Locally Risk-Free Asset Implied by a Finite Set of Assets

Abstract

This document is the translation of the thesis "Titolo localmente non rishioso implicito per un insieme finito di titoli" which I presented at the university of Padua, Italy, in November 1999, for the award of the degree in Pure and Applied Mathematics (Laurea in Matematica Pura ed Applicata). In this work, I provide an extension of the Fundamental Theorem of Asset Pricing (FTAP), which establishes the equivalence between the No Free Lunch with Vanishing Risk (NFLVR) condition and the Existence of an equivalent Separating Measure (ESM). I extend the theorem to the case where some of the assets are subject to short-selling constraints, by showing that the equivalence between NFLVR and ESM continues to hold, and that the assets subject to short-selling constraints turn out to be supermartingales under the equivalent separating measures. A key result allowing this extension, and which is also of independent interest, is a particular extension of Memin's theorem, which establishes the closure of the set of stochastic integrals with respect to a given semimartingale, within the Frechet space of real semimartingales starting from zero. I extend Memin's theorem to the case where the integrands are subject to certain geometric constraints. Finally, I use this extension of the FTAP to show that the NFLVR condition, together with the existence of cash (which is seen as a particular asset subject to short-selling constraints), implies that for every separating measure there exist a unique locally risk-free asset (or "implied bank account") which can be added to the initial set of assets in such a way that the final set also satisfies the NFLVR condition (and in this sense the locally risk-free asset is "implied" by the initial set).