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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

 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.
Citation: 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
##### 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. Google Scholar [2] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, J. Asymptotic Analysis, 4 (1991), 271-283.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] A. Conze and R. Vishwanathan, Path-dependent options: The case of lookback options, J. Finance, 46 (1991), 1893-1907. doi: 10.2307/2328577.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] M. B. Goldman, H. B. Sosin and M. A. Gatto, Path-dependent options, J. Finance, 34 (1979), 1111-1127. Google Scholar [15] J. Hamilton, A new approach to the economic analysis of non-stationary time series, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] Y. K. Kwok, Mathematical Models of Financial Derivatives, Second edition, Springer Finance, Springer, Berlin, 2008.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] R. Poulsen, Barrier options and their static hedges: Simple derivations and extensions, Quantitative Finance, 6 (2006), 327-335. doi: 10.1080/14697680600690331.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] P. Wilmott, Paul Wilmott on Quantitative Finance, Wiley, Chichester, 2000. Google Scholar [36] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, 1994. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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. Google Scholar [2] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, J. Asymptotic Analysis, 4 (1991), 271-283.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] A. Conze and R. Vishwanathan, Path-dependent options: The case of lookback options, J. Finance, 46 (1991), 1893-1907. doi: 10.2307/2328577.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] M. B. Goldman, H. B. Sosin and M. A. Gatto, Path-dependent options, J. Finance, 34 (1979), 1111-1127. Google Scholar [15] J. Hamilton, A new approach to the economic analysis of non-stationary time series, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] Y. K. Kwok, Mathematical Models of Financial Derivatives, Second edition, Springer Finance, Springer, Berlin, 2008.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] R. Poulsen, Barrier options and their static hedges: Simple derivations and extensions, Quantitative Finance, 6 (2006), 327-335. doi: 10.1080/14697680600690331.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] P. Wilmott, Paul Wilmott on Quantitative Finance, Wiley, Chichester, 2000. Google Scholar [36] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, 1994. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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