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Hedging in discrete time under transaction costs and continuous-time limit

Part of: Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Pierre-F. Koehl ,
Huyên Pham  and
Nizar Touzi
Pierre-F. Koehl*
Affiliation:
CREST and ENSAE
Huyên Pham*
Affiliation:
Université de Marne-la-Vallée and CREST
Nizar Touzi*
Affiliation:
CEREMADE and CREST
*
∗Postal address: CREST-ENSAE, Laboratoire de Finance, 3 av. Pierre Larousse, 92245 Malakoff Cedex, France.
∗∗Postal address: Université Marne-la-Vallée, Equipe d'analyse et de mathématiques appliquées, 2 rue de la butte verte, 93166 Noisy-le-grand Cedex, France.
∗∗∗Postal address: Université Paris Dauphine, Centre de recherche en mathématiques de la décision, Place du Maréchal de Lattre-de-Tassigny, 75016 Paris Cedex, France. Email address: [email protected].
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Abstract

We consider a discrete-time financial market model with L1 risky asset price process subject to proportional transaction costs. In this general setting, using a dual martingale representation we provide sufficient conditions for the super-replication cost to coincide with the replication cost. Next, we study the convergence problem in a stationary binomial model as the time step tends to zero, keeping the proportional transaction costs fixed. We derive lower and upper bounds for the limit of the super-replication cost. In the case of European call options and for a unit initial holding in the risky asset, the upper and lower bounds are equal. This result also holds for the replication cost of European call options. This is evidence (but not a proof) against the common opinion that the replication cost is infinite in a continuous-time model.

Keywords

Transaction costsreplicationsuper-replicationmartingalescontinuous-time limit

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 163 - 178
Copyright
Copyright © Applied Probability Trust 1999 

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