Completeness of bond market driven by L\'evy process

TL;DR

This paper investigates the conditions under which a bond market driven by Lévy processes is complete, showing that market completeness requires the Lévy measure's support to be finite, with explicit non-replicable claims constructed.

Contribution

It provides a necessary and sufficient condition for market completeness in Lévy-driven bond markets and constructs explicit examples of non-replicable claims.

Findings

Market is incomplete unless Lévy measure support is finite.

Explicit non-replicable claims are constructed.

Completeness depends on the support of the Lévy measure.

Abstract

The completeness problem of the bond market model with the random factors determined by a Wiener process and Poisson random measure is studied. Hedging portfolios use bonds with maturities in a countable, dense subset of a finite time interval. It is shown that under natural assumptions the market is not complete unless the support of the L\'evy measure consists of a finite number of points. Explicit constructions of contingent claims which can not be replicated are provided.