Until about the 1970s, financial mathematics has been rather modest compared with other mathematical disciplines. This changed rapidly after the path-breaking works of F. Black, M. Scholes, and R. Merton on derivative pricing, for which they received the Nobel prize of economics in 1997. Since 1973, the publication year of the famous Black and Scholes article, the importance of derivative instruments in financial markets has not ceased to grow. Higher risks associated with, for example, flexible instead of fixed exchange rates after the fall of the Bretton Woods system required a risk management and the use of hedging instruments for internationally active companies. More recently, globalization and the increasingly complex dependence of financial markets are reasons for using sophisticated mathematical and statistical methods and models to evaluate risks.
The necessity to improve and develop the mathematical foundation of existing risk management was emphasized in the turbulent 1990s with, for example, the Asian crisis, the hedging disasters of Metallgesellschaft and Orange County, and the fall of the Long-Term Capital Management hedge fund (controlled by Merton and Scholes!). This saw the legislator obliged to take action. In continental Europe, this development is mainly influenced by the Basel Committee on Banking Supervision, whose recommendations form the basis in the European Union for legislation, with which financial institutions are obliged to do a global, thorough risk management. As a result, there is an increasing demand for experts in financial engineering, who control risks internally, search for profitable investment opportunities and guarantee the obligations of legislation. In the future, such risk management is likely to become obligatory for other, deregulated markets such as telecommunication and energy markets. Being aware of the increasing price, volume, and credit risks in these markets, large companies usually have already created new departments dealing with asset and liability management as well as risk management.
The present text is supposed to deliver the necessary mathematical and statistical basis for a position in financial engineering. Our goal is to give a comprehensive introduction into important ideas of financial mathematics and statistics. We do not aim at covering all practically relevant details, and we also do not discuss the technical subtleties of stochastic analysis. For both purposes there is already a vast variety of textbooks. Instead, we want to give students of mathematics, statistics, and economics a primer for the modelling and statistical analysis of financial data. Also, the book is meant for practitioners, who want to deepen their acquired practical knowledge. Apart from an introduction to the theory of pricing derivatives, we emphasize the statistical aspects of mathematical methods, i.e., the selection of appropriate models as well as fitting and validation using data.
The present book consists of three parts. The first two are organized such that they can be read independently. Each one can be used for a course of roughly 30 hours. We deliberately accept an occasional redundancy if a topic is covered in both parts but from a different perspective. The third part presents selected applications to current practical problems. Both option pricing as statistical modelling of financial time series have often been topic of seminars and lectures in the international study program financial mathematics of Universität Kaiserslautern ( www.mathematik.uni-kl.de ) as well as in the economics and statistics program of Humboldt-Universität zu Berlin ( ise.wiwi.hu-berlin.de ). Moreover, they formed the basis of lectures for banking practitioners which were given by the authors in various European countries.
The first part covers the classical theory of pricing derivatives. Next to the Black and Scholes option pricing formula for conventional European and American options and their numerical solution via the approximation using binomial processes, we also discuss the evaluation of some exotic options. Stochastic models for interest rates and the pricing of interest rate derivatives conclude the first part. The necessary tools of stochastic analysis, in particular the Wiener process, stochastic differential equations and Itô's Lemma will be motivated heuristically and not derived in a rigorous way. In order to render the text accessible to non-mathematicians, we do not explicitly cover advanced methods of financial mathematics such as martingale theory and the resulting elegant characterization of absence of arbitrage in complete markets.
The second part presents the already classical analysis of financial time series, which originated in the work of T. Bollerslev, R. Engle, and C. Granger. Starting with conventional linear processes, we motivate why financial time series rarely can be described using such linear models. Alternatively, we discuss the related model class of stochastic volatility models. Apart from standard ARCH and GARCH models, we discuss extensions that allow for an asymmetric impact of lagged returns on volatility. We also review multivariate GARCH models that can be applied, for example, to estimate and test the capital asset pricing model (CAPM) or to portfolio selection problems. As a support for explorative data analysis and the search and validation of parsimonious parametric models, we emphasize the use of nonparametric models for financial time series and their fit to data using kernel estimators or other smoothing methods.
In the third part of the book, we discuss applications and practical issues such as option pricing, risk management, and credit scoring. We apply flexible GARCH type models to evaluate options and to overcome the Black and Scholes restriction of constant volatility. We give an overview of Value at Risk (VaR) and backtesting, and show that copulas can improve the estimation of VaR. A correct understanding of the statistical behavior of extremes such as September 11, 2001, is essential for risk management, and we give an overview of extreme value theory with financial applications. As a particularly popular nonparametric modelling tool in financial institutions, we discuss neural networks from a statistical viewpoint with applications to the prediction of financial time series. Next, we show how a principal components analysis can be used to explain the dynamics of implied volatilities. Finally, we present nonparametric extensions of conventional discrete choice models and apply them to the credit scoring problem.
We decided to collect some technical results concerning stochastic integration in the appendix. Here we also present Girsanov's theorem and the martingale representation theorem, with which dynamic portfolio strategies as well as an alternative proof of the Black and Scholes formula are developed. This appendix is based on work by Klaus Schindler, Saarbrücken.
In designing the book as e-book, we are going new ways of scientific publishing together with Springer Verlag and MD*Tech. The book is provided with an individual license key, which enables the reader to download the html and pdf versions of the text as well as all slides for a 60 to 90 hours lecture from the e-book server at www.quantlet.com . All examples, tables and graphs can be reproduced and changed interactively using the XploRe quantlet technology.
The present book would not exist without the cooperating contributions of P. Cízek, M. Fengler, Z. Hlávka, E. Kreutzberger, S. Klinke, D. Mercurio and D. Peithmann. The first part of the book arose from an extended vocational training which was developed together with G. Maercker, K. Schindler and N. Siedow. In particular, we want to thank Torsten Kleinow, who accompanied the writing of the text in all phases, developed the e-book platform and improved the presentation by various valuable contributions. Important impulses for an improved presentation were given by Klaus Schindler of the University of Saarbrücken, which we gratefully acknowledge. The chapter on copulas is based on a contribution by Jörn Rank, Andersen Consulting, and Thomas Siegl, BHF Bank, which we adopted with their kind approval. The quantlets for multivariate GARCH models were contributed by Matthias Fengler and Helmut Herwartz. All graphs were created by Ying Chen, who also led the text management. We would like to express our thanks to these colleagues. We also benefitted from many constructive comments by our students of the universities in Kaiserslautern, Berlin, and Rotterdam. As an example of their enthusiasm we depict the preparation sheet of a student for the exam at the front pages of the book. Graphs and formulae are combined to create a spirit of the ''art of quantitative finance''.
Finally, for the technical realization of the text we want to thank
Beate Siegler and Anja Ossetrova.
Kaiserslautern, Berlin and Rotterdam, July 2003