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

[1]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363
[2]

Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020179
[3]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113
[4]

Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65
[5]

Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347
[6]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[7]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020371
[8]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[9]

Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1405-1446. doi: 10.3934/dcdsb.2020167
[10]

Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367
[11]

Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170
[12]

Michal Beneš, Pavel Eichler, Jakub Klinkovský, Miroslav Kolář, Jakub Solovský, Pavel Strachota, Alexandr Žák. Numerical simulation of fluidization for application in oxyfuel combustion. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 769-783. doi: 10.3934/dcdss.2020232
[13]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[14]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115
[15]

Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103
[16]

Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020389
[17]

Linhao Xu, Marya Claire Zdechlik, Melissa C. Smith, Min B. Rayamajhi, Don L. DeAngelis, Bo Zhang. Simulation of post-hurricane impact on invasive species with biological control management. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4059-4071. doi: 10.3934/dcds.2020038
[18]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002
[19]

Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021002
[20]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328

 Impact Factor: 

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences