Analysis of Financial Time Series, Third Edition

6.7 Extension of Ito's Lemma

In derivative pricing, a derivative may be contingent on multiple securities. When the prices of these securities are driven by multiple factors, the price of the derivative is a function of several stochastic processes. The two-factor model for the term structure of interest rate is an example of two stochastic processes. In this section, we briefly discuss the extension of Ito's lemma to the case of several stochastic processes.

Consider a k-dimensional continuous-time process , where k is a positive integer and xit is a continuous-time stochastic process satisfying

(6.21)

where wit is a Wiener process. It is understood that the drift and volatility functions μi(xit) and σi(xit) are functions of time index t as well. We omit t from their arguments to simplify the notation. For i ≠ j, the Wiener processes wit and wjt are different. We assume that the correlation between dwit and dwjt is ρij. This means that ρij is the correlation between the two standard normal random variables ϵi and ϵj defined by Δwit = ϵi Δt and Δwjt = ϵj Δt. Assume that is a function of the stochastic processes xit and time t. The Taylor expansion gives

The discretized version of ...