Chebyshev Methods for Ultra-efficient Risk Calculations

TL;DR

This paper introduces Chebyshev interpolation techniques to significantly reduce the computational cost of risk calculations in finance, achieving high accuracy and efficiency across various complex valuation tasks.

Contribution

It extends Chebyshev interpolation methods to high-dimensional problems and demonstrates their practical application in industry-relevant risk computations.

Findings

Orders of magnitude reduction in computational effort

High accuracy maintained across multiple risk calculation examples

Applicable to various complex financial valuation problems

Abstract

Financial institutions now face the important challenge of having to do multiple portfolio revaluations for their risk computation. The list is almost endless: from XVAs to FRTB, stress testing programs, etc. These computations require from several hundred up to a few million revaluations. The cost of implementing these calculations via a "brute-force" full revaluation is enormous. There is now a strong demand in the industry for algorithmic solutions to the challenge. In this paper we show a solution based on Chebyshev interpolation techniques. It is based on the demonstrated fact that those interpolants show exponential convergence for the vast majority of pricing functions that an institution has. In this paper we elaborate on the theory behind it and extend those techniques to any dimensionality. We then approach the problem from a practical standpoint, illustrating how it can be applied to many of the challenges the industry is currently facing. We show that the computational effort of many current risk calculations can be decreased orders of magnitude with the proposed techniques, without compromising accuracy. Illustrative examples include XVAs and IMM on exotics, XVA sensitivities, Initial Margin Simulations, IMA-FRTB and AAD.