ST414: Advanced Topics in Statistics
Lecturer(s)
Dr John Aston, Dr Ben Graham & Dr Aleksandar Mijatovic
Prerequisite(s): ST218 Mathematical Statistics A, ST219 Mathematical Statistics B.
Content: Three self-contained sets of ten lectures. This module runs in Term 2.
Exotic Derivatives in Stochastic Volatility Models with Jumps (Aleksandar Mijatovic):
Aims: In equity and foreign exchange markets the risk-neutral dynamics of the underlying asset are commonly represented by a stochastic volatility model with jumps. In this short course we will study a dense subclass of such models (given by Markov additive processes) and describe analytically tractable formulae for the prices of a range of first-generation exotic derivatives. The following topics will be discussed:
- Fourier transforms of vanilla and forward starting options,
- Formula for the slope of the implied volatility smile for large strikes,
- Formula for a variance swap price,
- One-dimensional integral representation for volatility swaps and (if the time permits),
- Analytically tractable formula for the Laplace transform (in maturity) of the double-no-touch options based on complex-matrix Wiener-Hopf factorisation.
Recommended: A good understanding of continuous-time Markov chain theory (e.g. the content of J. Norris, Markov Chains) and of the fundamental properties of Brownian motion is essential. Some familiarity with integral transforms is desirable.
Course Material: See page.
Hidden Markov Models (John Aston)
Hidden Markov models (HMMs) provide a rich modelling structure for non-linear time series and sequence analysis. The aim of this course is to familiarise students with the fundamental ideas of HMMs including their setup, estimation and uses. In particular the following will be covered:
- The Viterbi Algorithm which is used to find the most likely underlying state sequence
- The Forwards-Backwards and Baum-Welsh Algorithms for parameter estimation
The course will be motivated by examples from many applications including engineering, economics and biological sciences.
Stein-Chen Method (Ben Graham):
Aims: The Stein-Chen method is a powerful modern technique which extends the Poisson 'law of small numbers' (given n independent events each of small probability p, the total number of event which occur is approximately Poisson of distribution np). There are many applications, for example in bioinformatics, insurance, and the study of extreme phenomena.
Objectives: By the end of the course students will be able to:
· Describe the principles of the Stein-Chen method;
· Apply it in a couple of central examples.
Prerequisites: ST217 Mathematical Statistics A & B.
Content: Three self-contained sets of ten lectures.
