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When terminal facelift enforces delta constraints

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  • Jean-François Chassagneux
  • Romuald Elie
  • Idris Kharroubi

Abstract

This paper deals with the superreplication of non-path-dependent European claims under additional convex constraints on the number of shares held in the portfolio. The corresponding superreplication price of a given claim has been widely studied in the literature, and its terminal value, which dominates the claim of interest, is the so-called facelift transform of the claim. We investigate under which conditions the superreplication price and strategy of a large class of claims coincide with the exact replication price and strategy of the facelift transform of this claim. In one dimension, we observe that this property is satisfied for any local volatility model. In any dimension, we exhibit an analytical necessary and sufficient condition for this property, which combines the dynamics of the stock together with the characteristics of the closed convex set of constraints. To obtain this condition, we introduce the notion of first order viability property for linear parabolic PDEs. We investigate in detail several practical cases of interest: multidimensional Black–Scholes model, non-tradable assets, and short-selling restrictions. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Jean-François Chassagneux & Romuald Elie & Idris Kharroubi, 2015. "When terminal facelift enforces delta constraints," Finance and Stochastics, Springer, vol. 19(2), pages 329-362, April.
    • Handle: RePEc:spr:finsto:v:19:y:2015:i:2:p:329-362
    DOI: 10.1007/s00780-015-0260-4
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    References listed on IDEAS

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    1. Emmanuel Gobet, 2004. "Revisiting the Greeks for European and American Options," World Scientific Book Chapters, in: Jiro Akahori & Shigeyoshi Ogawa & Shinzo Watanabe (ed.), Stochastic Processes And Applications To Mathematical Finance, chapter 3, pages 53-71, World Scientific Publishing Co. Pte. Ltd..
    2. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    3. repec:dau:papers:123456789/11551 is not listed on IDEAS
    4. Bouchard, Bruno & Chassagneux, Jean-François, 2008. "Discrete-time approximation for continuously and discretely reflected BSDEs," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2269-2293, December.
    5. 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.
    6. N'Zi, Modeste & Ouknine, Youssef & Sulem, Agnès, 2006. "Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1319-1339, September.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Kasper Larsen & Halil Soner & Gordan Žitković, 2016. "Facelifting in utility maximization," Finance and Stochastics, Springer, vol. 20(1), pages 99-121, January.
    2. Dylan Possamai & Nizar Touzi, 2020. "Is there a Golden Parachute in Sannikov's principal-agent problem?," Papers 2007.05529, arXiv.org, revised Oct 2022.
    3. Dirk Becherer & Todor Bilarev, 2018. "Hedging with physical or cash settlement under transient multiplicative price impact," Papers 1807.05917, arXiv.org, revised Jun 2023.

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    More about this item

    Keywords

    Superreplication; Portfolio constraints; Viability; Facelift; BSDEs; 93E20; 91G20; 60H30; C69; G11; G13;
    All these keywords.

    JEL classification:

    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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