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

Recently, not only academic researchers but also many practitioners have used the methodology so-called "an asymptotic expansion method" in their proposed techniques for a variety of financial issues. e.g. pricing or hedging complex derivatives under high-dimensional stochastic environments. This methodology is mathematically justified by Watanabe theory(Watanabe [1987], Yoshida [1992a,b]) in Malliavin calculus and essentially based on the framework initiated by Kunitomo and Takahashi [2003], 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 up to high orders in situations where the underlying processes are highly volatile as seen in recent financial markets. After Takahashi [1995,1999] and Takahashi and Takehara [2007] had provided explicit formulas for the expansion up to the third order, Takahashi, Takehara and Toda [2009] develops general computation schemes and formulas for an arbitrary-order expansion under general diffusion-type stochastic environments. In this paper, we describe them in a simple setting to illustrate thier key idea, and to demonstrate their effectiveness apply them to pricing long-term currency options under a cross-currency Libor market model and a general stochastic volatility of a spot exchange rate with maturities up to twenty years.