Function approximations for counterparty credit exposure calculations

ArXiv ID: 2507.09004 “View on arXiv”

Authors: Domagoj Demeterfi, Kathrin Glau, Linus Wunderlich

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.

Keywords: Exposure Measurement, Chebyshev Interpolation, XVA, Monte Carlo, Error Bounds, Derivatives/Counterparty Risk

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper is mathematically dense, featuring proofs of error bounds in L^p and uniform norms, exponential convergence for Chebyshev interpolation, and asymptotic efficiency gains, warranting a high math score. Empirical rigor is also high, with specific quantitative results from numerical experiments (speed-up factors up to 230, average 87) on complex derivatives, though it lacks full backtesting or public datasets.

  flowchart TD
    A["Research Goal:<br/>Resolve computational burden vs.<br/>oversimplification in exposure measurement"] --> B["Methodology:<br/>Function approximation via<br/>Chebyshev interpolation"]
    B --> C{"Inputs: Derivative Pricers<br/>& Risk Factors"}
    C --> D["Computational Process:<br/>Replace pricers with approximations<br/>Prove Lp and uniform error bounds"]
    D --> E["Key Findings & Outcomes"]
    E --> F["Exponential convergence<br/>& Efficient error bounds"]
    E --> G["Run-time reduction<br/>Speed-up: 230x (max), 87x (avg)"]
    E --> H["Application:<br/>Greeks approximation &<br/>Adaptive degree selection"]