Pricing European Option When the Stock Price Process Is Being Driven by Geometric Brownian Motion

Abstract

This report is about modeling a European Option in general when the stock price process is being driven by geometric Brownian motion (gBm). The volatility parameter is used as an example of a basic estimator and simulated values of geometric Brownian motion hence exploring some of the properties that improve the accuracy of an estimator. The theory is then extended to estimate the volatility from real data by using the Roger-Satchell Estimator. Hence the estimated volatility is used in the model developed in calculating the value on European option using the Donsker Delta Function approach and is compared with that of the Black-Scholes formula. A unique finding is an observation that the value of an option obtained from using the Donsker Delta Function approach is more of the European Put Option than European Call Option which uses the Black-Scholes formula, then this roughly leads to the conclusion that the Donsker Delta Function approach computes a European Put Option.