Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview
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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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