Abstract: In the
classical Black-Scholes model, the risk asset is taken in a standard Brownian
environment, where the risk is quantified by a constant volatility parameter. It
has been proposed by many authors that the volatilities should be modeled by a
stochastic process to obtain a more realistic model. For example, see Fouque et
al. [7, 8], Cotton et al. [5], andKallianpur
and Karandikar [15]. Precedent is singular perturbation analysis for financial
markets with stochastic volatility, which is a function of fast mean-reverting
Ornstein-Uhlenbeck process driven by a standard Brownian motion.
Here we
consider the European call option in a fractional Black-Scholes model in a
financial market that has two instruments: a risk-less asset and a risky asset.
A risky asset process Xis
governed by a standard Brownian motion W, whereas stochastic volatility
is a function of fast mean-reverting Ornstein-Uhlenbeck process Ywhich is influenced by a fractional Brownian motion with Hurst parameter We
are interested in three parameters describing Y: (i) the effective
volatility which is obtained by the average
with respect to the long-run distribution of Y, (ii) the rate of mean
reversion awhich is characterized in terms of with a small parameter e,
and (iii) the variance of the long-run distribution of Ywhich is dependent on
Hurst
parameter H.
Our aim is to obtain
asymptotics of the price of a European call option as We
can derive the pricing partial differential equation in terms of e, and obtain that the corrected Black-Scholes price
is given by sum of the classical Black-Scholes price with constant volatility
and the corrected term. Our theorem is an extension of the results in Fouque et
al. [7] and Kallianpur and Karandikar [15] to a fractional Black-Scholes model
with uncorrelated W and
Keywords and phrases: fractional Brownian motion, Ornstein-Uhlenbeck process, fractional Ito-integral, stochastic differential equation, Black-Scholes equation, European call option.
