## Advanced Course on Mathematical Finance, Cortona 2006

Lecturers: Location: Palazzone di Cortona>, Scuola Normale Superiore, Italy
Organization: Scuola Matematica Interuniversitaria (SMI)
Web Page/Announcement: http://www.matapp.unimib.it/smi/coursesCortona.html
Duration: July 24 - August 11, 2006 (three weeks)
Language: English

### Program:

#### Part I: Stochastic Integration (Lecturer: U. Schmock)

Motivation: Let S denote a stochastic process describing the evolution of the discounted price of an asset, and let H be the process describing the (possibly random) number of these assets at any given time in the investor's portfolio. The gains and losses of this investment strategy H is given by the stochastic integral of H with respect to S. It therefore lies at the heart of modern, continuous-time mathematical finance to clarify, for which investment strategies H and price processes S this stochastic integral is mathematically well defined and what its properties are.

Contents:

1. We will follow the approach given in Ph. Protter's textbook, developing the theory of general stochastic integration with respect to semimartingales, which includes the cases of Brownian motion and Lévy processes. Applications of the theory, in particular to the modelling to the stochastic evolution of the term structure of interest rates, will be given in Prof. Runggaldier's part of the course. Ph. Protter's book contains an extensive list of exercises, which can be discussed in the problem-solving sessions.
2. Depending on time and interest of the course participants,
• credit risk modelling with an emphasis on CreditRisk+ and its extensions,
• properties of expected shortfall, and
• allocation of risk capital by expected shortfall
will be treated in the seminars. Lecture notes for preparing these seminars are available upon request.

Prerequisites: The main topic of the course requires familiarity with measure theoretic probability theory and basic results about martingales, because these will be used without proofs. The textbook by D. Williams and Chapter 2 of the textbook by S. Ethier and T. Kurtz are certainly a good source.

Literature:

• Philip E. Protter: Stochastic Integration and Differential Equations, (2nd edition), Applications of Mathematics: Stochastic Modelling and Applied Probability, Vol. 21, 2004, Springer-Verlag, ISBN 3-540-00313-4.
• David Williams: Probability with Martingales, Cambridge Mathematical Textbooks, 1991, Cambridge University Press, ISBN 0-521-40605-6
• Stewart N. Ethier and Thomas G. Kurtz: Markov Processes, Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics, 1986, John Wiley & Sons, ISBN 0-471-08186-8
• Uwe Schmock: Modelling Dependent Credit Risks with Extensions of CreditRisk+, An Implementation-Orientated Presentation, Lecture Notes, 2006

#### Part II: Term Structure of Interest Rates, Hedging (Lecturer: W. Runggaldier)

Basic Structure:

1. Term structure of interest rates (lectures and problem-solving sessions and seminars)
2. Hedging of general claims by martingale representation (Mainly problem-solving sessions and seminars)

Specific Structure:

1. Term structure of interest rates
• Basic concepts and preliminaries;
• Martingale models for the short rate and their calibration;
• Forward rate models (HJM framework);
• Change of numeraire techniques;
• LIBOR and swap market models.
Remarks: The basic theory will be presented in a Brownian framework. As the lectures on the general integration theory (Prof. Schmock) progress, also settings beyond the Brownian framework will be envisaged.
2. Hedging: After a short basic introduction during the lectures, this will be mainly a topic for the problem-solving sessions and seminars. As for the term structure, here too we shall start from a Brownian framwork that will then be gradually generalized in line with the general integration theory.

Literature:

• T. Björk, Arbitrage Theory in Continuous Time. Oxford University Press 2004 (2nd edition)
• D. Brigo, F. Mercurio, Interest Rate Theory - Theory and Practice, Springer-Verlag 2005 (2nd edition)
Possible additional material for lectures and problem-solving sessions and related specific journal articles will be made available on site.

### Participants:

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• Dr. Beatrice Acciaio (Perugia, Italy)
• Francesco Blasi (Rome, Italy)
• Guglielmo D'Amico (Rome, Italy)
• Asli Deniz (Turkey)
• Barbara Dengler (Vienna, Austria)
• Fernanda D'Ippoliti (Italy)
• Eleonora D'Urzo (Perugia, Italy)
• Mariagrazia Fedele (Foggia, Italy)
• Salvatore Federico (Pisa, Italy)
• Camilla Ferretti (Florence, Italy)
• Vjaceslavs Geveilers (Hamburg, Germany)
• Paolo Giulietti
• Fabio Gobbi (Florence, Italy)
• Julien Hunt (Louvain, Belgium)
• Luana Lombardi
• Marcello Mastroleo (Perugia, Italy)
• Lorenzo Mercuri (Milan, Italy)
• Valentina Prezioso
• Mathias Rafler (Potsdam, Germany)
• Norbert Renz (Ulm, Germany)
• Emilio Russo (Bergamo, Italy)
• Istvan Vajda (Budapest, Hungary)
• Carla Valente
• Nele Vandaele
• Anastasia Zakharova (Russia)
• Ling Xu (Leipzig, Germany)
Secretary: Mrs Silvana Boscherini

### Picture Gallery:

Hopefully coming soon ...
 Please send comments and suggestions to Uwe Schmock, email: schmock@fam.tuwien.ac.at. Last update: August 21, 2006