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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A.N. Avramidis and P. L’Ecuyer. Efficient Monte Carlo and quasi-Monte Carlooption pricing under the Variance-Gamma model. Management Science, 52:1930–1944, 2006.
P. Boyle, M. Broadie, and P. Glasserman. Monte Carlo methods for securitypricing. Journal of Economic Dynamics and Control, 21:1267–1321, 1997.
M. Broadie and P. Glasserman. Estimating security price derivatives usingsimulation. Management Science, 42:269–285, 1996.
P. Carr, H. Geman, D.B. Madan, and M. Yor. The fine structure of assetreturns: An empirical investigation. Journal of Business, 75:305–332, 2002.
P. Carr, H. Geman, D.B. Madan, and M. Yor. Stochastic volatility for Lévy processes. Mathematical Finance, 2:87–106, 2003.
P. Carr and D. Madan. Option valuation using the fast Fourier transform. Journal of Computational Finance, 2:61–73, 1999.
G.S. Fishman. Monte Carlo Methods: Concepts, Algorithms, and Applications. Springer, 1996.
M.C. Fu. Stochastic gradient estimation. Chapter 19 in Handbooks in Operations Research and Management Science: Simulation, eds. S.G.Henderson and B.L. Nelson, Elsevier, 2006.
M.C. Fu and J.Q. Hu, Sensitivity analysis for Monte Carlo simulation of option pricing. Probability in the Engineering and Informational Sciences, 9:417–446, 1995.
M.C. Fu and J.Q. Hu. Conditional Monte Carlo: Gradient Estimation and Optimization Applications. Kluwer Academic, 1997.
M.C. Fu, S.B. Laprise, D.B. Madan, Y. Su, and R. Wu. Pricing Americanoptions: A comparison of Monte Carlo simulation approaches. Journal of Computational Finance, 4:39–88, 2001.
M.C. Fu, D.B. Madan and T. Wang. Pricing continuous Asian options: Acomparison of Monte Carlo and Laplace transform inversion methods. Journal of Computational Finance, 2:49–74, 1999.
P. Glasserman. Gradient Estimation Via Perturbation Analysis. Kluwer Academic, 1991.
P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer, 2003.
A. Hirsa and D.B. Madan. Pricing American options under variance gamma. Journal of Computational Finance, 7:63–80, 2004.
P. Jäckel. Monte Carlo Methods in Finance. Wiley, 2002.
S. Laprise. Stochastic Dynamic Programming: Monte Carlo Simulation and Applications to Finance. Ph.D. dissertation, Department of Mathematics, University of Maryland, 2002.
D.B. Madan, P. Carr, and E.C. Chang. The variance gamma process and option pricing. European Finance Review, 2:79–105, 1998.
D.B. Madan and F. Milne. Option pricing with V.G. martingale components. Mathematical Finance, 1:39–55, 1991.
D.B. Madan and E. Seneta. The Variance-Gamma (V.G.) model for sharemarket returns. Journal of Business, 63:511–524, 1990.
D.B. Madan and M. Yor. Representing the CGMY and Meixner processes astime changed Brownian motions. manuscript, 2006.
D.L. McLeish. A robust alternative to the normal distribution. Canadian Journal of Statistics, 10:89–102, 1982.
D.L. McLeish, Monte Carlo Simulation and Finance. Wiley, 2005.
P. Protter. Stochastic Integration and Differential Equations, 2nd edition. Springer-Verlag, 2005.
C. Ribeiro, and N. Webber. Valuing path-dependent options in the Variance-Gamma model by Monte Carlo with a gamma bridge. Journal of Computational Finance, 7:81–100, 2004.
K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, 1999.
E. Seneta. The early years of the Variance-Gamma process. In thisvolume, 2007.
M. Yor. Some remarkable properties of gamma processes. In this volume,2007.
B. Zhang. A New Lévy Based Short-Rate Model for the Fixed Income Market and Its Estimation with Particle Filter. Ph.D. dissertation (directed by Dilip B. Madan), Department of Mathematics, University of Maryland, 2006.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Birkhäuser Boston
About this chapter
Cite this chapter
Fu, M.C. (2007). Variance-Gamma and Monte Carlo. In: Fu, M.C., Jarrow, R.A., Yen, JY.J., Elliott, R.J. (eds) Advances in Mathematical Finance. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4545-8_2
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4545-8_2
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4544-1
Online ISBN: 978-0-8176-4545-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)