# MATH5816 Continuous Time Financial Modelling

MATH5816 is a Honours and Postgraduate Coursework 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: This recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus.

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 main goal of the course is a detailed study of 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.