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Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints

Part of: Stochastic analysis Stochastic processes Miscellaneous topics in calculus of variations and optimal control Stochastic systems and control Mathematical finance

Published online by Cambridge University Press:  04 January 2016

Christoph Czichowsky  and
Martin Schweizer
Christoph Czichowsky*
Affiliation:
University of Vienna
Martin Schweizer*
Affiliation:
ETH Zürich and Swiss Finance Institute
*
∗ Postal address: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A–1090 Vienna, Austria. Email address: christoph.czichowsky@univie.ac.at
∗∗ Postal address: Department of Mathematics, ETH Zürich, Rämistrasse 101, CH–8092, Zürich, Switzerland. Email address: martin.schweizer@math.ethz.ch
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Abstract

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We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L2 of the space of all gains from trade (i.e. the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad-hoc methods in specific frameworks.

Keywords

Mean-variance hedgingconstraintsstochastic integralconvex duality

MSC classification

Primary: 60G48: Generalizations of martingales 91G10: Portfolio theory 93E20: Optimal stochastic control 49N10: Linear-quadratic problems
Secondary: 60H05: Stochastic integrals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 1084 - 1112
Copyright
© Applied Probability Trust 

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