Entropic risk measure

About: Entropic risk measure is a research topic. Over the lifetime, 89 publications have been published within this topic receiving 6511 citations.

Stochastic Finance: An Introduction in Discrete Time

Abstract: This book is an introduction to financial mathematics. It is intended for graduate students in mathematics and for researchers working in academia and industry. The focus on stochastic models in discrete time has two immediate benefits. First, the probabilistic machinery is simpler, and one can discuss right away some of the key problems in the theory of pricing and hedging of financial derivatives. Second, the paradigm of a complete financial market, where all derivatives admit a perfect hedge, becomes the exception rather than the rule. Thus, the need to confront the intrinsic risks arising from market incomleteness appears at a very early stage. The first part of the book contains a study of a simple one-period model, which also serves as a building block for later developments. Topics include the characterization of arbitrage-free markets, preferences on asset profiles, an introduction to equilibrium analysis, and monetary measures of financial risk. In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk. This third revised and extended edition now contains more than one hundred exercises. It also includes new material on risk measures and the related issue of model uncertainty, in particular a new chapter on dynamic risk measures and new sections on robust utility maximization and on efficient hedging with convex risk measures.

Convex measures of risk and trading constraints

Abstract: We introduce the notion of a convex measure of risk, an extension of the concept of a coherent risk measure defined in Artzner et al. (1999), and we prove a corresponding extension of the representation theorem in terms of probability measures on the underlying space of scenarios. As a case study, we consider convex measures of risk defined in terms of a robust notion of bounded shortfall risk. In the context of a financial market model, it turns out that the representation theorem is closely related to the superhedging duality under convex constraints.

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.

Coherent multiperiod risk adjusted values and Bellman’s principle

Abstract: 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.

Conditional and Dynamic Convex Risk Measures

Abstract: We extend the definition of a convex risk measure to a conditional framework where additional information is available. We characterize these risk measures through the associated acceptance sets and prove a representation result in terms of conditional expectations. A suitable regularity property of conditional risk measures is defined and discussed. Finally, we introduce the concept of a dynamic convex risk measure as a family of successive conditional convex risk measures and characterize those satisfying some natural time consistency properties. As a reference example, illustrating all the proposed developments, we introduce a suitably defined dynamic version of the class of entropic risk measures.