Gijs Mast : A COS-tensor Framework for Credit Exposure Calculations

Monte Carlo (MC) simulation methods remain the predominant approach for computing credit exposures in the pricing and risk management practices of the financial industry, owing to their flexibility, implementation simplicity, and transparency. However, accurate computation of high-quantile exposure metrics for large portfolios remains time consuming due to the intrinsically slow convergence of MC simulation. This paper introduces a novel ``COS-tensor'' framework that transcends MC simulation. It has the potential to serve as a computationally efficient alternative, particularly for large, liquid portfolios, while maintaining sufficient flexibility and transparency. Our key insight is that the problem can be transformed and solved in the Fourier domain through two steps: First, rather than generating total exposure samples as in MC methods, we numerically compute the characteristic function (ch.f.) of the total exposure and subsequently recover the cumulative distribution function via the one-dimensional COS method, see Fang and Oosterlee (2009). Second, to circumvent the curse of dimensionality in ch.f. computation, we apply tensor decomposition to the Fourier-cosine coefficient tensor of the joint density function. This ``COS-tensor'' approach constitutes a general framework that generates distinct dimensionally reduced cosine expansions for different tensor decomposition techniques, effectively shifting the curse of dimensionality to the offline training phase for tensor decomposition. The main part of this paper builds and studies the COS-CPD method, resulted from inserting low-rank Canonical Polyadic (CP) decomposition into the COS-tensor framework. A secondary innovation herewith is our Fourier-domain training algorithm for offline CP decomposition, which demonstrates over 100-fold improvements in speed and accuracy compared to physical-domain backpropagation with gradient descent. Extensive testing on real-sized portfolios shows that achieving 0.1\% error for portfolios of thousands of trades under seven risk factors requires only a fraction of the computation time of MC simulation. Results confirm our theoretical error analysis: exponential convergence with respect to Fourier-cosine terms and quadrature points at netting-set level, versus algebraic convergence at counterparty level. Notably, the computational performance remains largely unaffected as portfolio size increases, which is a stark contrast to MC methods. The computational bottleneck lies in the offline training, where the dimensionality curse for risk factors persists. This limitation, combined with diverse extension avenues, indicates substantial potential for further research. For instance, as we will demonstrate in a companion paper, using Tensor Train decomposition can markedly alleviate high-dimensional training constraints. We further note that existing instrument-level acceleration methods remain compatible with our framework, and for portfolios with numerous risk factors, the COS-tensor methods can serve as effective variance reduction techniques for MC simulation.