Simulation of Lévy-Driven models and its application in finance

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2012, 2(4): 749-765. doi: 10.3934/naco.2012.2.749

Simulation of Lévy-Driven models and its application in finance

Rachel Chen ^1, , Jianqiang Hu ^1, and Yijie Peng ^1,

1.	School of Management, Fudan University, Guoshun Road 670, Shanghai 200433, China, China, China

Received May 2012 Revised September 2012 Published November 2012

Abstract
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Lévy processes have been widely used to model financial assets such as stock prices, exchange rates, interest rates, and commodities. However, when applied to derivative pricing, very few analytical results are available except for European options. Therefore, one usually has to resort to numerical methods such as Monte Carlo simulation method. The simulation method is attractive in that it is very general and can also handle high dimensional problems very well. In this survey paper, we provide an overview on various simulation methods for Lévy processes. In addition, we introduce two simulation based sensitivity estimation methods: perturbation analysis and the likelihood ratio method. Sensitivity estimation is useful in various applications, such as derivative pricing and parameter estimation. Finally, we provide a simple illustrative example of applying simulation and sensitivity estimation to parameter estimation of Lévy-driven stochastic volatility model.

Keywords: maximum likelihood estimation., Lévy processes, Simulation, gradient estimation.

Mathematics Subject Classification: Primary: 60G05, 68U2.

Citation: Rachel Chen, Jianqiang Hu, Yijie Peng. Simulation of Lévy-Driven models and its application in finance. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 749-765. doi: 10.3934/naco.2012.2.749

References:

[1]	O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, Journal Of The Royal Statistical Society, Series B, 63 (2001), 167-241.
[2]	J. Bertoin, "Lévy Processes," 2^nd edition, Cambridge University Press, Cambridge, 1996.
[3]	P. Carr and L. Wu, Time-changed Lévy processes and option pricing, Journal of Financial Economics, 71 (2004), 113-141. doi: 10.1016/S0304-405X(03)00171-5.
[4]	Z. Chen, L. Feng and X. Lin, Simulating Lévy processes from their characteristic functions and financial applications, ACM Transactions on Modeling and Computer Simulation, 22 (2012).
[5]	R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Chapman & Hall/CRC, Boca Raton, Florida, 2004.
[6]	P. Glasserman, "Gradient Estimation via Perturbation Analysis," Kluwer Academic, Boston, 1991.
[7]	P. Glasserman, "Monte Carlo Methods in Financial Engineering," Springer, New York, 2004.
[8]	P. Glasserman and Z. Liu, Estimating Greeks in simulating Lévy-driven models, Journal of Computational Finance, 14 (2010), 3-56.
[9]	M. C. Fu and J. Q. Hu, "Conditional Monte Carlo: Gradient Estimation and Optimization Applications," Kluwer Academic, Boston, 1997.
[10]	M. C. Fu, Variance-Gamma and Monte Carlo, Advances in Mathematical Finance, (2007), 21-35.
[11]	P. W. Glynn, Likelihood ratio gradient estimation: An overview, Proceedings of the 1987 Winter Simulation Conference, (1987), 366-374.
[12]	Y. C. Ho and X. R. Cao, "Perturbation Analysis of Discrete Event Dynamic Systems," Kluwer Academic, Boston, 1991. doi: 10.1007/978-1-4615-4024-3.
[13]	B. Mondelbrot, "The variation of certain speculative prices," The Journal of Business, 36 (1963), 394-419. doi: 10.1086/294632.
[14]	Y. J. Peng, M. C. Fu and J. Q. Hu, Gradient-based simulated maximum likelihood estimation for Lévy-Driven Ornstein-Uhlenbeck stochastic volatility models, Working Paper, (2012).
[15]	K. I. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge University Press, Cambridge, 1999.
[16]	W. Schoutens, "Lévy Processes in Finance: Pricing Financial Derivatives," Wiley, New York, 2003.

show all references

References:

[1]	O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, Journal Of The Royal Statistical Society, Series B, 63 (2001), 167-241.
[2]	J. Bertoin, "Lévy Processes," 2^nd edition, Cambridge University Press, Cambridge, 1996.
[3]	P. Carr and L. Wu, Time-changed Lévy processes and option pricing, Journal of Financial Economics, 71 (2004), 113-141. doi: 10.1016/S0304-405X(03)00171-5.
[4]	Z. Chen, L. Feng and X. Lin, Simulating Lévy processes from their characteristic functions and financial applications, ACM Transactions on Modeling and Computer Simulation, 22 (2012).
[5]	R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Chapman & Hall/CRC, Boca Raton, Florida, 2004.
[6]	P. Glasserman, "Gradient Estimation via Perturbation Analysis," Kluwer Academic, Boston, 1991.
[7]	P. Glasserman, "Monte Carlo Methods in Financial Engineering," Springer, New York, 2004.
[8]	P. Glasserman and Z. Liu, Estimating Greeks in simulating Lévy-driven models, Journal of Computational Finance, 14 (2010), 3-56.
[9]	M. C. Fu and J. Q. Hu, "Conditional Monte Carlo: Gradient Estimation and Optimization Applications," Kluwer Academic, Boston, 1997.
[10]	M. C. Fu, Variance-Gamma and Monte Carlo, Advances in Mathematical Finance, (2007), 21-35.
[11]	P. W. Glynn, Likelihood ratio gradient estimation: An overview, Proceedings of the 1987 Winter Simulation Conference, (1987), 366-374.
[12]	Y. C. Ho and X. R. Cao, "Perturbation Analysis of Discrete Event Dynamic Systems," Kluwer Academic, Boston, 1991. doi: 10.1007/978-1-4615-4024-3.
[13]	B. Mondelbrot, "The variation of certain speculative prices," The Journal of Business, 36 (1963), 394-419. doi: 10.1086/294632.
[14]	Y. J. Peng, M. C. Fu and J. Q. Hu, Gradient-based simulated maximum likelihood estimation for Lévy-Driven Ornstein-Uhlenbeck stochastic volatility models, Working Paper, (2012).
[15]	K. I. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge University Press, Cambridge, 1999.
[16]	W. Schoutens, "Lévy Processes in Finance: Pricing Financial Derivatives," Wiley, New York, 2003.

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