Applications of the Mellin transform in mathematical finance

In this thesis, we will be presenting a slew of mathematical finance scenarios where the Mellin transform and its associated techniques are incorporated to solve either a direct or inverse problem. Specifically, we will be investigating options pricing problems in both the European and American sense whereby the underlying asset is modelled by a jump-diffusion process. We exploit the elegant properties of the Mellin transform to elicit a result for the option valuation under a jump-diffusion model. Additionally, one of the main breakthroughs in this work is isolating and determining an expression for the jump term that is general and to our knowledge, has not been ascertained elsewhere. As an addendum to American options, we extend our Mellin transform framework to obtain a pricing formula for the American put in jump-diffusion dynamics. Furthermore, an approximate integro-differential equation for the optimal free boundary in the same aforementioned dynamics is also derived and we test the accuracy of this against the numerical finite difference method. The final area we investigated was the valuation of European compound options and particularly how to reformulate the pricing formulas by incorporating Mellin transform techniques and the put-call parity relationship that exists for vanilla European options.