Dynamic Asset Pricing Theory

By Darrell Duffie | Go to book overview
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8
Derivative Pricing

THIS CHAPTER APPLIES arbitrage-free pricing techniques from Chapters 6 and 7 to derivative securities that are not always easily treated by the direct PDE approach of Chapter 5. A derivative security is one whose cash flows are contingent on the prices of other securities, or on related indices. After summarizing the essential results from Chapter 6 for this purpose, we study the valuation of forwards, futures, European and American options, and certain exotic options. Option pricing with stochastic volatility is addressed with Fourier-transform methods.


A. Martingale Measures in a Black Box

Skipping over the foundational theory developed in Chapter 6, this section reviews the properties of an equivalent martingale measure, a convenient “black-box” approach to derivative asset pricing in the absence of arbitrage. Once again, 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). The standard filtration F = {Ft : 0 tT} of B is as defined in Section 5I.

We take as given an adapted short-rate process r, with

almost surely, and an Ito security-price process S in RN with

for appropriate μ and σ. It was shown in Chapter 6 that, aside from technical conditions, the absence of arbitrage is equivalent to the existence of a probability measure Q with special “risk-neutral” properties, called an equivalent martingale measure. For this chapter, we will use a narrow definition of equivalent martingale measures under which all expected

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