Bond market completeness and attainable contingent claims

A general class, introduced in [7], of continuous time bond markets driven by a standard cylindrical Brownian motion \(\bar{W}\) in \(\ell^{2}\) is considered. We prove that there always exist non-hedgeable random variables in the space \(\textsf{D}_{0} = \cap_{p \geq 1}L^{p}\) and that \(\textsf{D}_{0}\) has a dense subset of attainable elements, if the volatility operator is non-degenerate a.e. Such results were proved in [1] and [2] in the case of a bond market driven by finite dimensional Brownian motions and marked point processes. We define certain smaller spaces \(\textsf{D}_{s}\), s > 0, of European contingent claims by requiring that the integrand in the martingale representation with respect to \(\bar{W}\) takes values in weighted \(\ell^{2}\) spaces \(\ell^{s,2}\), with a power weight of degree s. For all s > 0, the space \(\textsf{D}_{s}\) is dense in \(\textsf{D}_{0}\) and is independent of the particular bond price and volatility operator processes.

A simple condition in terms of \(\ell^{s,2}\) norms is given on the volatility operator processes, which implies if satisfied that every element in \(\textsf{D}_{s}\) is attainable. In this context a related problem of optimal portfolios of zero coupon bonds is solved for general utility functions and volatility operator processes, provided that the \(\ell^{2}\)-valued market price of risk process has certain Malliavin differentiability properties.

Authors and Affiliations

1. Ecole International des Sciences du Traitement de l’Information, EISTI, Avenue du Parc, 95011, Cergy, France

   Erik Taflin

Corresponding author

Correspondence to Erik Taflin.

JEL Classification:

C61, C62, G10, G11

Mathematics Subject Classification (2000):

91B28, 49J55, 60H07, 90C46, 46E35

This article was partially prepared at CEREMADE, Université Paris IX - Dauphine, Place du Maréchal-de-Lattre-de-Tassigny, 75775 Paris (Cedex 16), France, taflin@ceremade.dauphine.fr.

The author would like to thank Ivar Ekeland and Nizar Touzi for fruitful discussions and the anonymous referees for constructive suggestions.

Manuscript received: February 2004; final version received: January 2005

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