On the use of geometric brownian motion in financial analysis

Abstract

Though geometric Brownian motion (GBM) is an essential tool in finance, a closed form solution for its transition density function has yet to be obtained. In option pricing, though Black and Scholes assumed GBM stock price dynamics, they transformed the problem to allow an option to be evaluated without the stock price’s transition density. This paper presents a closed form solution of Kolmogorov’s backward equation for GBM. As an application, the option price equation is derived directly.