The Economics of Continuous-Time Finance

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