A Gaussian Markov alternative to fractional Brownian motion for pricing financial derivatives

TL;DR

This paper introduces a Gaussian Markov process as a simplified alternative to fractional Brownian motion for pricing derivatives, aiming to incorporate past dependence while maintaining mathematical tractability.

Contribution

It develops a Markovian model that captures past dependence in asset prices, addressing limitations of fractional Brownian motion in derivative pricing.

Findings

The model retains the simplicity of Black-Scholes.

It improves pricing accuracy by incorporating past dependence.

The approach avoids arbitrage issues associated with fractional Brownian motion.

Abstract

Replacing Black-Scholes' driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the financial market. We present the development, testing, and implementation of a simplified alternative to using fractional Brownian motion for pricing derivatives. By relaxing the assumption of past independence of Brownian motion but retaining the Markovian property, we are developing a competing model that retains the mathematical simplicity of the standard Black-Scholes model but also has the improved accuracy of allowing for past dependence. This is achieved by replacing Black-Scholes' underlying process, Brownian motion, with a particular Gaussian Markov process, proposed by Vladimir Dobri\'{c} and Francisco Ojeda.