An exact and explicit formula for pricing lookback options with regime switching

Leunglung Chan; Song-Ping Zhu

doi:10.3934/jimo.2021203

Article Contents

2023, Volume 19, Issue 1: 723-729. Doi: 10.3934/jimo.2021203

This issue Previous Article An application of approximate dynamic programming in multi-period multi-product advertising budgeting Next Article The optimal product-line design and incentive mechanism in a supply chain with customer environmental awareness

An exact and explicit formula for pricing lookback options with regime switching

Leunglung Chan^1, , and
Song-Ping Zhu^2,

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
^* Corresponding author: Leunglung Chan

Received: April 2021

Revised: September 2021

Early access: November 2021

Published: January 2023

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Option pricing,
Markov-modulated Geometric Brownian motion,
Regime switching,
lookback options.

Mathematics Subject Classification: Primary: 91G99; Secondary: 91G80.

Citation:

Full Text(HTML)

Related Papers

Cited by

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. Goldman, H. 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. Yao, Q. 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. Zhu, A. 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.