This thesis explores the application of Kalman filtering techniques to enhance pairs trading strategies in financial markets. Pairs trading is a statistical arbitrage strategy that exploits temporary price divergences between historically correlated assets by taking opposite positions with the expectation of mean reversion. The study addresses a fundamental challenge in pairs trading: accurately modeling the underlying spread dynamics in the presence of market noise. The research implements a state-space model framework where the observed spread between asset prices is treated as a noisy measurement of a mean-reverting process. A default Kalman filter is applied to estimate the true underlying spread by filtering out market noise, with the goal of generating more reliable trading signals. To optimize the Kalman filter’s performance, the ExpectationMaximization (EM) algorithm is employed to estimate the model’s latent parameters, including process noise and observation noise covariances. The methodology is tested on a cryptocurrency pair (Ethereum-NEO) identified from existing literature using the distance method for pair selection, covering the period from January 2018 to December 2019. To test the performance of Kalman filtering three approaches are compared: trading on unfiltered spreads, trading on Kalman-filtered spreads with default parameters, and trading on spreads filtered using EM-optimized parameters. The empirical results reveal several key findings. Surprisingly, the unfiltered spread strategy initially outperformed the default Kalman filter approach, generating $725.73 in profits across 4 trades compared to $523.95 across 3 trades for the filtered approach. However, when EM optimization was applied, the Kalman filter strategy achieved the highest performance with $750.83 in profits across 4 trades. A notable discovery is that the estimated state transition coefficient consistently converged to 1, indicating random walk behavior rather than the expected mean-reverting dynamics. This suggests that the theoretical assumption of mean reversion may not always align with empirical data, highlighting the importance of model validation in quantitative finance applications. The study demonstrates that while Kalman filtering can enhance pairs trading strategies, parameter optimization through EM is crucial for achieving superior performance. The research contributes to the understanding of noise reduction techniques in financial time series and provides insights into the practical challenges of implementing statistical arbitrage strategies. Future work could explore larger asset universes, explicit mean-reversion constraints, and the incorporation of transaction costs and risk management considerations.

This thesis takes a step towards developing a vine-based regression method tailored to mixed-type (continuous & discrete) data. We visualize the difference between continuous conditional quantile functions of bivariate copulas and their discretized (discretized by binning the continuous variable into bins of equal probability) conditional quantile functions. This showed how discretization into a small number of bins loses dependence, especially in the tails. We show a different binning procedure can improve the preservation of dependence after discretization, motivating the development of methodologies which compute an ‘optimal’ (optimal in the sense of balancing tail dependence with dependence around the median) binning procedure given some data. For application in computing an ‘optimal’ binning procedure, we make an attempt at analytically quantifying the difference between discretized conditional quantiles and continuous conditional quantiles of bivariate copulas, but the analytical derivation is not possible for most settings (with exceptions for bivariate Clayton and bivariate Frank copulas, and the smallest conditioning value of discretized covariate).

From the aforementioned conclusion that discretization loses dependence, we infer a variable
selection measure tailored to mixed-type data should be biased against discretized covariates,
and that this bias should be monotone decreasing with the number of bins of the discretized
covariate. The bias for/against discretized covariates of various variable selection measures is
investigated, in a scenario with a single covariate and a scenario with two covariates. With a single covariate, Pearson’s/polyserial correlation and Kendall’s tau/tau-b were found unsuitable as variable selection measures for mixed-type data, due to a lack of bias against discretized covariates. Conditional log-likelihood and check-loss at the quantile level 0.05 seem nearly identically biased in the bivariate setting, although this should be said with the caveat that all simulation scenarios are homoskedastic. Finally, we show in a three-dimensional setting that correlation between covariates does not seem to affect the predictive performance when both covariates are continuous, but correlation between covariates has significant negative effects on the predictive performance when one of the covariates is discretized. This difference between the effect of discretization in two dimensions and three dimensions should be kept in mind when developing variable selection procedures for mixed-type data.

Bayesian Networks (BNs) are popular models that represent complex relationships between variables, which can be quantified by Conditional Probability Tables (CPTs) in the discrete case. If data are not sufficient, experts can be involved to assess the probabilities in the CPTs through Structured Expert Judgment (SEJ), which is often a burdensome task. To lighten the elicitation burden, several methods have been developed previously to construct CPTs using a limited number of input parameters, such as the Ranked Nodes Method (RNM), InterBeta and Functional Interpolation. These methods are first analyzed theoretically, where limitations and potential improvements are determined, which were used as inspiration to develop extensions to the methods. The methods and newly developed extensions, including "ExtraBeta" and "AutoRNM", were applied to reconstruct fully elicited CPTs. Finally, simulation studies are performed to find best practices for InterBeta. InterBeta with parent weights is determined as the best-performingmethod, and the AutoRNM and ExtraBeta extensions are worth exploring further.