Theory of Financial Decision Making Jonathan E. Ingersoll, Jr. Yale University Preface In the past twenty years the quantity of new and exciting research in finance has been large, and a sizable body of basic material now lies at the core of our area of study. It is the purpose of this book to present this core in a systematic and thorough fashion. The notes for this book have been the primary text for various doctoral-level courses in financial theory that I have taught over the past eight years at the University of Chicago and Yale University. In a11 the courses these notes have been supplemented with readings selected from journals. Reading original journal articles is an integral part of learning an academic field, since it serves to introduce the students to the ongoing process of research, including its mis-steps and controversies. In my opinion any program of study would be amiss not to convey this continuing growth. This book is structured in four parts. The first part, Chapters 1-3, provides an intro- duction to utility theory, arbitrage, portfolio formation, and efficient markets. Chapter 1 provides some necessary background in microeconomics. Consumer choice is reviewed, and expected utility maximization is introduced. Risk aversion and its measurement are also covered. Chapter 2 introduces the concept of arbitrage. The absence of arbitrage is one of the most convincing and, therefore, farthest-reaching arguments made in financial economics. Arbitrage reasoning is the basis for the arbitrage pricing theory, one of the leading models purporting to explain the cross-sectional difference in asset returns, Perhaps more impor- tant, the absence of arbitrage is the key in the development of the Black-Scholes option pricing model and its various derivatives, which have been used to value a wide variety of claims both in theory and in practice. Chapter 3 begins the study of single-period portfolio problems. It also introduces the student to the theory of efficient markets: the premise that asset prices fully reflect all information available to the market. The theory of efficient (or rational) markets is one of the cornerstones of modern finance; it permeates almost all current financial research and has found wide acceptance among practitioners, as well. In the second main section, Chapters 4-9 cover single-period equilibrium models. Chap- ter 4 covers mean-variance analysis and the capital asset pricing model - a model which has found many supporters and widespread applications. Chapters 5 through 7 expand on Chapter 4. The first two cover generalized measures of risk and additional mutual fund theorems. The latter treats linear factor models and the arbitrage pricing theory, probably the key competitor of the CAPM. Chapter 8 offers an alternative equilibrium view based on complete markets theory. This theory was originally noted for its elegant treatment of general equilibrium as in the models of Arrow and Debreu and was considered to be primarily of theoretical interest. More recently it and the related concept of spanning have found many practical applications in contingent-claims pricing. ii Chapter 9 reviews single-period finance with an overview of how the various models complement one another. It also provides a second view of the efficient markets hypothesis in light of the developed equilibrium models. Chapter 10, which begins the third main section on multiperiod models, introduces mod- els set in more than one period. It reviews briefly the concept of discounting, with which it is assumed the reader is already acquainted, and reintroduces efficient markets theory in this context. Chapters 11 and 13 examine the multiperiod portfolio problem. Chapter 11 introduces dynamic programming and the induced or derived singleperiod portfolio problem inherent in the intertemporal problem. After some necessary mathematical background provided in Chapter 12, Chapter 13 tackles the same problem in a continuous-time setting using the meanvariance tools of Chapter 4. Merton’s intertemporal capital asset pricing model is derived, and the desire of investors to hedge is examined. Chapter 14 covers option pricing. Using arbitrage reasoning it develops distribution- free and preference-free restrictions on the valuation of options and other derivative assets. It culminates in the development of the Black-Scholes option pricing model. Chapter 15 summarizes multiperiod models and provides a view of how they complement one another and the single-period models. It also discusses the role of complete markets and spanning in a multiperiod context and develops the consumption- based asset pricing model. In the final main section, Chapter 16 is a second mathematical interruption- this time to introduce the Ito calculus. Chapter 17 explores advanced topics in option pricing using Ito calculus. Chapter 18 examines the term structure of interest rates using both option techniques and multiperiod portfolio analysis. Chapter 19 considers questions of corporate capital structure. Chapter 19 demonstrates many of the applications of the Black-Scholes model to the pricing of various corporate contracts. The mathematical prerequisites of this book have been kept as simple as practicable. A knowledge of calculus, probability and statistics, and basic linear algebra is assumed. The Mathematical Introduction collects some required concepts from these areas. Advanced topics in stochastic processes and Ito calcu1us are developed heuristically, where needed, because they have become so important in finance. Chapter 12 provides an introduction to the stochastic processes used in continuous-time finance. Chapter 16 is an introduction to Ito calculus. Other advanced mathematical topics, such as measure theory, are avoided. This choice of course, requires that rigor or generality sometimes be sacrificed to intuition and understanding. Major points are always presented verbally as well as mathematically. These presentations are usually accompanied by graphical illustrations and numerical ex- amples. To emphasize the theoretical framework of finance, many topics have been left uncov- ered. There is virtually no description of the actual operation of financial markets or of the various institutions that play vital roles. Also missing is a discussion of empirical tests of the various theories. Empirical research in finance is perhaps more extensive than theoret- ical, and any adequate review would require a complete book itself. The effects of market imperfections are also not treated. In the first place, theoretical results in this area have not yet been fully developed. In addition the predictions of the perfect market models seem to be surprisingly robust despite the necessary simplifying assumptions. In any case an un- derstanding of the workings of perfect markets is obviously a precursor to studying market imperfections. The material in this book (together with journal supplements) is designed for a full year’s iii study. Shorter courses can also be designed to suit individual tastes and prerequisites. For example, the study of multiperiod models could commence immediately after Chapter 4. Much of the material on option pricing and contingent claims (except for parts of Chapter 18 on the term structure of interest rates) does not depend on the equilibrium models and could be studied immediately after Chapter 3. This book is a text and not a treatise. To avoid constant interruptions and footnotes, outside references and other citations have been kept to a minimum. An extended chapter- by-chapter bibliography is provided, and my debt to the authors represented there should be obvious to anyone familiar with the development of finance. It is my hope that any student in the area also will come to learn of this indebtedness. I am also indebted to many colleagues and students who have read, or in some cases taught from, earlier drafts of this book. Their advice, suggestions, and examples have all helped to improve this product, and their continuing requests for the latest revision have encouraged me to make it available in book form. Jonathan Ingersoll, Jr. New Haven November 1986 Glossary of Commonly Used Symbols a Often the parameter of the exponential utility function u(Z) = −exp(−aZ). B The factor loading matrix in the linear model. b Often the parameter of the quadratic utility function u(Z) = Z −bZ 2 /2. b k i = Cov(˜z i , ˜ Z k e )/(Cov(˜z k e , ˜ Z k e )). A measure of systematic risk for the ith asset with respect to the kth efficient portfolio. Also the loading of the ith asset on the kth factor, the measure of systematic risk in the factor model. C Consumption. E The expectation operator. Expectations are also often denoted with an overbar¯. e The base for natural logarithms and the exponential function. e ≈ 2.71828. ¯ f A factor in the linear factor model. I The identity matrix. i As a subscript it usually denotes the ith asset. J A derived utility of wealth function in intertemporal portfolio models. j As a subscript it usually denotes the Jth asset. K The call price on a callable contingent claim. k As a subscript or superscript it usually denotes the kth investor. L Usually a Lagrangian expression. m As a subscript or superscript it usually denotes the market portfolio. N The number of assets. N(·) The cumulative normal distribution function. n(·) The standard normal density function. O(·) Asymptotic order symbol. Function is of the same as or smaller order than its argument. o(·) Asymptotic order symbol. Function is of smaller order than its argument. p The supporting state price vector. q Usually denotes a probability. R The riskless return (the interest rate plus one). r The interest rate. r ≡ R − 1. S In single-period models, the number of states. In intertemporal models, the price of a share of stock. s As a subscript or superscript it usually denotes state s. T Some fixed time, often the maturity date of an asset. t Current time. t The tangency portfolio in the mean-variance portfolio problem. U A utility of consumption function. u A utility of return function. V A derived utility function. v The values of the assets. W Wealth. W (S , τ) The Black-Scholes call option pricing function on a stock with price S and time to maturity of τ. w A vector of portfolio weights. w i is the fraction of wealth in the ith asset. X The exercise price for an option. Y The state space tableau of payoffs. Y si is the payoff in state s on asset i. Z The state space tableau of returns. Z si is the return in state s on asset i. v ˆ Z w The return on portfolio w. z As a subscript it denotes the zero beta portfolio. ˜ z The random returns on the assets. ¯ z The expected returns on the assets. 0 A vector or matrix whose elements are 0. 1 A vector whose elements are 1. > As a vector inequality each element of the left-hand vector is greater than the corresponding element of the right-hand vector. < is similarly defined.  As a vector inequality each element of the left-hand vector is greater than or equal to the corresponding element of the right-hand vector, and at least one element is strictly greater.  is similarly defined.  As a vector inequality each element of the left-hand vector is greater than or equal to the corresponding element of the righthand vector.  is similarly defined. α The expected, instantaneous rate of return on an asset. β ≡ Cov(˜z, ˜ Z m ). The beta of an asset. γ Often the parameter of the power utility function u(Z) = Z γ /γ. ∆ A first difference. ˜ε The residual portion of an asset’s return. η A portfolio commitment of funds not nomalized. Θ A martingale pricing measure. ι j The Jth column of the identity matrix. ˜ Λ The state price per unit probability; a martingale pricing measure. λ Usually a Lagrange multiplier. λ The factor risk premiums in the APT. υ A portfolio of Arrow-Debreu securities. υ s is the number of state s securities held. π The vector of state probabilities. ρ A correlation coefficient. Σ The variance-covariance matrix of returns. σ A standard deviation, usually of the return on an asset. τ The time left until maturity of a contract. Φ Public information. φ k Private information of investor k. ω An arbitrage portfolio commitment of funds (1  ω = 0). ω A Gauss-Wiener process. dω is the increment to a Gauss-Wiener process. Contents 0.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Matrices and Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . 4 0.3 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.4 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 Utility Theory 1 1.1 Utility Functions and Preference Orderings . . . . . . . . . . . . . . . . . 1 1.2 Properties of Ordinal Utility Functions . . . . . . . . . . . . . . . . . . . 2 1.3 Properties of Some Commonly Used Ordinal Utility Functions . . . . . . . 4 1.4 The Consumer’s Allocation Problem . . . . . . . . . . . . . . . . . . . . 5 1.5 Analyzing Consumer Demand . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Solving a Specific Problem . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Expected Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . 9 1.8 Cardinal and Ordinal Utility . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9 The Independence Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.10 Utility Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.11 Utility of Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.12 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.13 Some Useful Utility Functions . . . . . . . . . . . . . . . . . . . . . . . 16 1.14 Comparing Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.15 Higher-Order Derivatives of the Utility Function . . . . . . . . . . . . . . 18 1.16 The Boundedness Debate: Some History of Economic Thought . . . . . . 19 1.17 Multiperiod Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Arbitrage and Pricing: The Basics 22 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Redundant Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Contingent Claims and Derivative Assets . . . . . . . . . . . . . . . . . . 26 2.4 Insurable States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 Dominance And Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Pricing in the Absence of Arbitrage . . . . . . . . . . . . . . . . . . . . . 29 2.7 More on the Riskless Return . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.8 Riskless Arbitrage and the “Single Price Law Of Markets” . . . . . . . . . 33 2.9 Possibilities and Probabilities . . . . . . . . . . . . . . . . . . . . . . . . 34 2.10 “Risk-Neutral” Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.11 Economies with a Continuum of States . . . . . . . . . . . . . . . . . . . 36 CONTENTS vii 3 The Portfolio Problem 38 3.1 The Canonical Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Optimal Portfolios and Pricing . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Properties of Some Simple Portfolios . . . . . . . . . . . . . . . . . . . . 41 3.4 Stochastic Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5 The Theory of Efficient Markets . . . . . . . . . . . . . . . . . . . . . . . 44 3.6 Efficient Markets in a “Riskless” Economy . . . . . . . . . . . . . . . . . 45 3.7 Information Aggregation and Revelation in Efficient Markets: The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.8 Simple Examples of Information Revelation in an Efficient Market . . . . . 48 4 Mean-Variance Portfolio Analysis 52 4.1 The Standard Mean-Variance Portfolio Problem . . . . . . . . . . . . . . 52 4.2 Covariance Properties of the Minimum-Variance Portfolios . . . . . . . . . 56 4.3 The Mean-Variance Problem with a Riskless Asset . . . . . . . . . . . . . 56 4.4 Expected Returns Relations . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.5 Equilibrium: The Capital Asset Pricing Model . . . . . . . . . . . . . . . 59 4.6 Consistency of Mean-Variance Analysis and Expected Utility Maximization 62 4.7 Solving A Specific Problem . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.8 The State Prices Under Mean-Variance Analysis . . . . . . . . . . . . . . 65 4.9 Portfolio Analysis Using Higher Moments . . . . . . . . . . . . . . . . . 65 A The Budget Constraint 68 B The Elliptical Distributions 70 B.1 Some Examples of Elliptical Variables . . . . . . . . . . . . . . . . . . . 72 B.2 Solving a Specific Problem . . . . . . . . . . . . . . . . . . . . . . . . . 75 B.3 Preference Over Mean Return . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Generalized Risk, Portfolio Selection, and Asset Pricing 78 5.1 The Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Risk: A Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.3 Mean Preserving Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.4 Rothschild And Stiglitz Theorems On Risk . . . . . . . . . . . . . . . . . 82 5.5 The Relative Riskiness of Opportunities with Different Expectations . . . . 83 5.6 Second-Order Stochastic Dominance . . . . . . . . . . . . . . . . . . . . 84 5.7 The Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.8 Solving A Specific Problem . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.9 Optimal and Efficient Portfolios . . . . . . . . . . . . . . . . . . . . . . . 87 5.10 Verifying The Efficiency of a Given Portfolio . . . . . . . . . . . . . . . . 89 5.11 A Risk Measure for Individual Securities . . . . . . . . . . . . . . . . . . 92 5.12 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 A Stochastic Dominance 96 A.1 Nth-Order Stochastic Dominance . . . . . . . . . . . . . . . . . . . . . . 97 viii CONTENTS 6 Portfolio Separation Theorems 99 6.1 Inefficiency of The Market Portfolio: An Example . . . . . . . . . . . . . 99 6.2 Mutual Fund Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 One-Fund Separation Under Restrictions on Utility . . . . . . . . . . . . . 103 6.4 Two-Fund Separation Under Restrictions on Utility . . . . . . . . . . . . . 103 6.5 Market Equilibrium Under Two-Fund, Money Separation . . . . . . . . . 105 6.6 Solving A Specific Problem . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.7 Distributional Assumptions Permitting One-Fund Separation . . . . . . . . 107 6.8 Distributional Assumption Permitting Two-Fund, Money Separation . . . . 108 6.9 Equilibrium Under Two-Fund, Money Separation . . . . . . . . . . . . . . 110 6.10 Characterization of Some Separating Distributions . . . . . . . . . . . . . 110 6.11 Two-Fund Separation with No Riskless Asset . . . . . . . . . . . . . . . . 111 6.12 K-Fund Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.13 Pricing Under K-Fund Separation . . . . . . . . . . . . . . . . . . . . . . 115 6.14 The Distinction between Factor Pricing and Separation . . . . . . . . . . . 115 6.15 Separation Under Restrictions on Both Tastes and Distributions . . . . . . 117 7 The Linear Factor Model: Arbitrage Pricing Theory 120 7.1 Linear Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.2 Single-Factor, Residual-Risk-Free Models . . . . . . . . . . . . . . . . . 120 7.3 Multifactor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.4 Interpretation of the Factor Risk Premiums . . . . . . . . . . . . . . . . . 122 7.5 Factor Models with “Unavoidable” Risk . . . . . . . . . . . . . . . . . . 122 7.6 Asymptotic Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.7 Arbitrage Pricing of Assets with Idiosyncratic Risk . . . . . . . . . . . . . 125 7.8 Risk and Risk Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.9 Fully Diversified Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.10 Interpretation of the Factor Premiums . . . . . . . . . . . . . . . . . . . . 130 7.11 Pricing Bounds in A Finite Economy . . . . . . . . . . . . . . . . . . . . 133 7.12 Exact Pricing in the Linear Model . . . . . . . . . . . . . . . . . . . . . . 134 8 Equilibrium Models with Complete Markets 136 8.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.2 Valuation in Complete Markets . . . . . . . . . . . . . . . . . . . . . . . 137 8.3 Portfolio Separation in Complete Markets . . . . . . . . . . . . . . . . . . 137 8.4 The Investor’s Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . 138 8.5 Pareto Optimality of Complete Markets . . . . . . . . . . . . . . . . . . . 139 8.6 Complete and Incomplete Markets: A Comparison . . . . . . . . . . . . . 140 8.7 Pareto Optimality in Incomplete Markets: Effectively Complete Markets . 140 8.8 Portfolio Separation and Effective Completeness . . . . . . . . . . . . . . 141 8.9 Efficient Set Convexity with Complete Markets . . . . . . . . . . . . . . . 143 8.10 Creating and Pricing State Securities with Options . . . . . . . . . . . . . 144 9 General Equilibrium Considerations in Asset Pricing 147 9.1 Returns Distributions and Financial Contracts . . . . . . . . . . . . . . . . 147 9.2 Systematic and Nonsystematic Risk . . . . . . . . . . . . . . . . . . . . . 153 9.3 Market Efficiency with Nonspeculative Assets . . . . . . . . . . . . . . . 154 9.4 Price Effects of Divergent Opinions . . . . . . . . . . . . . . . . . . . . . 158 CONTENTS ix 9.5 Utility Aggregation and the “Representative” Investor . . . . . . . . . . . 161 10 Intertemporal Models in Finance 163 10.1 Present Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.2 State Description of a Multiperiod Economy . . . . . . . . . . . . . . . . 163 10.3 The Intertemporal Consumption Investment Problem . . . . . . . . . . . . 166 10.4 Completion of the Market Through Dynamic Trading . . . . . . . . . . . 168 10.5 Intertemporally Efficient Markets . . . . . . . . . . . . . . . . . . . . . . 170 10.6 Infinite Horizon Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 11 Discrete-time Intertemporal Portfolio Selection 175 11.1 Some Technical Considerations . . . . . . . . . . . . . . . . . . . . . . . 187 A Consumption Portfolio Problem when Utility Is Not Additively Separable 188 B Myopic and Turnpike Portfolio Policies 193 B.1 Growth Optimal Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.2 A Caveat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.3 Myopic Portfolio Policies . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.4 Turnpike Portfolio Policies . . . . . . . . . . . . . . . . . . . . . . . . . 195 12 An Introduction to the Distributions of Continuous-Time Finance 196 12.1 Compact Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 12.2 Combinations of Compact Random Variables . . . . . . . . . . . . . . . . 198 12.3 Implications for Portfolio Selection . . . . . . . . . . . . . . . . . . . . . 198 12.4 “Infinitely Divisible” Distributions . . . . . . . . . . . . . . . . . . . . . 200 12.5 Wiener and Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . 202 12.6 Discrete-Time Approximations for Wiener Processes . . . . . . . . . . . . 204 13 Continuous-Time Portfolio Selection 206 13.1 Solving a Specific Problem . . . . . . . . . . . . . . . . . . . . . . . . . 208 13.2 Testing The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 13.3 Efficiency Tests Using the Continuous-Time CAPM . . . . . . . . . . . . 213 13.4 Extending The Model to Stochastic Opportunity Sets . . . . . . . . . . . . 213 13.5 Interpreting The Portfolio Holdings . . . . . . . . . . . . . . . . . . . . . 215 13.6 Equilibrium in the Extended Model . . . . . . . . . . . . . . . . . . . . . 218 13.7 Continuous-Time Models with No Riskless Asset . . . . . . . . . . . . . . 219 13.8 State-Dependent Utility of Consumption . . . . . . . . . . . . . . . . . . 220 13.9 Solving A Specific Problem . . . . . . . . . . . . . . . . . . . . . . . . . 221 13.10A Nominal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 14 The Pricing of Options 227 14.1 Distribution and Preference-Free Restrictions on Option Prices . . . . . . . 227 14.2 Option Pricing: The Riskless Hedge . . . . . . . . . . . . . . . . . . . . . 235 14.3 Option Pricing By The Black-Scholes Methodology . . . . . . . . . . . . 237 14.4 A Brief Digression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 14.5 The Continuous-Time Riskless Hedge . . . . . . . . . . . . . . . . . . . . 239 14.6 The Option’s Price Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 241 [...]... function F (x) of a vector x is said to be homogeneous of degree k to the point x0 if for all λ = 0 F (λ(x − x0 )) = λk F (x − x0 ) (16) If no reference is made to the point of homogeneity, it is generally assumed to be 0 For k = 1 the function is said to be linearly homogeneous This does not, of course, imply that F (·) is linear All partial derivatives of a homogeneous function are homogeneous of one smaller... unconditional expectation of both sides of (81) Then ˜ ˜ ˜ ˜ ˜ ˜ E[Xn+1 ] = E E[Xn+1 |Y0 , Y1 , , Yn ] = E[Xn ] ˜ ˜ Then by induction E[Xn+m ] = E[Xn ] for all m > 0 (85) Chapter 1 Utility Theory It is not the purpose here to develop the concept of utility completely or with the most generality Nor are the results derived from the most primitive set of assumptions All of this can be found in standard... intuition of the development Typically, goods are different consumption items such as wheat or corn Formally, we often label as distinct goods the consumption of the same physical good at different times or in different states of nature Usually, in finance we lump all physical goods together into a single consumption commodity distinguished only by the time and state of nature when it is consumed Quite often... Quantity of Good Two Increasing Utility Quantity of Good One x1 Figure 1.2 Strict Complements 1.3 Properties of Some Commonly Used Ordinal Utility Functions An important simplifying property of utility is preferential independence Two subsets of goods are preferentially independent of their complements if the conditional preference ordering when varying the amounts of the goods in each subset does not depend... scarcity of any good, and since it is bounded below the consumer cannot sell unlimited quantities of certain “goods” (e.g., labor services) to finance unlimited purchases of other goods 6 Utility Theory Theorem 2 Under Assumptions 1 and 2 the consumer’s allocation problem possesses a unique, slack-free solution x∗ for any positive price vector, p > 0, and positive wealth Proof The existence of a solution... concept of utility maximization to cover situations involving risk We assume throughout the discussion that the economic agents making the decisions know the true objective probabilities of the relevant events This is not in the tradition of using subjective probabilities but will suffice for our purposes The consumers will now be choosing among “lotteries” described generically by their payoffs (x1... an ordinal utility function defined over lotteries or an ordinal utility functional defined over probability distributions of payoffs The next three axioms are used to develop the concept of choice through the maximization of the expectation of a cardinal utility function over payoff complexes Axiom 5 (independence) Let L1 = {(x1 , , xv , , xm ), π} and L2 = {(x1 , , z, , xm ), π} If xv ∼... Properties of Wiener Processes 16.4 Derivation of Ito’s Lemma 16.5 Multidimensional Ito’s Lemma 16.6 Forward and Backward Equations of Motion 16.7 Examples 16.8 First Passage Time 16.9 Maximum and Minimum of Diffusion Processes 16.10Diffusion Processes as Subordinated Wiener Processes 16.11Extreme Variation of Diffusion... of marginal rates of substitution, Two subsets with more than one good each are preferentially independent if the marginal rates of substitution within each subset (and the indifference curves) do not depend upon the allocation in the complement A preferentially independent utility function can be written as a monotone transform of an additive form Υ(x) = θ[a(y) + b(z)] (1.7) since marginal rates of. .. found in standard textbooks on microeconomics or game theory Here rigor is tempered with an eye for simplicity of presentation 1.1 Utility Functions and Preference Orderings A utility function is not presumed as a primitive in economic theory What is assumed is that each consumer can “value” various possible bundles of consumption goods in terms of his own subjective preferences For concreteness it . Theory of Financial Decision Making Jonathan E. Ingersoll, Jr. Yale University Preface In the past twenty years the quantity of new and exciting research in finance. maturity of τ. w A vector of portfolio weights. w i is the fraction of wealth in the ith asset. X The exercise price for an option. Y The state space tableau of payoffs. Y si is the payoff in state. complete markets theory. This theory was originally noted for its elegant treatment of general equilibrium as in the models of Arrow and Debreu and was considered to be primarily of theoretical