Cited By
View all- Chen ZFeng LLin X(2012)Simulating Lévy Processes from Their Characteristic Functions and Financial ApplicationsACM Transactions on Modeling and Computer Simulation10.1145/2331140.233114222:3(1-26)Online publication date: 1-Aug-2012
Inverse transform method for simulating Levy processes and discrete Asian options pricing
Pages 444 - 456
Abstract
The simulation of a Lévy process on a discrete time grid reduces to simulating from the distribution of a Lévy increment. For a general Lévy process with no explicit transition density, it is often desirable to simulate from the characteristic function of the Lévy increment. We show that the inverse transform method, when combined with a Hilbert transform approach for computing the cdf of the Lévy increment, is reliable and efficient. The Hilbert transform representation for the cdf is easy to implement and highly accurate, with approximation errors decaying exponentially. The inverse transform method can be combined with quasi-Monte Carlo methods and variance reduction techniques to greatly increase the efficiency of the scheme. As an illustration, discrete Asian options pricing in the CGMY model is considered, where the combination of the Hilbert transform inversion of characteristic functions, quasi-Monte Carlo methods and the control variate technique proves to be very efficient.
References
[1]
Asmussen, S., and J. Rosiński. 2001. "Approximation of small jumps of Lévy processes with a view towards simulation". Journal of Applied Probability 38 (2): 482--493.
[2]
Avramidis, A. N., and P. L'Ecuyer. 2006. "Efficient Monte Carlo and quasi-Monte Carlo option pricing under the variance-gamma model". Management Science 52 (12): 1930--1944.
[3]
Barndorff-Nielsen, O. E. 1998. "Process of Normal Inverse Gaussian Type". Finance and Stochastics 2 (1): 41--68.
[4]
Benhamou, E. 2002. "Fast Fourier transform for discrete Asian options". Journal of Computational Finance 6 (1): 49--68.
[5]
Benth, F., M. Groth, and P. Kettler. 2006. "A quasi-Monte Carlo algorithm for the normal inverse Gaussian distribution and valuation of financial derivatives". International Journal of Theoretical and Applied Finance 9:843--867.
[6]
Bondesson, L. 1982. "On simulation from infinitely divisible distributions". Advances in Applied Probability 14 (4): 855--869.
[7]
Boyarchenko, S., and S. Levendorskii. 2002. Non-Gaussian Merton-Black-Scholes Theory. Singapore: World Scientific.
[8]
Carr, P., H. Geman, D. B. Madan, and M. Yor. 2002. "The fine structure of asset returns: an empirical investigation". Journal of Business 75 (2): 305--332.
[9]
Carverhill, A. P., and L. J. Clewlow. 1990. "Flexible convolution". Risk 3:25--29.
[10]
Chen, Z., L. Feng, and X. Lin. 2011. "Simulating Lévy processes from their characteristic functions and financial applications". Working paper, University of Illinois at Urbana-Champaign.
[11]
Cont, R., and P. Tankov. 2004. Financial Modelling with Jump Processes. Boca Raton, Florida: Chapman & Hall/CRC.
[12]
Devroye, L. 1981. "On the computer generation of random variables with a given characteristic function". Computers & Mathematics with Applications 7 (6): 547--552.
[13]
Feng, L., and X. Lin. 2011. "Inverting analytic characteristic functions and financial applications". Working paper, University of Illinois at Urbana-Champaign.
[14]
Feng, L., and V. Linetsky. 2008a. "Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: a fast Hilbert transform approach". Mathematical Finance 18 (3): 337--384.
[15]
Feng, L., and V. Linetsky. 2008b. "Pricing options in jump-diffusion models: an extrapolation approach". Operations Research 56 (2): 304--325.
[16]
Fu, M. C. 2007. "Variance-gamma and Monte Carlo". In Advances in Mathematical Finance, edited by M. C. Fu, R. A. Jarrow, J. Yen, and R. J. Elliott, 21--35. Boston: Birkhäuser.
[17]
Glasserman, P. 2004. Monte Carlo Methods in Financial Engineering. Berlin, Germany: Springer-Verlag.
[18]
Glasserman, P., and Z. Liu. 2007, December. "Sensitivity estimates from characteristic functions". In Proceedings of the 2007 Winter Simulation Conference, edited by S. G. Henderson, B. Biller, M.-H. Hsieh, J. Shortle, J. D. Tew, and R. R. Barton, 932--940. Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc.
[19]
Glasserman, P., and Z. Liu. 2010. "Sensitivity estimates from characteristic functions". Operations Research 58 (6): 1611--1623.
[20]
Hörmann, W., and J. Leydold. 2003. "Continuous random variate generation by fast numerical inversion". ACM Transations on Modeling and Computer Simulation 13:347--362.
[21]
L'Ecuyer, P. 2004, December. "Quasi-Monte Carlo methods in finance". In Proceedings of the 2004 Winter Simulation Conference, edited by R. G. Ingalls, M. D. Rossetti, J. S. Smith, and B. A. Peters. Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc.
[22]
L'Ecuyer, P., and C. Lemieux. 2002. "Recent advances in randomized quasi-Monte Carlo methods". In Modeling Uncertainty: an Examination of Stochastic Theory, Methods, and Applications, edited by M. Dror, P. L'Ecuyer, and F. Szidarovszki, 419--474. Boston: Kluwer Academic Publishers.
[23]
Madan, D. B., P. P. Carr, and E. C. Chang. 1998. "The variance gamma process and option pricing". European Finance Review 2 (1): 79--105.
[24]
Ribeiro, C., and N. Webber. 2003. "Valuaing path-dependent options in the variance-gamma model by Monte Carlo with a gamma bridge". Journal of Computational Finance 7:81--100.
[25]
Rosiński, J. 2001. "Series representations of Lévy processes from the perspective of point processes". In Lévy Processes--Theory and Applications, edited by O. Barndorff-Nielsen, T. Mikosch, and S. Resnick, 401--415. Boston: Birkhäuser.
[26]
Rydberg, T.H. 1997. "The normal inverse Gaussian Lévyprocess: simulation and approximation". Stochastic Models 13 (4): 887--910.
[27]
Schoutens, W. 2003. Lévy Processes in Finance: Pricing Financial Derivatives. Hoboken, New Jersey: John Wiley & Sons.
[28]
Shaw, W. T., and J. McCabe. 2009. "Monte Carlo sampling given a characteristic function: quantile mechanics in momentum space". Available from http://arxiv.org/abs/0903.1592.
[29]
Stein, E. M., and G. Weiss. 1971. Introduction to Fourier Analysis on Euclidean Spaces. Princeton, New Jersey: Princeton University Press.
Index Terms
- Inverse transform method for simulating Levy processes and discrete Asian options pricing
Recommendations
Pricing of early-exercise Asian options under Lévy processes based on Fourier cosine expansions
In this article, we propose a pricing method for Asian options with early-exercise features. It is based on a two-dimensional integration and a backward recursion of the Fourier coefficients, in which several numerical techniques, like Fourier cosine ...
A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options under Lévy Processes
A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-...
Pricing Bermudan Options in Lévy Process Models
This paper presents a Hilbert transform method for pricing Bermudan options in Lévy process models. The corresponding optimal stopping problem can be solved using a backward induction, where a sequence of inverse Fourier and Hilbert transforms needs to be ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
Sponsors
- IIE: Institute of Industrial Engineers
- INFORMS-SIM: Institute for Operations Research and the Management Sciences: Simulation Society
- SCS: Shanghai Computer Society
- SIGSIM: ACM Special Interest Group on Simulation and Modeling
Publisher
Winter Simulation Conference
Publication History
Published: 11 December 2011
Check for updates
Qualifiers
- Research-article
Conference
WSC'11
Sponsor:
- IIE
- INFORMS-SIM
- SCS
- SIGSIM
Acceptance Rates
WSC '11 Paper Acceptance Rate 203 of 270 submissions, 75%;
Overall Acceptance Rate 3,413 of 5,075 submissions, 67%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 48Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 25 Nov 2024
Other Metrics
Citations
Cited By
View all- Chen ZFeng LLin X(2012)Simulating Lévy Processes from Their Characteristic Functions and Financial ApplicationsACM Transactions on Modeling and Computer Simulation10.1145/2331140.233114222:3(1-26)Online publication date: 1-Aug-2012
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in