Abstract: In
a classical Black-Scholes model, option pricing is considered under a standard
Brownian environment, where the risk is quantified by a constant volatility
parameter. For a more realistic model, Fouque et al. [8, 9], Cotton et al. [5]
and Kallianpur and Karandikar [16] consider stochastic volatility as a function
of fast mean-reverting Ornstein-Uhlenbeck process in a standard Brownian
environment, and then obtain asymptotics for option pricing by singular
perturbation analysis.
We
consider the European call option in a fractional Black-Scholes model with two
instruments; a risk-less asset and a risky asset. A risky asset process X
is governed by a standard Brownian motion W,
whereas stochastic
volatility is a function of fast mean-reverting fractional Ornstein-Uhlenbeck
process Y which is driven by a fractional Brownian motion with Hurst parameter There are 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 a
which is characterized in terms of with a small parameter e, (iii) the variance of the long-run distribution of Y
which is dependent on Hurst parameter H.
In the case, where Narita [18] obtains asymptotics of
the price of a European call option as Here we extend the precedent result
by the author to the general case where For this purpose, we take the
stochastic integral for general integrands such that they are algebraically
integrable in the sense of Hu [12]. Such an integral enables us to get a
concrete Ito formula, and hence we can apply it to easy calculations of
financial derivatives in a fractional Brownian environment with arbitrary
Hurst
parameter
We can derive the
pricing partial differential equation in terms of e, and obtain the corrected Black-Scholes price as
sum of the classical Black-Scholes price with the effective constant volatility and the corrected term. Further, we
can obtain the explicit expression for the quantity appearing in the corrected
term. Our theorem is automatically an extension of the results in Fouque et al.
[9] and Kallianpur and Karandikar [16] to a fractional Black-Scholes model with
uncorrelated W and
Keywords and phrases: fractional Brownian motion, Ornstein-Uhlenbeck process, stochastic integral, Ito formula, stochastic differential equation, Black-Scholes equation, European call option, singular perturbation.
