Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview

3
The Dynamic Programming Approach

THIS CHAPTER PRESENTS portfolio choice and asset pricing in the framework of dynamic programming, a technique for solving dynamic optimization problems with a recursive structure. The asset pricing implications go little beyond those of the previous chapter, but there are computational advantages. After introducing the idea of dynamic programming in a deterministic setting, we review the basics of a finite-state Markov chain. The Bellman equation is shown to characterize optimality in a Markov setting. The first-order condition for the Bellman equation, often called the “stochastic Euler equation,” is then shown to characterize equilibrium security prices. This is done with additive utility in the main body of the chapter, and extended to more general recursive forms of utility in the exercises. The last sections of the chapter show the computation of arbitrage-free derivative security values in a Markov setting, including an application of Bellman’s equation for optimal stopping to the valuation of American securities such as the American put option. An exercise presents algorithms for the numerical solution of term-structure derivative securities in a simple “binomial” setting.


A. The Bellman Approach

To get the basic idea, we start in the T-period setting of the previous chapter, with no securities except those permitting short-term riskless borrowing at any time t at the discount dt > 0. The endowment process of a given agent is e. Given a consumption process c, it is convenient to define the agent’s wealth process Wc by

and

-49-

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Dynamic Asset Pricing Theory
Table of contents

Table of contents

  • Title Page iii
  • Contents vii
  • Preface xiii
  • I - Discrete-Time Models 1
  • 1 - Introduction to State Pricing 3
  • 2 - The Basic Multiperiod Model 21
  • 3 - The Dynamic Programming Approach 49
  • 4 - The Infinite-Horizon Setting 65
  • II - Continuous-Time Models 81
  • 5 - The Black-Scholes Model 83
  • 6 - State Prices and Equivalent Martingale Measures 101
  • 7 - Term-Structure Models 135
  • 8 - Derivative Pricing 167
  • 9 - Portfolio and Consumption Choice 203
  • 10 - Equilibrium 235
  • 11 - Corporate Securities 259
  • 12 - Numerical Methods 293
  • Appendixes 321
  • Bibliography 373
  • Symbol Glossary 445
  • Author Index 447
  • Subject Index 457
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