The Problem of Super-replication under Constraints

These notes present an overview of the problem of super-replication under portfolio constraints. We start by examining the duality approach and its limitations. We then concentrate on the direct approach in the Markov case which allows to handle general large investor problems and gamma constraints. In the context of the Black and Scholes model, the main result from the practical view-point is the so-called face-lifting phenomenon of the payoff function.Keywords: Super-replication, duality, dynamic programming, viscosity solutions, Hamilton-Jacobi-Bellman equation.AMS 1991 subject classification: Primary 49J20, 60J60; Secondary 49L20, 35K55.