Bond Market Completeness and Attainable Contingent Claims
Abstract
Description
A general class, introduced in [Ekeland et al. 2003], of continuous time bond markets driven by a standard cylindrical Brownian motion $\wienerq{}{}$ in $\ell^{2},$ is considered. We prove that there always exist non-hedgeable random variables in the space $\derprod{}{0}=\cap_{p \geq 1}L^{p}$ and that $\derprod{}{0}$ has a dense subset of attainable elements, if the volatility operator is non-degenerated a.e. Such results were proved in [Björk et al. 1997] in the case of a bond market driven by finite dimensional B.m. and marked point processes. We define certain smaller spaces $\derprod{}{s},$ $s>0$ of European contingent claims, by requiring that the integrand in the martingale representation, with respect to $\wienerq{}{}$, takes values in weighted $\ell^{2}$ spaces $\ell^{s,2},$ with a power weight of degree $s.$ For all $s > 0,$ the space $\derprod{}{s}$ is dense in $\derprod{}{0}$ 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 $\derprod{}{s}$ 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.
27 pages, Revised version to be published in Finance and Stochastics
27 pages, Revised version to be published in Finance and Stochastics