Function approximations for counterparty credit exposure calculations

TL;DR

This paper introduces a method using function approximation and Chebyshev interpolation to efficiently compute counterparty credit exposure measures, significantly reducing computational time while maintaining accuracy.

Contribution

It develops a systematic approach with error bounds and exponential convergence for exposure calculations, enabling faster and reliable risk assessment in finance.

Findings

Achieves up to 230-fold speed-up in computations

Provides error bounds and convergence guarantees

Demonstrates effectiveness across various derivative types

Abstract

The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing frequently called derivative pricers by function approximations covering all practically relevant exposure measures, including quantiles. We prove error bounds for exposure measures in terms of the $L^p$ norm, $1 \leq p < \infty$, and for the uniform norm. To fully exploit these results, we employ the Chebyshev interpolation and show exponential convergence of the resulting exposure calculations. As our main result we derive probabilistic and finite sample error bounds under mild conditions including the natural case of unbounded risk factors. We derive an asymptotic efficiency gain scaling with $n^{1/2-\varepsilon}$ for any $\varepsilon>0$ with $n$ the number of simulations. Our numerical experiments cover callable, barrier, stochastic volatility and jump features. Using 10\,000 simulations, we consistently observe significant run-time reductions in all cases with speed-up factors up to 230, and an average speed-up of 87. We also present an adaptive choice of the interpolation degree. Finally, numerical examples relying on the approximation of Greeks highlight the merit of the method beyond the presented theory.