The Martingale Representation Theorem

This chapter introduces the martingale approach to derivatives pricing. This approach consists of two major steps: the derivation of the martingale probability measure and the construction of the replication strategy. The derivation of the martingale probability measure is achieved by using the Cameron–Martin–Girsanov (CMG) theorem, while the construction of the replication strategy is based on the martingale representation theorem. The martingale representation theorem plays a critical role in the so-called martingale approach to derivatives pricing. Similar to the pricing of options on a single asset, the pricing of options on multiple assets consists of two steps: the construction of a martingale measure for the assets and the construction of the replication strategy. The CMG Theorem states that a Brownian motion with a drift is in fact a standard Brownian motion under a different measure.