Abstract
Lévy processes are popular models for stock price behavior since they allow to take into account jump risk and reproduce the implied volatility smile. In this paper, we focus on the tempered stable (also known as CGMY) processes, which form a flexible 6-parameter family of Lévy processes with infinite jump intensity. It is shown that under an appropriate equivalent probability measure a tempered stable process becomes a stable process whose increments can be simulated exactly. This provides a fast Monte Carlo algorithm for computing the expectation of any functional of tempered stable process. We use our method to price European options and compare the results to a recent approximate simulation method for tempered stable process by Madan and Yor (CGMY and Meixner Subordinators are absolutely continuous with respect to one sided stable subordinators, 2005).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asmussen S., Rosiński J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. Journal of Applied. Probability 38, 482–493
Boyarchenko S., Levendorskiĭ S. (2002). Non-Gaussian Merton-Black-Scholes theory. River Edge, NJ, World Scientific
Carr P., Geman H., Madan D., Yor M. (2002). The fine structure of asset returns: An empirical investigation. Journal of Business 75, 305–332
Carr P., Madan D. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance 2: 61–73
Chambers J., Mallows C., Stuck B. (1976). A method of simulating stable random variables. Journal of American Statistical Assiciation 71, 340–344
Cont, R., Bouchaud, J.-P., & Potters, M. (1997). Scaling in financial data: Stable laws and beyond. In: B. Dubrulle, F. Graner, & D. Sornette (Eds.), Scale invariance and beyond. Berlin: Springer.
Cont R., Tankov P. (2004). Financial modelling with jump processes. Boca Raton, FL, Chapman & Hall/CRC Press
Cont R., Voltchkova E. (2005). A finite difference scheme for option pricing in jump-diffusion and exponential Lévy models. SIAM Journal on Numerical Analysis 43, 1596–1626
Gradshetyn I., Ryzhik I. (1995). Table of integrals, series and products. San Diego, DA, Academic Press
Koponen I. (1995). Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Physical Review E 52: 1197–1199
Madan, D. B., & Yor, M. (2005). CGMY and Meixner subordinators are absolutely continuous with respect to one sided stable subordinators. Prépublication du Laboratoire de Probabilités et Modèles Aléatoires.
Rosiński, J. (2001). Series representations of Lévy processes from the perspective of point processes. In: O. Barndorff-Nielsen, T. Mikosch, & S. Resnick (Eds.), Lévy processes—theory and applications. Boston: Birkhäuser.
Rosiński, J. (2004). Tempering stable processes. Preprint (cf. www.math.utk.edu/∼rosinski/manuscripts.html).
Sato K. (1999). Lévy processes and infinitely divisible distributions. Cambridge, UK, Cambridge University Press
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Poirot, J., Tankov, P. Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes. Asia-Pacific Finan Markets 13, 327–344 (2006). https://doi.org/10.1007/s10690-007-9048-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10690-007-9048-7