Mathematical Methods of Finance
This module is not available to undergraduate students who have taken Stochastic Calculus.
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.
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 of a parabolic partial differential equation and solutions to boundary/initial/final value problems, within the class of equations arising in typical problems in finance.
- Fundamentals of probability: probability space and measure, algebras and sigma-algebras, random variables, probability distribution, expectation, variance, covariance, correlation.
- Lebesgue and Stieljes integrals, definition and basic properties.
- Radon-Nikodym theorem (without proof).
- Filtrations, partitions, their relationship, applications for modelling flow of information.
- Conditional expectation, conditional probability, dependence and independence.
- Stochastic processes in discrete time; random walk.
- Adapted processes, predictable processes.
- Martingales, submartingales, supermartingales.
- Central Limit Theorem and its financial application.
- Definition and construction of Brownian motion, properties of Brownian motion.
- Informal overview of Ito calculus: stochastic integrals, Ito formula.
- Quadratic variation and Ito processes.
- Informal overview of applications of stochastic processes and Ito calculus in finance (time allowing).
- 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.
- I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer 1991.
- Autumn Term
- Three one-hour lectures and one one-hour seminar per week for one term.
- Individual feedback and advice on assessed coursework will be offered to students during scheduled office hours.
Coursework (20%) and unseen written examination (80%).
Please note that this module forms part of the MSc in Mathematical Finance; please see the relevant Handbook at
http://maths.york.ac.uk/www/ForPostgrads for details on assessment rules, including penalties for late submission of coursework.
Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.
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