A High-Dimensional Pricing Framework for Fi nancial Instruments Valuation

Abstract

We present in this paper a method allowing to consider valuations of complex OTC products, including american exercising, written on a large number of underlyings. The motivation is to provide a framework able to quickly valuate big investment bank portfolios with more complex pricing algorithms, as those coming from risk measures, and we think particularly to CVA (counterpart risk valuation) ones, leading to consider high dimensional Optimal Stopping Problems.Indeed, it has already been noticed that current numerical methods devoted to pricing algorithms face an open mathematical problem, the curse of dimensionality (see for instance [5], [12], [4]), particularly in the context of CVA computations. We notice also that the econometric problem, consisting in fi nding a risk-neutral dynamic of the underlyings, faces the very same open problem. Indeed, the CVA approach, as others existing risk measures, although vertuous, can not be credible as long as they will be cursed.The framework introduced in this paper has two purposes: firstly to provide a fast calibration algorithm to determine a risk-neutral econometric, using market prices, able to capture correlation structures between underlyings. Secondly, to design a fast backwardation algorithms upon this econometry. Both algorithms break the curse of dimensionality wall. This is achieved in this paper in a particular context, given by martingales processes, a classical assumption in mathematical finance. From a mathematical point of view, these results are obtained blending together stochastic, partial differential equations, and optimal transport theories, and are a generalization to the multi-dimensional case of this paper. We believe that these tools should be the right candidates to provide a standardized framework for large portfolios valuations of risk measures or investment strategies, as well as a pedestal to tackle credible systemic risk measures.