An Introduction to Exotic Option Pricing

Chapter 3

Gaussian Random Variables

We have seen that one powerful method of pricing options in the BS-framework, on a dividend paying asset, is via the pair of formulae (1.51), repeated below for convenience:

$V\left(x,t\right)=e^{–r\tau }{\mathbb{E}}_Q\left\{F\left(X_T\right)|{ℱ}_t\right\};X_T=x{e}^{((r–q)–\frac{1}{2}{\sigma }^2)\tau +\sigma \sqrt{\tau }Z}.$

Essentially, this is the mathematical statement of the FTAP applied to the BS economy. The function F(XT) is the payoff of the derivative at expiry T, and the expression for XT is the random asset price at time T, given its current value x at time t < T, as seen under the EMM, Q. This random price depends only on a single Gaussian random variable Z, although for each t there will be a different such Z.

This chapter summarizes many ...