An asymptotic expansion scheme in finance initiated by Kunitomo
and Takahashi [15] and Yoshida[68] is a widely applicable methodology
for analytic approximation of the expectation of a certain functional of
diffusion processes. [46], [47] and [53] provide explicit formulas of conditional
expectations necessary for the asymptotic expansion up to the
third order. In general, the crucial step in practical applications of the
expansion is calculation of conditional expectations for a certain kind of
Wiener functionals. This paper presents two methods for computing the
conditional expectations that are powerful especially for high order expansions:
The first one, an extension of the method introduced by the
preceding papers presents a general scheme for computation of the conditional
expectations and show the formulas useful for expansions up to
the fourth order explicitly. The second one develops a new calculation
algorithm for computing the coefficients of the expansion through solving
a system of ordinary differential equations that is equivalent to computing
the conditional expectations. To demonstrate their effectiveness, the
paper gives numerical examples of the approximation for -SABR model
up to the fifth order and a cross-currency Libor market model with a general
stochastic volatility model of the spot foreign exchange rate up to the
fourth order.
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