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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 |
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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