Autumn School Courses
There will be five courses covering some basic theoretical aspects of financial
mathematics as well as some applications arising from business problems.
Courses, professors and summaries:
C1 Course 1 (5 hours)
Introduction to stochastic differential
equations
Carlos Braumann (University of Évora - Portugal)
Synopsis:
Stochastic processes in continuous time, martingales, Markov processes, diffusion processes, Kolmogorov equations, the Wiener process. Stochastic integrals with respect to the Wiener process, the Ito integral and its properties, examples. Ito's formula and applications. Stochastic differential equations, properties, examples, and applications.
C2 Course 2 (9 hours)
Arbitrage theory in continuous time
Tomas Björk (Stockholm School of Economics - Sweden)
Synopsis:
The Black-Scholes model, portfolios, self financing portfolios, arbitrage,
portfolio dynamics, risk neutral evaluation, Black-Scholes's formula. Martingale
measures. Completeness and hedging, completeness of the Black-Scholes model,
completeness versus arbitrage. Incomplete markets. An introduction to interest
rate theory.
Slides:
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[Download all (576 kB)]
C3 Course 3 (4 hours)
Extremes and risk management
Ivette
Gomes (FCUL, University of Lisbon - Portugal)
Synopsis:
Stochastic processes in continuous time, martingales, Markov processes. Extreme Value Theory (EVT) plays a relevant role in the area of financial risk modelling. As an example, the Value-at-Risk (VaR) is a parameter of extreme or even rare events: a high quantile of the loss distribution of a portfolio of investments over a fixed period of time. In this short course, after an introduction to the VaR problem, we shall review a few concepts of EVT, together with its application to financial time series, we shall discuss some of the peculiarities of financial data, advancing with the study of models which capture those peculiarities, and shall briefly address the issue of possible joint extreme movements in many underlying investments.
C4 Course 4 (1,5 hours)
Risk management
Joćo Duque (ISEG,
Technical University of Lisbon - Portugal)
Synopsis:
An introduction to the concept of risk will be presented. We shall refer to risk management,
namely, to some tools used in risk management. We shall show how relevant mathematics becomes in
finding solutions for problems related to the topics of the courses C1, C2, C3 and C5.
Presentation:
Presentation (393 kB);
Data (1375 kB)
C5 Course 5 (3 hours)
Introduction to statistical inference on
diffusion processes
Joćo Nicolau (ISEG, Technical University of Lisbon - Portugal)
Synopsis:
Stochastic continuous-time processes, especially diffusion processes generated by differential
equations have been widely used in financial economics. However, all models involve unknown
parameters or functions, which need to be estimated from observations of the process. The
estimation of diffusion processes is, thus, a crucial step in all applications and, in
particular, in applied finance. Parametric estimation based on continuous sampling observations
has been considered in the literature for many years. However, only recently has there been a
systematic approach to the parametric and nonparametric diffusion processes estimation based on
discrete time observations. It is our aim to revise some of the most important contributions in
this area and discuss some future challenges.
