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

## Author Info

• A. M. G. Cox
• David Hobson
• Jan Ob{\l}\'oj
Registered author(s):

## 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

## Bibliographic Info

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

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 Length: 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 Contact details of provider: Web page: http://arxiv.org/

## References

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1. Peter P. Carr & Robert A. Jarrow, 2008. "The Stop-Loss Start-Gain Paradox and Option Valuation: A new Decomposition into Intrinsic and Time Value," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 4, pages 61-84 World Scientific Publishing Co. Pte. Ltd..
2. 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.
3. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
4. Haydyn Brown & David Hobson & L. C. G. Rogers, 2001. "Robust Hedging of Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 285-314.
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