Pricing multi-asset American style options by low discrepancy mesh methods under levy processes

This paper discusses simulation methods for pricing American options under multivariate affine Levy process. Employing multivariate affine generalized hyperbolic (MAGH) and multivariate affine jump diffusion (MAJD) distributions, this paper derives their multidimensional Esscher transforms and employs the low discrepancy mesh (LDM) method proposed by Boyle et al. (2003) to price multidimensional American style derivatives. Stemming from our use of quasi-Monte Carlo techniques, the method is computationally efficient for high dimensional problems and is easy to implement. In addition, we first propose a new dimension reduction method referred as the directional control method (DC), which depends explicitly on the given function of interest to decompose covariance matrix in constructing multivariate affine models. The LDM methods based on DC method is highly more efficient than that based on Cholesky and PCA methods, which are widely used for covariance decomposition. Then, combining with a stratification method, for example using an inverse Gaussian bridge to simulate Normal Inverse Gaussian random variable, we could improve the efficiency both between (horizontally) and within (vertically) assets. Furthermore, the LDM estimates are higher than the corresponding estimates from the Least Square Method (LSM) of Longstaff and Schwartz (2001). This is consistent with the property that the LDM estimate is high bias while the LSM estimate is low bias. This property also ensures that the true option value will lie between these two bounds. Finally, many numerical examples are provided to support the comparative efficiency of the proposed method.