The Economics of Continuous-Time Finance

by Dumas, Luciano

ISBN: 9780262341424 | Copyright 2017

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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.

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Contents (pg. vii)
1 Introduction (pg. 1)
1.1 Motivation (pg. 1)
1.2 Outline (pg. 3)
1.3 How to Use This Book (pg. 4)
1.4 Apologies (pg. 5)
1.5 Acknowledgments (pg. 5)
I Discrete-Time Economies (pg. 7)
2 Pricing of Redundant Securities (pg. 9)
2.1 Single-Period Economies (pg. 10)
2.2 Multiperiod Economies (pg. 27)
2.3 Conclusion (pg. 50)
3 Investor Optimality and Pricing in the Case of Homogeneous Investors (pg. 53)
3.1 One-Period Economies (pg. 54)
3.2 A Benchmark Example (pg. 69)
3.3 Multiperiod Model (pg. 72)
3.4 Benchmark Example (continued) (pg. 85)
3.5 Conclusion (pg. 87)
4 Equilibrium and Pricing of Basic Securities (pg. 91)
4.1 One-Period Economies (pg. 91)
4.2 Competitive Equilibrium (pg. 94)
4.3 Incomplete Market (pg. 107)
4.4 Multiple-Period Economies (pg. 110)
4.5 Conclusion (pg. 117)
II Pricing In Continuous Time (pg. 119)
5 Brownian Motion and Itô Processes (pg. 121)
5.1 Martingales and Markov Processes (pg. 121)
5.2 Continuity for Stochastic Processes and Diffusions (pg. 123)
5.3 Brownian Motion (pg. 125)
5.4 Itô Processes (pg. 133)
5.5 Benchmark Example (continued) (pg. 135)
5.6 Itô’s Lemma (pg. 140)
5.7 Dynkin Operator (pg. 145)
5.8 Conclusion (pg. 146)
6 Black-Scholes and Redundant Securities (pg. 149)
6.1 Replicating-Portfolio Argument (pg. 151)
6.2 Martingale-Pricing Argument (pg. 155)
6.3 Hedging-Portfolio Argument (pg. 157)
6.4 Extensions: Dividends (pg. 160)
6.5 Extensions: A Partially Generalized Black-Scholes Model (pg. 163)
6.6 Implied Probabilities (pg. 165)
6.7 The Price of Risk of a Derivative (pg. 166)
6.8 Benchmark Example (continued) (pg. 168)
6.9 Conclusion (pg. 172)
7 Portfolios, Stochastic Integrals, and Stochastic Differential Equations (pg. 175)
7.1 Pathologies (pg. 176)
7.2 Stochastic Integrals (pg. 178)
7.3 Admissible Strategies (pg. 184)
7.4 Itô Processes and Stochastic Differential Equations (pg. 186)
7.5 Bubbles (pg. 191)
7.6 Itô Processes and the Martingale-Representation Theorem (pg. 193)
7.7 Benchmark Example (continued) (pg. 195)
7.8 Conclusion (pg. 196)
8 Pricing Redundant Securities (pg. 199)
8.1 Market Setup (pg. 200)
8.2 Changes of Measure (pg. 202)
8.3 Fundamental Theorem of Security Pricing (pg. 210)
8.4 Market Completeness (pg. 212)
8.5 Asset-Specific Completeness (pg. 218)
8.6 Benchmark Example (continued) (pg. 220)
8.7 Conclusion (pg. 223)
III Individual Optimality in Continuous Time (pg. 227)
9 Dynamic Optimization and Portfolio Choice (pg. 229)
9.1 A Single Risky Security (pg. 230)
9.2 A Single Risky Security with IID Returns and One Riskless Security (pg. 235)
9.3 Multiple, Correlated Risky Securities with IID Returns Plus One Riskless Security (pg. 242)
9.4 Non-IID, Multiple Risky Securities, and a Riskless Security (pg. 243)
9.5 Exploiting Market Completeness: Building a Bridge to Chapter 10 (pg. 250)
9.6 Benchmark Example (continued) (pg. 252)
9.7 Conclusion (pg. 254)
9.8 Appendix: The Link to Chapter 10 (pg. 255)
10 Global Optimization and Portfolio Choice (pg. 259)
10.1 Model Setup (pg. 260)
10.2 Solution (pg. 264)
10.3 Properties of the Global Approach (pg. 268)
10.4 Non-negativity Constraints on Consumption and Wealth (pg. 269)
10.5 The Growth-Optimal Portfolio (pg. 270)
10.6 Benchmark Example (continued) (pg. 271)
10.7 Conclusion (pg. 272)
IV Equilibrium in Continuous Time (pg. 277)
11 Equilibrium Restrictions and the CAPM (pg. 279)
11.1 Intertemporal CAPM and Betas (pg. 280)
11.2 Co-risk and Linearity (pg. 281)
11.3 Consumption-Based CAPM (pg. 282)
11.4 The Cox-Ingersoll-Ross Equilibrium (pg. 285)
11.5 Benchmark Example (continued) (pg. 293)
11.6 Conclusion (pg. 294)
11.7 Appendix: Aggregation Leading to the CAPM (pg. 295)
12 Equilibrium in Complete Markets (pg. 299)
12.1 Model Setup: Exogenous and Admissible Variables (pg. 300)
12.2 Definition and Existence of Equilibrium (pg. 301)
12.3 Obtaining Equilibrium (pg. 303)
12.4 Asset Pricing in Equilibrium (pg. 306)
12.5 Diffusive and Markovian Equilibria (pg. 309)
12.6 The Empirical Relevance of State Variables (pg. 310)
12.7 Benchmark Example (continued) (pg. 312)
12.8 Conclusion (pg. 315)
V Applications and Extensions (pg. 319)
13 Solution Techniques and Applications (pg. 321)
13.1 Probabilistic Methods (pg. 322)
13.2 Simulation Methods (pg. 327)
13.3 Analytical Methods (pg. 333)
13.4 Approximate Analytical Method: Perturbation Method (pg. 336)
13.5 Numerical Methods: Approximations of Continuous Systems (pg. 339)
13.6 Conclusion (pg. 344)
14 Portfolio Choice and Equilibrium Restrictions in Incomplete Markets (pg. 349)
14.1 Single-Period Analysis (pg. 350)
14.2 The Dynamic Setting in Continuous Time (pg. 354)
14.3 Portfolio Constraints (pg. 363)
14.4 The Minimum-Squared Deviation Approach (pg. 365)
14.5 Conclusion (pg. 368)
14.6 Appendix: Derivation of the Dual Problem 14.16 (pg. 369)
15 Incomplete-Market Equilibrium (pg. 373)
15.1 Various Concepts of Incomplete-Market Equilibrium, Existence, and Welfare Properties (pg. 373)
15.2 Obtaining Continuous-Time Incomplete-Market Equilibria (pg. 380)
15.3 Revisiting the Breeden CAPM: The Effect of Incompleteness on Risk Premia in General Equilibrium (pg. 385)
15.4 Benchmark Example: Restricted Participation (pg. 386)
15.5 Bubbles in Equilibrium (pg. 392)
15.6 Conclusion (pg. 398)
15.7 Appendix: Idiosyncratic Risk Revisited (pg. 398)
16 Interest Rates and Bond Modeling (pg. 403)
16.1 Definitions: Short Rate, Yields, and Forward Rates (pg. 404)
16.2 Examples of Markov Models (pg. 407)
16.3 Affine Models (pg. 412)
16.4 Various Ways of Specifying the Behavior of the Bond Market (pg. 414)
16.5 Effective versus Risk-Neutral Measures (pg. 421)
16.6 Application: Pricing of Redundant Assets (pg. 423)
16.7 A Convenient Change of Numeraire (pg. 425)
16.8 General Equilibrium Considerations (pg. 429)
16.9 Interpretation of Factors (pg. 430)
16.10 Conclusion (pg. 431)
16.11 Appendix: Proof of Proposition 16.3 (pg. 432)
17 Stochastic Volatility (pg. 437)
17.1 Motivation (pg. 437)
17.2 Examples of Markov Models with Stochastic Volatility (pg. 442)
17.3 Stochastic Volatility and Forward Variance (pg. 447)
17.4 VIX (pg. 453)
17.5 Stochastic Volatility in Equilibrium (pg. 454)
17.6 Conclusion (pg. 454)
17.7 Appendix: GARCH (pg. 455)
18 Heterogeneous Expectations (pg. 461)
18.1 Difference of Opinion (pg. 462)
18.2 The Value of Information (pg. 469)
18.3 Equilibrium (pg. 475)
18.4 Sentiment Risk (pg. 481)
18.5 Conclusion (pg. 484)
19 Stopping, Regulation, Portfolio Selection, and Pricing under Trading Costs (pg. 487)
19.1 Cost Functions and Mathematical Tools (pg. 490)
19.2 An Irreversible Decision: To Exercise or Not to Exercise (pg. 492)
19.3 Reversible Decisions: How to Regulate (pg. 497)
19.4 The Portfolio Problem under Proportional Trading Costs (pg. 505)
19.5 The Portfolio Problem under Fixed or Quasi-fixed Trading Costs (pg. 512)
19.6 Option Pricing under Trading Costs (pg. 514)
19.7 Equilibria and Other Open Problems (pg. 517)
19.8 Conclusion (pg. 518)
20 Portfolio Selection and Equilibrium with Habit Formation (pg. 521)
20.1 Motivation: The Equity-Premium and Other Puzzles (pg. 522)
20.2 Habit Formation (pg. 525)
20.3 Risk Aversion versus Elasticity of Intertemporal Substitution (EIS) (pg. 534)
20.4 Conclusion (pg. 536)
21 Portfolio Selection and Equilibrium with Recursive Utility (pg. 539)
21.1 Modeling Strategy (pg. 539)
21.2 Recursive Utility: Definition in Discrete-Time (pg. 542)
21.3 Recursive Utility in Discrete Time: Two Representations (pg. 543)
21.4 Recursive Utility: Continuous Time (pg. 546)
21.5 Individual Investor Optimality (pg. 550)
21.6 Equilibrium with Recursive Utility in Complete Markets (pg. 554)
21.7 Back to the Puzzles: Pricing under Recursive Utility (pg. 560)
21.8 Conclusion (pg. 561)
21.9 Appendix 1: Proof of the Giovannini-Weil Stochastic Discount Factor,Equation (21.5) (pg. 561)
21.10 Appendix 2: Preference for the Timing of Uncertainty Resolution (pg. 563)
An Afterword (pg. 569)
Basic Notation (pg. 571)
Appendix A: A Review of Utility Theory and Mean-Variance Theory (pg. 573)
A.1 Expected Utility (pg. 573)
A.2 Mean-Variance Utility Theory (pg. 578)
Appendix B: Global and Recursive Optimization (pg. 583)
B.1 Global Optimization (pg. 583)
B.2 Discrete-Time Recursive Optimality (pg. 585)
B.3 Continuous-Time Recursive Optimality: Classical Control (pg. 588)
B.4 Continuous-Time Optimality under Frictions: Singular Control (pg. 592)
References (pg. 595)
Author Index (pg. 609)
Index (pg. 613)
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