Multivariate risk: utility-based risk, risk subject to frictions, risk evolution in continuous time
Quantification of risk in multi-asset markets has recently gained interest in financial mathematics. In the existence of frictions in such markets, multiple portfolio vectors could possibly be used in order to compensate the risk of a financial position. These portfolios are said to form the set-valued risk of the position. In this talk, we will first discuss two classes of set-valued risk measures that are defined with respect to utility- based risk preferences. These risk measures can also be formulated as set optimization problems and the two classes are related via a recent Lagrange duality for such problems. Examples include the multivariate versions of the well-known entropic risk measure and average value at risk. Then, in a general convex market model, we will consider how market frictions such as transaction costs can be taken into account in the evaluation of set-valued risk measures. In the last part of the talk, we will discuss a continuous time framework where set-valued risk measures are defined via backward stochastic differential inclusions. As an analytical tool in the study of these inclusions, we will discuss a characterization theorem for the Aumann integrals (or expectations) of set-valued functions (or random sets).
Based on joint works with Birgit Rudloff and Andreas Hamel.
Cagin Ararat received his BS degree in 2010 from the Department of Industrial Engineering at Bilkent University. He is currently a PhD candidate in the Department of Operations Research and Financial Engineering at Princeton University. His research interests include financial mathematics, markets with transaction costs, risk measurement, multivariate risk, risk preferences, random sets and set optimization. He is a member of SIAM, INFORMS and AMS.