  This paper considers the valuation of a European call option under the Heston
stochastic volatility model. We present the asymptotic solution to the option
pricing problem in powers of the volatility of variance. Then we introduce the
artificial boundary method for solving the problem on a truncated domain, and
derive several artificial boundary conditions (ABCs) on the artificial boundary
of the bounded computational domain. A typical finite difference scheme and
quadrature rule are used for the numerical solution of the reduced problem.
Numerical experiments show that the proposed ABCs are able to improve the
accuracy of the results and have a significant advantage over the widely-used
boundary conditions by Heston in the original paper (Heston, 1993).
