Freddy Delbaen

Coherent Risk Measures on General Probability Spaces

We extend the definition of coherent risk measures, as introduced by Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how to define such measures on the space of all random variables. We also give examples that relates the theory of coherent risk measures to game theory and to distorted probability measures. The mathematics are based on the characterisation of closed convex sets Pσ of probability measures that satisfy the property that every random variable is integrable for at least one probability measure in the set Pσ.

Exponential Hedging and Entropic Penalties

We solve the problem of hedging a contingent claim B by maximizing the expected exponential utility of terminal net wealth for a locally bounded semimartingale X. We prove a duality relation between this problem and a dual problem for local martingale measures Q for X where we either minimize relative entropy minus a correction term involving B or maximize the Q-price of B subject to an entropic penalty term. Our result is robust in the sense that it holds for several choices of the space of hedging strategies. Applications include a new characterization of the minimal martingale measure and risk-averse asymptotics.

Coherent multiperiod risk adjusted values and Bellman’s principle

Starting with a time-0 coherent risk measure defined for “value processes”, we also define risk measurement processes. Two other constructions of measurement processes are given in terms of sets of test probabilities. These latter constructions are identical and are related to the former construction when the sets fulfill a stability condition also met in multiperiod treatment of ambiguity as in decision-making. We finally deduce risk measurements for the final value of locked-in positions and repeat a warning concerning Tail-Value-at-Risk.

Coherent risk measures

After some financial catastrophes for big firms it is obvious that existing risk measures are not reliable enough, therefore a “new” class of risk measures, which are called coherent, have been suggested. We will take a look at risk measures mathematically (both at general and coherent) and provide some examples. It appears that even coherent risk measures are not the answer to everything. Especially the subadditivity is a point of criticism and we discuss some examples where it cannot be fulfilled in the ”real world”. From that we conclude that there has no universally usable risk measure found yet and that the existing ones have to be used situation-dependent and with caution. The planning of the project and the analysis of the results have been done together. Reportwriting has been done individually with almost equally divided effort.

The Structure of m–Stable Sets and in Particular of the Set of Risk Neutral Measures

The study of dynamic coherent risk measures and risk adjusted values as introduced by Artzner, Delbaen, Eber, Heath and Ku, leads to a property called fork convexity. We give necessary and sufficient conditions for a closed convex set of measures to be fork convex. Since the set of martingale measures for price processes is fork convex, this leads to a characterisation of closed convex sets that can be obtained as the set of risk neutral measures in an arbitrage free model of security prices. We also relate the property of fork convexity or m–stability with the validity of Bellman’s principle. It turns out that the stability property investigated in this paper is equivalent to properties known as time-consistency and rectangularity as used in multiprior Bayesian decision theory.

The Existence of Absolutely Continuous Local Martingale Measures (1995)

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.

Weighted Norm Inequalities and Hedging in Incomplete Markets

Abstract. Let

Convergence of discretized stochastic (interest rate) processes with stochastic drift term

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Representation of the penalty term of dynamic concave utilities

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On Esscher Transforms in Discrete Finance Models

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