Option pricing

Abstract

Using mathematical techniques at undergraduate level, an introduction to axiomatic probability theory and stochastic calculus facilitates the classic derivation of the Black-Scholes-Merton approach in valuating a European option. Brownian motion is derived as the limit of a scaled symmetric random walk and its quadratic variation is determined. This serves to evaluate the Itô integral and the Itô-Doeblin change-of-variables formula. After employing these equations to arrive at the partial differential equation for the option value, the solution is determined by the use of an equivalent risk-neutral measure.