Research Focus
The Financial Mathematics group is committed to active research of international excellence. For the individual research interests of the members of the group, visit our Members page. An informal, overall outline of the group's research follows below in this page.
About Financial Mathematics
Financial Mathematics encompasses a wide range of topics. These include the development and analysis of rigorous stochastic models for asset price dynamics, the empirical analysis of financial time series data, as well as the practical implementation of risk management tools and the development of new derivatives pricing and hedging methodologies for use in an investment banking context and elsewhere in the financial sector. As a consequence the area is attractive in offering excellent research opportunities both to pure and applied mathematicians, and has benefited from cross-fertilisation with other disciplines such as economics, theoretical physics, and computer science.
Financial Mathematics is one of the few areas of academic research that is in a constantly active interaction with present developments in its domain of application. Indeed, it both draws from and has direct implications upon every-day practice in financial institutions. The research in this area aims at a better understanding of the stochastic evolution of financial markets through the formulation of appropriate mathematical models, as well as at the development of efficient new methodologies for the pricing and hedging of complex financial derivatives. The associated mathematical techniques come from a number of different branches of pure and applied mathematics, including probability, stochastic processes, analysis, partial differential equations, statistics, geometry, and numerical methods.
The research interests of the Financial Mathematics, Probability and Statistics group at King's College cover a number of different inter-related areas. Among others, these include stochastic processes, stochastic control and nonlinear filtering. In the mathematical finance sector research is active on derivatives pricing and hedging; asset price dynamics; interest rate, inflation and FX models; credit risk models; equity and commodities models; volatility smile modeling; counterparty risk (CVA) pricing; liquidity modeling; mortality risk; real options; risk management; portfolio optimisation under constraints; computational finance; information-based asset pricing; econophysics, differential geometric approach to statistics, information geometry. There is also an active research programme in collective phenomena in financial markets, including for example extreme phenomena such as hyper inflation, price bubbles, and market crashes, for which a better understanding is currently needed.
Probability and Stochastic Processes
The principal mathematical tools in finance derive from probability and the theory of stochastic processes. Indeed, it is becoming increasingly essential for risk managers and quantitative analysts that they should have a mastery of a variety of probabilistic techniques, along with knowledge of the theory of Brownian motion, martingales, and stochastic differential equations. The elementary geometric Brownian motion model for share-price movements was originally introduced by Samuelson based on an earlier theory of Bachelier. It was later incorporated into the extraordinary option pricing theory of Black, Scholes and Merton. This kind of price dynamics has subsequently been greatly generalised to situations where multiple asset prices are modelled by Itô processes. In these models, the drifts, volatilities and correlations of the various assets are "adapted", i.e., they are processes which do not reflect future events but can be dependent on the full history of the Brownian motion up to the time to which prices refer.
The condition of no arbitrage among the various assets, namely the assumption that financial markets offer no risk-free profit opportunities, has an elegant and useful characterisation in the language of martingale theory. Under certain additional assumptions (e.g. "market completeness", which is a kind of a non-degeneracy condition), martingale techniques offer a powerful tool for pricing and hedging derivatives, and provide a framework for the analysis of numerous other financial applications. Research in this area is actively evolving in two directions. The first one involves the relaxation of some of the underlying assumptions with a view to developing a theory that can account, e.g. for incomplete markets or for markets where transaction costs and other "frictions" are taken into consideration. The second direction focuses on the generalisation of the price dynamics of the traded assets to include broader classes of processes, such as the so-called Levy processes and their extensions.