American Institute of Mathematical Sciences

Advanced
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

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 & 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, 63 (2001), 167.   Google Scholar

[2]

J. Bertoin, "Lévy Processes,", 2nd edition, (1996).   Google Scholar

[3]

P. Carr and L. Wu, Time-changed Lévy processes and option pricing,, Journal of Financial Economics, 71 (2004), 113.  doi: 10.1016/S0304-405X(03)00171-5.  Google Scholar

[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).   Google Scholar

[5]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Chapman & Hall/CRC, (2004).   Google Scholar

[6]

P. Glasserman, "Gradient Estimation via Perturbation Analysis,", Kluwer Academic, (1991).   Google Scholar

[7]

P. Glasserman, "Monte Carlo Methods in Financial Engineering,", Springer, (2004).   Google Scholar

[8]

P. Glasserman and Z. Liu, Estimating Greeks in simulating Lévy-driven models,, Journal of Computational Finance, 14 (2010), 3.   Google Scholar

[9]

M. C. Fu and J. Q. Hu, "Conditional Monte Carlo: Gradient Estimation and Optimization Applications,", Kluwer Academic, (1997).   Google Scholar

[10]

M. C. Fu, Variance-Gamma and Monte Carlo,, Advances in Mathematical Finance, (2007), 21.   Google Scholar

[11]

P. W. Glynn, Likelihood ratio gradient estimation: An overview,, Proceedings of the 1987 Winter Simulation Conference, (1987), 366.   Google Scholar

[12]

Y. C. Ho and X. R. Cao, "Perturbation Analysis of Discrete Event Dynamic Systems,", Kluwer Academic, (1991).  doi: 10.1007/978-1-4615-4024-3.  Google Scholar

[13]

B. Mondelbrot, "The variation of certain speculative prices,", The Journal of Business, 36 (1963), 394.  doi: 10.1086/294632.  Google Scholar

[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).   Google Scholar

[15]

K. I. Sato, "Lévy Processes and Infinitely Divisible Distributions,", Cambridge University Press, (1999).   Google Scholar

[16]

W. Schoutens, "Lévy Processes in Finance: Pricing Financial Derivatives,", Wiley, (2003).   Google Scholar

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, 63 (2001), 167.   Google Scholar

[2]

J. Bertoin, "Lévy Processes,", 2nd edition, (1996).   Google Scholar

[3]

P. Carr and L. Wu, Time-changed Lévy processes and option pricing,, Journal of Financial Economics, 71 (2004), 113.  doi: 10.1016/S0304-405X(03)00171-5.  Google Scholar

[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).   Google Scholar

[5]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Chapman & Hall/CRC, (2004).   Google Scholar

[6]

P. Glasserman, "Gradient Estimation via Perturbation Analysis,", Kluwer Academic, (1991).   Google Scholar

[7]

P. Glasserman, "Monte Carlo Methods in Financial Engineering,", Springer, (2004).   Google Scholar

[8]

P. Glasserman and Z. Liu, Estimating Greeks in simulating Lévy-driven models,, Journal of Computational Finance, 14 (2010), 3.   Google Scholar

[9]

M. C. Fu and J. Q. Hu, "Conditional Monte Carlo: Gradient Estimation and Optimization Applications,", Kluwer Academic, (1997).   Google Scholar

[10]

M. C. Fu, Variance-Gamma and Monte Carlo,, Advances in Mathematical Finance, (2007), 21.   Google Scholar

[11]

P. W. Glynn, Likelihood ratio gradient estimation: An overview,, Proceedings of the 1987 Winter Simulation Conference, (1987), 366.   Google Scholar

[12]

Y. C. Ho and X. R. Cao, "Perturbation Analysis of Discrete Event Dynamic Systems,", Kluwer Academic, (1991).  doi: 10.1007/978-1-4615-4024-3.  Google Scholar

[13]

B. Mondelbrot, "The variation of certain speculative prices,", The Journal of Business, 36 (1963), 394.  doi: 10.1086/294632.  Google Scholar

[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).   Google Scholar

[15]

K. I. Sato, "Lévy Processes and Infinitely Divisible Distributions,", Cambridge University Press, (1999).   Google Scholar

[16]

W. Schoutens, "Lévy Processes in Finance: Pricing Financial Derivatives,", Wiley, (2003).   Google Scholar

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