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Super-replication in stochastic volatility models under portfolio constraints

Part of: Stochastic processes Stochastic analysis Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

Jakša Cvitanić ,
Huyên Pham  and
Nizar Touzi
Jakša Cvitanić*
Affiliation:
Columbia University
Huyên Pham*
Affiliation:
Université Marne-la-Vallée and CREST
Nizar Touzi*
Affiliation:
CEREMADE and CREST
*
∗Postal address: Department of Statistics, Columbia University, 2990 Broadway, New York, NY 10027. Email address: [email protected]
∗∗Postal address: Equipe d'Analyse et de Mathématiques Appliquées, Université Marne-la-Vallée, Cité Descartes, 5 Boulevard Descartes, Marne-la-Vallée Cedex, 77454, France.
∗∗∗Postal address: CEREMADE, Université Paris Dauphine, Place du Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France.
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Abstract

We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a ‘hedger’ in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.

Keywords

Stochastic volatilityportfolio constraintshedging optionsviscosity solution

MSC classification

Primary: 93E20: Optimal stochastic control 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 523 - 545
Copyright
Copyright © Applied Probability Trust 1999 

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