doi: 10.3934/jimo.2021203
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An exact and explicit formula for pricing lookback options with regime switching

1. 

School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

2. 

School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia

* Corresponding author: Leunglung Chan

Received  April 2021 Revised  September 2021 Early access November 2021

This paper investigates the pricing of European-style lookback options when the price dynamics of the underlying risky asset are assumed to follow a Markov-modulated Geometric Brownian motion; that is, the appreciation rate and the volatility of the underlying risky asset depend on states of the economy described by a continuous-time Markov chain process. We derive an exact, explicit and closed-form solution for European-style lookback options in a two-state regime switching model.

Citation: Leunglung Chan, Song-Ping Zhu. An exact and explicit formula for pricing lookback options with regime switching. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021203
References:
[1]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.

[2]

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

[3]

J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.

[4]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Applications of Mathematics (New York), 29. Springer-Verlag, New York, 1995. doi: 10.1007/978-0-387-84854-9.

[5]

M. B. GoldmanH. B. Sosin and M. A. Gatto, Path-dependent options buy at the low, sell at the high, J. Finance, 34 (1979), 1111-1127. 

[6]

X. Guo, Information and option pricings, Quant. Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.

[7]

X. J. He and S. P. Zhu, On full calibration of hybrid local volatility and regime-switching models, J. Futures Markets, 38 (2018), 586-606.  doi: 10.1002/fut.21901.

[8]

K. S. Leung, An analytic pricing formula for lookback options under stochastic volatility, Appl. Math. Lett., 26 (2013), 145-149.  doi: 10.1016/j.aml.2012.07.008.

[9]

S. J. Liao, Numerically solving non-linear problems by the homotopy analysis method,, Comput. Mech., 20 (1997), 530-540.  doi: 10.1007/s004660050273.

[10] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.  doi: 10.1016/C2013-0-11263-9.
[11]

H. Y. Wong and Y. K. Kwok, Sub-replication and replenishing premium: Efficient pricing of multi-state lookbacks, Review of Derivatives Research, 6 (2003), 83-106. 

[12]

D. D. YaoQ. Zhang and X. Y. Zhou, A regime switching model for European options, Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, 194 (2006), 281-300.  doi: 10.1007/0-387-33815-2_14.

[13]

S. P. Zhu, An exact and explicit solution for the valuation of American put options, Quant. Finance, 6 (2006), 229-242.  doi: 10.1080/14697680600699811.

[14]

S. P. ZhuA. Badran and X. Lu, A new exact solution for pricing European options in a two-state regime-switching economy, Comput. Math. Appl., 64 (2012), 2744-2755.  doi: 10.1016/j.camwa.2012.08.005.

show all references

References:
[1]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654.  doi: 10.1086/260062.

[2]

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

[3]

J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance, 5 (2002), 497-514.  doi: 10.1142/S0219024902001523.

[4]

R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Applications of Mathematics (New York), 29. Springer-Verlag, New York, 1995. doi: 10.1007/978-0-387-84854-9.

[5]

M. B. GoldmanH. B. Sosin and M. A. Gatto, Path-dependent options buy at the low, sell at the high, J. Finance, 34 (1979), 1111-1127. 

[6]

X. Guo, Information and option pricings, Quant. Finance, 1 (2001), 38-44.  doi: 10.1080/713665550.

[7]

X. J. He and S. P. Zhu, On full calibration of hybrid local volatility and regime-switching models, J. Futures Markets, 38 (2018), 586-606.  doi: 10.1002/fut.21901.

[8]

K. S. Leung, An analytic pricing formula for lookback options under stochastic volatility, Appl. Math. Lett., 26 (2013), 145-149.  doi: 10.1016/j.aml.2012.07.008.

[9]

S. J. Liao, Numerically solving non-linear problems by the homotopy analysis method,, Comput. Mech., 20 (1997), 530-540.  doi: 10.1007/s004660050273.

[10] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.  doi: 10.1016/C2013-0-11263-9.
[11]

H. Y. Wong and Y. K. Kwok, Sub-replication and replenishing premium: Efficient pricing of multi-state lookbacks, Review of Derivatives Research, 6 (2003), 83-106. 

[12]

D. D. YaoQ. Zhang and X. Y. Zhou, A regime switching model for European options, Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems, 194 (2006), 281-300.  doi: 10.1007/0-387-33815-2_14.

[13]

S. P. Zhu, An exact and explicit solution for the valuation of American put options, Quant. Finance, 6 (2006), 229-242.  doi: 10.1080/14697680600699811.

[14]

S. P. ZhuA. Badran and X. Lu, A new exact solution for pricing European options in a two-state regime-switching economy, Comput. Math. Appl., 64 (2012), 2744-2755.  doi: 10.1016/j.camwa.2012.08.005.

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