Mathematical Methods of Finance - MAT00020M

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Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23
B	Spring Term 2022-23

Module aims

The topics covered are selected because of their importance in quantitative finance theory and practice. Probability theory and stochastic processes provide the language in which to express and solve mathematical problems in finance due to the inherent randomness of asset prices. The introduction of more advanced tools will be preceded by a brief review of basic probability theory with particular focus on conditional expectation. Then the module will proceed to present the theory of martingales and the study of three basic stochastic processes in finance: random walks, Brownian motion, and the Poisson process. An informal overview of Ito stochastic calculus will be given and first financial applications indicated. The material will be illustrated by numerous examples and computer-generated demonstrations. By the end of this module students are expected to achieve a sufficient level of competence in selected mathematical methods and techniques to facilitate further study of Mathematical Finance.

Module learning outcomes

At the end of the module you should be able to:

use the language and tools of probability theory with confidence in the context of financial models and applications.

acquire an understanding of stochastic processes in discrete and continuous time and be familiar with the basic examples and properties of such processes appearing in financial modelling.

recognise the central role of Ito stochastic calculus for mathematical models in finance, and show familiarity with the basic notions and tools of stochastic calculus, at an informal level.

understand the notions and properties of martingale theory, their applications in stochastic calculus and relevance in quantitative finance.

Module content

Indicative Content:

1. Fundamentals of probability: probability space and measure, algebras and sigma-algebras, random variables, probability distribution, expectation, variance, covariance, correlation.
2. Lebesgue and Stieljes integrals, definition and basic properties.
3. Radon-Nikodym theorem (without proof).
4. Filtrations, partitions, their relationship, applications for modelling flow of information.
5. Conditional expectation, conditional probability, dependence and independence.
6. Stochastic processes in discrete time; random walk.
7. Adapted processes, predictable processes.
8. Martingales, submartingales, supermartingales
9. Central Limit Theorem and its financial application
10. Definition and construction of Brownian motion, properties of Brownian motion.
11. Informal overview of Ito calculus: stochastic integrals, Ito formula, Ito processes.

Informal overview of applications of stochastic processes and Ito calculus in finance (time allowing).

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

M. Capinski and T. Zastawniak, Mathematics for Finance: An Introduction to Financial Engineering, Springer 2003.

M. Capinski and T. Zastawniak, Probability Through Problems, Springer 2001.

Z. Brzezniak and T. Zastawniak, Basic Stochastic Processes, Springer 1999 http://libcatalogue.york.ac.uk/F/?func=direct&doc_number=001395188

I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer 1991.