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Volume 10/Number 2, Winter 2006/07 Research Papers A general dimension reduction technique
for derivative pricing Junichi Imai
Graduate School of Economics and Management, Tohoku University, 27-1 Kawauchi, Aoba-ku, Sendai Miyagi 980-8576, Japan Ken Seng Tan
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada and China Institute for Actuarial Science, Central University of Finance and Economics, 39 South College Road, Haidian, Beijing 100081, China For a trajectory simulated from s standardized independent normal variates
ε = (ε1, . . . , εs )1, the payoff of a European option can be represented as
max[g(ε), 0], where the function g(ε) is assumed to be differentiable and it
relates to the nature of the derivative securities. In this paper, we develop a
new simulation technique by introducing an orthogonal class of transformation
to ε so that the function g is instead generated from g(Aε), where A is an
s-dimensional orthogonal matrix. The matrix A is optimally determined so
that the effective dimension of the underlying function is minimized, thereby
enhancing the quasi-Monte Carlo (QMC) method. The proposed simulation
approach has the advantage of greater generality and has a wide range of
applications as long as the problem of interest can be represented by g(ε).
The flexibility of our proposed technique is illustrated by applying it to two
high-dimensional applications: Asian basket options and European call options
with a stochastic volatility model. We benchmark our proposed method against
well-known efficient simulation algorithms that have been advocated in these
applications. The numerical results demonstrate that our proposed technique
can be an extremely powerful simulation method when combined with QMC.
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