Abstract
Coherent and convex risk measures, Choquet expectation and Peng’s g-expectation are all generalizations of mathematical expectation. All have been widely used to assess financial riskiness under uncertainty. In this paper, we investigate differences amongst these risk measures and expectations. For this purpose, we constrain our attention of coherent and convex risk measures, and Choquet expectation to the domain of g-expectation. Some differences among coherent and convex risk measures and Choquet expectations are accounted for in the framework of g-expectations. We show that in the family of convex risk measures, only coherent risk measures satisfy Jensen’s inequality. In mathematical finance, risk measures and Choquet expectations are typically used in the pricing of contingent claims over families of measures. The different risk measures will typically yield different pricing. In this paper, we show that the coherent pricing is always less than the corresponding Choquet pricing. This property and inequality fails in general when one uses pricing by convex risk measures. We also discuss the relation between static risk measure and dynamic risk measure in the framework of g-expectations. We show that if g-expectations yield coherent (convex) risk measures then the corresponding conditional g-expectations or equivalently the dynamic risk measure is also coherent (convex). To prove these results, we establish a new converse of the comparison theorem of g-expectations.
Highlights
The choice of financial risk measures is very important in the assessment of the riskiness of financial positions
As an extension of coherent risk measures, convex risk measures in general probability spaces were introduced by Föllmer & Schied [3] and Frittelli & Rosazza Gianin [4]. g-expectations were introduced by Peng [5] via a class of nonlinear backward stochastic differential equations (BSDEs), this class of
We show that 1) in the family of convex risk measures, only coherent risk measures satisfy Jensen’s inequality; 2) coherent risk measures are always bounded by the corresponding Choquet expectation, but such an inequality in general fails for convex risk measures
Summary
The choice of financial risk measures is very important in the assessment of the riskiness of financial positions For this reason, several classes of financial risk measures have been proposed in the literature. Several classes of financial risk measures have been proposed in the literature Among these are coherent and convex risk measures, Choquet expectations and Peng’s g-expectations. We establish that if g-expectations are coherent (convex) risk measures, the same is true for the corresponding conditional g-expectations or dynamic risk. We are able to show, in the case of gexpectations, that coherent risk measures are bounded by Choquet expectation but this relation fails for convex risk measures; see Theorem 4.
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