Martingales and arbitrage: a new look

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Abstract

This paper addresses the equivalence between the absence of arbitrage and the existence of equivalent martingale measures. The equivalence will be established under quite weak assumptions since there are no conditions on the set of trading dates (it may be finite or countable, with bounded or unbounded horizon, etc.) or on the trajectories of the price process (for instance, they do not have to be right-continuous).

Besides we will deal with arbitrage portfolios rather than free-lunches. The concept of arbitrage is much more intuitive than the concept of free lunch and has more clear economic interpretation. Furthermore it is more easily tested in theoretical models or practical applications.

In order to overcome the usual mathematical difficulties arising when dealing with arbitrage strategies, the set of states of nature will be widened by drawing on projective systems of Radon probability measures, whose projective limit will be the martingale measure. The existence of densities between the “real” probabilities and the “risk-neutral” probabilities will be guaranteed by introducing the concept of “projective equivalence”. Hence some classical counter-examples will be solved and a complete characterization of the absence of arbitrage will be provided in a very general framework.

Keywords

Arbitrage martingale measure projective system 

Mathematics Subject Classifications

91B28 91B70 

Martingalas y arbitraje: un nuevo enfoque

Resumen

Analizaremos la equivalencia entre la ausencia de arbitraje y la existencia de una medida de martingala. Esta equivalencia se establecerá bajo supuestos débiles, puesto que no hay condiciones sobre el conjunto de fechas de negociación (puede ser finito o contable, con horizonte acotado o no acotado, etc.) ni sobre las trayectorias del proceso de precios (por ejemplo, no tienen que ser continuas por la derecha).

Trabajaremos con el concepto de arbitraje, y no con el defree-lunch. La noción de arbitraje es mucho más intuitiva y tiene una interpretación económica mucho más clara, además de ser más fácil de verificar en las aplicaciones prácticas.

Para salvar dificultades matemáticas, extenderemos el conjunto de estados de la naturaleza mediante el uso de sistemas proyectivos de probabilidades regulares (de Radon), cuyo láimite proyectivo será la medida de martingala. La existencia de densidades entre las «probabilidades reales » y las «neutrales al riesgo» se garantizará mediante la introducción del concepto de «equivalencia proyectiva». Algunos contra-ejemplos clásicos serán resueltos, y una caracterización completa de la ausencia de arbitraje será presentada en un contexto muy general.

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Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.Académico Correspondiente de la Real Academia de CienciasUniversidad Carlos III de MadridGetafe, Madrid(Spain)
  2. 2.Académico de Número de la Real Academia de CienciasUNED, Departamento de Matemáticas FundamentalesMadrid(Spain)

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