Dynamic programming and mean-variance hedging

HuyËn Pham () and Jean Paul Laurent ()
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HuyËn Pham: Equipe d'Analyse et de MathÊmatiques AppliquÊes, UniversitÊ Marne-la-VallÊe, CitÊ Descartes, 5 Boulevard Descartes, Champs-sur-Marne, F-77454 Marne-la-VallÊe Cedex 2, France and CREST, Laboratoire de Finance-Assurance Manuscript
Jean Paul Laurent: CREST, Laboratoire de Finance-Assurance, 15 bld. GabriÊl PÊri, F-92245 Malakoff Cedex, France

Finance and Stochastics, 1999, vol. 3, issue 1, 83-110

Abstract: We consider the mean-variance hedging problem when asset prices follow ItÆ processes in an incomplete market framework. The hedging numÊraire and the variance-optimal martingale measure appear to be a key tool for characterizing the optimal hedging strategy (see GouriÊroux et al. 1996; RheinlÄnder and Schweizer 1996). In this paper, we study the hedging numÊraire $\tilde a$ and the variance-optimal martingale measure $\tilde P$ using dynamic programming methods. We obtain new explicit characterizations of $\tilde a$ and $\tilde P$ in terms of the value function of a suitable stochastic control problem. We provide several examples illustrating our results. In particular, for stochastic volatility models, we derive an explicit form of this value function and then of the hedging numÊraire and the variance-optimal martingale measure. This provides then explicit computations of optimal hedging strategies for the mean-variance hedging problem in usual stochastic volatility models.

Keywords: Hedging; incomplete markets; dynamic programming; hedging numÊraire; variance-optimal martingale measure (search for similar items in EconPapers)
JEL-codes: G11 G12 (search for similar items in EconPapers)
Date: 1998-11-17
Note: received: June 1997; final version received: January 1998
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