In the original Black-Scholes model, the risky asset process is driven by a standard Brownian motion and the risk is quantified by a constant volatility parameter. The volatility that corresponds to actual market data for option prices in Black-Scholes model is called the implied volatility. Thus, if we may observe the market price of the option, then the implied volatility, that is, the volatility implied by the market price, can be determined by inverting the option formula.
A natural generalization is to model the constant volatility parameter by a stochastic process. There is precedent for the work where the risky asset process and the volatility-driving process are driven by standard Brownian motions. A typical situation is as follows: The risky asset process X is driven by a standard Brownian motion W and the volatility-driving process Y is driven by another standard Brownian motion so that Y is a fast mean-reverting Ornstein-Uhlenbeck process, under the assumption that the standard Brownian motions W and have constant correlation For instance, Fouque et al. [7] consider such a situation, derive an approximation for option prices by a singular perturbation expansion and obtain the implied volatility by an approximating price. However, Fouque et al. [6], Lee [15] and Sircar and Papanicolaou [26] show the need for introducing also a slowly varying factor in the model for the stochastic volatility. The fast mean-reversion approximation is particularly suited for pricing long-dated options, whereas the slow mean-reversion approximation is particularly suited for pricing short-dated options.
Here we consider a Black-Scholes model where the risky asset process X is driven by a standard Brownian motion W and the volatility-driving process Y is driven by a fractional Brownian motion (fBm) with arbitrary Hurst parameter so that Y is a mean-reverting fractional Ornstein-Uhlenbeck process (fOU process); we assume that W and are independent, that is, volatility shocks are uncorrelated with asset-price shocks.
The rate of mean-reversion a of a mean-reverting fOU process Y is characterized in terms of and d with small positive parameters e and d according to fast scale and slow one, respectively. In each case, we obtain the corrected Black-Scholes price for European call option and hence asymptotic expansion for the implied volatility. In the case of fast scale, the corrected Black-Scholes price is derived by a singular perturbation analysis of the pricing partial differential equation as and the asymptotic expansion for the implied volatility is obtained by a regular perturbation analysis as On the other hand, in the case of slow scale, both the corrected Black-Scholes price and the asymptotic expansion for the implied volatility are derived by a regular perturbation analysis as
In order to obtain a pricing partial differential equation, we shall need to apply fractional Ito formula to the differential of the total value of the portfolio influenced by fBm with arbitrary Hurst parameter For this purpose, we shall take the stochastic integral with respect to fBm for algebraically integrable integrands in the sense of Hu [11] and hence obtain a concrete and computable expression for fractional Ito formula.
Our theorems correspond to an extension of the results in Fouque et al. [6, 7], Lee [15] and Sircar and Papanicolaou [26] to a Black-Scholes model with fOU process as volatility-driving process under the uncorrelated condition such that volatility shocks and asset-price shocks are independent.
