This course gives a rigorous mathematical introduction to stochastic processes, stochastic differential equations, and their applications in finance. The first half of the course covers martingales, Poisson processes, Brownian motion, Ito integration, and stochastic differential equations driven by a Brownian motion. The second half of the course covers a range of more advanced topics to be selected by the students. This includes option pricing and investment optimisation, basics of Malliavin’s stochastic calculus of variations, Black-Scholes formula and hedging, as well as aspects of the relationship between stochastic analysis and partial differential equations. 

Stochastic Analysis with Financial Applications provides an accessible but mathematically rigorous introduction to financial mathematics and quantitative finance. The course provides a sound foundation for progress to honours and post-graduate courses in these or related areas.

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:

1. Explain the core mathematical tools and fundamental concepts of modern financial mathematics;
2. Use stochastic calculus in mathematical and financial problems, including option pricing;
3. Demonstrate capabilities for advanced mathematical reasoning, analysis and modeling linked to the theory of stochastic processes.

Research-Led Teaching

The first part of the course is a classic undergraduate course, emphasising depth of understanding, and going through the construction of Brownian motion and Ito integration, and through the resolution of stochastic differential equations, in full detail, before applying the theory to option pricing. The second part of the course, however, operates as a research seminar, emphasising breadth, and covering a large amount of advanced material in mathematics and/or finance through short presentations.

Recommended Resources

“Stochastic Calculus and Financial Applications” by J. Michael Steele

Recommended student system requirements 

ANU courses commonly use a number of online resources and activities including:

• video material, similar to YouTube, for lectures and other instruction
• two-way video conferencing for interactive learning
• email and other messaging tools for communication
• interactive web apps for formative and collaborative activities
• print and photo/scan for handwritten work
• home-based assessment.

To fully participate in ANU learning, students need:

• A computer or laptop. Mobile devices may work well but in some situations a computer/laptop may be more appropriate.
• Webcam
• Speakers and a microphone (e.g. headset)
• Reliable, stable internet connection. Broadband recommended. If using a mobile network or wi-fi then check performance is adequate.
• Suitable location with minimal interruptions and adequate privacy for classes and assessments.
• Printing, and photo/scanning equipment

For more information please see https://www.anu.edu.au/students/systems/recommended-student-system-requirements

Staff Feedback

Students are given feedback in this course in a variety of ways, matching the variety of assessment methods. This includes:

1. Oral feedback from the lecturer to the whole class regarding assignments and exams.
2. Written and oral feedback from the demonstrator on assignments.
3. Oral personal feedback from the lecturer through appointments.
4. Direct feedback on oral presentations just after the student’s talks.
5. Written feedback by the lecturer on the essay.

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Conditional expectation
2	Discrete time martingales
3	Dyadic martingales
4	Brownian motion and continuous time martingales
5	Ito integration
6	Ito integration 2	Assignment 1 due
7	Ito's formula and chain rule
8	Stochastic differential equations
9	Option pricing	Assignment 2 due
10	Practice and exam preparation	exam
11	seminar	oral presentation
12	seminar	oral presentation

Assessment Summary

Assessment task	Value	Due Date	Return of assessment	Learning Outcomes
Assignment 1	12 %	01/09/2021	08/09/2021	2,3
Assignment 2	12 %	29/09/2021	06/10/2021	2,3
Exam (take-home)	25 %	*	*	2,3
Oral presentation	25 %	*	*	1,2,3
Essay	25 %	05/11/2021	12/11/2021	1,2,3

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Policy and General Information
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure

Assessment Requirements

The ANU is using 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. For additional information regarding Turnitin please visit the ANU Online website Students may choose not to submit assessment items through Turnitin. In this instance you will be required to submit, alongside the assessment item itself, hard copies of all references included in the assessment item.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Assessment Task 1

Assignment 1

Prove several statements similar to the statements discussed in lectures and workshops. Hand in a complete typed or scanned personal solution.

Value: 12.5%

Assessment Task 2

Assignment 2

Prove several statements similar to the statements discussed in lectures and workshops. Hand in a complete typed or scanned personal solution.

Value: 12.5%

Assessment Task 3

Exam (take-home)

In 3 hours, prove several statements similar to the statements discussed in lectures and workshops. Hand in a complete scanned personal solution.

The exam will take place in week 10. Check course Wattle site for specific due date.

Assessment Task 4

Oral presentation

Produce a 4-7 minute video summarising a chapter (to be agreed upon with the lecturer) of an advanced textbook. The goal of the presentation is to maximise the learning of the audience (the rest of the class).

The oral presentations will take place in weeks 11-12.

Assessment Task 5

Essay

Write a 5 page summary of the material used for your oral presentation. The goal of this task is to demonstrate your understanding of the material to the lecturer.

Academic Integrity

Academic integrity is a core part of our culture as a community of scholars. At its heart, academic integrity is about behaving ethically. This means that all members of the community commit to honest and responsible scholarly practice and to upholding these values with respect and fairness. The Australian National University commits to embedding the values of academic integrity in our teaching and learning. We ensure that all members of our community understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with. The University has policies and procedures in place to promote academic integrity and manage academic misconduct. Visit the following Academic honesty & plagiarism website for more information about academic integrity and what the ANU considers academic misconduct. The ANU offers a number of services to assist students with their assignments, examinations, and other learning activities. The Academic Skills and Learning Centre offers a number of workshops and seminars that you may find useful for your studies.

Online Submission

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.

Hardcopy Submission

For some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.

Late Submission

Late submission of assessment tasks without an extension are penalised at the rate of 5% of the possible marks available per working day or part thereof. Late submission of assessment tasks is not accepted after 10 working days after the due date, or on or after the date specified in the course outline for the return of the assessment item. 

Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

Returning Assignments

Marked pieces of assessment are returned to the student via the Wattle site.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure The Course Convener may grant extensions for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

Resubmission of assignments is possible up to the submission deadline, but not after.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information. In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service — including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy. If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes. Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents undergraduate and ANU College students
• PARSA supports and represents postgraduate and research students