Robust no-free lunch with vanishing risk, a continuum of assets and proportional transaction costs

TL;DR

This paper develops a continuous-time financial market model with proportional transaction costs and a continuum of assets, extending fundamental no-arbitrage results and simplifying trading strategy definitions.

Contribution

It extends the Fundamental Theorem of Asset Pricing to markets with a continuum of assets and transaction costs, providing a natural framework for bond markets.

Findings

Equivalence of Robust No Free Lunch with Vanishing Risk and existence of a Strictly Consistent Price System.

Transaction costs facilitate well-defined trading strategies without complex stochastic integration.

Model applicable to bond markets with continuous maturities.

Abstract

We propose a continuous time model for financial markets with proportional transactions costs and a continuum of risky assets. This is motivated by bond markets in which the continuum of assets corresponds to the continuum of possible maturities. Our framework is well adapted to the study of no-arbitrage properties and related hedging problems. In particular, we extend the Fundamental Theorem of Asset Pricing of Guasoni, R\'asonyi and L\'epinette (2012) which concentrates on the one dimensional case. Namely, we prove that the Robust No Free Lunch with Vanishing Risk assumption is equivalent to the existence of a Strictly Consistent Price System. Interestingly, the presence of transaction costs allows a natural definition of trading strategies and avoids all the technical and un-natural restrictions due to stochastic integration that appear in bond models without friction. We restrict to the case where exchange rates are continuous in time and leave the general c\`adl\`ag case for further studies.