Research in Mathematical Finance
Mathematical Finance is a very active branch of Probability Theory and Mathematics in general. It is probably one of the few areas in academic research which interact constantly with their field of application and with a huge impact on the daily functioning of the world's financial institutions.
Mathematical Finance is born in 1900 with the doctoral dissertation Théorie de la speculation by Louis Bachelier. After a long gap, the Nobel laureate Paul Samuelson deepened and extended Bachelier's idea in the 1960s. Both works were based on or related to the Wiener process, the most important representant of a stochastic process. These ideas led finally to the central result of modern finance in 1973, the Black-Scholes formula which gives the price of a derivative and which was worth another Nobel prize in 1997.
One of the cornerstone in Mathematical Finance is Stochastic Calculus. Originally, Black and Scholes used partial differential equations to derive their pricing formula. Nowadays, tools from Stochastic Calculus, in particular martingales, are more commonly used to price derivatives.
Another cornerstone in Mathematical Finance is the theory of Stochastic Differential Equations. Not only in Mathematical Finance but also in many other disciplines the change of a state can not only be described by a deterministic equation but one also has to take into account some random perturbations.
Dr Ronnie Loeffen is particularly interested in stochastic processes appearing in Mathematical Finance that exhibit jumps like Lévy processes and affine processes. Several other members of the group (Moriarty, Peskir, Van Schaik and Zhang) have research interests in Mathematical Finance as well. For research activity in this area and people within the School of Mathematics see http://www.mims.manchester.ac.uk/research/mathematical-finance.