Mathematics for Finance
The importance of financial markets in the development of contemporary society can hardly be overstated. They make newspaper’s headlines when they crash, but their day to day operations often go unnoticed despite providing capital, liquidity, insurance and information to investors, governments and industries.
In the last half century mathematics has been the driving force of innovation in many aspects of finance.
For example, the Black, Scholes and Merton theory of option pricing has played an essential role in the development of derivative markets, which today are a multi-trillion dollars industry.
The application of mathematical methods to the solution of problems in finance, exploiting tools from applied mathematics, computer science, econometrics and statistics, is at the heart of financial mathematics.
Experts in this field, commonly known as quants, are now employed in banks, hedge funds, insurance companies, and regulatory agencies.
In the past two decades financial markets have undergone a remarkable transformation due to advances in technology and tremendous amounts of data are now available for investigation. This era of big data has brought many new benefits to investors, but also new risks together with new challenges to old assumptions: new mathematical models and tools need to be developed.
The research activity at the Scuola Normale Superiore is concentrated on the theoretical and applied study of financial markets at different time scales, using rigorous mathematical tools as well as quantitative methods, both analytically and empirically. It takes place not only with the involvement with the academic staff and research facilities of the Scuola Normale, but also of researchers from foreign universities (Imperial College London, Oxford University, École Polytechnique, Université Paris 1, City University of New York, Göthe-Universität Frankfurt, ETH Zürich), research centers, banks, investment companies, computer companies, and markets regulators.
Currently the main areas of research are:
- High frequency finance and market microstructure. The main focus is the mathematical modeling and empirical characterization of high (transactions) and very high (bid-ask, limit order book) frequency financial time series. The areas of interest are the modeling of liquidity, price formation, and the optimal execution of orders. Endogenous instabilities in financial markets, like the 2010 “flash crash”, a very rapid, deep, and volatile fall in security prices occurring within an extremely short time period, show how the speed and interconnectedness of markets and agents can have very critical consequences and result in the loss and recovery of billions of dollars in a matter of minutes and seconds.
- Dependencies between financial variables, correlation structures and networks. Modelling of dependence between financial variables such as stock returns, the returns of Credit Default Swaps, or trading activities of investors or brokerage firms.
- Mathematical modeling, computational and empirical analysis for the assessment of systemic risk and the analysis of the mechanisms of contagion in financial and interbank markets. The Financial Crisis of 2007-2009 and the sovereign debt crisis that followed have created renewed interest in this topic, which now extends from bank runs and currency crises to any broad-based breakdown in the financial system.
- Mathematical modeling, computational and empirical analysis to evaluate the efficiency of financial markets. According to the Efficient Markets Hypothesis, financial markets fully, accurately, and instantly incorporate all available information into market prices. This may be true most of the time but booms, busts, and financial crises suggest that this isn’t always the case.
Teaching is divided into lecture and seminar courses (all held in English). Graduate students in Financial Mathematics will annually agree with the PhD Coordinator a study plan to be presented to the Faculty Board. Such a document will specify the planned research and education activities for the relevant academic year. The courses will be chosen to enlarge the student background and deepen specific aspects related to the PhD Thesis project. PhD students are expected to take at least three courses and to pass the corresponding exams.At the end of the first year students are expected, in close consultation with the Coordinator and with approval from the Faculty Board, to choose the Thesis supervisor and project. At the end of the second year, PhD students should present a written report concerning the research done and the results achieved so far, together with any publications produced. The report will be discussed in an oral presentation in front of a panel of experts appointed by the Faculty Board. Upon successful performance, the student will be admitted to the third year.
According to research needs, students are encouraged to spend periods of study and research at Italian and foreign institutions.