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Discounted optimal stopping problems in first-passage time models with random thresholds

Part of: Markov processes Miscellaneous topics - Partial differential equations Mathematical economics Stochastic processes

Published online by Cambridge University Press:  27 June 2022

Pavel V. Gapeev  and
Hessah Al Motairi
Pavel V. Gapeev*
Affiliation:
London School of Economics
Hessah Al Motairi*
Affiliation:
Kuwait University
*
*Postal address: London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, UK. Email address: p.v.gapeev@lse.ac.uk
**Postal address: Kuwait University, Faculty of Science, Department of Mathematics, PO Box 5969, Safat 13060, Kuwait
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Abstract

We derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black–Merton–Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.

Keywords

Discounted optimal stopping problemgeometric Brownian motionfirst passage timesrunning maximum and minimum processesfree-boundary problemsmooth fit and normal reflectiona change-of-variable formula with local time on surfacesperpetual American optionsrandom dividends

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G44: Martingales with continuous parameter 60J65: Brownian motion
Secondary: 91B25: Asset pricing models 60J60: Diffusion processes 35R35: Free boundary problems
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 714 - 733
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Asmussen, S., Avram, F. and Pistorius, M. (2003). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79–111.10.1016/j.spa.2003.07.005CrossRefGoogle Scholar
Avram, F., Kyprianou, A. E. and Pistorius, M. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.10.1214/aoap/1075828052CrossRefGoogle Scholar
Baurdoux, E. J. and Kyprianou, A. E. (2004). Further calculations for Israeli options. Stochastics 76, 549–569.Google Scholar
Baurdoux, E. J. and Kyprianou, A. E. (2008). The McKean stochastic game driven by a spectrally negative Lévy process. Electronic J. Prob. 8, 173–197.Google Scholar
Baurdoux, E. J. and Kyprianou, A. E. (2009). The Shepp–Shiryaev stochastic game driven by a spectrally negative Lévy process. Theory Prob. Appl. 53, 481–499.10.1137/S0040585X97983778CrossRefGoogle Scholar
Baurdoux, E. J., Kyprianou, A. E. and Pardo, J. C. (2011). The Gapeev–Kühn stochastic game driven by a spectrally positive Lévy process. Stoch. Process. Appl. 121, 1266–1289.10.1016/j.spa.2011.02.002CrossRefGoogle Scholar
Bielecki, T. R. and Rutkowski, M. (2004). Credit Risk: Modeling, Valuation and Hedging, 2nd edn. Springer, Berlin.10.1007/978-3-662-04821-4CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion, 2nd edn. Birkhäuser, Basel.Google Scholar
Cvitanić, J. and Karatzas, I. (1996). Backward stochastic differential equations with reflection and Dynkin games. Ann. Prob. 24, 2024–2056.10.1214/aop/1041903216CrossRefGoogle Scholar
Detemple, J. (2006). American-Style Derivatives: Valuation and Computation. Chapman and Hall/CRC, Boca Raton.Google Scholar
Dubins, L., Shepp, L. A. and Shiryaev, A. N. (1993). Optimal stopping rules and maximal inequalities for Bessel processes. Theory Prob. Appl. 38, 226–261.10.1137/1138024CrossRefGoogle Scholar
Ekström, E. and Villeneuve, S. (2006). On the value of optimal stopping games. Ann. Appl. Prob. 16, 1576–1596.10.1214/105051606000000204CrossRefGoogle Scholar
Gapeev, P. V. (2007). Discounted optimal stopping for maxima of some jump-diffusion processes. J. Appl. Prob. 44, 713–731.10.1017/S0021900200003387CrossRefGoogle Scholar
Gapeev, P. V. and Li, L. (2022). Optimal stopping problems for maxima and minima in models with asymmetric information. Stochastics 94, 602–628.10.1080/17442508.2021.1979976CrossRefGoogle Scholar
Gapeev, P. V. and Al Motairi, H. (2018). Perpetual American defaultable options in models with random dividends and partial information. Risks 6, 127.10.3390/risks6040127CrossRefGoogle Scholar
Gapeev, P. V. and Rodosthenous, N. (2014). Optimal stopping problems in diffusion-type models with running maxima and drawdowns. J. Appl. Prob. 51, 799–817.10.1239/jap/1409932675CrossRefGoogle Scholar
Gapeev, P. V. and Rodosthenous, N. (2016). On the drawdowns and drawups in diffusion-type models with running maxima and minima. J. Math. Anal. Appl. 434, 413–431.10.1016/j.jmaa.2015.09.013CrossRefGoogle Scholar
Gapeev, P. V. and Rodosthenous, N. (2016). Perpetual American options in diffusion-type models with running maxima and drawdowns. Stoch. Process. Appl. 126, 2038–2061.10.1016/j.spa.2016.01.003CrossRefGoogle Scholar
Gapeev, P. V., Kort, P. M. and Lavrutich, M. N. (2021). Discounted optimal stopping problems for maxima of geometric Brownian motions with switching payoffs. Adv. Appl. Prob. 53, 189–219.10.1017/apr.2020.57CrossRefGoogle Scholar
Glover, K., Hulley, H. and Peskir, G. (2013). Three-dimensional Brownian motion and the golden ratio rule. Ann. Appl. Prob. 23, 895–922.10.1214/12-AAP859CrossRefGoogle Scholar
Graversen, S. E. and Peskir, G. (1998). Optimal stopping and maximal inequalities for geometric Brownian motion. J. Appl. Prob. 35, 856–872.10.1239/jap/1032438381CrossRefGoogle Scholar
Guo, X. and Shepp, L. A. (2001). Some optimal stopping problems with nontrivial boundaries for pricing exotic options. J. Appl. Prob. 38, 647–658.10.1239/jap/1005091029CrossRefGoogle Scholar
Guo, X. and Zervos, M. (2010). $\pi$ options. Stoch. Process. Appl. 120, 1033–1059.10.1016/j.spa.2010.02.008CrossRefGoogle Scholar
Kallsen, J. and Kühn, C. (2004). Pricing derivatives of American and game type in incomplete markets. Finance Stoch. 8, 261–284.10.1007/s00780-003-0110-7CrossRefGoogle Scholar
Kifer, Y. (2000). Game options. Finance Stoch. 4, 443–463.10.1007/PL00013527CrossRefGoogle Scholar
Kühn, C. and Kyprianou, A. E. (2007). Callable puts as composite exotic options. Math. Finance 17, 487–502.10.1111/j.1467-9965.2007.00313.xCrossRefGoogle Scholar
Kyprianou, A. E. (2004). Some calculations for Israeli options. Finance Stoch. 8, 73–86.10.1007/s00780-003-0104-5CrossRefGoogle Scholar
Kyprianou, A. E. and Ott, C. (2014). A capped optimal stopping problem for the maximum process. Acta Appl. Math. 129, 147–174.10.1007/s10440-013-9833-4CrossRefGoogle Scholar
Linetsky, V. (2006). Pricing equity derivatives subject to bankruptcy. Math. Finance 16, 255–282.10.1111/j.1467-9965.2006.00271.xCrossRefGoogle Scholar
Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes I, 2nd edn. Springer, Berlin.Google Scholar
Ott, C. (2013). Optimal stopping problems for the maximum process with upper and lower caps. Ann. Appl. Prob. 23, 2327–2356.10.1214/12-AAP903CrossRefGoogle Scholar
Pedersen, J. L. (2000). Discounted optimal stopping problems for the maximum process. J. Appl. Prob. 37, 972–983.10.1239/jap/1014843077CrossRefGoogle Scholar
Peskir, G. (1998). Optimal stopping of the maximum process: the maximality principle. Ann. Prob. 26, 1614–1640.10.1214/aop/1022855875CrossRefGoogle Scholar
Peskir, G. (2007). A change-of-variable formula with local time on surfaces. In Séminaire de Probabilités XL (Lecture Notes Math. 1899), pp. 69–96. Springer.10.1007/978-3-540-71189-6_2CrossRefGoogle Scholar
Peskir, G. (2012). Optimal detection of a hidden target: the median rule. Stoch. Process. Appl. 122, 2249–2263.10.1016/j.spa.2012.02.004CrossRefGoogle Scholar
Peskir, G. (2014). Quickest detection of a hidden target and extremal surfaces. Ann. Appl. Prob. 24, 2340–2370.10.1214/13-AAP979CrossRefGoogle Scholar
Peskir, G. and Shiryaev, A. N. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin.10.1007/978-3-662-06400-9CrossRefGoogle Scholar
Rodosthenous, N. and Zervos, M. (2017). Watermark options. Finance Stoch. 21, 157–186.10.1007/s00780-016-0319-xCrossRefGoogle Scholar
Shepp, L. A. and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631–640.10.1214/aoap/1177005355CrossRefGoogle Scholar
Shepp, L. A. and Shiryaev, A. N. (1994). A new look at the pricing of Russian options. Theory Prob. Appl. 39, 103–119.10.1137/1139004CrossRefGoogle Scholar
Shepp, L. A. and Shiryaev, A. N. (1996). A dual Russian option for selling short. In Probability Theory and Mathematical Statistics, eds I. A. Ibragimov and A. Yu. Zaitsev, pp. 209–218. Gordon and Breach, Amsterdam.Google Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance. World Scientific, Singapore.10.1142/3907CrossRefGoogle Scholar