37363 Stochastic Processes and Financial Mathematics
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Subject handbook information prior to 2025 is available in the Archives.
Credit points: 6 cp
Result type: Grade and marks
Requisite(s): ( 37262 Mathematical Statistics OR (35252 Mathematical Statistics AND 35363 Stochastic Models))
These requisites may not apply to students in certain courses. See access conditions.
Anti-requisite(s): 35361 Stochastic Processes
Description
This subject introduces the mathematics of random processes which are used to describe and predict the behaviour of complex systems. Applications arise across a very wide range of disciplines, from finance and economics, to physics and biology. Topics include: Gaussian-Markov processes including Brownian motion; Markov chains, birth-death processes; Compound Poisson processes, Levy processes; Kalman filtering, elements of time series; diffusion processes and their application to ruin probabilities and financial modelling; Black-Scholes formula.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | Define and illustrate the terms used in probability and stochastic processes; |
|---|---|
| 2. | Discuss and demonstrate the techniques of proof used in probability and some of the mathematical derivations that are important in the theory of stochastic processes; |
| 3. | State and apply the basic limit theorems of probability; |
| 4. | Demonstrate an ability to use mathematical techniques to analyse the behaviour of various stochastic processes, especially the long-run or steady state behaviour; |
| 5. | Formulate and solve applied and theoretical problems involving probability and stochastic processes; |
| 6. | Communicate clearly knowledge of the subject matter of probability and stochastic processes and solutions to problems involving these topics. |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Demonstrate theoretical and technical knowledge of mathematical sciences including calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management. (1.1)
- Evaluate mathematical and statistical approaches to problem solving, analysis, application, and critical thinking to make mathematical arguments, and conduct experiments based on analytical, numerical, statistical, algorithms to solve new problems. (2.1)
- Work autonomously or in teams to demonstrate professional and responsible analysis of real-life problems that require application of mathematics and statistics. (3.1)
- Use succinct and accurate presentation of reasoning and conclusions to communicate mathematical solutions, and their implications, to a variety of audiences, using a variety of approaches. (5.1)
Contribution to the development of graduate attributes
The material presented in this subject and the method of presentation are linked to the following Science graduate attributes:
1. Disciplinary knowledge.
Knowledge of mathematical sciences to demonstrate depth, breadth, application, and interrelationships of relevant discipline areas.
2. Research, inquiry and critical thinking.
The ability to frame conjectures and hypotheses using a scientific approach, to test current mathematics knowledge through critical evaluation and data analyses, and to solve problems through theoretical work and/or experimental observation.
3. Professional, ethical, and social responsibility.
A capacity to work ethically and professionally using technical, practical, and collaborative mathematical skills within the context of the workplace, and apply these to meet the current and future needs of society.
5. Communication.
Effective and professional communication skills for a range of scientific audiences.
Teaching and learning strategies
Each week, this subject involves four hours of classes comprising a lecture, a tutorial and a computer lab. The lecture will be the primary vehicle for communication of the theoretical aspects of the course, although practical examples will also be introduced and solved with participation of the students. The combined tutorial and lab will be practical in nature, with students asked to solve a variety of mathematical problems both analytically and computationally. This will allow students to apply what has been learned in lectures and to gain confidence and experience in using computational software. Students will be asked to work collaboratively in small groups on the tutorial and lab problems and will be asked to demonstrate their solutions to the rest of the class. The audience will be encouraged to ask questions or provide comments about these solutions. Students will also receive feedback on their work from the tutor.
The content of this course is quite challenging and will include many concepts and techniques that will be new to most students. Each week the content of the lectures, tutorials and labs builds on that presented previously, so it is essential that students prepare for each week’s activities by familiarising themselves with the previous week’s. Specifically, for the lecture each week students must learn the material from the previous lecture. For the lab and tutorial each week students must have attempted all questions from the previous week. For these reasons, attendance at all lectures, tutorials and labs is highly recommended.
Much use will be made of Canvas, from the distribution of subject material to communication of announcements, so students will be expected to regularly log onto this system and to remain up to date with its contents.
To clarify issues arising in the subject, students are encouraged to ask questions during lectures, labs and tutorials, and to persist with these questions until resolution of the issue. Provision will also be made for weekly consultation sessions and email or phone queries will also be accepted. E-mail messages will be responded to within two working days.
Content (topics)
A wide range of practical situations, including finance, biology and engineering, can be analysed by means of stochastic processes. In particular, methods of stochastic processes are extremely important for the pricing of options in finance. By discussing such situations and the corresponding mathematical models, this subject will further develop your ability to apply stochastic methods in practice.
The subject presents a number of methods, which are among the most commonly used in practice, together with the theoretical results justifying these methods. These theoretical results are important for the successful application of stochastic methods and will further develop your ability to learn new mathematical techniques independently.
Computational experiments will be aimed at achieving deeper understanding of the Monte-Carlo simulation technique and problems arising from its software implementation.
Assessment
Assessment task 1: Weekly Lab Worksheets
| Intent: | This assessment task contributes to the development of the following graduate attributes: |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 4, 5 and 6 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1 and 5.1 |
| Type: | Exercises |
| Groupwork: | Individual |
| Weight: | 25% |
| Criteria: | Marks to be awarded based on:
|
Assessment task 2: Assignment
| Intent: | This assessment task contributes to the development of the following graduate attributes: |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 4, 5 and 6 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1 and 5.1 |
| Type: | Exercises |
| Groupwork: | Group, group assessed |
| Weight: | 25% |
| Criteria: | Marks to be awarded based on:
|
Assessment task 3: Examination
| Intent: | This assessment task contributes to the development of the following graduate attributes: |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4, 5 and 6 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 3.1 and 5.1 |
| Type: | Examination |
| Groupwork: | Individual |
| Weight: | 50% |
| Criteria: | Marks to be awarded based on:
|
Minimum requirements
In order to pass this subject, students much achieve at least 50% of the total marks available.
Required texts
Lecture notes will be provided.
Recommended texts
K.Borovkov, Elements of stochastic modelling. World Scientific Publishing Co., Inc., River Edge, NJ, 2003
Second Edition (2014), ISBN-13: 978-9814571166 ISBN-10: 9814571164.
E. Platen, D. Heath, A benchmark approach to quantitative finance. Springer-Verlag Berlin Heidelberg, Berlin, Germany, 2006, ISBN 978-3-540-26212-1.