Variance-Gamma and Monte Carlo Michael C. Fu Robert H. Smith School of Business Department of Decision & Information Technologies University of Maryland College Park, MD 20742, USA mfu@rhsmith.umd.edu Summary. The Variance-Gamma (VG) process was introduced by Dilip B. Madan and Eugene Seneta as a model for asset returns in a paper that appeared in 1990, and subsequently used for option pricing in a 1991 paper by Dilip and Frank Milne. This paper serves as a tutorial overview of VG and Monte Carlo, including three methods for sequential simulation of the process, two bridge sampling methods, variance reduction via importance sampling, and estimation of the Greeks. Key words: Variance-Gamma process; L´ evy processes; Monte Carlo simulation; bridge sampling; variance reduction; importance sampling; Greeks; perturbation analysis; gradient estimation. 1 Introduction 1.1 Reflections on Dilip Dilip and I have been colleagues since 1989, when I joined the faculty of the Business School at the University of Maryland, and just one year after Dilip himself had returned to his (Ph.D.) alma mater after twelve years on the faculty at the University of Sydney, Australia, following the earning of his two Ph.D.s in pure math (1975) and economics (1971). However, because we were in different departments—he in the Finance Department and I in what was then called the Management Science & Statistics Department—we did not re- ally start collaborating until the mid-1990s when I first became interested in fi- nancial applications. Since then, Dilip and I have co-chaired five Ph.D. student dissertations (Tong Wang, Rongwen Wu, Xing Jin, Yi Su, Sunhee Kim), and each of us serves regularly on dissertation committees of the other’s Ph.D. stu- dents. We team-taught a course on computational finance twice (1997, 1999). We have led a research group on computational/mathematical finance since the late 1990s, which became known as the Mathematical Finance Research Interactions Team (RIT) under the interdisciplinary Applied Mathematics and