Option pricing with sparse grids
We investigate the suitability of sparse grids for solving high-dimensional option pricing and interest rate models numerically. Starting from the partial differential equation, we try to - at least partially - break the curse of dimensionality through sparse grids which will result from a multi-level splitting of the solution. We make use of an adaptive algorithm in spacetime exploiting the smoothness of the solution. In order to compute sensitivities (the so-calles "greeks"), we avail of interpolets as a smooth basis function leading to faster convergence. Finite differences allow us to adjust the order of consistency. The code providing the results of the paper was designed for fast solving making use of an efficient preconditioner and parallelization. The specific choice of boundary conditions is crucial to obtaining good approximations to the true solution. Different types will be compared here. Our findings suggest the usage of locally full grids in order to approximate the singularity in the initial data. However, this modification does not lead to a deterioration of the speed of convergence which will yield a rate of 4 for the solution. That means the Gamma sensitivity converges as a second derivative at a rate of 2
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