Stochastic root finding for optimized certainty equivalents

Global financial markets require suitable techniques for the quantification of the downside risk of financial positions. In the current paper, we concentrate on Monte Carlo methods for the estimation of an important and broad class of convex risk measures which can be constructed on the basis of optimized certainty equivalents (OCEs). This family of risk measures - originally introduced in Ben-Tal and Teboulle (2007) - includes, among others, the entropic risk measure and average value at risk. The calculation of OCEs involves a stochastic optimization problem that can be reduced to a stochastic root finding problem via a first order condition. We describe suitable algorithms and illustrate their properties in numerical case studies.