# SOME CONTRIBUTIONS TO STATISTICAL MODELING IN FINANCE

• Stephen Chan

Student thesis: Phd

### Abstract

The whole thesis contains 7 chapters. Chapter 1 is the introductory chapter of my thesis and themain contributions are in Chapter 2 through to Chapter 7. The theme of these chapters is developing and reviewing statistical distributions with financial applications.Value at Risk is the most popular measure of financial risk.It was introduced in the 1980s. Much theory have been developed since then. The developments have been most intensive in recent years. However, we are not aware of any comprehensive review of known estimation methods for Value at Risk. We feel it is timely that such a review is written. Chapter 2 (containing six sections and over one hundred and eighty references) attempts that task. We expect that this review could serve as a source of reference and encourage further research with respect to measures of financial risk.Value at Risk and Expected Shortfall are the two most popular measures of financial risk. But the available R packages for their computation have been limited. In Chapter 3, we introduce an R contributed package written by me and my supervisor. It computes the two measures for over one hundred parametric distributions, including all commonly known distributions. We expect that the R package could be useful to researchers and to the financial community.Bitcoin, the first electronic payment system, is becoming a popular currency. In Chapter 4, weprovide a statistical analysis of the log-returns of the exchange rate of Bitcoin versus the UnitedStates Dollar. Fifteen of the most popular parametric distributions in finance are fitted to thelog-returns. The generalized hyperbolic distribution is shown to give the best fit. Predictions aregiven for future values of the exchange rate.An extreme value analysis of electricity demand in the UK is provided in Chapter 5. The analysis is based on the generalized Pareto distribution. Its parameters are allowed to vary linearly and sinusoidally with respect to time to capture patterns in the electricity demand data. The models are shown to give reasonable fits. Some useful predictions are given for the Value at Risk of the returns of electricity demand.Chapter 6 answers the following question: given independent random variables, X1 and X2, what effect does the condition X1 less than X2 has?The question is answered for thirty commonly known families of distributions. A data application is given.Chapter 7 is a particular case of Chapter 6 for the case X1 and X2 are extreme value randomvariables. This chapter was motivated by a recent paper published by Adam and Tawn.Following up on the work of Nadarajah and Teimouri [Nadarajah, S., Teimouri, M., 2012. Onthe characteristic function for asymmetric exponential power distributions. Econometric Reviews31, 475-481], Chapter 8 derives explicit closed-form expressions for the characteristic function ofthe asymmetric Student's t distribution. The expressions involve hypergeometric and Bessel type functions.
Date of Award 31 Dec 2016 English The University of Manchester Saraleesan Nadarajah (Supervisor) & Michael Tso (Supervisor)

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