Mathematical Methods of Operations Research

, Volume 50, Issue 2, pp 297–320

Super-replication under proportional transaction costs: From discrete to continuous-time models

  • Nizar Touzi

DOI: 10.1007/s001860050099

Cite this article as:
Touzi, N. Mathematical Methods of OR (1999) 50: 297. doi:10.1007/s001860050099


In this paper, we study the problem of finding the minimal initial capital (i.e. super-replication value) needed in order to hedge (without risk) European contingent claims in a Markov setting under proportional transaction costs. The main result is that the cheapest (trivial) buy-and-hold strategy is optimal. Such a negative result has been derived previously in different contexts. First, we focus on discrete-time binomial models. We prove that the continuous-time limit of the super-replication value is the cost of the cheapest buy-and-hold strategy. Then, the result is proved in a multivariate continuous-time model with Brownian filtration. As a direct consequence, we obtain an explicit characterization of the hedging set, i.e. the set of all initial positions in the market assets from which the contingent claim can be hedged through some admissible portfolio strategy.

Key words: Transaction costshedging optionscontinuous-time limitdynamic programmingviscosity solutions

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Nizar Touzi
    • 1
  1. 1.Centre de Recherche de Mathématiques, Statistiques et Economie Mathématiques, Université Paris I, Panthéon-Sorbonne, 106-112 Bd de l'Hôpital, 75647 Paris Cedex 13, FranceFR