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.

An acceptability measure is a number that summarizes information on monetary outcomes of a given position in various scenarios, and that, depending on context, may be interpreted as a capital requirement or as a price. In a multiperiod setting, it is reasonable to require that an acceptability measure should satisfy certain conditions of time consistency. Various notions of time consistency may be considered. Within the framework of coherent risk measures as proposed by Artzner et al. [Artzner, Ph., Delbaen, F., Eber, J.-M., Heath, D., 1999. Coherent measures of risk. Math. Fin. 9, 203-228], we establish implication relations between a number of different notions, and we determine how each notion of time consistency is expressed through properties of a representing set of test measures. We propose modifications of the standard Tail-Value-at-Risk measure that have stronger consistency properties than the original.

We consider the optimal approximation of an observed multivariable time series by one that satisfies a set of linear, time-invariant diflerence equations, under a constraint on the number of independent equations and their total lag.

In this paper we present a novel approach for the modeling of multivariable time series. The model class consists of linear systems, i.e., the solution sets of linear difference equations. Restricting the model order, the aim is to determine a model with minimal la-distance from the observed time series. Necessary conditions for optimality are described in terms of state-space representations. These conditions motivate a relatively simple iterative algorithm for the nonlinear problem of identifying optimal models. Attractive aspects of the proposed method are that the model error is measured globally, it can be applied for multiinput, multi-output systems, and no prior distinction between inputs and outputs is required. We give an illustration by means of some numerical simulations.