The Basel Committee of Banking Supervision has recently set out the revised standards for minimum capital requirements for market risk. The Committee has focused, among other things, on the two key areas of moving from Value-at-Risk (VaR) to Expected Shortfall (ES) and considering a comprehensive incorporation of the risk of market illiquidity by extending the risk measurement horizon. The estimation of the ES for several trading desks and taking into account different liquidity horizons is computationally very involved. We present a novel numerical method to compute the VaR and ES of a given portfolio within the stochastic holding period framework. Two approaches are considered, the delta-gamma approximation, for modelling the change in value of the portfolio as a quadratic approximation of the change in value of the risk factors, and some of the state-ofthe-art stochastic processes for driving the dynamics of the log-value change of the portfolio like the Merton jump-diffusion model and the Kou model. Central to this procedure is the application of the SWIFT method developed for option pricing, that appears to be a very efficient and robust Fourier inversion method for risk management purposes.

No abstract

The SWIFT method for pricing European-style options on one underlying asset was recently published and presented as an accurate, robust and highly efficient technique. The purpose of this paper is to extend the method to higher dimensions by pricing exotic option contracts, called rainbow options, whose payoff depends on multiple assets. The multidimensional extension inherits the properties of the one-dimensional method, being the exponential convergence one of them. Thanks to the nature of local Shannon wavelets basis, we do not need to rely on a-priori truncation of the integration range, we have an error bound estimate and we use fast Fourier transform (FFT) algorithms to speed up computations. We test the method for similar examples with state-of-the-art methods found in the literature, and we compare our results with analytical expressions when available.

The Basel Committee of Banking Supervision has recently set out the revised standards for minimum capital requirements for market risk. The Committee has focused, among other things, on the two key areas of moving from Value-at-Risk (VaR) to Expected Shortfall (ES) and considering a comprehensive incorporation of the risk of market illiquidity by extending the risk measurement horizon. The estimation of the ES for several trading desks and taking into account different liquidity horizons is computationally very involved. We present a novel numerical method to compute the VaR and ES of a given portfolio within the stochastic holding period framework. Two approaches are considered, the delta-gamma approximation, for modelling the change in value of the portfolio as a quadratic approximation of the change in value of the risk factors, and some of the state-ofthe-art stochastic processes for driving the dynamics of the log-value change of the portfolio like the Merton jump-diffusion model and the Kou model. Central to this procedure is the application of the SWIFT method developed for option pricing, that appears to be a very efficient and robust Fourier inversion method for risk management purposes.

In this work, we investigate the challenging problem of estimating credit risk measures of portfolios with exposure concentration under the multi-factor Gaussian and multi-factor t-copula models. It is well-known that Monte Carlo (MC) methods are highly demanding from the computational point of view in the aforementioned situations. We present efficient and robust numerical techniques based on the Haar wavelets theory for recovering the cumulative distribution function (CDF) of the loss variable from its characteristic function. To the best of our knowledge, this is the first time that multi-factor t-copula models are considered outside the MC framework. The analysis of the approximation error and the results obtained in the numerical experiments section show a reliable and useful machinery for credit risk capital measurement purposes in line with Pillar II of the Basel Accords.