Application of a High-Order Asymptotic Expansion Scheme to Long-Term Currency Options

Recently academic researchers and practitioners have use the asymptotic expansion method to examine a variety of financial issues under high-dimensional stochastic environments. This methodology is mathematically justified by Watanabe theory (Watanabe, 1987), and Malliavin calculus (Yoshida, 1992a,b) and essentially based on the framework initiated by Kunitomo and Takahashi (2003) and Takahashi (1995, 1999) in a financial context. In practical applications, it is desirable to investigate the accuracy and stability of the method especially with expansion to higher orders in situations where the underlying processes are highly volatile. After Takahashi (1995,1999) and Takahashi and Takehara (2007) provided explicit formulas for the expansion to the third order, Takahashi, Takehara and Toda (2009) develop general computation schemes and formulas for an arbitrary-order expansion under general diffusion-type stochastic environments. In this paper, we describe these techniques in a simple setting to illustrate thier key ideas. To demonstrate their effectiveness the techniques are applied to pricing long-term currency options