Lectures on Mathematical Finance

ON THE CALIBRATION OF TERM STRUCTURE MODELS

Wolfgang J. Runggaldier

Department of Pure and Applied Mathematics

University of Padova

e-mail: runggal@math.unipd.it

The presentation will consist of three parts :

1. Review of the classical approach. We shall discuss some basic notions concerning the classical approach to model calibration. In particular, we shall review the notion of market price of risk, martingale models for the short rate as well as affine term structure models and,finally, the so-called inversion of the yield curve as one of the main, but also more demanding steps of classical model calibration.

2. The Heath-Jarrow-Morton (HJM) framework. We shall discuss advantages and disadvantages of this framework with respect to the classical setup, the major advantage being that the inversion of the yield curve can be avoided. On the other hand, the basic objects are now infinite-dimensional and so we discuss ways to obtain finite-dimensional (possibly Markovian) realizations as well as their possible interpretation in terms of economic quantities.

3. Filtering techniques as alternative to traditional model calibration. Even if the finite dimensional realizations in the HJM framework may have an economic interpretation, not all of the factors in this realization need to be directly observable and, in addition, there are unknown parameters. The filtering approach (combined filtering and parameter estimation) allows to obtain a recursive Bayesian estimate of the incompletely observed factors as well as of the parameters on the basis of noisy data available for the observable quantities. We show that this allows not only to adapt recursively the model to the market (via the parameter estimation), but also to obtain arbitrage-free prices of non traded bonds and bond derivatives that take into account all the market information which becomes successively available over time.

APPLICATIONS OF MALLIAVIN CALCULUS IN MATHEMATICAL FINANCE

Arturo Kohatsu-Higa

Department of Economics

University Pompeu Fabra - Barcelona,

e-mail: kohatsu@upf.es

In this introductory talk I will explain some of the basics of Malliavin Calculus and its applications in Finance such as the Monte Carlo calculation of greeks, calculation of replicating portfolios and the study of problems of asimetric information.

STOCHASTIC CONTROL WITH CONSTRAINTS: APPLICATION TO CALIBRATION

Claude Martini

INRIA Rocquencourt - projet Mathfi

e-mail: Claude.Martini@inria.fr

We discuss in this talk the approach proposed by Avellaneda, Friedman, Holmes and Samperi to handle the following calibration (or inverse) problem: given a finite number of Call market prices, find a local volatility such that these prices are explained by the corresponding diffusion model for the underlying. In a more recent paper, Samperi gives a regularized formulation of the problem more suitable for a neat mathematical analysis. Nevertheless the best result he can reach is a local one: the algorithm works as long as the initial prices lie in a small open subset of the set of all arbitrage-free price sheets. We shall try to relax this restriction by mixing a more probabilistic point of view and results of classical optimization theory.

MODELS WITH TRANSACTION COSTS

Yuri M. Kabanov

Laboratoire de Mathèmatiques Central Economics and Mathematics Institute Universitè de Franche-Comtè

e-mail: Youri.Kabanov@Math.Univ-FComte.FR