## Martingales Theory for Finance

 Unit code: MATH67201 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 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.

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#### 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.

#### 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.

#### Learning outcomes

On successful completion of the course students will:

• have a good understanding of the basic concept of integration with respect to a probability measure and the basic properties of fair games;
• be able to answer basic questions on martingales;
• experience applications of stochastic processes in discrete time financial models.

#### Assessment methods

• Written exam - 100%

#### 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]

• 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.

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

#### Teaching staff

Tusheng Zhang - Unit coordinator