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Financial Mathematics
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Research Activities of the Financial Mathematics Group

The Financial Mathematics group is committed to active research of international excellence. For the individual research interests of the members of the group, follow the links in the page Research Group. An informal, overall outline of the group's research follows below in this page.

The members of the Financial Mathematics group are keen to supervise able potential PhD candidates on a variety of topics within their research interests. For an informal inquiry, interested candidates are welcome to communicate with any of the members of the group. More information about the applications procedure can be found in the page Applications for PhD.

The group also organises and runs a series of seminars and conferences with the following aims:

(a) To support and stimulate the research environment within King's College London.

(b) To provide practitioners in financial institutions as well as the wider academic community with an access to the most recent advances in the area.

Information about the most recent and forthcoming events can be found in Seminars and Conferences. For past events, see Previous Financial Mathematics Seminars and Conferences.

A list of recent research papers can be found in Recent Publications.

Financial Mathematics and Applied Probability

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 and Applied Probability group at King's College cover a number of different inter-related areas. Among others, these include derivatives pricing, asset price dynamics, interest rate models, real options, portfolio optimisation under constraints or in the presence of transaction costs, and credit risk models. 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.

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.

Interest rate modelling is a very important area of finance. A discount bond is a note that promises to pay the bearer a fixed unit of cash at a specified date in the future. Owing to the variability of interest rates, the value of a bond will typically fluctuate over time as it converges towards the deliverable unit at maturity. Even small changes in interest rates can have significant economic consequences for governments, businesses and individuals. A good deal of effort is presently going into the development of models that can help us understand the dynamics of interest rates. A proper treatment of interest rates is essential for studying the behaviour of other classes of assets. The goal of an interest rate model is to characterise the arbitrage-free dynamics of the continuous system of assets consisting of bonds of all maturities. The first consistent model of this kind was put forward by Vasicek in 1977. Subsequently, a number of other important models have been developed, including that of Cox, Ingersoll and Ross in 1985. A further major step forward came in 1992 with the publication of the theory of Heath, Jarrow and Morton. These researchers showed how interest rate derivatives can be priced in a quite general context under the assumption that the discount bond system is driven by a multi-dimensional Brownian motion. Current research involves both the creation of new interest rate models, as well as extensions of the HJM theory. No definitive interest rate model has yet emerged, which is good news for those who wish to carry out research in this line. The absence of a generally agreed method for marking-to-market complex interest rate derivative positions is a source of concern to investment banks and their regulators, and there is a real need for further progress in this area.

The theory of "real options" is a relatively new framework for the pricing of an investment in a real asset such as an oil field or a copper mine. It has been well documented in the economics literature that the traditional discounted cash flow approaches to the problem of asset valuation results in prices that are significantly lower than actual quoted prices. It turns that the reason is because the traditional techniques attribute prices to investments in a static way. The real options approach aims at correcting this discrepancy by incorporating the value of managerial decisions through appropriate mathematical models. Furthermore, by considering portfolios of traded assets which are correlated with the uncertain cash flows resulting from the management of an investment, the theory of real options also provides techniques for hedging against the risks associated with investment ownership. Although the real options approach has attracted significant attention in the economics literature, the development of a rigorous mathematical framework is still at its early stages, and thus offers many interesting research opportunities.

The analysis of financial models frequently gives rise to various problems of stochastic optimal control, including optimal stopping. Examples occur in the pricing of financial derivatives in incomplete markets, in the pricing of real options, and in the pricing of a number of exotic derivatives, e.g. passport options. The pricing of American options also typically gives rise to optimal stopping problems. The solution of the resulting stochastic optimisation problems and the analysis of the properties of optimal strategies present challenging issues. These mathematical problems are often of considerable interest in themselves because they can give rise to developments in the theory of controlled stochastic differential equations which have applications to other areas.

Financial markets offer a rich body of empirical data for analysis, and one has the advantage of being able to compare the results of theoretical models to real data. Financial time series exhibit highly non-trivial statistical features which are hard to model and even harder to explain. Intermittent behaviour, volatility clustering (amplitudes of successive price movements are persistent, but not necessarily their signs), heavy tailed increments, and subtle dependence structures are important factors when assessing risk in financial markets and when pricing tailored risk-management products. One area of active research in this direction is the empirical analysis of credit data for the quantification of credit risk and the investigation of how to develop mathematically sound risk management and derivatives pricing models that can incorporate correlations and fat-tailed probability distributions for credit losses. The theory of credit risk management and credit derivatives has close links with interest rate theory, and makes subtle use of the interplay between point processes and Brownian motion.

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