American Institute of Mathematical Sciences

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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. 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, (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. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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, (2006), 1. doi: 10.1155/JAMSA/2006/18109. Google Scholar

[15]

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

[16]

S. R. Pliska, Introduction to Mathematical Finance: Discrete Time Models,, Blackwell Publishers, (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. 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. 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. 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. 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, (2011). Google Scholar

[22]

T. K. Siu, Integration by parts and martingale representation for a Markov chain,, Abstract and Applied Analysis, (2014). 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, 1 (2010), 177. 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), (1978), 575. doi: 10.1007/978-94-009-9941-1_24. Google Scholar

[25]

H. Tong, Threshold Models in Non-linear Time Series Analysis,, Springer-Verlag, (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. 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. 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. 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. 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, (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. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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, (2006), 1. doi: 10.1155/JAMSA/2006/18109. Google Scholar

[15]

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

[16]

S. R. Pliska, Introduction to Mathematical Finance: Discrete Time Models,, Blackwell Publishers, (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. 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. 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. 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. 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, (2011). Google Scholar

[22]

T. K. Siu, Integration by parts and martingale representation for a Markov chain,, Abstract and Applied Analysis, (2014). 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, 1 (2010), 177. 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), (1978), 575. doi: 10.1007/978-94-009-9941-1_24. Google Scholar

[25]

H. Tong, Threshold Models in Non-linear Time Series Analysis,, Springer-Verlag, (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. 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. 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. doi: 10.1016/j.cam.2009.09.019. Google Scholar

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