ADVANCED ASSET PRICING THEORY

Transcription

1 Series in Quantitative Finance -Vol. 2 ADVANCED ASSET PRICING THEORY Chenghu Ma Fudan University, China Imperial College Press

2 Contents List of Figures Preface Background Organization and Content Readership Acknowledgments Frequently Used Notation xvii xxiii xxiv xxvi xxxi xxxii xxxv Foundation 1 1. Introduction Historical Background Setup of Market Economy Marketplace States of Nature Economic Agents Flow Budget Constraints Optimal Choice Problem No-Arbitrage and the Positive Linear Pricing Rule Market Equilibrium Allocational Market Efficiency Aggregation and Representative Agent Informational Market Efficiency Price as Aggregator of Information 31

3 A Advanced Asset Pricing Theory Trading Volume and Strong REE Efficient Market Hypothesis Information Manipulation and EMH Remarks No-Arbitrage Asset Pricing Fundamental Theorem Primitive vs Derivative Securities Derivative Securities: \An Introduction Trading with Derivatives European Put-Call Parity Forward and Futures Prices Put-Call-Futures Parity Options and Futures Option Pricing: A Single-Period Model CRR Binomial Model Setup of the Model CRR Binomial Option Pricing Formula The Laplace Inverse Representation of Option Price Continuous-Time Limit: Black-Scholes Formula Option Pricing and State Prices Backward Procedure American Options Ho-Lee Model of the Term Structure of Interest Rates A No-Arbitrage Bond Pricing Model Risk-Free Interest Rate Yield Curve Forward Rates Interest-Rate Contingent Claims Estimating Unknown Parameters Remarks Risk and Risk Measures Stone's Family of Risk Measures Downside Risk and VaR Measures VaR 95

4 Contents vii C-VaR Put Options and Hedging the Downside Risk Attitudes Towards Risk and Risk Premium Arrow-Pratt Measure of Local Absolute Risk Arrow-Pratt Measure of Local Relative Risk Local vs Global Risk Measures and Comparative Risk Aversion. - v Downside Risk and Insurance Premium Stochastic Dominance and Risk Measures FSD and VaR FSD and Choice by EU Investor SSD and C-VaR SSD and Choice by EU Investors Mean-Preserving-Spread Remarks Portfolio Risk Management Portfolio Choice by Expected Utility Investors Comparative Risk Aversion and Portfolio Choice Wealth Effect Risk Effect Portfolio Choice by MPS Risk-Averse Investors Mean-Variance Analysis Portfolio Choice Efficient Frontier in Absence of a Risk-Free Asset Black's Separation Theorem Risk Decomposition and Risk-Return Relation Efficient Frontier in Presence of a Risk-Free Asset A Two-Fund Separation Theorem No-Arbitrage and Hansen-Jagannathan Bounds No-Arbitrage and a Factor-Based Linear-/? Model 140

5 nii Advanced Asset Pricing Theory Factor Return and Minimum-Length Portfolio An Orthogonal Decomposition Theorem Remarks MPS Risk Aversion and Equilibrium CAPM Setup Market Portfolio and a Derivation of CAPM Existence of Equilibrium CAPM and Multi-Factor Models Pricing Contingent Claims with the CAPM CRR Binomial Option Model and CAPM Multinomial Option Model with CAPM: The Case of Binomial Market Return Option Pricing with CAPM: The Case with Log-Normal Returns Representative Agent and Equity Premium Puzzle Comments on the CAPM Elliptical Distribution and CAPM Equity Premium Puzzle and CAPM Should the CAPM Price Options? Does CAPM Fit Data? APT and CAPM Remarks 175 Discrete-Time Modeling Preliminaries State Space Marketplace; Preference System and Recursive Utility Myopic Investor Intertemporal Recursive Utility Time Preference and Intertemporal Substitution Intertemporal Substitution and Risk Aversion Timing of Uncertainty Resolution 197

6 Contents ix Preferences for Information Remarks Equilibrium with MPS Risk-Averse Myopic Investors Portfolio Choice I-CAPM Interest Rate and Market Portfolio Myopic Representative Agent Power Utility.\.-.-/ A Second Look at the Equity Premium Puzzle Risk-Free Rate Puzzle W-CAPM and I-CAPM Kreps-Porteus Expected Utility On Testing the I-CAPM On Implementing the I-CAPM Remarks Dynamic Choice for Recursive Investors Sequential Choice Problem Dynamic Consistency and Optimal Trading Strategy Dynamic Programming Choices by Finite-Lived Agents Choices by Long-Lived Agents under Markov Uncertainty MPS Risk Aversion and Shadow MV Frontier On Risk Decomposition On Mutual-Fund Separation and Market Anomalies ^Conclusion Remarks Equilibrium Asset Pricing with Recursive Utility Investors MPS Risk Aversion and Shadow CAPM Shadow CAPM: Homogeneous Shadow Price Shadow CAPM: Heterogeneous MPS Risk- Averse Investors Shadow CAPM with Representative Agent.. 257

7 x Advanced Asset Pricing Theory Shadow CAPM vs I-CAPM Asset Pricing with RU Representative Agent Market Portfolio and Shadow Price Market Volatility Risk-Free Interest Rates Remarks Pricing Contingent Claims By Which Scheme? _ Scheme A Scheme B Scheme C Remarks Term Structure of Interest Rates Yield Curve and Expectations Hypothesis Yield-to-Maturity and Term Premium: A Parity On the Empirical Validity of the Expectations Hypothesis Equilibrium Bond Pricing with Recursive Utility A Simple Two-Factor Model A Multi-Factor Model On Coherent Models of the Term Structure of Interest Rates Risk-Neutral Density/M.G.F Option Pricing Rule Option-Based Asset Pricing Model: An Inversion Problem Options on Market Portfolio I Binomial Model: Example Multinomial Model: Example Discrete-Time Black-Scholes Model: Example Betweenness Option Pricing Model: Example Distinguishing Betweenness and KP Utilities with Options Estimating Risk-Neutral Density with Options 308

8 Contents xi Parametric Approach: Maximum Likelihood Estimation Moneyness Biases: Testing the Black-Scholes Model Testing the Betweenness Option Pricing Model Nonparametric Estimation of Risk-Neutral M.G.F, Options on Market Portfolio II Bond Options: A Closed-Form Formula Remarks 330 Continuous-Time Modeling Stochastic Processes and SDE Stochastic Processes in Continuous Time Rare Events and Poisson Process Poisson Point Process and Random Poisson Measure 343' Brownian Motion Processes Derived from Brownian Motion Levy Process Martingale and Semimartingale Stochastic Calculus Stochastic Integral for Brownian Motion Stochastic Integral for a Poisson Point Process Stochastic Integral for Semimartingale Stochastic Differential Equations Existence Backward-Forward SDE Ito Lemma Kolmogorov Equation Feynman-Kac Formula Change of Measure and Girsanov Theorem Examples Remarks 410

9 xii Advanced Asset Pricing Theory 12. An Arbitrage-Free Marketplace Preliminaries Market Span Truncated Markets Future Spot Markets and Trading Sessions Self-Financing Trading Strategies No-Arbitrage Condition Fundamental Theorem..' Remarks Black-Scholes Option Pricing Model Preliminary Black-Scholes Partial Differential Equation 429 " 13.3 Risk-Neutral Measure Pseudo-Price Process Black-Scholes Option Pricing Formula Some Static Analyses Option and Futures: The Black Formula Implied Volatility and Risk-Neutral M.G.F Black-Scholes PDE with Stochastic Coefficients ' Option Pricing with Time-Varying Coefficients State-Dependent Coefficients Jump Risk Remarks The American Option Preliminary American Option: A Submartingale Early Exercising Boundary: Preliminary American Option: A Free-Boundary Problem American Option Premium Optimal Exercising Boundary American Option: A Control Problem Hitting a Barrier European Barrier Options American Option: Supremum to Barrier Options Remarks 474

10 Contents xiii 15. No-Arbitrage Term Structure of Interest Rates Preliminaries Slope and Curvature of Yield and Forward Curves Expectations Hypothesis Empirical Evidence Interest Rate Modeling I: The Classical No-Arbitrage Approach Single-Factor Models Mis-Specification Error Coherency and Intertemporal Consistency Multi-Factor Affine Term Structure of Interest Rates Interest Rate Modeling II: HJM Approach Setup of the Model Induced SDEs for Bond Prices and Yield Curves Q-measure and Arbitrage-Free Coefficients Interest Rates under Measure Q Uniqueness of Q-Measure Interest Rate Modeling III: Jump Risks Preliminary Arbitrage-Free Coefficients with Jumps Interest Rates with Jumps under Q Uniqueness of Q-Measure with Jumps Interest-Rate Contingent Claims Coupon Bonds Bond Options Information Content of Bond Options Volatility, Moneyness Ratio and Bond Option in Diffusion Comments on the HJM Approach Remarks Stochastic Differential Utility Preliminaries Expected Additive Utility Continuous-Time Recursive Utility 551

11 xiv Advanced Asset Pricing Theory Utility Aggregator Certainty Equivalent Ordinal Utility Generator Existence of Recursive Utility Behavior Assumptions Underlying the SDU Gronwall Inequality Monotonicity, Concavity and Continuity Dynamic Consistency Intertemporal Risk Aversion Comparative Intertemporal Risk Aversion Preferences for Information Attitudes Towards the Timing of Uncertainty Resolution: An Example Remarks Sequential Choice and Optimal Trading Strategy Preliminaries Wealth-Maximizing EU Investor A Sequential Choice Problem First-Order Condition 590, Hamilton-Jacobi-Bellman Equation Example: Power Utility MPS Risk-Averse Investor MV Efficiency: An Optimal Tracking Problem Dynamic Consistency HJB Equation: MPS Risk-Averse Investor MV Efficiency: An Analytic Characterization Temporal vs Local Mutual-Fund Separation Risk-Return Relationship Risk Decomposition Temporal vs Instantaneous Efficient Frontier EU vs MPS Risk-Averse Investors Merton's Problem Flow Budget Constraint Sequential Choice Problem First-Order Condition: Euler Equation HJB Equation: Time-Additive Expected Utility 625

12 Contents xv Optimal Choice: Analytic Characterization Investor with SDU/Recursive Utility HJB Equation: SDU/Recursive Utility Optimal Cash Payout for SDU Investors Optimal Portfolio for SDU Investors Euler Equation for SDU Investors Sequential Choice under Constraints Remarks Equilibrium Asset Pricing: A General Theory Preliminary Aggregate Expenditure vs Aggregate Dividend Savings Rate, D/P Ratio and Plowback Ratio MRS, MRT and Equilibrium Interest Rate Pseudo-State and PV Pricing Rule Pseudo-State Process PV Pricing Rule Martingale Representation of Pseudo-Price Equilibrium PDE A Verification Theorem Remarks Applications Equity Premium Local vs Jump Premium Local Premium Decomposition Consumption-Based Local Premium ; Decomposition Consumption- and Market Portfolio-Based Local Premium Decomposition Equity Premium Puzzle (revisited) A Remark on Market Portfolio Equilibrium Term Structure of Interest Rates Risk-Free Interest Rate Pseudo-Drift Coefficient CIR Term Structure of Interest Rates 681

13 xvi Advanced Asset Pricing Theory CIR Term Structure of Interest Rates with Jumps Affine Term Structure of Interest Rates I: Diffusion Information Content of Forward Curve Affine Term Structure of Interest Rates II: Jump-Diffusion Equilibrium vs No-Arbitrage Interest Rate Models Information Role of Options Pseudo-M.G.F European Call Options and Pseudo-M.G.F Options on Market Portfolio Black-Scholes Formula Cox-Ross Formula Naik-Lee-Merton Formula Betweenness Option Pricing Model Distinguishing Betweenness and Expected Utility with Options Jump Risk and Moneyness Biases Remarks 712 Appendix A Probability Space 715 A.I Information Filtration 715 A.2 Probability 716 A.3 Random Variable 716 A.3.1 C.D.F. and P.D.F 716 A.3.2 Expectation and Conditional Expectation A.3.3 Independence 720 A.3.4 Characteristic Function 720 A.4 LP-Space 721 A.5 Discrete-Time Stochastic Processes 722 A.5.1 Convergence 722 A.5.2 Markov Process and Markov Chain 724 A.5.3 Markovian Uncertainty 725 Appendix B Bilateral Laplace Transform 727 B.I Preliminary 727

14 Contents xvii B.2 Laplace Inversion Theorem 728 B.3 Properties of { } and C' 1 { } 729 B.4 Laplace Transform for Generalized Functions 730 B.5 Special P.D.F.s and Characteristic Functions 731 Appendix C Real Analysis 735 C.I Preliminary : 735 C.I.I Vector Space 735 C.1.2 Metric Space ^ 736 C.I.3 Normed Vector Space 737 C.1.4 Hilbert Space 739 C.2 Riesz Representation Theorem 739 C.3 Separating Hyperplane Theorem 740 C.4 Contraction Mapping Theorem 741 C.5 Generalized Functions of Schwartz 742 C.5.1 L p (K) as a Subspace of S (R) 743 C.5.2 Dirac Functions 743 C.5.3 Generalized Derivative 744 C.5.4 Higher-Order Generalized Derivatives 745 Appendix D Optimization 747 D.I Weierstrass Theorem 748 D.2 Uniqueness 748 D.3 Kuhn-Tucker Theorem 749 D.4 Envelope Theorem 752 Bibliography 753 Subject Index 769 Author Index 111