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

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 due to Avramidis et al. (2003) and Ribeiro and Webber (2004), 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 both in plain Monte Carlo and QuasiMonte Carlo environments. It is tested in pricing lookback, barrier and Asian options and shown to provide consistent efficiency gains, compared to the sequential method and the difference-of-gammas bridge sampling due to Avramidis and L’Ecuyer (2006)