Unit of study_

# MATH4513: Topics in Financial Mathematics

## Overview

Securities and derivatives are the foundation of modern financial markets. The fixed-income market, for example, is the dominant sector of the global financial market where various interest-rate linked securities are traded, such as zero-coupon and coupon bonds, interest rate swaps and swaptions. This unit will investigate short-term interest rate models, the Heath-Jarrow-Morton approach to instantaneous forward rates and recently developed models of forward London Interbank Offered Rates (LIBORs) and forward swap rates. You will learn about pricing and hedging of credit derivatives, another challenging and practically important problem and become familiar with stochastic models for credit events, dependent default times and credit ratings. You will learn how to value and hedge single-name and multi-name credit derivatives such as vulnerable options, corporate bonds, credit default swaps and collateralized debt obligations. You will also learn about the most recent developments in Financial Mathematics, such as robust pricing and nonlinear evaluations. By doing this unit, you will get a solid grasp of mathematical tools used in valuation and hedging of fixed income securities, develop a broad knowledge of advanced quantitative methods related to interest rates and credit risk and you will learn to use powerful mathematical tools to address important real-world quantitative problems in the finance industry.

### Details

Academic unit Mathematics and Statistics Academic Operations MATH4513 Topics in Financial Mathematics Semester 2, 2021 Normal day Remote 6

### Enrolment rules

 Prohibitions ? None None None Students are expected to have working knowledge of Stochastic Processes, Stochastic Calculus and mathematical methods used to price options and other financial derivatives, for example as in MATH4511 or equivalent Yes

### Teaching staff and contact details

Coordinator Marek Rutkowski, marek.rutkowski@sydney.edu.au

## Assessment

Type Description Weight Due Length
Final exam (Take-home short release) Final examination
4 problem based questions
60% Formal exam period 3 hours
Outcomes assessed:
Assignment Assignment 1
Problem based questions
20% Week 07 2 weeks
Outcomes assessed:
Assignment Assignment 2
Problem based questions
20% Week 11 2 weeks
Outcomes assessed:
= Type D final exam

Detailed information for each assessment can be found on Canvas.

### Assessment criteria

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

Standard penalties

### Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

The Current Student website provides information on academic honesty, academic dishonesty, and the resources available to all students.

The University expects students and staff to act ethically and honestly and will treat all allegations of academic dishonesty or plagiarism seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic dishonesty. If such matches indicate evidence of plagiarism or other forms of dishonesty, your teacher is required to report your work for further investigation.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 An overview of the fixed-income market and interest rate derivatives. Market conventions and valuation principles for zero-coupon and coupon bonds. Lecture and tutorial (4 hr)
Week 02 Models of the short term rate: Vasicek’s model, the Cox-Ingersoll-Ross (CIR) model, the Hull and White model and the Black-Karasinski model. Lecture and tutorial (4 hr)
Week 03 Modelling of instantaneous forward rates through the Heath-Jarrow-Morton approach Lecture and tutorial (4 hr)
Week 04 Valuation of bond options and other fixed income derivatives in Gaussian, lognormal and the CIR models through the change of a numeraire method. Lecture and tutorial (4 hr)
Week 05 Modelling of forward LIBORs. Valuation and hedging of caps and floors in the LIBOR market model (LMM). Lecture and tutorial (4 hr)
Week 06 Modelling of co-terminal and co-initial forward swap rates. Valuation and hedging of swaptions in Jamshidian’s swap market model (SMM). Lecture and tutorial (4 hr)
Week 07 Vulnerable option and defaultable bonds. Structural approaches to the modelling of corporate credit risk: the Merton model of corporate debt and the Black and Cox approach. Lecture and tutorial (4 hr)
Week 08 Properties of the Poisson and Cox processes and their stochastic exponentials Lecture and tutorial (4 hr)
Week 09 Modelling of default times through hazard functions and hazard processes. Computations of conditional expectations and the study of the immersion property. Lecture and tutorial (4 hr)
Week 10 Valuation of credit default swaps (CDSs) in the hazard function and hazard process approaches Lecture and tutorial (4 hr)
Week 11 Models of conditionally independent default times and copula-based approaches to dependence of default events. Modelling of dependent default times: Jarrow and Yu model and Kusuoka’s approach. Lecture and tutorial (4 hr)
Week 12 Analysis of basket credit derivatives: valuation of first-to-default swaps, CDS indices and collateralized debt obligations (CDOs). Lecture and tutorial (4 hr)
Week 13 Exotic credit derivatives Block teaching (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

MATH4513 Topics in Financial Mathematics (course notes on Canvas)

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. Demonstrate familiarity with fundamental concepts in the area of financial markets with application to existing securities related to interest rates, such as: bonds, bond options and interest rate swaps and swaptions.​
• LO2. Develop stochastic models and solve qualitative and quantitative problems associated with the valuation and hedging of fixed income securities within the framework of short-term rate models.​
• LO3. Understand, explain and apply the principles of modelling of forward rates through different competing methods and analyse the relationships between alternative approaches​.
• LO4. Apply mathematical expertise to solve practical problems using various approaches and analyse the advantages and shortcomings of solutions obtained through different methods.​
• LO5. Examine the concept of credit risk and its impact on pricing problems for complex financial derivatives such as: corporate bonds, credit default swaps and collateralized debt obligations.​
• LO6. Analyse the issue of dependence between defaults of several credit-risky names and apply various probabilistic techniques for modelling of dependent defaults. ​
• LO7. Identify, formulate and solve original practical problems that can be addressed using mathematical techniques learnt in this unit, interpret these solutions and evaluate their implementations.​
• LO8. Demonstrate capability for independent learning using sources such as journal articles and working papers and use this information to evaluate recently developed approaches and mathematical tools.​

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.