Questions on the geometry of finite coherence

Abstract

In this article, we indicate that in any two-period model of financial markets with a finite state space, we may define a variety of coherent risk measures on the payoff space of the market. These risk measures depend on the Equivalent or on the Generalized Equivalent Martingale Measures of the market. We also present some examples which illustrate the multiplicity of the coherent risk measures, which depends on the extreme points of the set of the Generalized Martingale Measures of a market, if this market is arbitrage free. We actually present in a systematic way the previous connection jointly with the examples, while the origins of these ideas come from the work of Artzner et al. (1999). We also present an aspect of this connection related to the finite -dimensional Expected Shortfall, relying on the dual representation of it and we provide arithmetic examples, too.