The computation of multivariate expectations is a common task in various fields related to probability theory. This thesis aims to develop a generic and efficient solver for multivariate expectation problems, with a focus on its application in the field of quantitative finance, specifically for the quantification of Counterparty Credit Risk (CCR).

The proposed COS-CPD method utilizes the COS method to recover the exposure distribution by its Fourier-cosine series expansion, from which measures such as the PFE and EE can be obtained. The key insight is that the corresponding Fourier coefficients are readily available from the characteristic function, which can be solved using numerical integration methods. However, the efficiency of standard quadrature rules is limited to only a few risk factors, as the dimension of integration is determined by the number of risk factors involved.

To address this limitation, the COS-CPD method reduces the dimension of integration of the characteristic function through two steps. Firstly, the joint density function of the risk factors in the characteristic function is replaced by a dimension-reduced Fourier-cosine series expansion, which is obtained through CPD. With CPD, the computational complexity of computing the Fourier coefficient tensor is reduced to a linear growth with respect to the number of dimensions. Secondly, the portfolio is divided into segments that share the same risk factors. These two steps reduces the evaluation of the characteristic function to the calculation of only one- and two-dimensional integrals, which are solved by the Clenshaw-Curtis quadrature rule. As a result, the COS-CPD method is suitable for portfolios with more than three risk factors.

Numerical comparisons of the COS-CPD method and Monte Carlo (MC) method are made for netting-set PFE and EE profiles of multiple derivative portfolios up to five risk factors. For similar accuracy levels, the COS-CPD method greatly outperforms the Monte Carlo method in computation time. This difference increases for larger portfolios, which makes the COS-CPD method a much more efficient alternative for the MC method, especially for large portfolios.

Furthermore, the COS-CPD method is applied in the context of multi-asset option pricing. A six-dimensional basket option is considered, and the results are compared to a recently developed sparse grid method. The comparison shows that the COS-CPD method outperforms the sparse grid method in both accuracy and computation time. Moreover, the COS-CPD method allows the computation of the option value for multiple strike prices simultaneously, with no significant additional computational cost.

In dit onderzoek is een Monte Carlo algoritme beschreven voor het schatten van de dominante eigenwaarde en bijbehorende eigenvector van een niet-negatieve, irreducibele matrix. Naast de werking van het Monte Carlo algoritme is ook de convergentiesnelheid onderzocht. Tevens zijn een aantal simulaties in de praktijk uitgevoerd voor zowel het Monte Carlo algoritme als de power methode en aan de hand hiervan bekeken of het Monte Carlo algoritme een goede concurrent is van de power methode.