This course gives a rigorous mathematical introduction to stochastic processes and their calculus, stochastic differential equations, and their applications in finance. The first half of the course covers martingales, Brownian motion, Ito integration, stochastic differential equations driven by a Brownian motion, and their applications to pricing European options using the Black-Scholes model. The second half of the course covers a range of more advanced topics to be selected by the students. This includes change of measure theory, its application to pricing complex options, relationship between stochastic analysis and partial differential equations (and its computational uses), basics of Malliavin stochastic calculus of variations and its application in risk management, as well as topics in statistics (such as model fitting), economics (such as interest rates modelling) or other disciplines (such as generative AI in computer science, quantum field theory in physics, or insect behaviour in biology).
The first part of the course aims to develop a common perspective and language, anchored in mathematical analysis, and building on a large part of the undergraduate curriculum in Mathematics (calculus, differential equations, analysis in metric spaces, measure theory, probability theory).
In the second part of the course, students develop an ability to use their mathematical knowledge in collaborative transdisciplinary complex problem solving. This is done by working in groups on projects that are too complex to be fully completed in a short period of time, and with collaborators who have some common language, but different backgrounds (typically some collaborators with a primary training in mathematics, and some with a primary training in finance or economics). The projects highlight the different modes of communication and value systems of its potential stakeholders, including academics from different disciplines, as well as practitioners from different industries (e.g. what makes an argument convincing to a mathematician vs an economist, and what kind of solutions need to be achieved in investment banking vs in government).
The course operates as a flipped classroom, with technical content delivered through videos and lecture notes, and in person activities focusing on discussions. These discussions aim to develop communication skills as well as critical thinking skills. They involve technical discussions of the key parts of proofs and the nature of assumptions in theorems. They also involve historical and societal perspectives on the development of the ideas expressed through the theorems and their proofs, as well as their use in complex problems.
The course provides a sound foundation for progress to honours and post-graduate courses in probability, analysis, or differential equations. It also provides both technical and communication training for working in a range of professions (such as quantitative analyst or data scientist in government, or in finance, insurance, or information technology industries).
Note: This is an Honours Pathway Course. It continues the development of sophisticated mathematical techniques and their application begun in MATH3029 or MATH3320.
Learning Outcomes
Upon successful completion, students will have the knowledge and skills to:
- Explain the core mathematical tools and fundamental concepts of stochastic analysis, and their meaning in financial models;
- Use stochastic calculus in mathematical and financial problems, including option pricing;
- Demonstrate capabilities for advanced mathematical reasoning, analysis, and modeling linked to the theory of stochastic processes;
- Combine mathematical expertise with effective communication skills to work in a team with diverse background;
- Incorporate their mathematical perspective with other perspectives to collaboratively solve complex problems taking all perspectives and objectives into account.
Indicative Assessment
- Regular written assignments (25) [LO 1,2,3]
- Mid-semester examination (25) [LO 1,2,3]
- Small group work including pre-recorded oral presentation, large group discussion, and quizzes on rapid information retention (25) [LO 1,2,3,4,5]
- Final essay as a reflection letter to the lecturer, demonstrating depth of thought process in approaching complex problems (25) [LO 1,2,3,5]
The ANU uses Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. While the use of Turnitin is not mandatory, the ANU highly recommends Turnitin is used by both teaching staff and students. For additional information regarding Turnitin please visit the ANU Online website.
Workload
The expected workload will consist of approximately 130 hours throughout the semester. The course operates as a flipped classroom. This means that students first learn the content before meeting with the lecturer, and then they use the in-person class time to reach a deep level of understanding. In practice, this means the expected workload is:
• Approximately 82 hours of self-directed study which will include:
- watching approximately 33 pre-recorded videos and reading lecture notes covering the course content, to prepare for the in-person activities in Weeks 1 to 10,
- preparation for assessment tasks, and
- watching through other students' oral presentation videos in preparation for the discussions of them in Weeks 11 and 12;
• Face-to face component which will consist of approximately 48 hours of scheduled in-person activities during the semester. These in-person activities will include a 1 hour workshop most weeks plus 3 hours per week of other in person activities (detail in the class summary).
Inherent Requirements
No specific inherent requirements have been identified in this course.
Requisite and Incompatibility
Prescribed Texts
There is no prescribed textbook. Extensive lecture notes are provided.
Preliminary Reading
"Stochastic calculus and financial applications" by M. Steele (ISBN 978-1-4684-9305-4) is highly recommended.
Fees
Tuition fees are for the academic year indicated at the top of the page.
Commonwealth Support (CSP) Students
If you have been offered a Commonwealth supported place, your fees are set by the Australian Government for each course. At ANU 1 EFTSL is 48 units (normally 8 x 6-unit courses). More information about your student contribution amount for each course at Fees.
- Student Contribution Band:
- 1
- Unit value:
- 6 units
If you are a domestic graduate coursework student with a Domestic Tuition Fee (DTF) place or international student you will be required to pay course tuition fees (see below). Course tuition fees are indexed annually. Further information for domestic and international students about tuition and other fees can be found at Fees.
Where there is a unit range displayed for this course, not all unit options below may be available.
| Units | EFTSL |
|---|---|
| 6.00 | 0.12500 |
Offerings, Dates and Class Summary Links
ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities / tutorials so they can better plan their time. Find out more on the Timetable webpage.