Counterparty Credit Risk (CCR) refers to the risk that a counterpary involved in a financial contract will default before the final settlement of the contract, resulting in unrealized financial gains. One risk measure for managing counterparty credit risk is the Potential Future Exposure (PFE). The PFE is defined as the 97.5%-quantile of the exposure distribution. The traditional numerical method for computing the PFE is the Monte Carlo simulation method. Recent research has produced a semi-analytical method based on Fourier-cosine series expansion which produce PFE estimates with at least five times the accuracy of the Monte Carlo simulation in one-tenth of the CPU time. In this thesis this COS-PFE framework is combined with Monte Carlo simulation with the goal of reducing the variance of the PFE estimates.
In this thesis three methods are developed. Firstly, the Control Variate method that uses the COS-PFE framework to retrieve the CDF of the portfolio's exposure from which control variates are sampled. The second method developed uses Adaptive Importance Sampling. This method uses the COS-PFE framework to estimate the PFE and expected exposure (EE) of the portfolio. Using the risk factor that has the highest correlation with the exposure of the portfolio a shift is found such that the new joint probability distribution, from which the risk factors are sampled, has an EE that coincides with the portfolio's PFE. The third method developed in this thesis uses Adaptive Importance Sampling based on the Cross-Entropy method. This method aims to find an auxiliary probability distribution which minimizes the Kullbeck-Leibler divergence between itself and the theoretical zero-variance estimator.
The methods were tested using portfolios containing 100, 1000 and 10000 derivatives, both with and without collaterals. Testing showed that the control variate method was not able to produce a variance reduction compared to the straight forward Monte Carlo simulation. However, it did demonstrate the ability to produce a variance reduction for the EE.
The method that finds an optimal shift was successful in producing a variance reduction. The variance of the PFE estimates produced by this method was, on average, 3.5 times lower than the variance found using the straight forward Monte Carlo simulation. However, the algorithm required between 96 and 1773 seconds to produce a PFE estimate.
The adaptive importance sampling method using the cross-entropy approach was tested on all portfolio, both with and without collateral. This method was shown to produce PFE estimates with a significantly lower variance than the straight forward Monte Carlo simulation. For the uncollateralized portfolios containing 100, 1000 and 10000 derivatives the variance of the PFE estimates were on average 35.4, 38.6 and 37.2 times smaller compared to straight forward Monte Carlo simulation. For the portfolios with collateral this performance remains. We conclude that this method produces more accurate PFE estimations with significant lower variance than straight forward Monte Carlo simulation within the same CPU time.

Barrier options are fundamental financial tools that give rise to pricing challenges, particularly when embedded within stochastic models. This study directs its focus towards Lévy processes as a strategic approach to navigate and resolve these intricate complexities. The model assumption adopted in this thesis is that the underlying log-asset price follows a Levy process. This assumption, in combination with the COS method, enables us to reduce the dimensionality of the pricing problem for barrier options. To be more precise, the strike price and the log-asset price are both factored out from the calculation formula. The traditional COS method, known for rapid European option valuation, depends on the knowledge of the ch.f.. The COS method has been extended to pricing barrier options in [1] and [2]. However, both methods depend on a recursive calculation formula whereby the number of recursion equal to the number of monitoring dates, and thus, the calculation speed lags behind pricing European options , especially with quite a number of monitoring dates. Our key insight lies in the potential of using the traditional COS method for pricing barrier options. As the first step, we integrate the barrier hitting probability into the ch.f., thereby transforming it into a survival ch.f. Although the exact function of the survival ch.f. is unknown, its values on the grid used by the COS method can be accurately solved via Singular Value Decomposition (SVD). Testing results robustly reinforces our findings. Building on this, we work out two techniques to approximate the characteristic function through supervised learning. The model takes Lévy process model parameters and yields approximation function of the ch.f.. The first method, COS-GPR, combines Gaussian Process Regression (GPR) with the traditional COS method for ch.f. estimation. Assuming mutual independence among multivariate model outputs, COS-GPR addresses GPR’s limitations at boundaries with an additional Fourier expansion. Notably, COS-GPR demonstrates significant significantly faster calculation than the existing COS Barrier by seven times while maintaining heightened accuracy, with occasional outliers. The second method, the COS Fourier CPD (CFC) method, replaces the ch.f. in the COS method by a Fourier-series expansion of the ch.f., after which the dimension is strategically reduced using Canonical Polyadic Decomposition (CPD). CFC achieves superior overall accuracy compared to COS Barrier, with a speed increase of 180 times and minimal instances of extreme errors. In comparison to COS-GPR, CFC’s accuracy slightly decreases but exhibits a smaller maximum error. The CFC method has a 180- fold increase in speed compared to the COS Barrier method and maintains linear time complexity as training points per dimension increase. In conclusion, this study introduces efficient methods based on supervised machine learning for barrier option pricing under Levy processes. The fusion of conventional iii iv methods with advanced approaches leads to significant improvements in both speed and precision. These innovations hold the potential to transform financial derivative valuation, enhancing accuracy and efficiency in the process.

Barrier options, although highly liquid financial derivatives, present notable pricing challenges. In this thesis, we present a novel pricing approach for valuing continuously-monitored knock-out barrier options within the framework of stochastic volatility models.

The underlying process is firstly modelled under geometric Brownian motion and, subsequently, under Heston's stochastic volatility model. A key insight is that the value of a barrier option can be expressed as a single-dimensional integral, whereby the integrand involves the so-called survival density function, which captures the barrier-brea-ching information. Therefore, the option can be valued using the one-dimensional COS method for European options, once the Fourier series coefficients of the survival density are obtained.

The coefficients of the sine series expansion of the survival density function are, in fact, a continuous function, which is closely related to the characteristic function of the density. This motivates us to directly recover that function, which we refer to as the target function herewith, by selecting an appropriate series expansion for it. Thereafter, we insert this series expansion into the partial differential equation (PDE) that the target function should satisfy, which can be derived from the pricing PDE. This results in a linear system, solving which we obtain the coefficients needed to reconstruct the target function. Notably, this approach is particularly advantageous when the reference values of option prices are limited or unavailable, as it relies solely on the PDE to calibrate the series coefficients.

Our choice of series expansion is driven by the need for precise global and local approximations. Our research shows that a proper expansion for the target function can be built up from integrating a two-dimensional Fourier series of its first derivative with respect to time, marking our second pivotal insight. That results in a trigonometric expansion that significantly enhances the accuracy compared to a direct Fourier series expansion on the target function. Finally, our third pivotal insight: applying a change of variables further improves error convergence.

Extensive testing results suggest that, for similar accuracy levels, our approach greatly outperforms Monte Carlo simulations in terms of computation time. It also demonstrates superior computational efficiency and accuracy compared to other advanced numerical methods in the existing literature.
The benefits of this method become even more prominent when pricing a large number of options simultaneously.

Numerical tests reveal algebraic convergence for the series expansion reconstruction. For the option price, theoretical error analysis aligns with our findings, also predicting algebraic convergence.

The EAD metric is widely used in the calculations for the capital requirements concerning Counterparty Credit Risk (CCR). In this thesis we compare several methods for calculating this EAD. Basel III gives us two methods, the Standardized Approach for CCR (SA-CCR) and the Internal Model Method (IMM). Furthermore, we introduce an integrated benchmark model, whereby we estimate the EAD by integrating the Wrong Way Risk (WWR) directly, while in SA-CCR and IMM this WWR is captured by an alpha factor of 1.4. In this benchmark model, we derive the formula for backing out the probability of default using CDSs and then, via a Gaussian copula, we include a correlation between the exposures and default probability to model the WWR. The ultimate goal of this thesis is to find out how conservative the SA-CCR is compared to the IMM and the integrated benchmark model, and if the alpha factor of 1.4 is a reasonable value to account for WWR.

The test portfolios consist of interest rate swaps, cross currency swaps and FX forwards, which are the most liquid product types in the market. To value these products we model the interest rate using the Hull-White model and the exchange rate using a GBM, following industry standard.

Testing results suggest that the SA-CCR is at least a factor of 1.5 more conservative than the IMM in the presence of collateral, even with stressed parameters. The level of conservatism is even higher because no diversification is allowed between different asset classes by SA-CCR. Furthermore, we observe that using the parameters backed out from calibration and a default correlation of about 30 to 40\%, our integrated benchmark model based on copula returns more or less comparable EADs of IMM times the alpha factor of 1.4. This indicates that this value of 1.4 is a reasonable value to cover WWR in the IMM framework.