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# MMath Mathematics and Statistics / Course details

Year of entry: 2021

## Course unit details:Extreme Values and Financial Risk

Unit code MATH48181 15 Level 4 Semester 1 Department of Mathematics No

### Overview

The course will give some probabilistic and statistical details of univariate and bivariate extreme value theory. The topics covered will include: fundamental of univariate extreme value theory, the three extreme value distributions, various models for univariate extremes, fundamentals of bivariate extreme value theory, and various models for bivariate extremes. The course will contain a great deal material on applications of the models to finance. Software in R will be used.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Statistical Methods MATH20802 Pre-Requisite Compulsory
Probability 2 MATH20701 Pre-Requisite Compulsory
MATH48181 pre-requisites

Students are not permitted to take more than one of MATH38181 or MATH48181 for credit in the same or different undergraduate year.  Students are not permitted to take MATH48181 and MATH68181 for credit in an undergraduate programme and then a postgraduate programme.

### Aims

To introduce probabilistic fundamentals and some statistical models in extreme value theory with applications to finance.

### Learning outcomes

On successful completion of this unit students will be able to:

• Be able to construct the extreme value law given a univariate distribution;
• choose and fit appropriate extreme value models for a given data (univariate and bivariate);
• calculate probabilities associated with total portfolio loss, maximum portfolio loss and minimum portfolio loss;
• estimate financial risk measures;
• fit copulas to real data sets;
• fit GARCH type models to real data sets.

### Syllabus

I plan to cover all of the following topics:

• Fluctuations of univariate maxima: the theory [4]
• Fluctuations of univariate upper order statistics: the theory [3]
• Some statistical models for univariate extremes [4]
• Real data applications for univariate extremes using the R software [1]
• Portfolio theory [2]
• Real data applications [1]
• Financial risk measures and their estimation [3]
• Real data applications [1]
• Models for stock returns [2]
• Real data applications [1]
• Some models for bivariate extremes [4]
• Real data applications for bivariate extremes using the R software [1]
• Copulas [2]
• Real data applications [1]
• GARCH type models [2]
• Real data applications [1].

### Teaching and learning methods

Three lectures and one example class each week.  In addiiton students are expected to do at least four hours private study each week on this course unit.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• In-class test weighting within unit 20%;
• End of semester examination: weighting within unit 80%.

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  The in-class test also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Embrechts, P., KlÃ'ppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events: for Insurance and Finance, Springer-Verlag, Berlin.
• Leadbetter, M.R., Lindgren, G. and Rootz_en, H. (1983) Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, Berlin.
• Resnick, S.I. (1987) Extreme values, Regular Variation and Point Processes, Springer-Verlag, Berlin.
• Coles S. (2001) An Introduction to Statistical Modelling of Extreme Values, Springer-Verlag, London.
• Kotz, S. and Nadarajah, S. (2000) Extreme Value Distributions: Theory and Applications, Imperial College Press, London.

### Study hours

Scheduled activity hours
Lectures 24
Tutorials 12
Independent study hours
Independent study 114

### Teaching staff

Staff member Role