between 5 and 120 days. The call option prices are calculated on the basis of the estimated volatility surfaces, and the derivatives in equations (5.21), (5.22) and (5.23) are computed numerically. The moneyness boundaries .114i, = —0.15 and Mu = 0.10 and time to maturity boundaries TL = 5/365 and Tu = 120/365 were chosen such that the number of observations outside these intervals always suffices to ensure an accurate estimate of the implied volatilities within and at the boundaries. Since on many days option trades with a degree of moneyness greater than 0.10 or lower than —0.15 or a time to maturity of more than 120 days, respectively, do not occur, we were not able to enlarge the chosen boundaries. If any of the four no-arbitrage conditions is violated on a specific day, this day is excluded from 37 See,

e.g., Brunner/Hafner (2003) and Carr (2001).

92

5 Properties of DAX Implied Volatilities

the sample. In the overall sample, this happens on 42 days.39 The remaining

days are (re)numbered from 0 to N, with N now being equal to 1938. 39 5.3.2 Empirical Results Goodness of Fit For each day n E {0,...,N } , N = 1938, we estimate a regression of implied volatility on moneyness and time to maturity following the two-step procedure described above. As the result of the first regression we obtain time series of the daily coefficient estimates for the parameters 02, 03, 05, and )36 . These are used to estimate the model constants pl and 02 . We get: -P i = —1.6977 and -P.2 = —3.3768. The corresponding R2 values of 90.45% (Model 5.17) and 95.61% (Model 5.18) support the assumption of an almost deterministic, linear relationship between 02 and 05 , on the one hand, and between 05 and 06 , on the other hand. Repeating the estimation of pi and p2 , using different subsamples of the data, the estimates turn out to be quite stable. This suggests that the relations between )32 and 05 and between 05 and #6 are approximately time-invariant. Based on the parameter estimates -P i = —1.6977 and i*52 = —3.3768, we run regression (5.19). Across the 1939 days in the sample the average adjusted R2 value is 92.44% and the median adjusted R2 value amounts to 94.58%. 49 For comparison, the average adjusted R2 value obtained under the original regression model (5.16), using the same sample, is 93.00%. The loss in accuracy of 0.70% seems acceptable, when contrasted with the increase in model parsimony. All in all, the high R2 values suggest that our regression model captures most of the variation in implied volatilities attributable to moneyness and time to maturity. To further assess the quality of our model, the mean absolute error of the regression, i.e. the mean of the absolute deviations of the reported implied volatilities from the model's theoretical values, is computed each day. For almost all days in the sample, we find this measure to be well within the average bid-ask spread. 41 38 Of the four no-arbitrage relations, the butterfly spread relation is violated most frequently. 39 This results from 2010 — 29 — 42 — 1 = 1938. "It should be noted that in the case of a WLS regression model there exists no single generally accepted definition of R2 . The reported values are based on the non-weighted WLS regression residuals. The meaning of this R2 is not exactly the same as in an ordinary least-squares regression (OLS) regression. For more details, see, e.g., Greene (1993), p. 399. 41 As our database does not contain information on bid and ask prices, we use the average bid-ask spread of all liquid option contracts quoted on December 31, 2002 as a proxy for the bid-ask spread in the whole sample period. We find this value to be roughly 0.3 volatility points.

5.3 Structure of DAX Implied Volatilities

93

In a final analysis, we compute the estimated ATM volatility of DAX options with a time to expiration of 45 calendar days:

ATM„ = LZ(0, 45/365, ;3„),

n = 0, . . . , N,

(5.24)

where 73, denotes the estimated parameter vector on day n, and compare this variable with the German volatility index VDAX. This index represents the implied volatility of ATM DAX options with a remaining lifetime of 45 days. It is constructed as follows: for each DAX option's maturity traded at a given point in time, the Eurex calculates a volatility subindex based on the implied volatilities of the two calls and puts with strikes nearest to the DAX forward price for that maturity. The VDAX is then determined by linear interpolation between the two subindices representing times to maturity next to 45 days.42 Figure 5.5 shows that the ATM variable and the VDAX are almost identical although the estimation methods differ. The median of the difference between ATM and VDAX amounts to —0.0013. The strong correspondence between the two indices manifests itself in an almost perfect positive correlation of 0.9951 within the sample period."

Il 11 III 11111 1 IIIITT 1 1l 03 014 01 0 2 03 04 01 02 03 04 01 0 2 03 04 01 012 03 CH 01

11 11 111 1111111 Iuhlhhlhhlhh l

01

ca

1995

1996

1997

1996

1999

1111111111111,11111111111111

oa

03 04 01 02 03 04 01

2000

2001

oa

li

03 04 01

2002 2003

Date

Figure 5.5. VDAX and ATM on a daily basis over the sample period 1995-2002

Deutsche BtSrse (1999). largest difference between VDAX and ATM was observed on October 4, 2002 with 7.90 percentage points. A close examination of this day's data supports the correctness of ATM. 42 See

43 The

5 Properties of DAX Implied Volatilities

94

Average Parameter Estimates Table 5.1 reports the mean and the standard deviation of the daily coefficient estimates for each parameter, as well as the t-statistic for the mean. The (sample) standard deviation is calculated as

=

— N E

- ai)

2 ,

N

(i n=0

n=0

where

is the parameter estimate of parameter i on day n and

the mean of

denotes

kn. The t-value of 73i is then given by ---16i/sOi-

Table 5.1. Mean, standard deviation and t-value of the daily parameter estimates over the period January 1995 to December 2002

so , (t-value)

02 902 (t-value)

=

..

/33 s-43 (t-value)

134

so, (t-value)

1.4594 0.2361 —0.4966 0.0166 0.1031 (100.84) 0.1427 (-153.23) 1.0369 (61.98) 0.1397 (5.23)

Figure 5.6 gives a graphical representation of these results. It shows a

plot of the average estimated volatility surface, i.e. the function 4 - (M, 7- , 0), for different degrees of moneyness M E [ML, Mu] and times to maturity T E [TL,TU] For the interpretation of the regression results, it is convenient to recall the regression function:

-4(m, 7", 0) = 01 ± 02M (1 + ei ln (1 ± r)) +03M2 (1 + 22 ln (1 + 7-)) + 04 ln (1 + 7) .

(5.25)

The parameter 0 1 is common to all implied volatilities constituting the volatility surface. It may therefore be interpreted as the general level of volatility in the market. It should be closely related to the volatility of the underlying index. During the sample period, the average estimated value of 01 , i.e. 01 , amounts to 23.61%. The shape of the volatility smile is determined by the parameters 02 and 03 as the differentiation of the function ( M,- r , )3) with respect to moneyness shows:

ô-4(111 ' T ' fi) — /32( 1 + ln (1 + -7- )) + 203 M(1 + p2 ln (1 + r)), (5.26) 8M 492-4(m,T, /3 ) = 2,8 3 (1 ± 0 2 ln + 71)am-2

5.3 Structure of DAX Implied Volatilities

95

.0A5

Figure 5.6. Average estimated volatility surface --4(M, T, /3) for the sample period 1995-2002

Given el the parameter /32 reflects the common part of the slope of the volatility smile. As expected, its average estimated value of —0.4966 is negative. The curvature of the smile is represented by the parameter /33 . Since (1 + -02 ln (1 + T)) is positive for all T E [TL TU1 and the daily parameter estimates of 03 are mostly positive, the volatility smile is typically convex. The degree of convexity is however often small. On average, the curvature of the smile amounts to 1.4594. The minimum of the smile is almost always located at degrees of moneyness clearly above zero. Given 02 and /33, the estimates = —1.6977 and "02 = —3.3768 suggest that, in general, the volatility smile is steeper and more convex for shorter-term options. These features of the average volatility smile are also apparent from Figure 5.7, which plots the ,

function g'(M,•,#) for three different times to maturity. With regard to the smile patterns introduced in Section 3.3.1, the skew pattern is the predominant pattern in our sample." However, on some days,

44 Note, however, that a true skew, i.e. a linear function of implied volatility versus moneyness, hardly ever occurs.

96

5 Properties of DAX Implied Volatilities

Figure 5.7. Average estimated volatility smiles for 20, 60, and 100 days to expiration. Sample period: 1995-2002

for instance on October 6, 1995, a nearly literal smile pattern can be observed (see Figure 5.8). The slope of the volatility terra structure of ATM options is represented by the parameter /34 . Its mean estimated value of 0.0166 indicates that, on average, the implied volatilities of shorter-term ATM options are lower than those of longer-term ATM options. This implies that the average volatility term structure features a "normal shape". A detailed investigation of the reveals that the ATM volatility term structure exhibits coefficient estimates on 1316 days a normal shape (-4 4 > 0) and on 623 days ( 34 <0) an inverse shape. The standard deviations in Table 5.1 suggest that there is considerable variation in the coefficient estimates from day to day. This observation alone, however, cannot be taken as evidence for a strongly changing volatility surface. If the parameter estimates are highly correlated, the errors affecting them may cancel out when implied volatilities are looked at. To check this possibility, we compute the correlation among the parameter estimates across the 1939

&

5.3 Structure of DAX Implied Volatilities

97

0.28 -

0.26 -

>, 0.24 :12 0.22 -

ra.

E 0.20 -

0.18 -

0.16 -

0.14 -0.20

-0.04 1

-0.12 1

0.04

0.12

0.20

Log simple moneyness

Figure 5.8. Estimated volatility smiles for 20, 60, and 100 days to expiration on October 6, 1995

days in the sample and report them in Table 5.2. As the values show, the -1 correlations are generally quite low, except for the correlation between 4 and -al . This may serve as evidence that not only the coefficient estimates fluctuate, but also the volatility surface itself features considerable variation. Table 5.2. Correlation coefficients between the daily parameter estimates in the period 1995-2002 2

73 3

;3

4

—0.3149 —0.3161 —0.6758 0.4102 0.1191 0.1076 733 ;3

1

;3

2

Risk-Neutral Densities The estimated coefficients of the implied volatility functions can also be used to deduce the shape of the risk neutral density (RND) at the option expiration -

98

5 Properties of DAX Implied Volatilities

dates.45 However, as the function g(M, 0) is only known for degrees of moneyness ranging from ML to Mu, it has to be extrapolated beyond this range to fully recover the RND, using the Breeden/Litzenberger (1978) theorem. One such extrapolation function was proposed by Brunner/Hafner (2003). Given :-

the estimated volatility smile within the range of observable strike prices, and hence the middle part of the corresponding RND, the basic idea of the Brunner/Hafner (2003) method is to complete the RND by attaching nonnegative functions to the lower tail and to the upper tail such that the complete RND is consistent with the absence of arbitrage. As specific choices for the tail functions, Brunner/Hafner (2003) consider mixtures of two lognormal density functions. The implementation of their method involves only straightforward numerical procedures. Moreover, the method is robust, accurate, and fast. 46 For illustration purposes, we use the estimated volatility surface on October 31, 2001 and apply the extrapolation method of Brunner/Hafner (2003) to the volatility smiles of 20, 60, and 100 days to expiration. The current DAX index level is 4589.70. This implies the three RNDs shown in Figure 5.9.

0 .0010 -

- 20 days 60 days - - 100 days

0.0008 -

0.0006 -

0.0004 -

0.0002 -

0.0000 -

5500

6000

Figure 5.9. RNDs for 20, 60 and 100 days to expiration on October 31, 2001. DAX level: 4589.70. Range of observable strike prices: roughly 3440-5500 4 5 For an in-depth discussion of RND estimation methods, see Bahra (1997), Cont (1997), and Jackwerth (1999). 'For further details, see Bruruier/Hafner (2003).

5.3 Structure of DAX Implied Volatilities

99

The smooth continuation of the RNDs outside the range of observable strikes (here from approximately 3440 to 5500) is apparent. The wider variances for 60 and 100 days to expiration reflect the greater probability of large price moves over a longer time period. All distributions are skewed to the left, exactly the opposite of the right-skewness implied by the Black-Scholes assumption of lognormally distributed asset prices. The degree of skewness tends to be independent of time to maturity; on the other hand the degree of kurtosis obviously depends on time to maturity: the shorter the time to maturity the higher the kurtosis and vice versa. Compared with the kurtosis of the lognormal distribution, all distributions exhibit excess kurtosis, i.e. they are leptokurtic. The negative skewness and the excess kurtosis reflect the same deviations from the Black-Scholes world as are also observable from the pattern of implied volatilities. The above findings are not only true for the particular day considered, but are typical for the whole sample. 47

5.3.3 Identification and Selection of Volatility Risk Factors Original versus Abstract Risk Factors The discussion so far has shown that the DAX volatility surface evolves randomly over time. However, as the volatility surface forms a highly correlated complex multivariate system, it is difficult to model. To reduce complexity, we search for a smaller set of abstract risk factors which represents, in the best possible way, the set of original risk factors, i.e. the implied volatilities for different degrees of moneyness and times to maturity. In contrast to the original risk factors, abstract risk factors are not directly observable in the market, but are usually created by transforming the original risk factors in some manner. The set of possible transformations is limited to invertible functions, otherwise the original risk factors cannot be recovered. However, the recovery process may be approximate. In the following, we state some desirable properties of abstract risk factors: • • •

Abstract risk factors should be accurately estimable. The set of abstract risk factors should be parsimonious. Abstract risk factors should be easy to interpret.

Especially the last property, i.e. easy interpretability, is often crucial for a model to be accepted in practice. Fundamental versus Statistical Factors as Abstract Risk Factors Given our regression model, the volatility surface on day n is completely described by the four regression coefficients 0 (i =- 1, , 4) and the timeinvariant parameters ei and e2 . Since the coefficient estimates have proven to 47 Bliss/Panigirtzoglou

(2002) come to similar results for the FTSE 100 index.

100

5 Properties of DAX Implied Volatilities

be highly accurate, the regression coefficients are the most natural . candidates for being used as abstract risk factors. To investigate the issue of parsimony, we run a principal component analysis on the correlation matrix displayed in Table 5.2. Figure 5.10 shows a screeplot of the variances explained by the principal components. As can be seen, the first three principal components explain 92.90% of total variance. This value goes down to 85% when looked at different subsarnples, implying that the fourth factor still explains a substantial part of total variance. Consequently, any further reduction of the number of risk factors would lead to a significant loss in accuracy. The last section has shown that the regression parameters are easy to interpret. The parameter /31 represents the (overall) level of implied volatility, /32 and 03 stand respectively for the (overall) slope and the curvature of the implied volatility smile, and 04 represents the slope of the (ATM) term structure of volatility. As these parameters can be thought of to capture systematic risks an option's investor is facing, and are therefore directly linked to economic activity, they are commonly called fundamental risk factors.

0.501 2.0

1.5 0.782

0.929 0.5 1

0.0 Comp.1

Comp.2

Comp.3

Comp.4

Figure 5.10. Screeplot of variances explained by the principal components. Basis: correlation matrix of the time series

,i34 . Sample period: 1995-2002

5.3 Structure of DAX Implied Volatilities

101

As opposed to fundamental risk factors, one could alternatively use statistical methods such as factor analysis or principal component analysis to derive a set of statistical risk factors that characterize the dynamics of implied volatilities. From the typically nonparametrically estimated implied volatility surface, time series of implied volatilities for different times to maturity and degrees of moneyness are constructed by evaluating the implied volatility function at the respective grid points. Then, on the basis of these time series, a principal component analysis is performed. For instance, Skiadopoulos et al. (1999) analyze the volatility surface of S&P 500 for the years 1992-1995. Depending on the criterion used for factor selection, they find that at least two and at most six factors are necessary to capture the dynamics of S&P 500 implied volatilities. Cont (2001) and Cont/Fonseca (2002) also examine the dynamics of the S&P 500 volatility surface. Applying a Karhunen-Loève decomposition to the daily log-variations of the implied volatility, they report that the first three principal components account for more than 95% of the daily variance. As already mentioned, Fengler et al. (2000) perform a common principal component analysis based on the closing prices of DAX options during the year 1999. They conclude that three factors are sufficient to capture 95% of the daily variations in implied DAX volatilities. The main advantage of the statistical approach is the orthogonality of the obtained factors, i.e. the factors have a correlation of zero among each other. On the other hand, the factors are usually difficult to interpret. Moreover they are not unique, because a factor rotation can yield a different set of factors with the same degree of explanation. Mainly due to their better interpretability, we decide for the regression coefficient estimates )3 1 , ... 034 or transformations of them to serve as our abstract implied volatility risk factors or just volatility risk factors Y1 . • • , Y4. Concretely, we define ,

— hi 01,n)

Yi,n = fii n, i = 2,3,4, ,

(5.27)

for n = 0, , N. The variable 4 - 1 was normalized by taking the natural logarithm, because it represents the level of volatility in the market and as such it has to be positive in any economically meaningful model. Note that the log transformation is not appropriate for the variable 03, although 03 mainly assumes positive values. The reason is that in order to be consistent with no-arbitrage the smile needs not to be convex in moneyness (and strike), but can also be concave."

48 5ee

also Carr (2001).

102

5 Properties of DAX Implied Volatilities

5.4 Dynamics of DAX Implied Volatilities 5.4.1 Time-Series Properties of DAX Volatility Risk Factors Outline of the Analysis Having identified the set of risk factors characterizing the volatility surface, this section is concerned with the question of finding what process is most appropriate for each factor. 49 For that purpose, we individually examine the historical time series of the volatility risk factors {Yi,„ : n = 0, , NI, i = 1, , 4, and determine their main statistical properties so that we can afterwards propose models that are suitable to capture most of the historical features. 50 To reduce complexity, we restrict our search for models to the class of autoregressive integrated moving average (ARIMA) models. The correlation structure defining the relationships between the volatility risk factors and the DAX index will be studied in Section 5.4.2. 51 Our procedure for analyzing the data partly follows the model identification stage of the Box/Jenkings (1976) approach to time series analysis and involves the following four steps: 1. 2. 3. 4.

Graphical inspection of the data. Identification of nonstationarity: testing for unit roots. Analysis of the marginal distributions. Determination of model order.

Graphical Inspection of the Data We start our analysis with a graphical inspection of the data. Figure 5.11 has four panels containing plots of the time series of the volatility factors Y1, Y4 for the sample period January 1995 to December 2002. During the sample period, the values of the estimated volatility level -4 1 lie between 9.16% (August 10, 1996) and 63.13% (July 24, 2002). The values of Y1 , defined as the natural logarithm of OD therefore range between —2.39 and —0.46. As can be seen from the Y1-graph, the volatility levels in the years 1995 and 1996 are distinctly lower than they are in the period from July 49 In the general factor-based stochastic implied volatility model, the price process of the underlying asset is assumed to follow a GBM with stochastic volatility, and the volatility process is implicitly defined in terms of implied volatility. Consequently, the price process of the underlying is almost determined, and therefore not considered here. 5 0 A similar strategy for the specification of multivariate risk factor models is chosen by Algorithmics in their Mark to Future framework for scenario generation. See Reynolds (2001). 51 The relationships between the regression coefficient estimates ;3 1 , , ..;34, which are, except for Y1 equal to the volatility risk factors, have been analyzed before. See also Table 5.2. -

,

-

5.4 Dynamics of DAX Implied Volatilities Y1

103

Y2

c»

Figure 5.11. DAX volatility risk factors Y1,

, Y4 on a daily basis over the period

1995-2002

1997 to December 2002. To formally test on this, we compare the population mean /41, ) for the period January 1995 to December 1996 with the population mean py(21) for the period July 1997 to December 2002 by running a Welch modified two - sample t test. 52 More specifically, we test the null hypothesis Ho : /19) > pr against the alternative hypothesis H1 : 41) < 42). Since -

the sample mean YiP) = —1.3384 for the second period is significantly higher = —2.0691 for the first period, the null hypothesis than the sample mean is clearly rejected. Economically, the reason for this substantial increase in volatility in the first half of 1997 can be seen in the beginning of the Asian crisis. Reaching its peak in October 1997, the Asian crisis was followed by the Russian crisis just one year later in 1998 letting volatility levels — expressed in terms of 731 — climb to more than 60% (see top left graph in Figure 5.11). Subsequently, such high levels of volatility have only been reached during the WTC terrorists attacks on September 11, 2001 and during the stock market's turmoil in fall 2002. 32 This

test assumes normality. For more details see, e.g., Casella/Berger (2002).

104

5 Properties of DAX Implied Volatilities

As Figure 5.11 shows, the time series of the risk factors Y2, Y3, and Y4 tend to be mean-reverting, i.e. they tend to fluctuate around a long-run mean. Astonishingly, the structural break occurring in the Y1 time series does not appear here. Consistent with our previous findings, the slope of the volatility smile Y2 is always negative. On the other hand, the curvature of the volatility smile as represented by the factor Y3 is mostly, but not always, positive. Precisely, on 74 days Y3 assumes a slightly negative value. The Y4 time series alternates between positive and negative values. In 2001 and 2002 negative values clearly dominate. Correspondingly the volatility term structure changes between normal and inverse shapes. In comparison to the Y2 and Y4 series, the Y3 series looks quite erratic. The extreme points in the graphs of the Y2 and Y4 time series match quite well with the extreme points in the graph of the Y1 series, although there seems to be a time lag in some cases. Exceptionally high levels of volatility Y1 usually come along with a distinctly downward-sloping smile and a pronounced inverse term structure of volatility. An inverse term structure generally expresses the expectation of the investors of a quick end of the crisis.

Distributional Properties Table 5.3 presents descriptive statistics on the distribution of the volatility risk factors Y1 , . . . Y4. ,

Table 5.3. Summary statistics for the volatility risk factor time series Yi.,.. • ,Y4. Sample period: 1995-2002

Minimum 25% Quantile Mean Median

75% Quantile Maximum Standard deviation Skewness Excess Kurtosis

Y1

Y2

Y3

Y4

- 2.3896 -1.7962 -1.5297 -1.5289 -1.2872 -0.4599 0.4109 0.1912 -0.3655

-0.9276 -0.5881 -0.4966 -0.4992 -0.4012 -0.1549 0.1427 -0.0893 -0.2537

-0.7275 0.7127 1.4594 1.3057 2.0012 5.5113 1.0369 0.8447 0.6381

-0.7291 -0.0235 0.0166 0.0404 0.0954 0.5704 0.1397 -1.6864 4.7809

The distributions of Y1 and Y2 appear to be close to a normal distribution, as the skewness and excess kurtosis values are around zero and the median closely matches the mean in both cases." Performing a Jarque-Bera test and a Kolmogorov-Smirnov goodness-of-fit test, however, the null hypothesis that 53 The normal distribution exhibits a skewness of 0 and an excess kurtosis (defined as kurtosis minus 3) of 0.

5.4 Dynamics of DAX Implied Volatilities

105

Y1 follows a normal distribution is rejected at the 1% significance level in both tests. 54 In the case of Y2 normality cannot be rejected at the 5% level using the Kolmogorov-Smirnov test, but is rejected at the 1% level, using the JarqueBera test. The skewness and excess kurtosis values for the factors Y3 and Y4 indicate a nonnormal distribution — both empirical distributions exhibit a skewness that is distinctly different from zero and excess kurtosis. Figure 5.12 illustrates these findings graphically. The empirical densities (solid line) are computed as a smoothed function of the histogram using a normal kerne1. 55 Superimposed on the empirical density is a normal distribution having the same mean and the same variance as that estimated from the sample (dotted line).

2

4

6

Y3

Figure 5.12. Distributional properties of Y2 and Y3. Solid line: Kernel estimate of the density function; dotted line: Normal density having the same mean and the same variance as that estimated from the sample

54 For details on the Jarque-Bera test, see, e.g. Kmenta (1997), p. 265-266. More information on the Kolmogorov-Smirnov goodness-of-fit test can be found in Casella/Berger (2002). 5 5 See Silverman (1986), Chapter 3.

5 Properties of DAX Implied Volatilities

106

Although the risk factors Y3 and Y4 are not normally distributed, in both cases the normal distribution (and in part also the lognormal distribution) still represents the "best" approximation to the empirical distribution from a broad spectrum of potential continuous distributions. More specifically, computing the Chi-Square and the Kolmogorov-Smirnov goodness-of-fit measures for the beta, the chi-square, the exponential, the gamma, the inverse Gaussian, the normal, the lognormal, the Student's t, and the Weibull distribution, we find the normal distribution (and for Y3 also the lognormal distribution), to be the one where these measures are lowest. 56 Stationarity and Serial Correlation At a first glance, there is no evidence of a deterministic time trend, a seasonal component or a unit root in any of the four series. To further investigate the key issue of stationarity, we plot the sample autocorrelation function (sample ACF) for each of the series over the sample period 1995-2002 in Figure 5.13. Y1

Y2

6

ci

a

u.

a 6 O

5

10

15

20

25

30

0

10

Lag

15

20

25

30

20

25

30

Lag

Y3

Y4

6

o

<1,1

6 ' < •

ns

Ci

•

10

15 Lag

20

25

30

'

0

10

15 Lag

Figure 5.13. Sample ACF for each of the series Y1, , Y4 over the period 1995-2002 56 To fit a density, a first guess of parameters is made using maximum likelihood estimators. In a second step, the fit is optimized using the Levenberg Marquardt method. See Press et al. (1992), p. 683 ff., for more information on this method. -

5.4 Dynamics of DAX Implied Volatilities

107

The slow decay of the sample ACFs of Yi , Y2 and Y4 indicate either a large characteristic root, a unit root, or a trend stationary process. 57 For Y3 a unit root is not suspected from the sample ACF. Shocks to a unit root process are permanent so that the variance goes to infinity as time approaches infinity. The high positive autocorrelations that are observed for Y1 mean that high volatility levels follow high volatility levels and low volatility levels follow low volatility levels. This effect is well-documented for stock return volatilities and is commonly referred to as "volatility clustering" • 58 Clustering is also observed for the other three risk factors. Although plots of the sample ACF are useful tools for detecting the possible existence of unit roots, the method is necessarily imprecise. The problem becomes especially difficult for near unit root processes, because such processes will have the same shaped ACF as true unit root processes. For this reason, a number of formal unit root tests has been developed. 59 The most widely used tests are the augmented Dickey-Puller (ADF) and the Phillips-Perron (PP) tests. Dickey and Fuller (1979) consider three different regression equations that can be used to test for the presence of a unit root: (1) a pure random walk; (2) a random walk with drift; and (3) a random walk with drift and deterministic time trend. ° Philipps-Perron tests are similar to the ADF tests but entail less stringent restrictions on the error process. Assuming that the data generation process is given by a random walk with drift and deterministic time trend, we apply the ADF and PP test to Y . Y4. The results are reported in Table 5.4. The 5% and 1% critical values —3.4147 and —3.9681 are obtained from the MacKinnon response surface. 61

Table 5.4. ADF and PP test statistics for

,Y4. Specification: random walk

with drift and time trend, 20 time lags ADF t-stat. Yi

-3.2523

Y2 Y3 Y4

-4.1825 "

-6.0525" -4.2158"

PP t-stat. -3.9685" -5.0210" -18.9440" -7.4944"

• (**) significant at the 5% (1%) level "See Enders (1995), p. 211. 58 The volatility clustering effect dates back to Mandelbrot (1963) and Fama (1965). "For a comprehensive treatment of unit root tests, see Hamilton (1994), Chapters 15-17. 60 1f A is generated by the model X n = ao + aim + Xn-1 en, then X is said to follow a random walk with drift ao and deterministic time trend ai. A random walk is an example of a class of non-stationary processes known as integrated processes. See also Mills (2000), p. 41ff. Note that the ADF test additionally considers higherorder dependencies of X„ on its own lagged values. "See MacKinnon (1991).

108

5 Properties of DAX Implied Volatilities

For all variables, except from Y1 , we can reject the null hypothesis of a unit root at the 1% significance level. For Yi the null of a unit root can be rejected at the 1% level when we employ the PP test, but it cannot be rejected according to the ADF test. Yet, as is well known, the available tests have low power to distinguish between unit root and near unit root processes. 62 Our failure to reject the null hypothesis does not suffice to draw the conclusion that a unit root exists. As far as Y1 is concerned, economic theory does not support a unit root since the variable should be subject to an upper arbitrage bound even if severe violations of the Black-Scholes assumptions occur. For a similar economic argument, consider the distribution of the volatility level Y1 in 100 years. If Y1 were not mean-reverting, the probability of Y1 being between 1% and 150% would be rather low. Since we believe that it is more than likely that Y1 would in fact lie in this range, we deduce that Yi must be mean-reverting. 63 Similar economic arguments hold for the other three variables. To conclude, we suppose that all volatility risk factors follow stationary processes, i.e. we assume that they are integrated of order 0. Determination of Model Order The basic idea of the Box-Jenkins approach to ARIMA model identification is essentially to match the behavior of the sample ACF and the sample partial autocorrelation function (sample PACF) with that of various theoretical ACFs and PACFs. For each of the series, the sample ACF is shown in Figure 5.13 and the sample PACF is plotted in Figure 5.14. The Y1 , Y2 and Y4 time series exhibit partial autocorrelations which are close to zero for lags greater than 1. Therefore, these variables might be reasonably well modelled by AR(1) processes. For Y3 representing the curvature of the volatility smile, an AR(1) process might not be the best choice, because the partial autocorrelations of lags 2 and 3 are significantly different from zero. 64 Instead, an AR(3) process should be more appropriate. Unfortunately, an AR(3) process is incompatible with our continuous-time model framework, as it only allows for processes of serial dependence of at most order 1.65 For this reason, Y3 is also presumed to follow an AR(1) process.

Enders (1995), p. 261. also Gatheral (2000). In contrast, Dartsch (1999), p. 141-146, argues that the VDAX follows a random walk as he cannot reject the null hypothesis of a unit root at conventional significance levels. 64 For example, the sample partial autocorrelation for lag 2 is 0.2775. This is well above the 95% confidence level of 0.048. "In principle, there exists a continuous-time version of a discrete AR(3) process. However, the process becomes very complex and the Ito theory is no longer applicable. 62 See

63 See

5.4 Dynamics of DAX Implied Volatilities Y1

109

Y2

u. to 00

ig

Q.

I

a

6 O

5

J.

10

' 15

20

25

30

O

5

10

Lag

Y3

15 Lag

20

25

30

20

25

30

Y4

a

ci 1.eilt

O

5

10

15 Lag

20

25

30

O

5

10

15 Lag

Figure 5.14. Sample PACF for each of the series Y1, , Y4 over the period 1995-

2002 5.4.2 Relating Volatility Risk Factors to Index Returns and other Market Variables

The high first-order autocorrelations suggest that most of the variation of the volatility risk factors is explained by the factors itself, or more precisely by their own lagged values. For the unexplained part of the variance, a number of different market variables and fundamental factors may be responsible. With respect to the factor-based stochastic implied volatility model, one factor, the return of the underlying index, is thereby of particular interest. The relationship between index returns and both implied and realized volatility is usually negative: volatility increases when the index level falls. This well-established relationship is commonly referred to as the leverage effect. 66 A standard explanation ties the phenomenon to the effect a change in market valuation of a firm's equity has on the degree of leverage in its capital structure, with an increase in leverage producing an increase in stock volatil66 Among many examples in the literature, Schwert (1990a) and Bollerslev et al. (1994) give an extensive analysis of the leverage effect.

110

5 Properties of DAX Implied Volatilities

ity and hence also an increase in index volatility. In the following analysis we will examine the leverage effect for the German stock market, represented by the DAX index. The volatility level is represented by y1. 67 In addition to this analysis, we will also study the relation between the other three implied volatility risk factors and the DAX index level. Formally, we test on the relationship between the volatility risk factor Yi (i = 1, 2, 3, 4) and the DAX return R by running the ordinary least squares (OLS) regression = bo,i + bi,sYs,n—i + b2,iRn +

en,

n = 1,

, N.

(5.28)

The variable R, is defined as the continuously compounded DAX return on day n, i.e. Rn = in (Sn /Sn_1), where Sn denotes the last futures-implied DAX index leve168 on day n and E denotes, as usual, a random disturbance. The regression coefficients for the predictors of the volatility risk factor i (i = 1, , 4) are denoted by 60 ,i , 61 , i , and 6. The coefficient 6 2 ,i reflects the influence of the DAX return on the respective volatility factor. The lagged value Y2 ,n _i on the right-hand side of equation (5.28) is included to account for first-order autocorrelation in Y. The OLS coefficient estimates and the t-statistics (in parentheses) from the regressions are displayed in Table 5.5. Table 5.5. Relationship between volatility risk factors and DAX returns. Sample period: 1995-2002. Risk factor

30

31

32

R, 47

Y1

—0.0173 (-3.93")

0.9881 (353.62")

—2.3043 (-34.06**)

98.52%

—0.0164 —5.62'*)

0.9672 (171.82")

—0.2769 „ (-5.73 )

93.87%

Y3

0.3336 (12.90”)

0.7712 (53.40")

2.1630 (2.40")

59.64%

Y4

0.0006 (0.66)

0.9430 (135.71)

1.0598 " (18.15 )

90.66%

Y2 (

' (**) significant at the 5% (1%) level

Due to the high autocorrelations, the adjusted coefficients of determination R2a, d are high, too. The parameter estimates for 62 are all significant, at least 67 For a detailed study on the leverage effect for the US, see Figlewski/Wang (2000). "S,, means in fact the adjusted futures-implied DAX index level gn•

5.4 Dynamics of DAX Implied Volatilities

111

at the 5% level. The negative coefficient estimate of -2.3043 for b2 in the first regression supports the existence of a leverage effect: the volatility level Y1 increases when the DAX index falls. The coefficient estimates 32,1, j = 2,3, 4, are interpreted as follows: when the DAX index falls, the slope of the volatility smile (Y2) increases, the curvature of the volatility smile (Y3) decreases, and the slope of the (ATM) volatility term structure (Y4) decreases. To analyze the stability of the regression coefficients, we subdivide the sample period of 8 years into four subperiods of equal length and run regression model (5.28) for each period. The results reported in Table 5.6 suggest that the relationships between the volatility risk factors and the DAX return are relatively stable over time. Only the coefficient estimates 32,2 and 32,3 change their sign once over time. However, the impact of a change in index levels on the respective risk factors varies, in part, quite substantially. For instance, the coefficient estimate 32,1 for the period 2001-2002 is only half the coefficient estimate for the period 1995-1996. It is further interesting to observe that the "literal" leverage effect, i.e. the negative relation between Y1 and R, has become weaker over the years, as can be seen from the values in the first row of Table 5.6. Table 5.6. Influence of DAX index returns on the volatility risk factors

1995-1996

1997-1998

1999-2000

2001-2002

-3.5245 (-15.36")

-3.1202 (-21.51")

-2.0663 (-13.58")

-1.7897 (-19.42")

32 '2 1 ` 2J

-1.2812 (-8.21")

-0.6709 (-6.76")

-0.4225 (-3.80")

0.1764 (2.58")

32 , 3 [1'3]

-0.3036 (_0.06)

3.6017 (2.70")

1.0780 (0.54)

2.3676 (1.99`)

0.8079

1.9589 (16.01")

0.6226 (5.39")

0.7362 (6.78")

v.i

32 '4 [Y4 1 (9.98")

• (") significant at the 5% (1%) level

The high t-values for the coefficient estimate 32, 1 in the four subperiods indicate that there is a strong relationship between the implied volatility level Y1 and the DAX return R. The relationship between the DAX return and the slope of the volatility smile Y2 is also quite substantial. The same is true for the connection between the DAX return and the term structure of volatility, as represented by Y4. Rather weak, however, is the relationship between the curvature of the volatility smile, represented by Y3, and the DAX return. The explanation of variations in the smile pattern is also the subject to a number of empirical studies. For Germany, Hafner/Wallmeier (2001), using a subsample of our data, define 11 proxy variables for possible theoretical

112

5 Properties of DAX Implied Volatilities

explanations of the smile pattern and examine their impact on the slope of the DAX volatility smile. 69 The proxy variables can be divided into 5 groups. The first contains distributional parameters of intraday returns, the second distributional parameters of historical 1 day DAX returns. They also include proxy variables for parameters in stochastic volatility models and a variable for the overnight jump. As a proxy for market frictions they use the percentage trading volume of OTM puts. To test on the influence of the described factors, they run a time-series regression with the 11 proxy variables as the explanatory variables and the difference (the "span") between the implied volatility of a 45 calendar day 5% OTM put and the implied volatility of a 45 calendar day 5% ITM put as the variable to be explained. The hypothesis that none of the independent variables has an influence on the span is rejected at the 1% significance level. Estimated over the complete sample period (January 1995 to October 1999), the coefficients of several variables are significantly different from zero at least at the 5% level. Yet, only two of these coefficients have the predicted sign. The first of these variables is the volatility of intraday DAX return volatility. A high value for the volatility of volatility seemingly reflects great uncertainty of market participants and a rather high probability of a sharp stock market decrease. In addition, there is a positive relation between the percentage trading volume of OTM puts and the span. It is consistent with the hypothesis that a high demand for out-of-the-money puts pushes option prices, since market makers cannot easily neutralize the risk exposure from short put positions. This finding corresponds to the importance of transaction costs and liquidity effects for the pricing of index options reported in previous studies." Wallmeier (2003), using DAX option data from January 1995 to December 2000, extends the work of Hafner/Wallmeier (2001) and considers in addition to the span two further variables: the ATM volatility and the curvature of the smile. As before, the influences are either negligible or difficult to interpret. The preceding analysis has shown, that most of the variation in tomorrow's volatility surface is explained by the surface today. Apart from this, the DAX return is another important influence on the evolution of the DAX volatility surface. Other market variables have only little explanatory power.

69 Possible theoretical explanations of the smile pattern have been discussed in Section 3.4.2. 70 Long-staff (1995) for the US and Pena et al. (1999) for the Spanish stock market.

5.5 Summary of Empirical Observations

113

5.5 Summary of Empirical Observations Let us now summarize the main (statistical) properties of DAX volatility surfaces as observed in our data set: 1. Moneyness measures that depend on time to maturity, such as the Natenberg moneyness, are often not consistent with no-arbitrage when implied volatilities are stochastic. 2. The DAX volatility surface can be accurately estimated by a four-factor WLS regression model. On average, the variation of moneyness, which is defined as the natural logarithm of the ratio of strike price to futures price, and the variation of time to maturity explains about 92% of the cross-sectional variation of implied volatilities. 3. The four regression parameters or (implied) volatility risk factors may be interpreted as follows: factor one represents (the natural logarithm of) the overall level of implied volatility, factor two and factor three stand, respectively, for the (overall) slope and the curvature of the implied volatility smile, and factor four reflects the slope of the (ATM) term structure of volatility. 4. The RNDs corresponding to the estimated volatility smiles exhibit negative skewness and leptokurtosis. 5. The volatility risk factor time series display high (positive) autocorrelation and mean-reverting behavior. There is strong evidence that they are stationary, i.e. integrated of order O. 6. The distributions of the first two volatility risk factors, representing the level of volatility and the slope of the smile, are close to a normal distribution; the distributions of the other two risk factors are more skewed and leptokurtic. However, the normal distribution (and in part also the lognormal distribution) is still the best approximation to the empirical distributions of these factors from a broad spectrum of potential continuous distributions. 7. The univariate sample ACFs and PACFs support AR(1) processes for all volatility risk factors, except for the third. Nevertheless, for this factor, an AR(1) process is still a reasonable approximation. 8. The DAX volatility surface of tomorrow is mainly determined by the DAX volatility surface today and the return of the DAX index. Other market variables have only little explanatory power. 9. Shifts in the global level of DAX implied volatilities as represented by the first risk factor are negatively correlated with the DAX index level ("leverage effect"). 10. When the DAX index falls, the slope of the DAX volatility smile (factor two) increases, the curvature of the DAX volatility smile (factor three) decreases, and the slope of the (ATM) DAX volatility term structure (factor four) decreases.

6

A Four-Factor Model for DAX Implied Volatilities

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work. John von Neumann

Based on the empirical results of the previous chapter, we propose a fourfactor model for the stochastic evolution of the DAX volatility surface. The model fits into the general model framework of Chapter 4. Section 1 starts with the description of the model under the real-world measure. We suggest suitable processes for the DAX index and the DAX volatility surface, show how to estimate the model parameters from historical data, and perform various inand out-of-sample tests to assess the quality of the model. Section 2 describes the model under the risk-neutral or martingale measure. We derive the riskneutral dynamics of the DAX index and its instantaneous volatility, discuss the existence and uniqueness of a martingale measure, and show how to value and hedge contingent claims. Section 3 provides a brief model review and a conclusion.

6.1 The Model under the Objective Measure 6.1.1

Model Specification

Correlated Form The empirical study in the previous chapter has shown that the evolution of the DAX volatility risk factors Y1, . • • , Y4 may be well described by a system

116

6 A Four-Factor Model for DAX Implied Volatilities

of correlated AR(1) processes with normal innovations. Since the continuoustime analogue of an AR(1) process is a mean-reverting Ornstein-Uhlenbeck (OU) process l , we model the joint dynamics of the DAX volatility risk factors under the objective measure IF' by a system of correlated OU processes: 2

i = 1, . . . , 4, Vt E [0, T*} ,

dYt,t = at (cot — Yt,t)dt +

(6.1)

where Y1 ,0 , Y2,07 173,0) Y4,0 E R are the initial values of the processes, at E , 4) are constant parameters, and T* is, as R+ , Cj E R, -yi E R+ (i = 1, — horizon. The Brownian motions Wit and Wi are usual, some finite planning correlated with (instantaneous) correlation p, i.e. dlgt,tdWi,t =

, 4, and i j. According to (6.1), the volatility risk factor lit is for i, j = 1, pulled to a level et at rate at. When Yt is high, mean reversion tends to cause it to have a lower drift; when Yt is low, mean reversion tends to cause it to have a higher drift. The parameter ci is commonly referred to as the meanreversion level or the long-run average and the parameter at is called the speed of mean reversion (i = 1, . . . , 4). The parameter -yi may be interpreted as the "volatility" of the i-th volatility risk factor Yt (i = 1, , 4). x [ML, Mu ] x ETL , T u b T u < T* — t, we describe For any (t, M, T) E [0, the relative (DAX) volatility surface Frt (M,T) by3 at (111,r) = g(t, M,T 7 1t1,t) Y247 1734, 1744/

(6.2)

where g(.) is given by g(t,M,

T7

Y27Y3, Y4) =

with constant parameters

+ y2M (1 + el ln (1 + T)) (6.3) +y3M2 (1 + g2 In (1 + T)) + y4 ln (1 + T) ,

g2 E R, and M is the log simple moneyness, i.e.:

M = m(t, St , K,T,r)

In

t ) ( F(T)

ln (— st

(6.4) — r(T — t),

Since m(t, St , St , T, r) = 0, the dynamics of the underlying's instantaneous volatility is obtained from Theorem 4.2 by

vt = Frt (m(t, St , St , t, r), 0) = eY" ,

Vt E [0, 71.

(6.5)

'For details on the (mean-reverting) OU process, see Oksendal (1998), p. 74 and Karatzas/Shreve (1988). 2 For similar process specifications in the context of implied volatility modelling, see Cont/Fonseca (2002), Gonca1ves/Guidolin (2003), and Hafner/Wallmeier (2001). 3 For a motivation and empirical test of this representation of the DAX volatility surface, see Section 5.3.

6.1 The Model under the Objective Measure

117

Equation (6.5) shows that the (stochastic) instantaneous volatility of the underlying asset is equal to the level of implied volatility (in absolute terms). The other implied volatility risk factors, i.e. the slope and the curvature of the volatility smile and the slope of the term structure of volatility, have no influence. The intuition behind this result is that these factors represent specific properties of the options market and as such they have no impact on the stock price process. Note that in contrast to traditional stochastic volatility models, the stochastic volatility component vt = exp(Yi ,t ) is observable.4 Using relation (6.5) and assuming a constant drift rate II E R, the evolution of the (DAX) index level under P is given by dSt = Saidt + St eY1, tdWo, t

Vt E [0, T* ],

(6.6)

where the initial stock price So is supposed to be positive. To model the observed dependency between the DAX volatility surface and the DAX inWo is correlated dex, the Brownian motion that drives the DAX index — , W4 that drive the DAX volatility surwith the Brownian motions WI. , face. The instantaneous correlations poj (j = 1, , 4), with p0 ,3 defined by — dWo,tdW 3 ,t = N o ck are assumed to be constant. For completeness, we also state the price process of the money market account. It is given by the ODE: dB t= rBt dt,

Vt E [0, T*],

(6.7)

where Bo = 1 and the interest rate r is supposed to be constant and nonnegative. As can readily be shown, all processes fulfill the technical regularity conditions imposed in Chapter 4. Uncorrelated Form

In the general model of Chapter 4, the Wiener process W driving the stock price and the volatility risk factors was assumed to be a multi-dimensional standard Brownian motion. This implies that each component of W is an independent standard Brownian motion in one dimension. In contrast, the formulation above involves a vector Brownian motion W = — with correlation matrix E = (p. •)i ,.3=,...,4 To transform W into W we use .5 0 tO

a technique known as Cholesky decomposition or Cholesky factorization. 6 4 1n the estimation of the DAX volatility surface we have only included options with a time to maturity of 5 calendar days and more. Thus, the implied volatility for T = 0 represents an extrapolated value. For the ATM option that is relevant here, we suppose this estimate to be very precise. 6 1Nlote that pio = 1 for i = j, by definition. 6 The Cholesky decomposition and other related techniques can only create uncorrelated, but not statistically independent, processes. However, in the case of normality, which is considered here, independency is equivalent to uncorrelatedness. See Hartung et al. (1999).

6 A Four-Factor Model for DAX Implied Volatilities

118

is a symmetrical positiveSince the correlation matrix E = (p ,o) i3=0,., ,4 definite real matrix, it can be decomposed into its Cholesky factors 7 E -=--- EE',

(6.8)

where E = (et3=0,...,4 is a 5 x 5 lower triangular matrix with zeros in the upper right corners, i.e. E is of the form e0,0 0 0 0 e10 e11 0 0 E = ( e2,0 e2,1 e2,2 0 e3 ,0 e3 , 1 e3, 2 e3,3 e4,01 e4,2 e4,3

0 0 0 0

(6.9)

e4,4

For the matrix E to be rendered unique, we specify that all diagonal elements must be positive. Defining 4

E

dWt,t

i = 0, , 4,

(6.10)

or in vector notation dWt E • dWt , the stock price process and the processes of the volatility risk factors can be (re)expressed in terms of 5-dimensional standard Brownian motion W. We get

dSt = Studt + SteY"dWo,t, and

(6.11)

4

j=

dYi,t = ai (ci — Yt,t)dt +

1,

, 4,

(6.12)

i=o for all t E [0, Tl, with ^No defined by =

i = 1,...,4, j =- 0, .. • , 4.

(6.13)

For convenience, we summarize the elements -y i,3 in the matrix r = (7, i)i=1,...,4. Depending on the specific application, we will henceforth use the correlated or the uncorrelated form of the model. 6.1.2 Model Estimation Estimation Method Discretizing the explicit solution of the DAX price process (6.6) and applying the Euler method8 to the DAX volatility risk factor processes (6.1), the 7 See

Dhrymes (2000), p. 80-82. 'See Kloeden et al. (2003), pp. 91-97.

6.1 The Model under the Objective Measure

119

discrete-time approximation of the continuous-time model (6.6) and (6.1) the correlated form - is: 9 At + ln(Sn_i) + e Yi,.-1 V-6,760,n,

- - e2Y1, - -1 ln(Sn ) =p(1 2

= aic,At + (1 - at) Ys,n-1

(6.14)

i = 1, . . . , 4,

-Y i NŒtE1,n7

(6.15)

< tn < for a given time discretization 0 = to < tl < < tN• T* with equal time increments At = T*/N* = tn - tn _1 = 1 . , N*) . The index level and the value of the i-th volatility risk factor at time t n are denoted by Sn = Se n and Yi,n = Yi,t„, respectively. For each i = 0, ... 4, Ei follows a Gaussian white noise process with variance 1. This implies that = (go,gi, e2, 63,64) is a 5-dimensional white noise process - though not necessarily Gaussian Since the diagonal elements of the covariance matrix (Coy [Ei,n , Ei,n1) 2,3=0, ..,4 are equal to one, the covariance matrix equals the correlation matrix 'E.. Defining ,

ln(Sn )

(p

aiciAt

Yl,n

Yn =

Yz,n

ee''' ILEnO,n

ie2Y1, -) At

-

7

n=

a2c2At

/ En =

a3c3At a4c4At

Y3,n Y4,n

721/6

2,n

73 V'

1 't64,n 74 1./7

(6.16)

and 1 0 0 1 - aiAt cp = 0 0 1 0 0 (0 0

-

0 0 0 0 a2 At 0 0 1 - a3At 0 ) °0 0 1 - a4 At 0

(6.17)

(6.14)-(6.15) can be written more compactly as: Yn = en- 1 ± CPYn-1 + En /

n = 1, ... , N* ,

(6.18)

where e n lYn _i has zero mean and conditional covariance matrix (6.19)

EnlYn-i = (E ki,nei,nlYn-ip At (e2 YL-1) Atp0,1e Y1 ' n -1 71

... Atp1,27172

Atp0,4c Y1-1 74

••• At/11,47174 At•-d Atp2,37273 642,47274 ... At73 AtP3,47374

=

...

...

Atli

From (6.18) it can easily be seen that the variables Y1, Yz, Y3, and Y4 follow a vector autoregressive process of order 1, i.e. a VAR(1) mode1. 19 The parameter vector of the transformed model is denoted by 9 See 19 For

Taylor (1994) for a similar discrete-time specification. a thorough discussion of VAR models, see Liltkepohl (1993), Chapters 1-5.

•

6 A Four-Factor Model for DAX Implied Volatilities

120

O = (p,

, 62 63 1

64 (P 1 (P 2 (P3 (P4 71 72 .-Y3 '74 1 2 ) 1

= act and (pi = ço = 1 - at is the i-th diagonal element of (p , 4). Given observations {(Yo,n, Yin, /2,n, Y3,n, Y4,n) : n =- 0,... , N}, N < N*, where Yo,n = ln(Sn ) for n = 0, , N, we employ a maximum likelihood (ML) where

(i

1,

approach to estimate the model parameters. 11 Under mild regulatory conditions, the maximum likelihood estimator (MLE) based on the discrete-time approximation (6.14)-(6.15) is known to converge to the true maximum likelihood estimator as the sampling interval At goes to zero. 12 To make maximum IV i assumed likelihood estimation feasible, the conditional distribution r-n,-n—i .S to be multivariate normal with mean zero and covariance matrix En I Y_ i. If the normality assumption is violated, the estimation technique detailed below is known as quasi maximum likelihood estimation (QMLE). As a first step of ML estimation, it is necessary to calculate the likelihood function. Conditional on the values of Y observed through time n - 1, the value of Y for time n is equal to (6.20)

E[YnlYn-i] = en-i +

S-2) denotes the cumulative distribution plus a N(0, En ) variable, where N function of a multivariate normal random variable with mean vector a and covariance matrix Q. Thus: (6.21)

YnlYn-i N(en-i + (PYn-1, En).

Because Yn given Yn _1 is multivariate normally distributed, the conditional density of the n-th observation (vector) is (6.22)

f (YnlYn - i) = ( 27 ) -5/2 lE7T. 1 1 1 • exp{(-) (Yn - 6n _1 — (py

) '

E; 1 (Yn —

—

where 1E;11 denotes the determinant of the inverse conditional covariance matrix E. The joint density of observations 1 to N conditioned on the initial observation (vector) Yo satisfies

f ( 17n,Yn-i,Yn-2, • • • ,Yillro) =

H f ( 37n 1 17n-1)•

(6.23)

n=1

Taking logs yields the (conditional) log-likelihood function: "For simplicity, we use the same symbols to denote sample observations and process variables. 1 2 See Florens-Zmirou (1989).

6.1 The Model under the Objective Measure

5

1

2

2

L (0) = - (N - ) ln (271- ) + 1 N

E ln (1E;. 1 1)

121

(6.24)

- (pYn-ir E;; 1 (Yn -

- 49Yn-i)]

In the second step of ML estimation, values of 0 must be found that maximize the log-likelihood function L (0). It is well known that the MLE of the mean stock return is a very noise estimate of the true parameter i. Given that imprecision, we consider it more accurate simply to impose a value for the mean rather then trying to estimate the mean from the data. 13 This amounts to a kind of Bayesian approach, based on the notion that financial theory allows us to place tighter bounds on an asset's true return than classical statistics does. For example, we do not expect the DAX index should ever have an ex-ante equilibrium return that is negative, regardless of the sample mean in a given set of data. In contrast to traditional stochastic volatility models or GARCH models, the stochastic volatility component in the model as represented by eY1.- is observable. This eases parameter estimation for at least two reasons: first, the maximum of the log-likelihood function can be obtained analytically; and second, the estimation of and yo can be separated from the estimation of , -y4 and the correlation matrix E.' . In fact, it can be shown that for each and çoi are just the estimated i (i = 1, , 4) the MLE of the parameters regression coefficients from an OLS regression of on Yi „_ 1 . 14 Estimates of the original parameters a, and c, are then obtained from and (Pi (i = 1, , 4) using the transformations: a, =

c, -

1 - (,?)i

(6.25)

At

1-

.

Given the OLS residual vector î the MLE of the "volatility of volatility" parameter -y i is easily calculated as13 ,

Nikr

VEt.

i= 1,

, 4.

(6.26)

The expression under the square root in the numerator is just the average squared OLS residual. For the estimation of the correlation matrix F.. let us consider the (standardized) residual vectors ?0, ,g4, given by "See also Figlewski (1997), pp. 22-23. 14 See 15 See

Hamilton (1994), pp. 293-294, for a proof. also Hamilton (1994), pp. 295-296.

6 A Four-Factor Model for DAX Implied Volatilities

122

Yo ,n — Yo,n-i — ?0,n

(

— ie 21ti , n -1 ) At

eY1..-INFA7

(6.27)

îi,n

for

7/ =

1,... , N. Defining -Én = (?0,n,?1,11, --6.'2,n,.F3,a4,0 1 the MLE of E is ,

=

1 • EnE n• N 2.. n=1

(6.28)

In particular, the row i, column j element of E is given by

(6.29) which is the average product of the residual for variable i and the residual for , 4, i j). variable j (i, j = 0, Empirical Results In the sample period 1995-2002, the average number of trading days per year amounts to 251. 16 We thus set At = 1/251. The number of observations in the final sample is 1939; this implies N = 1938. The mean return of the DAX is supposed to be 8%, i.e. i = 8%. 17 Estimates of the parameters ei and soi (i = 1, , 4) are given in Table 6.1. Standard errors are reported in parentheses under the coefficient estimates. Additionally, the coefficient of determination R2 of each OLS regression is reported. All of the parameters, except for Z4 , are statistically significant at the 1% level. The estimated avtoregressive coefficients îpi , Ci32, and ;64 are, as expected, close to one. Since all autoregressive coefficients are strictly less than 1, the roots of the characteristic polynomial lie outside the unit circle, and thus the VAR(1) process followed by Y1 , , Y4 is stationary and invertible. 18

16 The

original sample consists of 2010 trading days, spread over 8 years. assumption is based on the results of a study of Claus/Thomas (2001). Using an implicit estimation method, they report a mean DAX return of 8% for the year 1997 (see p. 1649). For 1995 and 1996, the estimates range between 8 and 9%. "See Stier (2001), p. 70. 17 This

6.1 The Model under the Objective Measure Table 6.1. Estimates of the parameters

123

and (pi (i = 1, . , 4) for the sample

period 1995-2002. Risk factor

Z

‘71 )

R2

Y1

-0.0163 (0.0055)

0.9890 (0.0035)

97.64%

Y2

-0.0160 (0.0029)

0.9681 (0.0057)

93.77%

Y3

0.3337 (0.0259)

0.7715 (0.0145)

59.52%

Y4

0.0008 (0.0011)

0.9441 (0.0075)

89.07%

Using transformation rule (6.25), we obtain the estimates of the speed of mean reversion ai and the mean-reversion level c, (i = 1, , 4). We report them in Table 6.2. To illustrate the strength of the pull back to mean, we compute the "half-life" of each series. The (estimated) half-life HL, of process (i = 1, , 4) is defined as the number of trading days hi such that 19

(1 - at)' ‘= (X' = 0.5. Hence:

HL, -

ln(0.5) ln(0.5) = ln(ça) ln(1 - aaXt) .

(6.30)

(6.31)

a,

the shorter the half-life Clearly, the lower <77, or equivalently, the higher and thus the stronger the pull back to mean. In the last column of Table 6.2 we report the half-life of each of the processes Y1, • • • , Y. Apparently, the mean-reverting effect is most pronounced for Y3 with a half-life of roughly 3 trading days and least pronounced for Y1 with a half-life of 63 trading days. Furthermore, the values suggest that shocks to the structure of the volatility surface, represented by the variables .1/, Y3 and Y4, are rather temporary, whereas shocks to the volatility level Y1 tend to be much more persistent.

Table 6.2. Estimates of the parameters ai and c, (i = 1,...,4) for the sample period 1995-2002. In addition: estimated half-lifes HLi, ,

Y1 Y2

Y3 Y4 I9 See, e.g., Taylor

2.7575 7.9951 57.3609 14.0119

(2001), p. 58.

a

HL

-1.4797 -0.5013 1.4601 0.0144

63 21 3 12

6 A Four-Factor Model for DAX Implied Volatilities

124

Given the residuals of the OLS regressions, we use equation (6.26) to calculate the "volatility" estimates 71, • • • , 7 4 of the volatility risk factors Y1, Y4. We get

5,1 = 1.0006,

i2 = 0.5646,

ry3 =- 10.4561,

and

ry4 = 0.7317. (6.32)

Finally, the estimate of the correlation matrix E. is obtained from equation

(6.28):

2,3 = 0,...,4

1.0000 -0.6152 -0.1787 0.0315 0.3446 -0.6152 1.0000 0.0588 -0.1668 -0.8020 -0.1787 0.0588 1.0000 0.2041 0.0276 0.0315 -0.1668 0.2041 1.0000 0.0696 0.3446 -0.8020 0.0276 0.0696 1.0000

(6.33)

The negative correlation between bo and bi of -0.6152 is consistent with the leverage effect: when the DAX index (i.e. Y0) falls, the overall level of DAX implied volatilities as represented by the factor Y1 shifts upward. The other values in the first row of can be interpreted as follows: when the DAX index falls, the slope of the DAX volatility smile (Y2) increases, the curvature of the DAX volatility smile (Y3) decreases, and the slope of the (ATM) DAX volatility term structure (Y4) decreases. 2° The correlations among the volatility risk factors are in general quite low; however the estimate P I 4 = -0.8020 suggests that there exists a relatively strong negative relationship; between the volatility level and the slope of the ATM term structure. Applying the Cholesky decomposition to the matrix yields

1.0000 0 0 0 0 - 0.6152 0.7884 0 0 0 - 0.1787 -0.0648 0.9817 0 0 , (6.34) 0.0315 -0.1870 0.2013 0.9610 0 0.3446 -0.7483 0.0415 -0.0931 0.5574

=

and with the help of equation (6.13) we get the following estimate of the matrix F:

- (72,3 )1=1,...,4 = j =0,...,4

-0.6155 0.7889 0 0 0 -0.1009 -0.0365 0.5543 0 00) .35) . (6 0.3294 -1.9553 2.1043 10.0484 0.2522 -0.5476 0.0303 -0.0682 0.40790

6.1.3 Model Testing Diagnostic Checking After fitting the model to the data, the last step in the Box-Jenkings methodology to time series analysis involves diagnostic checking. 21 The standard prac20 These

results are consistent with the findings in Section 5.4.2.

'See Brockwell/Davis (1996), pp. 162-165 and Enders (1995), pp. 97-98.

6.1 The Model under the Objective Measure

125

tice is to plot the (standardized) residuals to look for outliers and evidence of periods in which the model does not fit the data well. If the fitted model is appropriate, the standardized residuals (i = 0, , 4) should have properties similar to those of univariate Gaussian white noise processes with variance 1. Plotting ?0, ,a4 , the graphs gives no indication of a nonzero mean or nonconstant variance, though some very few outliers can be observed. So on this basis there is no evidence for a violation of the white noise assumption. The next step is to check that the sample ACFs of go, ... behave as they should under the assumption that the fitted model is appropriate. Under the white noise assumption, it is well known that for large sample size N the sample autocorrelations are approximately i.i.d with distribution N (0,11N). We can therefore test whether or not the observed residuals are consistent with white noise by examining the sample ACF of the residuals and rejecting the white noise hypothesis if more than three out of 60 fall outside the confidence bound ±1.96/Nifi. Exemplarily, Figure 6.1 shows the sample ACF of ?2. As can be seen from the graph, only 3 or 4 out of 60 sample autocorrelations are significant. Therefore, we do not reject the null of a white noise process for The sample ACFs of Fo and "Fi look similar. On the other hand, the sample ACFs of 2.3 and "F4 indicate a violation of the white noise assumption, because more than 4 sample autocorrelations lie outside the critical bounds. Finally, the Box-Pierce and Ljung-Box Q-statistics can be used to test whether a group of autocorrelations is significantly different from zero. Under the null hypothesis of no significant autocorrelations, Q is approximately x2 distributed . Applying the Ljung-Box test to F.3 and 'al , the null hypothesis of no autocorrelation is rejected at the 1% level for almost all of the tested lag lengths. It is also rejected for -el , though only for some of the tested lag lengths. 22 A further examination reveals that for the residuals ê3 and 64 to become white noise AR models of order 10 and more would be needed. Obviously, this would present a conflict with model parsimony and also with other models which mainly advocate AR(1) processes for implied volatilities. 23 the hypothesis of normally distributed residuals is formally rejected Althoug at the 5% level, large deviations from normality are in general not observed for any of the series. 24 According to Stier (2001), p. 118, a "good" (V)AR model is characterized by three properties: 22 We

used different lag lengths of up to 200 lags. (2002), for instance, proposes a three-factor model for the S&P 500 implied volatility surface, where each factor follows an AR(1) process. For the DAX, Wallmeier (2003), pp. 194-195, suggests to model the ATM volatility and the difference between the implied volatility of a 5% OTM option and a 5% ITM option (called the "span") by an AR(1) process. These variables are similar to the risk factors Y1 and Y2 in our model. 24 The violation of the normality assumption means that the model is not able to fully match the empirical distributions of the DAX index and the DAX volatility risk factors. 23 Cont/Fonseca

126

6 A Four-Factor Model for DAX Implied Volatilities

..

I

o

1

.11

10

]

20

I

I '

I

.

1

II

I

1],

I

f

I

'

30

40

50

60

Lag

Figure 6.1. Sample ACF of ±1.96/.1Kr

2.

The parallel lines are the 95% confidence interval

1. The model is parsimonious. 2. The model fulfills the stationarity and invertibility conditions. 3. The model's residuals are (Gaussian) white noise. Judging our model with respect to these criteria, it shows the first and the second property and it partially exhibits the third property. Parameter Stability In order to examine the issue of parameter stability, we estimate each of the processes Y1, , Y4 using data for the periods 1995-1996, 1997-1998, 1999 2000, 2001-2002, and the whole period 1995-2002, for comparison. Table 6.3 reports the results. As the estimates indicate, the autoregressive coefficients ... ,(,o4 seem to be fairly stable over time, whereas the model constants , e4 are not. This implies that the mean-reversion levels, given by ci = C/ (1 — (pi ) , are time-varying, where, on the other hand, the speed of mean reversion, given by a, = (1 — cif) /t, is supposed to be rather constant for all factors, apart from a scaling effect that occurs from the division by At. ,)

6.1 The Model under the Objective Measure

127

The increase of Zi reflects the increase of volatility levels over the years. This effect has already been documented in Section 5.4.1.

and (pi Table 6.3. Estimates of the parameters subperiods and the total sample period 1995-2002

(i = 1,

, 4) for four different

1995-1996 1997-1998 1999-2000 2001-2002 1995-2002

(;32

-0.1267 0.9388

-0.0437 0.9679

-0.0577 0.9613

-0.0141 0.9874

-0.0163 0.9890

-0.0123 0.9729

-0.0267 0.9526

-0.0095 0.9789

-0.0305 0.9389

-0.0160 0.9681

0.4393

0.2055

0.4045

0.5031

0.3337

rp3

0.7725

0.7915

0.7097

0.6787

0.7715

ri34

0.0068 0.8580

0.0048 0.8770

0.0097 0.8927

-0.0061 0.9481

0.0008 0.9441

The level of implied volatility Y1 has a major influence on the evolution of the stock price. Hence, the model parameters -y i and pi" are of particular interest. To assess their stability, Table 6.4 reports estimates of -Yi and po for the four subperiods 1995-1996, 1997-1998, 1999-2000, and 2001-2002 as' well as for the whole sample period 2001-2002. As the values show, the fluctuations are rather moderate. Interestingly, the correlation between the DAX index and the DAX volatility level , Po,, is considerably higher for the second subperiod and the last than it is for the other two subperiods. This effect might be explained by the sharp decrease of stock prices during those periods. In such market situations it is often observed that correlations between stock prices and (implied) volatilities break down, i.e. they tend to -1.

Table 6.4. Estimates of the parameters ^yi and Am for four different subperiods and the total sample period 1995-2002

1995-1996 1997-1998 1999-2000 2001-2002 1995-2002 0.8157 -0.5816

1.2089 -0.6759

0.9346 -0.5344

0.9584 -0.6976

1.0006 -0.6152

All in all, the parameter estimates seem sufficiently stable so as to expect that the model is also useful in predicting the DAX volatility surface.

128

6 A Four-Factor Model for DAX Implied Volatilities

Out-of-Sample Forecasting So far, we have only investigated the in-sample behaviour of the model. In this section, we complete the analysis by examining the model's out-of-sample forecasting ability.25 Our forecasting procedure involves the following steps: 1. 2. 3. 4.

Divide the sample to an in-sample period and an out-of-sample period. Estimate the model using in-sample data. Obtain forecasts for the out-of-sample period. Compare the forecasts to the actual observations in the out-of-sample period.

We choose the (initial) in-sample period to include the first 250 daily observations or approximately data from the year 1995 (step 1). Using these data, we apply the procedure described in the previous section to estimate the model parameters (step 2). In step 3, we employ the estimated model to produce forecasts of the risk factor vector Y = (Y1, Y2, Y3, Y4) for three different forecasting horizons h: 1, 5, and 10 trading days. Due to specific structure of our VAR(1) model - the variables Y1, , Y4 do only depend on their own lagged values and € = (Et, €21 137 (4) is independent white noise so that En and em are independent for n m - it can readily be shown that the optimal h-days-ahead given the information at time n, equals the vector of the inforecast ,

dividually h-days-ahead optimal forecasts (il,n+h) P2,n+h, V-3,n+h 24,+h) - 26

, the optimal h-days-ahead forecast Conditional on 14,n, Yi,n- , , 4) is given by27 factor i (i = 1,

fti,n+h of (6.36)

= E 1Yi,n+h lYi,n, Yi,n-11 • • .1 = Ci (PNYi,n —

Cs).

For actually computing (6.36), the true parameters ci and cpi are substituted by their estimates ai and ri32 . In evaluating the forecasts, we conduct the forecasting in a ro lling window fashion, i.e. every period the latest data is added, and the oldest data is removed. The size of the rolling window is thereby kept fix at 250 observations. The advantage of a rolling-window scheme, as opposed to an extendingwindow scheme, is that always the most recent observations are used for estimation and forecasting. For h = 5 or h = 10, we have to decide whether to use overlapping or nonoverlapping windows. In general, overlapping windows give us more forecasts, however, they have the drawback that the forecasts are autocorrelated, by construction. Because our sample is sufficiently large -

2 5 Strictly speaking, the following tests are not truly out-of-sample as we have used all data from the sample to derive the functional form of the daily DAX volatility surface. 26 See Ltitkepohl (1993), pp. 28-30. "See Hamilton (1994), P. 80.

6.1 The Model under the Objective Measure

129

and h is suffciently small, we decide for nonoverlapping windows. To illustrate the procedure let us suppose that we have estimated the model using observations 0 through 249. Assuming that h = 5, we get the first 5-day ahead forecast. Next we update both the in-sample period and the forecasting period by rolling forward by 5 days. The updated in-sample period then comprises observations 5 through 254. Based on these data, the model is reestimated and the second 5-day ahead forecast is obtained. The procedure is repeated until the end of the sample period is reached. To compare the forecast values with the realized or actual values (step 4), we perform two analyses. In the first analysis, we run for each factor i (i = 1, , 4) and for each forecast horizon h E 11, 5, 101 the following regression: yi (kh)

= 14p,i)

u!,hk)

k =1,2, ...,mh.

(6.37)

where the k-th realization (forecast) of factor i for horizon h is denoted by yi (kh) ) u is a random disturbance and the sample size is mh. 28 At least three hypotheses can be tested using equation (6.37). First, if fr (kh) contains some information about

Yi(Z ) , the slope coefficient b(111? should be nonzero.

Second, if P., (:) is an unbiased forecast of Yi(:) , we should find

4:12 = 0

and

fek)

should be 14111 = 1. Finally, if the forecasts are efficient, the residuals white noise. For judging the overall out-of-sample performance of the model, the coefficient of determination R 2 is used. OLS estimates of (6.37) are shown in Table 6.5. Obviously, the slope estimate 'El is statistically different from zero in all cases. 28 Hence, the forecast values contain at least some information about the future actual values. Since the estimates for bi are morevover close to unity and the estimates for bo are close to zero — none of them is significant at the 5% level — we would expect the forecasts to be unbiased. To formally check on this, we perform F-tests. In all cases, the joint hypothesis H0 : bo = 0, b1 = 1 cannot be rejected; none of the F (2, m — 2)-statistics is significant at the 5% level. The Durbin-Watson statistic DW, reported in the last column of Table 6.5, is only in three cases significantly different from two. Therefore we conclude that the residuals u are not autocorrelated and hence that the forecasts are efficient. The high R2 values, especially for h = 1, suggest that the model forecasts explain most of the out-of-sample variations of the volatility risk factors. As expected, the out-ofsample R2 value is negatively correlated with the forecast horizon. However, the loss in out-of-sample performance turns out to be quite different for the different risk factors. So is the decrease in R2 relatively small for Yi , Y2 and Y4 but quite large for Y3. To conlude, our results suggest that the forecasts of the volatility risk factors are unbiased and efficient. 28 See

Christensen/Prabhala (1998), p. 134, for a similar test. "For convenience, we omit the risk factor index i and the forecast horizon index

h.

6 A Four-Factor Model for DAX Implied Volatilities

130

Table 6.5. Test on the out-of-sample performance of the model: regression results

t-ratio

F-ratio

(b = 0) (bo = 0 , h

=1

Yi Y2 Y3 Y4

h

=5

Yi Y2 Y3 Y4

h

=

10

Yi Y2 Y3 Y4

= 1)

R2

DW

0.0027 -0.0023 -0.0012 -0.0004

1.0008 244.02" 0.9969 146.48" 0.9867 46.54" 0.9997 116.88"

0.44 0.39 0.98 0.41

97.24% 92.71% 56.22% 89.01%

1.87 2.18 2.23 1.92

0.0027 -0.0084 0.0289 -0.0034

0.9971 0.9890 0.9307 1.0111

47.32" 29.26" 9.02" 24.07"

0.41 0.30 1.32 0.24

86.99% 71.89% 19.56% 63.38%

2.01 1.97 1.37 1.87

77.56% 56.40% 7.50% 51.50%

1.80 1.67 1.55 1.62

0.67 -0.0056 0.9855 23.95" -0.0197 0.9710 14.65" 0.34 0.4429 0.6502 3.66" 2.12 -0.0095 1.0568 13.15" 0.82 (") significant at the 5% (1%) level

Besides the magnitude of the forecasts, the number of correct direction predicted by a model may also be of interest. In the second analysis, we therefore compute the proportion of correct direction (PCD) for each risk ,4) and for each forecast horizon h E {1, 5,10 } , where the factor Yi (i = 1, PCD measure is defined by

PCD -

Number of correct direction forecasts

(6.38)

The results are shown in Table 6.6. The first observation is that all PCD values are above 50%, i.e. the model produces more correct direction forecasts than wrong direction forecasts. To check on their statistical significance, we test the null hypothesis H0 : PCD < 0.50 against the alternative hypothesis : PCD > 0.50. For all volatility risk factors but Y2 and for all forecast horizons h the null hypothesis of a PCD value lower than or equal to 0.50 (or 50%) can be rejected at conventional significance levels (1% and 5%).3 0 The second observation is that the PCD values increase with the length of the forecast horizon. Probably this is due to the mean-reverting property of the series: the longer the forecast horizon, the stronger the pull back to mean. This explanation is also supported by a third observation: for a given forecasting horizon h the PCD value is highest for the series with the shortest half-life (here: Y3), and lowest for the series with the longest half-life (here: Y1 and

Y2). 31

30 The

31

p-values for Y2 are 7.63% (h = 1), 5.67% (h = 5), and 10.79% (h = 10).

See Table 6.2.

6.2 The Model under the Risk-Neutral Measure

131

Table 6.6. Proportion of correct direction (PCD) values

Y1 Y2 Y3 Y4

h=1

h= 5

h=10

52.81%* 51.74% 57.90%** 55.06%**

58.16%** 54.30% 66.17%** 60.23%**

60.11%** 54.76%

69.64%" 68.45%**

(**) significant at the 5% (1%) level

The above analyses have shown that the model is able to forecast quite accurately the direction of the movements of the volatility risk factors as well as their future levels. 32

6.2 The Model under the Risk-Neutral Measure 6.2.1 Risk-Neutral

Stock

Price and Volatility Dynamics

This section focuses on the pricing and hedging of contingent claims. For the present, we shall assume that there exists a unique martingale measure Q. Then, according to Theorem 4.3 in Section 4.3, the arbitrage price of any Q-attainable claim H is given by the risk-neutral valuation formula where the expectation is taken with respect to the joint diffusion of the stock price and its volatility under the measure Q:

dSt = St rdt + Stvtdn t , dv t =

(6.39)

(iit (m(t, St , St , t, r), 0) — iit (m(t, St , St, t, r), 0) O t ) dt t (m(t,

(6.40)

St , St , t, r), 0) dWt*.

Here, W* is a (p + 1)-dimensional Q-standard Brownian motion, and ti) = (00 , , OpY is the (p + 1)-dimensional market price of risk process. The drift process Ti t = Tit (M, T) and the (p + 1)-dimensional volatility process j:9- t = t (M, 7- ) are given by33

ag l og

Fit = Tt (t,171,t, • • • , Yp,t) ±

± 32 For 33 See

1 — 2

P P

E

ag E ai,tav

(t, Yi,, • • • , Yp,t)

(6.41)

02 g (t,Yi,t, • • • ,

Yp,t)

i,k=1 j=0

a test on the economic significance of these forecasts, see Section 7.3.3. also equations (4.9) and (4.10).

6 A Four-Factor Model for DAX Implied Volatilities

132

and (6.42) with

ag

.3j,t = E -NJ t —

,

/ 14, • • •

(6.43)

,

'

Considering the specific structure of our model, p is equal to 4 and i = 1,...,4,

^/t,3,t =

-

j

= 0,...,4.

for all t E [0, T*] . By using (6.44) 9(t,M,T,YI,Y2,Y3,Y4) = es" + Y2M ( 1 + ln (1 + T)) +y3M2 (1 + p2 in (1+ 7)) + y4 in (1+ T) it follows that (6.45)

cri ,te Y" + a2,tM(1 + p ln(1 + 7))

Fit (M,

1 2

+3,M2 (1 + 6.2 1n(1 + 7)) + cx4, t In (1 + + —

4

0, jeYi•t,

and T9i, t (M,

7 1,3 e171 ' t --y 2 M(1 +

ln(1 + 7))

(6.46)

+73,i M2 (1 + p2 1n(1 + 7)) + -y40 ln (1 +

T),

with at, t = at(ct — for all t E [0, T* ] and for

j

, 4,

=-- 1,

(6.47)

= 0, , 4. Since the moneyness function

m(t, s, K ,T,r) = ln (serv—t) = ln ( 7 ) — r(T

—

t),

(6.48)

is 0 if evaluated at s = K = St , equation (6.40) becomes: dvt =

(0, 0) — (0, 0)1k t ) dt + .1-9 t(0,0)dWt* , 4

= ( a l (ci _ yi,t ) eYi.t + _ V -1,L eY"Le _ j=0

4

+ E -ri,, eY"to

dW3*,t•

3 =o

Using the fact that -y io = 0 for

j

= 2,3,4 and

E,,,Lieyi,t_,lp3 j=0

14

)

dt

6.2 The Model under the Risk-Neutral Measure e

Vt = e

Vt E [0,

133

,

the stock price volatility process under Q can finally be written as 1 f 2 dvt 2\ - = (a i (c i — ln (vt)) + — lY i o + 71,1) — 'Yi tolGo,t — -Y1,101,t)dt (6.49) 2

vt

'

+71,0dnt — 71,1dW11,t,

where the market price of stock price risk is given by

O o tt

/2— r = --• vt

(6.50)

At a first glance, it seems that the dynamics of the stock price volatility is completely determined by the evolution of Yi . However, this turns out not to be the case. The volatility risk factors Y2, Y3 and Y4 enter equation (6.49) through 0 1 , as 0 1 depends in general on the stock price and all volatility risk factors. Since the evolution of the volatility risk factors under the measure Q is given by 4

dYi, t =

[

ai (ci — Yi,t) —

E ,yi,itp,,t

4

i = 1,

dt +

J=0

, 4,

.J=0

(6.51) /P i also depends on ?,b(), 02)03 and 04'

for all t E [0,

6.2.2 The Market Price of Risk Process Recall from Section 4.2.1, that for the absence of arbitrage it is necessary that 19 t(K, T)/bt =

nt(K,T) 1 did2 +

+

2 crt (K,T)

(6.52)

19t(K ,T).0t(K,T)

1 1 2 2 , T)) 2 u t (K,T)(T t) ( v t crt(K d2

.\ o (K,T)T

-t vt19° 't(1 ' T) '

holds for all traded options at all times. To evaluate (6.52), we first compute qt (K,T) and 19t (K, T), generally defined in (4.20), (4.21) and (4.22). Differentiating the moneyness function m (given by equation (6.48)), once with respect to time and twice with respect to stock price, we get:

0m

= r,

Since in addition

0m

1

as

st '

— (t, St ) = --

and

02 m

1 as2 (t, St ) = R

.

6 A Four-Factor Model for DAX Implied Volatilities

134

au (M, r) in(i + T)) + 273,0m(1 + e 2 = 72,1)( 1 am ag Y2 tme1 + Y3,tM2 e2 + — (M,T)= ' 1+T Or ag

am

(M, T) = Y2,t (1 + p

+ T)),

ln(1+ -r)) + 2Y3, t M(1+ e2 In(1 + 7-)),

492 g

am 2

(M, 7") = 2173,t(1

g2 1n(1 + r)),

we finally obtain Me1 + }73,tM2 e2 Y2'

Y4,t

(6.53)

1+T

12

+- vt 2Y3 t(1+ e2 1n(1 + r)) 2 ' +—

± Iv?) [Y2,t(1 -1- e1 ln(1

—v t [72,o( 1 + Pi

r)) + 2Y3, t /t/g1+e2 1n4+

+ T)) + 2'y3,0M4 + e2

+

,

and

vo,t(K, T) =130,t(m, T) — vt [Y2,t(1+e 1 ln(1 + 7)) + 2Y3, t M(1-1-e2 1n4 + -0)1, 1, , 4, (6.54)

193 , t (K,T)= -1793 ,t (M,r), j =

for all t E [O , T*]. Because we assume that an equivalent martingale measure exists, equation (6.52) must hold especially for a specific option with strike price K1 and maturity date T1 and additionally

fim ?,bt <00,

(6.55)

Since —

11111 1.9 t T1

, 71) =

72,0M+1'3,0M2 +v7-1 (172,T1 +213,T' M) 1,1eY1 'T1 — /1,2-

2,01 2,2— 72,3 M

+ 73,1 )14.2 A/1'2

/3,2—

+ 73,3A/2 71,4 eY1 '71 — 72,4M + '73,4 M2 71,3 eY1 ' 7.1

<00,

—

(6.56)

P-a.s., the left-hand side of equation (6.52) is less than infinity P-a.s. 34 Let us consider the right-hand side now. As 34 Note that all processes appearing in equation (6.56) are under the real-world measure P. As such they are of course well-defined.

6.2 The Model under the Risk-Neutral Measure liM qt (K t—oTi

'TO =

135

(M 1) — Y2,T i M + YS,T 1 M 2 e2 j_ '4,T, + (r —

+ — 21 VT 2 1 ) [Y2,Ti

2173,TIM]

— VT1 [12,0 + 273,0 M] < oo P—a.s., but 1

di d2

+ d2

crt

WTI — t

1 2 o-t (Ki ,

1 (Ti — t)

— cr(K , Ti ))

vo90 t (Ki

explodes as t approaches T1, the right-hand side of equation (6.52) also explodes. This presents a contradiction to our initial assumption and thus gives rise to arbitrage opportunities in the model. One possibility to overcome this problem would be to modify the initially specified processes of the volatility risk factors such that the no-arbitrage condition (6.52) can be satisfied. But this would cause other problems, e.g., such processes are usually very complicated, and they often produce a worse fit to the observable implied volatility data. Therefore, we maintain our initial real-world model specification. To be still able to use risk-neutral pricing, we try to find a new market price of risk process 0 and thus a measure Q such that the stock's instantaneous volatility process y, defined by (6.49), exists, and the set of standard options is priced back correctly within a specified range of degrees of moneyness and times to maturity. 35 If we succeed, we can deduce that we have found a measure Q that at least avoids arbitrage opportunities "locally". In the following we show how to specify 0. This involves two basic steps: 1. Specify the functional dependence of 0i (i = 1, ables Y1, ... 2. Calibrate the model to market data.

, 4) on the state vari-

The remainder of this section is devoted to step 1. In order to specify the functional form of 0.i,t (i = 1, ... 4), we use a statistical approach that is based upon the observed market prices of risk. For this purpose, we select a set of options with degrees of moneyness

M=

35 This

:

=

Mu

—

9

z. = 0,...,9}

implies that we now are in an incomplete market.

136

6 A Four-Factor Model for DAX Implied Volatilities

and times to maturity T =- {10/365; 40/365; 70/365; 100/365 } . 36 The moneyness boundaries ML, and Mu are chosen to be —0.05 and 0.05. As for (M3 , r3 ) EMx T, the right-hand side of (6.52) exists, (6.52) is well-defined, and we want to keep close to the original model, we deduce the functional form of 0 (i = 1, , 4) from (6.52). Therefore, for each day n E {0, , N } in the sample, we estimate the market prices of risk ?/), n , j = 1, ... 4, from a cross-section of k = 40 options specified by (Mi, Ti) E M x T. Estimation is done by running the regression model 4

I in (Kg , Ti)-190 (K

= Ei9im (Ki , Ti) i,Gim -Fund

j = 1,

, k, (6.57)

where un is a random disturbance and fin (K 3 , T3 ) is defined as the righthand side of equation (6.52). The computation of I (Ks T3 ) ,19„ (Kg ,Ti ) , i = 0, , 4 and 1,bo n is based on the parameter estimates of Section 6.1.2 and the observation vector ,

(Sn, Yl,n1 172,n, 13,n, Y4,n, rn)

where rn is the risk-free rate on day n. Across the 1939 days in the sample, the average adjusted R2 is 80.93%. Given the time series of coefficient estimates 17/ • • • 7 ;1-1;4,w we next examine their statistical properties and their relationships to the state variables P 4 , the null Y1, , Y4 in more detail. Testing on the stationarity of "ti) i , ,:tof a unit root can be rejected at the 1% level for all variables, using the ADF and PP-test. Performing various statistical analyses and balancing the issues complexity and accuracy, we suppose 02, 03 and 04 to be constants. To find a suitable representation for 0 1 in terms of Y1, , Y4, we use stepwise regression.37 The set of explanatory variables consists of a constant, the variables Y1, , Y4, as well as all combinations of Y1 , , Y. The regression model that has the smallest value of Akaike's Information Criterion is considered the "best". Restricting the number of variables to five (including the constant), we find this to be: = tçl K2Y1,n K3Y12,n K4 171,nY4,n K5 372,073,n Cn)

n E [0, N] . (6.58)

where PC2 (i = 1, , 5) are the regression coefficients and ( is a random disturbance. The average adjusted R2 value amounts to 97.55%. All parameter estimates turn out to be significant, and an inspection of the residuals shows no severe violation of the assumptions underlying the regression model. To test 36 The specification of the set of options in (M, r) rather than (K ,T) has practical reasons: If we fix at time t = n • At an option with maturity date T this option ceases to exist at time T and has to be replaced by another option. Yet, it is unclear by which one. 37 For details on stepwise regression methods, see Miller (1990).

6.2 The Model under the Risk-Neutral Measure

137

the robustness of the model, we compare the model's goodness-of-fit among four different subperiods of equal length. The average adjusted R2 values of 96.04%, 98.90%, 99.12% and 98.19% suggest that the relationship between 0 1 and the variables on the right-hand side of (6.58) is sufficiently stable to allow us to model 11) 1,t

=

+ k2Y1,t K3 1712,t K4 171,0744 K5Y2,tY3,t,

t E [0, T*]. (6.59)

6.2.3 Pricing and Hedging of Contingent Claims Summing up the preceding results, we can state the following theorem: Theorem 6.1. Let Q be an equivalent martingale measure defined through

1)14 = k1 + N2Y1,t K3Y12,t

1

02,t = N62

k4 171,074,t N5Y2,073,t,

71) 34 = 1 71 04,t =

, Kg are real-valued constants and

where

4

dYi,t =

[ai(ci

4

— Yi,t) — E-y i,i 0j,t dt + E-yi ,i dW; t , i=o i=o

i = 1, ... , 4.

The process 01 is supposed to be well-defined. Then the arbitrage price Ilt (H) at time t E [0, 7 ] of any Q-attainable claim H on the stock S with maturity date T < T* in the factor-based stochastic implied volatility model is given by il(H) _

In [HI Ft] ,

Vt E [0,T] ,

(6.60)

where the joint evolution of the stock price and the stock price volatility under

Q is dSt = St rdt + StvtdK,t,

(6.61)

and

dvt vt

- =

1 (ai (ci — ln (vt )) + — 2

2

2 \

o + 71,1)

— 7 1 , 1 r,b 1 ,)dt (6.62)

+-Yi,odnt general, the expectation (6.60) cannot be solved in closed-form. Yet, it In general, can be accurately approximated with numerical methods such as tree methods, finite difference methods or Monte Carlo simulation. Due to the model's complexity, Monte Carlo simulation seems to be most appropriate. 38 Valuing the contingent claim H with maturity T at time t E [0, TI with the Monte Carlo method consists of the following steps: 38 See Hull (2000), pp. 406-415. For an in-depth discussion of Monte Carlo methods, see Jackel (2002).

6 A Four-Factor Model for DAX Implied Volatilities

138

1. Simulate J sample paths or trajectories of S under Q over the time horizon

[t,T] 2. Calculate the value of the contingent claim H at maturity T on each sample path, i.e. II3 ,T, J = 1, • , J. 3. Calculate the mean of the sample values at maturity II3 ,T to get an estimate of the expectation (6.60) and discount this value back to time t to get an estimate of the value of the contingent claim. In effect, this method computes an integral - the expected value of the discounted payoffs over the space of sample paths. Without loss of generality, we shall henceforth assume t = 0. To simulate the j-th path (j = 1, , J) followed by S under Q, we divide the life of the contingent claim into N short time intervals of length At = (T — t)IN =TIN and approximate the SDE for the stock price (6.61) by39

At+ln(43) 1 )+v,..,.(3) i vai43

ln(SV) = (r — —2 (v

n = 1, , N,

,

(6.63) where S;» denotes the value of S in path j at time t n . The simulation equation for the stock price volatility is based on the Euler discretization. It is given by

vW ) =t

1 {1 +

(6.64)

(c1 — ln (v± 1 )) + 1/2 (-yo

71,1 1trteii}

± with

oCi) =K 1,n

I

2 y(i) 1,n

-3

(0)) k 1,n

N 61

2

4_

-4

(i)y(i) 4_ y1 ,n 4,n

W3,n = k77

„h(j)

(6.65) (6.66)

=

and 4

Y U)

-

Y (i) + [at.(c.:- — Y (i)-1 )

-

E .., oi) .

k=0

4 1

dt +

E-yido/Ei41) k=0

(6.67) for any i = 1, ... , 4, and for n = 1, ... , N. For any k (k = 0, ... , 4) and any j (j =- 1, ... , J) , E is a random sample from a normal distribution with mean zero and standard deviation zero. The above equations enable the value of S in path j at time T = N - At to be obtained from the initial stock price = So and the initial instantaneous volatility 4 ) = exp(YP0) ). From the stock price 4 ) at time T = N • At, the value of the derivative in the j-path, II3,T, can easily be obtained by evaluating the payoff function. 39 This discretization is similar to (6.14).

6.2 The Model under the Risk-Neutral Measure

139

Carrying out the simulation J times, the Monte Carlo estimate Îlo of Ho can be computed as

fio = e-rT 1 V Ty . j

, 11

3,T.

(6.68)

j=1

The Monte Carlo estimate 110 can be shown to be unbiased and consistent. Its standard error is given by4° SE =

[eTHj,T]

(6.69)

In practical applications, the variance V [e- rTIli,T] is typically unknown and has to be estimated - most often by the sample variance. Since FI0 is the sum of independent random variables, it follows from the central limit theorem that for large J the estimator Îlo is normally distributed with mean 0 and standard deviation SE. This implies that with a probability of approximately 68% the true option price lies within the interval [I-10 - SE, Ho + SE] . It should be noted that using the SE alone to assess the accuracy of an option price can be quite misleading in some cases.41 The dynamic hedging of an option position requires knowledge of the various hedge parameters - the Greeks. For the calculation of the Greeks within the Monte Carlo framework, there are several methods awailable. 42 Here, we use what is called an explicit finite differencing approach with path recycling. According to this method, the option price is recalculated (on the basis of the already generated stock price paths) with varied inputs reflecting the potential changes in the respective parameters. For delta, we can just recalculate once with an upshift in the underlying stock, So -■ So + A So, resulting in a new estimated option value Îlo (So + ASO , and take

^ (5o

(So + ASo) - flo (So)

AS0

(6.70)

where Flo (So) denotes the original Monte Carlo estimate of the option price. As an alternative to the forward differencing approach in 6.70, we can recalculate twice, once for an upward and once for a downward shift, and approximate the delta by the central difference

= o (S o + ASo) Îlo (So - Aso) 2AS0

(6.71)

Using central differences has the additional advantage that we can then approximate gamma directly as "See, e.g., Campbell et al. (1997), p. 384. Jackel (2002) advises to additionally compute convergence diagrams. a detailed description of the different methods, see Jackel (2002), Chapter

41 Therefore, 42 For

11.

140

6 A Four-Factor Model for DAX Implied Volatilities

=

flo (So

+ ASO — 2110 (So) + flo (So — ASO Asa

(6.72)

The other Greeks can be computed in a similar fashion. Whilst the above calculations are straightforward, one important question has yet to be addressed: how do we choose ASo . Here, we follow the recommendation of Jackel (2002), p. 142, who suggests: ASO 6 1 So, (6.73) with E being a suitable representation of the machine precision. For applying the Monte Carlo method, we need an underlying random number generator that provides us with pseudo-random numbers, i.e. uniform numbers between 0 and 1. From the many generators available, we use the socalled "The Mersenne Twister" random number generator. Standard normal random numbers are obtained from pseudo-random numbers by inverting the cumulative probability function of the standard normal distribution." 6.2.4 Model Calibration Calibration Method Let us denote by K = (K i , K2 , t£3, K4, K5, K6, k7, Kg) the vector of parameters that together determine the market price of risk processes Oi (i = 1, , 4) and thus the martingale measure Q. In order to specify this measure, we use (current) price information from the options market and calibrate or fit the model to these data. Without loss of generality we shall henceforth assume that t = 0. Let the set of calibration instruments consist of n standard European call options. By observing today's stock price So, today's interest rate r = ro and today's volatility risk factors Y4,0, the (estimated) market price of , n) is given by the i-th option (i = 1,

(Ki , Ti) = CBs(0, So , Ki , Ti , r, cro(Ki,

TO),

(6.74)

where the option's implied volatility (estimate) o-0(K1 ,T1) is determined by 0 (K1,

K. T1 ) = eY") + Y2, 0 ln (sT +Y3,0 In

(1+ in (1+ TO)

( S÷ irT) 2 (1 -1-' 2 1n (1 + Ti )) +

Y4,0

(6.75) In (1 + Ti ).

On the other hand, given the vector of "historical" parameter estimates, we can compute today's theoretical prices of the options contracts. In general, they depend on the parameter vector K. We denote them by Cr y (K1 , T1 , te),

i =1,...,n. "For details see Jackel (2000), pp. 74-75 and pp. 100-102.

6.2 The Model under the Risk Neutral Measure

141

-

In order to calibrate the model to data, we choose the parameter k in such a way that the theoretical prices Cff iv (K2 , Tt , k) are "as close as possible" to the observed market prices Co (K, Ti ). This yields the estimated "implicit" parameter vector R. More precisely, we determine IC such that the mean absolute percentage error (MAPE), defined by"

MAPE(K) =

1E n I Cry (Ki,Ti,K) - Co(K„Tz ) I Co (Ki , T, )

(6.76)

-

n t_i

becomes minimal, i.e. R is the optimal solution to the problem' min MAPE(K).

(6.77)

KeR8 An Illustrative Example

In the following, we calibrate the model to options prices as of December 30, 2002. To keep the computational burden at a reasonable level, we select only two maturities 7 1 = T1 = 20/365 and T2 = T2 -=- 50/365, and for each maturity we consider only 5 degrees of moneyness: -0.05, -0.025, 0, 0.025 and 0.05. The Monte-Carlo estimates of the theoretical option prices are based on J =10,000 simulation paths. On December 30, 2002, the DAX index level So is 2885.73, the risk-free interest rate r is 2.936%, and the DAX volatility surface is characterized by

Y1,0 = -0.7199,

112,0 = -0.6173,

/73,0 = 0.8891,

Y4,0 = -0.1647.

With vo = exp(Y1,0) = 48.68%, the initial stock price volatility is exceptionally high. The estimates of the constants ei and 82 are respectively given by -1.6977 and -3.3768. Using data for 2002, we obtain the following estimates for the different components of the parameter vector 0:

•

i= 0.08;

• Zi = -0.0108; C2 = -0.0314; Z3 = 0.2900; -È-4 = -0.0091; • = 0.9883; rp2 = 0.9385; C,33 = 0.8006; Cf34 = 0.9439; • = 0.8814; 2 = 0.4854; 33 = 7.6036; '34 -= 0.9829; "-.)7

and

2. =

1.0000 -0.6688 0.0476 0.2058 0.2012 -0.6688 1.0000 -0.1517 -0.1457 -0.7748 0.0476-0.1517 1.0000 0.0007 0.1671 0.2058 -0.1457 0.0007 1.0000 -0.0436 0.2012 -0.7748 0.1671 -0.0436 1.0000

.

44 0ther commonly used error functions in this context are the mean absolute deviation and the sum of squared errors. 45 A solution IL iS only valid if the corresponding market price of risk process 0, (K) is well-defined. It turns out that this is almost always the case.

6 A Four-Factor Model for DAX Implied Volatilities

142

Equivalently,

• al •

= 2.9165; a2 = 15.4389; a 3 = 50.0483; a4 = 14.0686; = —0.9301; F2 = —0.5107; F3 = 1.4548; at = —0.1624;

and

-_ I —

( —0.5895 0.6553 0 0 0 0.0231 —0.0783 0.4786 0 0 1.5648 —0.9829 —0.0833 7.4399 0 ' 0.1977 —0.8465 0.0186 —0.0946 0.4485

The coefficient estimates of regression (6.58) are used as initial values for K5, and the sample means of 02 , 03 and 04 for 2002 are used as initial values for n6 , K7 and Ka. To solve the optimization problem (6.77), we employ the Downhill Simplex method, proposed by Nelder and Mead in 1965. 46 The method has the major advantage to require only function evaluations, but not derivatives. Solving the optimization problem (6.77), yields an optimal MAPE of 0.4498% or approximately 45 basis points. 47 This value indicates that the model is able to accurately price back the considered set of standard options. It is sufficiently low for most practical applications." The optimal solution is given by

(-8.1937,-8.6625,-0.1517,31.5799,-1.2344,-39.4111,-3.6882,-0.7633) . (6.78) Note that economic theory does not require the market price of risk vector to be positive in all components at all times. Negative values for some components, except for the market price of stock price risk 00 , are still consistent with economic intuition as the factors Y1, ... , Y4 are not actually traded. Figure 6.2 shows a typical simulation path of the DAX index S (left axis) and its volatility y (right axis) over one year (or 365 calendar days). The negative correlation is apparent. To illustrate the mean-reverting behaviour of the volatility process y, we additionally plot simulated values of y over the next five years (or 1825 calendar days) in Figure 6.3. The smooth behavior of are y in Figure 6.3 may also serve as an indicator that y and hence also indeed well-defined.

a description of this algorithm, see Press et al. (1992), Section 10.4. 'Note that this value is effectively an estimate, since option prices are obtained by Monte Carlo simulation. "When we calibrate the model against longer-term options, e.g., one-year options, the calibration error reduces significantly. This can be explained by the fact that the one-year stock price distribution is closer to a lognormal distribution than is, e.g., the 20-day stock price distribution. Further, we may also reduce the calibration error by increasing the number of free parameters, e.g., by including the parameter vector 0 in the calibration. 46 For

6.2 The Model under the Risk-Neutral Measure

143

Figure 6.2. Simulated path of the DAX index S and its volatility y over one year

Figure 6.3. Simulation path of the DAX volatility process y over five years

144

6 A Four-Factor Model for DAX Implied Volatilities

6.3 Model Review and Conclusion In an examination of the model's out-of-sample performance, we find the forecasts of the volatility risk factors to be unbiased and efficient. Moreover, the model produces significantly more correct direction forecasts than wrong direction forecasts. This justifies the modelling of the volatility risk factors as AR(1)/OU processes, although in-sample diagnostics might have suggested the use of higher-order models for some factors, especially for Y3. 49 The model turns out to be well-suited to capture the real-world dynamics of the DAX volatility surface and the DAX. But although the real-world processes fit the observable data very well, they give rise to arbitrage opportunities in the model as the market price of risk process and hence the stock's instantaneous volatility process do in general not exist. To be still able to use risk-neutral pricing, we have shown how to find a new market price of risk process and thus a new risk-neutral measure Q such that the stock's volatility process is wellbehaved. This measure Q at least avoids arbitrage opportunities "locally", because it ensures that a given set of standard options can be priced back nearly correct within a prespecified range of degrees of moneyness and times to maturity. In contrast to traditional stochastic volatility models such as the Heston model, the stock's instantaneous volatility process y in our model is observable (at least under the real-world measure) and the same set of parameters can be used under the real-world measure and the risk-neutral measure. As will be seen in the next chapter, this allows for the integrated pricing and hedging, risk managing, and trading of (DAX) index derivatives and derivatives on the index volatility.

49 To a similar conclusion come Goncalves/Guidolin (2003), p. 14, using data for S&P 500 implied volatilities.

7 Model Applications

The essence of knowledge is, having it, to apply it; not having it, to confess your ignorance. Confucius (551 BC - 479 BC)

This chapter presents applications of the factor-based stochastic implied volatility model (in short: SW model), as specified in the previous chapter, in the fields of pricing and hedging, risk management, and trading. For problems arising in the areas of risk management and trading, we apply the objective probability measure, and for questions concerning the valuation of contingent claims, we apply the risk-neutral measure. In Section 1, we discuss the pricing and hedging of selected exotic derivatives, including volatility derivatives. In Section 2, we apply the model to calculate the value at risk for option portfolios. Finally, Section 3 deals with volatility trading. We describe several ways of how to trade volatility, discuss the advantages and disadvantages of each strategy, and empirically test some of these strategies on their ability to generate abnormal trading profits.

7.1 Pricing and Hedging of Exotic Derivatives 7.1.1 Product Overview Exotic options were developed as advancements to standard options with specific characteristics tailored to particular investors' needs. They are also known as special-purpose or customer-tailored options. While standard options are often referred to as first-generation options, exotic options are also termed second-generation options. As there is no definition that fully describes exotic

7 Model Applications

146

options, they are usually understood to comprise all options that are nonstandard. Exotic options can be classified into two groups: path-independent options and path-dependent options.' The payoff of a path-independent option only depends on the terminal stock price, no matter whether the stock price at maturity is reached from above, below, or in a zigzag way. On the other hand, path-dependent options are designed to capture how the terminal stock price is reached, i.e. their payoff depends on the stock price path. A further distinction can be made for quantitative versus qualitative options.' In addition, there are one-factor as well as multi-factor exotic options. 3 Figure 7.1 lists some well-known (one-factor) exotic options and classifies them according to their path-dependency. 4

Exotic options

'

payoff oniy aepenas on terminal stock price

payon oepends on terminal stock

price and stock price path

• Digital options

• Forward-start options

• Power options

• Barrier options

• Contingent premium options

• Asian options

• Loolcback options

Figure 7.1. Classification of exotic options with examples The latest development in the market for exotic equity index products are volatility derivatives. 5 These contracts, written on realized or implied index volatility, provide direct exposure to volatility without inducing additional ex'Some authors further differentiate between weakly path-dependent and strongly path dependent-options. See, e.g., Wilmott (1998). 'For a qualitative option, the strike price only determines if the option is ITM, ATM or OTM but it does not influence the payout level in contrast to a quantitative option. 'Multi-factor options are options on several underlyings while one-factor options only have one underlying. The most popular multi-factor options are basket options and rainbow options. 4 For a comprehensive overview on exotic options, see Zhang (1998). 'In 1998, Eurex was among the first derivatives exchanges to launch a volatility derivative — the VOLAX future. The VOLAX futures was a future on the implied three-month ATM volatility of options on the DAX. Unfortunately the volume in

7.1 Pricing and Hedging of Exotic Derivatives

147

posure to the index level. So far, volume in this rapidly growing market is very much concentrated on variance swaps and volatility swaps. These are forward contracts on realized variance or realized volatility, respectively. 6 Recently, however, some investment banks (e.g., Merrill Lynch) started trading in options on variance and volatility swaps. 7 If we call exotic equity index options second-generation options, we may call volatility derivatives third generation -

derivatives. 8 Below, we will apply SW model to price and hedge two particular types of exotic equity index options: digital options (belonging to the class of pathindependent options) and barrier options (belonging to the class of pathdependent options). In addition, we discuss the pricing and hedging of variance and volatility swaps. The criteria that led to the selection of these instruments were the following: • • • •

Level of path dependency whereby path-dependent options are more complex than path-independent options; Relevance of implied volatility modelling in regard to the sensitivity of the respective option price; Importance for other derivative structures as building blocks or hedging instruments; Practical relevance in the sense that the selected instruments are regularly traded.

7.1.2 Exotic Equity Index Derivatives Digital Options Digital options, also known as binary or bet options, are very popular in the OTC marketplace for their simple payoff patterns. They are important building blocks for constructing more complex derivatives 9 and as hedging instruments for such structures. There are three basic types of digital options: 1. Cash or nothing options which pay a fixed amount of cash at maturity, 2. Asset or nothing options that have an asset as payoff, -

-

-

-

this contract was dissappointing and therefore the contract has been stopped. For more information on the VOLAX future, see Hafner (1998). 6 For convenience, we do not always strictly distinguish between the terms "volatility" and "variance". 7 See Patel (2003). 8 Some authors, e.g. Rodt/Schafer (1996), also use this term in the context of "standard" exotic options. 6 For example, digitals are used in the pricing of contingent premium options. These are plain vanilla options which cost nothing to initiate; the holder only pays the premium if the option expires ITM. In addition, Pechtl (1995) has shown that the payoff of a standard option can be duplicated with the sum of an infinite number of digital options.

148

7 Model Applications

3. Gap options where the payoff is determined as the difference between an

asset price and a prespecified level which is usually different from the strike price. Henceforth, we shall focus on the simplest and most commonly used digital option: the cash-or-nothing option (CON). A cash-or-nothing option is very much like a bet. If the underlying asset exceeds (falls below) the strike price K then a fixed amount of cash b is paid to the buyer of a cash-or-nothing call (put). Otherwise the option expires worthless. The payoff at maturity T from a long position in a (European-style) cash-or-nothing call option therefore equals1 ° if ST > K, CONCT = (7.1) if ST < K, f ob where ST denotes, as usual, the stock price at time T. Correspondingly, the payoff of a cash-or-nothing put option is

CONPT =

b 10

if ST K, if ST > K.

(7.2)

While the pricing of a cash-or-nothing option is fairly easy as will be seen later, the hedging of such an option can be difficult due to their discontinuous payoff profile. Digital options are highly sensitive to price changes of the underlying when they are ATM. The pin risk associated with these options is that if the price of the underlying oscillates around the strike price near expiration, the hedger would have to buy and sell large quantities of assets very quickly to replicate the option. The risk from small price changes may at some point exceed the maximum liability of the cash-or-nothing option. 11 A cash-or-nothing call option can be priced very conveniently. Its time t price CONCt is simply the cash amount b times the discounted risk-neutral probability of the option being ITM at maturity T (T> 0: 12

CONCt = e —r(T—t)— Q r[CONCH .F], _ b. e—ro--t) Q ( s7, > Kly) ,

(7.3) Vt G [O, T].

In the standard Black-Scholes framework, the probability Q( ST > KI Ft) is easily evaluated to N(d2), yielding an option price of be — r(T— t)N(d2 ). 13 In the SW model, there exist two (theoretically) equivalent methods to determine the cash-or-nothing call option price. Firstly, we can apply Monte Carlo simulation to approximate the probability Q( ST > KI.Ft ) and then use equation (7.3) to obtain the option price. Secondly, we can use arguments of static

1° Note

that at maturity the value of an option equals its payoff. "See Taleb (1997), Chapter 17, for more information. 12 Equation (7.3) follows directly from the general risk-neutral valuation formula and Theorem 6.1. "See Zhang (1998), p. 401.

7.1 Pricing and Hedging of Exotic Derivatives

149

option replication to derive an analytic formula for the option price. The second method is usually preferred since there is no estimation error and the computation is much faster. Following the second method, it is well-known that in any arbitrage-free model a cash-or-nothing call option can be perfectly replicated through an infinitesimal call spread. 14 This implies that the arbitrage price CONCt of a cash-or-nothing call option is equal to the first derivative of the standard call option price function with respect to the strike price. Taking the volatility smile crt (K,T) as a function of the strike price K, this can also be expressed as

CONCt = b. [CONCBs(ai(K,T)) - Ass(at(IC,T)) (9at(K'l OK

(7.4)

where

CONCBs(crt(K,T)) = e - r(T-t) N(d2(at (K,T)))

(7.5)

is the Black-Scholes price of a cash-or-nothing call option with cash payout 1, evaluated at y = at (K,T), and

ABs (at(K, T)) = St 7A.rin(di (cr t (K, T)))

(7.6)

is the Black-Scholes vega of a standard call option, also evaluated at y = at (K, T). 15 From the representation of the volatility surface in the SW model, i.e.

at(K,T)= eY1 't + Y2,t

( .F

,) ) (1 + p ln (1 + (T - t)))

(7.7)

2 +173,(ln

Ft(T) (1 + P2 K

ln (1+ (T - t))) + Y4,t ln (1+ (T

t)) ,

the slope of the smile at the strike price K is computed as:

crt (K,T)

ax

—

Y2 t ' (1 ± pi ln (1 +

T))

Y3 t

(7.8)

K Si

+2 (-r(T - t) + ln (— )) (1 +

ln (1 + 7)).

Equations (7.4), (7.5), (7.6), and (7.8) deliver an analytic solution for the cash-or-nothing call option price. Apparently, only the current smile but not the smile dynamics has an impact on the option price. Given the price CONCt of a cash-or-nothing call option, the arbitrage price of the corresponding cash-or-nothing put option CONPt can easily be derived from the call option price through the following put-call parity which holds in any arbitrage-free model: 14 See, 15 See

e.g., Taleb (1997), p. 281. equation (47) in Brunner/Hafner (2003).

150

7 Model Applications

CONCt + CONPt = be- r(T-t) .

(7.9)

To illustrate the price differences between the standard Black-Scholes model and the SW model, let us consider on December 30, 2002 (t = 0) cashor-nothing DAX calls with degrees of moneyness of -0.05, -0.025,0,0.025 and 0.05. The options' time to maturity is 2 months (or 41 trading days). For convenience, the cash amount b is set to 1,000 (Euro). The DAX index level So is 2885.73 and the risk-free interest rate r is 2.936%. Based on this information, the initial DAX forward price is calculated as Fo(T) = 2899.92. The DAX volatility surface on December 30, 2002 is characterized by the following parameters: = -0.7199,

173,0 = 0.8891,

Y2,0 = -0.6173,

Y4,0 = -0.1647,

as well as di = -1.6977 and --d2 = -3.3768. This implies an initial instantaneous stock volatility of vo = exp(Yi,o) = 48.68%. The volatility parameter in the Black-Scholes cash-or-nothing call pricing formula is set to the option's implied volatility. Table 7.1 reports the analytically computed SW price and the corresponding Black-Scholes (BS) price against moneyness. Table 7.1. Comparison of Black-Scholes (BS) and factor-based stochastic implied volatility (SW) cash-or-nothing call option prices. Time to maturity: 2 months M

SW price

BS price

-0.050 -0.025 0 0.025

639.07 588.52 533.90 476.06

558.03 510.49 460.17 407.87

0.050

416.19

354.67

Apparently, the SW prices are higher than the BS prices. This is also what one would expect from formula (7.4) in the presence of a volatility skew: since the Black-Scholes vega is always positive and the first derivative of cro (K,T) with respect to K is negative for M e [-0.05, 0.05] , the correction to the Black-Scholes price is positive in this moneyness region. In a further application, we compare the Black-Scholes cash-or-nothing call deltas, given by16 Delta = e- r(T-t) where

1

n(x)=e N/Tir 16 See

Wilmott (1998), pp. 103-105.

b St cr1/2 x2 ,

n (d2)

Vx E R,

(7.10)

7.1 Pricing and Hedging of Exotic Derivatives

151

with the deltas obtained from the SW model. Though the price of a cash-ornothing call option could have been derived analytically in the SW model, this is in general not possible for the SW delta. The reason is that the stock price and the implied volatility are not independent, but are correlated. To account for this correlation, we apply Monte Carlo simulation to equation (7.3) and calculate two option prices, one for the original underlying price and one for a slightly different underlying price. Using these prices, the delta is estimated according to equation 6.71. The Monte Carlo estimates are based on J =100,000 simulation paths and estimates of the parameters of the SW model are taken over from Section 6.2.4. 17 The results are shown in Table 7.2. The average standard error of the SW delta estimate is 0.008. For ITM options, the SW and BS deltas are similar in magnitude; however, for OTM options, the deltas differ significantly. Table 7.2. Comparison of Black-Scholes (BS) and factor-based stochastic implied volatility (SW) cash-or-nothing call option deltas. Time to maturity: 2 months

M

SW delta

BS delta

-0.050 -0.025 0 0.025 0.050

0.7031 0.7187 0.7508 0.7747 0.7924

0.6855 0.7111 0.7262 0.7283 0.7153

Barrier Opt ions The growing OTC-market for exotic options has greatly extended the range of marketable contingent claims. 18 Barrier options are among the most successful products. They correspond to standard European options with the additional feature that the existence or the continuance of the option depends on whether the price of the underlying exceeds or falls short of a contractual barrier. There are two types of barrier options: knock out options and knock in options. The former are voided once the underlying asset reaches or crosses the barrier. The latter, in contrast, only come to life if the barrier condition is met. In the following, we focus on European knock-out options. These can be subdivided into a down and out version characterized by a lower barrier and an up and out type knocked out when reaching an upper barrier. Knock-out -

-

-

17 Due

-

-

-

to the discontinuous payoff profile of a cash-or-nothing option, the option

price distribution is more difficult to approximate than it is for a standard option. To ensure a similar accuracy of the Monte Carlo estimate as in the case of a standard option, we increase the number of simulations to 100,000. For a detailed investigation of this issue, see Hafner (1997), pp. 138-146. "In the following, see Steiner et al. (1999b).

152

7

Model Applications

options sometimes provide a fixed payment (rebate) to the option buyer if the option expires prematurely. Usually, the rebate is due as soon as the barrier is crossed ("at hit"). An important determinant of the option value is how frequently the barrier condition is monitored. The barrier can be monitored continuously or discretely at specified dates, as is often the case for equity index and single stock options. 19 In the latter case, the barrier is effective only at discrete time intervals. The value of a knock-out option increases with the larger time span between successive monitoring dates due to the reduced knock-out probability. Let St denote, as usual, the price of the underlying asset at time t, K the option's exercise price, U the constant barrier and RE the fixed rebate. The rebate is payable as soon as the option is knocked out ("at hit"). The current time is to = 0 and the option expires at time T. The barrier is effective only at N points in time t 1 , t2, , tN with 0 = to < t i < < tN = T. At the beginning, the stock price lies above the barrier: So > U. The payment that the holder of a discrete down - and- out call receives at time tn (n = 1, . , N-1) is given by

RE

DOCt„ { 0

At expiration tN

if (St„ < U) A (St ..

= / RE 0

,

1 }) ,

(7 .11)

= T the payoff to the option's holder is

max {ST — K; 0} DOCT

> U Vz E {1,

otherwise

if St. > U Vz E {1, • • 'N } , if (StN < U)A (St .. > U Vz E {1, • • • , N — 1}), otherwise.

(7.12) In the following we assume monitoring of the barrier in equal discrete time intervals, i.e. 4, — = TIN -= constant for n = 1, , N. Without loss of generality, we additionally assume RE = O. The time t (T > t) arbitrage price of a discrete down-and-out call option in the SW model is given by

DOCt = e—r(T—t) in "'Q[DOCTITt],

Vt E [0, 7 ] ,

(7.13)

where Monte Carlo simulation can be used to compute the expectation

BQ [DOCTI.Ft ]. In the standard Black-Scholes framework, the arbitrage price of a barrier option with continuous monitoring can be analytically deduced. It consists of three additive components: the value of a standard European option, the reduction in value due to the barrier condition and the present value of the rebate." If the barrier is monitored discretely, however, an analytic solution is not available. Therefore, the use of numerical procedures is unavoidable. Among the most frequently applied methods are the binomial Continuous barrier options are frequently traded in the foreign exchange options markets. 20 See Cox/Rubinstein (1985). "

7.1 Pricing and Hedging of Exotic Derivatives

153

and trinomial models (tree methods). As is well known, the naive application of the binomial or trinomial model can result in erroneous prices, even if the number of time steps is large. Therefore, a number of modifications to the standard binomial and trinomial model has been suggested in literature. 21 Among those, the correction technique of Steiner et al. (1999a) and Steiner et al. (1999b) has proven to be very effective. Using this technique, a small number of time steps (about 200) suffices to obtain accurate option prices even if volatility is high and the barrier lies near the current stock price. Let us now apply the SW model to price down-and-out DAX call options with daily monitoring of the barrier as of December 30, 2002. In particular, we consider options with degrees of moneyness of —0.05, 0 and 0.05 and barrier levels of 90%, 95% and 99% of the current DAX index level So = 2885.73. All options have a time to maturity of 2 months (or 41 trading days). To account for the more complex payoff pattern of a down-and-out option compared to a cash-or-nothing option, the SW option price is calculated on the basis of 200,000 simulation paths. All other model and market parameters are the same as before. Table 7.3 shows the results. Standard errors are given in parentheses. For comparison, we also report the corresponding Black-Scholes values. These are computed using the modified trinomial tree model of Steiner et al. (1999a) and Steiner et al. (1999b) with a number of 488 time steps. 22 In addition to the SW and BS prices, we compute the options' deltas by recalculation of the respective option prices. These are also shown in Table 7.3. As expected, the higher the barrier level, the higher the knock-out probability, and hence the lower the option price. The differences between the SW prices and the BS prices are relatively small, with the largest differences being observed for a barrier level of 99%. The deltas for the SW model and the BS model are on average more different, though the absolute differences are still moderate. 7.1.3 Volatility Derivatives Variance Swaps Variance swaps provide exposure to (realized) variance without inducing additional exposure to the index leve1. 23 They took off as a product in the aftermath of the Asian and Russian crisis in 1998 when implied stock index volatility levels rose to unprecedented levels. Hedge funds took advantage of this by paying variance in swaps, i.e. they sold realized variance at high levels of implied variance. The key to their willingness to enter into a variance swap rather than to sell straddles was that a variance swap gives a constant exposure to realized volatility without the need to delta-hedge the position. 21 See Steiner et al. (1999a) and Steiner et al. (1999b) for a detailed discussion of the various approaches. 22 For this number of tree steps the pricing error is almost zero. 23 1n the following, see Brockhaus/Long (2000), Demeterfi et al. (1999a), Gatheral (2002), Lecture 6, and Mixon/Mason (2000).

154

7 Model Applications

Table 7.3. Comparison of Black-Scholes (BS) and factor-based stochastic implied volatility (SW) down-and-out call option prices and deltas. Time to maturity: 2 months

M

U

SW price (SE)

BS price

-0.05

90% 95% 99%

251.22 (0.81) 182.90 (0.76) 94.58 (0.61)

253.33 181.82 92.56

0.8225 0.9678 0.9623

0.7842 0.9300 0.9113

0

90% 95% 99%

189.92 (0.69) 142.40 (0.64) 75.97 (0.51)

191.48 143.05 74.23

0.6717 0.7772 0.7821

0.6260 0.7485 0.7685

0.05

90% 95% 99%

134.19 (0.57) 103.73 (0.53) 57.24 (0.41)

133.54 104.20 55.52

0.5236 0.5919 0.5998

0.4715 0.5770 0.6032

SW delta

BS delta

Besides the directional trading of volatility levels, variance swaps may also be used to trade the spread between realized and implied variance and to hedge portfolios that may be disrupted by a volatile market. 24 Given a stock price process S sampled on N equidistant points in time 0 = to
(7.14)

where 14(N) is the realized variance (quoted in annual terms) over the life of the contract [0, Tb KvARs is the delivery price for variance and Ar is the notional amount of the swap in Euros per annualized volatility point squared. The holder of a variance swap at expiry receives .AT Euros for every point by which the realized stock variance 4(N) has exceeded the delivery price for variance KvARs. The procedure for computing the realized variance is precisely specified in the contract. In a typical contract, the stock price is sampled each trading day at the official close, i.e. At = 1/251 or At = 1/252, and the mean of daily stock returns is assumed to be zero. More formally, v T (N) is usually defined as

4(N)

1 At (N

where Rt, = ln(St i ) - ln(St i , ) for i = 1,

1) i=1

'

(7.15)

, N. 25

24 Passive equity portfolio managers, for instance, may require more frequent rebalancing and greater transaction costs during volatile market periods. 25 Sometimes the maximum likelihood estimator of variance is E1 used instead of the sample variance at(k1) EN ]. (Rt.) 2 . The difference is, however, usually very small.

7.1 Pricing and Hedging of Exotic Derivatives

155

We now turn to the question of pricing and hedging variance swaps. 26 Suppose the stock price process S = {St : t E [0,1 1 is a diffusion (i.e. there are no jumps), with drift process z = : t E [0, 7 ] 1 and volatility process ]

= {vt : t E [0,2]1:27

dSt =hdt + StvtdWt,

Vt E [0, 7 ] ,

(7.16)

where W = {Wt : t G [0, 7 ] 1 is a one-dimensional standard Brownian motion. Volatility may be constant, deterministic or even stochastic. For a given price history, the realized continuously sampled variance iv2-. over the interval [0, T] is defined by 1 WT = —

T

T

v2dt,

(7.17)

t

2

o dt on the right-hand side of equation (7.17) is known where the integral f vt total variance over the interval [0, T]. The continuously sam(realized) as the pled variance wT is a good approximation to the realized (discretely sampled) variance 4(N) of daily returns used in the contract specifications of most 14(N).28 variance swaps, i.e. WT In principle, valuing a variance swap is no different from valuing any other contingent claim. According to the risk-neutral valuation formula, the arbitrage value of a variance swap with strike price K at time t E [0, T], VARS, is the discounted expected value of the future payoff under the risk-neutral measure Q: VARS t = e— r(T —0E41 [(WT K) • Afi-Ft] e —r(T—t) BQ

[

1 f V?Cit T 0

—

(7.18)

K) • .A.11.71 T

1 fo t V!Chl = e—r(T—t) EQ[(T

t

14,Citt — K)

1 f 1 —r(T—t) . A f . (_ 112 du + EQ — f V2 du — KITt •

=e

T 0 u

T t u

The term + fct, v2u du reflects the (known) realized variance from the start of the contract at time 0 up to the current valuation time t. For ease of exposition, we shall henceforth always assume t = O. The fair value of variance is the delivery price KvARs that makes the swap value zero today, i.e. for which VARS0 = O. It follows that29 KvARS = EQ

[WT] •

(7.19)

the following, see Demeterfi et al. (1999a). drift and the volatility process are required to be progressively measurable. 28 Exact equality is given in the limit: Niirrl i4(N) = WT. 26 1n

27 The

"Note that EQ [X I.7] = EQ [X].

7 Model Applications

156

In the case of the SW model, the volatility process is given by dvt 2 2 \ = (ai (ci – ln(vt)) + —21 vii,o -r71,000,t "71,101,t tv -

dt

VY1,0dnt

and the fair value of future variance KVARS can be computed directly as the following risk-neutral expectation (Method 1): KVARS = — 1MQ[f0 qdt .

(7.20)

In general, the expectation on the right-hand side of equation (7.20) cannot be solved analytically. Yet, it can be accurately approximated using a combination of Monte Carlo simulation and numerical integration. 30 First, Monte Carlo simulation is used to generate the volatility paths. Then, for each path, a numerical integration method (e.g., Romberg's method or Simpson's rule) is used to approximate the integral f0T V 2t dt ' Alternatively, it is possible to find a model-independent trading strategy that replicates variance exactly. 31 In fact, it can be shown that if the stock price process is a diffusion (as in equation (7.16)) and interest rates are constant, then the fair value of variance is given by the value of an infinite strip of European options (Method 2): Fo(T) 1

' 1 T)dK+f ,C0(K,T)dK , T o Fo (T) El (7.21) where Co (K, T) and Po (K,T), respectively, denote the current market price of a put and a call option of strike K and maturity T and Fo(T) = SoeT is the stock's T-maturity forward price. 32 Using the identities KVARS =

2 erT _

P (K' °

Co(K,T) = CBs(0, St , K,T,r, cro(K,T)) = CBS (K,T,cro(K,T)) , Po(K,T) = Pris(0, St , K,T,r, ao(K,T)) = PBS (K,T,a0(K,T)), equation (7.21) can also be written in terms of the implied volatility smile:

2

f Fo (T) 1

KVARS = T,er

— K2 PBs (K ,T, uo(K,T)) dK

0

(7.22)

±_ 2 erT foe T

— 1 CBs(K,T,o -o (K,T))dK, Fo(T) K 2

Provided that a continuum of European options with the same time to maturity and strike prices ranging from zero to infinity exists, we can use equation 30 For

details on numerical integration, see Press et al. (1992), Chapter 4. Carr/Madan (2002) and Demeterfi et al. (1999b). 32 For a proof, see Appendix B. 31

See

7.1 Pricing and Hedging of Exotic Derivatives

157

(7.21) or (7.22) to compute the fair value of variance KvARs in an easy and unique way. Yet, in practice, option contracts are only available for a discrete set of strike prices within a relatively small range around the ATM strike price. In order to apply equation (7.21) (or (7.22)), it is necessary to complete the market option price function (or smile function) by interpolating between available strike prices and extrapolating outside their range. As there exists in general an infinite number of extrapolation methods (and also interpolation methods) that is consistent with no-arbitrage, KvARs cannot be uniquely determined without presuming a particular model or method." Moreover, in many cases the extrapolation function is not available in closed-form, implying that the integral on the right-hand side of equation (7.21) (or (7.22), respectively) has to be computed numerically. In what follows, we apply the SW model to determine the fair value of a DAX variance swap with a time to maturity of 2 months and daily monitoring as of December 30, 2002. From the two methods available, we employ method 1, i.e. we compute KvARs according to equation (7.20). The Monte Carlo estimates are based on J =100,000 simulation paths. For numerical integration, we use the method of Romberg." The other model and market parameters are as before. Running the simulation, we obtain the following estimate for the fair value of variance:

kvARs = 0.226627. The standard error of the estimate is SE = 0.00026. The variance level of 0.226627 is higher than the implied variance of 0.46182 = 0.213259 of an option with a strike price equal to the current forward price Fo(T). As Gatheral (2002) and Demeterfi et al. (1999b) have shown, the difference can mainly be ascribed to the negative slope and the convexity of the DAX volatility smile. On the other hand, if the volatility surface is flat, the fair value of a variance swap equals the squared ATM implied volatility. Volatility Swaps Most market participants prefer to quote levels of volatility rather than variance. This has led to the development of volatility swaps. Conceptually, a volatility swap is similar to a variance swap; however, it refers to realized volatility instead of realized variance. Its payoff function is given by

VOLST = (0T(N) — Kvo Ls) • M,

(7.23)

where OT (N) = 1/4(N) is the realized volatility (quoted in annual terms)" over the life of the contract [0, Tb K ifo r.,s is the delivery price for volatility and ,Af is the notional amount. 33 Incorporating only price information of traded options, Demeterfi et al. (1999b) suggest a very powerful method to approximate the fair value of a variance swap. 34 See Press et al. (1992), pp. 140-141. 3 5 See also Definition (3.7).

158

7 Model Applications

In general, the fair delivery value of future (continuously sampled) volatility is the strike price KVOLS for which the contract value at inception is zero, i.e.: 36 1 volt] . (7.24) KvOLS = EQ [ “ri PT = EQ

[4, 0

A naive estimate of the fair price of future volatility would be the square root of fair variance KvARs : KvoLs -= /KVARS.

However, due to the Jensen inequality we know that this is in general not correct as37 KVOLS = EQ [017F]

[102 = N/KvARs]

(7.25)

Ki is known as the convezity adjustThe difference between KVOLS and N/ms ment. The magnitude of the convexity adjustment depends on the correlation between the underlying asset and the stock price volatility as well as on the volatility of volatility. In fact, there is no simple replication strategy for synthesizing a volatility swap and the magnitude of the convexity adjustment is highly model-dependent. 38 This makes pricing and hedging of volatility swaps much more complicated than pricing and hedging of variance swaps. Using the SW model, the fair price of a volatility swap can be obtained for in much the same way as for a variance swap: First, we compute each simulated volatility path using Romberg's method. Second, we take the average over all sample values. This yields the Monte Carlo estimate of fair volatility. Continuing the variance swap example of the last section, we calculate the fair price of the corresponding volatility swap. It is given by 0.469157 or 46.92% with a standard error of SE -= 0.00025. As expected, the fair price of volatility is lower than the square root of fair variance, given by 0.226627 = 0.476053 or 47.61%.

7.2 Value at Risk for Option Portfolios 7.2.1 The Value at Risk Concept The measurement of financial market risk, i.e. the risk that an institution incurs losses on its portfolio of financial assets due to unexpected changes in 36 Again, we assume that the realized continuously sampled volatility is a good approximation to the realized volatility of daily returns. 37 See Bamberg/Baur (1998), p. 121. 38 1n a comparison between a local volatility model and a stochastic volatility model, Brockhaus/Long (2000) find that the price of the former is always higher.

7.2 Value at Risk for Option Portfolios

159

prices or rates, is of primary importance for senior management and regulators. Broadly, there are four types of market risk: interest rate risk, exchange rate risk, equity risk, and commodity risk. Basic analytical tools apply to all of these markets. For example, sensitivity measures such as delta, gamma, and vega are used to describe different aspects of the risk in a portfolio consisting of equity index options. A financial institution usually computes each of these measures each day for every market variable to which it is exposed. This typically produces a huge number of different risk measures each day. Whereas these risk measures provide valuable information for traders who are responsible for managing various parts of the financial institution's portfolio, they are only of limited use to senior management. In contrast to market and instrument-specific risk measures, value at risk (VaR) provides an aggregated view of a portfolio's risk. It summarizes in a single, easy to understand number the total risk of an institution due to financial market variables. VaR has become widely used by financial institutions, corporates, and asset managers. Applications range from reporting and controlling of firm-wide risk up to determining the optimal allocation of risk. The Basle Committee on Banking Supervision (BIS) and other central bank regulators also use VaR as a benchmark risk measure in determining the minimum levels of capital a bank is required to maintain as reserves against market risk. In this context, the rules of BIS allow banks to use proprietary in-house models for measuring VaR as an alternative to a standardized measurement framework." A common intuitive definition of VaR is the following: "VaR summarizes the worst loss of a portfolio over a given period of time with a given level of confidence". More formally, given the complete probability space (S2, .F,P) , we define VaR as follows:40 Definition 7.1 (Value at Risk). Let 0 = (01 , ,0d)' be a portfolio of d assets held at current time t = 0, T > 0 be a time horizon, and (1—a) E (0,1) a given level of confidence. Let furthermore denote Vt (0) the value of the portfolio 0 at time t and AVT(q5 ) = VT(q) — Vo(0) the change of the portfolio value or the profit or loss over the time period [0, 7 ] . Then the value at risk VaR (a, 0,T) is defined as the negative a-quantile cAv,(0) (a) of the profit or loss distribution:41 VaR (a, 0,T) = — cAvr(0)(a) • (7.26) With FAv,,,(0) denoting the distribution function of AVT(0), c iw,( 0)(a) is given by (7.27) cAvT(0)(a) = sup {x E R : FAvT(0)(x) < ci}.

39 See

Basle Committee on Banking Supervision (1996). "See Huschens (2000) and Jorion (2000), p. 109. 41 I 1 the very rare case that Cpvi,(0) (a) is positive, VaR is zero, by definition.

160

7 Model Applications

Since we assume t = 0, the time horizon T is equal to the holding period T, defined by T = T — t. In the following, we will use the two terms interchangeably. In VaR calculations the time horizon (or equivalently the holding period) T is usually measured in trading days. We will follow this convention throughout this section and assume, as before, that a year consists of 251 trading days. In calculating a bank's capital, the BIS has set the confidence level (1—a) to 99% (i.e. actual losses on the portfolio should exceed the VaR estimate not more than once in every 100 cases on average) and the holding period T to 10 trading days (i.e. the maximum loss is calculated under the assumption that the portfolio remains unchanged for 2 weeks). For purposes of internal risk control most banks use a holding period of one day and a confidence level of 95%.42 In an investment environment, the time horizon, as measured by the portfolio holding period, is much longer, typically in the range of 1 month to

1 year. 43

7.2.2 Computing VaR for Option Portfolios Approaches to VaR basically can be divided into two groups. The first group uses local valuation. Local-valuation methods or analytical valuation methods measure risk by valuing the portfolio once, at the initial time t, and using local derivatives to deduce possible portfolio movements. The second group uses full valuation. Full-valuation methods measure risk by fully repricing the portfolio over a number of scenarios. 44 From the group of local valuation methods, the delta-normal method (henceforth used as a benchmark) is commonly used by practitioners to calculate the VaR of option portfolios. 45 It uses linear, or delta, derivatives and assumes normal distributions. To illustrate the approach, consider a portfolio 4) consisting of d options on a single stock whose time 0 price is So. The current value of the portfolio is Vo = V0(0). Suppose further that the portfolio delta is46 avo 64, =

aso

Then, using a first-order Taylor series expansion, it is approximately true that

AVT = (5,5AST,

(7.28)

42 This setup was first suggested by J.P. Morgan in their Risk Metrics framework. See Morgan (1996). 43 See Pallotta/Zenti (2000). 44 See Jorion (2000), p. 205. 45 Its popularity and its simplicity are the main reasons for selecting the deltanormal method as benchmark. Clearly, there exist more powerful methods to compute VaR for option portfolios. Some of them will be discussed later in the text. 46 The portfolio delta is the weighted sum of the deltas of the portfolio components. See also Section 3.1.4.

7.2 Value at Risk for Option Portfolios

161

where AST = ST So is the absolute change in stock price and AVT = VT — VO is the absolute change in portfolio value over the time period [0, T]. Denoting by RT

ST

—

SO

(7.29)

So

the discrete stock return in time period [0, 7 ] , equation (7.28) can be written as A VT = So(SoRT. (7.30) The expectation of

AVT under the real-world measure P is Ep [AVT] = Ep [S05 oRT] = SotR T

,

(

7.31)

where A RT = lEp [RT] is the P-expectation of RT . Similarly, the variance of AVT under P is VP [VT ] = V p [SO

RT] = Se5,v 2RT

,

(

7.32)

with yl = Vp [RT.] denoting the P-variance of RT . Under the assumption that RT is normally distributed, the portfolio VaR is obtained as

VaR (a, 0, T) = cN (0 , 1) (a) So I5 01 v R T — So4 /- R T (7.33) where cm om (a) is the a-quantile of the standard normal distribution. If the holding period T is short, A RT is often set to zero, in which case (7.33) simplifies to the well-known formula

VaR (a, 0,

= cmom (a) Soiboiv Rr •

(7.34)

When positions are constants and stock returns are small and i.i.d., it is approximately true that ART

=

VRT

= — N/Tg:

(7.35)

251 T,

(7.36)

where A and y are the annual expected return and the annual standard deviation of discrete stock returns, respectively.' Equations (7.35) and (7.36) are convenient to convert annual expected returns and volatilities into their

T-period equivalents. 48 47 Exact 48 1f

equality would only be given for log returns. the expected stock return is assumed to be zero, it follows that:

VaR(c, 0,T) = VaR (a, 0, 1 day)

VT.

This relationship, known as the square root of time adjustment, says that the VaR for a T-day holding period is the square root of T times the one-day VaR.

162

7 Model Applications

The full-valuation approach uses Monte Carlo simulation or historical simulation to generate the (unknown) probability distribution for AVT. 49 In the following, we focus on Monte Carlo simulation. Suppose we wish to calculate the T-horizon VaR for a portfolio of standard options. Using the SW model (as specified in Chapter 6) for scenario generation, the procedure is as follows: 1. Compute the value of the portfolio at time 0, Vo, using the current portfolio q5 and the current values of market variables. The time 0 index level is So and the current market prices of standard options are given in terms of their implied volatilities, which, in turn, are given in terms of the volatility risk factors Y1 ,0 , • • • Y4,02. Simulate one path or scenario of S, Y1, • • • , Y4 under IP over the time period [0,2] , using either specification (6.1) and (6.6) or specification (6.11) and (6.12). 3. Use the values ST7Y1,T, ,Y4,T to determine the value of each standard option. 4. Apply the portfolio function to revalue the portfolio at time T. The portfolio value is VT. 5. Compute the change in portfolio value, i.e. VT = VT — Vo. 6. Repeat steps two to five J times to build up the probability distribution for A VT. Let AVP ) denote the change in portfolio value over [0,2] in scenario j (j = 1, . . . , J) . Then the distribution function FAvT (x) of AVT can be approximated by

PAvr(x)=

E

(7.37)

j =1,...,J AV4? ) <x

Given PAvT (x), and using Definition 7.1, the a-quantile cAvT (4,)(a) and the VaR are straightforward to calculate. If the portfolio includes exotic options, the procedure described above has to be changed in the following way. In step 3, the model has to be calibrated to standard option prices using the information set STIY1,T1- • 1Y 4- ,T (see Section 6.2.4). The prices of the exotic options in the portfolio can then be obtained by using Monte Carlo simulation. 7.2.3 A Case Study In the following case study we will illustrate the differences between the deltanormal-method and the SW Monte Carlo simulation approach. Let us therefore consider two option portfolios on December 30, 2002 (t = 0). The level of the option's underlying DA)( index is 2885.73. The option portfolios consist of the following instruments: "For details, see Jorion (2000), Chapter 9. See also Hull (2000), pp. 355-356.

7.2 Value at Risk for Option Portfolios

• •

163

Portfolio I (PF1 ): a long position in 100 standard DAX call options with strike price K = 3000 and maturity date February 21, 2003. 5° Portfolio 2 (PF2): a short position in 100 ATM straddles (i.e. short positions in both a standard call and a standard put on the DAX with the same strike price and expiration date), with strike price K = 2900 and maturity date February 21, 2003. 51

The options' time to maturity is 38 trading days (or 53 calendar days). The relative DAX volatility surface on December 30, 2002 is given by

o (M, r) = e -0.7199 - 0.6173 • M (1 — 1.6977 ln (1 + r)) +0.8891 • M2 (1 — 3.3768 • In (1 + r)) —0.1647. ln (1 + r) . As before, estimates of the parameters of the SW model are taken over from Section 6.2.4. The mean return t of the DAX is assumed to be 8%. Based on this information, the implied volatilities and the market values (in Euro) of the options are computed as52

ao(3000, To) -= 44.79%, ao (2900, To) = 46.33%, cro (2900, To) = 46.33%,

C0(3000, To) = 157.38, C0(2900, To) = 206.62, P0(2900, To) = 208.02,

with To denoting the options' expiration date February 21, 2003. The time 0 portfolio values V0PF , i = 1, 2, are therefore

VoPF' = 100 • 157.38 = 15,738,

'VoPF2 = (-1) 100 • (206.62 + 208.02) = -41,464. The objective of the following analysis is to compute the VaR for each of the two portfolios assuming a time horizon of one day ("Risk Metrics") or 10 days ("BIS"), respectively. The confidence level (1—e) is set to 95% ("Risk Metrics"). Let us first consider the delta-normal method. Taking into account the current volatility smile (i.e. implied volatilities of 44.79% and 46.33%), the options' Black-Scholes deltas are calculated as 0.4561 for the call with strike 3000, 0.5348 for the call with strike 2900, and —0.4651 for the put with strike 2900. This yields the following portfolio deltas:

8

=

avoPFi aso —

100 0.4561 = 45.61

for portfolio 1, and was in fact a maturity date of Eurex-traded DAX options. Such a position was also run by Nick Leeson from Barings. In addition to a long position in Nikkei futures, Leeson sold huge amounts of straddles on Nikkei futures. For more details, see Jorion (2000). 52 For simplicity, we assume a contract multiplier of 1 Euro per index point. 50 This 51

164

7 Model Applications v_PF2

"

6 PF2

aso

— 100- (-0.5348 + 0.4651) =- —6.97

for portfolio 2. Using Ti = 8%, cmom (5%) = —1.645, and the options' implied volatilities for the volatility parameter y, the VaR for each portfolio and for each holding period is then computed according to formulas (7.33), (7.35), and (7.36). 53 The results are shown in Table 7.4, column "Delta-normal". Let us now consider SW Monte Carlo simulation. Following the procedure described in the preceding section, we generate 10,000 scenarios of S, Y4 under the real-world measure IP over the time horizon [0, 7 , where T equals 1 day or 10 days, respectively. For each scenario, we calculate the implied volatility, and based on that, we compute the portfolio value and the change in portfolio value. This provides us with a simulated distribution function P4,14,..(x). Figures 7.2 and 7.3 show the frequency distributions of AVTPF' and AVTPF2 Clearly, the distribution of VTPFL is positively skewed and the distribution of VTPF2 is negatively skewed. Both distributions show excess kurtosis. ]

-20000

o

80000

Change in portfolio value (in Euro)

Figure 7.2. Frequency distribution of 10-day changes in portfolio value for Phi 53 1n practice, historical volatility is commonly used as an estimate of the volatility parameter v. In the presence of volatility surfaces, however, this procedure may be quite misleading. For this reason, the VaR calculations here are based on the options' implied volatilities.

7.2 Value at Risk for Option Portfolios

165

VaR(5%,10 day) = 20,867 1200 -

1000 -

800 -

t so° 400 -

200 -

-40000

-20000

200100

400'00

Change in portfolio value (in Euro)

Figure 7.3. Frequency distribution of 10-day changes in portfolio value for PF2

Given PAv.,, (x), VaR is easily computed as the negative a-quantile of AVT. The results are shown in column "SW Monte Carlo" of Table 7.4. Table 7.4. Delta-normal VaR and SIV Monte Carlo simulation VaR (in Euro) for a confidence level of 95% Delta-normal SW Monte Carlo Holding period VaRPF1 VaRPF2 VaRfFi VaRPF2

1 day 10 day

6,611 20,618

1,010 3,150

4,887 13,392

5,390 20,867

Apparently, the VaR measures obtained under the delta-normal method differ substantially from the measures obtained under the SW Monte Carlo simulation approach. 54 For example, for portfolio 2 the 10-day SW Monte Carlo VaR of 20,867 Euro exceeds the delta-normal VaR of 3,150 Euro by 54 Note that in general the excess expected rate of return (over the risk-free rate) on a call equals the option's omega (i.e., delta times stock price divided by the option

7 Model Applications

166

17,717 Euro or 562%. This can be explained as follows: Since the straddle position is approximately delta-neutral, the value of the portfolio is relatively insensitive to small changes in the DAX and as such the delta-normal method produces a measure that suggests negligible risk. On the other hand, considering portfolio 1 and again the 10-day holding period the delta-normal VaR of 20,618 Euro is 54% higher than the VaR of 13,392 Euro obtained under the SW Monte Carlo simulation approach. In that case the delta-normal VaR is too high, because the maximum amount of money one can loose when holding portfolio 1 over 10 days is the initial option's premium, i.e. 15,738 Euro. It is further interesting to observe that under the delta-normal method the VaR for portfolio 2 is lower than the VaR for portfolio 1, whereas under the SW Monte Carlo simulation approach it is the opposite. 55 This implies that the delta effect must be dominated by other effects, that are captured within the SW Monte Carlo simulation approach, but not within the delta-normal method. In fact, the apparent problems of the delta-normal method can be traced back to a couple of key factors that are missing from the analysis:

•

Nonnormality: The delta-normal method assumes that the stock return RT is normally distributed. Empirical research on stock returns, however,

•

•

has accumulated convincing evidence for seemingly fat-tailed distributions, i.e. extremely low or high returns have greater probability than assigned by the normal distribution. 56 These fat-tails are particularly worrysome precisely because VaR attempts to capture the behavior of the portfolio return in the left tail. Nonlinearity: The delta-normal VaR inadequately measures the risk of nonlinear instruments. Under the delta-normal method, options are represented by their "deltas" relative to the underlying asset. This linear approximation, however, is only valid for small changes in the underlying asset. Asymmetries in the profit or loss distribution of options and option portfolios are not captured by the delta-normal VaR. Passage of time: The delta-normal method does not account for the time decay in option prices. For example, the long call position in portfolio 1 will loose in value over time and the short straddle position in portfolio 2 will gain in value over time, if everything else remains constant.

price) times the excess expected rate of return on the stock. Since omega is always greater than or equal to 1, the option's expected rate of return is never less than the expected rate of return on the stock, provided that the stock return exceeds the risk free rate. The same relationship holds for puts. Since the estimated stock return in our model exceeds the risk-free rate, long positions in options show a positive mean and short positions in options show a negative mean. See also Cox/Rubinstein

(1985). 5 5 This observation crucially depends on the chosen confidence level. For example, if we set the confidence level to 10%, the VaR for portfolio 2 is lower than the VaR for portfolio 1 under both the delta-normal and the 5 1V Monte Carlo simulation approach. This emphasizes again the role of VaR as a measure of extreme risk. "See also the discussion in Section 3.4.1.

7.2 Value at Risk for Option Portfolios

•

167

Changes in implied volatilities: The delta-normal method assumes a constant stock price volatility, and thus also constant implied volatilities. In fact, implied volatilities change over time. Considering portfolio 2, for example, an increase in implied volatilities will lower the value of the short straddle as the position shows a strong negative vega. In contrast, a decrease in implied volatilities will increase the value of the straddle.

To overcome some of these shortcomings the standard delta-normal method has been generalized in several aspects. For example, the delta-gamma method or quadratic Va.R includes the portfolio gamma to capture the nonlinearity of option positions with respect to the stock price. The delta-gamma-theta method additionally considers the passage of time by incorporating the option's theta. 57 Although these approaches undoubtedly present an improvement over the basic delta-normal method, they still have their limits. For example, they cannot capture all the nonlinearities the portfolio is exposed to and they still rely on the normal assumption. Since analytical methods to estimating VaR of option portfolios are often inadequate, consider instead a standard Monte Carlo approach (belonging to the group of full valuation approaches). 88 In this approach, the portfolio value is simulated over a set of scenarios containing instantaneous shocks to the relevant risk factors (here the stock price and eventually the volatility surface). A standard Monte Carlo approach captures the portfolio's nonlinearity but misses the passage of time effect. Moreover, it is not capable to adequately reflect the dynamic interplay between stock prices and (implied) volatilities. On the other hand, the SW Monte Carlo simulation approach presented here, accounts for the nonnormality in stock returns, the nonlinearity of option positions, the passage of time, and also for the (stochastic) change in implied volatilities. Hence, we would expect the approach to adequately measure the VaR of portfolios consisting of standard options, but also of portfolios that incorporate the underlying itself, futures on the underlying, exotic options, and also derivatives on the underlying's volatility.

7.2.4 Beyond VaR: Expected Shortfall VaR as a risk measure is heavily criticized for not being sub-additive. This means that the risk of a portfolio can be larger than the sum of the risks of its components. 58 Moreover, VaR does not take into account the severity of an incurred damage event. Often we do not only want to know the cutoff loss 57 A further generalization is the "delta-plus method", advised by the BIS. This method additionally accounts for vega risk. See Basle Committee on Banking Supervision (1996), A.5. For detailed information on all these methods, see Jorion (2000). "See Jorion (2000), Chapter 12. 9 See, e.g., Embrechts (2000).

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7 Model Applications

that will happen a percent of the time but also the average size of the loss, when it exceeds the cutoff value. A complementary measure that fulfills the sub-additivity property and that is able to quantify losses beyond VaR is expected shortfall, also called conditional expectation, conditional loss, or tail loss. It is defined as the expected value of VT (0), conditional that VT (0) exceeds the quantile cAvi,(0)(a) .60 Definition 7.2 (Expected Shortfall). The measure ES (a, , T) =

—

1Ep [AVT(0)1AVT(0) < cAv,(4,)(ce)]

(7.38)

is called the expected shortfall at confidence level (1 — a) of AVT(0)• Using the definition of conditional expectation, equation (7.38) can also be expressed as ES (a, 0, T) =

—

Ep [AVT(0)1AVT(0) < cAvr(0) (a)1

(7.39)

Ep {AVT (0)1 (A VT(0):AVr(0)
1

j

(7.40)

where

J* =

E

(2)

(7.41)

is the number of scenarios for which the condition A143) (0) < cAv,( 0) (a) is fulfilled. The results are shown in Table 7.5. For comparison, we also report the corresponding VaR numbers. Is is apparent that the size of the loss for portfolio 1, conditional on exceeding the VaR, is not much higher than the VaR itself (roughly 2.6% for a = 1% and 6.7% for a -= 5%). On the other hand, for portfolio 2, the VaR "In a very few, theoretical cases the definition of expected shortfall used here may violate the requirements of a coherent measure of risk in the sense of Arztner et al. (1999). For a definition of expected shortfall that always guarantees coherence, see Acerbi/Tasche (2002). Note that VaR is not a coherent risk measure.

7.2 Value at Risk for Option Portfolios a = 1% Portfolio VaR ES

PF1 PF2

VaR

169

ES

14,815 15,204 13,392 14,292 34,924 44,381 20,867 30,085

Table 7.5. SIV Monte Carlo simulation VaR and ES measures (in Euro) for portfolios 1 and 2. Holding period: 10 days

is significantly lower than the ES. For instance, considering a = 5%, the ES of 30,085 Euro exceeds the corresponding VaR of 20,867 Euro by more than 50% (see Figure 7.4 for a graphical illustration). This reflects the fact that the lower left tail of the profit or loss distribution of portfolio 1 decreases at a much faster rate than the lower left tail of the profit or loss distribution of portfolio 2.

ES

-60 1000

-401000

VaR

-20 1000

20000 1

400 00

Change in portfolio value (in Euro)

Figure 7.4. SW Monte Carlo simulation VaR and ES of 10-day changes in portfolio value for PF2. Confidence level (1—a) = 95%

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7 Model Applications

7.3 Volatility Trading 7.3.1 Definition and Motivation A substantial part of professional option trading focuses strictly on volatility and ignores the direction of the underlying market. 61 By speculating on whether realized volatility will exceed or fall short of implied volatility and whether implied volatility will rise or fall, volatility traders care more about how much the underlying price changes than about the direction. Why trade volatility? Just as stock investors think they know something about the direction of the stock market, so volatility traders think they can foresee the level of future (implied) volatility. If one thinks current volatility is low, for the right price one might want to take a position that profits if volatility increases. Part of the attractiveness of volatility trading may be explained by the well documented fact that forecasting directional changes in the underlying asset is very difficult 62 , whereas volatility can often be predicted with high accuracy. 63 The good forecasting ability of the SW model, documented in Section 6.1.3, also supports this assessment. Before we start discussing volatility trading strategies in more detail, we will give a definition of what we exactly mean by a (directional) volatility trade: Definition 7.3 (Volatility Trade). A position or portfolio whose value is only affected by changes in volatility (either actual or implied or both), but has no or only little exposure to the underlying asset is called a (directional) volatility trade or (directional) volatility trading strategy. Since volatility trades are delta-neutral by definition, they are also called market-neutral or undirectional (with respect to the underlying asset) trades. Unlike directional volatility trades, nondirectional volatility trades, also referred to as relative volatility trades or volatility arbitrage trades, attempt to profit by exploiting volatility differences on different markets. A typical trade that falls into this category is convertible arbitrage, a common hedge fund strategy. Convertible arbitrage strategies try to profit from underpriced implied volatilities that can often be observed for stock options embedded in convertible bonds relative to implied volatilities of standard options. 64 Below, we will focus on directional volatility trades.

"In a recent study, Chaput/Ederington (2002) have shown that much of the trading on option markets falls into the volatility trading category. 62 See Figlewski (1997), pp. 22-23, and Campbell et al. (1997), Chapter 2, among many others. 6 3 See, e.g., Ederington/Guan (1998), Ederington/Guan (1999), and Andersen et al. (1998). 64 For more details on convertible arbitrage, see Ineichen (2002), pp. 202-222.

7.3 Volatility Trading

171

7.3.2 Volatility Trade Design Trading Objects An investor will typically initiate a volatility trade when he has a certain view on volatility. For example, if he expects volatility to rise, he might be willing to "buy volatility". On the other hand, if he expects volatility to fall, he might be willing to "sell volatility". Since there exists a variety of different volatility concepts it is essential to precisely specify the type of volatility (henceforth called the trading object or the underlying) that should be traded. Possible trading objects are summarized in Figure 7.5. Basically, we can distinguish between two kinds of volatility trades. 65 The first class of trades focuses on the level of actual stock price volatility and depends on the relationship between ex-ante expected stock price volatility and ex-post realized stock price volatility. The success in this type of volatility trading depends on whether one is better than the market at forecasting the future volatility of the underlying asset. The second trade relies on changes in implied volatility (or the implied volatility surface). The most common trading objects in this kind of volatility trading are the implied volatility leve166 , the slope and the curvature of the volatility smile as well as the slope of the volatility term structure. Successful trading of the implied volatility surface requires a special understanding of the financial mechanisms that cause volatility surfaces and in particular the ability to forecast future volatility surfaces. The underlying of a volatility trade can be a spot or a forward quantity such as the spot smile or the forward smile. The volatility trading objects shown in Figure 7.5 are not strictly disjoint, but overlap. Most notably, the relationship between the actual volatility and the level of implied volatility (for a given maturity) is very close. To see this, consider, for example, the specification of the volatility surface in the SW model. Here, the actual stock price volatility vt is equal to exp(Yi,t). On the other hand, the level of implied volatility (for a given maturity r), -cit (0, r) , also depends crucially on Y1,t, since Fi t (0, r) = exp(Yi,6)-1-Y4,t ln (1 + r). This implies a high correlation between vt and'at (0, r) .

Trading Instruments In principle, all financial instruments with optional features such as standard options, exotic options, or structured products with embedded options provide exposure to volatility. However, the most natural way to trade directional views on volatility is by using volatility derivatives. For example, if one CBOT (1997), p. 93. this section, the "level of implied volatility" always refers to implied volatilities of options whose time to maturity T is greater than zero, because only such are actually traded. 65 See 66 1n

7

172

Model Applications

refers to the unaeriying asset

refers to the set of traosia options i

•

• Volatility level

Volatility level

Smile slope

• Smile curvature

Term structure slope

Figure 7.5. Basic volatility trading objects expects the future realized variance to be lower than the current implied variance (roughly the price of a variance swap), one might sell a variance swap. Alternatively, one could also sell a volatility swap or buy a put option on realized volatility with a strike price equal to the current implied volatility. 67 a convenient way to trade pure volatil-Althougvaiyderspo ity (either realized volatility or both realized and implied volatility, depending on whether the derivative is held until maturity or not), their liquidity is typically low as they are traded in the OTC market and only a few investment banks quote prices in these instruments. This makes it usually difficult to unwind positions at a fair price, especially in situations when liquidity is most needed, e.g., during a financial crisis. 68 But even in normal market situations the bid-ask spread is quite large. For example, the bid-ask spread of DAX variance swaps with maturities of up to one year is usually in the range of two to four volatility points." Exotic equity options may also be employed for trading volatility. The most prominent examples in this context are forward-start options and double barrier options (also known as dual barrier or corridor options). Considering, e.g., forward-start options, i.e. options which start at some prespecified time in the future with the strike price set to be the underlying price at the time when it starts, they are in effect bets on forward implied volatility (without Section 7.1.3 for details on these products. Liquidity also played a major role in the collapse of LTCM. LTCM had sold huge amounts of equity volatility and when equity volatility began to rise due to the Russian crisis, they were not able to timely close their positions at a reasonable price. For more details, see Lowenstein (2000), Chapters 7 and 8. 69 The bid-ask spreads of variance swaps on other liquid equity indices, such as the S&P 500 index or the Nikkei 225 index, are similar in magnitude. I am grateful to Dr. Jurgen Amendinger from HVB Equity Linked Products Group in Munich for providing me with this information. 67 See 68

7.3 Volatility Trading

173

any exposure to realized volatility)." Much like volatility derivatives, exotic options are also traded in the OTC market, and thus they also suffer from a lack of liquidity. Probably due to their high liquidity, most volatility traders prefer trading combinations of standard options when speculating for a rising or a falling volatility. To understand how to construct and manage volatility trades using standard options, we will first have a closer look at the sources of risk and return of standard options. Let us therefore consider a single standard European call option with strike price K and maturity date T at time t < T. We assume that the option's implied volatility ert (K ,T) and the volatility of the underlying stock yt follow arbitrary but well-defined Itô-type stochastic processes. The interest rate r is assumed to be constant. Then, by application of Itô's lemma to the Black-Scholes call option pricing formula (3.18), the dynamics of the call price Ct (K, T) under the real-world measure P isn

1 dCt (K, T) = O t dt St dSt + A tdat + — rd (s, s) t

(7.42)

2

1 ±- Vtd (a(K,T),u(K n t + ‘11t d (S,a(K,n t 2 where the Black-Seholes Greeks are given by (5t = 8Bs(t, St, K , T, r, t (K ,T)), rBs(t, St, K , T, r, (K, T)), At = Aas(t, St , K ,T, r, t (K , T)),

(7.43)

•=

et = eBs(t, St, IC, r, t(K T)), Vt = VBs(t, St , K , T, r, t (K ,T)), • = TBs(t, St, K,T, r, a t (K , T)). Since the Black-Scholes PDE

1

et + rts?6,1 (K, T)

+ Jt St r — rCt (K, T)

0

(7.44)

holds if the volatility parameter y equals the implied volatility cl? (K ,T) of the option and the quadratic variation of S is d (S, S) t =1.4S?dt, equation (7.42) can be written as

1 dCt (K, T) = r (Ct (K ,T) — 5 tSt )dt + — rtsT (14 — o (K, T)) dt td4t7 .45) 2 1 +Atdert(K,T) + — Vtd ((K, T), 0.(K ,T)) t + 'I'd (S,a(K t 2 70

See Langnau (1999).

71 See

also equation (4.24) in Chapter 4. Note that if interest rates were stochastic, the term ac drt would additionally appear on the right-hand side of equation (7.42). This term, however, is usually very small, at least for short-term and mediumterm options considered here, and is therefore neglected.

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7 Model Applications

Assuming zero interest rates (i.e. r = 0), for convenience, equation (7.45) can be further simplified to 1 u (K,T)) dt 6tdSt Atclut(K,T) dCt (K,T) =- rt s 2 1 +—Vt d (cr(K,T),cr(K,T)) t + Alf t d (S,cr(K,T)) t 2

(4 _

(7.46)

The discretization of (7.46) finally leads to the expression: ACt+At (K, T) = 6tASt-i-1t +

-11-sts? (4+4,t - cr? (K, T))

At (7.47)

+At Aat+At (K,T) + .1)t (Acrt+6,t(K,T)) 2 (K, T),

where At is a small time interval, ACt+At (K, T) = Ct-F-At (K,T)—Ct (K,T) , AS t+At = St+At — St, and Acrt+At(K, T) = t+At(K ,T) — crt(K,T). The realized volatility over At is defined as q+At = (ASt+pt/St) 2 . The term ACt+At (K, T) represents the profit or loss of the call option over the (small) time interval At. Equation (7.47) shows that a standard (call) option has five key exposures, each represented by a different (Black-Scholes) Greek, that, when combined with the value of their corresponding variables, determine the option's profit or loss over a small time period At." Table 7.6 shows the exposure, the associated variable, and the profit or loss for each of the components. 73 degree to which one is exposed to any of the five variables dependsThe on the strike price and the maturity date of the option. The interpretation of (Ao -t+At(K, T)) 2 and ASt+AtAat+6,t(K, T) as the "realized volatility of (implied) volatility" and the "realized correlation between implied volatility and stock price" results from the fact that for At approaching 0 the term (Aut+At (K, T)) 2 converges to the quadratic variation of the implied volatility process and ASt+AtAat+at(K, T) converges to the quadratic covariation of the stock price process and the implied volatility process.

Table 7.6. Profit or loss profile of a standard (European) call option Exposure

Variable

Profit or loss

Delta(bt ) Gamma(r) Vega(At) DVegaDVol(Vt) DDeltaDVol(W t )

Change in stock price Realized-implied volatility spread Change in implied volatility Realized volatility of volatility Realized correlation

6tASt+At

72 73

4 rts? (v4A , - 01 (K ,T)) At AtAat+At(K,T) 4 vt (Acrt+At(K,T)) 2 'I's ASt+AtAcrt+At(K,T)

See also CBOT (1997), p. 93. For convenience, we use the term "volatility" to denote both, volatility and

variance.

7.3 Volatility Trading

175

The exposures of a standard European call and a standard European put option (both long positions) are shown in Table 7.774 The DDeltaDVol measure is the only Greek that changes sign along strike prices. Long positions in ITM (OTM) call options have positive (negative) DDeltaDVols. Conversely, long positions in ITM (OTM) puts have negative (positive) DDeltaDVols.

Table 7.7. Exposures of a long call and a long put for a fixed maturity Exposure Delta Gamma Vega

DVegaDVol DDeltaD Vol

Long call positive positive positive positive strike-dependent

Long put negative positive positive positive strike-dependent

Of the five option characteristics, only one — the (Black-Scholes) delta — has nothing to do with volatility. Neglecting second-order terms, it is gamma and vega that allow for trading volatility. Traders seeking for exposure to actual or realized volatility should thereby take positions with large gammas and traders seeking for exposure to implied volatility should prefer positions with large vegas. For example, if one expects implied volatilities to rise (fall), a long vega (short vega) position should be established. By definition, volatility traders try to minimize pure stock price risk, implying that they try to keep their positions delta-neutral.75 To offset any net delta that an option position may have we can always buy or sell enough of the underlying stock or the corresponding future such that the overall position becomes delta-neutral. While it is theoretically possible to construct option portfolios (consisting of calls, puts, or a combination of the two) that have exposure to exactly one of the four volatility-related variables, such portfolios are usually difficult to manage. The reason is that such portfolios typically involve a large number of options and are only instantaneously insensitive to changes in the hedged variables. Since price changes, changes in volatility, and the passing of time affect the neutrality of the position, it is necessary to readjust some or all of the Greeks in order to keep the position neutral. If we rebalance often we will have a good hedge, but we will suffer from high monitoring and transaction costs. May be this is the reason why in practice most traders follow simpler strategies to take advantage of a view on volatility. Table 7.8 shows some of the most common strategies along with their primary trading objects. 76 74 75

Compare also Section 3.1.4. Note that unless DDeltaDVol is neutralized too, the option position is still

exposed to changes in the underlying stock. 76 1n fact, these strategies are officially supported by most derivatives exchanges, making it possible to trade them as an unit. For more information on these and other volatility trading strategies, see Steiner/Bruns (2002), Section 8.2.2.4, Natenberg

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7 Model Applications

Straddles are usually referred to as "combinations", because one buys (or sells) both puts and calls, while vertical spreads (also called bull or bear spreads, respectively), butterfly spreads and calendar spreads (also called horizontal spreads or time spreads) are termed "spreads" since some options are bought and others are sold. 77 For convenience, we will still use the term "volatility trade" or "volatility trading strategy" to refer to all of these strategies. The individual options making up a volatility trade are referred to as "legs".

Table 7.8. Volatility trading strategies. All descriptions are for long positions and are expressed as one combination unit. Unless otherwise stated, all options in a strategy have the same time to maturity Main Trading object Strategy name Definition Buy a call and a put with the same strike price (typically ATM) Vertical spread Buy a call (put) and sell a call (put) at Smile slope a higher (lower) strike price Butterfly spread Buy a call (put), sell two calls (puts) at Smile curvature a higher strike price and buy a call (put) at yet a higher strike Term structure slope Calendar spread Sell a call (put) and buy a call (put) at the same strike price but with a longer maturity Volatility level

Straddle

The assignment of a volatility trading strategy to a particular trading object is derived from the (implied) volatility risk factors (of our cross-sectional representation of the volatility surface) to which the strategy is exposed to at the outset of the trade. Consider, e.g., a long straddle with maturity date T and a strike price K equal to the current forward price Ft (T), i.e. an ATM long straddle. Since the call and the put have the same strike price and the same maturity date, they also have the same implied volatility:

at ( 0, 7) = ey,,i + Y4,t ln (1 ± r ) , where T = T — t. Obviously, Fi t (0, 7- ) only depends on the volatility level Yi, t and the slope of the volatility term structure Y4,t . Since the combination of the two is in this section referred to as the volatility level", a long straddle might be classified as a volatility trade whose (initial) primary trading object is the (implied) volatility level. Similar arguments apply to the other strategies. (1994), Chapters 8 and 10, Chaput/Ederington (2002), and McMillan (1996), Chapter 6. 77 Natenberg (1994), Chapter 8, classifies straddles, butterfly spreads and calendar spreads as "volatility spreads". 78 See also footnote 66.

7.3 Volatility Trading

177

We next characterize the four volatility trades on the two dimensions profit or loss patterns and Greeks. Profit or loss patterns are used to describe the position's risk/reward characteristics at expiry (or in the long run), whereas the Greeks are used to describe the position's risk/reward characteristics before expiry (or in the short run). The profit or loss patterns for a straddle, a vertical (call) spread, and a butterfly (call) spread as a function of the terminal stock price Sr are illustrated in Figure 7.6 for long positions (i.e. for positions involving an initial cash outflow)." The profit or loss patterns for the corresponding short positions are easily obtained by reflecting of the graphs for the long positions at the zero line. As is well known and illustrated in Figure 7.6, losses on bought straddles are bounded (to the paid price) while potential profits are unbounded. If a straddle is sold, the opposite is true, i.e. profits are bounded and losses are unbounded. In the case of butterfly spreads and vertical spreads, both potential profits and losses are bounded.

Figure 7.6. Profit or loss patterns (at expiration) of selected volatility trading strategies. Upper left graph: long straddle; upper right graph: long butterfly (call) spread; lower graph: long vertical (call) spread

79 Since a calendar spread involves options of two different maturities, the profit or loss pattern is not straightforward to construct. One leg is either still active or has already expired. For an illustration of the profit or loss pattern of a calendar spread at the expiration of the shorter-term option, see Natenberg (1994), pp. 149-150.

7 Model Applications

178

The Greeks of a volatility trading strategy are simple linear combinations of the Greeks for each of its legs.8 ° In all four volatility trades, the deltas of some legs have opposing signs. This implies that their (initial) position delta is small. For instance, in a straddle, the negative delta of the put is a natural offset for the positive delta of the call. On the other hand, gamma, vega and DVegaDVol are the same sign for both legs of a straddle, implying that these Greeks are sizable for the resulting combination. 81 Such a trade where the vegas and gammas are monotonically on the same side of the market is sometimes called a first-order volatility trade.82 On the other hand, a trade that is characterized by a gamma and a vega that always flips from positive to negative when the underlying moves is called a second-order volatility trade. Of the four volatility trades, butterfly spreads, vertical spreads, and calendar spreads have this property as there is some offset in terms of gamma, vega and DVegaDVol. In general, DVegaDVol is initially highest for vertical spreads and butterfly spreads, while it is almost zero for ATM straddles and ATM calendar spreads." First-order volatility trades have the advantages of being tractable, with a profit or loss that is easy to forecast. In relation to second-order volatility trades, they present positions with a smaller degree of complexity. When time passes and the market environment changes, the characteristics of a volatility trade may change, too. For example, a straddle that was initially ATM may loose this property when the stock price moves. To react on the changing market conditions, we may follow one of three basic strategies: 84 • • •

Keep-strike strategy: do nothing. Adjust-strike strategy: keep the old position and additionally buy new options such that the combined position has the desired exposure. Roll-over strategy: close the old position and build up a new one with the desired exposure.

In general, none of these strategies is dominant; it depends on the specific situation which strategy is most appropriate. 7.3.3 Profitability of DAX Volatility Trading Strategies Goal of the Study and Literature Review The goal of our study is to evaluate the profitability of applying the SW model to three of the volatility strategies described in the last section: a straddle, a also Section 3.1.4. that a straddle position involves exposure to actual/realized and implied volatility. 82 See Taleb (1997), pp. 263-264. 83 DVegaDVol is roughly zero for ATM options. 84 See Roth (1999), pp. 131-154. 80 5ee

81 Note

7.3 Volatility Trading

179

vertical call spread and a butterfly call spread. By checking whether the three volatility trading strategies are able to generate abnormal profits, i.e. profits that are not accounted for by the risk of the positions, this study presents a further test of the out-of-sample performance of the SW model. On the other hand, this study might also be seen as a test of the informational efficiency of the DAX index options market. If abnormal profits (after transaction costs) can be made, the efficient market hypothesis is rejected. A few existing studies have attempted to establish the potential profitability of volatility trading strategies. Harvey/Whaley (1992) analyze S&P 100 index options market efficiency using an implied volatility measure. They find that one-day ahead volatility forecasts are statistically quite accurate, but fail to deliver significantly positive profits after transaction costs. Engle et al. (1993) compute average option prices from 100 simulated sample paths of a GARCH model and assess profits from options trading for competing volatility forecasting methods. They conclude that profits earned by the GARCH model dominate those earned by any of three other alternatives. Noh et al. (1994) examine the mean daily profits of a straddle strategy derived from a GARCH(1,1) model against forecasts from a regression model applied to daily changes in weighted implied volatilities. Their trading strategies employ closest-ATM, short-term straddles. They report that the trading rule using the GARCH forecast method returns a greater profit than the rule based on an implied volatility regression model. In particular, the GARCH model earns significantly positive profits, even after accounting for transactions costs. In contrast, the implied volatility regression model produces negative returns after transactions costs. In a recent study, Goncalves/Guidolin (2003) propose a two-stage approach to modelling and forecasting the S&P 500 index options implied volatility surface which is conceptually similar to ours. Using daily closing CBOE prices of S&P 500 index options (calls and puts) covering the period from 1992 to 1996, they find that profitable delta-hedge positions can be set up to exploit the implied volatility dynamics predicted by the model. When plausible transaction costs are introduced, however, evidence turns somewhat mixed. Goncalves/Guidolin (2003) also examine the mean daily profits of a straddle strategy that is similar to that analyzed by Noh et al. (1994). Significantly positive before-transaction-costs profits are only generated by one out of their eight alternative models. When moderate transaction costs of 0.125$ per straddle are considered, the null hypothesis of a zero net return cannot be rejected. For a roundtrip cost of 0.50$ per straddle, the result becomes negative and significant. All of the above studies use US option data. For Germany, Schmitt/Kaehler (1997), using time-stamped daily transaction prices of DAX call and put options for the period 1993-1994, examine the profitability of applying different volatility forecasting models to the trading of one-day straddles with a timeto-maturity between 15 and 45 days. They find that an EGARCH model performs best in generating profits for market makers. Thereby they assume that market makers can always trade at the mid price, i.e. they do not incur

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7 Model Applications

any bid-ask spread. Forecasts based on historical volatility also produce statistically significant positive profits. The third model they consider is an AR(1) model for the volatility index VDAX. It generates insignificant or even negative returns. For non-market makers they report of larger transaction costs that imply that no significant profits can be gained with any of the three volatility forecasting models. Our study is different from the studies described above in that we do not restrict ourselves to the case of straddles but do also consider vertical spreads and butterfly spreads. With respect to the study of Schmitt/Kaehler (1997), our examination departs along several dimensions: First, we account for realistic bid-ask spreads that a large institutional investor (e.g., a hedge fund) usually faces; second, we consider a holding period of one week rather than one day; and third the option data we use is more recent and covers a longer period. Data Using DAX options and futures data 86 , we estimate for each trading day in the sample period 1997-2002 a regression of implied volatility on moneyness and time to maturity following the two-step procedure described in Section 5.3.1. 66 provides us with a daily estimate ari (M T) of the DAX implied volatilityThis surface. Assuming an equal probability for a particular options transaction to take place at the bid price or at the ask price, this estimate is expected to accurately reflect the mid implied volatilities of DAX options. 87 For each day n and for each option with strike price K and maturity date T we then compute the option's mid price at 4 p.m. (denoted by Cn (K, T) or Pn (K,T), respectively), by plugging in the futures-implied DAX index level at 4 p.m. (for convencience hereafter denoted by Sn ), the estimated mid implied volatility an (K, T), and the appropriate interest rate r(T) into the BlackScholes formula. Using mid option prices (constructed in the aforementioned way) instead of transaction prices brings two advantages: First, "off-market" prices, e.g., due to small trades with, presumably, unfavorable bid-ask spreads, are almost excluded. Second, given an accurately estimated mid price and given the bid-ask spread, it is easy to derive the actual bid and ask price of an option's quote. On each day n, the DAX futures contract most actively traded is used for delta-hedging. The maturity of this contract, which is normally the nearest available, is denoted by TF. Its (average) transaction price at 4 p.m. on day n is denoted by Fn (TF). For convenience, the contract multiplier of both DAX options and DAX futures is supposed to be 1. Moreover, we assume "See Section 5.2. 86 We exclude the 1995 and 1996 years' options data from our original sample to always guarantee a highly precise estimate of the DAX volatility surface. 87 Hereby we have assumed that the volatility surface is stable during the day. See Section 5.3.1 for a thorough discussion of this issue.

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181

that both index derivatives can be bought and sold in arbitrary units, and that funds may be freely invested at the risk-free interest rate. According to Keim/Madhavan (1998), among others, total transaction costs can be generally decomposed into explicit costs and implicit costs. Explicit costs (direct costs, processing costs) are the direct costs of trading, such as broker commissions, custodial fees, taxes etc. Implicit costs (indirect costs) comprise market impact costs (i.e. bid-ask costs and incremental market impact costs) and opportunity costs (i.e. the costs resulting from missed trading opportunities). In the case of DAX options and DAX futures, explicit costs are very sma1188 and are therefore neglected." Since we assume immediate order execution, opportunity costs are zero, too. The implicit costs of a trade are estimated to be 0.40 volatility points (round trip) for a (single) DAX option, and 4 basis points (round trip) for a DAX futures contract. These values primarily reflect the bid ask costs a typical institutional investor faces. If a combination or a spread is traded, the strategy's bid-ask spread is usually observed to be distinctly lower than the sum of the individual bid-ask spreads." The reason is that some options can be bought for a lower price than the prevailing ask price or sold for a higher price than the prevailing bid price. We assume implicit costs of 0.25 volatility points per leg in the case of vertical spreads and straddles and 0.125 volatility points in the case of butterfly spreads." Market makers, on the other hand, may be able to negotiate mid prices through "prearranged" trades, implying that their implicit costs are also zero." Note that the assumption of a constant bid-ask spread in terms of implied volatility means that the option's bid-ask spread expressed as a percentage of the call (put) price increases (decreases) when the moneyness increases (decreases). -

Trading Strategies The volatility trading strategies we consider are a straddle, a vertical (call) spread, and a butterfly (call) spread with a remaining time to maturity of at least 20 calendar days (at the time of closing the position). This always guarantees sufficient liquidity. The strategies' strike prices are chosen to be closest to: •

ATM (i.e.

M = 0), for the straddle,

99 As of 2003, explicit costs are in fact zero for most market makers. For large investors the contract fee is below 1 Euro. See Eurex (2003). "This is consistent with other studies that investigate the profitability of volatility trading strategies as, e.g., Goncalves/Guidolin (2003) and Noh et al. (1994). "See also Chaput/Ederington (2002), p. 7. 91 1 am grateful to Luc 01linger, DAX options trader at HVB Equity Linked Products Group in Munich, for his advice. "See Schmitt/Kaehler (1997).

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7 Model Applications

5% ITM and ATM (i.e. M = —0.05 and M = 0), for the vertical spread, and 5% ITM, ATM, and 5% OTM (i.e. M = —0.05, M = 0, and M = 0.05), for the butterfly spread.

• •

All three volatility trading strategies are based on out-of-sample forecasts of the volatility surface. More precisely, on trading day n, we apply the SW model to get an estimate of the DAX volatility surface one week or 5 trading days ahead. If the price of the volatility trading strategy is predicted to increase (decrease) due to the predicted change in the implied volatility surface, the investor goes long (short) the strategy. For example, if the straddle price forecast is greater than the market straddle price, the straddle is bought (long position). Conversely, if the straddle price forecast is less than the market straddle price, the straddle is sold (short position). Each week, we separately invest 1,000 Euro in each of the three strategies, which are then held for one week ("keep-strike strategy") with nonoverlapping time windows). When we go short in a strategy, the strategy generates an initial cash inflow. This cash flow plus the additional 1,000 Euro available are then invested in the money market account at the interpolated one-week riskless interest rate. Each volatility trading strategy is delta-hedged (however, only the first-order delta effect measured by the Black-Scholes delta 5 is considered) on a daily basis using DAX futures. 93 To determine the daily delta exposure of an option position, we compute once a day at 4 p.m. the Black-Scholes delta of the position on the basis of the observed DAX volatility surface for that day. To hedge at time t = tn a long (short) position in one unit of the underlying DAX index, we sell (buy) e—r,,(TF)(TF—t) DAX futures contracts. 94 Delta-hedging ought to guarantee that predicted volatility changes would translate in option price changes, rather than being offset by movements in the underlying asset. Implied volatility forecasts are obtained as in Section 6.1.3. In particular, on day n we use the last 250 observations of the time series of the DAX volatility risk factors to estimate the model parameters and to obtain the one-week-ahead optimal forecast of the DAX volatility risk factors Pn+5 = ( 91,n+5) 9.2,n+5) P3,n+5, Y4,+5). The forecast kn+5 is then used to predict next week's DAX volatility surface. Since the value of the DAX index and the interest rate at time n +5 are unknown as of time n, we assume that today's values are tomorrow's best forecasts. 93 The model is reestimated every week in a rolling-window fashion. The size of the rolling window is kept fix at 250 observations or trading days. The one-week profit or loss (before transaction costs) P&L, of volatility trading strategy s E {straddle,vertical spread, butterfly spread} is determined Margin payments are not considered. "See Hull (2000), p. 318. "Note that the assumption of a DAX return of zero differs from our previous assumption of a DAX return of 8%. Because the holding period is short, this has only a minute impact on the results. 93

7.3 Volatility Trading

183

as follows:

psri,

psri, Os ptzon

pszLInterest pgri, Future

where PM-, 8°Pticm is the one-week profit or loss of strategy s resulting from ,s is the interest earned of strategy s in one changes in option prices, NJInterest week, and P.S&P"ure is the weekly gain of strategy s that is due to the daily delta-hedging with DAX futures. If a strategy is sold, psdia nterest defined by p1zL8/ntere8t

1,000 - r

where r is the one-week riskless interest-rate, is positive; otherwise it is zero. The other two profit or loss components may show a positive or a negative sign. The net profit or loss P&L:, i.e. the profit or loss net of transaction costs TA, occurring for options and futures transactions, is defined by

P&L: = P&L, —TA,. Finally, the weekly rate of return before transactions costs (after transaction costs) on buying or selling strategy s, R, (R:), is computed as

R8

P&L, -

1,000

( 19. P&L: rt.

s

—

1,000 ) •

To protect against strong and erratic price movements towards the end of an options or futures contract, we exclude all weeks where either the option's time to maturity falls below 20 calendar days and therefore the option has to be rolled over to the next longer maturity or the most-liquid futures contract expires. In addition, we correct for outliers. Empirical Results The trading experiment begins at January 2, 1998, and ends at December 30, 2002. Table 7.9 shows the weekly trading results for the three volatility trading strategies during the full sample period before and after transaction costs. The ratio of long to short signals is 0.93 for straddles, 1.01 for vertical spreads, and 1.09 for butterfly spreads. The average weekly rates of return before transaction costs are 0.21% for the straddle, 0.35% for the vertical spread and 1.72% for the butterfly spread. These profits are far from certain, since the corresponding standard deviations (SD) are 12.80%, 3.79%, and 15.56%, respectively. And, in fact, t-ratios of 0.21, 1.19, and 1.41 indicate that profits are not significantly greater than zero at significance levels of 1% and 5%. 96 The low standard deviation of vertical spread returns compared to 96 Since weekly rates of return are assumed to be independent, the t-ratio is computed as a ratio of mean to standard deviation divided by the square root of the number of trades.

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7 Model Applications

Table 7.9. Weekly rate of return from trading 1,000 Euro worth of straddles, vertical spreads, and butterfly spreads Mean return (%) SD return (%) t-ratio Sharpe ratio before transaction costs ("market maker") Straddle Vertical spread Butterfly spread

0.21 0.35 1.72

12.80 3.79 15.56

0.21 1.19 1.41

0.01 0.07 0.10

after transaction costs ("institutional investor") Straddle Vertical spread Butterfly spread

-0.68 -1.36 -2.08

12.80 3.82 15.29

-0.68 -4.65 -2.35

-0.06 -0.37 -0.19

the standard deviation of straddle returns and butterfly spread returns can be explained by the fact that delta-hedging is more effective for vertical spreads than it is for straddles and butterfly spreads. This is because the gamma of a vertical spread tends to be lower (in absolute terms) than the gamma of a straddle or a butterfly spread, at least for near-the-money options considered here. Interestingly, the profitability of a strategy (before transaction costs) is positively correlated with its complexity. The average return is lowest for straddles, and highest for butterfly spreads, with the vertical spread return in between. This may be explained as follows: the more complex a strategy, the less market participants actually perform this strategy, and therefore the higher the potential profitability. A further explanation is related to mean reversion. According to Table 6.2 the mean-reverting effect is most pronounced for the curvature of the smile (Y3) and least pronounced for the volatility level (Y1 ), with the slope of the smile variable Y2 in between. Strikingly, this ranking matches exactly with the profitability (before transaction costs) of the three volatility trading strategies. Butterfly spreads, focusing on changes in the curvature of the smile, achieve the highest return; they are followed by vertical spreads and straddles focusing on changes in the smile slope and the volatility level, respectively. After transaction costs the mean rate of return turns negative (partly significant) for all three strategies and the ranking of the strategies changes. The options' average round trip transaction costs, expressed as a percentage of the average price of the strategy, are 0.90% for the straddle, 1.76% for the vertical spread and 4.25% for the butterfly spread, and therefore rather conservative. The future's transaction costs are almost negligible. In order to assess the economic significance of the trading results, we compute Sharpe ratios. The Sharpe ratio is defined as the ratio of the excess return of a strategy (defined as the strategy's average one-week return minus the average oneweek interest rate) to its corresponding standard deviation. The Sharpe ratios before transaction costs are positive for all three strategies, however, they are

7.3 Volatility Trading

185

rather low in absolute terms. In the presence of transaction costs, the Sharpe ratios of all three strategies become negative. This suggests that the risks of these strategies are not (sufficiently) rewarded by the returns they generate. An analysis of the time series of weekly returns reveals that there is little serial correlation and virtually no correlation with DAX returns. All in all we may conclude, that although profits before transaction costs are positive, they are neither statistically nor economically significant. Moreover, profits turn negative when we account for realistic transaction costs. Thus, the hypothesis of an inforrnationally efficient market is not rejected. 97 Since the model's out-of-sample forecasting ability was shown to be rather good in Section 6.1.3, this observation might be explained by the fact that the mean-reverting property of volatility is a well-known feature that many market participants are actually betting on.

97 Similar results for volatility trading strategies that are based on implied volatility forecasting models are reported by Goncalves/Guidolin (2003) and Noh et al. (1994) for the US options market, and Schmitt et al. (1997) for the German options market.

8

Summary and Conclusion

If the Black-Scholes model could be regarded as a good description of reality, then all options on the same underlying asset should provide the same implied volatility. Yet, on many markets, a strike price structure (known as the volatility "smile") and a term structure of implied volatilities is observed. Their combination is commonly referred to as the volatility surface. The existence of a volatility surface implies that the implied volatility of an option is not necessarily equal to the expected volatility of the underlying asset's rate of return. It rather also reflects determinants of the option's value that are neglected in the Black-Scholes formula. In general, smile patterns may be either due to market frictions (e.g., transaction costs or taxes) or to deviations of the underlying asset's price process from a geometric Brownian motion (e.g., when actual volatility varies over time, either deterministically or stochastically, or jumps do occur). In attempting to explain the volatility surface, neither (one-factor) stochastic volatility models nor simple jump-diffusion models can reproduce correctly the empirically observed implied volatility patterns. Further problems with these models arise from their incompleteness and the fact that infinitesimal quantities such as the local or stochastic volatility or the jump intensity, are not directly observable. These problems along with the increasing liquidity in the market for standard options, especially in the area of equity index options considered here, have recently brought up a new modelling approach. In directly taking as primitive the implied volatility (surface), market-based models of implied volatility have the advantage to be complete and automatically fitted to market option prices. In difference to fundamental quantities such as an (unobservable) instantaneous volatility or a jump intensity, implied volatilities are easy to interpret and continuously monitored by market participants. Due to the noticeable standard deviation found in time series of implied volatilities, Cont/Fonseca (2002) argue that deterministic implied volatility models are in general not able to fully capture the observable implied volatility dynamics. Chapter 4 develops a general mathematical model of a financial market in continuous time where in addition to the usual underlying securities stock

188

8 Summary and Conclusion

and money market account, a continuum of standard European call options is traded. Because the volatility surface can be more easily parameterized and estimated as a function in the option's moneyness and time to maturity, we rather consider the continuum of relative call prices which is represented by the relative volatility surface. Due to the high correlation of implied volatility across degree of moneyness and time to maturity, the volatility surface can be reasonably well described by a small number of risk factors. Given an initial volatility surface and a mechanism which describes how it fluctuates, we derive a condition that has to be imposed on the drift coefficients of the options' implied volatilities to ensure absence of arbitrage in the model. As this so-called drift or no-arbitrage condition has to be fulfilled for options of all strike prices and maturity dates at all times, it presents a serious constraint on the implied volatility processes. To guarantee that there exists a unique equivalent martingale measure simultaneously making all relative call option prices martingales, we show that the market prices of risk must be independent of the vector of call options chosen. Under the assumption that a unique equivalent martingale measure exists, the arbitrage price of a general stock price dependent claim can be obtained from the risk-neutral valuation formula where the expectation is taken with respect to the joint diffusion of the stock price and its volatility under the risk-neutral measure. Because the atthe-money implied volatility converges to the stock price volatility when the time to maturity goes to zero, the stock's instantaneous volatility (process) is implicitly defined. In particular, this means that if implied volatilities are stochastic the stock price volatility is also required to be stochastic in order for the model to be consistent. In the empirical part of this work (Chapters 5 and 6), we propose a twostage approach to modelling the DAX volatility surface. In the first stage we model the daily DAX volatility surface along the dimensions degree of moneyness and time to maturity ("structura"). To compute implied volatilities it is crucial to use synchronized prices of the option and the underlying asset. We achieve this by properly matching transaction data for the DAX option and future. The current index level is distorted by tax effects of dividend payments. We solve this problem by deriving a market implied correction of the underlying prices which has not yet been presented in the literature. Using all call and put prices of each trading day in the sample period from 1995 to 2002, we estimate the DAX volatility surface via regression analysis. More precisely, we formulate a four-factor regression model and apply the method of weighted least squares. The factors may be interpreted as (the natural logarithm of) the overall level of implied volatility (factor one), the overall slope of the volatility smile (factor two), the curvature of the volatility smile (factor three), and the slope of the at-the-money term structure of volatility (factor four). The results show a very accurate fit to the data. On average, the variation of moneyness, which is defined as the natural logarithm of the ratio of strike to futures price, and the variation of time to maturity explains about 92% of the cross-sectional variation of implied volatilities. In this context it was shown that moneyness

8 Summary and Conclusion

189

measures that depend on time to maturity, such as the commonly used Natenberg moneyness, are in general not consistent with no-arbitrage when implied volatilities are stochastic. The vast majority of all smile patterns appear as a skew. Typically, implied volatilities decrease monotonically with increasing moneyness beyond at-the-money until, at the right border, the function rises slightly. The time series of at-the-money implied volatilities obtained from our regression model is almost perfectly correlated with the German DAX volatility index, VDAX. Using the approach of Brunner/Hafner (2003) to obtain the risk-neutral density from the estimated volatility smile, we find the distribution of DAX log returns as implied by DAX option prices to be negatively skewed and leptokurtic. In the second stage we investigate and model the dynamics of the surface by applying time series techniques. We find the time series of the four risk factors representing the DAX implied volatility surface to display high (positive) autocorrelation and mean-reverting behavior. Their marginal distributions may be reasonably well described by normal distributions. The univariate sample autocorrelation functions (ACFs) and partial autocorrelation functions (PACFs) support stationary autoregressive processes of order 1 — AR(1) processes — for all factors but one. However, this factor, representing the curvature of the volatility smile, may still be reasonably well approximated by an AR(1) process. In analyzing the relationship between volatility risk factors and DAX returns we find that shifts in the global level of DAX implied volatilities as represented by the first risk factor are negatively correlated with index returns ("leverage effect"). Furthermore, we find that when the DAX index falls, the slope of the DAX volatility smile (factor two) increases, the curvature of the DAX volatility smile (factor three) decreases, and the slope of the (atthe-money) DAX volatility term structure (factor four) decreases. Based on these findings, we propose a four-factor model for the stochastic evolution of the DAX implied volatility surface in continuous time. The DAX volatility risk factors are modelled by mean-reverting Ornstein-Uhlenbeck (OU) processes, 1 is given by a geometric Brownian motion with constantandtheDAXix drift and stochastic volatility. The processes are correlated and can easily be estimated using quasi maximum likelihood estimation (QMLE). In an examination of the model's out-of-sample performance, we find the forecasts of the volatility risk factors to be unbiased and efficient. Moreover, the model produces significantly more correct direction forecasts than wrong direction forecasts. But although the processes fit the observable data quite well, they give rise to arbitrage opportunities in the model as the market price of risk process and hence the stock's instantaneous volatility process do in general not exist. To be still able to use risk-neutral pricing, we have shown how to find a new market price of risk process and thus a new risk-neutral measure such that the stock's instantaneous volatility process is well-behaved. This 'An OU process is the continuous-time analogue of an AR(1) process in discrete time.

190

8 Summary and Conclusion

measure at least avoids arbitrage opportunities "locally", because it ensures that a given set of standard options can be priced back almost correctly within a prespecified range of degrees of moneyness and times to maturity. However, in doing so, the model loses its completeness and has to be calibrated to market option prices, usually by minimizing some error function. In general, the arbitrage price of a contingent claim in the factor-based stochastic volatility model cannot be obtained analytically. Yet, it can be accurately approximated with numerical methods such as tree methods, finite difference methods, or Monte Carlo simulation. Due to the model's complexity, we use Monte Carlo simulation. In Chapter 7, we present applications of the model in three different fields. In the first application, we price several exotic derivatives, including a cashor-nothing option, a discrete down-and-out barrier option, a variance swap, and a volatility swap. The differences between the Black-Scholes prices of these instruments and the prices obtained from the factor-based stochastic implied volatility model are, in part, quite large. Substantial deviations are also detected for the derivatives' deltas. In the second application, we apply the model to compute the value at risk (VaR) and the expected shortfall for an options portfolio. When comparing the VaR estimates produced by the factor-based stochastic implied volatility model with the VaR estimates generated by a method that is commonly used in practice (the delta-normal method), we find large deviations. These can be traced back to a couple of key factors that are missing from the delta-normal analysis but are accounted for by the factor-based stochastic implied volatility model: the nonnormality in stock returns, the nonlinearity of option positions, the passage of time, and the (stochastic) change in implied volatilities. In the last application, we discuss and empirically test selected volatility trading strategies on their ability to generate abnormal trading profits. In particular, we consider a straddle, a vertical call spread, and a butterfly spread. The factor-based stochastic implied volatility model is used to obtain forecasts of the future DAX volatility surface. Using DAX options data from 1997 to 2002, we find that profits before transaction costs are positive. However, they are neither statistically nor economically significant. Moreover, profits turn negative when we account for realistic transaction costs. Thus, the hypothesis of an informationally efficient market is not rejected. Since the model's out-of-sample forecasts were shown to be quite accurate, this observation might be explained by the fact that the mean-reverting property of volatility is a well-known feature that many market participants are actually betting on. All in all, the proposed factor-based stochastic implied volatility model turns out to be well-suited to capture the real-world dynamics of the DAX volatility surface and the DAX index. However, the suggested parametrization gives rise to arbitrage opportunities in the model. In general, it may be very hard to find a specification of a stochastic implied volatility model that leads to an arbitrage-free and complete market and is at the same time consistent with

8 Summary and Conclusion

191

the empirical observations. 2 Although we have not been able to find an implied volatility parametrization that is consistent with no-arbitrage, we have shown how to find a new market price of risk process and thus a new risk-neutral measure such that the stock's instantaneous volatility process is well-behaved. In contrast to classical stochastic volatility models, the instantaneous volatility in our model is observable and the same set of parameters can be used under the real-world measure and the risk-neutral measure. As shown, this allows for the integrated pricing and hedging, risk management, and trading of index derivatives as well as derivatives on the index volatility.

2 Filipovic (2000) comes to a similar conclusion in the context of HJM-type interest rate models. One idea towards such a specification could be the inclusion of a jump component in the stock price process and the volatility risk factor processes.

A Some Mathematical Preliminaries

The mathematical language of modeling financial markets in continuous time involves that of probability and continuous-time stochastic processes. In this section, we recall some basic notions from the two fields. 1

A.1 Probability Theory Definition A.1 (Sigma algebra). Given a non-empty set S-2, a sigma-algebra (in short: a--algebra) .F on S2 is a family of subsets of f2 with the following properties: -

1. 0 E .F; 2. A E F = A c E F , where Ac = EF= U 1 A i E .F. 3. Ai, A2, .

Q\A is the complement of A in f2;

As a special case, the family of Borel sets F = B (R) is a a--algebra on R. The pair (II, .F) is called a measurable space. Definition A.2 (Probability Measure). A probability measure P on a measurable space (I2, .F) is a function P: F —+ [0, 1 ] such that

1. P (0) = 1, P(1l) = 1; 2. if Ai, A2, ...are pairwise disjoint sets (i.e.

A nA = 0 if i j) then

P (0 Ai ) 'An excellent introductory measure-theoretic text on probability theory is Williams (1995). An introduction to the theory of stochastic processes can be found in Brzezniak/Zastawniak (1998). Besides, a number of textbooks on mathematical finance contain the material presented here. For example, Bingham/Kiesel (1998), Korn/Korn (1999), Lamberton/Lapeyre (1996), and Musiela/Rutkowski (1997).

194

A Some Mathematical Preliminaries

Two probability measures P and Q defined on the same measurable space P (A) = 0 Q (A) = O. The triple (St, .F) are equivalent, if for any A E (St, F, P) is called a probability space.2 The sets belonging to F are called events. An event is said to occur P-almost surely (in short: P-a.s.) whenever

P(A) = 1. If F is a a-algebra on f2, then a function X : T-measurable, in abbreviated form X E T, if

X -1 (B) = {w

Rn is said to be

c S2 : X (w) E BI e F ,

for every Borel set B E B (Ra). If (5.2, F P) is a probability space, then such a function X is called a (n-dimensional) random variable. If X is a random variable, then the a-algebra .Fx generated by X is the smallest a-algebra on St containing all the sets X -1 (B), B E B (R 3 ) . A random variable X is said to be integrable if fo (XI dP < oo. In that case ,

E [X] =

XdP

exists and is called the expectation of X with respect to P. If there is ambiguity on the measure, we write Ep [X]. Let g be a a-field contained in F. Then, the conditional expectation of X given g is defined to be a random variable E pogi such that E[xig] is „F.-measurable and for any set A E g

JA E [xig] dP = f X A

A.2 Continuous-time Stochastic Processes Definition A.3 (Stochastic Process). A (n-dimensional) stochastic process X is a parametrized collection of random variables {X t : t E T} , T C R, defined on a probability space (f2, F, P) and assuming values in Rn . We shall say indifferently: the stochastic process X or X. X is called a process in discrete time if T = Z or T =- N and in continuous time if T is an interval in R. A stochastic process can also be regarded as a function Rn of two variables. For each fixed t E T, we have a random X: T Rn . On the other hand, the function X(w) : T Rn for variable Xt : St arbitrary but fixed w E S2 is called a path (or trajectory). Below, we assume that the parameter space T is the halfline [0, oo). Definition A.4 (Filtration). Considering the probability space (S2, F P) , a : filtration IF = {.Ft : t > 0 } is a nondecreasing family of sub a-algebras of ÇF CF for 0 < s < t < co. ,

,

2 1t is called complete, if 1- contains all subsets G of Il with IF-outer measure zero. Any probability space can be made complete. See Taylor (1996).

A.2 Continuous-time Stochastic Processes

195

We assume that the filtration IF also satisfies the "usual conditions":

1. Fo contains all P-null sets of .F, i.e. if A E .F and P (A) = 0, then for any t, A 2. IF is right continuous, i.e. Ft = ns>t.Fs• Condition 1 ensures that if X = Y P-a.s. 3 for random variables X and Y, and Y is Ft-measurable, then X is also Ft -measurable.' In a financial model, the g-algebra Ft represents the information available at time t, and the filtration IF represents the information flow evolving with time. We say that a stochastic process X is adapted to the filtration F, or equivalently, Fradapted, if for any t > 0, Xt E Ft, i.e. Xt is Ft-measurable. Thus Xt is known when Ft is known. We can build a filtration Fx generated by a process X and we write Tic = (X, : s < t) . We call it the natural filtration. Obviously, a process is adapted to its natural filtration. In general the natural filtration does not fulfill condition 1. However, if we extend Fi" by the a-algebra containing all P-null sets of F, we obtain a filtration with the desired property. In this work, we only deal with filtrations satisfying the usual conditions. The class of progressively measurable processes is a slight enlargement of the class of adapted processes. We define: Definition A.5 (Progressively Measurable Process). A (n-dimensional) process X = {Xt : t > 0} on some probability space (Q, F, P) is called progressively measurable with respect to a filtration IF if, for any t > 0, the map (s, (w) from [0, t] x Q —■ Rn is measurable on the product a-algebra

B ([0 , t]) ®F. If all paths of a stochastic process X are right-continuous, then the process X is also progressively measurable. 5 Two classes of stochastic processes are fundamental to continuous-time financial models: martingales and Markov processes. Definition A.6 (Martingale). A stochastic process X = {X t : t > 0} on some probability space (Q, F P) is a martingale relative to a filtration F and relative to a probability measure F if ,

1. X is Ft -adapted, and E [iXtl] < co for all t > 0; 2. E [Xt IF8] = X. for 0 < s < t. The second property is called the martingale property. In a financial context, saying that the price process X = {Xt : t > 0 } of an asset is a martingale implies that, at each time s, the best estimate (in the least square means: P (X (w) = Y (co)) = 1 for all co G O. Lamberton/Lapeyre (1996), p. 30. 3 For a proof, see Korn/Korn (1999), p. 37. 3 This 4 See

196

A Some Mathematical Preliminaries

sense) of Xt is given by X,. The martingale concept is sometimes too restrictive and one needs to generalize it: A process is said to be a local martingale X = {X t : t > 0} if there exists an increasing sequence of stopping times T k -> 00 such that each stopped process X(k) = (X(t A rk) is a martingale. —4 10, cc] is called a stopping time with respect to a A random variable T : every t > 0 the event {7- < t} belongs to the cy-algebra filtration IF, if for martingale is a martingale. Trivially, each local Definition A.7 (Markov Process). A stochastic process X = {X t : t> 0} on some probability space (f ,P) is a Markov process, if for each t, each set A E a (X, : s > t) (the "future") and B E a (X8 : s < t) (the "past"), the Markov property

P (Al X t

,

P ( Xt)

holds. The future conditional distribution of a Markov process, occasionally called "no-memory" process, does only depend on the present, not on the past equivalently, past and future are conditionally independent given the present. If capital markets are informationally efficient 6 , no investor can earn excess returns by developing trading rules based on historical price information. This implies that asset prices have to follow Markov processes. A particularly important example of stochastic process in finance is the (standard) Brownian motion. It belongs to the classes of Markov and martingale processes' and will underlie most continuous-time financial models. Definition A.8. A stochastic process W = {W t : t > 0} is a (one-dimensional) standard Brownian motion on some probability space (S2, .T, P) , if 1. Wo = 0, P 2. W has continuous paths: W(t) is a continuous function oft for all w E St; 3. W has independent increments: Wt+u — Wt is independent of .F tw for all u > 0; 4. w has stationary increments: the law of Wt+, — Wt only depends on u; 5. W has Gaussian increments: Wt-Fu — Wt is normally distributed with mean 0 and variance u, Wt+, — Wt N (0, u) . Standard Brownian motion is also termed Wiener process. 8 . Standard Brownian motion in n dimensions (multi-dimensional standard Brownian mo6 For an extensive discussion on capital market efficieny, see Copeland/Weston (1988), p. 330 ff. 'It also belongs to the large class of Lévy processes, characterized by stationary independent increments. Another important Levy process which is not considered here in detail is the Poisson process. In financial models, it is used to incorporate jumps, e.g, in the price of a stock. For a modern textbook reference on Lévy process see Bertoin (1996). 8 Although the two processes are defined differently, they are equivalent according to a theorem of Levy. See Neftci (1996), p. 149.

A.2 Continuous-time Stochastic Processes

197

tion) is defined by W = (W1 , W2 , , W,)', where W1, , Wn are independent standard Brownian motions in one dimension. To describe the evolution of asset prices in continuous time we introduce the concept of Ito processes.9 Definition A.9 (Ito Process). Let W = (W 1 ,W2, . . , Wm )' be standard mdimensional Brownian motion defined on a filtered probability space (fl, .F, P, IF) . Then,

1. X = {Xt : t > 0} is a mal valued Ito process, if for any t > 0, it has the unique representation

X t = X0 + f as ds f 14,dW s ,

t = Xo + f as ds + Ef b3,3dw,, 8 ,

3=1 0

: t > 0} with where X 0 is To -measurable and a = {as : t > O}, b = bt = (bi,t , , bm,t) are progressively measurable processes satisfying I as I ds < oo and f

o

sds < oc P-a.s. Vt > 0, j = 1, . . . , m.

, X n ) is a n-dimensional Ito process, if X 1 ,. , X n are realvalued Ito processes in one dimension.

2. X = (X 1 ,

Sometimes, a one-dimensional Ito process is defined with respect to a onedimensional standard Brownian motion, i.e. m = 1. 10 The expression f; bs dW3 is usually referred to as the stochastic integral. If X is an Ito process, it is often written in the shorter differential form

dX t = at dt + bt dWt ,

(A.1)

although only the integral notation is meaningful from a mathematical point of view. If at = a(t, X t ) and bt = b(t, Xt ) are deterministic functions of t and Xt for all t > 0, equation A.1 is called a stochastic differential equation (SDE). The solution of an SDE is called a diffusion. In stochastic analysis, the equivalent to the chain rule in ordinary calculus is the Itô formula, also referred to as Itô's Lemma.' Theorem A.10 (1-dimensional Itt5 Formula.). Let X be an Ito process given by

dX t = at dt + btdWt 9 See Korn/Korn (1999), pp. 48-49. "See, e.g., Oksendal (1998), p. 44. II See Oksendal (1998), Chapter 4.

198

A Some Mathematical Preliminaries

and f(t,x) E C2 ([O, co) x R) (i.e. f is twice continuously differentiable on 10, co) x R). Then Vt E [0, oo), Yt = f(t,xt), is again an ItO process, and

a

a at

f 1 a2 f f dYi = —(t, X i )dt — (t, X i )dX i + – — (t, X i )d < X, X >t, Ox 2Ox2 where d < X, X > i = (dXi) 2 , the quadratic variation of X, is computed according to the rules

dt dt = dt • dWi = dWi • dt = 0, dWi •dWt = dt. Proof See Korn/Korn (1999), pp. 51-59. When f is a function of several It6 processes, the general Ito formula may be applied. Theorem A.11 (General Itel Formula.). Let X = , X n ) be an n, f,n (t,x)) be a C2 map from dimensional ItO process and f (t,x) = [0, oo) x Rn into Rm. Then the process

Yt = f (t, Xt),

Vt E [0, 00 ),

is again an Ito process, whose component number k,

Ykt,

is given by

a

k ,t = dy fkt (t,x t )dt

+ Ê aOxfki (t, Xi)dXi,t i=1

1 r Nv ti=1 j=1

a2 fk OXiaXj

(t,Xi)d<Xi,Xi>t,

where d < Xi, Xi >i= dXi,idXj, t , the quadratic covariation (or crossvariation) of Xi and Xi, is computed according to the rules

dt, if i • dWi,t • dWi,t { 0, otherwise. Proof See Korn/Korn (1999), p. 59. In this text, we will also need a generalization of the It6 formula: the generalized It6-Venttsel formula." 12 See

Venttsel (1965) and Brace et al. (2002), Appendix A.

A.2 Continuous-time Stochastic Processes

199

Theorem A.12 (Generalized Itto-Venttsel Formula).

Let W = (W1,W2,... ,W, n )' be an m-dimensional standard Brownian motion. Suppose G(t,u) is twice differentiable with respect to the parameter u and satisfies the SDE dG(t,u) = A i (t,u)dt + Bi(t,u)dWt• If ti t satisfies the SDE

dut = A2 (t,ut)dt + B2 (t, ut)dWt, then an SDE for G(t, ut ) is

dG(t,ut ) = Ai(t,u t )dt + Bi(t,ut )dWt

1 , ., _ ,

aG , t ,)aut , + --kt,u01012 a2G +—kt,u kt,u t .)i12 - a,t Ou 2 Ou2 0B1 +--67--1 (t,ut )B2 (t,ut )dt. Proof. See Venttsel (1965).

0

Pricing of a Variance Swap via Static Replication

This appendix contains the proof of equation (7.21) on p. 156 in the main text. 1 Theorem B.1. Suppose the interest rate r is constant and suppose further the stock price process S = {St : t E [0,7 ] 1 is a diffusion with progressively mea: t E [0,7 ] 1 and progressively measurable volatilsurable drift process p = t : t E [0, T]}: ity process v = {v dSt = StA t dt + StvtdWt,

Vt E [0,7] ,

(B.1)

where W = {Wt : t E [0, T]} is a one-dimensional standard Brownian motion. Then, provided that a continuum of standard European options with maturity date T and strike prices ranging from zero to infinity exist on the stock S, a T-maturity variance swap can be statically replicated and its fair price at time t =- 0 is given by (B.2)

KVARS = BQ [WT] 2

= —T erT o(f0

Fo(T) ---P (K,T)dK + K

1 .2 Co(K ,T)dIC) . Fool n

Here, Co (K, T) and Po (K ,T) , respectively, denote the current market price of a put and a call option of strike K and maturity T, Fo (T) = SoerT is the stock's T -maturity forward price, and wg- , is the realized continuously sampled variance over the interval [0, T] : WT =-

1 f — T

v 2 dt. t

(B.3)

Proof. By the application of Itel's lemma to ln(St), we obtain: 'See also Carr/Madan (2002) and Demeterfi et al. (1999b), for a similar proof.

202

B Pricing of a Variance Swap via Static Replication

1

d (ln (St)) = (ktt - -2 v0 dt +vtdi'Vt,

Vt E [0,T].

Subtracting equation (B.4) from equation (B.1), we obtain 1 dSt — - d (ln (S t )) = .14dt,

Vt E [0,7].

St

(B.5)

Integrating equation (B.5) from 0 to T yields:

foT dsStt ln (ST )

so

=

1

0 qdt.

Dividing (B.6) by T, we finally obtain the continuously sampled variance over [0,7 ] : dS t (ST) ] 2 (B.7) WT = f qdt = --

[fT

Tj o

T o

St

0)

.

Identity (B.7) dictates the replicating strategy for variance. The first term in represents the payoff of a trading brackets on the right-hand side, LT d '—5g strategy which involves the continuous rebalancing of a stock position that is always instantaneously long 1/St units of stock worth 1 dollar. The second

t,

term, - ln (k) 0 , represents a static short position in a contract that pays off the continuously compounded stock return over the contract's lifetime [0, T]. This contract is known as the log contract.2 To obtain the fair delivery price of variance Kvjuis , we compute the riskneutral expectation of the right-hand side of (B.7):

115_ [ f T dSt ln ST)] h st so )

(B.8)

KVARS = BQ [WT] = T

(EQ [foT d,'Stt]

EQ [ln (It)]

.

The first expectation is easily evaluated to

EQ

[fT Jo

fT st

Jo

[dSt i st

f

rdt = rT.

o

(B.9)

The second expectation represents the fair value of the log contract. Unfortunately, the log contract is not a market-traded security. To make the replicating strategy viable in practice, we need to duplicate the log contract with traded standard options. For convenience, let us write the log payoff as: 2 The

log contract was first discussed by Neuberger (1994).

B Pricing of a Variance Swap via Static Replication

ln

— ST (

So

) = ln ( T ST ) + ln

.

GI) , o

203

(B.10)

where S* > 0 is an arbitrary stock price. Since the second term on the righthand side, ln (e), is constant and known at time 0, only the first term, ln (k) , has to be replicated. Assuming that a complete collection of standard European options with strikes K > 0 and maturity T E (t, T*] is traded, it

--) s can be rewritten as:3 can be shown that the payoff ln (s ln

(ST ) ST

— S*

f S. max {K — ST;0} dK 0 K2

S*

S* —

(B.11)

00 1 — max {Sr — K;0} dK. s. K 2

The first term on the right-hand side of equation (B.11) can be interpreted as a long position in (1/S* ) forward contracts struck at S. The second term arises from a short position in (1/K2 ) put options struck at K, for all strikes less than S. Similarly, the last term arises from a short position in (1/K2 ) call options struck at K, for all strikes greater than S. All contracts mature at time T. Taking the risk-neutral expectation of (B.10), where In (k) is given by

(B.11), yields ° B[

( ST : T0

)]

=1

3

(S

_erT

°

S*) — erT f

erT —

S.

1 —2 Po(K,T)dK (B.12)

0 K ' 1 — C0(K T)dK + ln fs. K2 '

( S* ) So

.

Substituting (B.12) and (B.9) into (B.8), we finally obtain the fair value of future variance:

KVARS

2

( so erT — s*) — ln ( S*)

(rT —

+err f

-5T1

S. — K2

_FerT

(B.13)

Po(K T)dK '

oo

Is.,Co(K,T)dif) .

If we choose the arbitrary parameter S* , defining the boundary between call and put strikes, to be the current forward price F0(T) = soerT, equation (B.13) can be further simplified to: 3 See Demeterfi et al. (1999b), p. 18. More general, it can be shown that any twice differentiable payoff can be statically replicated in such a way. For a proof, see Carr/Madan (2002).

204

B Pricing of a Variance Swap via Static Replication

2 err (f F°(T) 1 °.° — Po(K T)dK + f —Co(K T)dK KvARS = — T K2 ' 0 Fo(T) K2 (B.14) This proves the claim.

List of Abbreviations

a.s. ACF ADF AR ARIMA ATM BIS BS CBOE CEV DAX DTB e.g. etc. FIA FTSE GARCH GBM HJM i.e. i.i.d. MAPE ML MLE ITM NFLVR ODE OLS OTC OTM

almost surely autocorrelation function augmented Dickey-Fuller (test) autoregressive (process) autoregressive integrated moving average (process) at-the-money Basle Committee on Banking Supervision Black-Scholes Chicaco Board Options Exchange constant elasticity of variance model Deutscher Aktienindex Deutsche Terminboerse for example (lat: exempli gratia) etcetera (lat: et cetera) Futures Industry Association Financial Times Stock Exchange generalized autoregressive conditional heteroscedasticity geometric Brownian motion Heath-Jarrow-Morton model that is (lat: id est) independent and identically distributed mean absolute percentage error maximum likelihood maximum likelihood estimator or estimation in-the-money no free lunch with vanishing risk ordinary differential equation ordinary least sqaures over the counter (market) out-of-the money

206

List of Abbreviations

OU PACF PCD PDE PP

QMLE RND SD

SDE SW S&P

SOFFEX VaR VAR

VDAX WLS

Ornstein-Uhlenbeck (process) partial autorcorrelation function proportion of correct direction partial differential equation Phillips-Perron (test) quasi maximum likelihood estimation risk-neutral density standard deviation stochastic differential equation factor-based stochastic implied volatility model (specified for DAX options) Standard and Poors Swiss Options and Financial Futures Exchange Value at Risk vector autoregressive (process) DAX volatility index weighted least squares

List of Symbols

Roman case: a, ATM

Bt cx (a) C2

CBS()

Ct (K,T)

b-t(M,T)

CONCt CONPt CONCBs•)

d c/10,d2(-)

D Dt DW

speed of mean reversion of risk factor process Y, ATM volatility of DAX options with a time to maturity of 45 calendar days on day n fixed cash payout of a cash-or-nothing option money market account process value of the money market account at time t mean reversion level of risk factor process Y, a-quantile of distribution (or random variable) X set of twice continuously differentiable functions arbitrage price of a standard European call option at time Black-Scholes call option pricing formula market price at time t of a standard European call option with strike price K and maturity date T; also: call price as a function of K and T market price at time t of a European call option with moneyness M and time to maturity r; also: call price as a function of M and T arbitrage price of a cash-or-nothing call option at time t arbitrage price of a cash-or-nothing put option at time t Black-Scholes price function of a cash-or-nothing call option (d+1) is the number of primary traded assets in the financial market auxiliary functions in the Black-Scholes formula price process of numéraire asset price of numéraire asset at time t Durbin-Watson test statistic

208

List of Symbols

DIV DOC, AD/Vt ,T

ES F,(T) F,1 (TF)

Fx (x)

X.) (G* ( 0)) G(0)

(G;(0)) G(ç5) h H HL, kd

km KVARS

KvoLs m(.) m-10

M = Mt ML Mu MAPE(K) n(x)

N* N(x) N (p ,a2 )

R(A)

gross dividend arbitrage price of a discrete down-and-out call option time T terminal value of the difference dividend incurred between dates t and T the Euler number 2.71... expected shortfall price of a futures or forward contract (on a stock) with maturity date T at time t price of a DAX futures contract with maturity date TF on day n at minute 1 distribution function of X volatility surface function regression function or approximate DAX volatility surface function (discounted) gains process of portfolio 0 (discounted) gains of portfolio accumulated up to and including time t forecasting period in days contingent claim or derivative security half-life of process i time-independent random amplitude of a jump corporate income tax rate for distributed gains marginal investor's tax rate strike price or exercise price of an option delivery price for variance; fair value of variance delivery price for volatility; fair value of volatility number of trading minutes per day moneyness function inverse function of m(.) with respect to the strike price moneyness of an option (at time t) lower and upper moneyness boundary mean absolute percentage error with respect to the parameter vector IC probability density function of a standard normal random variable sample size terminal time horizon measured in units of At cumulative distribution function of a standard normal random variable cumulative distribution function of a normal random variable with mean ti and variance cr2 Poisson process with intensity A

List of Symbols 9)

Pt PBS•)

Pt(K,T)

P&L, P&L; PCD 4.57. (s) r(t) R, R; Rt, R(N) p2 R "act ,) "

RE (S*) S (Sr) St (

n,l) Sn,1

209

cumulative distribution function of a multivariate normal random variable with mean vector it and covariance matrix S2 (p+1) is the number of random sources in the financial market model arbitrage price of a standard European put option at time Black-Scholes put option pricing formula market price at time t of a standard European put option with strike price K and maturity date T; also: put price as a function of K and T one-week profit or loss of strategy s one-week profit or loss of strategy s net of transaction costs proportion of correct direction risk-neutral density for ST constant risk-free interest rate time-dependent risk-free interest rate continuously compounded one-day DAX return on day n one-week return of strategy s one-week return of strategy s net of transaction costs continuously compounded stock return in period i N-period sample mean of continuously compounded stock returns at time t (adjusted) coefficient of determination rebate (discounted) stock price process (discounted) stock price at time t (adjusted) futures-implied DAX level on day n at minute

(DAX) index level on day n; also: underlying price at time t a standard error SE time trading date corresponding to trading day n ta maturity date or expiration date maturity date of a futures contract TF maturity date of an option To terminal time horizon T* barrier level initial value Vo (q5) (discounted) value process of the portfolio 0 (V 3 (95)) 17(0) (discounted) value of the portfolio at time t (Vt3 (95)) Vt (0) value at risk VaR VARS t arbitrage value of a variance swap at time t Sn

List of Symbols

210

VOL St WT

W (W*) Wt (Wt* ) ( 1474t)

w Wi

X Yi YO,n Yi tn n+h

(Z*) Z (Zt) Zt (Zn (ZZt ) Zi,t

arbitrage value of a volatility swap at time t continuously sampled variance over the time interval [0,7] standard Brownian motion (one- or multidimensional) under P (Q) standard Brownian motion (one- or multidimensional) at time t under P (Q) standard Brownian motion i at time t under P (Q) vector Brownian motion i-th component of a vector Brownian motion a general random variable or general stochastic process i-th implied volatility risk factor (process) natural logarithm of stock price Sn value of the i-th volatility risk factor (i = 1. 4) on day n optimal h-days-ahead forecast of volatility risk factor i given the information on day n value of the i-th volatility risk factor at time t (discounted) asset price vector process (Z0, .••, Zd)' (discounted) asset prices (Zo,t, •••, Zd,t)' at time t (discounted) price process of asset i (discounted) price of asset i at time t

Greek case:

fin i,n

rBs rt Sas () St 50 1, E

fi,n

7g(K,T) qt (K, T)

drift rate of Yi at time t vector of regression coefficients on day n regression coefficient i on day n in the DAX volatility surface regression volatility of volatility risk factor Yi variance rate of volatility risk factor Y, with respect to Brownian motion Wi at time t covariance matrix Black-Scholes gamma (function) gamma of a standard European option at time t Black-Scholes delta (function) delta of a standard European option at time t delta of portfolio 0 (vector) Gaussian white noise process; random disturbance value of Gaussian white noise process i on day n random disturbance risk-neutral drift rate of crt (K,T) at time t real-world drift rate of crt (K,T) at time t

List of Symbols

Fit(M)r)

0 efts (.)

et (K ,T) tjt(M,r)

X

ABS (•)

At IL

en

(II* (H)) II(H)

211

real-world drift rate of -6-" t (M,T) at time t parameter vector Black-Scholes theta (function) theta of a standard European option at time t real-world variance rate (vector) of at (K,T) at time t real-world variance rate (vector) of at (m,r) at time t vector of parameters that together determine the market price of risk process speed of mean reversion in the Heston model intensity of Poisson process Black-Scholes vega (function) vega of a standard European option at time t constant instantaneous rate of return (process) from the stock instantaneous rate of return from the stock over [t, t +dt] vector of time-dependent model constants in VAR(1) process model constant in AR(1) process of correlation matrix volatility of volatility in the Heston model (discounted) arbitrage price process of contingent claim

H (11;(11 )) Ilt(H) fi

ei,e2 at(K .) at( . , T) at(K,T)

at tf,r) (

an(m, r) (Ti T2)

En TL, TU

(discounted) arbitrage price of contingent claim H at time t Monte carlo estimate of II correlation coefficient instantaneous correlation between Wi and vvi surface parameters volatility smile volatility term structure time t implied volatility of a standard European option with strike price K and maturity date T; also: (absolute) implied volatility surface at time t time t implied volatility of a standard European option with moneyness M and time to maturity T; also: (relative) implied volatility surface at time t implied volatility of a DAX option with moneyness M and time to maturity T on day n; also: (relative) implied DAX volatility surface on day n forward implied volatility (curve) conditional covariance matrix in VAR(1) model time to maturity, defined as T — t lower and upper time to maturity boundary instantaneous stock volatility (process)

212

List of Symbols

vt v(t) v(t,St) v2 00 0(t, T)

ût (N) 14(N)

T

(TT)

TT(Q*) (ki,t

Sht (Pi 4'(')

,b

1

Ijt

WErs

instantaneous stock volatility at time t time-dependent but deterministic instantaneous volatility local volatility long-run variance in the Heston model average volatility over [0, T] realized or historical stock price volatility over the last N periods at time t realized or historical stock price variance over the last N periods at time t class of all tame strategies (over [0, T]) class of (r-admissible strategies number of units of asset i held at time t vector of portfolio holdings {0 04 ,014 , ..., 0,14 ) at time t trading strategy or portfolio process vector of autoregressive coefficients in VAR(1) model autoregressive coefficient in AR(1) process of Yi payoff or contract function market price of risk process market price of risk vector {0 04 ,0, 4 ,...,07, 4 ) i at time t market price of risk of random source i at time t Black-Scholes DDeltaDVol (function) DDeltaDVol of a standard European option at time t state sample space

Other symbols: the Borel a-algebra unconditional expectation of X with respect to the measure Q conditional expectation of X with respect to the measure

g

Radon-Nikod3'rm density I at time t Ft or-algebra or-algebra representing information at time t the a-algebra generated by the process {X, : s < t} filtration constant elasticity parameter in the CEV model likelihood function nominal value of a variance or volatility swap set of natural numbers set of equivalent martingale measures objective or real-world probability measure -

List of Symbols Q Qs, Qi, Q2

R (Rn) R+ (R7) R++ (R7+ ) Vss() Vt

NIQ [X] VQ[XITt] Z 0

PO

PC / X) t --X,

lA

213

equivalent martingale measure or risk-neutral measure particular equivalent martingale measures set of real numbers (n-dimensional) set of nonnegative real numbers (n-dimensional) set of positive real numbers (n-dimensional) Black-Scholes DVegaDVol (function) DVegaDVol of a standard European option at time t unconditional variance of X with respect to the measure Q conditional variance of X with respect to the measure Q set of integers empty set the absolute value of X; also: the determinant of the matrix X the quadratic variation of X the quadratic covariation or cross-variation of X and Y distributed as approximately equal to the transpose of the vector x indicator function returning 1 if the set A is non-empty and 0 otherwise

Further comments: •

•

A function f(x) is said to be increasing (nondecreasing) on an interval I if f(b) > f (a) (f(b) ? 1(a)) for all b > a, where a,b E I. Conversely, a function f(x) is said to be decreasing (nonincreasing) on an interval I if 1(b) < f (a) (1(b) _< 1(a)) for all b> a with a, b E I. A quantity x is said to be positive (nonnegative) if x > 0 (x > 0) and negative (nonpositive) if x < 0 (x < 0).

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-

-

-

-

-

Index

4-sigma rule, 90 American-style, 16 AR, 108, 116 arbitrage free, 13 opportunity, 13 price, 18, 26 pricing theory, 18

ARIMA, 102 augmented Dickey-Fuller test, 107 autoregressive process univariate, see AR vector, see VAR barrier option delta, 153 down-and-out, 151 pricing, 152 types, 151 bid-ask costs, 181 binomial model, see tree methods

Black-Scholes call pricing formula, 28 market, 24 model, 1, 24 partial differential equation, 29, 67,

173 put pricing formula, 29 sensitivities, see Greeks Box-Jenkins approach, 102 Box-Pierce statistic, 125 Brownian motion deviation from, 43

fractional, 50 geometric, 24 multi-dimensional, 117, 197 standard, 196 vector, 117 butterfly spread, 176 calibration, 57, 140 cash-or-nothing option, see digital option CEV model, 46 Cholesky decomposition, 117 conditional expectation, 194 contingent claim, see derivative security contract function, see payoff function convexity adjustment, 158 crash risk, 48 cross-variation, 198 DAX adjusted index level, 79 correction for taxes and dividends, 78 future, 74, 78 index, 73 option, 74-76, 82 volatility index, see VDAX DAX implied volatility model applications, 145 calibration, 140 correlated form, 115 diagnostic checking, see diagnostic checking

discretization, 118 estimation method, 120

226

Index

hedging, 139 market price of risk, 133 out-of-sample test, 128 parameter estimates, 122 parameter stability, 126 pricing, 137 real-world dynamics, 116 review, 144 risk-neutral dynamics, 131 risk-neutral measure, 137 simulation, see Monte Carlo specification, 115 uncorrelated form, 117 volatility of volatility factor, 116 DAX volatility risk factors

ACF, 106 distributional properties, 104 forecasts of, 129 level, 94 model order, 108

PACF, 108 relation to market variables, 109 smile curvature, 95 smile slope, 95

stationarity, 107 term structure slope, 96 DAX volatility surface choice of moneyness, 86 empirical results, 92 outlier detection, 90 overall estimation procedure, 84 regression model, 89 risk factors, see DAX volatility risk factors shape, 94 stochastic model, see DAX implied volatility model two-step estimation, 89 delta-gamma method, 167 delta-normal method description, 160 drawbacks, 166 derivative security, 15 attainable, 18, 20 forward, see forward contract future, see futures contract Option, see option swap, see swap third-generation, 147

diagnostic checking, 124 difference dividend, 78 diffusion, 155, 197 digital option delta, 150 pricing, 148 types, 147 distribution fat-tailed, 44

leptokurtic, 99 normal, see normal distribution down-and-out option, see barrier option drift condition, 66, 133 Euler method, 118, 138 exotic derivative Asian option, 17 barrier option, see barrier option digital option, see digital option forward-start option, 172 log contract, 202 lookback option, 17 overview, 145 path-dependent, 146 path-independent, 146 variance swap, see variance swap volatility swap, see volatility swap expectation, 194 expected shortfall, 168

Feynman-Kac formula, 29 filtration, 194 financial market

Black-Scholes, 24 complete, 21, 55 general implied volatility model, 60 incomplete, 3, 21 finite difference approach, 139 flattening-out effect, 39, 88 forecast efficient, 36, 129 unbiased, 129 forward contract, 16, 26 futures contract, 17 gains process, 11

GARCH, 179 general implied volatility model assumptions, 60

Index drift condition, see drift condition hedging, 72 instantaneous volatility, 70 interpretation of drift terms, 68 market price of risk, 63 pricing, 72 real-world dynamics, 62 risk-neutral dynamics, 63 risk-neutral measure, 68 Girsanov's theorem, 14 Greeks, 30, 65, 139, 173, 175

DDeltaDVol, 31 delta, 31, 139, 150, 153

DVegaDVol, 31 gamma, 31, 139 theta, 31

vega, 31 half-life, 123 Heston model, 3, 47 Hyperbolic model, 50 implicit method, 80 implied tree, 46 implied volatility as primitive, 4, 45 calculation, 34 concept, 2, 32 deterministic, 53 forward, 40 general model, see general implied volatility model interpretation, 35 risk factors, see DAX volatility risk factors skew, 38 smile, 2, 38, 87 stochastic, 55 surface, 55 surface of DAX options, see DAX volatility surface term structure, 39, 55, 87 indicator function, 168 informational efficiency, 179, 185, 196 Ito formula, 197 process, 197 It&Venttsel formula, 64, 198

227

Jensen inequality, 158 jumps, 3, 44, 48 keep-strike strategy, 178, 182 leverage effect, 48, 109 liquidity, 172 Ljung-Box statistic, 125 mark to market, 17 market frictions, 3, 43, 50 market maker, 179, 181 market price of risk, 14, 63, 133 market risk, 158 market-based approach, 4, 45 Markov process, 25, 196 martingale, 195 local, 196 property, 195 representation theorem, 20 martingale measure, 13, 15, 26, 68, 137 maximum likelihood estimation, 120 likelihood function, 120 quasi, 120 mean absolute percentage error, 141 mean reversion level, 116 speed, 116 Merton model, 49 meta-theorem, 21

moneyness concept, 42 fixed-strike, 42 log simple, 86

Natenberg, 85 simple, 86 standardized, 85 valid, 61, 86 Monte Carlo estimate, 139 simulation, 137 simulation for value at risk, 162, 167 simulation path, 142 standard error, 139

Newton-Raphson procedure, 34 NFLVR condition, 15 no-arbitrage condition, see drift condition

228

Index

relations, 91 nonparametric methods, 84 normal distribution

multivariate, 120 univariate, 28 Novikov condition, 14, 63 numéraire, 11 option at-the-money (ATM), 17, 86 contract, 17 exotic, see exotic option exposures, 175 in-the-money (ITM), 17, 86 maturity, 17 out-of-the-money (OTM), 17, 86 profit or loss, 174 standard, 17, 27, 28 strategies, 176 strike price, 17 Ornstein-Uhlenbeck process, 116 parameter vector, 119 parsimonious, 99 payoff function, 16 Philips-Perron test, 107 pin risk, 148 Poisson process, 49 portfolio delta, 160

delta-neutral, 32 strategy, see trading strategy principal component analysis, 100, 101 probability measure, 193 progressively measurable, 195 proportion of correct direction, 130 put-call parity, 28, 78 quadratic variation, 198 random variable, 194 rebate, 152 risk factors abstract, 4, 99 fundamental, 100 of DAX volatility surface, see DAX volatility risk factors original, 99 statistical, 101

risk-neutral density, 51 density of DAX options, 97 measure, see martingale measure valuation formula, 19, 26, 72 rolling window, 128 sigma-algebra, 193 smile effect, see implied volatility smile models comparison, 56 overview, 43 square-root process, 47 stepwise regression, 136 sticky-implied tree rule, 54 sticky-moneyness rule, 54 sticky-strike rule, 54 stochastic differential equation, 197 stochastic process, 194 straddle, 163, 176 swap, 17 tax effects, 51 tax system

Halbeinktinfte-Verfahren, 80 KtSrperschaftssteueranrechnungsVerfahren, 80 trading strategy, 11 admissible, 18, 20 replicating, 18 self-financing, 12 tame, 13 transaction costs, 50, 181 tree methods, 137, 152 trimmed regression, see 4-sigma rule value at risk computation methods, 160 concept, 159 delta-normal method, see deltanormal method full-valuation method, 162 with DAX implied volatility model,

162 value process, 11 VAR, 119, 125 variance fair value of, 155, 203 realized, 155 swap, see variance swap

Index total, 155 variance gamma process, 50 variance swap description, 154 pricing, 155, 201 replicating strategy, 202

VDAX, 93 vertical spread, 176 volatility actual, 33 average, 36 clustering, 107 derivative, 171 derivatives, 146 deterministic, 2, 3, 45 fair value of, 158 historical, 33 implied, see implied volatility instantaneous, 33 local, 2, 45 of stock price, 24

229

realized, 33 smile, see implied volatility stochastic, 47 surface, see implied volatility swap, see volatility swap term structure, see implied volatility time-varying, 44 trading, see volatility trade volatility swap description, 157 pricing, 158 volatility trade definition, 170 empirical analysis, 180 first-order, 178 second-order, 178 trading instruments, 171 trading objects, 171 weighted least squares, 89

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