Consider the American basket call option in the case where there are N underlying assets, the number of possible exercise times prior to maturity is finite, and the vector
of N asset prices is modeled using a Levy process. A numerical method based on regression and numerical integration is proposed to estimate the price of the American option. In the proposed method, we first express the asset prices as nonlinear functions of N uncorrelated standard normal random variables. For a given set of time-t asset prices, we next determine the time-t continuation value by performing a numerical integration along the radial direction in the N-dimensional polar coordinate system for the N uncorrelated standard normal random variables, expressing the integrated value via a regression procedure as a function of the polar angles, and performing a numerical integration over the polar angles. The larger value of the continuation value and the time-t immediate exercise value will then be the option value. The time-t option values over the N-dimensional space may be represented by a quadratic function of the radial
distance, with the coefficients of the quadratic function given by second degree polynomials in N-1 polar angles. Partitioning the maturity time T into k* intervals of
length Δt, we obtain the time-(k-1)Δt option value from the time-kΔt option values for k= k*, k*-1,…, 1. The time-0 option value is then the price of the American option. It is
found that the numerical results for the American option prices based on regression and numerical integration agree well with the simulation results, and exhibit a variation of
the prices as we vary the non-normality of the underlying distributions of the assets. To assess the accuracy of the computed price we may use estimated standard error of the
computed American option price. The standard error will help us gauge whether the number of selected points along the radial direction and the number of selected polar angles are large enough to achieve the required level of accuracy for the computed American option price
