Finance and Stochastics

, Volume 19, Issue 2, pp 329–362 | Cite as

When terminal facelift enforces delta constraints

  • Jean-François ChassagneuxEmail author
  • Romuald Elie
  • Idris Kharroubi


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.


Superreplication Portfolio constraints Viability Facelift BSDEs 

Mathematics Subject Classification

93E20 91G20 60H30 

JEL Classification

C69 G11 G13 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jean-François Chassagneux
    • 1
    Email author
  • Romuald Elie
    • 2
  • Idris Kharroubi
    • 3
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Laboratoire d’Analyse et de Mathématiques AppliquéesUniversité de Marne-la-ValléeMarne-la-Vallée cedex 2France
  3. 3.Université Paris DauphineParis cedex 16France

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