An Introduction to Financial Mathematics, 2nd Edition

Chapter 13

Martingales in the Black-Scholes-Merton Model

In Chapter 9 we described option valuation in the binomial model in terms of discrete-time martingales. In this chapter, we carry out a similar program for the Black-Scholes-Merton model using continuous-time martingales. The first two sections develop the tools needed to implement this program. The central result here is Girsanov’s Theorem, which guarantees the existence of risk-neutral probability measures. Throughout the chapter, $\left(\text{Ω,}ℱ\text{,}\mathbb{P}\right)$ denotes a fixed probability space with expectation operator $\mathbb{E}$, and W is a Brownian motion on $\left(\text{Ω,}ℱ\text{,}\mathbb{P}\right)$.

13.1  Continuous-Time Martingales

Definition and Examples

A continuous-time stochastic process (M_t)_t≥0 on $\left(\text{Ω,}ℱ\text{,}\mathbb{P}\right)$ adapted to a filtration ${({ℱ}_{}}_{}$