Coherent Acceptability Measures in Multiperiod Models

Abstract

The coherent risk framework has been introduced by Artzner et al. (1999) Concerning the problem of computing hedges that optimize the degree of acceptability of a given position, we provide sufficient conditions under which an algorithm of dynamic programming type can be applied. For the special case of a derivative on a single underlying with convex payoff, and for a particular class of acceptability measures, we show that this algorithm simplifies considerably and we give explicit formulas for hedges that maximize the degree of acceptability.

Conditional and dynamic convex risk measures

“…In fact, it requires that the riskiness ρ n (X) of a final payoff X today equals the riskiness of the capital requirement ρ n+1 (X) that has to be set aside tomorrow. The property of recursiveness has been introduced by Roorda et al [12] and Riedel [11] and it is actually equivalent to time consistency. Indeed, assume that (ρ n ) N n=0 is time consistent and fix n; by conditional translation invariance of ρ n+1 we have ρ n+1 (−ρ n+1 (X)) = ρ n+1 (X), so that ρ n (−ρ n+1 (X)) = ρ n (X).…”

Coherent Measurement of Factor Risks

“…4], [55]. Much of the current research in this area deals with defining properly a dynamic risk measure (let us mention in this connection the papers [13], [23], [35], [50], [54]). Currently, more and more research in the theory of coherent risk measures is related to applications to problems of finance rather than to the study of "pure" risk measures.…”

“…Currently, more and more research in the theory of coherent risk measures is related to applications to problems of finance rather than to the study of "pure" risk measures. In particular, the problem of capital allocation was considered in [6], [14], [20], [21], [26], [38], [49], [61]; the problem of pricing and hedging was investigated in [8], [11], [12], [14], [18], [20], [33], [35], [42], [48], [54], [56], [59]; the problem of the optimal portfolio choice was studied in [15], [51], [53]; the equilibrium problem was considered in [7], [8], [15], [32], [37], [44]. This list is very far from being complete; for example, on the Gloria Mundi web page over two hundred papers are related to coherent risk measures.…”

Time-Consistent and Market-Consistent Evaluations

“…See Riedel (2004), Roorda et al (2005), Cheridito et al (2006), Rosazza Gianin (2006), Artzner et al (2007). In a dynamical context time-consistency is a natural approach to glue together static evaluations.…”