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Optimal time to invest when the price processes are geometric Brownian motions. A tentative based on smooth fit

Author

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  • GAHUNGU, Joachim

    (Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium)

  • SMEERS, Yves

    (Université catholique de Louvain, CORE, B-1348 Louvain-la-Neuve, Belgium)

Abstract

This paper considers the problem of the optimal timing of the exchange of the sum of n geometric Brownian motions for the sum of m others. We propose a closed form determinable stopping time based on the heuristic principle of smooth fit. We cannot prove that this stopping time is optimal. However, we show numerically on examples that it is a potentially useful candidate: letting S^Ø denote the stopping region induced by our stopping time we show that (i) S^- c S^Ø c S^+ where S^- and S^+ are well-known subset and superset of the optimal stopping region; (ii) stopping at the first entry time of S^Ø offers a better payoff than stopping at the first entry time of S^- or S^+, especially when assets are correlated.

Suggested Citation

  • GAHUNGU, Joachim & SMEERS, Yves, 2011. "Optimal time to invest when the price processes are geometric Brownian motions. A tentative based on smooth fit," LIDAM Discussion Papers CORE 2011034, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2011034
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    References listed on IDEAS

    as
    1. Yaozhong Hu & Bernt Øksendal, 1998. "Optimal time to invest when the price processes are geometric Brownian motions," Finance and Stochastics, Springer, vol. 2(3), pages 295-310.
    2. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    3. Robert McDonald & Daniel Siegel, 1986. "The Value of Waiting to Invest," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 101(4), pages 707-727.
    4. Olsen, Trond E. & Stensland, Gunnar, 1992. "On optimal timing of investment when cost components are additive and follow geometric diffusions," Journal of Economic Dynamics and Control, Elsevier, vol. 16(1), pages 39-51, January.
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    Cited by:

    1. GABSZEWICZ, Jean J. & VAN YPERSELE, Tanguy & ZANAJ, Skerdilajda, 2011. "Does the seller of a house facing a large number of buyers always decrease its price when its first offer is rejected?," LIDAM Discussion Papers CORE 2011049, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Støre, Kristian & Fleten, Stein-Erik & Hagspiel, Verena & Nunes, Cláudia, 2018. "Switching from oil to gas production in a depleting field," European Journal of Operational Research, Elsevier, vol. 271(2), pages 710-719.
    3. Christensen, Sören & Irle, Albrecht, 2020. "The monotone case approach for the solution of certain multidimensional optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1972-1993.
    4. GAHUNGU, Joachim & SMEERS, Yves, 2011. "Sufficient and necessary conditions for perpetual multi-assets exchange options," LIDAM Discussion Papers CORE 2011035, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. VAN VYVE, Mathieu, 2011. "Linear prices for non-convex electricity markets: models and algorithms," LIDAM Discussion Papers CORE 2011050, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

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    More about this item

    Keywords

    optimal stopping; geometric Brownian motion; smooth fit;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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