ford University Arexa. Walton Street. Oxford oxa Oxford New' York
FINANCE THEORY AND ASSET PRICING
6DP
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Frank Milne
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f&3# 0-19-877397-8 f&2# 0-19,877398-6 4#:kJ 13 5 7 9 l 8 6 4 2
CLARENDON
1995
PRESS
'
OXFORD
Typeset by Jkre Tech Corporation. 'oatcerry. 'rlfed a Great Britain on acid-free NNr by &ocrc/f (Bath) Ltd., ffoaler Norton,
India
zvon
-*1.
Contents
Introduction
1
Introduction
A Brief History of Finan
Theory
2 Two-Date Models: Complete Markets Incomplete Markets with Production 4 Arbitrage and Assetricing:
12 27
Indudreferen
Approach
37
Martingale Pricing Methods
50 60
6 Representative Consumers Diverscation
and Assetrking
8 Multiperiod Asset-pricing: Complete Markets 9 General Assetricing
in Complete Markets
10 Multiperiod Assetricing:
Incomplete Asset-Marke?
70 78 89 100
Conclusion
114
Bibliography
117
Index
123
nis book is based on a set of lectures that I gave at the Institute of Advanced Studies in Vienna in 1992, and subsequently taught 1I1 the Economics Department, Queen'sUniversity. My brief was to provide a series of ten lectures that surveyed and introdud recent asset-pricing models in Finance, using mathematical techniques and microeconomic theory at the level of Varian's Microeco,4 Course in Microeconomic nomic Analysis, or Kreps's Theory. With these prerequisites the book should be acgraduate student who has a good cessible to any srst-year grounding in microeconomics. Nessarily this book is not complte, and omits much that is important in terms of generality and detail. To make the book complete in that and increase the level of sense would triple its length mathematical diculty substantially. The rst chapter provides a brief histoq of modern fman theory, emphasizing the main contnbutions and sketching the role of application in the development of the theory. Chapter 2 introduces the two-date model with complete markets and uncertainty. The chapter recalls standard microeconomic arguments and introdus some of the geometric arguments that are developed more fully later in the book. Chapter 3 generalizes the model by allowingfor incomplete markets and non-trivial asset markets. In Chapter 4 the incomplete market onomy equilibrium is analysed using the idea of induced preferences and production. sets over assets. This idea allows us to construct geometric proofs of rbitrage results and relate them The to familiar microeconomic theoretical arguments. Modigliani-Miller
arguments on capital structme and the
2
Introduction
binomial option-pricing model are introduced as illustrations of the general argu/ent. Chapter 5 covers the same vound but uses the techmque of personalized martingale pricing as an alternative method for analysing the same problems. Chapter 6 considers asset-pricing models with consumer aggregation when arbitrage arguments are not ossible. The arguments are geometric and avoid functlonal forms except as illustrations from the literature. Chapter 7 discusses diverscation arguments using inand establishes the equilibrium arbitduced greferences, rage-pncing theory when there is a nite number of assets. ne capital asset-pricing model is dedud as a special case of the general theorem. Chapter 8 extends the two-date model to a multi-date complete market structure, and results. Chapter 9 explores introduces some yreliminary ln arbitrage pricing the complete market model and illustrates the ideas with the multiperiod binomial optionpricing model. ne last chapter allows for incomplete markets and shows how previous results can be extended into a multi-geriod incomplete-markets framework. The book ends wlth a brief conclusion diKussing extensions ud models in of the developments asset-pricing rent models. I wish to thank my students and colleagues at Queen's for many comments on earlier drafts of this book. Also I would like to thank the Institute of Advanced Studies for suggesting tMs project; and the Canadian SSHRCC for funding. I would like to thank Linda Freeman for her help in prepahng and typing this manuscript in Wordperfect, and those at OUP involved in its publication.
1 A Brief History of Finance Theory
The history of nance theory is an interesting exnmple of the interaction between abstract theorizing and practical application. Many of the original contributions in fmance theory began as theoretical abstractions that appeared to be of limited or no practical use. But with additional assumptions and restrictions, these snme theories have become commonplace in the major fnancial markets as standard frames of reference in analysing fmancial decisions and the functioning of markets. In addition, what had once been seen as a group of related theories can now be uned within a general frnmework. These develop ments have taken place in a relatively short space of time: the original ideas were developed in the 1950s, and culminated in the general theoretical structures published in the
1980s.
THE IMPORTANT CONTRIBUTIONSOF THE 1950s
To understand the current state of fman theory, we should go back to the fundamental contributions of Arrow (1963)-:rst published in French in 1953-and Debreu (1959). Their contribution was fundamental in showing how the economic model under certainty could be adapted to incorporate uncertainty. ne basic idea was
4
W Brief History of Finance Theory
very simple: the commodity space was expanded to incorporate possible future states of the world. The market system was complete in the sense that there was a set of contingent markets for all commodities. Standard theorems on the existen and Pareto optimality of commtitive eqllibria could be reinterpreted, so that one could have an ecient allocation of resours under unrtainty. Although not recognized at the time, tMs abstract economy was the foundation for much of what was to follow. Two other important theoretical developments ourred in the 1950s. In 1958, Modigliani and Miller published a controversial pamr arglling that the nancial structure of firms was a matter of indxerence for all agents in the economy. Their proof relied upon the idea that individuals could employ a riskless arbitrage to undo the variation in the irm's fmancial structure. Although originally couched in terms of the flrm's choi over debt and equity, it becam apparent that the argmnent was general and could be applied to changes in dividend policy, debt structure, or other fmancial decisions. (See Miller, of these ideas.) The major 1988 for a detailed aount novelty in the Modigliani-Miller paper was the use of nancial arbitrage. In the cming decades, arbitrage arguments were to play an important role in understanding a whole array of complex sset-pricing problems. The other major development was the publication of
Markowitz's (1959)monograph on mean-varian portfolio selection. ne basic idea was qtlite straightforward: if consumers were concerned about the average, and variability of portfolio returns, then one could obtain a simple analysis of portfolio choice in terms of the means and covadances of the original assets. nis contribution was the flrst step in the development of portfolio analysis and asset pricing based on mean-variance
analysis.
W Brief History of Finance Theory
5
THE 1960s:THEORY ANDTHE BEGINNINGSOF APPLICATION
nere were two major developments in Mance theory in the 1960s. The srstextended the Arrow-Debreu theory to markets in more detail. Hlshleifer (1965, explore snancial 1966)made an important contribution by showing how the Arrow-Debreu theory could be applied to basic nan problems. ln particular, he proved the Modigliani-Miller fmancial irrelevance result in the Arrow-Debreu frnmework. nis was the flrst time that Arrow-Debreu had been linked to arbitrage theory. These papers were followed quickly by Diamond's (1967) paper investigating the implications of incomplete asset markets. Diamond showed, in a two-date model that with exogenously speced asset under unrtainty, equilibrium is a constrained opticommtitive markets, the obtain the mum. Furthermore, he showed that one could Modigliani-Miller theorem so long as the bonds did not have default risk. The second major development in the 1960s was the extension of the Markowitz mean-vahance analysis to a competitive economy. Sharpe (1964),Lintner (1965),and
all that, with market clearan, linear consumers would choose portfolios that were a combination of the risk-free asset and the market portfolio. A direct consequence of that observation is that equilibrium asset pris can be written as a linear combination of the bond price and the market value of the market portfolio. Or, in more fnmiliar terms, the expected rate of return on any asset can be written as the risk-free rate of interest plus the asset's normalized covarian with the market times the diferen between market's exlxcted rate of return and the risk-free rate. nis model and the pricing result became known as the capital asset-pricing model (CAPM). For the &st time snan theory had created a simple model relating asset returns that could (in Mossin
(1966)observed
6
:4
Brief Htory
of Finance Theory
principle) be tested with econometric methods. By the late 1960s these tests were being canied out at the University of CMcago using the newly acquired CRSP share price data. ne full Qoweling of this empirical research was to come in the next decade.
THE I9DS:THEORETICAL AND EMPIRICALFINANCE COME OF AGE
There were a number of major developments in fmance theory in the 1970s. ne frst was a continuation of the CAPM research programme, extending the model to a multiperiod economy (Merton, 1973*, introducing restrictions on borrowing (Black, 1972), introducing transaction costs tMilne and Smith, 1980), and applying it to a range of empirical problems in nance. As an empirical model CAPM began to have a major impact on the way investors and mutual fund managers controlled portfolios and assessed their performance. (For an informal discussion of the impact of these ideas see Bernstein, 1992.) The second major contribution grew out of dissatisfac-' tion with empirical tests of the CAPM. Although initial testing of CAPM appeared to show that the theory provided good Ets to the data, subsequent work (Rol1, 1977) showed that the predictive power of CAPM was exaggtrated by the test methodology. Ross (1976) introduced the arbitrage-pricing theory (AF1) as a generalized competitor to CAPM. By amalgnmating pure arbitrage and diverscation arguments he showed that one could obtain asset prices as a linear function of a few basic factors. Potentially, the model aypeared more iexible and robust than CAPM, and posslbly immune to the testing problems associated with CAPM. As we shall see, the AFI- played a more important role in asset-pricing theory in the following decade.
W Brief History of Finance Theory
The third advan in nan theory has had a drnmatic impact on theory, and practical Mancial decisions in capital markets. Black and Scholes (1972)and Merton (1973:) showed that one could explpit an arbitrage argu.ment to obtain a relatively simple formula for a call stock option. This result led to the rapid development of a whole range of variations on this model. (See Smith, 1976 for > survey of the advans of that period.) Finance traders and bankers were interested in the models for providing pricing formulae for an ever-increasing array of derivative nancial assets being traded in fmancial markets. Because these models exploited techniques used in physics (i.e. stock returns follow a difusion pross, Ito's lemma is used to obtain the arbitrage hedge, and the solution to a heat exchange equation is employed to derive the formula) there arose a mystique about derivative asset-pricing associated scientist' image. In an important with a popular and Rubinstein (1979)showed Ross, contribution Cox, that the Black-scholes logic and pricing derivation could be greatly simpled. Assuming an elementary binomial stochastic process for the stock it is easy to use arbitrage argllments to derive a binomial option-pricing formula. In addition they showed that by taking appropriate limits, one could obtain the Black-scholes formula. Although not stressed in the paper, the underlying model used arbitrage arguments to derive Arrow-Debreu grices,so that the pricing formula was a discounted martmgale with Arrowfrocket
Debreu prices acting as probabilities. Another interesting development was the derivation by Rubinstein (1976)of the Black-scholes formula from a discrete-time incomplete markets equilibrium model. By assuming consllmer aggregation, the economy achieved a trivial Pareto optimal allocation and the Arrow-Debreu prices suppoted the consumer optmlm. This was the flrst representative consllmer model where the martingale pricing result was obtained, albeit in a restricted form. In the next decade this general insight was exploited in Mance,
9
W Brief History of Finance Theory
W Brief History ofFinance Theory
and particularly in macroeconomic representative consnmer models following Lucas (1978).
reveal private information. These ideas were explored in detail by a number of writers. (See Huang and Litzenberger, 1988 for a brief reviem) Asymmetric information ideas were introduced to explore the theory of corporate Enan when there were diFerences in information between shareholders and management. These theories examined the robustness of the Modigliani-Miller theorem, when nancial structure could act as a signal, or as an incentive mhanism. (See Huang and Litzenberger, 1988; or Bhattacharya and Constantinides, 1989: ii for a review of this literatme.) Because this book concentrates on competitive symmetric information models we VII not discuss this large and interesting research topic of asymmetric information and game-theoretic models in fmance.
8
The idea of martingale pricing was exploited in detail by Hanison and Ueps (1979).They showed that the martingale binomial logic could be generalized to a more abstract setting with continuous or discrete asset-price processes. This abstract approach was to have a big impact on nance theory in the following decade in sorting out lmbiguity that had arisen over the ecient-markets hypothesis (EMH). The idea of the EMH was flrst introdud by Fama (1970).Building on the earlier work of Samuelson (1965)and earlier writers, he argued that, in fnancial markets with free entry, no agent could make abnormal returns by exploiting publicly available information. This simple idea was to have a profound imp>ct on empirical nance and the way agents in fmancial markets viewed their role and mrformance (seeBernstein, 1992). One of the early problems with the theory was its lack of coheren m making a link with asset-pricing models. This ambiguity was clared in the 1980s using the theoretical ideas of martingale pricing. nere were two flzrther signifcant developments. The flrst was the elaboration and analysis of complete and incomplete asset markets with multiple commodities and 6nite and in6nite time-horizons. The work of Radner (1972) and Hart (1974,1975) was important in clarifying the promrties of incomplete markets. Unfortunately this work and related work on transaction costs in asset trading, introducing money into the model, the objective function of the 61-m with incomplete markets, and other generalizations, were largely ignored by fmance theorists for nearly two decades. The other major innovation was the introduction of rently develomd ideas in asymmetric information into fmance theory. Grossman (1976)analysed stock markets where agents had asymmetric information, and explored the idea that stock prices could completely or partially
THE 1980s AND BEYOND: THEORETICAL CONSOLIDATIONAND UNIFICATION
In the 1980s the advans in theory were largely nnifying and extending the existing theories. The various ideas were unifed under the general Arrow-Debreu framework, and shown to be very qexible in application. This :exibility proved to be imgortant in understanding the rapidly expanding market ln derivative securities. ln particular, the hedging and pricing of a whole array of securities became example of a major industry. Perhaps the most spectacular portfolio of a derivative security was the development insurance. This was an application of option-hedging ideas to portfolio management. Although simple in principle, the idea was developed into a signcant Mancial product b# two Berkeley fnance theorists-Hayne Leland and Mark Rubinstein (seeBernstein, 1992). On the theoretical front, the martingale idea became a central tool in characterizing asset-pricing in arbitrage or
10
W Brief Httory ofFinance Theory
Arrow-Debreu economies. Using the general idea of stochastic intepals, the models of Black-scholes and Merton were generalized signcantly by Harrison and Pliska (1981), Due and Huang (1985), and Due (1986). A more slxcialized version of those models was introdud by Cox, Ingersoll, and Ross 1985:) to (19854, explore the implications of stochastic intertst rates for asset-priczg. This model stimulated a series of papers extending the hedging idea to derivative securities dehned over bonds, or associated with bonds-see Heath, Jarrow, and Morton (1992)or Jarrow (1992). Recalling the Rubinstein (1976)equilibrillm approach to the Black-scholes pricing formtlla, Turnbull and Milne (1991) were able to construct an equilibrium (possibly incomplete market) model that paralleled the Heath, Jarrow, and Morton results and applications. This provided a striking illustration of a more general idea that martingale asset-pricing could be obtained via equilibrium or arbitrage arguments (seeMilne and Turnbull, 1994). For practical asset-pricing it is important to construct an argument (either arbitrage or equilibrizzm) that redus the general martingale measure to a simpler density that can be written as a function of a small nllmber of observable variables, simulated mlmerically on a latti (fora survey see Jarrow, 1992), or approximated by polynomial methods (Madan and Milne, 1992). Another advance was the clari6cation of Ross's Am'. Two alternative approaches were taken: the flrst exploited an approximation argument (Chamberlain, 1983; Chamberlain and Rothschild, 1983; Huberman, 1983); the second used general equilibrium arguments to provide an exact or approximate (Connor, 1984; Milne 1988). The AFI- idea of pricing factors has assetpricing models, so that many models can mrmeated be seen as static or dynamic factor-pricing theories. In particular, dynamic asset-pricing models based on diFusion processes can be viewed as a special case of a more general dynarnic factor AFIN
11 ,4 Brief History of Finance Theory model. Furthermore, by taking an aypropriate basis, simple discrete models can mimic thelr more complex continuous-thne counterparts. Tllis discrete model provides an accessible and highly qexible frnmework for integrating asset-pricing theory (seeMilne and Turnbull, 1994for a detailed discussion of this model and its applications.) In addition the model can be adapted td incorporate fat money and nominal asset returns, multiple currencies and exchange rates, transaction costs, taxes, and developed many other features. These variations have been of development. recently, or are in the process This tmication of fmance theory has fotmd a parallel in modern macroeconomics, where representative agent economies have been analysed to investigate real and pricing variables. Clearly macroeconomics and fmance theory exploit the same underlying Arrow-Debreu model. It is hardly surprising that the same Modighani-Miller type of results reappear in discussions of government fnancin and open-market operations (in the guise of Ricardian equivalence theorems). lncreasingly tllis literature and Enance have become integrated so that the bouzidaries of the two disciplines are blurred. o
.
'
SUMMARY The development of fnance theory has been rapid. Not only has it provided higMy iexible models, but they have found wide application in fmancial markets. nese developments have been important in providing a coherent frnmework for thinking about existing fmancial markets and decision-making; and for creating ways of thinking about new Enancial products. It is ironic that abstract ideas developqd in the 1950s and 1960s,which once were thought to have llmited application, should become the common language of fnancial markets.
Two-Date Models: Complete Markets
consumer's problem. Each consumer i problem'.
2
(Max Ujtxnj, xlj, lxfe l
Two-Date Mod:ls: Complete Markets
!tt'nxaf
+
.
.
.
kpsxsi >
,
1,
=
.
.
.
13 ,
1, has the
xsij
Wif
#
where: (i) the utility function is standard neoclassical (ike. strictly increasing, qllnqi-concave, dxerentiable (if necessary) ); (ii) pn is the price at t 0 of the commodity; ps is the pri at t 0 of the contingent commoty s and (iii)
=
In the early 1950s Arrow and Debreu introduced simple extension to the existing theory of competitivea
a equilibrium. Consider two dates: today there is certainty and tomorrow there is uncertainty, with = 1, S, states of s the world. To make life simple, asslzme there is only one physical commodity, at each date state. By expanding the denition of the commodity or spa to include dates and states, we can use a1l the standard tools of pri theory under certainty to analyse an economy with contingent consumption and production. We begin with the .
,
J=1
2
=
We can analyse the conszxmer problem using the snme tools as the certainty theory. For example, we can derive an indirect utility function, expenditure function, and obtain a Slutsky decomposition of consumer demand. (For details see Varian, 1992.) Using the same idea we can analyse the frm's problem. Consider flrm f= 1, F, to have the problem: .
.
,
.
S
Z psysj
Max
(l'j
e Yjl s
l
=
-
poyzj - #y?,
where p and y) are the price and production vectors respectively. The 61.1= maximizes its net present value by choosing the most protable contingent production plan in the production set 5., where yg is the flrst-date input and contingent commodity s. Again this is sj the output of ldentical to the standard theory of the :rm, and can be analysed with the same tools (e.g. prot function, cost functions. For details see Varian, 1992). We can close the system by requiring market-clearing prices for commodity markets. equilibrium
DEFINITION xv()
Fig. 2.1
& '= 1
c/lm a price
W competitiye economy yector
(#;, #!,
.
.
.
,
#:);
for
the contingent
Two-Date Models: Complete Markets
Two-Date Models.. Complete Markets
14
.
(
x: solves:
where 1 > 0# > 0 and 51-mprofts; .y%
J
.
,
here.
Max Uix s t psxi
@)
.
<
p'-k'i + je
gq
=
f
J
uqp),
1, are consumer shares in
=
i i
ne two theorems can be illustrated in two commodity examples. Indd, one can thlnk of the theorems as abstrad generalizations of the geometric examples 2.2, 2.3, and 2.4. 1. Pure Exchange:
solves: Max p'yj; n e 19.
(c) Ei xz
Single Consumer
y1.
+ j
U
There are two important observations one can make about this equilibrium: 1. You can show that under reasonable conditions an equilibrillm for this economy exists. (See Vadan, 1992 or Debreu, 1959 for more detailed discussions) 2. You can show by standard techniques the following important results linking a competitive equilibrium with the concept of an ecient, or Pareto optimal allocation. FIRST
15
Comment. Varian (1992)or Debreu (1959)prove this result using weaker assumptions on preferences than we assume
x'w)i e f; consumer demands x: * (xlj, that.. such plans production J,hrm @;),j e .x!f,
FUNDAMENTAL
THEOREM
OF WELFARE
P
E
h
l I I I l I l I 1 I 1 I 1 1 1
ECONOMICS:
Given a competitive equilibrium with neoclassical consumers, then the equilibrium is a Paeto optimum. Comment. Varian (1992)or Debreu (1959)prove this result using weaker assumptions on consllmer preferens than our convenient neoclassical assumptions. Secoe FIJNDAMBNTAL THEOREM Giyen a Pareto optimal allocation E(J;:),
oF WELFARE
ECONOMICS:
(J/9)1,
neoclassical preferences, and convex :., then there exists prices p. and a general equilibrium which will support that allocation.
*
.:1
Fig. 2.2 Noter
.11
>
Pure Exchange: Single Consumer
supporting prices depend upon the endowment and the utility function. Change either and you change the prices. (2) The allocation E is Pareto optimal trivially. (3) We can think of the two commodities as commodities at t 0 and t l (with rtainty); or as two contingent
(1)
'rhe
=
=
16
Two-Date Models: Complete Markets
Two-Date Models.. Complete Markets
commoditiesat s flrst date.
=
1, 2, iroring consumption at the
h
2. Pure Exchange: Two Consumers p*
B
#
*
E
UA
E
UA
11.k
Ua Contract curve
.
Fig. 2.3
p
21
Fig. 2.4
A
.
Consumption and Production
Pure Exchange: Two Consumers CHARACTERIZINGPRICES
Notesl
(1) Now the pris and consumption allocations depend and the endowments, of both upon the preferens altering By the division of the aggregate consllmers. endowment, the commtitive allocation and pris will change (1 general). (2) ne competitive allocation E is on the contract curve, i.e. it is Pareto optimal.
Unlike much of the standard microeconomic theory, which emphasizes comparative static results, nance theory emphasizes conditions on preferences, production thnology, and endowments that restrict relative pris so that we can predict asset prices with some formula. There are two basic tricks that are used.
3. Consumption and Production Noter
that the t 0 commodity is not consumed, and a1l of the endowment is used as an input to produce contingent production (yI, yg. This contingent production is consumed (x!, xg. (2) ne commtitive allaotion E is Pareto optimal trivially.
(1) Assllme
=
1. Arbitrage between Pedect Subslitute Assets ne idea is very simple: two commodities or assets that are viewed as perfect substitutes will, in equilibrium, sell for the same pri.
19
Two-Date Models: Complete Markets
Two-Date Models: Complete Markets
Consider the case of a single consumer (or identical consumers) with linear indxerence curves (Figure 2.5).
consumerl; or assume that they dihkr in a restricted sense that wealth redistributions will not alter relative prices. For more details of this approach see Varian (1992,sect. 9.4).
18
the economy Flc a neoclassiwhich is quasi-homothetic, Le.
Ifeachconsumer in
THEOREM:
cal utilityfunction
U).@J*
+
uiai
JS.YJ
where ui( ) ul ) is homothetic, then there ex/w a reprsentatiye consumer with preferences
UA
=
p
*
W(.tX M u
Zi a/ + I5.ta
and a budget constraint which i the budget constraints. E
I l I I I I l 1 l I
ullpl
of the f?WfW#=/
Consider the case of two consumers in an exchange economy (Figure 2.6). B
.
45* r
11
Fig. 2.5 Notesl
F
r the consllmer views (xI, x2) as perfect substitutes ihen eqlzilibrium prices have the property /4 #!. nus if we know #!, automatically we can
(1) Given
p* EM
=
yyjx:
pkp
p*
(2) Although the example looks trivial, we will see many sophisticated applications of this principle throughout this book.
Contract Curve
A
2. Consumer Aggregation
Fig. 2.6
'rhis thnkue is used when arbitrage arguments are not applicable. The fundamental idea is very simple: assnme that either all consumers are identical (a representative
that relative pris Notel Under consumer aggregation noti of endo not change when there is a redistribution dowments between A and B. The reason for this is simple:
20
Two-Date Modelr Complete Markets
Two-Date Models.. Complete Markets
bause the conditions imply a representative consumer with qunqi-homothetic utility, we get the diavam of a singleconsumer and an endowment (seeFigure 2.7).
21
that assume a much more restrictive type of preferen risk. explicit probabilities and attitudes incomorates to
EXPECTED UTILITYFUNCTIONS Assume that our consumer has preferences that satisfy the expected utility form,
p*
U(xn, xl
.
.
.
X,)
,
=
Zs lztxa,
x2l,,
J=l
xrx- + #
=X
,
E
-.--.-------.-
where
I I I I I 1 I 1
1, u((aj+a+
p (xa))
l
I 1 I I 'lA
. ... . ..
r
xj
= 5 IA+ :f 1#
0;
>
=
eVv(2a)
S
S
Now, consider the implications of agvegation on relative of u ), we can pris. Given smoothness (dxerentiability) compute prices by the slom of i.e. p.
1 and u() is neoclassical.
Clearly a, is a probability of state J; and the consumer texpeded evaluates contingent bundles aording to utility'. Often this utility function is restricted further to be additive-separable over time (as well as over states), i-e. J=1
,
=
S
Fig. 2.7
uhj
Z ls
,
where Tu is the gadientvector of utility at the consumer optimum, and 8 ls just a constant of proportionality. We will see how this thnique is applied with more restrictive
utility functions.
SPECIAL UTILITYFUNCTIONS
So far we have assumed that conszlmers have neoclassical utility functions. But it has been traditional in nance to
lz(xn,x,)l,
=
l/txp
+
Z v(x,)x,,
J=1
where u' ) > 0, and u'' ) < 0. Given this set of preferens we can show conditions on u ) that will imply neoclassical indxerence curves. THEOREM.
Given u ) is strictly increasing and concaye, then
the preferred sets are convex. Proof Milne
(1974).
By using a quadrant diagram (Figure 2.8), we can illustrate tllis result. Given any point on the constant expected utility locus in the south-west quadrant, we can tra through to its commodity bundle in the north-east quadrant. nus we curve in the contingent can trace out an indi/eren commodity quadrant. Noti that it inherits the familiar neoclassical shape.
Twomate Models.. Complete Markets
Twomate Models: Complete Markets
22
23
WHYDO FIRMSMAXIMIZEPROFIT? Previously we assumed that the 5n): mnximiv.e.d prot. Here we will show sllcient conditions that imply that all shareholders want the rm to maximize prots. markets Given Competitiye and no externalitiesfowing between the owner and thejrm, then a1I owners will desire #roJ/ (or net present value) FISHER SEPARATION
I
u(xz)
1 1
1 I
I
I I I
1 1
EU xl
THEOREM
maximization. that the consllmer has Proof We know from Varian (1992) utility function an indirect Fk.(#*, lpf),
F&=u(xI)A'l+I4>z)mz
u(xj) Fig. 2.8
Now consider the joint assumptions of quasi-homotheity and additivity. We obtain the following remarkable result. THEOREM.
Utility is additiyely separable and
(
E
2
1 l l l l I 1 l I I
lzai-
homothetic.
8(x, + ()
z/lxf)
=
lnt8x,
af)C
+
lexptxs),
for
0
aJ,
<
af <
c >
0;
Statement ( is equivalent to statement Proof Milne
(1979),Brennan
and Kraus
<
1,
0k <
I
2
0.
(1976).
E
1
nI
1 1 l I I 1 I
fb).
This theorem is important, because it reveals, in conjunction with the representative consumer theorem, the importan of log, power, and exponential utility in creating a representative consllmer. We shall see variations on this theorem throughout this book.
--
*
yl
I I I 1 1 I 1
.
*
71
Flg. 2.9 Notes.. (1) Ek is the proft-maximizing production plan (y!, #!). (2) Ez is the utilitpmnximizing consumption plan (x!, x!).
24
Two-Date Models: Complete Markets
Two-Date Models: Complete Markets
25
which is strictly incregsing in Gcf. Because
+
j
nqpj
1 > 64/ > 0,'
:#
=
1
7
then each consumer (with04/ > 0) prefers yJ to any other y7e Yj if ff J > pj. ne Fisher theorem can be illustrated with a simple twocommodity diagrnm, showing that an owner will want her wealth (or share of prots) to be as large as possible (Figure 2.9).
ya
FAILURE OF THE FISHER THEOREM
yx
It is not dicult to show that without the two assumptions, the Fisher theorem can fail, and owners will disagree Fig. 2.11
t;s
over the production plan. Consider the case where there are two owners with 50 per cent ownership of the thnology. Assume that the firm has monopoly power such that for production plan y' there is an associated price vtor and for plan y'' there is a price vector p' pfj; pfj. In Figure 2.10 we illustrate how consumer A p'' and consumer B will rank the two plans diFerently and disagree over which production plan to implement. A similar conclusion can be reached if there are externalities qowing between the 11= and the owners. A slxcial when there are missing case of an externality ours where Consider markets. the case there are no markets at all. Given the two consumers, A and B, with diFerent preferens, we can illustrate the coniict in terms of Figure and 2.11. Consumer A chooses production bundle consumer B chooses production ya * yx. Clearly, missing =
=
#
f
tr 4
:
Flg. 2.10
.ya;
26
Two-Date Models: Complete Markets
markets can create owner covicts over production and Mancing decisions. (For a more detailed analysis of these problemssee Milne (1975,1981!8, and Milne and Shefrin
(1984).)
3 Incomplete Markets with Production
CONCLUSION @
This chapter has outlined the basic results for the two-date, complete market economy. ln later chapters we will draw upon these results as we modify and extend the economy to incorporate incomplete markets and/or multiperiod dision-making.
Our flrst modihcation towards realiiy is to realize that we do not see a complete set of markets. There are reasons why we do not see markets for some states of the world. One example is the existence of transaction costs in operating markets. For the moment we will not model these costs explicitly, but just assume that certain asset markets exist, and some do not. (A more complete theory would incorporate transaction costs and deduce market activity endogenously.)
THE BASIC MODEL , K with pay-oFs Zj, Consider a set of assets k 1, ZK where Zk e R.9. We present two exnmples of asset pay-ofs when there are three states of the world. Assllme that the states are ordered aording to economic activity such that st is high activity, Ja is moderate activity, and ss is a recession. =
s
=
1
2 3
.
.
.
.
Riskless bond
Equity
1 1 1
3
;
1
.
.
,
Now we can prove the following theorem.
THE CONSUMER
At time t 0, the consumer buys ak units of asset k, at prices pk; and xaf units of the consumption good at pri =
#Q.
At date f = 1, the consumer's contlgent consumption is constrained by the contingent endowment (.f,) and the pay-oFs for the portfolio, i.e. K
xsi
=
Z Zskaki +
k
ksi,
s
1,
=
.
.
.
S.
,
x If &/4.x':/, ,
-
.
xs'
,
.
the conslxmer's problem can be written as: (Max Ujtxj, x lj, (.*'f> 01 K
=
X Zskaki + k=l +
r
pnxni k=1
.
.
.
x-,f,
pkau
1,
=
.
.
.
,
S
(3.1)
akl.> 0 tk. Note that the non-negative constraint on asset holdings eliminates borrowing or short sales. Later in the chapter we will explore relaxing this constraint. We can rewrite the problem, by eliminating the contingent conslxmption constraints tb obtain: MM qij > 0J (.x'(,f> s-t. 'nxaf +
where
aj
Fjtxtjf,' aij
O
Uf
Ff(xaj,'aij
Xpkaki
Zskaki
k (b) (xaf.lf) e X4t -
RX+ +
=
(3.2)
<()/
' .
,
THE PRODUCER'S PROBLEM
By similar methods we can construct an indud producer's problem over fzrst-date input and asset sales. (We rule out, for the moment, the possibility of the 15- holding the assets of other firms.l The basic 6= problem is: Max
E pkakj
pzynj
-
k
S.t
.
Zk Zskakj
ysj,
=
(.1'v',ytp
.
.
.
=
q
1
,
.
.
.
,
S. (3.3)
,
ys? e Yj,
akj > 0, Vk. Now, by substituting for ysj, we obtain'.
k
.Xnf,'
strictly increasing. Lemms Lemma 1, or Milne (1988) Proof See Milne (1976) 1, 2. This result derives induced preferences over t 0 consumption, and the asset portfolio ai, such that the utility function inherits the stapdard neoclassical properties. As a consequence a1l the standard results of consumer theory (Slutsky, etc.) can be used on the consllmer portfolio problem.
xs
,
J
=
gzzzf3.1. Given Ui ) is continuous, dtrentiable, neoclassical, strictly increasing, Le. it and concave, Zsk > 0 and Zk has at least one strictly positive component; then K.() is continuous, d@entiable, quasi-concave, qnd
THEOREM
=
I
-
Now p'ven the consumer's utll'ity function
s t. xsi
29
Incomplete Markets with Production
Incomplete Markets with Production
28
1nd
Max
k t. yv;
pkakj
:
k
-
pnytj
Zskaki
e Yj
(3.4)
30
Incomplete Markets with Production
Incomplete Markets with Production
akj > 0, Yk. Analogous to the induced-preferen argument for the cnsllmer, we can construct an induced asset-production set for the produr:
r, l(yw;ap
eR! +K
-
1.pe FJ-,ys)
=
k
THEOREM
akj > zakakp.
0,
3.2. Giyen :. is closed, conyex, and 0 e 17 closed, convex and 0 e V. Proof See Milne (1976)Lemma 2.
vkJ.
k
pkakj
pnynj
-
(3.5)
yzj, &) B
V.
Again, standard neoclassical techniques of comparative statics can be applied to the Grm's problem. To close the model, we assume the commodity and asset markets clear. DEFINITION
3.1. competitiye equilibriumfor an incomplete c.ze/ economy a price yector
((xlf,-
.
.
(y*w;#) is the
X xlf X
.k
)V;
E i
a) ./
.
3.2. Given a set of consumers
1(K.(); A?; 2f)l and a set of producers ( F./J then a constrained allocation f. a feasibleallocation feasible allocation
(y%; )V./)
to the producer
X.F*ty; ./
f
=
solution +
=
at,
DEFINITION
ltxlf
optimal
's
problem
(3.3),.
); (>%,#)l
,
for
ltxlf
(xk; ) is the solution to the consumer's problem (3.2);
f
With the Arrow-Debreu complete market model w know that the equilibrhzm is Pareto optimal. But with incomplete markets this is no longer true. Nevertheless, for ihe single commodity/two-date model we can dene the concept of a constrained optimum that is due to Diamond (1967). ne trick is defne an optimum in terms of our asset economy,
,
suchthat
(c)
OPTIMALITY AND THE WELFARETHEOREMS
.p1:)
.
and an allocation
bj
Given that the reduced-form asset economy has the same structure as the standard Debreu economy, then it is easy to modify the standard existence proofs. If we allow short sales/borrowing, then there are some complexities. We *1 deal with these shortly.
,4
(#:,#!,
(
EXISTENCEOF EQUILIBRIUM
5.;then
Thus the Grm's problem collapses to the neoclassical form (assuminga diFerentiable production function). Max
31
,
J1
which there exists
); y
aJ3
,
no other
)
for which Fftxlf
,
J,f)
Fktxk, x1)
>
for each i Ja# with strict inequalityfor at /eaf one coztlwler. Noti that the set of feasible allocations is constrained Zxl which is treated as by the set of asset returns IZ) of in constnzction the a ,
ftechnology'
.
.
.
,
K'(); X/;
and
Y).
Now by using standard techniques, we can prove the two fundnmental theorems for our reduced-form asset economy. (For proofs based on Debreu see Milne, 1988.)
to an exchange economy with only two consllmers. (The general economy is discussed in Milne, 1976 and 1980.) Consider two consumers i = W, B, with consumer problems with unlimited short-selling: (Max Ui(xjf
,
.
.
.
,
xsij
K
S.t.
THE FISHER SEPARATION THEOREM IN THE ASSET ECONOMY
Given that the reduced-form asset economy has the same structure as an Arrow-Debreu economy we can simply adapt the proof of the Fisher separation theorem to our new economy by relabelling variables. Clearly all owners will want the f5= to maximize proft, if they are initial shareholders of the rm. Noti that if the 61-m introduced an asset with returns that were not included in the current set of assets (or a linear combination of the current set) then it would be a monopolist, and the Fisher theorem fails. For more on this see Milne (1976)(on leverage) or Milne and Shefrin (1984)(for production decisions). Indeed with non-compditive behaviour and/or externalities there may be no constitution for the 11n:1 (see Milne, 198184.
xsi
=
Z Zskaki,
k
=
S
=
1,
.
.
.
,
S;
l
pkaki Z'lf. k =
k
For simplicity we have irored t 0 consumption, and introduced endowments of the assets. For illustrative purposes, consider just two assets, i.e. K = 2. We can now draw an Edgeworth box. Assllming no short-sales, the old constraints are represented by the dashed box so that short-sales are outside the dashed box. In Figure 3.1, the equilibrilzm allocation E is outside the box. Mter Asset trades, consllmer A is long in both assets, but consumer B is long in asset 2 but short in asset 1. =
l l I
l
....
J
I
. 1 1
p
I
l 1 I I I I
INTRODUCINGSHORT-SELLING AND BORROWING
l
So far we have avoided discussing short-selling and borrowing, by constraining the consumer's asset purchases to Rf and the rm's asset issuing to R+f. We can extend the analysis to allow for short-selling by consumers and Nnns. This introduces some additional complications that we will address here brieiy. To keep the argument as simple as possible, we restrict our discussion
33
Incomplete Markets with Production
Incomplete Markets with Production
32
l I
1 I I
l
I I
+*
A
'
*
l l 1 1 1 I I 1 I
B
j
1 1 I I 1 1 l I I I l
-
FB E
F4
Fig. 3.1
So long as the indurwl gains from trade line botmde (it is closed, clearly) then we do not nm into any problems in proving the existence of an equilibrium. (For a proof along these lines see Hart, 1974, or more rvzmtly Werner, 1987.) But there are examples where such a proof fails. For exnmple, consider the case where both consumers are risk-neutral, but have diferent probability distributions. In jeneral, this will imply dxerent slopes for their linearlnduced preferences over assets. We can check that there will be no equilibrillm in this economy. If we postulate a gricevector p' and an allocation E', both consumers w11lwish to trade away from f'-a contradiction. This will be true for any postulated qrice p' that has a price line in the gains from trade cone FsJ Px. Does this imply that there is no equilibrium for an economy with risk-neutral consumers, diferent expectations, and short-selling? A possible solution to this problem is to recall the underlying constraints on contingent consumption, and se if they impose constraints on the asset-constraint sets that w111bound the asset economy. q
--
1 I l 1 l I I I 1
-
1 I I I l 1 1 I I 1 1 I 1 I
p
Intuitively, unbounded short-selling wolves consumers believingthat there is unbounded contingent consumption in at least one state of the world. This assertion follows directly from the observation that any allocation sequence where the consumer is increasingly better oF, must imply increasing contingent consumption. (For a more detailed discussion see Milne, 1980.) Thus, if we dehne asset constraint sets, Ai
=
(,f e
I
Rf xi e
m;xsi Zk Zskak =
ZK is of rank K, then standard and assume that z1, constructions of feasible asset trades will imply that the set of obtainable allocations is compact, and standard existence proofs can be used (seeMilne 1976, 1980). Cnsider the example shown in Figure 3.3. .
.
.
,
MORE GENERAL CONTINGENT COMMODITY SPACES'
For the sake of simplicityyweRN.have restricted the continBut the construction of gent commodity space to be
B
l l l 1 1 I I l
J
35
Incomplete Markets with Production
Incomplete Markets with Production
34
I 1 I l I 1 1 I I l 1 l I
. 1 I
l
-
I
1 l
a
I ....
V
1 l 1 l I
l
I
r
1 1 4 I I I
l
I I
l . I 1 1 l I 1 l I 1 I l I 1 I l l I 1 I 1
-
F P
*
E
A Infeasible FA
Fig. 3.2
Fig. 3.3
Incomplete Markets with Production is sllciently iexible for it to be induced preferens in6nite-dimensional contingent comwith extended to deal modity spas. Btcause such an extension rtquires more complex mathematics, we will omit any formal analysis and direct the reader to Milne (1981J, 1988) for a more complete discussion.
36
4 Arbitrage and Asset-pricing: Induced-preference Approach
CONCLUSION
In tllis chapter we have introduced incomplete asset markets and explored some basic results conrning an equilibrium and its optimality properties. ln the next chapter, we will begin our discussion of asset-pricing methods by introducing arbitrage pricing using our construction of induced preferences and induced-production sets.
Given the basic asset economy outlined in the last chapter, we can now analyse the role of arbitrage in asset allocations and asset prices. By using induced preferences, we have a simple tool for constnzcting geometric proofs. ln this chapter, we will illustrate the ideas using examples, but the ideas generalize in a straightforward fashion (see Milne, 19814, 1988). Consider an economy where asset 1 has contingent returns that are a linear combination of the asset returns K, i.e. of k = 2, .
.
.
,
K
zj Assume that Zz,
kzk
=
k=2 ZK .
.
.
,
for non-trival fa1). are linearly independent. Dene K
Zp
ukzk
=
k=2
to be the returns on a portfolio with returns that afe identical to Z1. Consider a consllmer at an optkmlm i-e.
(a1f,
J1)
solves
Max F).(xaj, lf)
s.t. yllxaf +
pkaki
=
eaj.
At the optimal (xTf, ) the indiFerence surface in asset spaces will be defned by:
5: -
lJj
e
RA1
Fk.(x:j,
)
=
Pf).
By construction it will contain the linear manifold dehned
by K a!ZI
-
E ukzk k =
=
0,
.
.
then the consumer will want to take unbounded trades, violating the assumption of a consllmer optkmlm. TMs is illustrated in Figure 4.2, where the consumer w111hold increasingly larger amounts of asset 1 and short-sell the portfolio. This is the case where < x X p, k=2
2
which is independent of the neoclassical utility function Ui ). ne reader can check this assertion by calculating the marginal rate of substitution between the portfolio and asset 1, given that Uf() and Ff( ) are dxerentiable. But we can dismnse with dferentiability of V ), and the indiFerence manifold 5: will still exist. Thus dxerentiable utility is not nezvssary for our argument. In other words, if we projed down on to the asset subspace we will obtain a linear indiflkrence curve over asset 1, apd the composite ax) (seeFigure 4.1). portfolio (az, K are not If the asset prices of asset 1 and assets 2, rdation in the .
39
Arbitrage and Asset-pricing
Arbitrage and Asset-pricing
38
With the reverse inequality the consumer will want to take the ppposite position. Notice that the consllmer will ot be constrained in the contingent commodity spa by such trades, because their impact is zero, as a perfect hedge (see Figure 4.2). The only covguration of asset prices conjistent with a consumer optimum is where J1
,
.
.
.
,
K
Z ukpk rl k=2
kpk.
=
x a#. X k=2
This price relation is known as an arbitrage relation. To summarize:
(free)pricing
=
J
p
p
p* F
Jl
aj
*p
Fi -vf F
Fig. 4.1
Fig. 4.2
40 THEOREM
4.1. lfthe
uef
returns are
linearly dependent,Le.
there exi/. non-trivial (akJ such that
Z
kzk
and short-selling constraints
then the absence of arbitrage implies
Z
0,
=
kpk
0.
=
k
Having deduced restrictions on prices, we turn now to consider the non-uniqueness of asset allocations, Fhen there is linear dependence on future asset returns. Consider an eqllilibrium where Theorem 4.1 holds. Atl allocation for this economy will be
RxTf,J:)V, (yR,altjj. The opportuzlity sets of the consumers and 6rms can be written as x*si
'pxp/+ E#t4/ k
=
=
Ek Z df VJ ,
for each consumer
<
(@1)V; (Z)W1. For details of the proof, see Milne (1988). The intuition of this result can be illustrated by an exchange economy, unbounded Edgeworth box. Consumers A and B have parallel indxerence curves through the The ajset prics p. will be detennined asset endowment by arbitrager i.e. orthogonal to the linear indiflkrence curves (seeFlgure 4.3). Notice that the result can be obtained for the case with short-selling constraints, except that the equilibrium asset allocation is a subset of the linear manifold, deMed by the short-selling constraints. In Figure 4.3 this would be all the asset allocations on the line sepnent ap. In the case of a production economy, we can illustrate the result with a simple consumer/rm diagrnm (seeFigure 4.4). In this Figure the consumer and firm can take .
J
J 1% =
=
linear algebra to prove that there is a linear subspa of asset allocations that solve the opportunity set/marketclearing system, and gives a constant contingent consumption/production allocation
,
and
y*,/
41
Arbitrage and Asset-pricing
Arbitrage and Asset-pricing
k
pkakj
Z Z.aLj Vz
-
pzfz.i
I I l l I
1
l I 1 l 1 1 I I 1 1 I I 1
for each Erm j.
k
ne market-clearing conditions for asset, are
Xi Jt
=
X Jk,
Yk.
j
Notice that in asset space Rf we have here a set of linear equations with linear dependence in the asset return (Z..) and, by arbitrage (free) pricing, the same linear dependence in the asset prices. It is an easy exercise in
1I I
..-
fl
p .
:y
1 I 1 1 I l I l I 1 l l I I 1 1 1 l l I
p -
I
F VB
Fi g. 4 3 .
Arbitrage and Asset-pricing
Arbitrage and Asset-pricing
42 ('
.
43
$ I I 1 J
Firm 2
j I l I 1 1 1
p
*
1 I l l I 1 l I 1 1 1 1 l I I I I I $
I
I
I
j ,
YJA
-
FA
1 I 1 l
l
#
**
j
P
+
FA
:)
.
.
FA l
.
I I I I I I 1 I 1 I I I I 1 1 l I l
Flrm l
Fig. 4.4
Fig. 4.5
oFsetting asset positions to each other to obtain the same contingent conszlmption/production plans. Notice that the constrained short-selling result obtains here also on the sepnent a$. These two examples cover consllmers taking ofsetting positions to a change in a frm's asset portfolio, or a consllmer portfolio. But we can illustrate the case where a f1r1ntakes the ofsetting position to a firm's portfolio by considering two Erms (Figure 4.5). ln this case, if Firm 1 alters its portfolio, Firm 2 takes the ofsetting position. Indeed, if Firm 2 has y! 0, we can treat it as a costless Mancial intermediary that ofsets the fmancial structure change of Firm 1. To sllmmarize, we have the following theorem: =
THEOREM
4.2
(MODIGLIANI-MILLER).
GiMen
returns Le. 3 non-trivial
(a)
such that
Z ukzk k
=
0
dependent
JJ',d/
then in equilibrium
Z tkpk
=
0;
k
and there a linear manifold ofequilibrium asset allocations over which a1l agents are indfrent. We called this theorem the Modigliani-Miller (1958) theorem because it expresses the central idea of that famous paper. For a discussion of the evolution of the theorem and its applications see Miller (1988). To illustrate the power of the Modigliani-Miller theorem,
WC Consider
tWO CXAmPICS.
1. Firm Leverage with Default Risk Consider Firm 1 to have unlevered equity with pay-oF per share of:
Arbitrage and Asset-pricing
44
Arbitrage and Asset-pricing
Zz
s
1 10 2 6 3 4
=
So long as there is a ompetitive market for this defaulting bond (i.e.there is a compditive market for this cash Cow)then Modigliani-Miller applies and:
'
s
1
=
Zz
10 6 4
2 3
1
Unlevered equity
Za
6 2 0
=
L1-1 --
Levered equity
Riskless
debt
In the new structure, the firm has issued four bonds and one new levered share. shares, and Given competitive markets for the llnlevered riskless bond, levered share pay-oF can be replithe the ZE Za.4. cated: i.e. Zz By the Modigliani-Miller theorem we have: (i) #z PE #a.4. (ii) AII agents are indxerent to the nancial restructuring as they can take oFsetting asset allocations. We can extend the example to allow for default risk. =
=
(i) pL pz pna5. (ii) All agents are indiFeret to the nancial restructuring. It is important to realize that if the firm changes its fmancial structure, and creates a security that is not traded by the original assets, then the finn is a or monopolist, who perturbs the whole equilibrium. that is, the Fisher separation and Modigliani-Miller theorems fail. (For more on this see Milne, 1975 and 19814, Milne and Shefrin, 1984.) =
Given that there is a market for such pay-oF streams which is compditive, the flrm has a current market price of its shares #s. If the 514n restructures by choosing to issue debt and equity, then it will retire some of the existing equity. For simpllcity let the firm issue only one share of the unlevered equity. ZE
45
-
Kspanned'
2. Call Option Pricing Consider the two-date model introduced by Cox, Ross, and Rubinstein (1979).Let there be two states of the world, and two securities: a stock and a riskless bond: Zs
Za
1 ups
R R
-
s
=
2 dps
.
-
ZE
s
=
1
zz
10
2 6 3 4
1
=
5 1 0
zoa 1+
1 1 4/5
5.
In this case, the 151-mhas issued ve bonds; and in state 3 the hrm's bond commitments exceed its cash Cow, and it defatllts. The bond-holders divide up the cash ;ow earning only 4/5 of the face value.
The current price of the stock is ps and in state 1 the stock in price to give ups; and in state 2 it can go can go <down' in price to dps. The lketurn to the bond (current price pz 1) is R e (1 + r) in both states of the world. Let r be the rate of interest. To avoid arbitrage possibilities, we require u > R > d. Notice that there are two securities and only two states of the world. In simple linear algebra tenns, the asset returns span the states of the world. Given that the asset returns span the states, we can create two portfolios that have primitive Arrow-Debreu security returns. That
=
is:
46
Arbitrage and Asset-pricing
s
AD1 1 1
=
2
S -
dps
0
B
1
(#,(lz #))-
ups
=
Arbitrage and Asset-pricing
R dfRlu R
-
-
#))-
PAnI
1.
su d)1tmits of the stock; and bocows #.EA(lf
'
J))-'
-
tmits of the bond. By the Modigliani-Miller theorem, we can price AD1 by arbitrage, and the introduction of the AD1 security is a matter of indiFeren to all agents. If the price of AD1 is pm, then by arbitrage:
pxm=pssu =y =
J)-1
-
1
2
0
-
=
-
1
-
F)
dj-
=
we can create the second Arrow-
(#a(lz #)J
+
-
:)1-1
-pssu
l = ( 1 + r) -
-
u
(1 +
-
u
-
d
+
d+ u u d
(1 + r)
-
-
'
+ F)-
.
=
s
#c0 il/#s
.
B R R
lzklz
-
djRj
.
By arbitrage:
'xw
(1
r)
In words, a simple portfolio of one AD1 and one AD2, will have a current market valu (by arbitrage) of (1 + r)- 1. Mter all, the portfolio has the same pay-oF as a riskless bond. Using the device of creating replicating portfolios one can pri any asset by arbitrage. For example, consider a call option on a stock. Given the strike price K such that ups > K > 0, the option will be exercised in s 1, and w111 lapse in s = 2. ne call's pay-oF is given by:
=
u-d
ups dps
+ l (1 + r) - (1
1 2
=
'
d
-
=
47
ps
-
K
0
'
By arbitrage, we can price the call:
S
Am =
1.#IA(u
-
#..R-1)
(1
+ 1 (1
(1 + r)-
By similar reasoning, Debreu security:
s
d)1-1
-
Pxo,
=
The replicating portfolio for this asset takes -
+
-
1&.Ru #).&-' -
r)
= (1 +
-
Al#lo?
r) -
l
ps
=
u(1
-
+
u
r -
-
d
dj
(1+
+ r)-1
Lups f1(1 -
A1
r
#)
-
u-d
+ r
u
-
-
d
dj
.
Notice that the call option has been priced in two stages: (i) by replicating Arrow-Debreu securities; (ii)by replicating the option using the Arrow-Debreu securities. Bgt of course, we can collapse both steps into one, and think of the replication in terms of the underlying stock and bond. Tllis is the intuition underlying the binomial option pricing model, and its continuous time counterparq the BlackScholes (1973) option pricing model.
.
Observe the symmetry in the Arrow-Debreu prices. First they involve a present-value term (1 + r)- 1, and then an undiscounted term in square brackets. By constnzction both prices are strictly positive. Furthermore, observe that
COMPLETE ASSET MARKETS
The call option example suggests a more general restllt: given a set of assets that span the full spa of contingent
Arbitrage and Asset-pricing
Arbitrage and Asset-pricing
consumption, then we can reproduce Arrow-Debreu securities by forming portfolios of the original securities. This is formalized in the following extension of Theorem 4.2.
where rank (Z1, z'xl S will also have a Pareto optimal allocation of resources. It is important to observe that complete markets are sucient for Pareto optimality, but they are not necessary. Consider the following trivial exsmple. Let there be an and economy with one consumer with an endowment one asset Jl If we set the consumet's optimal choice
48
Corollary 4.2.1. .Given the matrix of security returns (zl zxl has rank S K then there exists a matrix of portfolio holdings (a,k)such that ,
.
.
.
=
,
(ZI
,
.
.
.
,
72
laskl =
1.
Furthermore
pa Z k =
.
=
,
.
.
.'%
.
(x1
Arrow securities.
Having constructed the Arrow securities from the spanning set of assets, we apply Theorem 4.2 again to value any asset introduced into the economy. Corollary 4.2.2. Given an asset economy with assets that contain the Arrow-Debreu set, then any asset k introduced into the economy can be priced as
s #k Xpazsk; =
J=1
and its introduction is armatter of indiFerence to a11agents. Although these two corollaries appear trivial in a twodate model they can be extended to multidate models (a we VII see) to obtain a whole range of derivative assetpricing formulae. One last comment: we know from Chapter 2 that an Arrow-Debreu economy with a complete set of Arrow-
Debreu securities achieves a Pareto optimal allocation of contingent commodities. By applying Corollary 4.2.1, we know that an economy with an arbitrary set of assets
.%
=
,
J!
=
'
and have supporting prices pn, #) for the indxerence curve defmed by
skpk,
and a11 agents are indxerent to the introduction of the
49
F(A, J1)
,
then we have constructed a trivial competitive equilibrium. By construction
5s x*, z,l tq =
=
=
z,l a'l, for
all s.
Given smooth preferences for U ) we have supporting
prices
for U5z, Rsj p pn, #1 and this is a trivial competitive equilibrilzm in the contingent commodity space. By the flrst fundamental theorem of welfare economics it is a Pareto optimum. In Chapter 6 we will generalize this example to a class of representative agent models, showing that there are two major routes to Pareto optimality in asset economies, spanning and the construction of a representative consumer. .k'l
,
.
.
.
,
,
.
.
.
,
51
Martingale Pricing Methods
4 Ui
5
!
-
xsi
.
;
and p is the vector of asset prices. From the strict monotonicity of the utility function, we know that V V > 0 (i.e.a11components are strictly positive). Now 1etus introduce the following normalizmtion tricks.
Martingale Pricing Methods
First defme
An alternative and more popular method for asset-pricing is the so-called martingale pricing or risk-neutral pricing method. ne idea is very simple particularly if Ui ) is dxerentiable. ne non-diFerentiable case is treated at the end of this chapter. Consider the consumers' problem:
#nxaf+
(5.1)
Zai)
Maxltxnf;
Z Pkaki
=
1,
.
.
Viz
=
where VI4 is a vector with components 4V Jx s
Vtl
'
is a vector with components
I
-
-
xsi
li
j .
=
1.
Thus the normalized marginal utility vector has the same properties as a probability vector. Furthermore yf represents the marginal rate of substitution between consumption at f 0 and a vector of unit contingent consumptions. Notice the i jubscripts which denote the marginal conditions for the fth consumer. Returning to condition (5.3),we obtain: =
(5.24) .
,
K.
(5.2:)
Conditions (5.2:)can be written more compactly in matrix form: or
-
Clearly qsi > 0, for a1l s; and
J=1
%Uiz= kp
: Ui
qsi>
#
3 Ui kp. :xaf = k
'
.
then desne
k
#,
j
-
*
a
J=l
Z qsi
maximlzm:
=
s (j Z aXsi
M
lf%f.
=
Assuming Ui ) is neoclassical (and dtfl-erentiable), we obtm'n the following necessary conditions for an interior
s aUi Z, X 3xaf
li
p,
Zs xizsqsi
=
#k,
k
=
1,
.
.
.
,
(5.4)
K.
#=l
If we interpret Zsk as the future contingent price of asset k, then (5.4)says that the consumer will optimize, treating the asset prices as a subjectively discounted martingale. (A martingale is a stochastic process where the current value equals the expected future values.) 1 for Given there exists a riskless sset R such that ZsR 1, = + r)becomes: and S; then (5.4) a11s 1, pp (1 =
.
.
.
,
Martingale Pricing Methods
Martingale Pricing Methods
52
Zs (1 +
r)-' Zskqsi
k
=pk,
=
1,
.
.
.
,
(5.5)
K.
Lemma 5.1. If a11consumers have utility functions
uftxfl
J=1
THEOREM
such that
5.1. Giyen a set of
JJle/l
Z akzk
=
0,
=
0.
with non-trivial
(a)
k
then
Z a# k
Proof From
(5.4)
akpk =
k
li Zskqsi
a s
=
E li Zk ukzsk
w
=
0.
s
Notice that neorem5.1 reprodus the reslzlt of neorem 4.1. Of course, we can obtain the same result by considering 111-mj and deriving %, qsj, as the 61-m shadow prices op contingent productions. Most theoretical and applied papers and texts irore the flrm, and concentrate on the consllmer-optimizing problem, even though arbitrage activity in Mancial markets is undertaken largely by frms.
RISK-NEUTRAL PRICING?
Given the characterizmtion of asset pris from conditions (5.4) and (5.5),it is tempting to ask: can we exploit the characterization to price assets as if' there were riskneutral consumers? nis is false in general. It is true that riskif a11 consumers have von Neumann-Morgenstem neutral preferens, then we have:
Z uixs%s,
+
uix
=
The subjective discotmt rates have been equalized because al1 consumers fa the same riskless asset, and each consumer is at an interior maximum. Using this technology, we can reproduce our arbitrage pricing results of Chapter 4:
53
with
lzftx,fl a =
thqn qsi
=
zr, and
pk
=
Z (1 +
+
r)-
pxwf, l
k
Zskzs,
=
1,
.
.
.
,
K.
w
Proof. From the defmition qsi follows by direct substitution.
=
a,,. and the pricing rule
satisfying This result fails if consumers had preferens probsubjective and axioms disagreed on the the Savage abilities, i.e. aaj. # a,f,,. ln this case, we have linear 111diserence curves and corner solutions to avoid unbounded asset holdings with short-selling. But without risk-neutrality, the q vector may bear little relation to the probabilities A. For example, if there is a consllmer with von Neumann-Morgenstern risk-averse
preferences
uftxa)+ such that then
UJ()
>
0,
lz/txall, #
uJ'() <
0,
qsi-vsi) NJ(X1f)Xx(XfW)=
*
'.
Clearly probabilities, attitudes to risk and intertemporal substitution are intertwined in the personplized ArrowDebreu shadow prices. Later we shall see how in various applications further assumptions are introduced to relate all the qi and m.
SPANNING,PARETO OPTIMALITY, AND MARTINGALE PRICING We know from the flrst fundamental theorem of welfare economics that a competitive equilibrium in an Arrow-
Martingale Pricing Methods
54
Martingale Pricing Methods
Debreu economy will achieve a Pareto optimal allocation. In Chapter 4, we showed using the induced-preferen method that complete asset markets allowed us to construct an Arrow-Debreu set of securities; and that by appealing to the flrst welfare theorem the allocation was Pareto optimal in contingent commodities. Here, we will prove essentially the same result using the martingale approach, showing how the personaled prices yf, f can be charactelized in complete asset markets. Z2 5.2 Giyen an asset return matrix (Z1 R'V, allocation will achieve then the equilibrium which spans optimum Pareto in the contingent commodity space; and a THEOREM
,
Tf Xj =
=
w qi qj q, for all pk Z lzskqs. =
=
=
j;
.
.
.
,
q
Condition
-
'ap;
The importance of this result is that in statements (.6) and (5.7) the personalized Arrow-Debreu prices are equalized. Furthermore, from (5.7)we can dedu y and q from market data. That is, given K S assets with rank Zx1 S, then p yqZ, or Wj, =
,
=
(5.8) zq =pZ-3. Now by the Modigliani-Miller theorem we can constnzct ZR 1 from the spanning set of assets, and a riskless asset its equilibrium price #a (1 + r)-1. By (5.5) we deduce 1, know that y (1 + r)- so that (5.8)becomes: =
=
=
=
(1 +
r)-
-
-
=
.
-
-
Desning
-
-
=
=
(5.9)provides
1 + r) d l ( u d l u (1 + r) Pxn2 (1 + rj u d (l + r) # ql u d (j + p.) qz - u u d l 'azv = (1 + r)- qk
(5.6) (5.7)
From the Modigliani-Miller theorem 4.2 we can construct a full set of Arrow-Debreu securities, in addition to the original spanning set, and al1 agents are indiFerent to their introduction. Denoting the Arrow-Debreu prices by Xqs #,, statements (5.6)and (5.7)follow by the Modigliani-Miller theorem. Eliminating the original spanning set of assets (the elimination is welfare-irrelevant), we are left with an Arrow-Debreu asset economy equilibrium. By the &st fundamental theorem it is Pareto optimal.
.
(5.9)
.
a method for deducing ArrowDebreu prices from the t 0 pris, and contingent returns matrix of the K assets. Indeed, it is a generalization of the two-state example in Chapter 4 where we analysed binomial option-pricing, and an alternative derivation of the Arrow-Debreu pricing result. nere we obtained the result with induced preferences. The general Theorem 5.2 can be illustrated with a familiar example. Returning to the binomial option-pricing example of Chapter 4, recall:
Proof
.
r)pz-
=
S
.
= (1 +
55
1
.
-
sve have
'xoz = (1 + and
qk+ qz
=
r)-
i
qz
1.
That is, qk qz are the martingale prices before discounting, We can provide a further example of deriving ArrowDebreu prices, when a stock and a bond do not span the state spa, but the introduction of a third asset will allow spanning. Consider there to be three states of the world (N 3). Let there be a stock, s, a riskless bond, b, and a call option, c, on the stock with an exercise price K psm. Let the states be u (up),m (middle),and d (down).The matrix Z takes the form show in Table 5.1. ,
=
=
56
Martingale Pricing Methods
Martingale Pricing Methods
TABLE 5.1
Asset
b
s State
c
u
ps u
1+
m
ps m ps d
1+ r
0
1+ r
0
.
.
d
-
r
psu
-
K
Clearly the three assets span the three-dimensional state space and the returns are llearly independent. Notice that the call option cannot be priced by arbitrage by taking a portfolio of the stock and the bond, but the call is required to span the asset-return space. We know from (5.9)that the Z matrix can be inverted, and by observin! the current prices of the stock, ps, bond (pri equal to unlty), and the call option, #c, then we can solve for the Arrow-Debreu prices, Given the Arrow-Debreu prices, then we can price any other security. For example, consider another call option on the stock with an exercise price K ps #. The price of tltis option, pcz, will be given by:
57
%.() is non-diferentiable. This appears to be a rninor issue, but recently there has been work on preferences under uncertainty, which are not diferentiable (seeEpstein and Wang, 1992, and Kelsey and Milne, 199248. As we will show, the generalation is relatively easy, so that we can incorporate this recent work on preferens. To begin, consider contingent consumption over two states, and preferences which, apart from non-dxerentiability, are neoclassical. At any consumption bundle x: we can draw an indiFerence curve lxf e Rt lVxij
>
Uf@1)l.
In Figure 5.1( we illustrate the diFerentiable case with a unique gradient V&j@1);
.
=
.
3 zcgs) #cz Z 1 + r qs, =
J1
where Zcz(z)
=
Max
(#x .
s
-
K,
0J, s
=
u, m, d.
This exnmple is a special case of an argument employed by a number of authors (e.g.Banz and Miller, 1978). The most recent version of this idea is contained in Madan and Milne (1992),where the asset-return space is a Hilbert space that allows for ivnite states, but exploits the linear pricing structure.
V U (.e) *
1 l I I I l l
l
l I 1 I I I 1 X* li
NON-DIFFERENTIABLE UTILITY
At the beginning of this chapter we observed that martingale pricing methods can be adapted to the case where
Fig. 5.1a
UjAj l l
r
Martingale Pricing Methods
Martingale Pricing Methods
58
59
consumption bundles, using x: as the reference consump tio bundle. We can write a generalization of
is a subgradient VC@:)
Subgradient cone
N 1 I l I 1 I I l 1 l I I l 1 I * li
(yjxA) l
l
(5.2)and (5.3),where of
(21).
The normalizations to yi and qi can be applied to each subgradient vector in the usual way. Furthermore, Theorem 5.1 remains true: this is not surprising because neorem 4.1 did not require di/erentiable utility. If the assets span the state space, then neorem 5.2 holds. Notice that a11 consumers will have the ArrowDebreu prices lying in their optimal subgradient. This implies that arbitrage arguments on asset-pricing do not require dxerentiable utility-an observation we made in Chapter 4.
r-
Fig. 5.1b
CONCLUSION
and in Figure 5.1(:) we illustrate the non-dxerentiable case where there is a closed, convex cone of subgradients. Now we can modify otzr analysis tp deal with subgradients. Because the preferred set at x:,
This chapter introduced martingale pi icing methods and showed how they could be used to obtain arbitrage pricing methods. The next chapter exploits induce.d preferen and martingale pricing arguments when spanning is not
.t.d@:) lxf e =
I
A'k' Uixib
>
Uf(x:)1
is closed, convex, and non-empty, and x1 is on the boundary of that set, we can use the Minkowski separating hymrplane theoyem (seeVarian, 1992 for a discussion) to show that there exists a Aj
e R'9+1 such that mixi
>
=/x1
for a11xf e
.71(x1).
) is dxerentiable at x: then =i is unique; othemise will be a closed, convex cone of mls. there In economic theory terms, the vector =i is the conmarginal rate of substitution. Or, we sllmer's (generalized) of xi as the consumer's shadow prices over can think If Ui
feasible.
Representative Consumers
6 Representative
Consumers
the case of this economy with a single consumer, treating the replicas in an identical fashion. In what follows we will merely assume I = 1. Consider the optimum problem for the contingent commodity problem'. Max U3(xaj, x1l
at An altemative route to asset-pricing can be obtained when spanning is not feasible. nis subclass of models has evolved in parallel with the arbitrage models, and in many cases the slme pricing formulae can be derived. Consider a simple exnmple of three states of the world and two securities.
s
=
1
2 3
Stock
Bond
3 2 1
1
1
1
Clearly these two securhy returns span a linear subspace of dimension 2 and do not span the whole of R3. Therefore if a security is introduced that is not spanned by the stock and the bond, it cannot be priced by arbitrage. Furthermore, its introduction will not be welfare-neuiral. The reason is straightforward: the new asset will increase the opportunities for contingent trades perturbing the equilibrbzm asset, and contingent commodity allocations, and the asset and personalized contingent prices. Nevertheless, there are cases where the introduction of the new asset will be welfare-neutral and the asset can be prid by the (tmdisturbed) contingent pris. IDENTICALCONSUMERS Assllme that each of the I consumers has identical preferens and endowments. It is obvious that we can analyse
61
A',!
x:,
(yzj, y 1./,
.
.
.
,
.
.
.
xsj).
,
z
x-a, +
=
.V.J.
.j
Z ysj,
=
,
s
1,
=
ysp e Y),
.
.
.
S.
,
1
./
=
,
.
.
.
J
,
By the second fundamental theorem, we know that tMs trivial Pareto optimum can be supported by contingent prices pn, JI ps) such that thre is competitive equilibrium at the optmlm allocation. By constnzction: ,
.
.
.
,
*
.*Y ()l
*
=
X- 1
x-(1I
Z y ()J
+
.
*
5%J .
=
.
J.
x-s ,
x-0
>
s
=
1
,
.
-
-
,
S
.
Now introdgce assets into this economy. Assume that the allocations Rx:j),(J4.)) 1ie in the span of IZj Zk.). Notice that there is no reason why the asset returns should span the full state space in general. In particular, if the economy is an exchange economy with no production, asset returns merely require that there exists (JJ such that ,
x*,l
.
.
.
,
kskk.
=
k
We can introduce the asset markets without altering the consumer's contingent allocation (or welfare) so long as asset-prices are given by S
pk
S
pszsk
=
>a1
=
Z lkzskqsk.
#=l
62
Representative Consumers
Furthermore,
we know from Chapter 5, that (yl, are derived from the gradient of %() valuated at (.%,
.:1
,
.
.
.
,
Xsj.
Therefore, if we can evaluate this gradient at the equilibrillm aggregate consumption, then we have a method for pricing assets. Noti that we can price any asset if we make the assnmption that any additional asset added is in zero net supply. Trivially x,
when
=
*
xs
K =
k=I
z a a-k + z ax+1 '
JI+l
=
0
.
a-x+1
Proof Consider any consumer's indpced problem. Because Uft ) is homothetic, then it is easy to prove that the indud utility function F;.(xaf,a is homothetic. From standard consumer theory, we know that the consumer demands can be written as
a:(#:,
.
.
.
#1)
,
k
=
0, 1,
.
.
.
,
K.
Z akak(#*a,
=
k
.
.
.
,
p*
-
Hij,
.
THEOREM
6.1. Giyen an asset exchange economy, JJJI/ZAIe each that consumer /14..' (i) Neoclassical preferences Ui ) #f(I ) ), where gf I a dfrentiable, strictly increasing funtion; and f7( ) homogeneous of tfegree J,' then the equilibrium c/a/fagent allocation I Pareto optimal Ja# yiqsi ps for all i
for a11
s
=
0, 1
,
.
=
.
.
,
1,- or
x*af= x*.v ()
.
Taking the gradient of Rx*)we obtain a unique vector #,) e # Of Arrow-Debreu prices. Consider the (#:, #1, <#x:, new consumer problem: Max Uifx, subject to p.xi fundamental flrst each By theorem, for th: consumer. Rx)))is a Pareto optimal allocation. By the standard normalization we have yiqsi=ps and yf qsi qs, for al1 i. of Note that the case identical consumers (with nonhomothetic preferences) is really a degenerate case, where no diferences in wealth are allowed. Now we can allow for nne translations of utility to obtain a generalization of the theorem. .
.
.
,
=y,
=
Corollary 6.1. Given the hypotheses of the theorem, let the utilityfunction be Uixij
=
#(P@f+
xij
)
whemthere exists pkfsuch that 5si Z Z,p, k =
=
J.
<ef for
-
Notice that ak( ) is independent of i because 17 is common for a1l consumers. Thus
x* ,f
Although tllis model is ridiculously simple, it has been used extensively to provide pricing for assets that cannot be reprodud by arbitrage. (For example, Rubinstein, 1976; Brennan, 1979; Stapleton and Subrahmanyan, 1984; Madan and Milne, 1992.) In fact, we can generalize this result a little to allow for changes in proportional endowments by assuming that consllmers have ane homothetic preferences, such that apart from the ane component, they are identical. The intuition is straightfomard. Ayart from an endowment scale eFect, the consllmer are ldentical, so that a representative consllmer can be constructed, and the allocation is Pareto optimal. For expository ease we will prove the homothetic case fzrst for an exchange economy, treating the amne homothetic case as a simple corollary.
Ja#
63
Representatiye Consumers wI)
then the conclusions of the theorem follow. Proof By treating the for consumer i, let
pfas a defacto
endowment of assets
65
Representative Consumers
Representatiye Consumers
ak( ) - pf + uki ) where akf is the net trade of assets. The theorem now applies trivially.
that there is an aggregate consumer, we will obtain a unique martingale pricing formula.
As we observed before, we can add assets in zero net supply to tllis economy in a welfare-neutral fasllion withprices. Forout disturbing the Arrow-Debreu (martingale) mally, we have the following obvious result:
The Aggregate
64
Corollary 6.2. Given a representative consumer then any asset can be priced as
pk Z Xzskqs. =
Consumer and Option Pricing
Rubinstein (1976)considers the case where a11consumers have utility functions of the form:
ln(xn)+ lzl(x,)
PREFERENCES AND
We observed in Chapter 2 that in an Arrow-Debreu preferences economy, with von Nenmann-Morgenstern and ane homothetic preferences, then the utility function must be of the fonn:
(x, + a). c 0 ufx
=
lltx,
<
c
<
p j
1 -
j-a
xs
s
.
pk Z Zskliqsi. J
But, we know that with von Neumann-Morgenstern preferences, intertemporal additive separability, and the existence of a riskless asset, qsi
0;
=
'x.
xi
si
'; )a s (z y).
and
= UJ(x1J, assuming pn - l y (1 + r)
i'hus
qsi
and <
+
xn
=
.
=
ulx*si) ?w(1 ultx() .
+ r)
k
uf'(x*f)s (1 +
1
+ Gf)y
lexptxs),
1-a
s
-
=
neorem6.1.
VON NEUMANN-MORGENSTERN AGGREGATION
1
Now recall our pricing equation'.
J
Proof Obvious implication of
1
=
<
0.
Because this utility function is sne homothetic with Arrow-Debreu securities, then more general securities may reduce the applicability of this class. As it turns out, the class is barely reduced; and if we include a riskless asset, the class is not reduced at all. (See Cass-stiglitz, 1970,. Milne, 1979, neorem5.) Therefore, if we mssume that all consumers have utility functions in this class, and the parameters are a1lchosen so
pk Z(1+ =
lz
r)-
.,
'
NJIXQf).
J
=
=
S
z.o
uJ(x*,).as u (x-n) -:
.
.
' ulx*i)
#
z.w (x*()1, ul 'x.
) u. (xa.
ui = Ex Zsk
zf
,
.
.
'a
r)
=
1;
66
Because of aggregation, we obtain'.
l/,(2a) pk sx zskutr (s aj =
& = pEx Zw# #
.
.z
Now Rubinstein wishes to obtain a formula for pricing #), wherep,txl is the call options, i.e. Zg Max (0,#(J) price of a share in state s, pc is the price of the call at t 0, and K is the exercise price. This gives us =
67
Representative Consumers
Representative Consumers
-
able to derive the snme formula, using discrete time and specilk restrictions on a representative consumer. Notice that the Rubinstein derivation does not mssume spanning, or arbitrage, but relies merely on aggregation and the judiciouschoice of utility and probabiliq distributions. Although the derivation focuses on the pncing of a call option, it relies on the constnzction of an Arrowof Debreu measure over the future share price (the claim contingent the world'). Therefore, in principle, any can be priced. <state
=
'c
p E x Max
=
(0,#!(.)
-
.v A') M 5s
The Aggregate
,
Unfortunately, this still depends upon the true probability measure which is yet unspeced. Rubinstein assumes that plsj and Xs are bivariate lorormally distributed, i.e. 1n) and ln# are bivariate normally distributed. Notice that this speccation requires an iMnite set of states, but otlr fmite arguments can be extended in a straightforward fashion by replacing sums with integrals. With some tedious calculations (seeRubinstein, 1976; Huang and Litzenberger, 1988: 163-6) it can be shown that: (5.1) pc ##(Z + cs) (1 + r)- 'KNCZ), =
-
where
Stapleton and Subrahmanyam (1984)extended the models of Rubinstein and Brennan (1979)by using the representative agent model to price complex contingent laims that go beyond the simple call option. The reader should observe that the following theorem is a numerically tractable example of our pore general Theorem 6.1 and its two
corollaries.
6.2. (Stapleton and Subrahmanyam). Given a representatiye consumer with u(x) exptx), and that the Jnfe set of underlying stochastic variables and wealthJ.de/are multiyariate normally distributed, then any complex 1 price pk (1 + r)- EZk), where the Arrow-Debreu prices are derived asfunctions of the means and coyariances of the THEOREM
=
=
#(Z)
=
fztxv-uzlztdu,
l
V(2JQ
--'
is the normal distribution function; cs is the variance of the share price; and Z
Now equation
Consumer and General Contingent Claims
.
e
LnplK)
+
1n(1 + r)
cs
(5.1)is the celebrated
-
2 ns.
Black-scholes (1973) option-pricing equation. Whereas Black-scholes derived the equation using continuous-time Brownian motion for the stock pri and arbitrage arguments, Rubinstel was
underlyingdistribution.
Proof For a prpof and examples, see Stapleton Subrahmanyam (1984).
and
BEYONDVON NEUMANN-MORGENSTERN REPRESENTATIVE AGENTS Recently there has been interest in preference structtlres utility. Epstein
that go beyond von Neumann-Morgenstern
68
Representatiye Consumers
Representative Consumers
and Zinn (1989,1991) analyse preferences tat are more Qexible than von Neumann-Morgenstern and allow for diFerent speccations of risk-aversion and intertemporal rates of substitution. (The additively separable von Neumann-Morgenstern model of Rubinstein, and Stapleton and Subrahmanynm could not allow such a distinction.) Although strictly speaking, Epstein and Zinn have a multiperiod model, the intuition of their result is captured by our two-date homothetic representative consumer: asset prices are derived as the gradient of their recursive utility at the endowment point. An alternative approach has been to avoid choice under risk with objective probabilities, and consider variants of podels under uncertainty. Our framework with preferences over contingent consumption is iexible enough to incorporate either speccation. Consider the recent work of Epstein and Wang (1992).They draw upon work by Gilboa and Scbmeidler (1989),specifying a utility function which is a Choquet integral. In short, this is a utility function with that do not add up to one, and reqect uncertainty aversion. (We will not discuss the details here as they would require an extended treatment.) Bt for our puloses, we observe that with a representative consnmer wlth homothetic preferences of this form, the utility function will not be di/erentiable at the endowment. lt follows directly from our discussion of non-diferentiable preferences in Chapter 5, that we can extend Theorem 5.1 and its corollaries to the non-dxerentiable case and obtain non-unique supporting Arrow-Debreu prices. This is the essen of Epstein-Wang's indetenninancy of asset-pricing.
general homothetic preference relation and its associated supporting Arrow-Debreu prices. Now, decompose the q i.e. vector as a linear combination of a few
69
Efactors',
Z azy= f
q.
With a Enite state space this is relatively trivial, but in innite dimensional spaces the idea of factor decomposition is more complex. Madan and Milne (1992) explore this idea with Hilbert space techniques, using Hennite polynomials as an example. One interpretation of this approach is to assllme the existence of a representative consumer so that the addition of assets in zero net supply will be welfare-neutral. Using a subset of the existing assets to estimate the coecients (ay), and identify the factors qp one can price any new assets as a linear function of a few Arrow-Debreu factors.
.
Kprobabilities'
APPROXIMATINGMARGINAL UTILITY OF A REPRESENTATIVE AGENT
lnstead of exploring diFerent functional forms of utility for the representative agent, we can assume the existence of a
CONCLUSION
In this chapter we have set out the basic tools of the representative agent model for pricing assets. As we showed, the central idea is straightforward, but more complex in application. It is important in these models not to lose sight of the centr>l simple story that drives the pricing result.
71
Divers6cation and Asset-pricing
&
7
=
kfzk'
Z
f=
k
1,
.
.
.
,
A
By arbitrage pricing, factor prices are related to the underlying asset-prices by
Diversification and Asset-pricing
Pf
Z %kfpk.
=
k
Now we can relate factor and tmderlying asset returns by the simple construction of or idiosyncratic returns
e: zk -
So far, we have conntrated on two major methods for asset-pricing: arbitrage and consumer aggregation. In both discussions we have not mentioned one of the nportant traditional topics in fmance theory-oiverscation. In this chapter, we will explore the idea of portfolio diverscation and its implications for asset-pricing. The analysis will be at a fairly abstract level; but it will become apparent that traditional arguments can be included as special cases of the more general frnmework. ne abstract discussion focuses on the essential issues without the distracting clutter of more restrictive assumptions that hide the simple logic that drives the central results. (For a more detailed discussion, see Milne, 1988.)
Consider a consumer with induced preferences over assets and a budget constraint. Assume that the asset returns (Z1 z'xl are linearly independent. (Although the reader can think of these returns to be in R&,the argument is general and can be applied to more general vector spaces.) Now we can create portfolios, priced by arbitrage, Factor to create another set of assets called will be constructed by: returns .
.
.
Z pyk-y+ e
=
,
Kfactors'.
(7.1)
.
z
Pk
(7.2)
+ psk. Z q.tkpf
=
f
So far we have done nothing other than make repeated applications of the generalized Modiglian-Miller theorem. Now to introduce the ide of diverscation across assets, let us introduce some additional assnmptions on the structure of the ek's; and consumer attitpdes io portfolios collections of the e's. that include Let us assnme that there are portfolios which are diverfundiversised'
that is:
la()
A.1. There exists portfolios
ARBITRAGE AND DIVERSIFICATION
f
Again, by arbitrage pricing we can pri the asset k in prices, i.e. deviation the prices and factor of the tenns
sed,
,
Zk
Or
pykFy;
-
thatBV #
0 for-f= 1,
where B
..#7
.
.
.
such that
,
=
E ekal
k
=
0 and
(pv).
The idea here is that consllmers are able to create portfolios which are composed solely of the factors. Indeed, they are able to create portfolios that are any linear .A combination of the factors 1, the idea that the The second assumption introdus undiversifed portfolios. Or, more predislikes consumer with portfolios factor given returns, the isely, snme two .
.
.
,
72
the consumer will desire the portfolio which es away' the e terms. Before we do that it will be convenient to create an ZK,. Fl Fy; el, economy with returns (Zj and pris deMed by arbitrage. Now e we can consider portfolios that are equivalent', but diFer in the addition of nontrivial e risk'.
,
.
.
.
,
,
.
.
.
.
,
.
.
,
A.2. For any consllmer, and any (a) such that
Z alez
*
(uJ),given
non-trivial
F;. xaf;
ale
ayvy
+
0,
k
<
f
UXFF;
Fk.xaf;
one fmal
THEOREM
7.1. Given a consumer that satishes A.2 and A.3; and returns satisfy A.1, and #' the consumer Jlo/ a divers/edporolio at the optimum, then =
f
jjyp).
Proos Given a diversed portfolio (J./.(), then the consllmer will redu utility by holding an undiversed (d/). Thus the gradient of Fk.() with respect to Jf at the optkmlm is zero, i-e.
a Fk.x lf;
Z ae e f
k
apF
k +
Fy
./'
auef
But at the consumer ,s optimum > 0. which implies
= 0, for a11k. : K.()
3JL
=
=
T
qkfpy.
lngersoll (1983). Although this result is interesting in itself, it does not explain why a consumer will hold a diversed portfolio. With one more assumption we can show that every consumer will hold a diversihed portfolio.
XZk ekki
=
0.
i
With this assumption in hand, we can show that all portfolios. consumers will choose well-diversed 7.2. Given A.1, A.2, and A.4 then in a competitive equilibrium, all consumers will hold fully divers6ed port-
We are now in a position to prove the following theorem.
0yfor all k,. and #)
f
then pk qhfpy-pzk,
THEOREM
W.J. F;.() is dxerentiable.
=
=
Note. That this result is a generalization of Chen and
.
f
To derive some pricing reslts we will introdu assllmption:
pk
pzk= 0, for a1l k. Because pk
A.4. Given an asset exchange economy, then the aggregate endowment' of assets is well diversed, i.e.
k
then
73
Diversecation and Asset-pricing
Diyersecation and Asset-pricing
ktpw, with
folios.
Proof We know that an equilibrium is a (constrained) Pareto optkrmm. Assume the equilibrium has consumers holding non-diversiEed portfolios. Then we can always fmd a set of consllmers who can exchange portfolios of idiosyncratic assets and improve their welfare. This is feasible given that the aggregate endowment is fully diver-
siable. 7.1 and 7.2, we obtain the general arbitrage-pricing theorem IAFr).
gvenTheorems
Now
THEOREM
and Jale/. satisfy
7.3. Giyen conwmers who satisfy A.2 and A.3 A.1 and A.4, fAen asset prices will that Ju/f/z
pk Z pyk#y. f =
Proo.f
'rivial
application of
neorem7.1 and neorem7.2.
74
Divers6cation and Asset-pricing
Divers6cation and Asset-pricing
With this general structure in pla, we can show how it tool of analysis. As an illustration can be used as a sexible will introduce the mean-variance analysis of traditional we portfolio theory and show how it can be fhted into our
variables Fm and E, and the riskless asset. In addition, the random variables have been constructed s that Ezj IFmj = 0. It is not dictllt to check that these returns have been constructed to satisfy A.1 and that
general frnmework.
Ek'k
(i.e.A.4).
0
=
k Mean-variance Analysis and CAPM
Consider a consnmer who faces K + 1 assets. The &st assd is riskless, and the remainder are multivariate normally distributed. Given that the consnmer has von NeumannMorgenstern preferences over t 1 consumption (we ignore t = 0 consllmption), we can write: =
Via
=
.
.
.
akizk
ui
+
k=l
1 aoi
(zj
.
,
.
.
,
.
ZdzL
,
.
.
.
,
m
-
E Zka-k k=l
,
where
a-k =
a-ki
where pkp, =covtz.
=
Zk
-
i
fsa
'
1
-
Zk
=
I3ka1 +
qkmFm +
'
pk where pvk 0 and =
'(Z)#0
=
and Fm x Zm-Ezm4.
In other words, every asset return Zk can be written as a linear combination of two normally distributed random
=
Ez
+
qkmFm +
k.
Recalling Fm - Zm
-
then
pFm
By defmition, 1et r
pk (1 + =
r)-
=
1
+#q, qkmpt,m
+
is the price of the market portfolio. ECZm4,and that the price of m is pms
pFm
=
Pm
(,p(f(Z1)
l
+
-
E (Zm)#a.
-
1. Thus
(#,a(1+
r)
-
'(Z,?,)1paJ. (7.3)
If we desne the rates of return,
qkmFm,
zml/vartzm);)a)=EZkj;
Ek
Thus by Theorem 7.3, we deduce that
.
By considering the market portfolio m as one factor, and tlte riskless asset as another factor, we can write E
Kmarket'
dZK,
where/t ) is the multivariate normal density function, with a K vector of means and a covariance matrix of dimension K. The normal distribution has the convenient property that linear combinations of multivariate normally distributed random variables are normally distributed. Furthermore we can construct a market portfolio of the risky assets such that Z
With some extr eflbrt you can show that the induced preferences satisfy A.2 and 3. Thus we deduce from Theorem 7.2 that all consumers will hold diversed portfolios. That is, they will hold only combinations of the riskless portfolio, i.e. the portfolio of a11 asset and the result This is known as the two fund separrisky assets. ation theorem for mean-variance portfolios. To obtain our msset-pricing result, notice that
Rk
=
Zk
-pk
'
then it is possible to show that (7.3)becomes: ER
=
Rz
+
ERmj
-
Aalpm.
This is the celebrated CAPM formula.
(7.4)
76
Diyers6cation and Asset-pricing
CONNOR'S (1984)APT The CAPM is mrhapsthe most famous, and restrictive of the AN family of models: it assumes von Neumann-Morgenstern preferens and multivariate normally distributed returns for the risky assets. In the 1970s Ross (1976) extended the theory by asszlming von Neumann-Morgenstern preferences, but a factor structure on returns, such that the idiosyncratic risks could be diversed away. Subsequently, a nnmber of authors attempted to place tllis argument on a rigorous and more general basis. Connor (1984) assumed dxerentiable von Neumann-Morgenstern preferens and a fador structure of returns. His theory exploited the properties of general equilibrium to show a special case of our general theorems above. In addition, he allowed for a countable number of assets, so that diversifw cation could be achieved by a law of large numbers, rather than nite linear dependence.
APTTHEOREMS WITHOUTVON NEUMANN-MORGENSTERN PREFERENCES In our general proofs (Theorems 7.1-7.3) we did not rely upon von Neumann-Morgenstern preferences, but upon properties of induced preferences over assets. In Kelsey and Milne (1992:)the general framework is extended to an in6nity of assets and introduces non-expected utility prefof the notion of erens. With a careful speccation risk-aversion, and the assumption of a factor stnzcture of returns, they were able to show that the induced preferences w111inherit the properties assumed above. nerefore, Kelsey-Milne have extended the APT to classes of nonexpected utility theory. The reader should note that pure arbitrage results are robust to such generalizations; the
Divers6cation and Asset-pricing
77
complexities are introduced when we try to characterize risk or uncertainty avoidance.
APPROXIMATE APT
The reader should recall (7.2)where an application of the Modigliani-Miller theorem to the factor structme gave us an exact pricing equation withpu not necessarily zero. One can consider this equation and provide conditions under which p.k is small, but not necessarily zero. A simple example of this argument would be where the economy has a representative consumer and A.4 does not hold, i.e. the endowment of the consumer is not diverszed given the assumed factor structure of remrns. (For a more detailed discussion of this argument and referens to related literature, see Milne, 1988.)
CONCLUSION In this chapter we have shown that arbisrage pricing arguments can be extended to allow for diversifable risks. and In equilibriltm, these risks will be ftllly diversable have zero price. I'hus every asset can be priced exactly (or approximately) as a linear combination of a relatlvely small number of common factors.
Multiperiod Asset-pricing
8 Multiperiod Asset-pricing: Complete Markets
79 12
11 22 0
21 32
So far, we have restricted attention to two-date models. Although this is instructive for introducing basic ideas of arbitrage, aggregation, and diverscation, we require a multimriod framework to capture a range of intertemporal problems. For example, we would like to investigate the term structlzre of interest rates, complicated multiperiod derivative securities, the dynamics of stock prices, and dynnmic hedging strategies. It will turn out that our two-period analysis has laid an important foundation for this analysis. By choosing an appropriate dpmmic frnmework, we can generalize our two-date results, and obtain obvious sophisticated reinterpretations of familiar results.
l
Fig. 8.1
at 12, then he/she knows with certainty that 32 will our with certainty, but that ( 12, 21) is impossible. Formally, we can dene the evints lef) as a pafiially ordered tree with a unique ev Now using the same trick as in the two-date model, we can dene contingent commodities, trades, returns, etc., according to which node of the tree we are considering. In this formal sense, a1l we are doing is enlarsng the dimension of the commodity space to E + 1, Efoot'
where E
Et
r
=
t
Z =
1 e
et, =
and Et is the number of events at
f.
1
MULTIPERIOD UNCERTAINR
Recall that we introduced uncertainty by defming continjent conslxmption, production, and asset returns on a simple tree. nis tree structure can be extended to include many dates and states (or events) (Figure 8.1). The events et are partial histories of the world up to time suerAedipg event /. In our exnmple, the tree has ( 12, 22) 11, but the sufvnssor to 21 is only 32. In the flrst case, any agent at 11 w111not know which event ( 12, 21) will occur, but knows that 23 is impossible. Conversely, if'the agent is
THE CONSUMER
Consider a consumer who has a consumption set Xf
=
Rf+1 + A
and a utility function Ui : Xi R. We will asszlme that %.l) is continuous, quasi-concave, strictly increasing, and di/erentiable. -+
Multeriod
80
Multeriod
Asset-pricing
Notice that these assllmptions include some special cases. For example, we collld have von Neumann-Morgenstern preferences; but this is not necessary for much of our discussion. With direct analogy with the two-date problem, consider a full set of Arrow-Debreu markets. That is, we assume that each consumer can buy and sell, on competitive markets, claims for each contingency. Let the pri at t 0, for a unit claim if and only if event et ours, be #e,. Thus, the consllmer's problem is: =
81
Asset-pricing
(i) x: a solution to f/le consumer's problem for all t' (ii) y; a solution to the producer's problem for allj; (iii) x: zf + Z y). =
j
f
./
In general we can apply standard arguments to the existence and optimality of the equilibrblm. Mter all, we have merely expanded the dimension of the commodity spa without altering any of the key assnmptions.
Max Uix
fxfem) pxi < #2f where RZ+ &e
i
l
+
X
(8.1)
qpyj,
1 e Rf+ is the consumer's is hrm j's production vector.
endowment;
and
By direct analogy with the two-date problem we defne the irm's production set Xj c
RZ6
Model: Certainty
xe X
s.t.
The Erm's problem can be written as: Max pyj. 6
Max Uxj
.
=
yj
First, consider a Pareto optimum:
'
Often we will characterize the srm's production teclmology by an implicit production function L.yjj 0. Notice that positive components of yj will be considered outputs, and negative components inputs.
Yj
(8.2)
Putting this all together, we have: 8.1. W competitiye equilibrium in a multeriod 1 economy is a price yector p. e Rf+ and an allocation ((x1),(#J))such that-' Arrow-Debr-
A Single-consumer
Consider a simple economy with a single consumer and certainty, so that the commodity space is Rr+1. We can exploit the second fundamental theorem of welfare economics to characterize the equilibrium of this economy.
THE FIRM
DEFmITION
SPECIAL CASES
;;
=
(8-3)
# + #;
y e F. Notice that we have lumped together all the production activities into one rm'. With standard assumptions on the consllmption
and
production sets we can show that the set of feasible allocations f(x,
.y)
e
RAT+I'
lx
e X, y e F; x
.%
=
+
J,)
is closed and bounded. Given a continuous utility function then, by the Weierstrass theorem, an optimllm exists. Cfhis is just a smcial case of the more general eistence theorem for the existence of Pareto optima.)
82
Multeriod
Multiperiod Asset-pricing
Asset-pricing
Now, if #, F are both convex, and U ) is continuous, increasing, and quasi-concave, then we can apply the second fundmental theorem of welfare economics to show that we can support the optimum @*,y*) with competitive prices: p > 0: i.e.: Max &(x)
px
='.k
(8.5)
For exmple, consider the consumer's problem when we assume U ) is dxerentiable and additively separable over
time:
Max
'u(x,)
z
l=0
s.t.
r
E
z'
ptxt
l=0
=
Z
ptst
+
l=0
r
E ptyt,
f=9
where 0 < < 1, and u' ) > 0, lz''4 ) < 0. From the rst-order conditions for an interior maximum we have:
rt
>
(j
+ t. f, t + s
)-s .
we have that Pt+& pt
or (1 +
Max'.y # e F.
T
By denitin
(8.4)
+py
pt +,1 Pt
Long rates
83
, +
s)-'
=
=
pt+% #f+s-1
(1 +
rf
+
*
s
-
#f+s-l Pt+x-2 1, 1+
s)-1
Pt+L .
.
,.
.
.
Pt .
y
(1 + rt. t +
1)- 1.
(8.6)
This simple relation connects the long rate for any interval to the product of the short rates for the same interval. As we shall see later, this result can be obtained as an arbitrage result from an asset economy trading long and
short bonds. An important question is the determination of the shortand long-term interest rates. Because these rates are simply derived from present value prices (pf), then their determiand nation relies on the properties of the preferens of and production endowment the consumer the set. It is where examples interest in a11 rates construct to vary easy of rises, time, that the structure ways sorts tenn over so falis, or oscillates.
u'(x1) kpt. =
If we postulated a particular form for the subutility function (1n() or expt ) say), mlmerical values for the subjective discount factor e (0, 1) and aggregate consumption' solve numerically for the relative prices. (x1), then we can Of course, this is very restrictive. But what are the relative prices? ne price pt is the price at t 0 for the delivery of a unit of the commodity at time t How is tllis sequen of prices related to interest rates? We can derive interest rates via the following defmi=
.
tions:
2.
A SINGLE CONSUMER MODEL: UNCERTAINTY
A similar analysis can be carried out when the previous model is expanded to incorporate uncertainty. For simplicity, we will dispense with production and concentrate on the consumer's problem, and the interpretation of the flrst-order conditions for an optinmm. The problem now looks trivial: Max Uxj
.
x=7
Short rates
#f + l pt
-
(1 +
rt
'
t
+
j)
-
j
The price vector is proportional to the gradient of th4 consumer's utility function calculated at .
Multeriod
84
Asset-pricing
V&(2)
=
Multeriod
kp.
(8.7)
If we introdu further restrictions on preferences then we obtain some simple flrst-order conditions. Assume that the consumer preferens are von Neumann-Morgenstern and additively separable over time. nat is, the consllmer's
problem is:
T
'x(x(e,))a(e,)
Max t
rt.
=
0 e e Et
XX#(eJx(eJ f
Et
(8.8) #(e,).7(e,)
=
t
=
H$.
Et
It is nportant to realize that the probabilities a can be manipulated as conditional probabilities (formore on this see Huang and Litzenberger, 1988: ch. 7). Now consider the fzrst-orderconditions (8.7)when utility is of the form (8.8):
tu'set) )a(ef) kpetj. =
Of course tMs is just a special case of (8.7),where prices l;(el)l will depend upon
85
Asset-pricing
STOCHASTICINTEREST RATES Returning to our general formulation, we will construct a series of interest rates from the Arrow-Debreu prices @@). Recall the way in which riskless short- and longterm rates were derived in the riskless economy. This derivation did not rely on the simple nature of the economy, but were merely denitions desned on prices. ln the same way we will defme contingent or stochastic interest rates. Given an event et, consider an immediate suessor event et + 1. Dene the set of suessor events to et to be St + 1 Ie,), so that et+ I e St + 1 1e,). Defme the contingent short rate of return as 1)
#(e,+ (1+ petj =
r(e,+
,
le,))
-1
.
Notice that this rate is not riskless in the sense that it represents the rate of return at et for an asset that pays one unit if and only if et + l e St + 1 Ietj occurs. To obtain a riskless interest rate contingent on et, we deee ',) pt + 1 I
#(e,+l),
x
st
+
I
1 ep
and
ll(eJl, lstel of the subutility function u ). By and the speccation imposing more structure on u ), the probabilities, and endowments, it is possible to obtain closed-form solutions for prices. For exnmple, by assuming that u(x) lnx and of the probability density it is particular speccations possible to obtain simple formulae for prices (seeRubinstein, 1976). As an aside, we should observe that many authors treat the consumer problem via dynamic programming techniques. nis is a clumsy way of tackling this problem as it disguises the elementary nature of the consllmer's problem. =
pt + 1 Ie,) (j + #(e) denition rt + 1 le,) is
,t
+
j.
j e,)l-l.
In this a conditional short-term interest rate. As we move through the tree it will hange in fashion, i.e. given et, then w will know the a short rate. But prior to et the short rate will depend upon which future node eventuates. In the same way, contingent long rates can be deMed. Given et, consider a bond that pays oll- a unit of the commodity in successor events St + T Ie. Defme <predictable'
pt + Iet) ':
-
E
el..se st
pet+xj. +
s IeJ
86
Multeriod
Asset-pricing
Then the contingent long rate for t +
Multeriod given et will be
:,
dehned by:
pt
le,) (1+ . #(eJ +
-r
r(f +
s je,)!
-v
.
r(f +
PRICE NORMALIZATION
(1+
rt
l
l e,)-
+
l
=
st
E l
+
1)
#(e,+ Iep Pe
f)
defnes the short rate of interest between et and t
1.
+
87
Our derivation of q ) was deMed recursively over successive dates; but it is easy to extend the argument to non-suessive dates. Consider et and a date t + %, < > 1. Recall St + 1 le as all those suezvssor events at t + I that can be reached from :f; and
(1+ Given the Arrow-Debreu prices, we can generalize our martingale pricing idea, by appropriate normalization rules. To begin, consider an event et with t < F, with immediate suessor events St + 1 Iet). Now recall
Asset-pricing
then
qet I +
s e
s jeflj -s .
st
z
#(e,+s)
+ Tlep
#(eJ
;
+s)
-
pet pet
(1+
r(f +
*
1e,)! s
.
Clearly the construction of q ) follows a11 the nlles of conditlonal probability, although q ) may not match th underlying probabilities. Having defmed short- and long-term interest rates, and derived martingale prices from Arrow-Debreu prices, we close this chapter by discussing two special economies that provide closed-fonn solutions for interest rates.
Now de6ne l
#(e,+ (1+ # (e,) 1)
qet l +
et)
l
r(f + 1 e,)l
for
ef + 1 e
st
+
l
1 e,).
If pet are strictly positive then the derived q ) is pe well de6ned and strictly poyive. Also, by construction +
A REPRESENTATIVEAGENT KONOMY
I),
st
Z
+
l
1 ep
qet-v
l
Ie,)
=
1,
so that q4 has al1 the promrties of a conditional probability measure. Notice that we have met a smcial case of this construction before when we discussed the two-d>te martingale pricing methods with complete markets. We observed that, in general, there did not have to be any simple relation between the q ) prices and true probabilities over the states, i.e. the true stochastic process. Of course, that observation applies to our more general multidate construction.
Earlier in this chapter we discussed a single (representative) consumer economy, where the Arrow-Debreu qrices were obtained from the gradient of the utility functlon at the endowment point. By restricting preferens and the endowment, i.e. the stochastic process describing the endowment, it is possible to obtain closed-form formulae for the interest rates and the q's. Turnbull and Milne (1991) assume the HARA class of utility, and that the endowment growth follows a Gaussian autoregressive process. Using this model they are able to obtain the Arrow-Debreu pris, interest rates, and q's in terms of parameters of the stochastic process. Of course, if the preferens are not diflkrentiable, then the Arrow-Debreu prices, interest will be non-unique. Epstein and Wang rates, and (1992) 's
88
Multiperiod Asset-pricing
provide a mtlltiperiod example of such a model where the representative consllmer has non-exmcted utility.
General Asset-pricing Complete Markets
PRICES AND PRODUCTION @
The representative agent model of asset pricing has been used widely in the literature. But the alternative route of pricing from the production side has seldom been used. Here we will skeych how such a pricing argument can be
constructed. Consider an economy where the 6r1:1 (or firmsl has constant returns to scale teclmology that satises the non-substitution theorem (see Varian (1992),ch. 18). At any event et assllme that there are coecients atef + j letj of production that $ve the productivity of a umt of the commodity at et, in producing the commodity at et. !. ln equilibrium, frms will earn zero prohts and the ArrowDebreu prices *1 be determined by the production coecients. From the defmition of the interest rates and the q's, one obtains a pricing and interest rate structure independent of the consumption decisions of consumers. By specifyingthe evolution of the a( )'s, one can determine the stochastic pross of interest rates and q's.
in
To introduce more complex assets with multiple pay-oFs, we require a construction that introduces asset markets explicitly. At the same time, the construction shotlld be relativelysimple to reveal the basics of the general argument. For expositional easej consider a three-date world with an event tree that describes a binolnial process (Figure 9.1). Notice that there are seven nodes. It will become clear that our rgument is quite general and can be extended to multinomial and multidate information trees. To keep the argument simple, consider an exchange economy where consumers can trade in Arrow-Debreu assets at t 0. Thus a consumer's problem is: =
11 12
22
CONCLUSION Having introdud the multiperiod normalization and schemas nomy rates and martingale prices, we turn a more complicated asset structure
9
0
Arrow-Debreu ecofor deriving interest in the next chapter to in a complete market
32
21 t
=
0
Fig. 9.1
42
90
CompleteAsset-Markets
CompleteAsset-Markets
Max Uixij
By the construction of R, %Uiwill have a unique solution (9.3).Denote MUi p, wllich is the vector of ArrowDebreu pris. It follows immediately that because asset markets are complete, the economy has an equilibrium which satisfes the flrst fundamental theorem of welfare economics. That is, (9.3)simply says that all consumers equate their marginal rates of substitution-a necessary condition for a Pareto optmlm.
s-t. xf(0)
kpetjaietj
.k'j(0)
=
-
(9.1)
ef
xf(e,) kietj
aietj, Yet.
+
=
Problem (9.1)can be written in a more compact form, using vector and matrix notation'.
&f(xJ,
Max
s.t. xi
(9.2)
ADDITIONAL ASSETS
a
=
(cf(11),
.
.
.
.
uf(42)1
and R is a matrix (7 x 6 in our example), where the colllmns represent the positive or negative pay-oFs across events, and the columns represent the dxerent assets
(Table 9.1). 9.1
Asset
11 0 11 Pay-off
over events
=
k'i + Rai,
=
where
TABLE
in
91
21 12 22
32 42
-
p(11) 1 Q 0 0 0 0
21 -
p(21) 0
-
p(12)
1 0 0 0
0 Q 1 0 0
0
0
-
p(22) 0 Q 0 1 0 0
32 -
p(32) 0 (1 0 0 1
0
=
0.
9.2
42 -
p(42) 0 Q 0 0 0 1
It is an easy exercise to show that the consumer's sequence of linked budget constraints can be collapsed to a single budget constraint, so that the consumer's problem redus to the consumer's problem discussed in Chapter 8. Now, consider the flrst-order condition: for an interior solution to the consllmer's problem:
YUA
-
TABLE
22
12
We can introduce more complex securities,by adding them to our base Arrow-Debreu asset economy. As an exnmple, consider an asset k' which is traded for the flrst time at event 11 with a price pk. (11),and has yay-ofat events 12 and 22 of Rk.(12) and Ak.(22) respectlvely (Table 9.2). This asset k' has a pay-of vector
(9.3)
##
0 11 21 12 22 32 42
-
0 pk,(11) 0 Rk-(12)
Rk-(22) 0 0
Enlarging the R matrix to include asset k' we can write down the hrst-order conditions for the consumer as in (9.3). lt follows nmediately that foy asset k' we have
# TRk, g; =
or, expanding the equation:
92
Complete Asset-Markets
#(11)#'(11) =
#(12)A'(12)
CompleteAsset-Markets
+#(22)R,(22);
0r
93
THEOREM
9.1. Giyen a competitiye equilibriumfor the tzxe/exchange economy Defnition 9.14, then 4/ the column rank of R less than the zllwl:er of columns (i.e. there are dependent a-e/ returns), then there a linear vltla//fz of equilibriumasset allocations, and any dependent tzue/.& K can be priced by .
#ke(11) =
#412)Rk. (12)+ #(11)
#(22) A.(22). ,411)
Recnlling our construction of the discounted ArrowDebreu q ) from the previous chapter, we have: pk'
1 (1 + r(2 11))
=
l
((12 ll 1)Rk.(12)
+
qll
ll 1)Ak'(22)).
This particular pricing formula is a generalization of our binomial pricing formula from Chapters 4 and 5. Later in this chaptey, we will show how such analysis can be undertaken recursively to generate the multiperiod binomial option-pricing formula. We can genernlize our model to allow for additional assets with complex pay-oFs by the simple exmdient 6 adding asset return vectors to the R matrix and adding asst trades to the asset vector ai. nus a mpditive equilibrium in our asset economy can be defmed ms follows: DEHNITION
9.1. W competitiye equilibrium in a complete asset-exchange economy isprice-returns matrix R, consumption plans
X, and an
(,:,
(i) (ii)
E
i
=
1,
.
.
.
,
I)
:
solves =
MM
s-t. xi
TRg
=
(),
whereP the vector ofArrow-Debreu prices. Proof From the defmition of the equilibrillm economy and the rank condition on R, then clearly there are non-unique equilibrium asset allocations (a linear manifold) solving x:
=
ki
+
Rq3
and
=
i
0 for al1 (a:).
From the hrst-order conditions for any consnmer (9.3)we have pTRk = 0.
ASSET ECONOMIESWITHFIRMS
This theorem can be extended to allow for firms and production. ln an obvious fashion, consider 6rms to have the problem: Max pTxj
allocation
JJJe/
such that
i e f),
#
Vxi)
=
Tf + Rai;
0.
i
Given a commtitive equll'ibrium, we can prove an exchange version of our Modigliani-Miller theorem.
s.t. i) x, yl + Raj (0 h e Yj'. =
(9.4)
ln words, the hrm maximizes its net present value given its constraints on production and its asset portfolio. Noti that because in equilibrilxm pTR 0, problem (9.4)collapses to the standard Arrow-Debreu problem: =
Max pj s.t
.
yj e Yj.
95
Complete Asset-Markets
CompleteAsset-Markets
Given the 6rm's problem, we can generalize our Defnition 9.1 to a production economy.
perfectly hedge the risks of an asset or cash Qow. ln a two-date model, this is a relatively simple procedure, but for multiple dates we may require complicated contingent undertake the hedge. The second P ortfolio strategies to aspect of Theorem 9.2 is the deduction of pris from the hedging procedure. To illustrate these arplments consider the event tree in Figure 9.1. Let the events be described by uncertainty about the return on a share. At each node (beforethe nodes at t 3) a share price can go either up u', or down d'. We can set out the return matlix for the case of shares that follow an umdown process; and a.sequence of riskless bonds with a constant rate of interest. For the moment we will ignore the Arrow-Debreu securities as represented in Table 9.1.
94
competitive equilibrium in a complete asset economy with production, a price matrix R, an asset
9.2.
DEFINITNN
,4
.
allocation
(, and ctlzl-lwlpfion
i e Ija;
j
,
andproduction
J)
,
vectors
@1,i e I )(7Jt
such that'.
B
.f
,
e J)
(i) @1, ) solves the consumer problem;
(J,J,#) solyes (iii) 41 E JJ; (ii)
the Jr?'nproblem;
=
=
./
i
(iv) E xl i
=
TABLE 9.3
Zi si Zn%. ./
S(0)
As an elementary extension of Theorem 9.1, we obtain the full Modigliani-Miller theorem'. Given a competitive the (Dejnition 9.23, then 4T th-ef economy equilibriumfor rank the number ofcolumns (i.e. ofR is less than the column there are dependent tz-ef returns), then there is a linear manfold ofequilibrium asset allocations, and any dependent avez. k cla be priced by THEOREM
Asset
+
9.2 (Modigliani-Miller).
#
ra,
=
gj
wherep is the Arrow-Debreu price vector. MULTIPERIOD HEDGINGSTRATEGIES AND ARBITRAGE PRICING
There are two important aspects to Theorem 9.2. The flrst involves the constnzction of arbitrage-free portfolios that
Pay-off
over events
0 11
21 12 22
32 42
-
ps ups dps
S(1 1) S(21) -
0
0
ups
0 dps 0 0
0
0 0 0
u2p udps
0
0
0
-
djps ddps
B(0) -
B(11) B(21)
1
(1 + r) (1 + r) 0 0
0 0'
-
0 1 0
(1 + r) (1 + r) 0
0
0 0 -
1 0 0
(1 + r) (1 + r)
The reader can check that the Arrow-Debreu return matrix (Table 9.1) and the share-bond matrix (Table 9.3), both span the pay-ol tvents. Indeed, we can rtlate the Arrow-Debreu prices to the given garnmeters u, d, and r, in a generalization of our discusslon in Chapter 4. ne easiest way to see this is by applying a backward-recursive pricing argument. Consider the node 11. ne conditional Arrow-Debreu price at 11 to buy 1 unit of the commodity at 12 is given by:
Complete Asset-Markets
96
(1 + r) d 11) (1 + r)-' (12 1 # N- d
Complete Asset-Markets
-
=
>
(1 + r)
-
j
l.
'Clearly, the hedge is almost exactly as in our discussion in Chapter 4. (As the reader can check, there is an additional u term.l By the snme hedging argumnt we deduce u (1 + r) u d -
,(22 l11) (1 + =
1
r)-
.1
>
-
(1 + r) qz.
it is easy to see that the ArrowBecause qb + qz Debreu prices have the same prmrties as a binomial stochastic process. The spnmetry of the process, and the description of events by the stock pri imply that the events are dened independently of the order in which tTl1is is called path indeand the our. pendence.) Therefore, we have ,(22) #(32). It is not hard to show (seeCox, Ross, and Rubinstein, 1979; or Huang and Litzenberger, 1988: ch. 8) that our three-date model can be extended to many periods, t 0, F.' Furthermore, the snple binomial argument ex. and F n tends so that the event of n
<downs'
=
=
Now consider the event node 21. By the same argument we
deduce
.
.
,
Edowns'
Eups'
I
#432 21)
I
#(42 21)
=
=
(1
1 (1 + r)-
(1 +
u u
1
r)-
+ r) -
d
(1 +
-
u
-
d
-
r)
d
=
r)-ll,
=
r)-
(1 +
r)
-
(1 +
r)
-
qt
,
j
qz
.
l
r)-
l1)2
=
r)-2g).
(1 +
Similarly, we deduce that
yvhereq -
(1+
r)-
qb and
(1
rl-zjz,
=
rj-lqzqt
=
r)-2l.
psundT-nj
F!
T
a!(F -
-
a)! q
n
(1 qj r-a -
,
q4 - qz.
So far we have considered Arrow-Debreu securities and shares and bonds. But our general framework allows us to price any pay-oF stream. Recall the call option introduced in Chapter 4. We can extend that simple two-date analysis to many dates. From the general dehnitiop of a call option, the return to a call option at matmity date F, with exercise undT-n is: price K, in fnal state RC
=
#(22) (1 + #(32) (1 + ,(42) (1 +
price is
jhare
OPTION PRICING
'a.
I
=
-
has an Arrow-Debreu price of -
Finally we can chain these arguments together by the obserkatlon that a dynsmic portfolio of the (11 10) and (12 11) securities will provide the same tmit pay-oF as the ori/nal Arrow-Debreu security (12)in Table 9.1. nus by neorem 9.1 or 9.2 we deduce that
#(12) ( (1 +
(i.e. the
..j
.
-
Having analysed the contingent Arrow-Debreu prices at 11 and 21, we can retreat to the initial node 0. By the same argument we obtain:
,(11 l0) (1 + ,(21 l0) (1 +
97
=.1,
,
undT-nj
-
Max
(0 >
psundT-n
-
A!
*
From Theorem 9.1 or 9.2 we know that the call option can be priced by arbitrage and its return replicated by a dynamic hedge of the share and bond. Thus the price of a call at f = 0 is:
Complete Asset-Markets
98 #e(0) (1 + =
r)- TEq
(.X)
r
T
= (1 + r)-
r
a=0
Complete Asset-Markets
a!(F
r! -
a)! q
n
ndp-nj (1 qj r-a Ar(u -
.
Now it is not dicult to show (see Cox, Ross, and Rubinstein, 1979,*or Huang and Litzenberger, 1988: ch. 8) that this formula can be rewritten as:
rc(0) #,*(./; F. 3 #(1 =
-
r)-F*(./;
+
r, *,
( 9.5)
where (1) j is the minimum positive integer such that K
y 2> 1n T
(2) *(./; F, qj (3) q
'
-
(1 +
>
Z
n=j
lls6lz
r!
!(w
-
a)j q
- 1
.@
1n Kl
;
n
) r.- a.
(1
-
.
r)-1u.
Equation (9.5) is the binomial option-pricing formula derived by Cox, Ross, and Rubinstein (1979). It is possible to show that ap the number of trials per unit time increases the ntral limit theorem can be invoked to show that (9.5)converges in an appropriate sense to the lebrated Black-scholes formula (see Cox, Ross, and Rubinstein, 1979).
MULTINOMIAL MOQELS
It should be obvious that the binomial option pricing formula can be leneralized to a multinomial pricing forShefrin Mllne, and Madan, mula. (1989)consider the case suerAssors for any non-terminal where there are exactly n node et. By assuming a independent asset returns, one can derive the Arrow-Debreu prices for the tree. Assllming
99
constant interest rates, it is not dicult to construct a multinomial version of the binomial formula (9.5).By taking appropriate limits it is possible to obtain Brownian motion or Poisson jump limits to obtain a generalization of the Cox, Ross, and Rubinstein analysis.
CONCLUSION Having constructed a complete market economy that is supported by Arrow-Debreu prices, we turn to the case of multiperiod, incomplete asset-markets.
Incomplete Asset-Markets
101
10
12 11
Multiperiod Asset-pricing: Incomplete Asset-Markets
22 0
21 32
So far, we have discussed a multiperiod economy with complete asset or Arrow-Debreu markets. In this chapter we will introduce a general structure that allows us to charactelize incomplete or incomplete asset-markets. Wi do this by introducing asset marke' ts that may or may not span the full event space CE+ 1). Let there be a #) of assets that can be traded at diferent set K ( 1, events. By representing asset returns as a matrix R of dimension CE+ 1) x K, we can represent returns or dividends as positive components, and pris paid as negative components of the matrix. In other words, we can use our asset return matrix R from Chapter 9, even though the asset markets are incomplete. Consider the tree structure we discussed in Chapter 8, Figure 10.1. Assnme that there are only three assets; asset k = 1 can be bought at 0 for a price #j(0) and held until f = 2 where it pays (Rl (12),R1(22))and zero elsewhere. At date t 1 it pays nothing: this can be interpreted as saying that the market for asset 1 is closed at that date, and reomns at t 2. The second asset, k = 2, is not traded at 0, but opens for trade at t 1, event 11, where it can be bought for #c(11). Subsequently it pays returns (Rz(12), Ra(22)) at t = 2. Finally, asset k 3 can be bought at t = 0 for #a(0) and returns Aa(11) at event ell, and zero =
.
.
.
t
=
0
Fig. 10.1 This pay-of
matrix R can be represented
in Table 10.1.
TABLE
10.1 k 2
,
1
0 11
et
as set out
-
p, (0) 0
-
21
0
12 22 32
R, (12) R) (22) 0
0 p2(11) 0 8a(12) R222)
-
3 pa(0) 8a(1 1)
0 0 0 0
0
Now #ven this general structure we set out the consumer's problem:
=
=
=
=
elsewhere.
Max Uf@fl R'i+ Rai rt. K. .ri e Xi ; a i e R .r/
=
( 10.1)
Assuming an interior optimum, xt, then the flrst-order conditions for a maximum are: ,
MVR
=
0.
(10.2)
102
Incomplete Asset-Markets
Incomplete Asset-Markets
Notice that R has positive and negative elements. For example, asset 2 in Table 10.1 has flrst-order conditions :V : Ui : Uf Rz(12) + Rz(22) ,2(11) ?xf(12) :xf(11) 3xf(22) Later we will interpret these fzrst-order conditions to constnzct martingale-type conditions on asset-prices and returns. To avoid problems in dening the objective function of the f5rm with incomplete markets, we will restrict our discussion to an exchange economy.
Now if you recall our example set out in Table 10.1, and rewrite condition (10.2)for the three assets, we obtnin:
=
.
DEFmITION
10.1. In a multiperiod asset-exchange economy an equilibrium returns matrix R and an allocation (@:, ) Vl1 satisfes:
(i) (x:, ) satisjes the consumer's problem (10.13: (1 (111)
i
,
i Ai az
i
J: f
.
i
ARBITRAGE-FREE ASSET RETURNS
The equilibrium asset-returns matrix R excludes arbitrage possibilities, otherwise, we would not have a competitive equilibrium. That is, there is no portfolio a e Rf such that Aa > 0. But we can introduce the idea of perfect substitute portfolios, and provide a generalization of the
Modigliani-Miller theorem. Assumption. For some k' there exists G
such that Rk.
e Rf-1 akRk,
=
k
y
(Asset 1)
(Asset 3)
4 3 V .Rl (12) V + Al(22) 3xf(12) 3xj(0)
Jl(0)
(Asset 2) #z(11)
=
? Ui 3.Yf(22)
3V 3V 4 Ui + R2(22) Ac(12) 3xf(22) 3x/(12) 3xf(11) =
? V R3(11) 3 Uf 3xf(0) axi (11)
#3(0)
=
'
If the asset returns for 1 and 2 are equal (i.e. R!(12) Rz(12), RI (22)= Rz(22) ) then manipulating the tllree conditions, we obtain: =
#z(11) ,1(0) =
Ra(l1)(#z(0) )-1.
.
Thus, the price of asset 2 at 11 equals the price of asset 1 times the gross rate of rett!rn for asset 3. Clearly this is an arbitrage result, in the sense that if this equality did not hold, an arbitrage oportunity would our. As the reader can check, with the above equality, there are two eqivalent portfolios: aj hold one unit of asset lj b) hold one upit of asset 2 and one unit of asset 3. Thls implies that our matrix R in Table 10.1 is of rank 2. Given this example to motivate the idea of arbitraqe we ca generalize our two-date and multidate Modighani-
Miller theorems:
asset economy, giyen rf7zlk 10.1. In a multeriod non-trivial linear manfold of equiliis then there R K a which all consmers are inbrium asset allocations oyer
THEOREM <
difrent.
Proof. Given tl* consumer opportunity sets
x:
=
k'i + Rai?
JI j
i
1,
=
.
.
.
,
conditions
and the asset market-clearing
k'
where Rk denotes the ktb colllmn of R.
103
*. =
0
,
f;
104
,
.
.
.
,
10.2. Given an asset market equilibrium with rank E, then the equilibrium contingent c/l?n allocation is
THEOREM
we have a set of linear equations in (J1
105
Incomplete Asset-Markets
Incomplete Asset-Markets
(R) Pareto tpp/l'al/. Proof From condition (10.2)MVR 0. By the rank conditions, there is only one solution p to PR 0, i.e. I and scalar j. But we know MV. kip, for a1l i = 1, condition for Rx:))to and sllcient that this is a necessary be Pareto optimal (seeVarian, 1992: ch. 17). =
c:).
As the reader can check, the null space of these equations is non-trivial. Clearly tis generaliyrs our Modigliani-Miller and provides a framework for any dynnmic arbitrage/pricing result. Notice that over time redundant or derivative asset markets can be opened or closed without altering the contingent consumptions @:) of the consumers, or their welfare.
=
=
=
.
.
.
,
From the theorem we have the condition that pRk 0, K. As we will see a little later, this for every k 1, reinterpreted condition can be as martingale pricing, using normalized Arrow-Debreu prices as the probabilities and discount factors. =
=
.
.
.
,
PARETO OPTIMALITY We have from our discussions of the Arrow-Debreu economy that Svenstandard assllmptions on preferences etc., any Pareto ogtimal allocation can be supported by a commtitive pnce system. ln particular, we have %Ui = kip 1. Now the interesting question is: what for all i 1, ari sllcient conditions to ensure Pareto optnal allocations? We will show that a full set of asset markets appropriately dened, or consumer aggregation are sllcient to enstlre Pareto optimality. Given these conditions then the asset-pricing condition V%.R 0, reduces to PR 0. =
.
.
.
,
=
=
CONSUMER AGGREGATION AND PARETO OPTIMALITY
An alternative route to Pareto optimality is to consider a representative consnmer. The following' theorem generalizes our aggregation result in the two-date economy. 10.3. Given a multeriod each consumer has:
THEOREM
#/ > (i) &f@f) #f(P@f)), =
(ii) ki
=
Mf .
Jxzef
) homothetic;
0,
Vroportionalendowmentsj
then there is an equilibrium where.. ( xl Mf2 for alI i; =
b4 2, FULL SPANNING AND PARETO OPTIMALITY
We know from the two-date and multidate problems with spanning of the contingent claims spa, that the equilibrium in the asset economy is a full Pareto optinmm. For completeness we report what we discovered in Chapter 9.
=
0. for all f;
(c) V(/(.)
=
p ;
(#) the allocation
Proof From
((x:)Jis Pareto
(10.2)we ;
optimal.
have
MlhRai
economy where
+
h4R
=
0.
xt, Rai
But
=
+
ki
=
existing set of assets, but it does not change the endowment of the representative consumer # = x*.
Mjz,
which implies MhRai
+
2f)
=
VP(a'l
V-)
=
.p.
Clearly the allocation is an equilibrium with the properties
aj-dt.
We can extend the theorem to the case where the endowments 2j can be considered as an endowment of assets, i.e. there exists Rf such tllat si Ri. fe Corollary 10.3. Given the hypothesis of the theorem, but =
U@f) f
=
kilhh
+ xf)
), Tf Af, f0r some =
f
e Rf,
then the conclusion of the theorem follows.
INTRODUCINGASSETS
Given that the contingent allocation is Pareto optimal (either because of full spanning or consumer aggregation), consider the introduction of new asset K + 1 with return vector #x+ 1 and in zero net supply. Intuitively the introduction of this asset will not perturb the allocation, so that in the new equilibrium the asset will be priced such that PRK. 0,. and a11 consumers will continue to consume ! (19. Noti that the sour of this result diFers between the spanning formulation and the representative consumer formulations. With full spanning, the K + 11 asset is redundant because its pay-oF structure can be replicated by arbitrage. ne representative consumer version allows an asset to be introduced which is not spanned by the =
107
Incomplete Asset-Markets
Incomplete Asset-Markets
106
PERSONALIZEDMARTINGALEPRICING Just as in the two-date model, we can introdu a normalization trick on the marginal utilities to convert them into personaled probabilities'. That is, the normalized marginal utilities will have the same structure s personalized conditional probabilities or Arrow-Debreu prices. To begin, divide Vf by the flrst component
: Uf :xf(0)
Ui.
,
so that we get:
4 Ui axi(e;
where Ufe,and
Jje, .
where
V.z
-
-
l
ftef
.
.
,
&ftl
.
dene
X sfl,u
,
NOW
Xit 0)
Uis + g fa
Ui()
- x V&/
re, (yf(f 0) ) I v:
l0)
Uiet =
n tlin
,
Ufe, Uin l
-
I
-
1
Xit 0),
-
zit lojqfgtj(p,
(yf(fl0) )
-
I .
Given strictly incremsing utility, then by construction qiet l0) > 0, and
f(e,I0) 15 1()
=
1.
st
<prob-
(ftdf I0)J is a personalized Arrow-Debreu ability' price over the nodes of the tree at t. By similar teclmiques we can construct conditional mrsonalized Arrow-Debreu prices. Consider a date s < t and et is a node reached from beginning at es. Dene
That is,
108
Incomplete Asset-Markets
lit Ie
le,
EIe; Ve, .s'(f
-
Incomplete Asset-Markets
,
wherest Ie,4 is the set of events reached at t, from the evente,. By monotonicity of preferences and deEnition, &/e, qiet Ie - tazn i ea (:f(f IeJ) > 0, and s(f jey)qiet Ie,) 1. j
-
obtain the results by embedding the ssme argument in an incomplete market structure, where the share pris are insensitive to many of the events observed. For example, consider the share at eac node to have a pay-oF strucmre, at suessive nodes, as follows: state
=
1 2 !
Clearly, qiet le,) has the required conditional Yrobability' structure. Recalling the condition %UiRk 0, then it can be written as
state
=
Rke
+
X Xit Ie
t
>
s
Rketjqiet lrtflea)
Ie,)
=
0,
gation,
vitles)
=
zt le,),
qiet
Iea)
=
qet
Iea) for a1l f.
That is, the personalized element disappears from the Arrow-Debreu prices, and we obtain the Arrow-Debreu martingale prices of Chapter 9. Using our general construction, we will sketch a series of basic models that exploit arbitrage or aggregation to price assets. ne &st two examples assume that interest rates are non-stochastic; the second two allow for stochastic interest
rates.
1. Constant Interest Rates and Arbitrage ne
snplest exnmple of this idea is the formulation of Cox, Rqss, and Rubinstein (1979). In Chapter 9 we outtMs option-pricing model Ened binomial in the context of with complete market binomial event tree. But we can a a
s'
s +
psu psu Psu psd
1
!
! S
(10.3)
where e, is the flrst time/event that the asset is traded. Notice that if there is flzll spanning or consumer agve-
109
Up states Down states
psd
Technically, we say that the share price is measurable with respect to the coarse information (x, dt rather than S(. nus the binomialthe ner infonnation ( 1, option pricing technology can be used even though the background uncertainty, and other asset pay-ofs, are more detailed than the coarse (u, dj information. Of course this type of argllment can be generalized to multinpmial spanning arguments, where there is a bmsic set of assets that span a linear subspa of asset returns. Any new asset in that span can be priced by arbitrage. This is just an application of the Modigliani-Miller neorem 10.1. .
.
.
,
2. Constant Interest Rates and the Representative Cosumer
In Chapter 6 we discussed a series of models with a representative consumer in a two-date incomplete market setting. In the original papers, the models were multidate so that one can obtain multidate versions of asset pricing as a natural extension of the two-date model.
110
Incomplete Asset-Markets
Incomplete Asset-Markets
3. Stochastic Interest Rates and Aggregation Theorem 10.3 and the asspciated asset-pricing equation (10.3) can be exploited to obtain a general asset-pricing equation with stochastic interest rates:
#k(e,)
=
X (1 +
t
>
s
I
r(f ea))-'
R(el)(el .%/
Ie,)
lea). (10.4)
Noti that by aggregation, the Arrow-Debreu prices ) q are indemndent of i, and deducible in principle from the representative consumer's mar/nal rate of substitution. Because we have assllmed stochastic interest rates, and (implicitly) a full set of contingent long bonds, the Nrsonalized intertemporal marginal rates of substitution (1 + rt ld,))-*, for all consumers. 'it l/,) In general (1.4) can be applied to a wide variety of homothetic preferens, but the easiest case to consider is the HARA class of von Neumann-Morganstern utility. By choosing an appropriate member of the HARA class and a stochastic process generating the aggregate endowment Turnbull-Milne (1991)were able to produ an ArrowDebreu measure q ) that is sllciently tractable to generate closed-form solutions for a range of asset prices. For exnmple, they provided closed-form pricing formulae for options on long-term bonds and securities. =
111
simultaneous development of these pricing equations parallel the work of Rubinstein (1976)in obtaining the BlackScholes option-pricing model from consllmer aggrgation.)
DISCRETE NUMERICAL METHODS We have argued that it is possible to obtain closed-form equations for asset valuation from our general fmite-timeevent tree model by appropriate modcations and simplcations. But it has become apparent that closed forms are the exception rather than the rule, and that many pricing problems defy closed-form solution concepts. This is where the discrete model is useful as a procedure that can be programmed on a computer. Much of the most rezvnt work on asset-pricing on Wall Street uses dixrete tree methods to simulate arbitrage strategies, and compute Arrow-Debreu asset prices.
FACTOR PRICING AND DIVERSIFICATION IN A MULTIPERIOD ASSET ECONOMY
<exotic'
4. Stochastic Interest Rates and Arbitrage We know from Theorem 10.2 that we can dedu the pricing condition (1Q.4). ne two important issues lying behind this pricing relationship are: (i)the derivation of the ortfolio that replicates the return stream of asset K; and (1i)the dehvation of q4 to price the asset. Heath, Jarrow, and Morton (1992)obtained closed-form pricing formulae almost identical to those derived by Turnbull and Milne (1991) by using a continuous time dxusion model. Cfhis
In the last three chapters we have discussed arbitrage and
representative agent models to obtain general pricing formulae, but we have not discussed factor pricing and diverscation. This modifcation is straightforward in that we can adapt our general AFI* discussion from Chapter 7 by the simple trick of redoing the analysis conditional on event node et. That is, we can obtain conditional AFI' results at each event node. This methodology has been used widely to analyse stochastic interest-rate models, derivative pricing, and representative consumer models where pris are bmsed upon dynnmic factors. For exnmple, the models of Heath, Jarrow, and Morton (1992)and
Incomplete Asset-Markets
Incomplete Asset-Markets
Turnbull and Milne (1991)exploit this factor intemretation in discussing stochastic interest rates and derivatives based upon bond prices. (For a general survey and synthesis of these ideas, see Milne and Turnbull, 1994.)
standard fmance models. Much of the analysis can be seen as a natu' ra1 extension and generalization of ollr two-date models.
112
INTRODUCINGTHE FIRMINTOINCOMPLETE ASSET-MARKETS So far we have avoided introducing firms into the incomplete market setting. With incomplete markets, proht is no longer well defmed and the objective function of the firm is problematic. Nevertheless, one lnight argue that the 11ny1 has an objective function that we can encapsulate via a utility function Uyxp over the net (cash);ow xi e Rf + 1. ne net cash ;ow is a residual from the production decisions and msset trades of the finn. As a natural extension of the model of complete markets, in Chapter 9, we obtain: Max Uyxp
s.t. (8
xj
=
yj
+ RJy;
(f0 & e Y). lf the rm's preferences are neoclassical then it is easy to extend our analysis in this chapter to incorporate arbitrage arguments emanating from 6rms. Of course, in the special case of a representative-consumer ecopomy, U) ) should be interpreted as p h, the present value of the fll'm where the Arrow-Debreu prices are derived as supporting prices for the optimal allocation. .
CONCLUSION
With the multimriod, incomplete market economy, we have constructed the most general asset economy used in
113
Conclusion
Conclusion
In this book. we have surveyed the central ideas underlying rezvnt asset-pricing models: arbitrage, consumer aggregation, and portfolio diversifcation. We have discussed these ideas in a sequence of increasingly sophisticated models, showing how they can be adapted and illustrated with well-known hnance models. In addition we have indicated in passing how recent papers can be incorporated into the frnmework and can be considered special cases of general
theorems.
It is possible to extend the basic ideas in a number of directions to incorporate more sophisticated interactions. For example, we have mssumed a single commodity, but it is relatively straightfomard to allow for many commodities and spot commodity markets. This allows for: ( real production decisions that incorporate labour and physical capital goods in the flrm's planning problem, including discussions of real options; and b) labour, education, and other interpretations of the consumer's intertemporal decision-making. A ond extension introdus money so that the assetpricing results can be adapted to nominal returns. This allows dired comparisons with macroonomic models with representative consumers. In addition one can analyse asset returns using arbitrage arguments with nominal returns and the existen of index bonds. A variation on this model introduces multiple currencies so that by taking a partition over agents (countries)the model can be treated
115
as a framework for analyslg intemational nance and asset-pricing. In discussing arbitrage asset-pricing we used the idea of decomposition. TMs vector decomposition can be factor a generaled by introducing a more sophisticated notion of a factor that incomorates an underlying probability measure. ne idea is to construct an orthonormal basis such that random returns can be written as a linear combination of the orthonormal basis. nis construction provides a direct discrete analogue with continuous-time formulations, and taking appropriate limiting arguments one can obtain the continuous-time models in the literature. (For a detailed discussion see Milne and Turnbull (1994).) Other variations can be introduced that generalize the model in a more realistic and complex manner. Some of the general results are altered or destroyed, but the basic geometric tools continue to be applicable. Two exnmples of these variations are: a) the introduction of taxation; and b) the possibility of transaction costs in trading assets. These topics are active research areas, but it is possible at tMs stage to discern how the general models can be adapted to provide useful and instructive results.
B ib Ii o g ra p h y
Arrow, K. (1963),'The Role of Securities in the Optimal Allocation of Risk Bearing', AeWew ofEconomic s'fefez,31: 91-6. for State Contingent Banz, R., and Miller, M. (1978),Tris Claims: Some Estimates and Applications', Journal of Wlzfnesh 51: 621-52. Bernstein, P. (1992),Capital Ideas: The Improbable Origins of Modern Wall Street (New York: Free Press). Financial MarBhattacharya, S., and Constantinides, G. (1989), kets and Incomplete Information: Frontiers ofModern Financial Theory, ii (Totowa, NJ: Rowman & Littleeld). Market Equilibrium with Restrided Black, F. (1972), Borrowing', Journal ofll/xfaeu., 45: 44*55. Pricing of Options and and Scholes, M. (1973), Journal ofpolitical Liabilities', Economy, 3: 637-54. Corporate Contingent Claims in Pricing of Brennan, M. J. (1979),d'Fhe of Finance, 34 (March); Discrete Time Models', Journal 53-68. and Kraus, A. (1976),The Geometry of Separation and Analysis, 11(2): Myopia', Journal ofFinancial and Quantitatiye 171-93. Cass, D., and Stiglitz, J. (1970),K'T'heStructlzre of Investor and Asset Returns, and Separability in Polfolio Preferen Allocation: A Contribution to the Purc Theory of Mutual Funds', Journal ofEconomic Theory, 2: 122-60. Chamberlain, G. (1983),'Ftmds, Factors and Diversication in Arbitrage Pricing Models', Econometrica, 51: 1305-23. and Rothschild, M. (1983),EArbitrage, Factor Stnzcture Analysis on Large Asset Markets', Ecoand Mean Varian 1281-30k. nometrica, 51: fExact Pricing in Lhear Chen, N. F., and Ingersoll, J. (1983),' Factor Models with Finitely Many Assets: A Note', Journal of Finance, 38: 985-8. Ecapital
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Economic Theory, 34: 13-31. Cox, J., Ross, S., and Rubinstein, M. (1979), Pricing: A Simplifed Approach', Journal of Financial Economics, 7:
229-63. lngersoll,J., and Ross, S. (1985*,W Theory of the Term
Structure of Interest Rates', Econometrica, 53: 385-408. (1985*. fAn lntertemporal General Equilibrium Model of Asset Pris', Ecbnometrica, 53: 363-84. Debreu, G. (1959),Theory of Value (New Haven, Conn.'. Yale University Press). Dismond, P. (1967),ne Role of A Stock Market in a General Equilibrium Model with Teclmological Unrtainty', Amer-
iclzlEconomic Aevfew, 57: 759-76. Dllme, D., and Huang, C. (1985), Arrow-Debreu Equilibria by Continuous Trading of Few Long-laived Securhies', Econometrica, 53: 1337-56. Eqllilibria: Existen, Spanning Nlzmber (1986), stochastic the RNo Finaniial Gain from Trade'' HypoExpected and Econometrica, 54(5): 1161-84. thesis', Epstein, L., and Tan Wang (1992), Asset Pricing under Knightiaa Unrtainty' (University of Toronto, Dept. of Economics, WP No. 9211). and Zinn, S. (1989), Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A neoretical Framework', Econometrica, 5744): 937-69. (1991), substitution, msk Aversion and the Temporal Behavior of Consllmption and Asset Returns: An Empirirml Analysis', Journal ofpolitical Economy 99(2): 261-86. Fama, E. (1970), <Ecient Capital Markets: A Review of Theory and Empirical Work', Journal ofFinance, 25(2): 383-417. Gilboa, l., and Schmeidler, D. (1989),EMnxmin Expected Utility w1:11non-unique Prior', Journal of Mathematical Economich 16: 141-53. Grossman, S. (1976),'On the Eciency of Competitive Stock Markets where Traders Have Diverse Information', Journal of Finance, 31: 575-85. Hanison, J., and Kreps, D. (1979),VMartingales and Arbitrage in Multimriod Securities Markets', Journal ofEconomic Theory, 20: 381-408. flmplementing
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Markowitz, H. (1959),Porfolio Selection: Escient Dfverl/cltion oflnvestments @ewYork: Jolm Wiley & Sons). Merton, R. (1973, 'An Intertemporal Capital Asset Pricing Model', Econometrica, 41: 867-87. (1973*, W neory of Rational Option Pricing', Bell Jourpa/ ofEconomics and Management Science, 4(1): 141-83. Modigliani-Miller Propositions after Miller, M. H. (1988), Journal ofEconomic Perspectives, 2(4): 99-120. Years', 30 tcorporate and Finance Theory in lnvestment F. MMe, (1974), Record, 50: 511-33. Equilibllm', Economic Commtitive Default Risk and Economies: Over Asset (1975), of Financial Economich 2(2): Comorate Leverage', Journal Ene
165-85. (1976),VDefault Risk in a General Fzplilibrium Asset Economy with Incomplete Markets', International Economic Re#few 17: 613-26. ,
Preferens, Linear Demand Functions (1979), and Ag#egation in Competitive Asset Markets', Review of Economic Studies, 4643): 407-17. Selling, Defalt Risk and the Existen of (1980), Equilibrium in a Secmities Model', International Economic Review, 21(2): 255-67. . tlndud Preferens and the Theory of the Con(1981, sumer', Journal ofEconomic Theory, 24(2): 205-17. (1981, Vhe Firm's Objective Function as a Collective Choi Problem', Public Choice, 37: 473-86. (1988),EArbitrage and Diversication in a General Equilibrium Asset Economy', Econometrica, 56: 813-40. and Shefrin, H. (1984),clarifyingsome Misconptions abput Stock-Market Economies', QuarterlyJournal of Economich99(3): 615-27. and Smith, C. (1980)rapital Asset Pricing with Proportional Transaction Costs', Journal of Financial and QuantitativeWv/ylfa, 15(2): 253-65. and Turnbull, S. M. (1994), neoretical Methods for Security Pricing', rnimeo (Queen's University, Kingston, Ont.). Modigliani, F., and Miller, M. (1958),t'I'he Cost of Capital, and the neory of Corporation Finance', Corporate Finan American Economic Review, 48: 261-97. fconsumer
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fne
Eoption
t'rhe
INDEX
aggregation: consumer 18-20; and multimrie asxt-pridng 105-6 and interest rates 110-1 1 and reprexntative consllmers 65-7 and von Neummm-
Morgenstern pmferens *-7 under arbitrage AFI'. s e -pricing
arbitrage: and asset-pricing: induced preferen approach 37-499 *11 option pricing 45-7; complete asset markets 47-9; Iirm leverage with default risk 43-5 and asset-pricing, general 94-7 betwn perft substitute assets 17-18 and diversication 70-6 -free asxt returns 102-4 and history of nance theory 6-7, 9-l 1 and interest rates 108-9 and martingale pricing methes 52 -pricing theory (AFI) 6, 10-11, 73, 7*7, 111 Jee also Modigliani-Miller Arrow-Debreu theory lArrow, K. and Debreu. G.): and arbitrage and asset-pridng 45-8 and general asset-pricing 89,
91-9
and history of snance theory 3, 5, 7, 9-10, 11 and incomplete markets with preuction 31, 32 and martingale pricing methes 53-6, 59 and multimrie asxt-pricing 8X1, 85-.8 and multimriM ncqet-pricing in incomplete markets 1*, 1(*-5, 107-8, 11*12 and reprexntative consumers 63-4, 67, 68-9 and two-date meels 12 asset-pricing, see arbitrage; diverscation; general asset-pricing; incomplete markets; multimriod asset-pricing; representative consumers; two-date models asymmetric information 12-13 Banz, R. 56 Bernstein, P. 6, 8, 9 Bhattacharya, S. 9 Black, F. 6 Black-Aholes formula 7, 10, 47, 66, 98, 111 borrowing and incomplete marke? 32-5 Brerman, M. J. 22, 62, 67 option pricing 45-7 capital asset pricing model ICAPMI 7, 74-6 Cass, D. fW *11
124
Index
rtsinty 12, 83-4 Chamlxrlaizls G. 10 Chen, N. F. 73 Goquet intep'al 68 commodity spas, see
diverszcation and asxt-pricing 70-7 AFI- theorems 76-7 arbitrage 70-6 mean-varip analysis and
contingency commtitive equilibrium 13-14 Jlo Arrow-breu see complete markets 8 and arbitrage and nmqet-pridng47-9 see also general asset-pricing; two-date models and under Connor, G. 10, 76 Constantinides, G. 9
asRt-pricing
also preferens;
representative consumers contingency 60-1, 67, 79 Cox, J. 45, 108 and general asset-pridng
97-8,
99 and history of nance theory 7, 10 Debrem G. 14-15. 31-2 theory see cllo Arrow-breu default risk, Erm leverage with
43-5
Diamond, P. 5, 31
discrete mlmerical
methods
Due,
D. 10
ecient market.s hypothesis (EMH) 8 Epstein, L. 57, 67-8, 87 equilibrium 31 see also commtitive
factor pricing and multiperiod msset-pricing111-12 Fama, E. 8
105-6 and general asset-pricing 89-90 incomplete markets w1t1: preuction 28-9 martingale pricing methods 50 multimrie a-t-pricing 79.-80,81-4 two-date meels 13, 15-17 Jee
CAPMPG multimfie 111-12
equilibrillm expted utility function 21-2
consumers: agpegation: in complete markets model 18-20; and multimriM
lndex
111
nan theory, see history rms: asset conomies with 93-4 leverage with default risk 4>5 multimrie asset-pridng 80-1, 112 two-date meels 13 Fisher separation theorem 23-4, 32. 45 failure of 24-5 general asset-pricing in complete markets 89-99 additional assets 91-3 asxt onomies with l'mK 93-4 multinomial models 98-9 multimriod hedging stratees and arbitrage pricing 94-7 option pricing 97-8 general contingent commodity spas 35-6
Gilboa, 1. 68 Grossman, S. 8
Harrison, J. 8, 10 Harq 0. D. 8, M Heath, D. 10, 110, 111 hedging stzatees 94-7 HilYrt spa 56, 69 Hirshleifer, J. 5 history of nan theory 3-1 1 in 1950: 3-4 in 19*s 5-6 in 1970s 6-9 in 1980s 9-11 Huang, C. 9, 10, 66, 84, 97-8 Huberman, G. 10
identical representative consumers 60-4 incomplete markets 7-8 with production 27-369 basic meel 27; consumer 28-9; equilibrium 31; Fisher separation theorem in asset onomy 32; more general contingent commodity spas 35-6; optimality and welfare theorems 31-2,. preucer's problem 29-30; short-sellingand borrowing 32-5 see also under multimriod asset-pricing induced preferen approach, Jee asset-pricingtmder arbitrage Ingersoll, J. 10, 73 interest rates 10 and aggregation 110 and arbitrage 108-9 and multimriM asxt-pricing
85-6 and representative 109
consllmer
Jarrow. R. l0, 110, l 11
l25
Kelsey, D. 57, 76 Kralzs, A. 22 Kreps, D. 1, 8 Leland H. 9
leverage with default risk 43-5 Lintner, J. 5 Litzenberger, R. 9, 66, 84, 97-8 Lucas, R. 8 Madnn, D. 10, 56, 62, 69, 98
margnl utility of reprexntative consllmers, approximating 68-9 Markowitz, H. 4, 5 martinale pricing methes 50-9, 86 and history of 6nsnce theory 8, 9-10 non-diFerentiable utility 56-9 personnlized 107-11 risk-neutrality 52-3 spalming and Pareto optimality 53-6 mean-varianm analysis 4, 5,
74-6 Merton, R. 6, 7, 10 Miller, M. 43, 56 Jee also Modigliani-Miller Milne, F.: and arbitrage and asset-pricing 37, 41 and diverscation and asxt-pricing 70, 76-7 and general asset-pdcing 98 and history of flnance theory 6, 10, 11 and incomplete markets witil preuction 29, 32-3, 35-6 and martingale pridng methes 56, 57 and multimriM asxt-pricing 87, 110, 112, 1l5
126
lndex
Milne, F.: contdj and repreatative consumers 62, 69 and two-date meels in complete marke? 21-2, 26 Minkowski thxrem 58 Modigliani-Miller theorem tModigliani, F. and Miller, M.)
and diverscation and a-t-pricing 71, 77 and general asset-pridng 92-3, 94 and history of nance theory 4, 5, 9, 11 and multimrie aat-pricing in Zcomplete marke? 103-5, 109 Morton, A. 10, 110, 111 Mossin, J. 5 multinominal models of general
amqet-pricing98-9 mttltilxrie asxt-pricing 8 in mpl marke? 78-88,. twfainty 81A nmlmer 79-80, 81-* 6r1n 80-19 pri nommlizntion 86-7,. pri= and preuction 88,. reprvntative agent economy 87-% ngle nmlmer meel 81% sthmstic intert rat 78-9, 83-4 8s-6; llavzertainty in incomplete markets 100-13; arbitrage-free asset returns 102.% discrete ntunerical methes 111; factor pricing 111-12; and diverstion flrm 112; intrGblcing asxts 106-79Pareto optimality ItW-6; Ierstmnliyitd
martingale pricing 107-1 1 multimrie heging sate/es and 94-7 general n-t-pricing
lndex
non-dferentiable utility 5G9 normaliyation, pri 86-7 optimality: and incomplete markets with preuction 31-2 also Pareto optimality option pricing 9, 55 call 45-7 and general asset-pridng 97-8 and reprexnutive consumers Jee
65-7
Pareto optimality: and arbitrage and asset-pridng 49 and diverszcation and asxt-p rid g 73 and histo? of finnnc- thxry 4, 7 and martingale pricing niethes M and multimrie asxt-pricing 8l, 104 6 and reprexntative conszlmers 61. 63 and s-nning 53-6, 1*-5 and two-date meels 14, 16 mrfect substitute assets, arbitrage betwn 17-18 personalizzd malingale pricing
107-11
Pliska, S. 10
preferens, Jee von Neumann-Morgenstern price, see asset-pricing production: and consumption 1*17 and pris 88 produr's problem and incomplete markets with preuction 29-30 see also incomplete markets with production
127
proft muimization reasons 23-4 plzre exchange 15-16
short-xlling and inmplete markets with prblction single nmlmer mYel and
Radner, R. 8
Slutsky 13, 29 Smith, C. 6, 7 spanning and Pareto optimality 53-6, 1*-5 special utility function 20-1 Stapleton, R. 62, 67, 68 Stiglim J. 64 sthastic interest rates 10 and aggregation 110 and arbitrage 110-11 aad multimrie msxt-pricing 85-6 Subrahmanyam, M. 62, 67, 68 substitute mssets, perfect, arbitrage betwn 17-18
representative agent economy and multimriod assetpricing 87-8 representative consumers 60-9 approximnting margnal utility of 68-9 identical 60-4 and interest rates 1 von Neumann-Morgenstem preferens and aggregation CW-7;beyond 67-8 risk-neutrality, see martingale pricing Roll, R. 6 Ross, S. 45, 76, 108 and general asset-pricing 97-8,
99 and history of Enance theory 6, 7, 10 .
Rotbmhild, M. 10 Rubinstein, M.: and arbitrage and asset-pricing 45 and general asset-pricing 97-8,
99 and history of nance theory 7. 9, 10 asxt-pricing and multimrie 84, 108, 111 and reprexntative consllmers 62, 65-7 Samuelson, P. 8 Schmeidler, D. 68 Scholes, M., see Black-scholes separation theorem, see Fisher Sharpe, W. 5 Shefrin, H. 25, 32, 45, 98
mlzltimritxlRmqet-pridng
32-5 81-4
Tan Wang 57, 68, 87 Tllrnbull, S. M. 10, 11. 87, 110, 112, 115 two-date models: complete markets 12-26 characterizing prices 17-20 failure of Fisher theorem 24-5 proft maximiyntion reasons 23-4 see also utility function unrtainty 12 multimrie 78-9, 83-4 utility: nondxerentiable 56-9 and two-date mMels: exlvted 21-2; smcial 20-1 Jlao marginal util see Varian, H. R. 1, 58, 88, 105 and two-date meels in complete market.s 13-15, 19, 23
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