Options pricing using combination of Brownian motion with a random time component

Abstract

This paper provides an efficient option pricing technique for European call options. The technique is a process that combines Geometric Brownian motion with random time component. The additional parameter to the Geometric Brownian motion is the volatility of the time change. This additional parameter provides control over the skewness and kurtosis of the return distribution. Partial integro-differential equation (PIDE) for pricing options under the proposed process was derived. Its values were priced numerically and the pricing performance of the new model is compared on option data sourced from Bloomberg on 28th April, 2023 with the Geometric Brownian motion. The result revealed that the additional parameter was able to address the pricing biases of the Geometric Brownian motion. Recommendation was made that more work should be done in the advancement of quick and effective procedures for calculating PIDE’s having in view to calibrating stochastic processes to a surface of option prices.