# Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview

4
The Infinite-Horizon Setting

THIS CHAPTER PRESENTS infinite-period analogues of the results of Chapters 2 and 3. Although this setting requires additional technicalities and produces few new insights, it sometimes simplifies results and it serves the large-sample theory of econometrics, which calls for an unbounded number of observations. We start directly with a Markov dynamic programming extension of the finite-horizon results of Chapter 3, and only later consider the implications of no arbitrage or optimality for security prices without using the Markov assumption. Finally, we return to the stationary Markov setting to review briefly the large-sample approach to estimating asset pricing models. Only Sections A and B are essential; the remainder could be skipped on a first reading.

A. Markov Dynamic Programming

Suppose X = {X0, X1X2, … } is a time-homogeneous Markov chain of shocks valued in a finite set Z = {1, … , k}, defined exactly as in Section C, with the exception that there is an infinite number of time periods. Given a k × k nonnegative matrix q whose rows sum to 1, sources given in the Notes explain the existence of a probability space (Ω, F, Pi), for each initial shock i, satisfying the defining properties Pi(X0 = i) = 1 and

As in Chapter 3, Ft denotes the tribe generated by {X0, …, Xt}. That is, the source of information is the Markov chain {Xt}. This is the first appearance in the book of a set Ω of states that need not be finite, but because there is only a finite number of events in Ft for each t, most of this chapter can be easily understood without referring to Appendix C for a review of general probability spaces.

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Dynamic Asset Pricing Theory

• 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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