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Barrier option hedging under constraints: A viscosity approach

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  • Bentahar, Imen
  • Bouchard, Bruno

Abstract

We study the problem of finding the minimal initial capital needed in order to hedge without risk a barrier option when the vector of proportions of wealth invested in each risky asset is constraint to lie in a closed convex domain. In the context of a Brownian diffusion model, we provide a PDE characterization of the super-hedging price. This extends the result of Broadie, Cvitanic and Soner (1998) and Cvitanic, Pham and Touzi (1999) which was obtained for plain vanilla options, and provides a natural numerical procedure for computing the corresponding super-hedging price. As a by-product, we obtain a comparison theorem for a class of parabolic PDE with relaxed Dirichet conditions involving a constraint on the gradient.

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  • Bentahar, Imen & Bouchard, Bruno, 2006. "Barrier option hedging under constraints: A viscosity approach," SFB 649 Discussion Papers 2006-022, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    • Handle: RePEc:zbw:sfb649:sfb649dp2006-022
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    References listed on IDEAS

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    1. Broadie, Mark & Cvitanic, Jaksa & Soner, H Mete, 1998. "Optimal Replication of Contingent Claims under Portfolio Constraints," The Review of Financial Studies, Society for Financial Studies, vol. 11(1), pages 59-79.
    2. Uwe Wystup & Uwe Schmock & Steven E. Shreve, 2002. "Valuation of exotic options under shortselling constraints," Finance and Stochastics, Springer, vol. 6(2), pages 143-172.
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    2. Christopher W. Miller, 2016. "A Duality Result for Robust Optimization with Expectation Constraints," Papers 1610.01227, arXiv.org.

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    Keywords

    Super-replication; barrier options; portfolio constraints; viscosity solutions;
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