The classical Black-Scholes formula gives the price of European call options when the risky asset process is a geometric Brownian motion with a constant volatility. A natural generalization is to model the constant volatility parameter by a stochastic process. There is precedent for the work of the option pricing where the risky asset process and the volatility process are in the standard Brownian motion environment. For instance, Fouque et al. [6] take a fast-mean reverting Ornstein-Uhlenbeck process as a volatility process, derive an approximation for option prices by a singular perturbation expansion, and then obtain the implied volatility by an approximating price. On the other hand, Fouque et al. [5], Lee [16] and Sircar and Papanicolaou [24] show the need for introducing also a slowly varying factor in the model for the stochastic volatility.
Inspired by these works, Narita [18-20] investigates a class of models where the volatility process is driven by either fast or slow mean-reverting fractional Ornstein-Uhlenbeck process with arbitrary Hurst parameter and obtains a corrected price of the European call option, and hence shows an asymptotics of the implied volatility. These are given under the uncorrelated condition such that volatility shocks and asset-price shocks are independent.
Here, we introduce a class of multiscale stochastic volatility models driven by fractional Brownian motions. More precisely, we consider the volatility process which is simultaneously driven by two fractional Ornstein-Uhlenbeck processes with arbitrary Hurst parameter one fluctuating on a fast time-scale and the other fluctuating on a slow time-scale. We shall show that it is possible to combine a singular perturbation expansion with respect to the fast scale, and with a regular perturbation expansion with respect to the slow scale. Then, under the assumption that volatility shocks are uncorrelated with asset-price shocks, we shall obtain a corrected price of the European call option and the asymptotics of the implied volatility. As a result, a corrected price is expanded around the classical Black-Scholes price with an effective volatility which depends on the slow factor. In this case, a leading order term of implied volatility is also expanded around the same effective volatility.
For these purposes, we shall consider the total value of the portfolio influenced by fractional Brownian motion with arbitrary Hurst parameter appeal to the fractional Ito formula as given by Hu [10], and hence derive the pricing partial differential equation.
As mentioned above, Narita [18-20] treats a restricted case where the rate of mean-reversion of fractional Ornstein-Uhlenbeck process is either fast or slow. Our theorems extend these to multiscale stochastic volatility model where both fast and slow time scales are involved.
On the other hand, Fouque et al. [5] investigate multiscale stochastic volatility model in standard Brownian motion environment where volatility shocks and asset-price shocks are correlated. Our theorems extend their results to fractional Brownian motion environment although volatility shocks and asset-price shocks are uncorrelated.
