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Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

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

Published online by Cambridge University Press:  17 March 2021

Pavel V. Gapeev ,
Peter M. Kort  and
Maria N. Lavrutich
Pavel V. Gapeev*
Affiliation:
London School of Economics
Peter M. Kort*
Affiliation:
Tilburg University and University of Antwerp
Maria N. Lavrutich*
Affiliation:
Norwegian University of Science and Technology
*
*Postal address: London School of Economics, Department of Mathematics, Houghton Street, LondonWC2A 2AE, United Kingdom.
**Postal address: Tilburg University, CentER, Department of Econometrics and Operations Research, PO Box 90153, 5000 LE Tilburg, The Netherlands; University of Antwerp, Department of Economics, Prinsstraat 13, 2000 Antwerp 1, Belgium.
***Postal address: Norwegian University of Science and Technology, Department of Industrial Economics and Technology Management, 7491Trondheim, Norway.
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Abstract

We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.

Keywords

Discounted optimal stopping problemgeometric Brownian motionrunning maximum processcontinuous-time Markov chainfree-boundary probleminstantaneous stopping and smooth fitnormal reflectionperpetual American and real optionschange-of-variable formula with local time on surfaces

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G44: Martingales with continuous parameter 60J65: Brownian motion
Secondary: 91B25: Asset pricing models 60J27: Continuous-time Markov processes on discrete state spaces 35R35: Free boundary problems
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 1 , March 2021 , pp. 189 - 219
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Adkins, R. and Paxson, D. (2016). Subsidies for renewable energy facilities under uncertainty. Manch. Sch. 84, 222–250.CrossRefGoogle Scholar
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.CrossRefGoogle 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.Google Scholar
Boomsma, T. K. and Linnerud, K. (2015). Market and policy risk under different renewable electricity support schemes. Europ. J. Operat. Res. 89, 435–448.Google Scholar
Boomsma, T. K., Meade, N. and Fleten, S.-E. (2012). Renewable energy investments under different support schemes: a real options approach. Europ. J. Operat. Res. 220, 225–237.CrossRefGoogle 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.CrossRefGoogle Scholar
Baurdoux, E. J. and van Schaik, K. (2014). Predicting the time at which a LC)vy process attains its ultimate supremum. Acta Appl. Math. 134, 21–44.CrossRefGoogle Scholar
Beibel, M. and Lerche, H. R. (1997). A new look at optimal stopping problems related to mathematical finance. Statistica Sinica 7, 93–108.Google Scholar
Chronopoulos, M., Hagspiel, V. and Fleten, S-E. (2016). Stepwise green investment under policy uncertainty. Energy J. 37, 87–108.CrossRefGoogle Scholar
Dalang, R. C. and Hongler, M.-O. (2004). The right time to sell a stock whose price is driven by Markovian noise. Ann. Appl. Prob. 14, 2167–2201.CrossRefGoogle Scholar
Dalby, P. A. O., Gillerhaugen, G. R., Hagspiel, V., Leth-Olsen, T. and Thijssen, J. J. J. (2018). Green investment under policy uncertainty and Bayesian learning. Energy 161, 1262–1281.CrossRefGoogle Scholar
Dixit, A. K. and Pindyck, R. S. (1994). Investment under Uncertainty. Princeton University Press.CrossRefGoogle 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.CrossRefGoogle Scholar
Elliott, R. J., Aggoun, L. and Moore, J. B. (1995). Hidden Markov Models: Estimation and Control. Springer, New York.Google Scholar
Eryilmaz, D. and Homans, F. R. (2016). How does uncertainty in renewable energy policy affect decisions to invest in wind energy? Electricity J. 29, 64–71.CrossRefGoogle Scholar
Gapeev, P. V. (2007). Discounted optimal stopping for maxima of some jump-diffusion processes. J. Appl. Prob. 44, 713–731.CrossRefGoogle 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.CrossRefGoogle Scholar
Gapeev, P. V. and Rodosthenous, N. (2015). On the drawdowns and drawups in diffusion-type models with running maxima and minima. J. Math. Anal. Appl. 434, 413–431.CrossRefGoogle 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.CrossRefGoogle Scholar
Gerber, H. U., Michaud, F. and Shiu, E. S. W. (1995). Pricing Russian options with the compound Poisson process. Insurance Math. Econom. 17, 79.CrossRefGoogle Scholar
Glover, K., Hulley, H. and Peskir, G. (2013). Three-Dimensional Brownian motion and the golden ratio rule. Ann. Appl. Prob. 23, 895–922.CrossRefGoogle Scholar
Graversen, S. E. and Peskir, G. (1998). Optimal stopping and maximal inequalities for geometric Brownian motion. J. Appl. Prob. 35, 856–872.CrossRefGoogle Scholar
Guo, X. (2001). An explicit solution to an optimal stopping problem with regime switching. J. Appl. Prob. 38, 464–481.CrossRefGoogle 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.CrossRefGoogle Scholar
Guo, X. and Zervos, M. (2010). $\pi$ options. Stoch. Process. Appl. 120, 1033–1059.CrossRefGoogle Scholar
Guo, X. and Zhang, Q. (2004). Closed-Form solutions for perpetual American put options with regime switching. SIAM J. Appl. Math. 64, 2034–2049.Google Scholar
Hassett, K. A. and Metcalf, G. E. (1999). Investment with uncertain tax policy: does random tax policy discourage investment? Economic J. 109, 372–393.Google Scholar
Jiang, Z. and Pistorius, M. R. (2008). On perpetual American put valuation and first-passage in a regime-switching model with jumps. Finance Stoch. 12, 331–355.CrossRefGoogle Scholar
Jobert, A. and Rogers, L. C. G. (2006). Option pricing with Markov-Modulated dynamics. SIAM J. Control Optimization 44, 2063–2078.CrossRefGoogle Scholar
Kyprianou, A. E. and Ott, C. (2014). A capped optimal stopping problem for the maximum process. Acta Appl. Math. 129, 147–174.CrossRefGoogle Scholar
Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes I, 2nd edn. Springer, Berlin.Google Scholar
Mordecki, E. and Moreira, W. (2001). Russian options for a diffusion with negative jumps. Publ. Mat. Uruguay 9, 37–51.Google Scholar
Ott, C. (2013). Optimal stopping problems for the maximum process with upper and lower caps. Ann. Appl. Prob. 23, 2327–2356.CrossRefGoogle Scholar
Pedersen, J. L. (2000). Discounted optimal stopping problems for the maximum process. J. Appl. Prob. 37, 972–983.CrossRefGoogle Scholar
Peskir, G. (1998). Optimal stopping of the maximum process: the maximality principle. Ann. Prob. 26, 1614–1640.CrossRefGoogle Scholar
Peskir, G. (2007). A change-of-variable formula with local time on surfaces. In Séminaire de Probabilité XL (Lecture Notes Math. 1899), Springer, Berlin, pp. 69–96.Google Scholar
Peskir, G. (2012). Optimal detection of a hidden target: the median rule. Stoch. Process. Appl. 122, 2249–2263.CrossRefGoogle Scholar
Peskir, G. (2014). Quickest detection of a hidden target and extremal surfaces. Ann. Appl. Prob. 24, 2340–2370.CrossRefGoogle Scholar
Peskir, G. and Shiryaev, A. N. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Rodosthenous, N. and Zervos, M. (2017). Watermark options. Finance Stoch. 21, 157–186.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin.CrossRefGoogle Scholar
Ritzenhofer, I. and Spinler, S. (2016). Optimal design of feed-in-tariffs to stimulate renewable energy investments under regulatory uncertainty—a real options analysis. Energy Econom. 53, 76–89.CrossRefGoogle Scholar
Shepp, L. A. and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631–640.CrossRefGoogle Scholar
Shepp, L. A. and Shiryaev, A. N. (1994). A new look at the pricing of Russian options. Theory Prob. Appl. 39, 103–119.CrossRefGoogle Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance. World Scientific, Singapore.CrossRefGoogle Scholar