Walter Schachermayer

The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes.

F. Delbaen, W. Schachermayer
Mathematische Annalen, Vol. 312 (1998), pp. 215-250. 

Abstract:

The Fundamental Theorem of Asset Pricing states - roughly speaking - that the absence of arbitrage possibilities for a stochastic process $S$ is equivalent to the existence of an equivalent martingale measure for $S$. It turnsout that it is quite hard to give precise and sharp versions of this theorem in proper generality, if one insists on modifying the concept of ``no arbitrage" as little as possible. It was shown in [DS94] that for a locally bounded $\R^d$-valued semi-martingale $S$ the condition of No Free Lunch with Vanishing Risk is equivalent to the existence of an equivalent local martingale measure for the process $S$. It was asked whether the local boundedness assumption on $S$ may be dropped.

In the present paper we show that if we drop in this theorem the local boundedness assumption on $S$ the theorem remains true if we replace the term equivalent local martingale measure by the term equivalent sigma-martingale measure. The concept of sigma-martingales was introduced by Chou and Emery --- under the name of ``semimartingales de la classe $(\Sigma _m)$".

We provide an example which shows that for the validity of the theorem in the non locally bounded case it is indeed necessary to pass to the concept of sigma-martingales. On the other hand, we also observe that for the applications in Mathematical Finance the notion of sigma-martingales provides a natural framework when working with non locally bounded processes $S$.
The duality results which we obtained earlier are also extended to the non locally bounded case. As an application we characterize the hedgeable elements.

[DS94] --- F. Delbaen, W. Schachermayer: A General Version of the Fundamental Theorem of Asset Pricing. Math. Annalen 300 (1994): pages 463 - 520.

Keywords:

Sigma martingale, Arbitrage, fundamental theorem of asset pricing, Free Lunch

1991 Mathematics Subject Classification:

Primary 60G44; Secondary 46N30,46E30,90A09, 60H05

Preprints:

Publications marked with have appeared in refereed journals.

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