# American Institute of Mathematical Sciences

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

## Simulation of Lévy-Driven models and its application in finance

 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.
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," 2nd 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," 2nd 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.
 [1] Johnathan M. Bardsley. A theoretical framework for the regularization of Poisson likelihood estimation problems. Inverse Problems and Imaging, 2010, 4 (1) : 11-17. doi: 10.3934/ipi.2010.4.11 [2] Johnathan M. Bardsley. An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Problems and Imaging, 2008, 2 (2) : 167-185. doi: 10.3934/ipi.2008.2.167 [3] Esmail Abdul Fattah, Janet Van Niekerk, Håvard Rue. Smart Gradient - An adaptive technique for improving gradient estimation. Foundations of Data Science, 2022, 4 (1) : 123-136. doi: 10.3934/fods.2021037 [4] Yang Yang, Kaiyong Wang, Jiajun Liu, Zhimin Zhang. Asymptotics for a bidimensional risk model with two geometric Lévy price processes. Journal of Industrial and Management Optimization, 2019, 15 (2) : 481-505. doi: 10.3934/jimo.2018053 [5] Xingchun Wang, Yongjin Wang. Hedging strategies for discretely monitored Asian options under Lévy processes. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1209-1224. doi: 10.3934/jimo.2014.10.1209 [6] Mingshang Hu, Shige Peng. G-Lévy processes under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 1-22. doi: 10.3934/puqr.2021001 [7] A. Settati, A. Lahrouz, Mohamed El Fatini, A. El Haitami, M. El Jarroudi, M. Erriani. A Markovian switching diffusion for an SIS model incorporating Lévy processes. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022072 [8] Thi Tuyet Trang Chau, Pierre Ailliot, Valérie Monbet, Pierre Tandeo. Comparison of simulation-based algorithms for parameter estimation and state reconstruction in nonlinear state-space models. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022054 [9] Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100 [10] Kamil Rajdl, Petr Lansky. Fano factor estimation. Mathematical Biosciences & Engineering, 2014, 11 (1) : 105-123. doi: 10.3934/mbe.2014.11.105 [11] Aihua Fan, Jörg Schmeling, Weixiao Shen. $L^\infty$-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363 [12] Huiqing Zhu, Runchang Lin. $L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1493-1505. doi: 10.3934/dcdsb.2013.18.1493 [13] Kaikai Cao, Youming Liu. Uncompactly supported density estimation with $L^{1}$ risk. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4007-4022. doi: 10.3934/cpaa.2020177 [14] Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282 [15] Wei Zhong, Yongxia Zhao, Ping Chen. Equilibrium periodic dividend strategies with non-exponential discounting for spectrally positive Lévy processes. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2639-2667. doi: 10.3934/jimo.2020087 [16] Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial and Management Optimization, 2022, 18 (2) : 795-823. doi: 10.3934/jimo.2020179 [17] Huijie Qiao, Jiang-lun Wu. Path independence of the additive functionals for stochastic differential equations driven by G-lévy processes. Probability, Uncertainty and Quantitative Risk, , () : -. doi: 10.3934/puqr.2022007 [18] Jie Huang, Xiaoping Yang, Yunmei Chen. A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation. Inverse Problems and Imaging, 2011, 5 (3) : 645-657. doi: 10.3934/ipi.2011.5.645 [19] Saroja Kumar Singh. Moderate deviation for maximum likelihood estimators from single server queues. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 2-. doi: 10.1186/s41546-020-00044-z [20] Ben A. Vanderlei, Matthew M. Hopkins, Lisa J. Fauci. Error estimation for immersed interface solutions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1185-1203. doi: 10.3934/dcdsb.2012.17.1185

Impact Factor: