ABSTRACT
IN THIS CHAPTER, we introduce the martingale approach to derivativespricing. This approach consists of two major steps: the derivation of the martingale probability measure and the construction of the replication strategy. The mathematical foundation of this approach is formed by two theorems, namely, the Cameron-Martin-Girsanov (CMG) theorem and the martingale representation theorem. The derivation of the martingale probability measure is achieved by using the CMG theorem, while the construction of the replication strategy is based on the martingale representation theorem. A significant portion of this chapter is devoted to establishing these two theorems. We motivate our discussion with a simple binomial model for tradable assets and then proceed to establish the two theorems in continuous time for a complete market with multiple underlying securities. Once the pricing formula for general options has been established, we price call options as an important example and derive the
famous Black-Scholes formulae. Throughout this book, we limit ourselves to a discussion of a complete market for which every source of risk can be traded. The proof of the martingale representation theorem is provided in the Appendix. At the end of the chapter, we give some references on derivative pricing in incomplete markets.
