Measuring financial risks under uncertainty

Abstract

Coping with the uncertainties of future outcomes is a fundamental theme in finance in a stochastic environment. In the field of stochastic programming, which grown from the traditions of linear and quadratic programming, constrains on future outcomes have commonly been relaxed to the penalty expressions. Probabilistic constrains, requiring that a condition only to be satisfied up to a given probability. Objectives have usually taken the form of maximizing of expected utility or minimizing an expected cost. In financial optimization where uncertainties are likewise unavoidable, approaches of stochastic programming have prevailed. An important example is constrains and objective based on the notion value-atrisk, which related closely to probabilistic one unfortunately it suffer from similar mathematical shortcomings. Value-at-risk suffers from financial inconsistencies, which have led to axiomatic development of coherent risk measures, so we add also the robust alternative called conditional value-at-risk. We cope also with some connection between CVaR and stochastic dominance.stochastic programming, coherent risk measures, value-at-risk, conditional value-at-risk, stochastic dominance