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

Article type: Research Article

Authors: Bensoussan, Alain^; | Touzi, Nizar | Menaldi, José Luis

Affiliations: CEREMADE, Université Paris IX, 75775 Paris cedex 16, France E‐mail: [email protected] | Centre de Recherche en Economie et Statistique and CEREMADE, 15 Bd Gabriel Péri, 92245 Malakoff cedex, France E‐mail: [email protected] | Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA E‐mail: [email protected]

Note: [] Corresponding author.

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.

Keywords: hedging under portfolio constraints, penalization, viscosity solutions

Journal: Asymptotic Analysis, vol. 41, no. 3-4, pp. 311-330, 2005