Appl Math Optim 52:279–296 (2005)
2005 Springer Science+Business Media, Inc.
Uniform Asymptotic Expansions for Pricing European Options
Rafail Z. Khasminskii and G. Yin
Department of Mathematics, Wayne State University,
Detroit, MI 48202, USA
Communicated by A. Bensoussan
Abstract. Starting with a stochastic volatility model, in which the volatility de-
pends on a nonlinear function of a fast varying diffusion, and assuming the fast
diffusion is mean reverting, the problem of pricing European options is considered
in this paper. Uniform asymptotic expansions of the option price are obtained. The
formal expansions are justiﬁed and the uniform error bounds are derived using outer
and inner expansions of the option prices.
Key Words. Singular perturbation, Two-time scale, Diffusion, European option.
AMS Classiﬁcation. 34E05, 60J27, 60F05.
Financial markets are sometimes quite calm and at other times much more volatile.
The celebrated Black–Scholes theory provides an important tool for pricing options.
On the other hand, it has been recognized that the assumption of constant volatility,
which is essential in their theory, is a less-than-perfect description of the real world. To
capture the complicated behavior of stock prices and other derivatives, it is necessary
to take into consideration frequent changes of the volatility. It is more suitable to use a
stochastic process to model the variation of the volatility; see . In lieu of the usual
GBM (geometric Brownian motion) model or log-normal model, a second stochastic
differential equation is used to describe the random environments. Such a formulation
is known as a stochastic volatility model.
This research was supported in part by the National Science Foundation under Grants DMS-9971608