# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Martingale Theory for Finance

Unit code MATH47201 15 Level 4 Semester 1 Department of Mathematics No

### Overview

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.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Foundations of Modern Probability MATH20722 Pre-Requisite Compulsory
Random Models MATH20712 Pre-Requisite Compulsory

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.

### Aims

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.

### Learning outcomes

On successful completion of the unit, students will be able to:

1. compute integrals of simple random vairables with respect to a probability measure and apply the dominated and monotone convergence theorems;
2. use the computational rules of conditional expectation and  prove that a given stochastic process is a martingale;
3. apply Doob's optional stopping theorem to calculate interesting probabilities concerning some practical problems, e.g. the gambler ruin problem;
4. state and apply basic properties of Brownian motion and Poisson processes;
5. compute the quadratic variation of simple semimartingales;
6. state and apply properties of self-financing strategies and complete markets, and determine when a financial market is arbitrage-free;
7. find the fair price and replicating strategices for simple financial claims.

### Syllabus

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]

### Assessment methods

Method Weight
Written exam 100%

### 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.

• 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.

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 126

### Teaching staff

Staff member Role
Sebastian Andres Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.