Dimension-Reduced Cosine Expansions via Tensor Decomposition: Methodology and Applications to Credit Exposure Quantification

TL;DR

The core methodology of dimension-reduced cosine expansions is established, together with the efficient training algorithm, and applies it to netting-set-level MtM and exposure calculations, which improves both training speed and accuracy by more than two orders of magnitude.

Abstract

This paper initiates a series of studies on a COS-tensor framework, as an efficient alternative to MC for large and liquid portfolios characterized by a modest number of dominant risk factors but a large number of trades. The framework is built on three key insights. First, the cumulative distribution function (CDF) of portfolio level mark-to-market (MtM) values and exposures can be recovered in the Fourier domain by first numerically evaluating their characteristic functions and subsequently applying the one-dimensional COS method. Second, the curse of dimensionality arising in the evaluation of netting-set characteristic functions is mitigated by dimension-reduced cosine expansions derived by combining Fourier-cosine series representations with tensor decomposition techniques. This reformulation shifts the main computational burden from online evaluation to an offline training stage. Third, the offline training can be performed directly in the Fourier domain by reapplying the core idea of the COS method. This improves both training speed and accuracy by more than two orders of magnitude compared with gradient-based training in the physical domain. This paper establishes the core methodology of dimension-reduced cosine expansions, together with the efficient training algorithm, and applies it to netting-set-level MtM and exposure calculations. Among several tensor decomposition techniques, we focus on Canonical Polyadic Decomposition (CPD) for CCR quantification due to its simplicity and transparency, which are particularly appealing in practical risk-management applications. Numerical experiments on netting sets comprising tens of thousands of trades driven by seven risk factors demonstrate that the proposed method achieves relative errors below 0.1% while requiring only a small fraction of the runtime of MC simulation.

Figures (10)

• Figure 1: Schematic of the Monte Carlo framework for CCR quantification. Solid arrows represent full re-evaluation, while dashed arrows indicate accelerated calculations, for example via Chebyshev interpolation as in laris2018chebyshev.
• Figure 2: The proposed COS--tensor (deterministic) framework for CCR quantification. Solid black lines denote components introduced in this paper, while dashed lines indicate compatibility with existing pricing acceleration techniques. When the number of risk factors exceeds three, the joint density is approximated by a low-rank tensor decomposition obtained through an offline training procedure.
• Figure 3: Efficiency and accuracy of the principal--COS method. Left: CPU time percentage compared to the original COS formulation and fraction of retained principal frequencies (in percentage of the full set of frequencies) versus the coefficient-magnitude tolerance. Right: approximation error $\|\epsilon\|_\infty$ for the recovered three-dimensional Gaussian density with correlations up to $60\%$. The full COS expansion uses $150$ terms per dimension.
• Figure 4: Convergence behavior of Algorithm \ref{['algo:our_training_method']} with and without principal--COS filtering. The error is measured in absolute terms using the infinity norm, $|\epsilon|_\infty$, across iterations in a three-dimensional setting.
• Figure 5: Computation pipeline from offline-trained factor matrices to exposure distribution, for a fixed time point in the future.

Theorems & Definitions (9)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6: Computational complexity
• Remark 7
• Remark 8
• Remark 9