Penalty approximation and analytical characterization of the problem of super‐replication under portfolio constraints

Abstract

In this paper, we consider the problem of super‐replication under portfolio constraints in a Markov framework. More specifically, we assume that the portfolio is restricted to lie in a convex subset, and we show that the super‐replication value is the smallest function which lies above the Black–Scholes price function and which is stable for the so‐called face lifting operator. A natural approach to this problem is the penalty approximation, which not only provides a constructive smooth approximation, but also a way to proceed analytically.