ADVANCED ASSET PRICING THEORY

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

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 3 1.1 Historical Background 4 1.2 Setup of Market Economy 6 1.2.1 Marketplace 8 1.2.2 States of Nature 9 1.2.3 Economic Agents 11 1.3 Flow Budget Constraints 13 1.4 Optimal Choice Problem 16 1.5 No-Arbitrage and the Positive Linear Pricing Rule... 20 1.6 Market Equilibrium 22 1.7 Allocational Market Efficiency 24 1.8 Aggregation and Representative Agent 27 1.9 Informational Market Efficiency 30 1.9.1 Price as Aggregator of Information 31

A Advanced Asset Pricing Theory 1.9.2 Trading Volume and Strong REE 37 1.9.3 Efficient Market Hypothesis 40 1.9.4 Information Manipulation and EMH 43 1.10 Remarks 45 2. No-Arbitrage Asset Pricing 49 2.1 Fundamental Theorem 49 2.2 Primitive vs Derivative Securities 54 2.2.1 Derivative Securities: \An Introduction... 54 2.2.2 Trading with Derivatives 55 2.2.3 European Put-Call Parity 57 2.2.4 Forward and Futures Prices 59 2.2.5 Put-Call-Futures Parity 62 2.2.6 Options and Futures 63 2.2.7 Option Pricing: A Single-Period Model... 64 2.3 CRR Binomial Model 67 2.3.1 Setup of the Model 67 2.3.2 CRR Binomial Option Pricing Formula... 68 2.3.3 The Laplace Inverse Representation of Option Price 70 2.3.4 Continuous-Time Limit: Black-Scholes Formula 71 2.3.5 Option Pricing and State Prices 74 2.3.6 Backward Procedure 76 2.3.7 American Options 79 2.4 Ho-Lee Model of the Term Structure of Interest Rates. 82 2.4.1 A No-Arbitrage Bond Pricing Model 85 2.4.2 Risk-Free Interest Rate 86 2.4.3 Yield Curve 87 2.4.4 Forward Rates 87 2.4.5 Interest-Rate Contingent Claims 88 2.4.6 Estimating Unknown Parameters 89 2.5 Remarks 90 3. Risk and Risk Measures 93 3.1 Stone's Family of Risk Measures 94 3.2 Downside Risk and VaR Measures 95 3.2.1 VaR 95

Contents vii 3.2.2 C-VaR 97 3.2.3 Put Options and Hedging the Downside Risk 98 3.3 Attitudes Towards Risk and Risk Premium 101 3.3.1 Arrow-Pratt Measure of Local Absolute Risk 102 3.3.2 Arrow-Pratt Measure of Local Relative Risk. 103 3.3.3 Local vs Global Risk Measures and Comparative Risk Aversion. - v 104 3.4 Downside Risk and Insurance Premium 106 3.5 Stochastic Dominance and Risk Measures 108 3.5.1 FSD and VaR 108 3.5.2 FSD and Choice by EU Investor 109 3.5.3 SSD and C-VaR 110 3.5.4 SSD and Choice by EU Investors 112 3.6 Mean-Preserving-Spread 115 3.7 Remarks 118 4. Portfolio Risk Management 121 4.1 Portfolio Choice by Expected Utility Investors 121 4.1.1 Comparative Risk Aversion and Portfolio Choice 122 4.1.2 Wealth Effect 123 4.1.3 Risk Effect 124 4.2 Portfolio Choice by MPS Risk-Averse Investors 127 4.2.1 Mean-Variance Analysis 127 4.2.2 Portfolio Choice 129 4.2.3 Efficient Frontier in Absence of a Risk-Free Asset 130 4.2.4 Black's Separation Theorem 131 4.2.5 Risk Decomposition and Risk-Return Relation 132 4.2.6 Efficient Frontier in Presence of a Risk-Free Asset 135 4.2.7 A Two-Fund Separation Theorem 137 4.2.8 No-Arbitrage and Hansen-Jagannathan Bounds 137 4.2.9 No-Arbitrage and a Factor-Based Linear-/? Model 140

nii Advanced Asset Pricing Theory 4.2.10 Factor Return and Minimum-Length Portfolio 141 4.2.11 An Orthogonal Decomposition Theorem... 144 4.3 Remarks 146 5. MPS Risk Aversion and Equilibrium CAPM 147 5.1 Setup. 147 5.2 Market Portfolio and a Derivation of CAPM 148 5.3 Existence of Equilibrium 150 5.4 CAPM and Multi-Factor Models 156 5.5 Pricing Contingent Claims with the CAPM 157 5.5.1 CRR Binomial Option Model and CAPM.. 159 5.5.2 Multinomial Option Model with CAPM: The Case of Binomial Market Return 160 5.5.3 Option Pricing with CAPM: The Case with Log-Normal Returns 161 5.6 Representative Agent and Equity Premium Puzzle... 168 5.7 Comments on the CAPM 170 5.7.1 Elliptical Distribution and CAPM 171 5.7.2 Equity Premium Puzzle and CAPM 171 5.7.3 Should the CAPM Price Options? 171 5.7.4 Does CAPM Fit Data? 172 5.7.5 APT and CAPM 173 5.8 Remarks 175 Discrete-Time Modeling 177 6. Preliminaries 181 6.1 State Space 182 6.2 Marketplace; 182 6.3 Preference System and Recursive Utility 186 6.3.1 Myopic Investor 187 6.3.2 Intertemporal Recursive Utility 187 6.3.3 Time Preference and Intertemporal Substitution 192 6.3.4 Intertemporal Substitution and Risk Aversion 194 6.3.5 Timing of Uncertainty Resolution 197

Contents ix 6.3.6 Preferences for Information 202 6.4 Remarks 205 7. Equilibrium with MPS Risk-Averse Myopic Investors 207 7.1 Portfolio Choice 207 7.2 I-CAPM 208 7.3 Interest Rate and Market Portfolio 211 7.3.1 Myopic Representative Agent 212 7.3.2 Power Utility.\.-.-/ 214 7.3.3 A Second Look at the Equity Premium Puzzle 215 7.3.4 Risk-Free Rate Puzzle 215 7.3.5 W-CAPM and I-CAPM 216 7.3.6 Kreps-Porteus Expected Utility 218 7.4 On Testing the I-CAPM 219 7.5 On Implementing the I-CAPM 220 7.6 Remarks 222 8. Dynamic Choice for Recursive Investors 223 8.1 Sequential Choice Problem 223 8.2 Dynamic Consistency and Optimal Trading Strategy.. 225 8.3 Dynamic Programming 227 8.3.1 Choices by Finite-Lived Agents 229 8.3.2 Choices by Long-Lived Agents under Markov Uncertainty 236 8.4 MPS Risk Aversion and Shadow MV Frontier 244 8.4.1 On Risk Decomposition 246 8.4.2 On Mutual-Fund Separation and Market Anomalies 248 8.4.3 ^Conclusion 250 8.5 Remarks 250 9. Equilibrium Asset Pricing with Recursive Utility Investors 253 9.1 MPS Risk Aversion and Shadow CAPM 254 9.1.1 Shadow CAPM: Homogeneous Shadow Price. 254 9.1.2 Shadow CAPM: Heterogeneous MPS Risk- Averse Investors 255 9.1.3 Shadow CAPM with Representative Agent.. 257

x Advanced Asset Pricing Theory 9.1.4 Shadow CAPM vs I-CAPM 259 9.2 Asset Pricing with RU Representative Agent 260 9.2.1 Market Portfolio and Shadow Price 261 9.2.2 Market Volatility 262 9.2.3 Risk-Free Interest Rates 264 9.3 Remarks 266 10. Pricing Contingent Claims 269 10.1 By Which Scheme? _ 269 10.1.1 Scheme A 269 10.1.2 Scheme B 270 10.1.3 Scheme C 272 10.1.4 Remarks 272 10.2 Term Structure of Interest Rates 273 10.2.1 Yield Curve and Expectations Hypothesis.. 273 10.2.2 Yield-to-Maturity and Term Premium: A Parity 275 10.2.3 On the Empirical Validity of the Expectations Hypothesis 276 10.2.4 Equilibrium Bond Pricing with Recursive Utility 277 10.2.5 A Simple Two-Factor Model 278 10.2.6 A Multi-Factor Model 285 10.2.7 On Coherent Models of the Term Structure of Interest Rates 288 10.3 Risk-Neutral Density/M.G.F 292 10.4 Option Pricing Rule 294 10.5 Option-Based Asset Pricing Model: An Inversion Problem 295 10.6 Options on Market Portfolio I 297 10.6.1 Binomial Model: Example 1 300 10.6.2 Multinomial Model: Example 2 301 10.6.3 Discrete-Time Black-Scholes Model: Example 3 301 10.6.4 Betweenness Option Pricing Model: Example 4 303 10.6.5 Distinguishing Betweenness and KP Utilities with Options 307 10.7 Estimating Risk-Neutral Density with Options 308

Contents xi 10.7.1 Parametric Approach: Maximum Likelihood Estimation 310 10.7.2 Moneyness Biases: Testing the Black-Scholes Model 312 10.7.3 Testing the Betweenness Option Pricing Model 314 10.7.4 Nonparametric Estimation of Risk-Neutral M.G.F, 316 10.8 Options on Market Portfolio II. 324 10.9 Bond Options: A Closed-Form Formula 328 10.10 Remarks 330 Continuous-Time Modeling 333 11. Stochastic Processes and SDE 337 11.1 Stochastic Processes in Continuous Time 337 11.1.1 Rare Events and Poisson Process 339 11.1.2 Poisson Point Process and Random Poisson Measure 343' 11.1.3 Brownian Motion 346 11.1.4 Processes Derived from Brownian Motion... 351 11.1.5 Levy Process 360 11.1.6 Martingale and Semimartingale 363 11.2 Stochastic Calculus 365 11.2.1 Stochastic Integral for Brownian Motion... 365 11.2.2 Stochastic Integral for a Poisson Point Process 367 11.2.3 Stochastic Integral for Semimartingale... 371 11.3 Stochastic Differential Equations 372 11.3.1 Existence 373 11.3.2 Backward-Forward SDE 375 11.3.3 Ito Lemma 375 11.3.4 Kolmogorov Equation 379 11.3.5 Feynman-Kac Formula 382 11.3.6 Change of Measure and Girsanov Theorem.. 386 11.3.7 Examples 396 11.4 Remarks 410

xii Advanced Asset Pricing Theory 12. An Arbitrage-Free Marketplace 411 12.1 Preliminaries 411 12.1.1 Market Span 412 12.1.2 Truncated Markets 413 12.1.3 Future Spot Markets and Trading Sessions.. 414 12.1.4 Self-Financing Trading Strategies 416 12.2 No-Arbitrage Condition 416 12.3 Fundamental Theorem..'...- 419 12.4 Remarks 425 13. Black-Scholes Option Pricing Model 427 13.1 Preliminary 427 13.2 Black-Scholes Partial Differential Equation 429 " 13.3 Risk-Neutral Measure 430 13.4 Pseudo-Price Process 431 13.5 Black-Scholes Option Pricing Formula 431 13.6 Some Static Analyses 432 13.7 Option and Futures: The Black Formula 435 13.8 Implied Volatility and Risk-Neutral M.G.F 436 13.9 Black-Scholes PDE with Stochastic Coefficients... 437' 13.10 Option Pricing with Time-Varying Coefficients 439 13.11 State-Dependent Coefficients 439 13.12 Jump Risk 441 13.13 Remarks 441 14. The American Option 443 14.1 Preliminary 443 14.2 American Option: A Submartingale 445 14.3 Early Exercising Boundary: Preliminary 446 14.4 American Option: A Free-Boundary Problem 452 14.5 American Option Premium 460 14.6 Optimal Exercising Boundary 463 14.7 American Option: A Control Problem 466 14.7.1 Hitting a Barrier. 467 14.7.2 European Barrier Options 470 14.7.3 American Option: Supremum to Barrier Options 472 14.8 Remarks 474

Contents xiii 15. No-Arbitrage Term Structure of Interest Rates 475 15.1 Preliminaries 476 15.1.1 Slope and Curvature of Yield and Forward Curves 478 15.1.2 Expectations Hypothesis 481 15.1.3 Empirical Evidence 487 15.2 Interest Rate Modeling I: The Classical No-Arbitrage Approach 488 15.2.1 Single-Factor Models-.- 489 15.2.2 Mis-Specification Error 495 15.2.3 Coherency and Intertemporal Consistency.. 496 15.2.4 Multi-Factor Affine Term Structure of Interest Rates 500 15.3 Interest Rate Modeling II: HJM Approach 503 15.3.1 Setup of the Model 503 15.3.2 Induced SDEs for Bond Prices and Yield Curves 504 15.3.3 Q-measure and Arbitrage-Free Coefficients.. 508 15.3.4 Interest Rates under Measure Q..- 510 15.3.5 Uniqueness of Q-Measure 513 15.4 Interest Rate Modeling III: Jump Risks 515 15.4.1 Preliminary 516 15.4.2 Arbitrage-Free Coefficients with Jumps... 520 15.4.3 Interest Rates with Jumps under Q 522 15.4.4 Uniqueness of Q-Measure with Jumps 524 15.5 Interest-Rate Contingent Claims 529 15.5.1 Coupon Bonds 529 15.5.2 Bond Options 534 15.5.3 Information Content of Bond Options 534 15.5.4.Volatility, Moneyness Ratio and Bond Option in Diffusion 536 15.6 Comments on the HJM Approach 540 15.7 Remarks 541 16. Stochastic Differential Utility 543 16.1 Preliminaries 543 16.2 Expected Additive Utility 544 16.3 Continuous-Time Recursive Utility 551

xiv Advanced Asset Pricing Theory 16.3.1 Utility Aggregator 552 16.3.2 Certainty Equivalent 555 16.3.3 Ordinal Utility Generator 557 16.3.4 Existence of Recursive Utility 560 16.4 Behavior Assumptions Underlying the SDU 568 16.4.1 Gronwall Inequality 568 16.4.2 Monotonicity, Concavity and Continuity... 571 16.4.3 Dynamic Consistency 576 16.4.4 Intertemporal Risk Aversion 577 16.4.5 Comparative Intertemporal Risk Aversion.. 579 16.4.6 Preferences for Information 582 16.4.7 Attitudes Towards the Timing of Uncertainty Resolution: An Example 584 16.5 Remarks 586 17. Sequential Choice and Optimal Trading Strategy 587 17.1 Preliminaries 587 17.2 Wealth-Maximizing EU Investor 589 17.2.1 A Sequential Choice Problem 589 17.2.2 First-Order Condition 590, 17.2.3 Hamilton-Jacobi-Bellman Equation 592 17.2.4 Example: Power Utility 598 17.3 MPS Risk-Averse Investor 600 17.3.1 MV Efficiency: An Optimal Tracking Problem 602 17.3.2 Dynamic Consistency 603 17.3.3 HJB Equation: MPS Risk-Averse Investor.. 606 17.3.4 MV Efficiency: An Analytic Characterization 607 17.3.5 Temporal vs Local Mutual-Fund Separation. 610 17.3.6 Risk-Return Relationship 614 17.3.7 Risk Decomposition 615 17.3.8 Temporal vs Instantaneous Efficient Frontier. 616 17.3.9 EU vs MPS Risk-Averse Investors 618 17.4 Merton's Problem 621 17.4.1 Flow Budget Constraint 621 17.4.2 Sequential Choice Problem 622 17.4.3 First-Order Condition: Euler Equation... 622 17.4.4 HJB Equation: Time-Additive Expected Utility 625

Contents xv 17.4.5 Optimal Choice: Analytic Characterization.. 628 17.5 Investor with SDU/Recursive Utility 632 17.5.1 HJB Equation: SDU/Recursive Utility... 634 17.5.2 Optimal Cash Payout for SDU Investors... 635 17.5.3 Optimal Portfolio for SDU Investors 636 17.5.4 Euler Equation for SDU Investors 636 17.5.5 Sequential Choice under Constraints 640 17.6 Remarks 643 18. Equilibrium Asset Pricing: A General Theory 647 18.1 Preliminary 647 18.1.1 Aggregate Expenditure vs Aggregate Dividend 649 18.1.2 Savings Rate, D/P Ratio and Plowback Ratio 650 18.1.3 MRS, MRT and Equilibrium Interest Rate.. 651 18.2 Pseudo-State and PV Pricing Rule 655 18.2.1 Pseudo-State Process. 656 18.2.2 PV Pricing Rule 657 18.3 Martingale Representation of Pseudo-Price 658 18.4 Equilibrium PDE 659 18.5 A Verification Theorem 661 18.6 Remarks 664 19. Applications 665 19.1 Equity Premium 665 19.1.1 Local vs Jump Premium 665 19.1.2 Local Premium Decomposition 667 19.1.3 Consumption-Based Local Premium ; Decomposition 668 19.1.4 Consumption- and Market Portfolio-Based Local Premium Decomposition 669 19.1.5 Equity Premium Puzzle (revisited) 670 19.1.6 A Remark on Market Portfolio 671 19.2 Equilibrium Term Structure of Interest Rates 674 19.2.1 Risk-Free Interest Rate 674 19.2.2 Pseudo-Drift Coefficient 680 19.2.3 CIR Term Structure of Interest Rates 681

xvi Advanced Asset Pricing Theory 19.2.4 CIR Term Structure of Interest Rates with Jumps 684 19.2.5 Affine Term Structure of Interest Rates I: Diffusion 689 19.2.6 Information Content of Forward Curve... 691 19.2.7 Affine Term Structure of Interest Rates II: Jump-Diffusion 693 19.2.8 Equilibrium vs No-Arbitrage Interest Rate Models 695 19.3 Information Role of Options 696 19.3.1 Pseudo-M.G.F 697 19.3.2 European Call Options and Pseudo-M.G.F.. 698 19.4 Options on Market Portfolio 699 19.4.1 Black-Scholes Formula 703 19.4.2 Cox-Ross Formula 704 19.4.3 Naik-Lee-Merton Formula 705 19.4.4 Betweenness Option Pricing Model 707 19.4.5 Distinguishing Betweenness and Expected Utility with Options 709 19.4.6 Jump Risk and Moneyness Biases 710 19.5 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.. 717 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

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