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Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions

Author

Listed:
  • Pavel V. Gapeev

    (London School of Economics)

  • Peter M. Kort

    (Tilburg University CentER
    University of Antwerp)

  • Maria N. Lavrutich

    (Norwegian University of Science and Technology)

  • Jacco J. J. Thijssen

    (University of York)

Abstract

We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima of a geometric Brownian motion. It is shown that the optimal stopping times are th first times at which the underlying process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original double optimal stopping problems to sequences of single optimal stopping problems for the resulting three-dimensional continuous Markov process. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state space. We show that the optimal stopping boundaries are determined as the extremal solutions of the associated first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual real double lookback options with floating sunk costs in the Black-Merton-Scholes model.

Suggested Citation

  • Pavel V. Gapeev & Peter M. Kort & Maria N. Lavrutich & Jacco J. J. Thijssen, 2022. "Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 789-813, June.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:2:d:10.1007_s11009-022-09959-w
    DOI: 10.1007/s11009-022-09959-w
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    References listed on IDEAS

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    1. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    2. S. E. Graversen & G. Peškir, 1998. "Optimal Stopping and Maximal Inequalities for Linear Diffusions," Journal of Theoretical Probability, Springer, vol. 11(1), pages 259-277, January.
    3. Kristoffer Glover & Hardy Hulley & Goran Peskir, 2011. "Three-Dimensional Brownian Motion and the Golden Ratio Rule," Research Paper Series 295, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Carl Chiarella & Boda Kang, 2009. "The Evaluation of American Compound Option Prices Under Stochastic Volatility Using the Sparse Grid Approach," Research Paper Series 245, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Erik Brynjolfsson & Daniel Rock & Chad Syverson, 2018. "Artificial Intelligence and the Modern Productivity Paradox: A Clash of Expectations and Statistics," NBER Chapters, in: The Economics of Artificial Intelligence: An Agenda, pages 23-57, National Bureau of Economic Research, Inc.
    6. René Carmona & Nizar Touzi, 2008. "Optimal Multiple Stopping And Valuation Of Swing Options," Mathematical Finance, Wiley Blackwell, vol. 18(2), pages 239-268, April.
    7. Gapeev, Pavel V., 2020. "Optimal stopping problems for running minima with positive discounting rates," Statistics & Probability Letters, Elsevier, vol. 167(C).
    8. René Carmona & Savas Dayanik, 2008. "Optimal Multiple Stopping of Linear Diffusions," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 446-460, May.
    9. Pavel V. Gapeev, 2022. "Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 749-788, June.
    10. Erik Brynjolfsson & Lorin M. Hitt, 2003. "Computing Productivity: Firm-Level Evidence," The Review of Economics and Statistics, MIT Press, vol. 85(4), pages 793-808, November.
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    Cited by:

    1. Zbigniew Palmowski & Paweł Stȩpniak, 2023. "Last-Passage American Cancelable Option in Lévy Models," JRFM, MDPI, vol. 16(2), pages 1-14, January.
    2. Pavel V. Gapeev, 2022. "Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 749-788, June.

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    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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