École Nationale Supérieure de
Techniques Avancées

Abstract

We study stochastic volatility as a Volterra process. We start by shortly presenting this precise type of process and simulating it numerically. We focus on asset Pricing in the fractional Heston model. This model has been lately in the spotlight, which is the reason why we use the literature to expose it. In order to respond to the Pricing problem we solve a family of fractional differential equations called fractional Riccati equations. We use a so-called \textit{hybrid} method to do so. Within this method, the solution takes the form of a fractional power series (with a finite convergence radius) on the convergence domain. Out of this domain, the solution is numerically calculated using an Euler scheme. We then test this method by reproducing numerical experiences. Our tests show a real efficiency and stability of the method. However, we need to be cautious about the influence of numerical values of the problem's parameters on the conclusions drawn at the end of the paper.