Course coordinator
Dr Kazutoshi Yamazaki
Mondays 12:30pm-1:30pm in my office 67-744 and on Zoom (link to be provided on the Announcement page)
Course overview
- Study period
- Semester 1, 2025 (24/02/2025 - 21/06/2025)
- Study level
- Undergraduate
- Location
- St Lucia
- Attendance mode
- In Person
- Units
- 2
- Administrative campus
- St Lucia
- Coordinating unit
- Mathematics & Physics School
Topics from financial calculus including financial derivatives & arbitrage, asset prices, price dynamics, continuous-time hedging, Brownian motion, Martingales, stochastic integration, solving stochastic differential equations & stochastic control.
The course introduces the mathematics of continuous-time derivative security pricing models. ﾠWe first use the binomial model to introduce theﾠrisk-neutral approach to derivative security pricing. ﾠA substantial part of the course is then devoted to developing the mathematics of Brownian motion,ﾠstochastic calculus, stochastic differential equations, and partial differential equations (PDE), which is needed forﾠcontinuous-time pricingﾠmodels. The mainﾠgoal is toﾠdevelop the two mainﾠapproaches to deriving continuous-time pricing models in the standard Black-Scholes market model, namely the PDEﾠand the risk-neutral approaches. Weﾠinvestigateﾠthe link between these approaches via the Feynman-Kac theory, and finishﾠwith presenting basic short-rate models of the yield curve.
In the School of Mathematics and Physics we are committed to creating an inclusive and empowering learning environment for all students. We value and respect the diverse range of experiences our students bring to their education, and we believe that this diversity is crucial for fostering a rich culture of knowledge sharing and meaningful exploration. We hold both students and staff accountable for actively contributing to the establishment of a respectful and supportive learning environment.
Bullying, harassment, and discrimination in any form are strictly against our principles and againstﾠUQ Policy,ﾠand will not be tolerated. We have developed aﾠsuite of resourcesﾠto assist you in recognising, reporting, and addressing such behaviour. If you have any concerns about your experience in this course, we encourage you to tell a member of the course teaching team, or alternatively contact an SMP Classroom Inclusivity Champion (see Blackboard for contact details). Our Inclusivity Champions are here to listen, to understand your concerns, and to explore potential actions that can be taken to resolve them. Your well-being and a positive learning atmosphere are of utmost importance to us.
Course requirements
Assumed background
This course assumes familiarity with calculus and linear algebra atﾠthe level of MATH2000/2001. ﾠAn understanding of probability and stochastic processes at the level ofﾠSTAT3004 is also assumed. ﾠConcepts such as probability space, random variables, expected value,ﾠmoment generating and characteristic functions,ﾠconditional expectation, and stochastic processes will oftenﾠ be discussed without detailed definition.ﾠﾠ Knowledge of finance atﾠthe level of MATH3090 would also be useful.ﾠ It is theﾠresponsibility of the student to fill in any gaps in their assumed knowledge. ﾠThe student may need to undertake background reading in orderﾠ to understand the lecture material and answer questions in learning activities.
Prerequisites
You'll need to complete the following courses before enrolling in this one:
STAT3004
Recommended prerequisites
We recommend completing the following courses before enrolling in this one:
MATH3090
Incompatible
You can't enrol in this course if you've already completed the following:
MATH7091 (co-taught, last offered 2022)
Course contact
Course staff
Timetable
The timetable for this course is available on the UQ Public Timetable.
Additional timetable information
All classes will be conducted on campus at the times and location advertised in your personal timetable.
Important: If you are ill, then do not attend any classes in person. Alternative arrangements can be organised – consult Blackboard for details.
Aims and outcomes
MATH4091 is primarily concerned with the mathematics of continuous-time derivative securityﾠpricing models. It moves from theﾠbinomialﾠmodel toﾠmore sophisticated continuous-time models driven by Brownian motion.ﾠﾠAﾠmajor portion of the course is devoted to developing the mathematical backgroundﾠneeded in quantitative finance such as martingales and Brownian motion,ﾠstochastic calculus,ﾠstochastic differential equations, and partial differential equations. ﾠThe main aimﾠof the course is to deriveﾠthe Black-Scholes option pricing model viaﾠthe two standard approachesﾠin quantitative finance: ﾠthe partial differential equation (PDE) and the risk-neutral approaches. ﾠThe course also investigates the Feynman-Kac theory and basic short-rate models of the yield curve. Finally, the course also focuses on solving a range of practical problems arising in financial mathematics via stochastic calculus tools.
Learning outcomes
After successfully completing this course you should be able to:
LO1.
Define and use stochastic processes in financial mathematics, in particular martingales and Brownian motion.
LO2.
Derive, understand, and work with the properties of the Ito integrals. Understand and work with basic notions surrounding the concept of an Ito process.
LO3.
Explain with rigour the key notions of arbitrage, risk-neutral or equivalent martingale measure, market completeness, and the fundamental theorems of asset pricing.
LO4.
Analyse the link (via the Feynman-Kac theory) and the differences between the Partial Differential Equation and the risk-neutral pricing approaches for financial contracts.
LO5.
Use the Partial Differential Equation and the risk-neutral pricing approaches to European options in the Black-Scholes model.
LO6.
Construct basic short-rate models of the yield curve, and explain their key properties, and formulate a range of related pricing problems.
LO7.
Apply Ito stochastic calculus tools to solve a range of problems arising in financial mathematics.
Assessment
Assessment summary
| Category | Assessment task | Weight | Due date |
|---|---|---|---|
| Computer Code, Tutorial/ Problem Set | Assignment 1 | 20% |
3/04/2025 1:00 pm |
| Computer Code, Tutorial/ Problem Set | Assignment 2 | 20% |
28/04/2025 1:00 pm |
| Computer Code, Tutorial/ Problem Set | Assignment 3 | 20% |
26/05/2025 1:00 pm |
| Examination |
Final Exam
|
40% |
End of Semester Exam Period 7/06/2025 - 21/06/2025 |
A hurdle is an assessment requirement that must be satisfied in order to receive a specific grade for the course. Check the assessment details for more information about hurdle requirements.
Assessment details
Assignment 1
- Mode
- Written
- Category
- Computer Code, Tutorial/ Problem Set
- Weight
- 20%
- Due date
3/04/2025 1:00 pm
- Learning outcomes
- L01
Task description
A list of questions and problems to be answered and solved.
Submission guidelines
An electronic copy of your assignment solutions must be uploaded as one PDF document to the Blackboard site by the due date.
Deferral or extension
You may be able to apply for an extension.
The maximum extension allowed is 7 days. Extensions are given in multiples of 24 hours.
A maximum of seven calendar days is allowed for an extension because the solutions need to be released.
See ADDITIONAL ASSESSMENT INFORMATION for extension/deferral information relating to this assessment item.
Late submission
A penalty of 10% of the maximum possible mark will be deducted per 24 hours from time submission is due for up to 7 days. After 7 days, you will receive a mark of 0.
Assignment 2
- Mode
- Written
- Category
- Computer Code, Tutorial/ Problem Set
- Weight
- 20%
- Due date
28/04/2025 1:00 pm
- Learning outcomes
- L02, L03, L04, L05
Task description
A list of questions and problems to be answered and solved.
Submission guidelines
An electronic copy of your assignment solutions must be uploaded as one PDF document to the Blackboard site by the due date.
Deferral or extension
You may be able to apply for an extension.
The maximum extension allowed is 7 days. Extensions are given in multiples of 24 hours.
A maximum of seven calendar days is allowed for an extension because the solutions need to be released.
See ADDITIONAL ASSESSMENT INFORMATION for extension/deferral information relating to this assessment item.
Late submission
A penalty of 10% of the maximum possible mark will be deducted per 24 hours from time submission is due for up to 7 days. After 7 days, you will receive a mark of 0.
Assignment 3
- Mode
- Written
- Category
- Computer Code, Tutorial/ Problem Set
- Weight
- 20%
- Due date
26/05/2025 1:00 pm
- Learning outcomes
- L04, L05, L06, L07
Task description
A list of questions and problems to be answered and solved.
Submission guidelines
An electronic copy of your assignment solutions must be uploaded as one PDF document to the Blackboard site by the due date.
Deferral or extension
You may be able to apply for an extension.
The maximum extension allowed is 7 days. Extensions are given in multiples of 24 hours.
A maximum of seven calendar days is allowed for an extension because the solutions need to be released.
See ADDITIONAL ASSESSMENT INFORMATION for extension/deferral information relating to this assessment item.
Late submission
A penalty of 10% of the maximum possible mark will be deducted per 24 hours from time submission is due for up to 7 days. After 7 days, you will receive a mark of 0.
Final Exam
- Hurdle
- Mode
- Written
- Category
- Examination
- Weight
- 40%
- Due date
End of Semester Exam Period
7/06/2025 - 21/06/2025
- Learning outcomes
- L01, L02, L03, L04, L05, L06, L07
Task description
The final examination will test all topics presented in the course.
The examination will be closed-book and invigilated on-campus. Alternative arrangements will be advised on Blackboard should the campus be closed for any reason.
Hurdle requirements
Please see ADDITIONAL COURSE GRADING INFORMATIONExam details
| Planning time | 10 minutes |
|---|---|
| Duration | 120 minutes |
| Calculator options | (In person) Casio FX82 series only or UQ approved and labelled calculator |
| Open/closed book | Closed Book examination - specified written materials permitted |
| Materials | One A4 sheet of handwritten notes, double sided, is permitted |
| Exam platform | Paper based |
| Invigilation | Invigilated in person |
Submission guidelines
Deferral or extension
You may be able to defer this exam.
See ADDITIONAL ASSESSMENT INFORMATION for extension/deferral information relating to this assessment item.
Course grading
Full criteria for each grade is available in the Assessment Procedure.
| Grade | Description |
|---|---|
| 1 (Low Fail) |
Absence of evidence of achievement of course learning outcomes. Course grade description: Student will achieve a final mark of at least 1% and less than 20% by demonstrating a poor knowledge/understandingﾠof the basic concepts in the course material. |
| 2 (Fail) |
Minimal evidence of achievement of course learning outcomes. Course grade description: Students must achieve a final mark of at least 20% and less than 45% by demonstrating some knowledge of the basic concepts of the stochastic calculus and its applications to finance as discussed in the course and its associated materials. This includes attempts at expressing their deductions and explanations as well as attempts to answer a few questions accurately. |
| 3 (Marginal Fail) |
Demonstrated evidence of developing achievement of course learning outcomes Course grade description: Students must achieve a final mark of at least 45% and less than 50% OR have a final mark greater than 50% but scored less than 40% in the final exam by demonstrating some knowledge of the basic concepts of stochastic calculus and its applications to finance as discussed in the course and its associated materials. This includes occasionally expressing their deductions and explanations clearly, the use of a few appropriate and efficient mathematical techniques and attempts to answer a few questions and tasks accurately and with appropriate justification. They will have demonstrated knowledge of techniques used to solve problems. |
| 4 (Pass) |
Demonstrated evidence of functional achievement of course learning outcomes. Course grade description: Students must achieve a final mark of at least 50% and less than 65% by demonstrating an understanding of the basic concepts of stochastic calculus and its applications to finance as discussed in the course and its associated materials. This includes occasionally expressing their deductions and explanations clearly, the occasional use of appropriate and efficient mathematical techniques and accurate answers to a few questions and tasks with appropriate justification. They will have demonstrated knowledge of techniques used to solve problems and successfully applied this knowledge in some cases. |
| 5 (Credit) |
Demonstrated evidence of proficient achievement of course learning outcomes. Course grade description: Students must achieve a final mark of at least 65% and less than 75% by demonstrating an adequate understanding of the course theory and its applications to finance as discussed in the course and its associated materials. This includes clear expression of some of their deductions and explanations, the use of appropriate and efficient mathematical techniques in some situations and accurate answers to some questions and tasks with appropriate justification. They will have demonstrated knowledge of techniques used to solve problems and successfully applied this knowledge in several cases. |
| 6 (Distinction) |
Demonstrated evidence of advanced achievement of course learning outcomes. Course grade description: Students must achieve a final mark of at least 75% and less than 85% by demonstrating a comprehensive understanding of the theory of stochastic calculus and its applications to finance as discussed in the course and its associated materials. This includes clear expression of most of their deductions and explanations, the general use of appropriate and efficient mathematical techniques and accurate answers to most questions and tasks with appropriate justification. They will have demonstrated knowledge of and be able to apply most of the results from partial differential equations to substantially solve both theoretical and practical problems. |
| 7 (High Distinction) |
Demonstrated evidence of exceptional achievement of course learning outcomes. Course grade description: Students must achieve a final markﾠof at least 85%ﾠby demonstrating an excellent understanding of all the theory of stochastic calculus and its applications to finance as discussed in the course and its associated materials. This includes clear expression of nearly all their deductions and explanations, the use of appropriate and efficient mathematical techniques and accurate answers to nearly all questions and tasks with appropriate justification. They will have demonstrated knowledge of and be able to apply all results from partial differential equations to completely solve both theoretical and practical problems. |
Additional course grading information
HURDLE: ﾠStudents must achieve a mark of 40% or more on the final exam to pass the subject.
ﾠ
Supplementary assessment
Supplementary assessment is available for this course.
Should you fail a course with a grade of 3, you may be eligible for supplementary assessment. Refer to my.UQ for information on supplementary assessment and how to apply.
Supplementary assessment provides an additional opportunity to demonstrate you have achieved all the required learning outcomes for a course.
If you apply and are granted supplementary assessment, the type of supplementary assessment set will consider which learning outcome(s) have not been met.
Supplementary assessment in this course will be a 2-hour examination similar in style to the end-of-semester examination. To receive a passing grade of 3S4, you must obtain a mark of 50% or more on the supplementary assessment.
Additional assessment information
Artificial Intelligence
Assessment tasks in this course evaluate students' abilities, skills and knowledge without the aid of generative Artificial Intelligence (AI) or Machine Translation (MT). Students are advised that the use of AI or MT technologies to develop responses is strictly prohibited and may constitute student misconduct under the Student Code of Conduct.
Applications for Extensions to Assessment Due Dates
Extension requests are submitted online via my.UQ – applying for an extension. Extension requests received in any other way will not be approved. Additional details associated with extension requests, including acceptable and unacceptable reasons, may be found at my.UQ.
Please note:
- Requests for an extension to an assessment due date must be submitted through your my.UQ portal and you must provide documentation of your circumstances, as soon as it becomes evident that an extension is needed. Your application must be submitted on or before the assessment item's due date and time.
- Applications for extension can take time to be processed so you should continue to work on your assessment item while awaiting a decision. We recommend that you submit any completed work by the due date, and this will be marked if your application is not approved. Should your application be approved, then you will be able to resubmit by the agreed revised due date.
- If an extension is approved, you will be notified via your my.UQ portal and the new date and time for submission provided. It is important that you check the revised date as it may differ from the date that you requested.
- If the basis of the application is a medical condition, applications should be accompanied by a medical certificate dated prior to the assignment due date. If you are unable to provide documentation to support your application by the due date and time you must still submit your application on time and attach a written statement (Word document) outlining why you cannot provide the documentation. You must then upload the documentation to the portal within 24 hours.
- If an extension is being sought on the basis of exceptional circumstances, it must be accompanied by supporting documentation (eg. Statutory declaration).
- For extensions based on a SAP you may be granted a maximum of 7 days (if no earlier maximum date applies). See the Extension or Deferral availability section of each assessment for details. Your SAP is all that is required as documentation to support your application. However, additional extension requests for the assessment item will require the submission of additional supporting documentation e.g., a medical certificate. All extension requests must be received by the assessment due date and time.
- Students may be asked to submit evidence of work completed to date. Lack of adequate progress on your assessment item may result in an extension being denied.
- If you have been ill or unable to attend class for more than 14 days, you are advised to carefully consider whether you are capable of successfully completing your courses this semester. You might be eligible to withdraw without academic penalty - seek advice from the Faculty that administers your program.
- There are no provisions for exemption from an assessment item within UQ rules. If you are unable to submit an assessment piece then, under special circumstances, you may be granted an exemption, but may be required to submit alternative assessment to ensure all learning outcomes are met.
Applications to defer an exam
In certain circumstances you can apply to take a deferred examination for in-semester and end-of-semester exams. You'll need to demonstrate through supporting documentation how unavoidable circumstances prevented you from sitting your exam. If you can’t, you can apply for a one-off discretionary deferred exam.
Deferred Exam requests are submitted online via mySi-net. Requests received in any other way will not be approved. Additional details associated with deferred examinations, including acceptable and unacceptable reasons may be found at my.UQ.
Please note:
- Applications can be submitted no later than 5 calendar days after the date of the original exam.
- There are no provisions to defer a deferred exam. You need to be available to sit your deferred examination.
- Your deferred examination request(s) must have a status of "submitted" in mySI-net to be assessed.
- All applications for deferred in-semester examinations are assessed by the relevant school. Applications for deferred end-of-semester examinations are assessed by the Academic Services Division.
- You’ll receive an email to your student email account when the status of your application is updated.
- If you have a medical condition, mental health condition or disability and require alternative arrangements for your deferred exam you’ll need to complete the online alternative exam arrangements through my.UQ. This is in addition to your deferred examinations request. You need to submit this request on the same day as your request for a deferred exam or supplementary assessment. Contact Student Services if you need assistance completing your alternative exam arrangements request.
Learning resources
You'll need the following resources to successfully complete the course. We've indicated below if you need a personal copy of the reading materials or your own item.
Library resources
Find the required and recommended resources for this course on the UQ Library website.
Additional learning resources information
Students gain most benefit from lectures if they have completed recommended reading (to be assigned weekly), listen carefully, think critically, and then ask questions if they do not understand.
Students gain most benefit from tutorials if they haveﾠread the relevant section of the notes first andﾠare prepared to contribute to the solution of the problems.
Students should regularly check theﾠBlackboard website https://learn.uq.edu.au/ for course materials and additional resources.
Learning activities
The learning activities for this course are outlined below. Learn more about the learning outcomes that apply to this course.
Filter activity type by
Please select
| Learning period | Activity type | Topic |
|---|---|---|
Multiple weeks From Week 1 To Week 13 |
Lecture |
Stochastic Calculus and Financial Mathematics Introduce students to the mathematical tools used to derive continuous-time derivative security pricing models in quantitative finance. Learning outcomes: L01, L02, L03, L04, L05, L06, L07 |
Multiple weeks From Week 2 To Week 13 |
Practical |
Exercise Questions Exercises, problems, and questions to consolidate knowledge and understanding of the material. Practicals start the second week of class. Learning outcomes: L01, L02, L03, L04, L05, L06, L07 |
Policies and procedures
University policies and procedures apply to all aspects of student life. As a UQ student, you must comply with University-wide and program-specific requirements, including the:
- Student Code of Conduct Policy
- Student Integrity and Misconduct Policy and Procedure
- Assessment Procedure
- Examinations Procedure
- Reasonable Adjustments - Students Policy and Procedure
Learn more about UQ policies on my.UQ and the Policy and Procedure Library.