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

Bruno Bouchard (1, 2) , Emmanuel Lépinette (1) , Erik Taflin (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CREST - Centre de Recherche en Économie et Statistique
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

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, Ràsonyi and Lépinette (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 càdlàg case for further studies.

Mots clés: No-arbitrage; transaction costs; continuous time bond market

Domaines: Probabilités [math.PR]

Bruno Bouchard, Emmanuel Lépinette, Erik Taflin. Robust no-free lunch with vanishing risk, a continuum of assets and proportional transaction costs. Stochastic Processes and their Applications, 2014, 124, pp.3231-3259. ⟨hal-00783977⟩