# CoDeFi, a new algorithm for risk measurement

In this Preprint, we present a new algorithm (CoDeFi) to tackle the Curse Of Dimensionality In Finance and deal with a broad class of Partial Differential Equations (PDE’s) including the Kolmogorov equations as, for instance, the Black and Scholes equations. As a main feature, our method allows one to solve the Kolmogorov equations in large dimensions and provides a very general framework to deal with risk measurements. In financial applications, the number of dimensions corresponds to the number of underlyings or risk sources.

The Curse of Dimensionality (CoD – This term comes from a 1957 paper by R.E. Bellman.) refers to the fact that the computational time increases exponentially with the number of risk sources, and in the present preprint we propose a solution to this long-standing open problem. Our approach is based on a combination of techniques from the theory of partial differential equations (PDE’s): Monte-Carlo trajectories, a classical numerical method that is not cursed (not affected by CoD) are used as moving-in-time grids. These grids allow solving Kolmogorov equations, using unstructured mesh PDE techniques together with optimal transport theory, to overcome the cursing. We are also using zero-error calibration techniques, for a perfect fit of replication instruments or complex financial modeling. All these techniques are bundled together in a framework that we call CoDeFi: it can be seen as a general risk measurement framework based on the algorithm presented in [4]-[6].

This preprint reviews this technology and proposes a first benchmark to the multi-dimensional case, filling it up to 64 dimensions (or risk sources). Indeed, to our knowledge, little other technology could benchmark CoDeFi framework: optimal quantization [1] or wavelet analysis [9] might be used for up to, let say 10 dimensions. Above this limit, American-Monte Carlo methods [7] might provide lower bounds, as these methods are known to compute sub-optimal exercising. We emphasize that the same computational framework is used to provide simulations for nonlinear hyperbolic problems [3].

Our perception is that this technology can already confidently compute risk measurements. There are numerous potential applications linked to the curse of dimensionality. We identified some of them in the insurance industry, but mainly in the financial sector. Above pricing and hedging, allowing to issue new financial products, more adapted to client’s needs, or market arbitraging, we are addressing risk measurements: indeed, the test realized in this paper shown that CoDeFi could compute accurate Basel regulatory measures based on VAR (Value at Risk) or CVA (Credit Value Adjustment) on a single laptop, whereas farm of thousand of computers are used on a daily basis today with approximation methods. Finally, such a framework could be helpful in systemic risk measurement that can be accurately modeled through high dimensional Kolmogorov equations like (4). This could notably be useful for developing tools dedicated to economical crisis supervision.

[1] V. Bally, G. Pages, J. Printems, First order schemes in the numerical quantization method, Rapport de recherche INRIA N 4424.

[3] P.G. LeFloch, J.M., Mercier, Revisiting the method of characteristics via a convex hull algorithm, J. Comput. Phys. 298 (2015), 95-112.

[4] P.G. LeFloch, J.M., Mercier, A Framework to solve high-dimensional Kolmogorov equations. In preparation.

[6] J.M., Mercier, A high-dimensional pricing framework for financial instruments valuation, April 2014, available at http://ssrn.com/abstract=2432019.

[7] Longstaff, Francis A., Schwartz, Eduardo S, Valuing American Options by Simulation: A Simple Least-Squares Approach, The Review of Financial Studies, 14(1):113-147, 2001

[9] Matache A.M., Nitsche P.A. and Schwab C.,Wavelet Galekin Pricing of American Options on Levy Driven Assets, Research reports N 2003/06, (2003).