Efficient Monte Carlo and quasi-Monte Carlo option pricing under the variance gamma model

We develop and study efficient Monte Carlo algorithms for pricing path-dependent options with the variance gamma model. The key ingredient is difference-of-gamma bridge sampling, based on the representation of a variance gamma process as the difference of two increasing gamma processes. For typical payoffs, we obtain a pair of estimators (named low and high) with expectations that (1) are monotone along any such bridge sampler, and (2) contain the continuous-time price. These estimators provide pathwise bounds on unbiased estimators that would be more expensive (infinitely expensive in some situations) to compute. By using these bounds with extrapolation techniques, we obtain significant efficiency improvements by work reduction. We then combine the gamma bridge sampling with randomized quasi-Monte Carlo to reduce the variance and thus further improve the efficiency. We illustrate the large efficiency improvements on numerical examples for Asian, lookback, and barrier options.