Fractional stochastic volatility: F-Ornstein–Uhlenbeck and F-CIR processes

Abstract

We consider fractional Ornstein–Uhlenbeck process as well as fractional CIR-process with Hurst index H ∈ (0,1), and several approaches to the exact and approximate option pricing of the asset price model that is described by the geometric linear model with stochastic volatility, where volatility is driven by fractional Ornstein–Uhlenbeck process. We assume that the Wiener process driving the asset price and the fractional Brownian motion driving stochastic volatility are correlated. We consider three possible levels of representation and approximation of option price, with the corresponding rate of convergence of discretized option price to the original one. We can rigorously treat the class of discontinuous payoff functions of polynomial growth. As an example, our model allows to analyze linear combinations of digital and call options. Moreover, we provide rigorous estimates for the rates of convergence of option prices for polynomial discontinuous payoffs f and Hölder volatility coefficients, a crucial feature considering settings for which exact pricing is not possible