Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview

6
State Prices and Equivalent
Martingale Measures

This chapter summarizes arbitrage-free security pricing theory in the continuous-time setting introduced in Chapter 5. The main idea is the equivalence between no arbitrage, the existence of state prices, and the existence of an equivalent martingale measure, paralleling the discrete-state theory of Chapter 2. This extends the Markovian results of Chapter 5, which are based on PDE methods. For those interested mainly in applications, the first sections of Chapters 7 and 8 summarize the major conclusions of this chapter as a “black box,” making it possible to skip this chapter on a first reading.

The existence of a state-price deflator is shown to imply the absence of arbitrage. Then a state-price “beta” model of expected returns is derived. Turning to equivalent martingale measures, we begin with the sufficiency of an equivalent martingale measure for the absence of arbitrage. Girsanov’s Theorem (Appendix D) gives conditions under which there exists an equivalent martingale measure. This approach generates another proof of the Black-Scholes formula. State prices are then connected with equivalent martingale measures; the two concepts are more or less the same. They are literally equivalent in the analogous finite-state model of Chapter 2, and we will see that the distinction here is purely technical.


A. Arbitrage

We fix a standard Brownian motion B = (B1 . .. , Bd) in Rd, restricted to some time interval [0, T], on a given probability space (Ω, F, P). We also fix the standard filtration F = {Ft : t ∈[0, T]} of B, as defined in Section 5I. For simplicity, we take F to be FT. Suppose the price processes

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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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