The area of mathematical finance covered by this Euro Summer School has experienced a remarkable development in the recent years, to the point that it constitutes one of the most active branches in applied mathematics and economical sciences. A fundamental research line in this area is the study of modifications of the classical Black and Scholes model that are able to capture modern market phenomena like heavy-tailed log-returns and stochastic volatility. These new developments are based on advanced methods of stochastic calculus and require the use of high-level mathematical tools.
The objective of this Summer School is to present, in a series of lectures given by renowned experts in the field, a detailed account of the following significant new developments:
1) Advances in hedging of contingent claims.
2) Change of time and change of measures to the modelling in finance.
3) Analysis of alternative models for stock prices based on processes with independent increments.
Topic 1) will be covered in a series of lectures given by I. Karatzas. Particular problems presented in these lectures will include the hedging theory under constraints for both European and American contingent claims, problems of partial hedging, hypothesis testing and least-square approximation of random variables by stochastic integrals.
Topic 2) will be delivered by Prof. A. N. Shiryaev that will include time change in basic processes, integral transformations and effects on characteristics. The lectures will also cover applications of these concepts to filtering, statistical analysis of financial data, de-volatilisation and stochastic volatility models.
Topic 3) will be covered in two short courses given by O. Barndorff-Nielsen and D. Madan and will include the study of models where the volatility is driven by a Lévy process. Along the School some lectures from practitioners and young academics will be organized where participants will present their research progress in stochastic calculus and financial mathematics. The aim of the Summer School is to provide young researchers with the necessary tools to tackle open problems in the subject area that rise from applied point of view, and to give them the opportunity to learn from leader experts in financial mathematics and stochastic calculus.