ford University Arexa. Walton Street. Oxford oxa Oxford New' York
FINANCE THEORY AND ASSET PRICING
6DP
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* Frcnk Milne. 1995 All rhu relervez No Nrf of t/1d.,publication zn/y & reproduced, stored f?la refrfeu/ system. or trumitted. a anyform or by ly? means. without1& prior permission fn writing ofoxford (arnfveryflyPress. Bsl/ll the UK, exceptiona are allowed in respect tl-fly fairdealingfor the J/udy. or criticism or review. J. permitted purN'e ofresearch or private .zazlr the Copyright, Delfg'm * Patents Act, 1988, or 1 /& case of ofthe licences l accordance with the fere reprographicreprogction by 1/1e Copyright Liceuing Agency. Enquiries concerning z'ttxz-d repro*ction tll/f.lkfe these lerzn. tz?ld a other countries should be Oxford Uniyersity 'rer, sent ftl the AgAt. Department. nrlrltess feye at JAe
Frank Milne
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J-'JAr/rAof Congress Cl/fl/tlgag n Publication Data Milne. Frtza. F''nnnce theory J?ltf =et Jrcag / Frcn Milne. #. cm. l'aclzzzH bibliographical rferences tz?ld index. mo&ls. 2. Cclf/cl tt.re/a pricing 1. F''v.zre--Afcle-tfca/ f. Title. 5G774.M554 1995 #4-JJOJK 332-ak20
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)
=
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,
01 K
=
X Zskaki + k=l +
r
pnxni k=1
.
.
.
x-,f,
pkau
1,
=
.
.
.
,
S
(3.1)
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)
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
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
=/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 () 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 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
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
(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
'l'he
118
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<Martingales and Stochastic Integrals and Pliska, S. (1981), in the Theory of Continuous Trading', Stochtic Processes and Their Applicationh 11: 215-60. of Equilibrium in a Hart, 0. D. (1974), g d
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