On the Second Fundamental Theorem of Asset Pricing

TL;DR

This paper establishes a fundamental link between market completeness and the uniqueness of equivalent sigma-martingale measures, providing a key theoretical result in asset pricing.

Contribution

It proves that market completeness is equivalent to the uniqueness of the equivalent sigma-martingale measure, extending the second fundamental theorem of asset pricing.

Findings

Market completeness iff unique ESMM

Characterization of bounded martingale representation

Connection between sigma-martingales and market completeness

Abstract

Let $X^1,\ldots, X^d$ be sigma-martingales on $(\Omega,{\cal F}, P)$. We show that every bounded martingale (with respect to the underlying filtration) admits an integral representation w.r.t. $X^1,\ldots, X^d$ if and only if there is no equivalent probability measure (other than $P$) under which $X^1,\ldots,X^d$ are sigma-martingales. From this we deduce the second fundamental theorem of asset pricing- that completeness of a market is equivalent to uniqueness of Equivalent Sigma-Martingale Measure (ESMM).