Pricing financial derivatives using Brownian motion and a Gaussian Markov alternative to fractional Brownian motion

Abstract

This thesis examines different models for pricing financial options. Instead of using Brownian motion as the underlying process, as is done in the Black-Scholes model, fractional Brownian motion is introduced and discussed. Then the Dobri-Ojeda process, a Gaussian Markov alternative, and a modified version of it will be presented as an alternative to fractional Brownian motion, based on the analysis of Conus and Wildman. In contrast to Brownian motion, fractional Brownian motion and its alternatives incorporate past dependencies, using the Hurst index. The Black-Scholes and the Conus-Wildman model will be tested on options of the S&P 500 index, where the implied volatility and the implied Hurst index are estimated. The pricing accuracy of the two models will be compared using the obtained estimators. We find that the Conus-Wildman model estimates option prices better than the Black-Scholes model, concluding that past dependencies matter and should be incorporated when pricing options.