Dirichlet Bridge Sampling for the Variance Gamma Process: Pricing Path-Dependent Options

Abstract

The authors develop a new Monte Carlo-based method for pricing path-dependent options under the variance gamma (VG) model. The gamma bridge sampling method proposed by Avramidis et al. (Avramidis, A. N., P. L'Ecuyer, P. A. Tremblay. 2003. Efficient simulation of gamma and variance-gamma processes. Proc. 2003 Winter Simulation Conf. IEEE Press, Piscataway, NJ, 319–326) and Ribeiro and Webber (Ribeiro, C., N. Webber. 2004. Valuing path-dependent options in the variance-gamma model by Monte Carlo with a gamma bridge. J. Computational Finance 7(2) 81–100) is generalized to a multivariate (Dirichlet) construction, bridging “simultaneously” over all time partition points of the trajectory of a gamma process. The generation of the increments of the gamma process, given its value at the terminal point, is interpreted as a Dirichlet partition of the unit interval. The increments are generated in a decreasing stochastic order and, under the Kingman limit, have a known distribution. Thus, simulation of a trajectory from the gamma process requires generating only a small number of uniforms, avoiding the expensive simulation of beta variates via numerical probability integral inversion. The proposed method is then applied in simulating the trajectory of a VG process using its difference-of-gammas representation. It has been implemented in both plain Monte Carlo and quasi-Monte Carlo environments. It is tested in pricing lookback, barrier, and Asian options and is shown to provide consistent efficiency gains, compared to the sequential method and the difference-of-gammas bridge sampling proposed by Avramidis and L'Ecuyer (Avramidis, A. N., P. L'Ecuyer. 2006. Efficient Monte Carlo and quasi-Monte Carlo option pricing under the variance gamma model. Management Sci. 52(12) 1930–1944).