A Monte Carlo approach to fractional Brownian motion

Used widely in physics, from the study of crystals to activity of solar flares, Fractional Brownian Motion (fBM) has witnessed an outburst of research in the last decade in the field of financial mathematics, giving birth to the field of rough volatility models. These models have the ability to account for a range of stylized facts of financial markets which Brownian motion based alternatives simply fail to replicate. However, rough volatility models are challenging to implement due to the non-Markovian and non-semimartingale nature of fBM.In this thesis we are leveraging Monte Carlo methods to investigate the efficiency of the Euler discretization scheme for the Rough Heston model. Our analysis reveals that the Euler implementation is not as effective as moment matching schemes, such as the Quadratic Exponential approach, in the classical Heston context, particularly when the Feller condition is not met. Despite this, it is capable of efficiently pricing European, Asian and Lookback options in the short term in the Rough Heston model. We also investigate convergence of the Euler scheme to the Rough Heston characteristic function and comment on its range of usability.