Boeker, Maria Elisabeth; (2021) Scaling and mean reversion in foreign exchange time series. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis aims to further our understanding of the statistical properties of foreign exchange rate time series. We propose a model of nominal exchange rates as following an Ornstein-Uhlenbeck process with a time-dependent reversion level and test this model against real-world data with a particular focus on the scaling properties of the series. In doing so, we aim to contribute towards the development of a stochastic model of FX rates which combines the mean-reverting behaviours predicted by economic models and trading strategies with the multifractal properties of foreign exchange time series well-known in econophysics. Our findings may be used to improve models of foreign exchange time series as well as trading strategies. The research analyses real-world foreign exchange time series and synthetic data sets and comprises the following three parts: - Part 1 explores the scaling of the volatility of exchange rates under the premise of the series having a mean-reverting component. Using simulated data, we show that our model is as good at reproducing the volatility scaling of the real data as a random walk. - Part 2 examines two aspects of Ornstein-Uhlenbeck parameter estimation. First, the accuracy of the estimators in the case of incomplete or irregular finite samples is numerically investigated. Second, a novel parameter estimation method for Ornstein-Uhlenbeck processes with unknown time-dependent reversion level is proposed, and this and an existing method are tested. Neither of these topics are well-represented in the literature, yet they have numerous applications, within and beyond the field of finance. - Part 3 calibrates our model to real-world FX data using the method proposed in Part 2. We give an empirical relationship between the calibrated parameters of the model and the way the underlying trend is defined. We show that this dependence can be described by the Hurst exponent of the series. This finding may help inform the construction of models of FX rates as well as parameter estimation and calibration methods where unknown time-dependent reversion levels are involved. This thesis contributes to science in a number of ways: - we have extended the range of intervals for which the scaling of FX volatility has been shown to hold; - we have shown that this scaling law may be compatible with a mean reversion based model of FX rates; - we have gathered some insight into the reliability of parameter estimation of the Ornstein-Uhlenbeck process under imperfect conditions; - a novel calibration method for an Ornstein-Uhlenbeck process with time-dependent mean has been proposed and an existing method has been tested; - a relationship between the calibrated parameters of a mean-reverting model of FX rates and the self-similarity of the series was found.

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