by Ralf Korn
With so much science and so many applications focused on quantitative output, mathematics plays a key role in the design and evaluation of quantitative models. While in some cases simple or standard mathematics is enough to formulate models and support theories, sometimes methods are required that are very recent or even yet to be developed.
An application that falls into the second case is modern financial mathematics. From its first tentative steps in the early seventies it has undergone extensive development, via a great deal of hype in the eighties and nineties to the critical attention it is currently receiving in the face of the credit crisis.
Financial mathematics can be divided roughly into four areas:
- modelling of price movements (for stocks, commodities, interest rate products etc) and underlying factors (such as interest rates, exchange rates, inflation)
- portfolio optimization, ie the determination of an optimal investment strategy in a financial market
- option pricing, ie the determination of a price for a derivative, a security that consists of an uncertain future payment stream (such as simple call or put options or more complicated, so-called exotic variants)
- risk management, ie the measurement and management of risk connected with an investment in securities (such as credit risk, market risk or operational risk).
While the other three areas seem to be the more natural ones when one is thinking about mathematical applications in finance, it is option pricing which is the star area of modern financial mathematics. Here, the Black-Scholes formula for the prices of European call and put options earned Myron Scholes and Robert E. Merton the Nobel prize. On the theoretical side, the area is one of the rare cases in which very recent mathematical methods such as stochastic processes, stochastic calculus and stochastic differential equations are indispensable. What is more, option pricing actually motivated theoretical research in these areas of probability. On the application side, tools were developed that allowed the pricing - at least in principle - of all kinds of derivative securities, which could be shaped according to the demands of the market. This then called for efficient numerical methods for calculating option prices, and as there is a close relation between stochastic differential equations and parabolic partial differential equations, numerical mathematicians and physicists entered the scene.
For some years, therefore, the inventive activity of the market in creating new derivative products resulted in new option pricing problems. At the same time, the huge influx of researchers from other areas into financial mathematics led to new theoretical results and to more complicated models and valuation methods. The climax of this was the introduction in this decade of credit derivatives. These are nothing more than insurance against the defaults of creditors. With the reasons for a default already hard to model, the financial market made the problem even harder by putting many credits together and then selling tranches of different riskiness in this package. This technique resulted in the now infamous collateralized debt obligation (CDO). In principle this is a package of credits, but it is divided into pieces: the equity tranche (the highly risky piece from which all default losses are paid as long as this tranche still exists), the mezzanine tranche (the mildly risky piece from which default losses are paid when the equity tranche has completely disappeared), and the senior tranche (relatively safe pieces that only suffer losses due to credit defaults if the other tranches are already used up).
As CDOs are very complicated securities, many investors bought them without fully understanding the mechanism how they work. In addition, the credits used in the packages were sometimes of questionable quality, the main issue that led to the current credit crisis. On the modelling side, determining the risk of a CDO investment requires an understanding of the correlation structure among the credits within it (at least the correlation when it comes to defaults). Here, very simplistic models became popular in the industry and, on top of that, were regarded as perfect images of the real world. This proved to be wrong as soon as the first defaults occurred, and as a consequence the values of CDOs dropped dramatically. Since credit derivatives and in particular CDOs had been enormously popular in recent years, the losses were huge and the consequence is the current financial crisis.
Examples of more off-the-shelf applications of mathematics in economics include operations research methods applied to revenue management problems, queuing theory applied to storage problems, and dynamical systems used in macro-economic theory.
Although financial mathematics has been the most popular application of mathematics in economics in recent decades, this special theme contains reports on a variety of applications. The contributions can be grouped in three blocks, plus three survey-type papers:
- an article by Paul Wilmott that sheds a slightly controversial light on 'Financial Modelling' and points out some important open problems
- a paper by Marlene Müller that surveys what modelling can do for the finance and insurance industry
- a block of papers on theoretical finance, with topics such as option pricing, scenario generation and price modelling
- a block containing applications of operations research in economics, dealing with topics such as revenue management, sales forecasting, CO2 emissions and fiscal fraud
- a block on software/hardware for applications in finance and economics, describing software projects and recent hardware design for problems in finance
- a closing paper by Pierluigi Contucci and Francesca Romiti that gives a statistical mechanics perspective of the application of mathematics in economics.
While reading these contributions, one should keep in mind a key difference between this area of applied mathematics and more classical physics and engineering. The quality of models in finance and economics is not fixed: their correctness can change with time, since it is not nature being modelled but the much more changeable and unpredictable interactions of humans .
University of Kaiserslautern and Fraunhofer ITWM, Germany