In this paper, we study a classical extension of the Black and Scholes model for asset prices and option pricing, generally known as the Heston model. In our specification, the volatility is a fractional integral of a Cox, Ingersoll, Ross process (also known as an "affine" model): this implies that it is not only stochastic but also admits long memory features. We study the volatility and the integrated volatility processes and prove their long memory properties. We address the issue of option pricing and we study discretizations of the model. Lastly, we provide an estimation strategy and simulation experiments in order to test this methodology.

This paper proposes semi-closed-form solutions to value derivatives on mean-reverting assets. We consider a very general mean-reverting process for the underlying asset and two stochastic volatility processes: the Square-Root process and the Ornstein-Uhlenbeck process. For both models, we derive semi-closed-form solutions for Characteristic Functions, in which we need to solve simple Ordinary Differential Equations, and then invert them to recover the cumulative probabilities using the Gaussian-Laguerre quadrature rule. As benchmarks, we use our models to value European Call options within Black-Scholes (1973) (representing constant volatility and no mean-reversion), Longstaff-Schwartz (1995) (representing constant volatility and mean-reversion), Heston (1993) and Zhu (2000) (representing stochastic volatility and no mean-reversion) frameworks. These comparisons show that we only need polynomials with small degree for convergence and accuracy. Indeed, when applied to our processes (representing stochastic volatility and mean-reversion), the Gaussian-Laguerre rule is very efficient and very accurate. We also show that the mean-reversion could have a large impact on option prices even though the strength of the reversion is small. As applications, we value credit spread options, caps, floors and swaps.