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Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping

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  • A. M. G. Cox
  • David Hobson
  • Jan Ob{\l}\'oj
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    Abstract

    We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero and $M_t$ a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois' Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form $\sup_{\tau}\mathbb{E}[F(L_{\tau})-\int _0^{\tau}\beta(B_s) ds]$. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques.

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    File URL: http://arxiv.org/pdf/math/0702173
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number math/0702173.

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    Date of creation: Feb 2007
    Date of revision: Nov 2008
    Publication status: Published in Annals of Applied Probability 2008, Vol. 18, No. 5, 1870-1896
    Handle: RePEc:arx:papers:math/0702173

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    Web page: http://arxiv.org/

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    References

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    1. Carr, Peter P & Jarrow, Robert A, 1990. "The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value," Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 469-92.
    2. Haydyn Brown & David Hobson & L. C. G. Rogers, 2001. "Robust Hedging of Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 285-314.
    3. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    4. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
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    Cited by:
    1. 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.
    2. Pierre Henry-Labordere & Nizar Touzi, 2013. "An Explicit Martingale Version of Brenier's Theorem," Papers 1302.4854, arXiv.org, revised Apr 2013.
    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. Beatrice Acciaio & Mathias Beiglb\"ock & Friedrich Penkner & Walter Schachermayer, 2013. "A model-free version of the fundamental theorem of asset pricing and the super-replication theorem," Papers 1301.5568, arXiv.org, revised Mar 2013.
    5. Pierre Henry-Labordere & Nizar Touzi, 2013. "An Explicit Martingale Version of Brenier's Theorem," Working Papers hal-00790001, HAL.

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