American Institute of Mathematical Sciences

Advanced
April  2016, 12(2): 529-541. doi: 10.3934/jimo.2016.12.529

On a Markov chain approximation method for option pricing with regime switching

Kun Fan 1, , Yang Shen 2, , Tak Kuen Siu 3, and  Rongming Wang 4,

1. 

School of Finance and Statistics, East China Normal University, Shanghai, 200241, China
2. 

School of Risk and Actuarial Studies and CEPAR, UNSW Business School, University of New South Wales, Sydney, NSW 2052, Australia
3. 

Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia
4. 

School of Finance and Statistics, Research Center of International Finance and Risk Management, East China Normal University, Shanghai 200241

Received  June 2013 Revised  March 2015 Published  June 2015

In this paper, we discuss a Markov chain approximation method to price European options, American options and barrier options in a Markovian regime-switching environment. The model parameters are modulated by a continuous-time, finite-state, observable Markov chain, whose states represent the states of an economy. After selecting an equivalent martingale measure by the regime-switching Esscher transform, we construct a discrete-time, inhomogeneous Markov chain to approximate the dynamics of the logarithmic stock price process. Numerical examples and empirical analysis are used to illustrate the practical implementation of the method.
Keywords: Option pricing, Markov chain approximation., regime switching.
Mathematics Subject Classification: Primary: 91G20, 91G60; Secondary: 60J1.
Citation: Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529
References:
[1]

P. Boyle and T. Draviam, Pricing exotic options under regime switching, Insurance: Mathematics and Economics, 40 (2007), 267-282. doi: 10.1016/j.insmatheco.2006.05.001.  Google Scholar

[2]

W. K. Ching, X. Huang, M. Ng and T. K. Siu, Markov Chains: Models, Algorithms and Applications, 2nd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-6312-2.  Google Scholar

[3]

J. C. Duan, E. Dudley, G. Gauthier and J. G. Simonato, Pricing discretely monitored barrier options by a Markov chain, The Journal of Derivatives, 10 (2003), 9-31. Google Scholar

[4]

J. C. Duan and J. G. Simonato, American option pricing under GARCH by a Markov chain approximation, Journal of Economics and Dynamic Control, 25 (2001), 1689-1718. doi: 10.1016/S0165-1889(00)00003-8.  Google Scholar

[5]

R. J. Elliott, L. Aggoun and J. Moore, Hidden Markov Models: Estimation and Control, Springer, New York, 1994.  Google Scholar

[6]

R. J. Elliott, L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432. Google Scholar

[7]

R. J. Elliott, T. K. Siu and A. Badescu, On pricing and hedging options in regime-switching models with feedback effect, Journal of Economic Dynamics and Control, 35 (2011), 694-713. doi: 10.1016/j.jedc.2010.12.014.  Google Scholar

[8]

R. J. Elliott, C. C. Liew and T. K. Siu, Characteristic functions and option valuation in a Markov chain market, Computers and Mathematics with Applications, 62 (2011), 65-74. doi: 10.1016/j.camwa.2011.04.050.  Google Scholar

[9]

R. J. Elliott and T. K. Siu, A note on differentiability in a Markov chain market using stochastic flows, Stochastic Analysis and Applications, 33 (2015), 110-122. doi: 10.1080/07362994.2014.975359.  Google Scholar

[10]

K. Fan, Y. Shen, T. K. Siu and R. Wang, Pricing foreign equity option with regime-switching, Economic Modelling, 37 (2014), 296-305. doi: 10.1016/j.econmod.2013.11.009.  Google Scholar

[11]

K. Fan, Y. Shen, T. K. Siu and R. Wang, Pricing annuity guarantees under a double regime-switching model, Insurance: Mathematics and Economics, 62 (2015), 62-78. doi: 10.1016/j.insmatheco.2015.02.005.  Google Scholar

[12]

S. M. Goldfeld and R. E. Quandt, A Markov model for switching regressions, Jounal of Econometrics, 1 (1973), 3-16. doi: 10.1016/0304-4076(73)90002-X.  Google Scholar

[13]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.  Google Scholar

[14]

R. H. Liu, Q. Zhang and G. Yin, Option pricing in a regime-switching model using the fast Fourier transform, Journal of Applied Mathematics and Stochastic Analysis, Article ID 18109 (2006), 1-22. doi: 10.1155/JAMSA/2006/18109.  Google Scholar

[15]

R. Norberg, The Markov chain market, ASTIN Bulletin, 33 (2003), 265-288. doi: 10.2143/AST.33.2.503693.  Google Scholar

[16]

S. R. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell Publishers, United States, 1997. Google Scholar

[17]

R. E. Quandt, The estimation of parameters of linear regression system obeying two separate regimes, Journal of the American Statistical Assocation, 55 (1958), 873-880. doi: 10.1080/01621459.1958.10501484.  Google Scholar

[18]

Y. Shen and T. K. Siu, Pricing bond options under a Markovian regime-switching Hull-White model, Economic Modelling, 30 (2013), 933-940. doi: 10.1016/j.econmod.2012.09.041.  Google Scholar

[19]

Y. Shen, K. Fan and T. K. Siu, Option valuation under a double regime-switching model, Journal of Futures Markets, 34 (2014), 451-478. doi: 10.1002/fut.21613.  Google Scholar

[20]

J. G. Simonato, Computing American option prices in the lognormal jump-diffusion framework with a Markov chain, Finance Research Letters, 8 (2011), 220-226. doi: 10.1016/j.frl.2011.01.002.  Google Scholar

[21]

T. K. Siu, Regime Switching Risk: To Price or Not To Price? International Journal of Stochastic Analysis, Article ID 843246 (2011), 14 Pages.  Google Scholar

[22]

T. K. Siu, Integration by parts and martingale representation for a Markov chain, Abstract and Applied Analysis, Article ID 438258 (2014), 11 Pages. doi: 10.1155/2014/438258.  Google Scholar

[23]

N. Song, W. K. Ching, T. K. Siu, E. S. Fung and M. K. Ng, Option valuation under a multivariate Markov chain model, Proceedings of CSO2010, Huangshan, IEEE Computer Society Proceedings, 1 (2010), 177-181. doi: 10.1109/CSO.2010.73.  Google Scholar

[24]

H. Tong, On a threshold model, in Pattern Recognition and Signal Processing (ed. C. H. Chen), NATO ASI Series E: Applied Sc. No. 29, The Netherlands: Sijthoff & Noordhoff, (1978), 575-586. doi: 10.1007/978-94-009-9941-1_24.  Google Scholar

[25]

H. Tong, Threshold Models in Non-linear Time Series Analysis, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4684-7888-4.  Google Scholar

[26]

J. van der Hoek and R. J. Elliott, American option prices in a Markov chain market model, Applied Stochastic Models in Business and Industry, 28 (2012), 35-59. doi: 10.1002/asmb.893.  Google Scholar

[27]

J. van der Hoek and R. J. Elliott, Asset pricing using finite state Markov chain stochastic discount functions, Stochastic Analysis and Applications, 30 (2012), 865-894. doi: 10.1080/07362994.2012.704852.  Google Scholar

[28]

F. L. Yuen and H. Yang, Option pricing with regime switching by trinomial tree method, Journal of Computational and Applied Mathematics, 233 (2010), 1821-1833. doi: 10.1016/j.cam.2009.09.019.  Google Scholar

show all references

References:
[1]

P. Boyle and T. Draviam, Pricing exotic options under regime switching, Insurance: Mathematics and Economics, 40 (2007), 267-282. doi: 10.1016/j.insmatheco.2006.05.001.  Google Scholar

[2]

W. K. Ching, X. Huang, M. Ng and T. K. Siu, Markov Chains: Models, Algorithms and Applications, 2nd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-6312-2.  Google Scholar

[3]

J. C. Duan, E. Dudley, G. Gauthier and J. G. Simonato, Pricing discretely monitored barrier options by a Markov chain, The Journal of Derivatives, 10 (2003), 9-31. Google Scholar

[4]

J. C. Duan and J. G. Simonato, American option pricing under GARCH by a Markov chain approximation, Journal of Economics and Dynamic Control, 25 (2001), 1689-1718. doi: 10.1016/S0165-1889(00)00003-8.  Google Scholar

[5]

R. J. Elliott, L. Aggoun and J. Moore, Hidden Markov Models: Estimation and Control, Springer, New York, 1994.  Google Scholar

[6]

R. J. Elliott, L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching, Annals of Finance, 1 (2005), 423-432. Google Scholar

[7]

R. J. Elliott, T. K. Siu and A. Badescu, On pricing and hedging options in regime-switching models with feedback effect, Journal of Economic Dynamics and Control, 35 (2011), 694-713. doi: 10.1016/j.jedc.2010.12.014.  Google Scholar

[8]

R. J. Elliott, C. C. Liew and T. K. Siu, Characteristic functions and option valuation in a Markov chain market, Computers and Mathematics with Applications, 62 (2011), 65-74. doi: 10.1016/j.camwa.2011.04.050.  Google Scholar

[9]

R. J. Elliott and T. K. Siu, A note on differentiability in a Markov chain market using stochastic flows, Stochastic Analysis and Applications, 33 (2015), 110-122. doi: 10.1080/07362994.2014.975359.  Google Scholar

[10]

K. Fan, Y. Shen, T. K. Siu and R. Wang, Pricing foreign equity option with regime-switching, Economic Modelling, 37 (2014), 296-305. doi: 10.1016/j.econmod.2013.11.009.  Google Scholar

[11]

K. Fan, Y. Shen, T. K. Siu and R. Wang, Pricing annuity guarantees under a double regime-switching model, Insurance: Mathematics and Economics, 62 (2015), 62-78. doi: 10.1016/j.insmatheco.2015.02.005.  Google Scholar

[12]

S. M. Goldfeld and R. E. Quandt, A Markov model for switching regressions, Jounal of Econometrics, 1 (1973), 3-16. doi: 10.1016/0304-4076(73)90002-X.  Google Scholar

[13]

J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.  Google Scholar

[14]

R. H. Liu, Q. Zhang and G. Yin, Option pricing in a regime-switching model using the fast Fourier transform, Journal of Applied Mathematics and Stochastic Analysis, Article ID 18109 (2006), 1-22. doi: 10.1155/JAMSA/2006/18109.  Google Scholar

[15]

R. Norberg, The Markov chain market, ASTIN Bulletin, 33 (2003), 265-288. doi: 10.2143/AST.33.2.503693.  Google Scholar

[16]

S. R. Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell Publishers, United States, 1997. Google Scholar

[17]

R. E. Quandt, The estimation of parameters of linear regression system obeying two separate regimes, Journal of the American Statistical Assocation, 55 (1958), 873-880. doi: 10.1080/01621459.1958.10501484.  Google Scholar

[18]

Y. Shen and T. K. Siu, Pricing bond options under a Markovian regime-switching Hull-White model, Economic Modelling, 30 (2013), 933-940. doi: 10.1016/j.econmod.2012.09.041.  Google Scholar

[19]

Y. Shen, K. Fan and T. K. Siu, Option valuation under a double regime-switching model, Journal of Futures Markets, 34 (2014), 451-478. doi: 10.1002/fut.21613.  Google Scholar

[20]

J. G. Simonato, Computing American option prices in the lognormal jump-diffusion framework with a Markov chain, Finance Research Letters, 8 (2011), 220-226. doi: 10.1016/j.frl.2011.01.002.  Google Scholar

[21]

T. K. Siu, Regime Switching Risk: To Price or Not To Price? International Journal of Stochastic Analysis, Article ID 843246 (2011), 14 Pages.  Google Scholar

[22]

T. K. Siu, Integration by parts and martingale representation for a Markov chain, Abstract and Applied Analysis, Article ID 438258 (2014), 11 Pages. doi: 10.1155/2014/438258.  Google Scholar

[23]

N. Song, W. K. Ching, T. K. Siu, E. S. Fung and M. K. Ng, Option valuation under a multivariate Markov chain model, Proceedings of CSO2010, Huangshan, IEEE Computer Society Proceedings, 1 (2010), 177-181. doi: 10.1109/CSO.2010.73.  Google Scholar

[24]

H. Tong, On a threshold model, in Pattern Recognition and Signal Processing (ed. C. H. Chen), NATO ASI Series E: Applied Sc. No. 29, The Netherlands: Sijthoff & Noordhoff, (1978), 575-586. doi: 10.1007/978-94-009-9941-1_24.  Google Scholar

[25]

H. Tong, Threshold Models in Non-linear Time Series Analysis, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4684-7888-4.  Google Scholar

[26]

J. van der Hoek and R. J. Elliott, American option prices in a Markov chain market model, Applied Stochastic Models in Business and Industry, 28 (2012), 35-59. doi: 10.1002/asmb.893.  Google Scholar

[27]

J. van der Hoek and R. J. Elliott, Asset pricing using finite state Markov chain stochastic discount functions, Stochastic Analysis and Applications, 30 (2012), 865-894. doi: 10.1080/07362994.2012.704852.  Google Scholar

[28]

F. L. Yuen and H. Yang, Option pricing with regime switching by trinomial tree method, Journal of Computational and Applied Mathematics, 233 (2010), 1821-1833. doi: 10.1016/j.cam.2009.09.019.  Google Scholar

[1]

Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control & Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237
[2]

Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042
[3]

Lin Xu, Rongming Wang, Dingjun Yao. Optimal stochastic investment games under Markov regime switching market. Journal of Industrial & Management Optimization, 2014, 10 (3) : 795-815. doi: 10.3934/jimo.2014.10.795
[4]

Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079
[5]

Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072
[6]

Baojun Bian, Nan Wu, Harry Zheng. Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1401-1420. doi: 10.3934/dcdsb.2016002
[7]

Lin Xu, Dingjun Yao, Gongpin Cheng. Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax. Journal of Industrial & Management Optimization, 2020, 16 (1) : 325-356. doi: 10.3934/jimo.2018154
[8]

Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132
[9]

Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control & Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007
[10]

Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100
[11]

Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021074
[12]

Ralf Banisch, Carsten Hartmann. Addendum to "A sparse Markov chain approximation of LQ-type stochastic control problems". Mathematical Control & Related Fields, 2017, 7 (4) : 623-623. doi: 10.3934/mcrf.2017023
[13]

Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021077
[14]

Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial & Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158
[15]

Na Song, Ximin Huang, Yue Xie, Wai-Ki Ching, Tak-Kuen Siu. Impact of reorder option in supply chain coordination. Journal of Industrial & Management Optimization, 2017, 13 (1) : 449-475. doi: 10.3934/jimo.2016026
[16]

Samuel N. Cohen, Lukasz Szpruch. On Markovian solutions to Markov Chain BSDEs. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 257-269. doi: 10.3934/naco.2012.2.257
[17]

Kai Zhang, Song Wang. Convergence property of an interior penalty approach to pricing American option. Journal of Industrial & Management Optimization, 2011, 7 (2) : 435-447. doi: 10.3934/jimo.2011.7.435
[18]

Tak Kuen Siu, Howell Tong, Hailiang Yang. Option pricing under threshold autoregressive models by threshold Esscher transform. Journal of Industrial & Management Optimization, 2006, 2 (2) : 177-197. doi: 10.3934/jimo.2006.2.177
[19]

Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial & Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783
[20]

Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (212)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences