Seminario Dottorato: The explicit Laplace transform for the Wishart process

Wednesday 2 November 2011 - Alessandro Gnoatto

ARGOMENTI: Seminari Dottorato

Wednesday 2 November 2011 h. 15:00, room 2BC30
Alessandro Gnoatto (Dip. Mat.)
"The explicit Laplace transform for the Wishart process"

The first part of the talk will provide an introduction to mathematical finance. We start with a brief historical and philosophical perspective on the field and briefly review the main problems one tries to solve. We concentrate on the valuation of simple derivatives and review the famous Black-Scholes formula. After that we recall how the weaknesses of this standard approach motivated the introduction of more advanced models, in particular stochastic volatility (SV) models. In the context of SV models we review the role played by characteristic functions and the fast Fourier transform. This last point will serve as an introduction to the results of the paper, where we derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru (1991). We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast, accurate and very useful for applications when dealing with stochastic volatility and stochastic correlation modelling.

Rif. int. C. Marastoni, T. Vargiolu

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