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

Abstract

Abstract. 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, rectangularity or m-stability. 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 m-stable, 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 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. Finally we characterise the set of risk neutral measures for continuous price processes.

Representation results for law invariant time consistent functions

“…The limiting cases γ = 0 and γ = ∞ are defined as [13]. In a continuous time framework, under a filtration for which all martingales are continuous, it is shown that the only law invariant, time consistent, dynamic coherent risk measure (a dynamic convex risk measure, which additionally satisfies ρ t (λX) = λρ t (X) for all λ ∈ (L ∞ t ) + ) is either the negative of the expected value or the worst case risk measure.…”

Dynamics of state price densities

“…These examples all belong to the class of law-invariant convex risk measures. The failure of dynamic consistency for law-invariant coherent risk measures has already been pointed out by Delbaen [9]. Example 3.4 (Shortfall risk) Let : R → R be convex, increasing, and nonconstant and take x in the interior of (R).…”

Robust Preferences and Robust Portfolio Choice

“…Moreover, since the level sets {dQ/dP | H(Q|P) ≤ c} are weakly compact (this follows, e.g., by combining Theorem 1.2 with the straightforward fact that the entropic monetary utility functional (2.1) is continuous from below), also γ must have weakly relatively compact level sets. In fact, one can show that the level sets of γ are weakly closed (see DELBAEN [2006] for the coherent and HERNÁNDEZ-HERNÁNDEZ and SCHIED [2007a, Lemma 4.1] for the general case), so that γ is the minimal penalty function of φ, and φ is continuous from below. In particular, φ and γ satisfy the assumptions of Sections 4 and 7.…”