MATH5816 is a Mathematics Level V course. See the course overview below.

Units of credit: 6

Prerequisites: MATH5965 and MATH5975

Cycle of offering: each year in Session 2.

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. (This pdf will usually be updated by the end of the first week of the session.)

The Online Handbook entry contains up-to-date timetabling information.

If you are currently enrolled in MATH5816, you can log into the My eLearning Vista instance of this course.

The course focuses on continuous-time modelling of financial market 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 claim 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 maximization 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.