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G.L. Mast
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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. ...
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. ...
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.
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.
This thesis produces a pre-characterization numerical model capable of handling and calculating electromagnetic fields within a rectangular reverberation chamber near its lowest usablefrequency at 200 MHz. As reverberation chambers strive to have high electric field uniformityto meet field uniformity standards, high electric fields, having the property of being naturallymore uniform, are out of the scope of interest. Additionally high frequencies require more computational memory and CPU time.The model is made using a finite element method based modelling software called Comsol Multiphysics. The modelled reverberation chamber consisting of an antenna, a reflectiveshielded chamber and a Z-fold mode-stirrer is gradually build up. This means that first ananalysis of only the antennas will be made, thereafter the antennas will be put into a reflec-tive shielded chamber environment and finally the antennas are put into a reflective shieldedchamber environment with a z-fold mode stirrer. Furthermore, an extra situation will be considered in which a dielectric object will be added to a shielded chamber environment excited by an antenna without mode stirrer. The effect of an added dielectric object will be studied because of physical interest and completeness and not as added intermediate step to build a reverberation chamber environment. As all situations have similar difficulties in modelling and measuring, the gradual development of the reverberation chamber will allow for the best error analysis of the model. The model replicates the setup of a real reverberation chamber located at Comtest, Zoeterwoude. The reliability and accuracy of the model is studied by comparing the modelled electric field to the measured one. It was found that all models showed a good resemblance between simulated and measured electric fields above 60 MHz except the most complex reverberation chamber model. The simulated field uniformity expressed as standard deviationis twice as high as measurements suggest. The model can therefore only be used as worst-case scenario prediction for the field uniformity. Two kinds of antennas are used during the modelling and measuring phase, a 3104c biconical antenna used in the frequency range 25-200 MHz and the 3146a log-periodic dipole array antenna in the frequency range 200-1000 MHz. The electromagnetic radiation pattern in the far field domain was modelled and corresponded to the expected omni-directional and directional field pattern respectively. Next the antennas were placed in a highly reflective shielded chamber with dimensions (4.05 m×2.55 m×2.925 m). By taking data in specific slices from the model and comparing these to the measurements at the same points it was concluded that the model does not ac-curately predict the electric fields below 60 MHz. Above 60 MHz the model does predict the general electric field pattern in the chamber. It however does not predict local maxima or minima of the electric field.The same comparison was done for the situation in which a dielectric object was added to the setup. The dielectric object was chosen to be a container filled with water (0.27 m×0.565 m×0.269 m). This container is placed in the formerly empty shielded chamber to change the inner electric field. Water was used due to its favourable properties for reflectivity. The same behaviour of the model was observed with this added dielectric object. Finally the electric field in the reverberation chamber at Comtest (5.03 m×3.97 m×2.85 m) was modelled and solved using a GMRES algorithm with geometric multigrid preconditioning.The preconditioning in the GMRES allows for faster convergence. The field uniformity wascomputed for both the model and the measurements as outlined by IEC 61000-4-21. This wasdone for 4 and 12 stirrer rotation positions. Both showed that the model had a less uniform field compared to the Comtest reverberation chamber. The Comtest reverberation chamber complied with the electromagnetic compatibility requirements for measurements using 12 stirrer positions whereas the model did not. However the 4 stirrer position model, which made use of perfect electric conductor boundary conditions, showed a maximum of 13% increase in field uniformity when steel walls were used instead. To show a glimpse of an innovation that put the Delft University of Technology on the map, the shifted laplacian preconditioner is briefly discussed. As an intermediate step in solving a problem with little to no damping, a complex preconditioning matrix is used. It is shown that for an increasing imaginary shift in the Helmholtz problem, expressed as an increasing electrical conductivity σ, the number of iterations needed to reach a relative tolerance smaller than 0.01 decreases. At last, the effect of a shifted laplacian contribution to a multigrid preconditioning on the convergence speed is studied for an non-damped pressure acoustic Helmholtz problem. It is shown that the added contribution slows convergence in a simple geometry, while a more complex geometry cannot be solved without this contribution.
...
This thesis produces a pre-characterization numerical model capable of handling and calculating electromagnetic fields within a rectangular reverberation chamber near its lowest usablefrequency at 200 MHz. As reverberation chambers strive to have high electric field uniformityto meet field uniformity standards, high electric fields, having the property of being naturallymore uniform, are out of the scope of interest. Additionally high frequencies require more computational memory and CPU time.The model is made using a finite element method based modelling software called Comsol Multiphysics. The modelled reverberation chamber consisting of an antenna, a reflectiveshielded chamber and a Z-fold mode-stirrer is gradually build up. This means that first ananalysis of only the antennas will be made, thereafter the antennas will be put into a reflec-tive shielded chamber environment and finally the antennas are put into a reflective shieldedchamber environment with a z-fold mode stirrer. Furthermore, an extra situation will be considered in which a dielectric object will be added to a shielded chamber environment excited by an antenna without mode stirrer. The effect of an added dielectric object will be studied because of physical interest and completeness and not as added intermediate step to build a reverberation chamber environment. As all situations have similar difficulties in modelling and measuring, the gradual development of the reverberation chamber will allow for the best error analysis of the model. The model replicates the setup of a real reverberation chamber located at Comtest, Zoeterwoude. The reliability and accuracy of the model is studied by comparing the modelled electric field to the measured one. It was found that all models showed a good resemblance between simulated and measured electric fields above 60 MHz except the most complex reverberation chamber model. The simulated field uniformity expressed as standard deviationis twice as high as measurements suggest. The model can therefore only be used as worst-case scenario prediction for the field uniformity. Two kinds of antennas are used during the modelling and measuring phase, a 3104c biconical antenna used in the frequency range 25-200 MHz and the 3146a log-periodic dipole array antenna in the frequency range 200-1000 MHz. The electromagnetic radiation pattern in the far field domain was modelled and corresponded to the expected omni-directional and directional field pattern respectively. Next the antennas were placed in a highly reflective shielded chamber with dimensions (4.05 m×2.55 m×2.925 m). By taking data in specific slices from the model and comparing these to the measurements at the same points it was concluded that the model does not ac-curately predict the electric fields below 60 MHz. Above 60 MHz the model does predict the general electric field pattern in the chamber. It however does not predict local maxima or minima of the electric field.The same comparison was done for the situation in which a dielectric object was added to the setup. The dielectric object was chosen to be a container filled with water (0.27 m×0.565 m×0.269 m). This container is placed in the formerly empty shielded chamber to change the inner electric field. Water was used due to its favourable properties for reflectivity. The same behaviour of the model was observed with this added dielectric object. Finally the electric field in the reverberation chamber at Comtest (5.03 m×3.97 m×2.85 m) was modelled and solved using a GMRES algorithm with geometric multigrid preconditioning.The preconditioning in the GMRES allows for faster convergence. The field uniformity wascomputed for both the model and the measurements as outlined by IEC 61000-4-21. This wasdone for 4 and 12 stirrer rotation positions. Both showed that the model had a less uniform field compared to the Comtest reverberation chamber. The Comtest reverberation chamber complied with the electromagnetic compatibility requirements for measurements using 12 stirrer positions whereas the model did not. However the 4 stirrer position model, which made use of perfect electric conductor boundary conditions, showed a maximum of 13% increase in field uniformity when steel walls were used instead. To show a glimpse of an innovation that put the Delft University of Technology on the map, the shifted laplacian preconditioner is briefly discussed. As an intermediate step in solving a problem with little to no damping, a complex preconditioning matrix is used. It is shown that for an increasing imaginary shift in the Helmholtz problem, expressed as an increasing electrical conductivity σ, the number of iterations needed to reach a relative tolerance smaller than 0.01 decreases. At last, the effect of a shifted laplacian contribution to a multigrid preconditioning on the convergence speed is studied for an non-damped pressure acoustic Helmholtz problem. It is shown that the added contribution slows convergence in a simple geometry, while a more complex geometry cannot be solved without this contribution.