Black Scholes the Martingale Approach

This paper establishes the Black Scholes formula in the martingale, risk-neutral valuation framework. The intent is two-fold. One, to serve as an introduction to expectation pricing and two, to examine this framework in explicit mathematical detail. The reader is assumed to have fluent background in the mathematical theory of stochastic processes and calculus, but is not assumed to have background in finance. The relevant financial definitions are given in section 2. Section 2 also establishes the main result (proposition 2.3) that links equivalent martingale measures (EMMs) to expectations. The key mathematical tool at work here is the martingale representation theorem, which guarantees the existence of a hedging strategy under an EMM. In section 3, we exploit this result by explicitly finding an EMM for geometric brownian motion (proposition 3.3). Here the key mathematical tool is Girsanov’s theorem, which tells us how to convert to and from an EMM. Readers with financial background will recognize this conversion as substracting the market price of risk. For general contingent claims we of course cannot reduce valuation to a simple formula. But the techniques used here indicate how we ought to proceed numerically. It is a two-step process. First, we back out an EMM from underlying market prices. Then we compute the value of the claim as its discounted expectation under the risk neutral measure. There is a potential obstacle to this procedure: an EMM might not exist. For this we appeal to a powerful guarantee called the fundamental theorem of asset pricing, which asserts the existence of an EMM if and only if the market is arbitrage-free.