Aims:
This module introduces the applied mathematical and computational aspects of Quantitative Finance;Learning outcomes:
On successful completion of the module, a student will be able to successfully apply the necessary probability and differential equation based approach to the pricing of financial derivatives; using both quantitative and numerical techniques;

Course content:

A practical mathematical finance course centering on probability and differential equation approaches to price financial derivatives; starting with an overview of financial products and markets. Delta hedging and no arbitrage is introduced via the Binomial model. Stochastic calculus is presented which serves as a starting point for the Black-Scholes framework; both European options and early exercises features are detailed. The relevant PDE and simulation based computational schemes are studied. Further advanced topics in fixed income (using stochastic interest rates) and exotic options are also presented.

The representation theory of finite groups, which solidifies one&#039;s knowledge of group theory, is perhaps the easiest part of the general theory of...

The representation theory of finite groups, which solidifies one's knowledge of group theory, is perhaps the easiest part of the general theory of symmetry. It goes back to F. Klein who considered the possibility of representing a given abstract group by a group of linear transformations (matrices) preserving the group's structure, leading mathematicians such as G. Frobenius, I.Schur, W. Burnside and H. Maschke to follow and develop the idea further. Essentially, it is a formal calculus designed to give an explicit answer to the question "What are the different ways (homomorphisms) a finite group G can occur as a group of invertible matrices over a particular field F?". The link between group representations over a field F and modules is obtained using the concept of a group ring F[G], thus an essential step is the systematic study and classification of group rings (the so-called semisimple algebras) which behave like products of matrix rings. Therefore, the story of the representation theory of a group is the theory of all F[G]-modules, viz modules over the group ring of G over F. The ultimate goal of this course is to teach students how to construct complex representations for popular groups as well as their character tables which serve as invariants for group rings. In addition to the obvious applications to physical symmetry, the theory leads to significant insights into the structure theory of finite groups.