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A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options

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  • Alfred Galichon

    (Département d'économie)

  • Pierre Henri-Labordère
  • Nizar Touzi

    (Centre de Mathématiques Appliquées - Ecole Polytechnique)

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 to recover previously known results about Lookback options. In particular, our methodology induces a new presentation of the Azema-Yor solution of the SEP.

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Paper provided by Sciences Po in its series Sciences Po publications with number info:hdl:2441/5rkqqmvrn4tl22s9mc0ck8ecp.

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Date of creation: 2013
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Publication status: Published in Journal of Applied Probability, 2013, vol. 24, pp.312-336
Handle: RePEc:spo:wpmain:info:hdl:2441/5rkqqmvrn4tl22s9mc0ck8ecp

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  1. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
  2. RØdiger Frey, 2000. "Superreplication in stochastic volatility models and optimal stopping," Finance and Stochastics, Springer, vol. 4(2), pages 161-187.
  3. 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.
  4. Dylan Possama\"i & Guillaume Royer & Nizar Touzi, 2013. "On the Robust superhedging of measurable claims," Papers 1302.1850, arXiv.org, revised Feb 2013.
  5. 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.
  6. 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.
  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. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
  9. Karandikar, Rajeeva L., 1995. "On pathwise stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 11-18, May.
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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.

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