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.
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