Applications of copulas in optimisation

Abstract

The methods for modelling uncertainty and assessing the risk of financial markets were placed under scrutiny after the 2008 crisis. To protect against the worst possible scenario, in a problem of asset allocation, robust optimisation is required. Still, within this framework, assumptions about the uncertainty set have to be made. In our work, we expand the possible options for describing uncertainty sets, through the use of copulas. Copulas are a useful tool for describing uncertainty because of the modelling flexibility that they provide. They are able to easily describe asymmetric dependence structures and tail risk. Both are vital for emulating the financial markets behaviour, during periods of extreme shocks and comovements. Also, copulas are associated with robust measures of dependence. We introduce copulas into the robust optimisation framework by following two different approaches. At first, we derive a Worst Case Conditional Value at Risk optimisation problem, in which the uncertainty set consists of a selection of copulas. We formulate the problem into a convex optimisation problem. The advantages of such a model are supported by numerical examples using real data. In the second approach, copulas are used as means for creating non-symmetric, convex uncertainty sets, in the form of domains. We present examples where these sets can be used in a robust optimisation problem.