Marco Frittelli

Putting order in risk measures

Abstract This paper introduces a set of axioms that define convex risk measures. Duality theory provides the representation theorem for these measures and the link with pricing rules.

On the Existence of Minimax Martingale Measures

Embedding asset pricing in a utility maximization framework leads naturally to the concept of minimax martingale measures. We consider a market model where the price process is assumed to be an R-semimartingale X and the set of trading strategies consists of all predictable, X-integrable, R-valued processes H for which the stochastic integral (H.X) is uniformly bounded from below. When the market is free of arbitrage, we show that a sufficient condition for the existence of the minimax measure is that the utility function u: R - R is concave and nondecreasing. We also show the equivalence between the no free lunch with vanishing risk condition, the existence of a separating measure, and a properly defined notion of viability.

Risk Measures and Capital Requirements for Processes

In this paper we propose a generalization of the concepts of convex and coherent risk measures to a multiperiod setting, in which payoffs are spread over different dates. To this end, a careful examination of the axiom of translation invariance and the related concept of capital requirement in the one-period model is performed. These two issues are then suitably extended to the multiperiod case, in a way that makes their operative financial meaning clear. A characterization in terms of expected values is derived for this class of risk measures and some examples are presented.

Law invariant convex risk measures

As a generalization of a result by Kusuoka (2001), we provide the representation of law invariant convex risk measures. Very particular cases of law invariant coherent and convex risk measures are also studied.

Introduction to a Theory of Value Coherent with the No-Arbitrage Principle

Abstract. This paper defines the value of a general claim based on agents preferences and coherent with the No Arbitrage Principle. This Value is a non trivial extension of the certainty equivalent since it takes into consideration the possibility of partially hedging the risk carried by the claim. When the market is complete this Value is the unique no arbitrage price. When the risk may not even be partially covered, this Value is the certainty equivalent. Between these two cases just some of the risk may be hedged and the no arbitrage principle requires the price to lie in the “arbitrage interval”. The Value we propose is exactly designed to satisfy this condition.

On the extension of the Namioka-Klee theorem and on the Fatou property for Risk Measures

This paper has been motivated by general considerations on the topic of Risk Measures, which essentially are convex monotone maps defined on spaces of random variables, possibly with the so-called Fatou property.

Utility maximization in incomplete markets for unbounded processes

Abstract.When the price processes of the financial assets are described by possibly unbounded semimartingales, the classical concept of admissible trading strategies may lead to a trivial utility maximization problem because the set of stochastic integrals bounded from below may be reduced to the zero process. However, it could happen that the investor is willing to trade in such a risky market, where potential losses are unlimited, in order to increase his/her expected utility. We translate this attitude into mathematical terms by employing a class

A Unified Framework for Utility Maximization Problems: an Orlicz space approach

\mathcal{H}^{W}

Dual Representation of Quasi-convex Conditional Maps ∗

of W-admissible trading strategies which depend on a loss random variable W. These strategies enjoy good mathematical properties and the losses they could generate in trading are compatible with the preferences of the agent.We formulate and analyze by duality methods the utility maximization problem on the new domain

CONDITIONAL CERTAINTY EQUIVALENT

\mathcal{H}^{W}