On fractional volatility modelling

Abstract

In this thesis, we investigate the roughness feature within realised volatility for different financial markets by using the multifractal detrended fluctuation approach and microstructure noise index technique, and we confirm that the Hurst parameter H 6= 1/2. To include this feature in stochastic volatility modelling, we construct an arbitrage-free financial market model that con sists of two assets, the risk-free and the risky assets. The price of a risk-free asset is described by an exponential function while the one for a risky as set is driven by a geometric Brownian motion with its stochastic volatility described as a function of fractional Cox-Ingersoll-Ross process defined by Yt = Z 2 t , where the process (Zt)t≥0 satisfies a singular stochastic differential driven by fractional Brownian motion (WH t )t≥0, H∈(0,1). The stochastic pro cess (Zt)t≥0 verifies dZt =