Title:

Model free optimisation in risk management

Following the financial crisis of 2008, the need for more robust techniques to quantify the capital charge for risk management has become a pressing problem. Under Basel II/III, banks are allowed to calculate the capital charge using internally developed models subject to regulatory approval. An interesting problem for the regulator is to compare the resulting figures against the required capital under worst case scenarios. The existing literature on the latter problem, which is based on the marginal problem, assumes that no apriori information is known about the dependencies of contributing risks. These problems are linear optimisation problems over a constrained set of probability measures, discretisation of which leads to large scale LPs. But this approach is very conservative and cannot be implemented robustly in practice, due to the scarcity of historical data. In our approach, we take a less conservative strategy by incorporating dependence information contained in the data in a form that still leads to LPs, an important feature of such problems due to their high dimensionality. Conceptually, our model is the discretisation of an infinite dimensional linear optimisation problem over a set of probability measures. For some specific cases we can prove strong duality, opening up the approach of discretising the dual instead of the primal. This approach is preferable, as it yields better numerical results. In this work we also apply our model to modelfree pathdependent option pricing. Use of delayed column generation techniques allows us to solve problems several orders of magnitude larger than via the standard simplex algorithm. For highdimensional LPs we also implement Nesterov's smoothing technique to solve the problems.
