The numeraire portfolio in semimartingale financial models

TL;DR

This paper characterizes the existence of the numeraire portfolio in general semimartingale financial models with convex constraints, using predictable characteristics and weaker no-arbitrage conditions, facilitating utility optimization.

Contribution

It provides necessary and sufficient conditions for the numeraire portfolio's existence based on predictable characteristics and introduces the weaker NUPBR condition as sufficient for utility maximization.

Findings

Characterization of the numeraire portfolio via predictable characteristics.

Establishment of NUPBR as the minimal assumption for utility optimization.

Simplification of checking conditions compared to NFLVR.

Abstract

We study the existence of the numeraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numeraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numeraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free-lunch-type notion. In particular, the full strength of the "No Free Lunch with Vanishing Risk" (NFLVR) is not needed, only the weaker "No Unbounded Profit with Bounded Risk" (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks.