## Extreme Values and Financial Risk

 Unit code: MATH48181 Credit Rating: 15 Unit level: Level 4 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

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.

#### 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.

#### 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.

#### 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.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• In-class test weighting within unit 20%;
• Three hours end of semester examination, weighting within unit 80%.

#### 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].

• 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.

#### 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.

#### Study hours

• Lectures - 33 hours
• Tutorials - 11 hours
• Independent study hours - 106 hours