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June  2015, 5(2): 237-258. doi: 10.3934/mcrf.2015.5.237

Lookback option pricing for regime-switching jump diffusion models

Zhuo Jin 1, and  Linyi Qian 2,

1. 

Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia
2. 

School of Finance and Statistics, East China Normal University, 500 Dongchuan Road, Shanghai, 200241

Received  June 2014 Revised  August 2014 Published  April 2015

In this paper, we will introduce a numerical method to price the European lookback floating strike put options where the underlying asset price is modeled by a generalized regime-switching jump diffusion process. In the Markov regime-switching model, the option value is a solution of a coupled system of nonlinear integro-differential partial differential equations. Due to the complexity of regime-switching model, the jump process involved, and the nonlinearity, closed-form solutions are virtually impossible to obtain. We use Markov chain approximation techniques to construct a discrete-time Markov chain to approximate the option value. Convergence of the approximation algorithms is proved. Examples are presented to demonstrate the applicability of the numerical methods.
Keywords: floating strike, Lookback option, Markov chain approximation..
Mathematics Subject Classification: 60H35, 91G20, 91G80, 65C0.
Citation: Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control and Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237
References:
[1]

L. Andersen and J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), 231-262.

[2]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, J. Asymptotic Analysis, 4 (1991), 271-283.

[3]

G. K. Basak, M. K. Ghosh and A. Goswami, Risk minimizing option pricing for a class of exotic options in a Markov-modulated market, Stochastic Analysis and Applications, 29 (2011), 259-281. doi: 10.1080/07362994.2011.548665.

[4]

S. Boyarchenko and S. Levendorskii, Barrier options and touch-and-out options under regular Lévy processes of exponential type, Ann. Appl. Probab., 12 (2002), 1261-1298. doi: 10.1214/aoap/1037125863.

[5]

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.

[6]

M. Broadie, P. Glasserman and S. G. Kou, Connecting discrete and continuous path-dependent options, Finance Stochastics, 3 (1999), 55-82. doi: 10.1007/s007800050052.

[7]

P. Buchen and O. Konstandatos, A new method of pricing lookback options, Mathematical Finance, 15 (2005), 245-259. doi: 10.1111/j.0960-1627.2005.00219.x.

[8]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081. doi: 10.1287/mnsc.1110.1393.

[9]

P. Carr and J. Crosby, A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options, Quantitative Finance, 10 (2010), 1115-1136. doi: 10.1080/14697680903413605.

[10]

A. Conze and R. Vishwanathan, Path-dependent options: The case of lookback options, J. Finance, 46 (1991), 1893-1907. doi: 10.2307/2328577.

[11]

M. Dai and Y. K. Kwok, Characterization of optimal stopping regions of American Asian and lookback options, Mathematical Finance, 16 (2006), 63-82. doi: 10.1111/j.1467-9965.2006.00261.x.

[12]

J. DiCesare and D. Mcleish, Simulation of jump diffusions and the pricing of options, Insurance: Mathematics and Economics, 43 (2008), 316-326. doi: 10.1016/j.insmatheco.2008.06.001.

[13]

H. U. Gerber and E. S. Shiu, Pricing lookback options and dynamic guarantees, North American Actuarial Journal, 7 (2003), 48-67. doi: 10.1080/10920277.2003.10596076.

[14]

M. B. Goldman, H. B. Sosin and M. A. Gatto, Path-dependent options, J. Finance, 34 (1979), 1111-1127.

[15]

J. Hamilton, A new approach to the economic analysis of non-stationary time series, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.

[16]

C. He, J. S. Kennedy, T. F. Coleman, P. A. Forsyth, Y. Li and K. R. Vetzal, Calibration and hedging under jump diffusion, Review of Derivatives Research, 9 (2006), 1-35. doi: 10.1007/s11147-006-9003-1.

[17]

S. Jaimungal and T. Wang, Catastrophe options with stochastic interest rates and compound Poisson losses, Insurance: Mathematics and Economics, 38 (2006), 469-483. doi: 10.1016/j.insmatheco.2005.11.008.

[18]

Z. Jin, G. Yin and F. Wu, Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods, Insurance Mathematics and Economics, 53 (2013), 733-746. doi: 10.1016/j.insmatheco.2013.09.015.

[19]

Z. Jin, G. Yin and H. L. Yang, Numerical methods for dividend optimization using regime-switching jump-diffusion models, Mathematical Control and Related Fields, 1 (2011), 21-40. doi: 10.3934/mcrf.2011.1.21.

[20]

Z. Jin, G. Yin and C. Zhu, Numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation, Automatica, 48 (2012), 1489-1501. doi: 10.1016/j.automatica.2012.05.039.

[21]

Z. Jin, H. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica, 49 (2013), 2317-2329. doi: 10.1016/j.automatica.2013.04.043.

[22]

K. I. Kim, H. S. Park and X. S. Qian, A mathematical modeling for the lookback option with jump-diffusion using binomial tree method, Journal of Computational and Applied Mathematics, 235 (2011), 5140-5154. doi: 10.1016/j.cam.2011.05.002.

[23]

S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101. doi: 10.1287/mnsc.48.8.1086.166.

[24]

S. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192. doi: 10.1287/mnsc.1030.0163.

[25]

H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.

[26]

H. Kushner and P. Dupuis, Numerical Methods for Stochstic Control Problems in Continuous Time, Second edition, Stochastic Modelling and Applied Probability, 24, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0007-6.

[27]

H. J. Kushner and L. F. Martins, Numerical methods for stochastic singular control problems, SIAM J. Control Optim., 29 (1991), 1443-1475. doi: 10.1137/0329073.

[28]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, Second edition, Applications of Mathematics (New York), 35, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2003.

[29]

Y. K. Kwok, Mathematical Models of Financial Derivatives, Second edition, Springer Finance, Springer, Berlin, 2008.

[30]

R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2.

[31]

H. Pham, Optimal stopping, free boundary, and American option in a jump-diffusion model, Applied Mathematics and Optimization, 35 (1997), 145-164. doi: 10.1007/BF02683325.

[32]

R. Poulsen, Barrier options and their static hedges: Simple derivations and extensions, Quantitative Finance, 6 (2006), 327-335. doi: 10.1080/14697680600690331.

[33]

Q. S. Song, G. Yin and Z. Zhang, Numerical method for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42 (2006), 1147-1157. doi: 10.1016/j.automatica.2006.03.016.

[34]

Q. S. Song, G. Yin and Z. Zhang, Numerical solutions for stochastic differential games with regime swiching, IEEE Transaction on Automatic Control, 53 (2008), 509-521. doi: 10.1109/TAC.2007.915169.

[35]

P. Wilmott, Paul Wilmott on Quantitative Finance, Wiley, Chichester, 2000.

[36]

P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, 1994.

[37]

C. Xu and Y. K. Kwok, Integral price formulas for lookback options, Journal of Applied Mathematics, (2005), 117-125. doi: 10.1155/JAM.2005.117.

[38]

H. Yang and G. Yin, Ruin probability for a model under Markovian switching regime, in Probability, Finance and Insurance (eds. T. L. Lai, H. Yang and S. P. Yung), World Scientific, River Edge, NJ, 2004, 206-217.

[39]

G. Yin, H. Jin and Z. Jin, Numerical methods for portfolio selection with bounded constraints, J. Computational Appl. Math., 233 (2009), 564-581. doi: 10.1016/j.cam.2009.08.055.

[40]

G. Yin, Q. Zhang and G. Badowski, Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit, Adv. Appl. Probab., 35 (2003), 449-476. doi: 10.1239/aap/1051201656.

[41]

Z. Zhang, G. Yin and Z. Liang, A stochastic approximation algorithm for American lookback put options, Stochastic Analysis and Applications, 29 (2011), 332-351. doi: 10.1080/07362994.2010.503473.

[42]

X. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583.

show all references

References:
[1]

L. Andersen and J. Andreasen, Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Review of Derivatives Research, 4 (2000), 231-262.

[2]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, J. Asymptotic Analysis, 4 (1991), 271-283.

[3]

G. K. Basak, M. K. Ghosh and A. Goswami, Risk minimizing option pricing for a class of exotic options in a Markov-modulated market, Stochastic Analysis and Applications, 29 (2011), 259-281. doi: 10.1080/07362994.2011.548665.

[4]

S. Boyarchenko and S. Levendorskii, Barrier options and touch-and-out options under regular Lévy processes of exponential type, Ann. Appl. Probab., 12 (2002), 1261-1298. doi: 10.1214/aoap/1037125863.

[5]

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.

[6]

M. Broadie, P. Glasserman and S. G. Kou, Connecting discrete and continuous path-dependent options, Finance Stochastics, 3 (1999), 55-82. doi: 10.1007/s007800050052.

[7]

P. Buchen and O. Konstandatos, A new method of pricing lookback options, Mathematical Finance, 15 (2005), 245-259. doi: 10.1111/j.0960-1627.2005.00219.x.

[8]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081. doi: 10.1287/mnsc.1110.1393.

[9]

P. Carr and J. Crosby, A class of Lévy process models with almost exact calibration to both barrier and vanilla FX options, Quantitative Finance, 10 (2010), 1115-1136. doi: 10.1080/14697680903413605.

[10]

A. Conze and R. Vishwanathan, Path-dependent options: The case of lookback options, J. Finance, 46 (1991), 1893-1907. doi: 10.2307/2328577.

[11]

M. Dai and Y. K. Kwok, Characterization of optimal stopping regions of American Asian and lookback options, Mathematical Finance, 16 (2006), 63-82. doi: 10.1111/j.1467-9965.2006.00261.x.

[12]

J. DiCesare and D. Mcleish, Simulation of jump diffusions and the pricing of options, Insurance: Mathematics and Economics, 43 (2008), 316-326. doi: 10.1016/j.insmatheco.2008.06.001.

[13]

H. U. Gerber and E. S. Shiu, Pricing lookback options and dynamic guarantees, North American Actuarial Journal, 7 (2003), 48-67. doi: 10.1080/10920277.2003.10596076.

[14]

M. B. Goldman, H. B. Sosin and M. A. Gatto, Path-dependent options, J. Finance, 34 (1979), 1111-1127.

[15]

J. Hamilton, A new approach to the economic analysis of non-stationary time series, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.

[16]

C. He, J. S. Kennedy, T. F. Coleman, P. A. Forsyth, Y. Li and K. R. Vetzal, Calibration and hedging under jump diffusion, Review of Derivatives Research, 9 (2006), 1-35. doi: 10.1007/s11147-006-9003-1.

[17]

S. Jaimungal and T. Wang, Catastrophe options with stochastic interest rates and compound Poisson losses, Insurance: Mathematics and Economics, 38 (2006), 469-483. doi: 10.1016/j.insmatheco.2005.11.008.

[18]

Z. Jin, G. Yin and F. Wu, Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods, Insurance Mathematics and Economics, 53 (2013), 733-746. doi: 10.1016/j.insmatheco.2013.09.015.

[19]

Z. Jin, G. Yin and H. L. Yang, Numerical methods for dividend optimization using regime-switching jump-diffusion models, Mathematical Control and Related Fields, 1 (2011), 21-40. doi: 10.3934/mcrf.2011.1.21.

[20]

Z. Jin, G. Yin and C. Zhu, Numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation, Automatica, 48 (2012), 1489-1501. doi: 10.1016/j.automatica.2012.05.039.

[21]

Z. Jin, H. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica, 49 (2013), 2317-2329. doi: 10.1016/j.automatica.2013.04.043.

[22]

K. I. Kim, H. S. Park and X. S. Qian, A mathematical modeling for the lookback option with jump-diffusion using binomial tree method, Journal of Computational and Applied Mathematics, 235 (2011), 5140-5154. doi: 10.1016/j.cam.2011.05.002.

[23]

S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101. doi: 10.1287/mnsc.48.8.1086.166.

[24]

S. Kou and H. Wang, Option pricing under a double exponential jump diffusion model, Management Science, 50 (2004), 1178-1192. doi: 10.1287/mnsc.1030.0163.

[25]

H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.

[26]

H. Kushner and P. Dupuis, Numerical Methods for Stochstic Control Problems in Continuous Time, Second edition, Stochastic Modelling and Applied Probability, 24, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0007-6.

[27]

H. J. Kushner and L. F. Martins, Numerical methods for stochastic singular control problems, SIAM J. Control Optim., 29 (1991), 1443-1475. doi: 10.1137/0329073.

[28]

H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, Second edition, Applications of Mathematics (New York), 35, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2003.

[29]

Y. K. Kwok, Mathematical Models of Financial Derivatives, Second edition, Springer Finance, Springer, Berlin, 2008.

[30]

R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2.

[31]

H. Pham, Optimal stopping, free boundary, and American option in a jump-diffusion model, Applied Mathematics and Optimization, 35 (1997), 145-164. doi: 10.1007/BF02683325.

[32]

R. Poulsen, Barrier options and their static hedges: Simple derivations and extensions, Quantitative Finance, 6 (2006), 327-335. doi: 10.1080/14697680600690331.

[33]

Q. S. Song, G. Yin and Z. Zhang, Numerical method for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42 (2006), 1147-1157. doi: 10.1016/j.automatica.2006.03.016.

[34]

Q. S. Song, G. Yin and Z. Zhang, Numerical solutions for stochastic differential games with regime swiching, IEEE Transaction on Automatic Control, 53 (2008), 509-521. doi: 10.1109/TAC.2007.915169.

[35]

P. Wilmott, Paul Wilmott on Quantitative Finance, Wiley, Chichester, 2000.

[36]

P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, 1994.

[37]

C. Xu and Y. K. Kwok, Integral price formulas for lookback options, Journal of Applied Mathematics, (2005), 117-125. doi: 10.1155/JAM.2005.117.

[38]

H. Yang and G. Yin, Ruin probability for a model under Markovian switching regime, in Probability, Finance and Insurance (eds. T. L. Lai, H. Yang and S. P. Yung), World Scientific, River Edge, NJ, 2004, 206-217.

[39]

G. Yin, H. Jin and Z. Jin, Numerical methods for portfolio selection with bounded constraints, J. Computational Appl. Math., 233 (2009), 564-581. doi: 10.1016/j.cam.2009.08.055.

[40]

G. Yin, Q. Zhang and G. Badowski, Discrete-time singularly perturbed Markov chains: Aggregation, occupation measures, and switching diffusion limit, Adv. Appl. Probab., 35 (2003), 449-476. doi: 10.1239/aap/1051201656.

[41]

Z. Zhang, G. Yin and Z. Liang, A stochastic approximation algorithm for American lookback put options, Stochastic Analysis and Applications, 29 (2011), 332-351. doi: 10.1080/07362994.2010.503473.

[42]

X. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583.

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