An Introduction to Financial Mathematics, 2nd Edition

Chapter 14

Path-Independent Options

In Chapter 13 the price of a standard European call option in the BSM model was determined using the risk-neutral probability measure given by Girsanov’s Theorem. There are a variety of other European options that may be similarly valued. In this chapter we consider the most common of these. We also give a brief analysis of American claims in the BSM model.

Throughout, $\left(\Omega ,ℱ,\mathbb{P}\right)$ denotes a fixed probability space with expectation operator $\mathbb{E}$, where $\mathbb{P}$ is the law governing the behavior of the price of an asset $\mathcal{S}$. Risk-neutral measures on Ω will be denoted, as usual, by the generic notation ${\mathbb{P}}^{*}$, with ${\mathbb{E}}^{*}$ the corresponding expectation operator. As before, $\left({ℱ}_t^s\right)$ denotes the natural filtration for the price process S