Generalised geometric Brownian motion: Theory and applications to option pricing

TL;DR

This paper introduces a generalized geometric Brownian motion model incorporating memory effects, providing new analytical tools and applying it to improve European call option pricing based on empirical performance.

Contribution

It develops a generalized GBM with memory kernels, deriving analytical expressions and demonstrating its application to option pricing with empirical validation.

Findings

Kernel performance varies with option maturity

Generalized GBM captures empirical asset dynamics better

Model improves option pricing accuracy

Abstract

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics due to irregularities found when comparing its properties with empirical distributions. As a solution, we develop a generalisation of GBM where the introduction of a memory kernel critically determines the behavior of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and obtain the corresponding probability density functions by using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.