A Lazy, Eclectic Derivation of the Black Scholes Formula

Abstract

The classical Black Scholes option-pricing formula which has been an indispensable tool in the hands of the practitioners of quantitative finance has been derived numerous ways. Some of these derivations have been very theoretical with the full machinery of Girsanov's theorem, some informal and based on practical hedging arguments. We suggest yet another derivation (Actually, we just do certain things while computing the drift in the changed measure very differently), which is based on sound theoretical framework, but does not use much of advanced theory. Instead the derivation suggested here makes repeated use of the technique of measure change that is rather a universal and standard tool in derivative pricing. The approach is lazy, because it defers doing certain things, unless absolutely necessary, while at the same time elegant enough to be pedagogically interesting in its own right.

Keywords: Measure Change, Martingale, Girsanov's Theorem, FX Options, Drift in Spot Measure