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A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options

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  • A. Galichon
  • P. Henry-Labord\`ere
  • N. Touzi

Abstract

We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset, and statically trade European call options for all possible strikes with some given maturity. This problem is classically approached by means of the Skorohod Embedding Problem (SEP). Instead, we provide a dual formulation which converts the superhedging problem into a continuous martingale optimal transportation problem. We then show that this formulation allows us to recover previously known results about lookback options. In particular, our methodology induces a new proof of the optimality of Az\'{e}ma-Yor solution of the SEP for a certain class of lookback options. Unlike the SEP technique, our approach applies to a large class of exotics and is suitable for numerical approximation techniques.

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File URL: http://arxiv.org/pdf/1401.3921
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Paper provided by arXiv.org in its series Papers with number 1401.3921.

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Date of creation: Jan 2014
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Publication status: Published in Annals of Applied Probability 2014, Vol. 24, No. 1, 312-336
Handle: RePEc:arx:papers:1401.3921

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  1. Karandikar, Rajeeva L., 1995. "On pathwise stochastic integration," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 57(1), pages 11-18, May.
  2. 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.
  3. 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.
  4. RØdiger Frey, 2000. "Superreplication in stochastic volatility models and optimal stopping," Finance and Stochastics, Springer, vol. 4(2), pages 161-187.
  5. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 11(3), pages 215-260, August.
  6. Dylan Possama\"i & Guillaume Royer & Nizar Touzi, 2013. "On the Robust superhedging of measurable claims," Papers 1302.1850, arXiv.org, revised Feb 2013.
  7. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
  8. Kreps, David M., 1981. "Arbitrage and equilibrium in economies with infinitely many commodities," Journal of Mathematical Economics, Elsevier, vol. 8(1), pages 15-35, March.
  9. A. M. G. Cox & David Hobson & Jan Ob{\l}\'oj, 2007. "Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping," Papers math/0702173, arXiv.org, revised Nov 2008.
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Cited by:
  1. Y. Dolinsky & H. M. Soner, 2014. "Martingale optimal transport in the Skorokhod space," Papers 1404.1516, arXiv.org.
  2. Yan Dolinsky & H. Soner, 2014. "Robust hedging with proportional transaction costs," Finance and Stochastics, Springer, vol. 18(2), pages 327-347, April.
  3. Erhan Bayraktar & Gu Wang, 2014. "Quantile Hedging in a Semi-Static Market with Model Uncertainty," Papers 1408.4848, arXiv.org.

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