École Nationale Supérieure de
Techniques Avancées

Abstract

The study of semi-stationary Brownian processes (BSS) introduced by Barndorff-Nielsenet Schmiegel [1] [2] which are stochastic processes (not typically semi-martingales) leads us to explore discretization schemes to simulate these types of processes using the kernel function. These processes are particularly useful since they have been used to study turbulence in physics and finance especially as models of energy prices. These processes are interesting since we can see similarities with time series such as roughness parameter and stochastic volatility. In this project, we use the hybrid scheme that is introduced in the article [3] to discretize the stochastic integral representation of the semi-stationary Brownian process. We propose to take a closer look at the theoretical study of the mean squared error associated with this scheme. There is a considerable reduction in the mean square error when using this scheme correctly. We use this scheme in particular to price an option according to the rough bergomi model introduced by [4], and to estimate the roughness parameter using the COF (change of frequency statistics). We also manage to find the implied volatility curve as a function of strike and maturity and the related volatility surface. We find the usual shape of a smile. This scheme guarantees a reduction of the mean square error as well as the complexity thanks to the FFT method (Fast Fourier Transform).