DYNAMIC INDIFFERENCE VALUATION VIA CONVEX RISK MEASURES

The (subjective) indifference value of a payoff in an incomplete financial market is that monetary amount which leaves an agent indifferent between buying or not buying the payoff when she always optimally exploits her trading opportunities. We study these values over time when they are defined with respect to a dynamic monetary concave utility functional, that is, minus a dynamic convex risk measure. For that purpose, we prove some new results about families of conditional convex risk measures. We study the convolution of abstract conditional convex risk measures and show that it preserves the dynamic property of time-consistency. Moreover, we construct a dynamic risk measure (or utility functional) associated to superreplication in a market with trading constraints and prove that it is time-consistent. By combining these results, we deduce that the corresponding indifference valuation functional is again time-consistent. As an auxiliary tool, we establish a variant of the representation theorem for conditional convex risk measures in terms of equivalent probability measures.     

ONE-PARAMETER FAMILIES OF DISTORTION RISK MEASURES

This paper introduces parametric families of distortion risk measures, investigates their properties, and discusses their use in risk management. Their derivation is based on Kusuoka's representation theorem of law invariant and comonotonically additive coherent risk measures. Our approach is to narrow down a tractable class of risk measures by requiring their comparability with the traditional expected shortfall. We make numerical comparison among them and propose a method of estimating the value of the distortion risk measures based on data. Their use and interpretation in risk management will also be discussed.     

A SHORT NOTE ON SECOND-ORDER STOCHASTIC DOMINANCE PRESERVING COHERENT RISK MEASURES

We improve results on law invariant coherent risk measures satisfying the Fatou property due to Kusuoka (2001; Adv. Math. Econ . 3, 83–95) by considering risk measures which are in addition second order stochastic dominance preserving. In particular, we derive a representation result for such risk measures.     

AN OLD-NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT

The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper re-examines this fundamental concept, studies and extends its main properties, and puts it in perspective to recent concepts of risk measures. We show that the negative of the OCE naturally provides a wide family of risk measures that fits the axiomatic formalism of convex risk measures. Duality theory is used to reveal the link between the OCE and the φ-divergence functional (a generalization of relative entropy), and allows for deriving various variational formulas for risk measures. Within this interpretation of the OCE, we prove that several risk measures recently analyzed and proposed in the literature (e.g., conditional value of risk, bounded shortfall risk) can be derived as special cases of the OCE by using particular utility functions. We further study the relations between the OCE and other certainty equivalents, providing general conditions under which these can be viewed as coherent/convex risk measures. Throughout the paper several examples illustrate the flexibility and adequacy of the OCE for building risk measures.     

EFFICIENT HEDGING OF EUROPEAN OPTIONS WITH ROBUST CONVEX LOSS FUNCTIONALS: A DUAL‐REPRESENTATION FORMULA

Motivated by numerical representations of robust utility functionals, due to Maccheroni et al., we study the problem of partially hedging a European option H when a hedging strategy is selected through a robust convex loss functional L(·) involving a penalization term γ(·) and a class of absolutely continuous probability measures . We present three results. An optimization problem is defined in a space of stochastic integrals with value function EH(·) . Extending the method of Föllmer and Leukerte, it is shown how to construct an optimal strategy. The optimization problem EH(·) as criterion to select a hedge, is of a “minimax” type. In the second, and main result of this paper, a dual‐representation formula for this value is presented, which is of a “maxmax” type. This leads us to a dual optimization problem. In the third result of this paper, we apply some key arguments in the robust convex‐duality theory developed by Schied to construct optimal solutions to the dual problem, if the loss functional L(·) has an associated convex risk measure ρ^L(·) which is continuous from below, and if the European option H is essentially bounded.     

OVERLAPPING SETS OF PRIORS AND THE EXISTENCE OF EFFICIENT ALLOCATIONS AND EQUILIBRIA FOR RISK MEASURES

The overlapping expectations and the collective absence of arbitrage conditions introduced in the economic literature to insure existence of Pareto optima and equilibria with short‐selling when investors have a single belief about future returns, is reconsidered. Investors use measures of risk. The overlapping sets of priors and the Pareto equilibrium conditions introduced by Heath and Ku for coherent risk measures are respectively reinterpreted as a weak no‐arbitrage and a weak collective absence of arbitrage conditions and shown to imply existence of Pareto optima and Arrow–Debreu equilibria.     

MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF DERIVATIVE INSTRUMENTS

Uncertainty on the choice of an option pricing model can lead to "model risk" in the valuation of portfolios of options. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk management of derivative instruments, we introduce a quantitative framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure compatible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of "model risk."