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.

In this paper we started by explaining what a Markov chain is. After this we defined some key concepts such as stationarity, reversibility and ergodicity which were used throughout the rest of the paper. Next, the classical central limit theorem was stated in order to refresh the reader of this theorem and to show that, because of the dependence, this theorem is not applicable for the Markov chains we discussed earlier. In the next section we started working on the Kipnis-Varadhan central limit theorem. We defined the Martingale central limit theorem and used it to show that the additive functions can be approximated by a martingale. This had the result that the Martingale central limit theorem implies the central limit theorem for the additive functions. Furthermore, we also derived an equation for calculating the limiting variance. For this we used the key concepts discussed in section 2 and another concept called spectral measures. The section is finished by a simple example in order to illustrate what we have discussed previously. In the next section we used simulations in one-dimension and two-dimensions to show that, in a random environment, the distribution of the additive functions indeed converges to that of the normal distribution. Because the probability distributions defined on the random environment was a symmetric distribution the Kipnis-Varadhan central limit theorem was not yet needed since the Martingale central limit theorem already applied. After this we defined some key concepts for a Markov chain working in continuous time. The Kipnis-Varadhan central limit theorem was proven for these Markov chains as well. Furthermore, a different equation was derived to find the limiting variance for continuous time Markov chains. In section 7 the random conductance model was explained. In this chapter the random conductance model was used in order to make simulations to show that the Kipnis-Varadhan central limit theorem works. After simulating the Markov chain in continuous time we indeed saw that the histograms show convergence towards the normal distribution. This shows that for the simulations the Kipnis-Varadhan central limit theorem works.