Financial Risk Measurement with Imprecise Probabilities

Although financial risk measurement is a largely investigated research area, its relationship with imprecise probabilities has been mostly overlooked. However, risk measures can be viewed as instances of upper (or lower) previsions, thus letting us apply the theory of imprecise previsions to them. After a presentation of some well known risk measures, including Value-at-Risk or VaR, coherent and convex risk measures, we show how their definitions can be generalized and discuss their consistency properties. Thus, for instance, VaR may or may not avoid sure loss, and conditions for this can be derived. This analysis also makes us consider a very large class of imprecise previsions, which we termed convex previsions, generalizing convex risk measures. Shortfall-based measures and Dutch risk measures are also investigated. Further, conditional risks can be measured by introducing conditional convex previsions. Finally, we analyze the role in risk measurement of some important notions in the theory of imprecise probabilities, like the natural extension or the envelope theorems.