Application of the Fast Gauss Transform to Option Pricing

In many of the numerical methods for pricing American options based on the dynamic programming approach, the most computationally intensive part can be formulated as the summation of Gaussians. Though this operation usually requires O(NNʹ) work when there are Nʹ summations to compute and the number of terms appearing in each summation is N, we can reduce the amount of work to O(N + Nʹ) by using a technique called the fast Gauss transform. In this paper, we apply this technique to the multinomial method and the stochastic mesh method, and show by numerical experiments how it can speed up these methods dramatically, both for the Black-Scholes model and Mertonʹs lognormal jump-diffusion model. We also propose extensions of the fast Gauss transform method to models with non-Gaussian densities.