MMath Mathematics and Statistics / Course details

Martingales are a special class of random processes which are key ingredients of the modern probability and
stochastic calculus. They can be used as mathematical models for fair games. In recent years, the martingale
theory also plays a vital role in the area of mathematical finance. This course will introduce a circle of
ideas and fundamental results of the theory of martingales and provide some applications in the discrete time
financial models.

You must have completed MATH20722 Foundations of Modern Probability or MATH20712 Random Models.

Students are not permitted to take more than one of MATH37001 or MATH47201 for credit in the same or different undergraduate year.  Students are not permitted to take MATH47201 and MATH67201 for credit in an undergraduate programme and then a postgraduate programme.

The unit aims to provide a firm grasp of a range of basic concepts and fundamental results in the theory of
martingales and to give some simple applications in the rapid developing area of financial mathematics.

On successful completion of the course students will be able to: 

• compute integrals of simple random variables with respect to a probability measure and apply the dominated, monotone convergence theorems;
• use the computational rules of conditional expectation and prove that a given stochastic process is a martingale;
• apply Doob Optional Sampling Theorem to calculate interesting probabilities concerning some practical problems, e.g. the gambler ruin problem;
• answer simple questions regarding Brownian motion and Poisson processes;
• compute brackets of simple semimartingales;
• answer simple questions about self-financing portfolios and complete markets;
• determine when a financial market is arbitrage-free;
• find out a replicating strategy for a financial claim;
• calculate the fair price of a financial claim in a financial market.

 An introduction to a circle of ideas and fundamental results of the theory of martingales, which play a vital
role in stochastic calculus and in the modern theory of Finance.

• Probability spaces, events, sigma-fields, probability measures and random variables. Integration with respect to a probability measure. Convergence theorems (dominated, monotone and Fatou). [4]
• Conditional expectations. Fair games and discrete time martingales, submartingales and supermartingales. Stopping times and the optional sampling theorem. The upcrossing inequality and the martingale convergence theorem. The Doob maximal inequality. [13]
• Continuous time martingales, Doob stopping theorem. Processes of bounded variation, semimartingales. Doob-Meyer decomposition, quadratic variation processes and brackets . Brownian motion and Poisson processes. [10]
• Discrete time random models in financial markets. Price processes, self-financing portfolio and value processes. Arbitrage opportunities and equivalent martingale measures. Completeness of the markets. Options and option pricing. [9]

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.
 

• Bingham, N. H. and Kiesel, R. (1998). Risk-Neutral Valuation. Springer.
• Williams, D. (1991). Probability with Martingales. Cambridge Univ. Press.
• Shiryaev, A. N. (1996). Probability. Springer.

End of semester examination: 3 hours, weighting 100%