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On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale

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  • Cox, Alexander M.G.
  • Obłój, Jan

Abstract

We study the joint laws of the maximum and minimum of a continuous, uniformly integrable martingale. In particular, we give explicit martingale inequalities which provide upper and lower bounds on the joint exit probabilities of a martingale, given its terminal law. Moreover, by constructing explicit and novel solutions to the Skorokhod embedding problem, we show that these bounds are tight. Together with previous results of Azéma & Yor, Perkins, Jacka and Cox & Obłój, this allows us to completely characterise the upper and lower bounds on all possible exit/no-exit probabilities, subject to a given terminal law of the martingale. In addition, we determine some further properties of these bounds, considered as functions of the maximum and minimum.

Suggested Citation

  • Cox, Alexander M.G. & Obłój, Jan, 2015. "On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3280-3300.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:8:p:3280-3300
    DOI: 10.1016/j.spa.2015.03.005
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    References listed on IDEAS

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    1. Jose Scheinkman & René Carmona & Erhan Cinlare & Ivar Ekeland & Elyès Jouini & Nizar Touzi, 2010. "Paris-Princeton Lectures on Mathematical Finance," Post-Print halshs-00706281, HAL.
    2. B. Acciaio & M. Beiglbock & F. Penkner & W. Schachermayer & J. Temme, 2012. "A trajectorial interpretation of Doob's martingale inequalities," Papers 1202.0447, arXiv.org, revised Jul 2013.
    3. Jacka, S. D., 1988. "Doob's inequalities revisited: A maximal H1-embedding," Stochastic Processes and their Applications, Elsevier, vol. 29(2), pages 281-290, September.
    4. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    5. Laurent Carraro & Nicole El Karoui & Jan Ob{l}'oj, 2009. "On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation," Papers 0902.1328, arXiv.org, revised Sep 2012.
    6. Cox, A. M. G. & Hobson, D. G., 2004. "An optimal Skorokhod embedding for diffusions," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 17-39, May.
    7. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    8. Haydyn Brown & David Hobson & L. C. G. Rogers, 2001. "Robust Hedging of Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 285-314, July.
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