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Hedging Under Gamma Constraints By Optimal Stopping And Face‐Lifting

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  • H. Mete Soner
  • Nizar Touzi

Abstract

A super‐replication problem with a gamma constraint, introduced in Soner and Touzi, is studied in the context of the one‐dimensional Black and Scholes model. Several representations of the minimal super‐hedging cost are obtained using the characterization derived in Cheridito, Soner, and Touzi. It is shown that the upper bound constraint on the gamma implies that the optimal strategy consists in hedging a conveniently face‐lifted payoff function. Further an unusual connection between an optimal stopping problem and the lower bound is proved. A formal description of the optimal hedging strategy as a succession of periods of classical Black–Scholes hedging strategy and simple buy‐and‐hold strategy is also provided.

Suggested Citation

  • H. Mete Soner & Nizar Touzi, 2007. "Hedging Under Gamma Constraints By Optimal Stopping And Face‐Lifting," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 59-79, January.
    • Handle: RePEc:bla:mathfi:v:17:y:2007:i:1:p:59-79
    DOI: 10.1111/j.1467-9965.2007.00294.x
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    Cited by:

    1. Bruno Bouchard & G Loeper & Y Zou, 2016. "Almost-sure hedging with permanent price impact," Post-Print hal-01133223, HAL.
    2. René Carmona & Kevin Webster, 2019. "The self-financing equation in limit order book markets," Finance and Stochastics, Springer, vol. 23(3), pages 729-759, July.
    3. Rene Carmona & Kevin Webster, 2019. "Applications of a New Self-Financing Equation," Papers 1905.04137, arXiv.org.
    4. B Bouchard & G Loeper & Y Zou, 2015. "Almost-sure hedging with permanent price impact," Working Papers hal-01133223, HAL.
    5. Gregoire Loeper, 2013. "Option pricing with linear market impact and non-linear Black and Scholes equations," Papers 1301.6252, arXiv.org, revised Aug 2016.
    6. B. Bouchard & G. Loeper & Y. Zou, 2015. "Almost-sure hedging with permanent price impact," Papers 1503.05475, arXiv.org.
    7. Dylan Possamai & Nizar Touzi, 2020. "Is there a Golden Parachute in Sannikov's principal-agent problem?," Papers 2007.05529, arXiv.org, revised Oct 2022.

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