The Economics of Continuous-Time Finance

Description

This book introduces the economic applications of the theory of continuous-time finance, with the goal of enabling the construction of realistic models, particularly those involving incomplete markets. Indeed, most recent applications of continuous-time finance aim to capture the imperfections and dysfunctions of financial markets—characteristics that became especially apparent during the market turmoil that started in 2008. The book begins by using discrete time to illustrate the basic mechanisms and introduce such notions as completeness, redundant pricing, and no arbitrage. It develops the continuous-time analog of those mechanisms and introduces the powerful tools of stochastic calculus. Going beyond other textbooks, the book then focuses on the study of markets in which some form of incompleteness, volatility, heterogeneity, friction, or behavioral subtlety arises. After presenting solutions methods for control problems and related partial differential equations, the text examines portfolio optimization and equilibrium in incomplete markets, interest rate and fixed-income modeling, and stochastic volatility. Finally, it presents models where investors form different beliefs or suffer frictions, form habits, or have recursive utilities, studying the effects not only on optimal portfolio choices but also on equilibrium, or the price of primitive securities. The book strikes a balance between mathematical rigor and the need for economic interpretation of financial market regularities, although with an emphasis on the latter. Front Cover......Page 1 Contents......Page 8 1.1 Motivation......Page 20 1.2 Outline......Page 22 1.3 How to Use This Book......Page 23 1.5 Acknowledgments......Page 24 I DISCRETE-TIME ECONOMIES......Page 26 2 Pricing of Redundant Securities......Page 28 2.1.1 The Most Elementary Problem in Finance......Page 29 2.1.2 Uncertainty......Page 32 2.1.3 Securities Payoffs and Prices, and Investors’ Budget Set......Page 33 2.1.4 Absence of Arbitrage and Risk-Neutral Pricing......Page 35 2.1.5 Complete versus Incomplete Markets......Page 41 2.1.6 Complete Markets and State-Price Uniqueness......Page 42 2.1.7 Benchmark Example......Page 43 2.1.8 Valuation of Redundant Securities......Page 44 2.2 Multiperiod Economies......Page 46 2.2.1 Information Arrival over Time and Stochastic Processes......Page 47 2.2.2 Self-Financing Constraint and Redundant Securities......Page 52 2.2.3 Arbitrage, No Arbitrage, and Risk-Neutral Pricing......Page 54 2.2.5 Statically versus Dynamically Complete Markets......Page 59 2.2.6 Benchmark Example......Page 61 2.3 Conclusion......Page 69 3 Investor Optimality and Pricing in the Case of Homogeneous Investors......Page 72 3.1.1 Investor Optimality and Security Pricing under Certainty......Page 73 3.1.2 Investor Optimality and Security Pricing under Uncertainty......Page 76 3.1.3 Arrow-Debreu Securities......Page 77 3.1.4 Complex or Real-World Securities......Page 82 3.1.5 Relation with the No-Arbitrage Approach......Page 85 3.1.6 The Dual Problem......Page 87 3.2.1 Isoelastic Utility......Page 88 3.2.2 Securities Pricing......Page 89 3.2.3 From Security Prices to State Prices, Risk-Neutral Probabilities, and Stochastic Discount Factors......Page 90 3.3 Multiperiod Model......Page 91 3.3.2 Recursive Approach......Page 93 3.3.3 Global Approach......Page 96 3.3.4 Securities Pricing......Page 100 3.4 Benchmark Example (continued)......Page 104 3.5 Conclusion......Page 106 4.1 One-Period Economies......Page 110 4.2 Competitive Equilibrium......Page 113 4.2.1 Equalization of State Prices......Page 118 4.2.2 Risk Sharing......Page 120 4.2.3 Security Pricing by the Representative Investor and the CAPM......Page 122 4.2.4 The Benchmark Example (continued)......Page 125 4.3 Incomplete Market......Page 126 4.4.1 Radner Equilibrium......Page 129 4.4.2 State Prices and Representative Investor: From Radner to Arrow-Debreu Equilibria......Page 131 4.4.3 Securities Pricing......Page 132 4.4.4 Risk Sharing......Page 134 4.4.5 A Side Comment on Time-Additive Utility Functions......Page 135 4.5 Conclusion......Page 136 II PRICING IN CONTINUOUS TIME......Page 138 5.1 Martingales and Markov Processes......Page 140 5.2 Continuity for Stochastic Processes and Diffusions......Page 142 5.3.1 Intuitive Construction......Page 144 5.3.2 A Financial Motivation......Page 149 5.3.3 Definition......Page 150 5.4 Itô Processes......Page 152 5.5.1 The Black-Scholes Model......Page 154 5.5.2 Construction from Discrete-Time......Page 157 5.6 Itô’s Lemma......Page 159 5.6.1 Interpretation......Page 161 5.6.2 Examples......Page 163 5.7 Dynkin Operator......Page 164 5.8 Conclusion......Page 165 6 Black-Scholes and Redundant Securities......Page 168 6.1.1 Building the Black-Scholes PDE......Page 170 6.1.2 Solving the Black-Scholes PDE......Page 173 6.2 Martingale-Pricing Argument......Page 174 6.3 Hedging-Portfolio Argument......Page 176 6.3.1 Comparing the Arguments: Intuition......Page 178 6.4.1 Dividend Paid on the Underlying......Page 179 6.4.2 Dividend Paid on the Option......Page 181 6.5.1 Replicating-Portfolio Argument......Page 182 6.6 Implied Probabilities......Page 184 6.7 The Price of Risk of a Derivative......Page 185 6.8 Benchmark Example (continued)......Page 187 6.9 Conclusion......Page 191 7 Portfolios, Stochastic Integrals, and Stochastic Differential Equations......Page 194 7.1.1 Doubling Strategies......Page 195 7.1.2 Local Martingales......Page 196 7.2 Stochastic Integrals......Page 197 7.3 Admissible Strategies......Page 203 7.4.2 Stochastic Differential Equations......Page 205 7.4.3 When Are Itô Processes Markov Processes? When Are They Diffusions?......Page 207 7.5 Bubbles......Page 210 7.6 Itô Processes and the Martingale-Representation Theorem......Page 212 7.7 Benchmark Example (continued)......Page 214 7.8 Conclusion......Page 215 8 Pricing Redundant Securities......Page 218 8.1 Market Setup......Page 219 8.2 Changes of Measure......Page 221 8.2.1 Equivalent Measures and the Radon-Nikodym Derivative......Page 222 8.2.2 Girsanov’s Theorem: How to Shift to the Risk-Neutral Measure in Black-Scholes Economies......Page 225 8.2.3 Change of Measure, Stochastic Discount Factors, and State Prices......Page 228 8.3 Fundamental Theorem of Security Pricing......Page 229 8.4.1 Necessary and Sufficient Condition......Page 231 8.4.2 Martingale Measure, Stochastic Discount Factor, and State-Price Uniqueness......Page 236 8.5 Asset-Specific Completeness......Page 237 8.6 Benchmark Example (continued)......Page 239 8.7 Conclusion......Page 242 III INDIVIDUAL OPTIMALITY IN CONTINUOUS TIME......Page 246 9 Dynamic Optimization and Portfolio Choice......Page 248 9.1 A Single Risky Security......Page 249 9.1.1 Budget Constraint......Page 250 9.1.2 IID Returns and Dynamic Programming Solution......Page 251 9.1.3 The Marginality Condition......Page 252 9.1.4 Subcases and Examples......Page 253 9.2 A Single Risky Security with IID Returns and One Riskless Security......Page 254 9.2.2 Examples Revisited......Page 256 9.3 Multiple, Correlated Risky Securities with IID Returns Plus One Riskless Security......Page 261 9.4 Non-IID, Multiple Risky Securities, and a Riskless Security......Page 262 9.4.1 Myopic and Hedging Portfolios......Page 265 9.4.2 Fund Interpretation......Page 267 9.4.3 Optimization and Nonlinear PDE......Page 268 9.5 Exploiting Market Completeness: Building a Bridge to Chapter 10......Page 269 9.6 Benchmark Example (continued)......Page 271 9.7 Conclusion......Page 273 9.8 Appendix: The Link to Chapter 10......Page 274 10 Global Optimization and Portfolio Choice......Page 278 10.1 Model Setup......Page 279 10.1.1 Lifetime versus Dynamic Budget Constraint......Page 281 10.2 Solution......Page 283 10.2.1 Optimal Wealth......Page 285 10.2.2 Portfolio Mix......Page 286 10.3 Properties of the Global Approach......Page 287 10.4 Non-negativity Constraints on Consumption and Wealth......Page 288 10.5 The Growth-Optimal Portfolio......Page 289 10.6 Benchmark Example (continued)......Page 290 10.7 Conclusion......Page 291 IV EQUILIBRIUM IN CONTINUOUS TIME......Page 296 11 Equilibrium Restrictions and the CAPM......Page 298 11.1 Intertemporal CAPM and Betas......Page 299 11.2 Co-risk and Linearity......Page 300 11.3 Consumption-Based CAPM......Page 301 11.4.1 Model Setup......Page 304 11.4.2 The Riskless Security......Page 307 11.4.3 The Risky Securities......Page 309 11.5 Benchmark Example (continued)......Page 312 11.6 Conclusion......Page 313 11.7 Appendix: Aggregation Leading to the CAPM......Page 314 12 Equilibrium in Complete Markets......Page 318 12.1 Model Setup: Exogenous and Admissible Variables......Page 319 12.2 Definition and Existence of Equilibrium......Page 320 12.3.1 Direct Calculation......Page 322 12.3.2 The Representative Investor......Page 323 12.4 Asset Pricing in Equilibrium......Page 325 12.4.1 The Riskless Security......Page 326 12.4.3 The Risky Securities......Page 327 12.5 Diffusive and Markovian Equilibria......Page 328 12.6 The Empirical Relevance of State Variables......Page 329 12.7 Benchmark Example (continued)......Page 331 12.7.2 Equilibria with Heterogeneous Power-Utility Investors......Page 332 12.8 Conclusion......Page 334 V APPLICATIONS AND EXTENSIONS......Page 338 13 Solution Techniques and Applications......Page 340 13.1.1 Probabilities of Transitions as Solutions of PDEs......Page 341 13.1.2 Integral Representation of the Solution (Characteristic Functions)......Page 344 13.1.3 Other Uses of Integral Representations......Page 345 13.2 Simulation Methods......Page 346 13.2.2 Mil’shtein’s Scheme......Page 347 13.2.4 The Doss or Nelson-and-Ramaswamy Transformation......Page 349 13.2.5 The Use of “Variational Calculus” in Simulations......Page 350 13.3.1 Solutions of Linear PDEs......Page 352 13.3.2 The Affine Framework and Solutions of Riccati Equations......Page 353 13.4 Approximate Analytical Method: Perturbation Method......Page 355 13.5.1 Lattice Approximations......Page 358 13.5.2 Finite-Difference Approximations......Page 360 13.6 Conclusion......Page 363 14 Portfolio Choice and Equilibrium Restrictions in Incomplete Markets......Page 368 14.1.1 Model Setup......Page 369 14.1.3 Consumption and Portfolios......Page 370 14.2.1 Model Setup......Page 373 14.2.2 Dual......Page 374 14.2.3 Implied State Prices......Page 375 14.2.5 Portfolios......Page 378 14.3 Portfolio Constraints......Page 382 14.4.1 The One-Period Case......Page 384 14.4.2 Continuous Time: The “Minimal” Martingale Measure......Page 385 14.5 Conclusion......Page 387 14.6 Appendix: Derivation of the Dual Problem 14.16......Page 388 15.1.1 One-Good, Static Setting......Page 392 15.1.2 Problems in More General Settings......Page 394 15.1.3 Example: The Role of Idiosyncratic Risk......Page 395 15.1.4 Incomplete Markets andWelfare: Equilibrium and Constrained Pareto Optimality in the Static Setting......Page 397 15.2.1 Direct Calculation......Page 399 15.3 Revisiting the Breeden CAPM: The Effect of Incompleteness on Risk Premia in General Equilibrium......Page 404 15.4 Benchmark Example: Restricted Participation......Page 405 15.4.1 Endowment and Securities Markets......Page 406 15.4.2 Consumption Choices and State Prices......Page 407 15.5 Bubbles in Equilibrium......Page 411 15.5.1 Benchmark Example (continued)......Page 412 15.5.2 Bubble Interpretation......Page 415 15.7 Appendix: Idiosyncratic Risk Revisited......Page 417 16 Interest Rates and Bond Modeling......Page 422 16.1 Definitions: Short Rate, Yields, and Forward Rates......Page 423 16.2.1 Vasicek......Page 426 16.2.2 Modified Vasicek Models......Page 429 16.2.3 Cox, Ingersoll, and Ross......Page 430 16.3 Affine Models......Page 431 16.4 Various Ways of Specifying the Behavior of the Bond Market......Page 433 16.4.2 Specifying the Behavior of Forward Rates......Page 434 16.4.3 Specifying the Behavior of the Short Rate......Page 436 16.4.4 Condition for the Short Rate to Be Markovian......Page 437 16.5 Effective versus Risk-Neutral Measures......Page 440 16.6 Application: Pricing of Redundant Assets......Page 442 16.7 A Convenient Change of Numeraire......Page 444 16.7.1 Change of Discounting Asset as a Change of Measure......Page 445 16.7.2 Using Bond Prices for Discounting: The Forward Measure......Page 446 16.8 General Equilibrium Considerations......Page 448 16.9 Interpretation of Factors......Page 449 16.10 Conclusion......Page 450 16.11 Appendix: Proof of Proposition 16.3......Page 451 17.1.1 Empirics......Page 456 17.1.2 Time-Varying Volatility with Market Completeness......Page 459 17.2.1 Hull and White......Page 461 17.2.2 Heston......Page 463 17.3 Stochastic Volatility and Forward Variance......Page 466 17.3.2 Definition of Forward Variance......Page 467 17.3.3 Interpretation of Forward Variance......Page 469 17.3.4 Summary of the Option Valuation Procedure......Page 471 17.4 VIX......Page 472 17.6 Conclusion......Page 473 17.7 Appendix: GARCH......Page 474 17.7.1 Parameter Specification and Estimate......Page 475 17.7.2 GARCH versus Continuous-Time Stochastic Volatility......Page 476 18 Heterogeneous Expectations......Page 480 18.1 Difference of Opinion......Page 481 18.1.1 Endowment and Securities Markets......Page 482 18.1.2 The Several Risk Premia......Page 484 18.1.3 Investor Optimization......Page 486 18.1.4 Comparative Analysis of a Change in Disagreement......Page 487 18.2.1 Bayesian Updating and Disagreement between Investors......Page 488 18.2.2 Information and Portfolio Choice......Page 490 18.3 Equilibrium......Page 494 18.3.1 Equilibrium Consumption and Price Parameters......Page 495 18.3.2 Consensus Beliefs......Page 497 18.4 Sentiment Risk......Page 500 18.5 Conclusion......Page 503 19 Stopping, Regulation, Portfolio Selection, and Pricing under Trading Costs......Page 506 19.1 Cost Functions and Mathematical Tools......Page 509 19.2 An Irreversible Decision: To Exercise or Not to Exercise......Page 511 19.2.1 The American Put......Page 513 19.3 Reversible Decisions: How to Regulate......Page 516 19.3.1 Base Case: Calculating the Value Function......Page 517 19.3.2 Boundary Conditions: Value Matching......Page 518 19.3.3 Optimizing the Regulator via Smooth-Pasting Boundary Conditions: The Case of Impulse Control......Page 519 19.3.4 Optimizing the Regulator via Smooth-Pasting Boundary Conditions: The Case of Instantaneous Control......Page 520 19.3.5 Why Not Classical Control?......Page 522 19.4 The Portfolio Problem under Proportional Trading Costs......Page 524 19.4.1 A Semi-explicit Policy Case: Power Utility......Page 526 19.4.2 Comment on Quadratic Costs......Page 530 19.5 The Portfolio Problem under Fixed or Quasi-fixed Trading Costs......Page 531 19.6.1 The Inadequacy of the Replication and Super-Replication Approaches......Page 533 19.6.2 Option Pricing within a Portfolio Context......Page 534 19.7 Equilibria and Other Open Problems......Page 536 19.8 Conclusion......Page 537 20 Portfolio Selection and Equilibrium with Habit Formation......Page 540 20.1 Motivation: The Equity-Premium and Other Puzzles......Page 541 20.2 Habit Formation......Page 544 20.2.1 Internal Habit......Page 545 20.2.2 External Habit......Page 547 20.3 Risk Aversion versus Elasticity of Intertemporal Substitution (EIS)......Page 553 20.4 Conclusion......Page 555 21.1.1 The Restriction: Time Consistency......Page 558 21.1.2 The Motivation......Page 559 21.2 Recursive Utility: Definition in Discrete-Time......Page 561 21.3.1 Aggregator Representation......Page 562 21.3.2 Discount-Factor Representation......Page 564 21.4.1 Stochastic Differential Utility......Page 565 21.4.2 Variational Utility......Page 566 21.5.1 Choice of the Consumption Path in Complete Markets......Page 569 21.5.2 Benchmark Example......Page 571 21.6.1 Direct Calculation of Equilibrium......Page 573 21.6.2 Calculating a Pareto Optimum Conveniently......Page 575 21.6.3 The Markovian Case......Page 576 21.6.4 The Market Prices of Risk......Page 577 21.7 Back to the Puzzles: Pricing under Recursive Utility......Page 579 21.9 Appendix 1: Proof of the Giovannini-Weil Stochastic Discount Factor, Equation (21.5)......Page 580 21.10 Appendix 2: Preference for the Timing of Uncertainty Resolution......Page 582 An Afterword......Page 588 Basic Notation......Page 590 A.1 Expected Utility......Page 592 A.1.2 Risk Aversion and Prudence......Page 594 A.1.3 The HARA Class......Page 596 A.2 Mean-Variance Utility Theory......Page 597 A.2.1 The Mean-Variance Frontier......Page 598 A.2.2 The Mean-Variance CAPM......Page 600 B.1 Global Optimization......Page 602 B.2.1 Statement of the Optimization Problem......Page 604 B.2.2 Bellman’s Principle in Discrete Time......Page 605 B.3.1 Statement of the Optimization Problem......Page 607 B.3.2 Bellman’s Principle and Its Verification Theorem......Page 608 B.3.3 Perturbation Reasoning or Variational Calculus......Page 609 B.4 Continuous-Time Optimality under Frictions: Singular Control......Page 611 B.4.1 Bellman’s Principle......Page 612 References......Page 614 Author Index......Page 628 Index......Page 632