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The Combination Method for Multidimensional Black-Scholes Partial Differential Equations
A fundamental challenge in the numerical solution of multidimensional partial differential equations (PDEs) is the exponential increase of the number of unknowns as the number of dimensions increases; on an isotropic grid with d dimensions and n unknowns in each dimension, the number of unknowns is O(n^d). This exponential increase is known as the curse of dimensionality, which creates difficulties in obtaining accurate solutions for even moderate-dimension problems. In the context of computational finance, multidimensional PDEs arise from the pricing of multiasset options or options with multiple risk factors considered, as each option/risk factor leads to a spatial variable. We use a sparse grid combination method to alleviate the curse of dimensionality and present computationally efficient results for multidimensional pricing problems for European and American options in the Black-Scholes model.