MATH5816 is an honours and postgraduate mathematics course. See the course overview below.

Units of credit: 6

Prerequisites: MATH5965, MATH5975

Cycle of offering: Term 3

Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.

More information:  The Course outline will be made available closer to the start of term - please visit this website: www.unsw.edu.au/course-outlines

The course outline contains information about course objectives, assessment, course materials and the syllabus.

Important additional information as of 2023

UNSW Plagiarism Policy

The University requires all students to be aware of its policy on plagiarism.

For courses convened by the School of Mathematics and Statistics no assistance using generative AI software is allowed unless specifically referred to in the individual assessment tasks.

If its use is detected in the no assistance case, it will be regarded as serious academic misconduct and subject to the standard penalties, which may include 00FL, suspension and exclusion.

The online handbook entry contains information about the course. The timetable is only up-to-date if the course is being offered this year. 

If you are currently enrolled in MATH5816, you can log into UNSW Moodle this course.

Course overview

The course focuses on continuous-time modelling of financial markets under deterministic interest rates. The aim of the course is to study in detail the classical Black-Scholes model and its variants. We introduce the concept of a continuously rebalanced portfolio, and we examine the arbitrage-free property of the model by examining the existence and uniqueness of a martingale probability measure (using Girsanov's theorem). We provide two alternative proofs of the Black-Scholes option pricing formula. The first relies on the calculation of the replicating strategy; it thus requires solving the Black-Scholes PDE. The second method is based on probabilistic considerations and it makes direct use of the risk-neutral valuation formula.

We introduce and study the notions of historical and implied volatilities. Subsequently, we present the approach known as the implied local volatility modelling. In this approach, the observed market prices at a given date (and thus the observed smiles and skews) are taken as inputs. We show the existence of a diffusion-type model which, by construction, is fitted to the observed term structure of volatility smile.

In the second part of the course, we study contingent claims of American style in the Black-Scholes set-up. We explain that the valuation of American claims is closely related to specific optimal stopping problems. We show that for the purpose of arbitrage valuation, the maximisation of the expected discounted payoff should be done under the martingale measure. The value of an American put option is compared to the value of a corresponding European put.

The last part of the course is devoted to cross currency derivatives.