Pricing barrier and average options in a stochastic volatility environment

A new approximation method for pricing barrier and average options in a stochastic volatility environment by applying an asymptotic expansion approach is proposed in this paper. In particular, a high-order expansion scheme for general multidimensional diffusion processes is successfully applied. Moreover, the paper combines a static hedging method with the asymptotic expansion method for pricing barrier options. Finally, numerical examples show that the fourth-order or fifth-order asymptotic expansion scheme provides sufficiently accurate approximations using the stochastic alpha, beta, rho (SABR) and lambda-SABR models.