Function approximations for counterparty credit exposure calculations
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.