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Superreplication under gamma constraints. (English) Zbl 0960.91036
In a financial market with the underlying price process given by a Markovian Itō diffusion, this paper studies the problem of superreplicating a given payoff under a gamma constraint on the hedging strategy. This constraint imposes a boundedness condition on the space derivative of the hedging strategy or on the second space derivative of the pricing function. For the general case, the authors prove a verification theorem characterizing the superreplication cost as the solution of a quasi-variational inequality; this is then solved explicitly in the Black-Scholes model with constant coefficients. The main tools are a new dynamic programming principle and viscosity solution techniques.

MSC:
91G10 Portfolio theory
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K55 Nonlinear parabolic equations
49J20 Existence theories for optimal control problems involving partial differential equations
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