Replacing the Monte Carlo Simulation with the COS Method for PFE (Potential Future Exposure) Calculations

Abstract

To fulfil the need in the industry for fast and accurate PFE calculations in practice, a new, semi-analytical method of calculating the PFE metric for CCR has been developed, tested and analyzed in this thesis. Herewith we focus on the calculation of PFEs for liquid IR and FX portfolios involving up to three correlated risk-factors: a domestic and foreign short rate and the exchange rate of this currency pair. Both netting-set level and counterparty level PFEs are covered in our research. The short rates are modelled under the one-factor Hull-White (HW1F) model and for the exchange rate we assume they follow geometric Brownian motion. The key insight is that the cumulative distribution function (CDF) can be recovered semi-analytically using Fourier-cosine expansion, whereby the series coefficients are readily available from the characteristic function of the total exposure. The characteristic function in turn can be solved numerically via quadrature rules. Risk metrics, such as the potential future exposure (PFE), can be attained once the CDF is reconstructed using the Fourier series.

Our theoretical error analysis predicts stable convergence of the COS method and observed exponential convergence of the COS method for both netting-set and counterparty level PFE calculations. For three artificial portfolios of different sizes, it was observed that the COS method is at least five times more accurate than the Monte Carlo (MC) simulation method but takes only one-tenth of the CPU time of the MC method. The advantage of the COS method becomes even more prominent when the number of derivatives in a portfolio increases. We conclude that the COS method is a much more efficient alternative for MC method for PFE calculations, at least for portfolios involving three risk factors.
Our theoretical error analysis predicts stable convergence of the COS method and observed exponential convergence of the COS method for both netting-set and counterparty level PFE calculations. For three artificial portfolios of different sizes, it was observed that the COS method is at least five times more accurate than the Monte Carlo (MC) simulation method but takes only one-tenth of the CPU time of the MC method. The advantage of the COS method becomes even more prominent when the number of derivatives in a portfolio increases. We conclude that the COS method is a much more efficient alternative for MC method for PFE calculations, at least for portfolios involving three risk factors.
We conducted theoretical analysis on the error convergence and observed exponential convergence of the COS method for both netting-set and counterparty level PFE calculations. For three artificial portfolios of different sizes, it was observed that the COS method is at least five times more accurate than the Monte Carlo (MC) simulation method but takes only one-tenth of the CPU time of the MC method. The advantage of the COS method becomes even more prominent when the number of derivatives in a portfolio increases. We conclude that the COS method is a much more efficient alternative for MC method for PFE calculations, at least for portfolios involving three risk factors.