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Financial Mathematics Program

Introduction

Research in Financial Mathematics is obviously interdisciplinary, but it primarily hinges on sophisticated mathematical tools such as: theory of probability, theory of martingales, Ito's stochastic calculus, stochastic differential equations, stochastic backward differential equations, partial differential equations, variational inequalities, optimisation methods, stochastic optimal stopping, stochastic optimal control, Dynkin's games, stochastic differential games, statistics of stochastic processes, time series and, last but not least, various computational methods used in financial applications.

The School offers a variety of specialised units of study in the broad area of Financial Mathematics and Statistics covering most of the abovementioned areas of mathematical knowledge and ranging from introductory units for undergraduates to advanced units for honours and masters students. We thus give you an opportunity to complete high-quality BSc, BAS (honours and advanced coursework) and masters teaching programs capable of competing with analogous programs at other universities in Australia. Our graduates with specifically strong mathematical and statistics backgrounds are in high demand by the finance industry in Australia and overseas.

Currently Available Financial Mathematics Units

We offer the following undergraduate, fourth year (honours) and fifth year (masters) units of study.

• Year 2

STAT2011 Probability and Estimation Theory and STAT2911 Probability and Statistical Models

The unit STAT2011 provides an introduction to univariate techniques in data analysis and the most common statistical distributions that are used to model patterns of variability. Common discrete random variable models, like the binomial, Poisson and geometric, and continuous models, including the normal and exponential, will be studied. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The unit will have weekly computer classes where candidates will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method. STAT2911 is an advanced version of STAT2011 with an emphasis on the mathematical techniques used to manipulate random variables and probability models. Common random variables including the Poisson, normal, beta and gamma families are introduced. Probability generating functions and convolution methods are used to understand the behaviour of sums of random variables. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The unit has weekly computer classes where students will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

MATH2070 Optimisation and Financial Mathematics and MATH2970 Optimisation and Financial Mathematics (advanced)

Problems in industry and commerce often involve maximising profits or minimising costs subject to constraints arising from resource limitations. The first part of this intermediate unit looks at programming problems and their solution using the simplex algorithm, nonlinear optimisation and the Karush-Kuhn-Tucker conditions. The second part includes an introduction to some problems and techniques from Financial Mathematics, such as: the pricing of riskless bonds and risky securities, the modern portfolio theory (MPT) due to H. Markowitz, the capital asset pricing model (CAPM), and the dynamic programming principle (DPP) due to R. Bellman. Some understanding of probability theory including distributions and expectations is required in this part. Theory developed in lectures will be complemented by computer laboratory sessions using MATLAB. The prerequisites for Optimisation and Financial Mathematics are junior mathematics units (taken at either mainstream or advanced level) covering differential calculus, linear algebra, integral calculus, mathematical modelling and statistics. The unit is offered at both advanced (MATH2970) and mainstream (MATH2070) levels.

DATA2002 Data Analytics: Learning from Data and DATA2902 Data Analytics: Learning from Data (advanced)

This is an intermediate level unit in statistics and data sciences, focusing on learning data analytic skills for a wide range of problems and data. How should the Australian government measure and report employment and unemployment? Can we tell the difference between decaffeinated and regular coffee ? In this course, you will learn how to ingest, combine and summarize data from a variety of data models which are typically encountered in data science projects as well as reinforcing their programming skills through experience with statistical programming language. You will also be exposed to the concept of statistical machine learning and develop the skill to analyse various types of data in order to answer a scientific question. From this unit, you will develop knowledge and skills that will enable you to embrace data analytic challenges stemming from everyday problems. The unit is offered at both advanced (DATA2902) and mainstream (DATA2002) levels.

• Year 3

STAT3021 Stochastic Processes and STAT3921 Stochastic Processes (advanced)

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. After setting up basic elements of stochastic processes, such as time, state, increments, stationarity and Markovian property, this unit develops important properties and limit theorems of discrete-time Markov chain and branching processes. You will then establish key results for the Poisson process and continuous-time Markov chains, such as the memoryless property, super positioning, thinning, Kolmogorov's equations and limiting probabilities. Various illustrative examples are provided throughout the unit to demonstrate how stochastic processes can be applied in modeling and analyzing problems of practical interest. By completing this unit, you will develop the essential basis for further studies, such as stochastic calculus, stochastic differential equations, stochastic control and financial mathematics. This unit is offered at both advanced and mainstream levels.

MATH3075 Financial Derivatives and MATH3975 Financial Derivatives (advanced)

This senior year unit focuses on arbitrage-free pricing of modern financial derivatives, such as equity options of either a European or an American style. The theory of arbitrage-free pricing hinges on the concept of perfect hedging (or superhedging) through a dynamic trading in primary securities. The main mathematical tools taught in this unit are: conditioning with respect to a filtration, the concept of a discrete-time martingale originated by J. L. Doob, an equivalent change of a probability measure through the Radon-Nikodym density, which leads to the risk-neutral valuation formula, and solution to the optimal stopping problem through the backward induction based on Bellman's dynamic programming principle. The unit provides a thorough study of discrete-time market models, such as the Cox-Ross-Rubinstein (CRR) binomial model of the stock price. It also gives an introduction to the celebrated continuous-time Black-Scholes model, which is based on the Wiener process (Brownian motion) and shows how the continuous-time Black-Scholes model can be approximated by a sequence of discrete-time CRR models. This unit is offered at both advanced (MATH3975) and mainstream (MATH3075) levels.

FMAT3888 Projects in Financial Mathematics

Mathematics and Statistics are powerful tools in finance and, more generally, in the world at large. To really experience the power of mathematics and statistics at work, students need to identify and explore interdisciplinary links. Engagement with other disciplines also provides essential foundational skills for using mathematical and statistical ideas in financial contexts and in the world beyond. In the first part of this unit, you will commence by working on a group project in an area of Financial Mathematics or Statistics. From the disciplinary project, you will acquire skills of teamwork, research, writing and project management, as well as disciplinary knowledge in the area of computational finance. In the second part, you will have the opportunity to apply your mathematical knowledge to identify and solve real-world problems and communicate your findings through an interdisciplinary report and oral presentation.

• Year 4

STAT4528 Probability and Martingale Theory

This unit introduces the students to modern probability theory that was developed by A. N. Kolmogorov. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that brings together both the theory of discrete random variables and the theory of continuous random variables that were introduced earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, conditional expectation, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and to thoroughly study discrete-time martingales. Originally used to model betting strategies, martingales are a powerful generalisation of random walks that allow us to prove fundamental results such as the Strong Law of Large Numbers or analyse problems such as the gambler's ruin. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability, stochastic systems or financial mathematics.

MATH4511 Arbitrage Pricing in Continuous Time

This unit is foundational for honours and masters programs in Financial Mathematics. Its aim is to introduce the basic concepts and problems of securities markets and to develop theoretical frameworks and computational tools for pricing financial products and hedging the risk associated with them. It covers mathematical techniques for pricing derivative securities and exotic options in continuous-time models for equities and foreign exchange and provides a detailed study of the problem of pricing and exercising of an American put option in the Black-Scholes model. A substantial part of this unit is devoted to various generalizations of the classical Black-Scholes model, such as: the constant elasticity of variance (CEV) model, the local volatility model due to Dupire, modelling of stochastic volatility (in particular, the Heston model) and the issues of a model risk and the robustness of the Black-Scholes model.

MATH4512 Stochastic Analysis

This unit will introduce an important class of stochastic processes, using the theory of martingales. You will study concepts such as the Ito stochastic integral with respect to a continuous martingale and related stochastic differential equations. Special attention will be given to the classical notion of the Brownian motion (also known as the Wiener process), which is the most celebrated and widely used example of a continuous martingale. This unit explores some fundamental concepts and results from the Ito stochastic calculus, such as: conditional expectations, filtrations, martingales, stopping times, the Wiener process and its properties, the Ito stochastic integral, semimartingales, the Ito lemma, the Levy characterization theorem, martingale representation property of a Wiener process, stochastic differential equations and stochastic exponentials, the Feynman-Kac theorem, the Girsanov theorem, and distributions of first passage times for the Wiener process. It is essential that students have a very good command of the contents of STAT3021/3921 Stochastic Processes and STAT4528 Probability and Martingale Theory.

MATH4513 Topics in Financial Mathematics

The fixed-income market 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 modelling of instantaneous forward interest rates and models of forward LIBORs and swap rates. You will also learn about pricing and hedging of credit derivatives and become familiar with stochastic models for credit events. This unit will cover the pricing and hedging of single-name and multi-name credit derivatives, such as: vulnerable options, corporate bonds and credit default swaps. You will also learn about the most recent developments in financial mathematics, such as the concepts of robust pricing and nonlinear evaluation in imperfect markets.

FMAT4103, FMAT4104, FMAT4105, FMAT4106: Honours Project in Financial Mathematics and Statistics

A significant part of the honours program is the completion of a research project by each student. An honours project involves intensive research, analysis or computation. It may cover a classical problem of acknowledged importance and mathematical depth with the student providing his/her own critical evaluation. Each student must choose a project supervisor who is willing to supervise the student’s chosen topic for the project before applying for admission to honours program in November for admission in Semester 1 of the next year (or in June for admission in Semester 2 of the same year). For a list of potential topics of honours projects, see the handbook.

• Year 5

MATH5550 Optimal Control and Game Theory (not currently offered)

The theory of stochastic optimal control and games is an indispensable tool in many areas of applied mathematics. In the first part of this unit, you will be familiarised with the dynamic programming principle and learn how to show that it provides a unified approach to a large number of seemingly unrelated problems. The second part is devoted to backward stochastic differential equations and their applications to stochastic optimal control and game theory. You will learn how to solve continuous time problems based either on the Wiener process or more general classes of stochastic processes. After completing this unit, you will be able to formulate a diverse suite of problems arising in finance, applied sciences and engineering as stochastic optimal control problems and solve them using the concepts of the Bellman principle, Hamilton-Jacobi-Bellman equation and backward stochastic differential equations.

MATH5551 Stochastics and Finance

The theory of stochastic phenomena has applications in engineering systems, the physical and life sciences and economics, to give just a few examples. Applications of stochastic processes arise particularly naturally in finance where there are fluctuations in stock prices and practitioners are required to solve different types of optimisation problems in stochastically driven systems. For this reason, it is particularly important that mathematicians in general and especially mathematicians specialising in problems in the financial industry are equipped with tools to analyse and quantify random phenomena. This unit will expose you to critical topics in the theory and application of stochastic processes and analysis in mathematical finance. You will learn how to identify problems that require the application of stochastic theory, how to rigorously describe such problems using appropriate mathematical frameworks and how to tackle and solve the problem once it has been phrased in terms of stochastic theory.

BSc Major in Financial Mathematics and Statistics

Students in the BSc and BAS degrees can choose to complete a major in Financial Mathematics and Statistics. The best students who complete a major in Financial Mathematics and Statistics may go on to Bachelor of Advanced Studies (Honours) in Financial Mathematics and Statistics. Admission to Honours requires the prior completion of all requirements of the Bachelor of Science, including Open Learning Environment (OLE) units. If you are considering applying for admission to honours year embedded in your BAS degree, ensure your degree planning takes into account the completion of a second major and all OLE requirements prior to honours commencement. Notice that entry requirements to the honours program in Financial Mathematics and Statistics vary slightly depending on whether the candidate is completing a Bachelor of Science (BSc) degree or a Bachelor of Advanced Studies (BAS) degree. For more details on entry requirements, see the handbook or the Faculty websites for Bachelor of Advanced Studies (Honours) and Bachelor of Science (Honours).

Major in Financial Mathematics and Statistics (48 cp)

For a full description of a major and a minor in Financial Mathematics and Statistics, see the handbook FMS major and minor.

Core Junior units (12 cp) Note that either MATH1005/1905 or DATA1001/1901 should be completed.

  • MATH1002/1902 Linear Algebra (3cp, Sem 1)
  • MATH1021 Calculus of One Variable (3cp, Sem 1 + Sem 2)
  • MATH1921/1931 Calculus of One Variable (3cp, Sem 1)
  • MATH1023 Multivariable Calculus and Modelling (3cp, Sem 1 + Sem 2)
  • MATH1923/1933 Multivariable Calculus and Modelling (3cp, Sem 2)
  • MATH1005 Statistical Thinking with Data (3cp, Sem 1 + Sem 2)
  • MATH1905 Statistical Thinking with Data (3cp, Sem 2)
  • DATA1001/1901 Foundations of Data Science (6cp, Sem 1 + Sem 2)

Core Intermediate units (18 cp)

  • STAT2011/2911 Probability and Estimation Theory/Probability and Statistical Models (6cp, Sem 1)
  • MATH2070/2970 Optimisation and Financial Mathematics (6cp, Sem 2)
  • DATA2002/2902 Data Analytics: Learning from Data (6cp, Sem 2)

Core Senior units (12 cp)

  • STAT3021/3921 Stochastic Processes (6cp, Sem 1)
  • MATH3075/3975 Financial Derivatives (6cp, Sem 2)

Project unit (6 cp) Note that either FMAT3888 or SCPU3001 should be completed.

  • FMAT3888 Projects in Financial Mathematics (6cp, Sem 2)
  • SCPU3001 Science Interdisciplinary Project (6cp, Sem 1 + Sem 2)

Honours in Financial Mathematics and Statistics (48 cp)

Admission to Honours requires the prior completion of all requirements of the Bachelor of Science, including Open Learning Environment (OLE) units. If you are considering applying for admission to honours year embedded in your BAS degree, ensure your degree planning takes into account the completion of a second major and all OLE requirements prior to honours commencement. Notice that entry requirements to the honours program in Financial Mathematics and Statistics vary slightly depending on whether the candidate is completing a Bachelor of Science (BSc) degree or a Bachelor of Advanced Studies (BAS) degree. For more details on entry requirements, see the handbook or the Faculty websites for Bachelor of Advanced Studies (Honours) and Bachelor of Science (Honours). For a full description of admission and completion requirements for honours program in Financial Mathematics and Statistics, see the handbook and Table A

Core units (24 cp) MATH4511, STAT4528 and at least one of the units MATH4512 and MATH4513 should be completed.

  • STAT4528 Probability and Martingale Theory (core, 6cp, Sem 1)
  • MATH4511 Arbitrage Pricing in Continuous Time (core, 6cp, Sem 1)
  • MATH4512 Stochastic Analysis (6cp, Sem 2)
  • MATH4513 Topics in Financial Mathematics (6cp, Sem 2)

Research project (24 cp)

  • FMAT4103-4106 Research project in Financial Mathematics and Statistics (24cp, Sem 1 + Sem 2)

Most of the Mathematics and Statistics units of study mentioned above have their own web pages.
Core units: MATH1002, MATH1902, MATH1021, MATH1921, MATH1931, MATH1923, MATH1933, MATH1005, MATH1905, DATA1001, DATA1901, MATH2070, MATH2970, STAT2011, STAT2911, MATH3075, MATH3975, FMAT3888.
Electives: MATH3971, MATH3076, MATH3976, MATH3078, MATH3978, MATH3969, MATH3974, STAT3021, STAT3911, STAT3022, STAT3922, STAT3023, STAT3923, STAT3888, DATA3404.