## Extreme Values and Financial Risk

 Unit code: MATH68181 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

None

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.

#### Learning outcomes

On successful completion of this unit students will:

• have some understanding of the probabilistic fundamentals of univariate and bivariate extreme value theory; and,
• be able to choose and fit appropriate extreme value models for a given data (univariate and bivariate).

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework, 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]
• A point process characterization for extreme values [3]
• Some statistical models for univariate extremes [4]
• Fluctuations of bivariate extremes: the theory [3]
• Some models for bivariate extremes [4]
• Time series models for extremal processes [3]
• Financial risk management [2]
• Extremal index, large claim index, the longest-success run, reinsurance treaties [3]
• Other applications to problems in finance [3]

• 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.  Coursework or in-class tests (where applicable) also provide 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 - 22 hours
• Tutorials - 22 hours
• Independent study hours - 106 hours