Chebyshev Reduced Basis Function applied to Option Valuation

This paper concerns the design of a Reduced basis function approach to mitigate the impact of the "Curse of Dimensionality" which appears when we deal with multidimensional interpolation, in particular, when we price financial derivatives employing multivariable models, like GARCH, or whose price may depend on multiple assets which follow different stochastic processes. In this kind of problems, multidimensional models appear and, very often, explicit formulas are not available. Numerical approximations of the value of the functions for different parameter values must be computed and one approach is to perform polynomial interpolation. For a high number of dimensions, the memory requirements for storing the interpolant, and possibly the evaluation time, grows drastically. Employing information from the interpolant, the technique that we propose builds a set of orthonormal polynomials that are employed to construct a new polynomial which gives similar accuracy as the interpolant but where the memory requirements (and the evaluation time) are greatly reduced