Walter Schachermayer

Law Invariant Risk Measures Have the Fatou Property

S. Kusuoka [K 01, Theorem 4] gave an interesting dual characterizationof law invariant coherent risk measures, satisfying the Fatou property.The latter property was introduced by F. Delbaen [D 02]. In thepresent note we extend Kusuokas characterization in two directions, thefirst one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M. Fritelli and E. Rossaza Gianin [FG05] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. F¨ollmer and A. Schied [FS 04]. Secondly — and more importantly — we show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance.We also introduce the notion of the Lebesgue property of a convex risk measure, where the inequality in the definition of the Fatou property is replaced by an equality, and give some dual characterizations of this property.

Utility maximization in incomplete markets with random endowment

Abstract. This paper solves the following problem of mathematical finance: to find a solution to the problem of maximizing utility from terminal wealth of an agent with a random endowment process, in the general, semimartingale model for incomplete markets, and to characterize it via the associated dual problem. We show that this is possible if the dual problem and its domain are carefully defined. More precisely, we show that the optimal terminal wealth is equal to the inverse of marginal utility evaluated at the solution to the dual problem, which is in the form of the regular part of an element of

A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time

({\bf L}^\infty)^*

The Existence of Absolutely Continuous Local Martingale Measures (1995)

(the dual space of

Arbitrage possibilities in Bessel processes and their relations to local martingales

{\bf L}^\infty

The no-arbitrage property under a change of numéraire

).

Weighted Norm Inequalities and Hedging in Incomplete Markets

Abstract Dalang, Morton and Willinger (1990) have proved a beautiful version of the Fundamental Theorem of Asset Pricing which pertains to the case of finite discrete time: In this case the absence of arbitrage opportunities already characterizes the existence of an equivalent martingale measure. The purpose of this paper is to give an elementary proof of this important theorem which relies only on orthogonality arguments. In contrast, the original proof of Dalang . uses heavy functional analytic machinery, in particular measurable selection and measure-decomposition theorems. We feel that the theorem (as well as its proof) should be accessible to a wider public and we therefore made an effort to keep the arguments as selfcontained as possible. In a final chapter we review and prove the necessary tools for our presentation of the theorem.

A Model‐Free Version of the Fundamental Theorem of Asset Pricing and the Super‐Replication Theorem

We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also rephrase Radon-Nikodym theorems for predictable processes. 1.Introduction. In our paper Delbaen and Schachermayer (1994a) we showed that for locally bounded finite dimensional stochastic price processes S, the existence of an equivalent (local) martingale measure – sometimes called risk neutral measure – is equivalent to a property called No Free Lunch with Vanishing Risk (NFLVR). We also proved that if the set of (local) martingale measures contains more than one element, then necessarily, there are non equivalent absolutely continuous local martingale measures for the process S. We also gave an example, see Delbaen and Schachermayer (1994a) Example 7.7, of a process that does not admit an equivalent (local) martingale measure but for which there is a martingale measure that is absolutely continuous. The example moreover satisfies the weaker property of No Arbitrage with respect to general admissible integrands. We were therefore lead to the investigation of the relation between the two properties, the existence of an absolutely continuous martingale measure (ACMM) and the absence of arbitrage for general admissible integrands (NA). From an economic viewpoint a local martingale measure Q, that gives zero measure to a non negligible event, say F , poses some problems. The price of the contingent claim that pays one 1991 Mathematics Subject Classification. 90A09,60G44, 46N10,47N10,60H05,60G40.

A Super-Martingale Property of the Optimal Portfolio Process

SummaryWe show that, if we allow general admissible integrands as trading strategies, the three dimensional Bessel process, Bes3, admits arbitrage possibilities. This is in contrast with the fact that the inverse process is a local martingale and hence is arbitrage free. This leads to some economic interpretation for the analysis of the property of arbitrage in foreign exchange rates. This notion (relative to general admissible integrands) does depend on the fact, which of the two currencies under consideration is chosen as numéraire. The results rely on a general construction of strictly positive local martingales. The construction is related to the Föllmer measure of a positive super-martingale.

Transaction Costs, Trading Volume, and the Liquidity Premium

For a price process that has an equivalent risk neutral measure, we investigate if the same property holds when the numeraire is changed. We give necessary and sufficient conditions under which the price process of a particular asset-which should be thought of as a different currency can be chosen as new numeraire, I he result is related to the characterization of attainable claims that can be hedged. Roughly speaking: the asset representing the new currency is a reasonable investment (in terms of the old currency) if and only if the market does not permit arbitrage opportunities in terms of the new currency as numeraire. This rough but economically meaningful idea is given a precise content in this paper. The main ingredients are a duality relation as well as a result on maximal elements. The paper also generalizes results previously obtained by Jacka, Ansel-Strieker and the authors