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Seminari di Calcolo delle Probabilita'

Martedi' 21 Giugno 2011 - A. Cipriani, A. Gnoatto, J.M. Montes

ARGOMENTI: Seminari

SEMINARI DI CALCOLO DELLE PROBABILITÀ
Martedi' 21 giugno 2011 alle ore 15:00, in aula 2BC60 si terranno tre seminari di Calcolo delle Probabilita'.

-Programma
15:00 Alessandra Cipriani
"Pinning problem for the membrane model"

16:00 Alessandro Gnoatto
"A Multifactor Libor Market Model"

16:30 Juan Miguel Montes
"Computable pricing in continuous-time Markov chain term structure models: a combined analytic and simulations - based approach"

-Abstract Cipriani
In order to describe the behaviour of a random interface in terms of its curvature we will consider a lattice-based model given by a Gaussian field with the discrete biharmonic Green's function as covariance matrix. Our interest is focused primarily on the introduction of a pinning force which gives a reward every time the interface touches a hyperplane. The comparison of Green's functions which solve the same discrete PDE but with different boundary conditions will enable us to obtain good bounds on the covariances and the positivity of the free energy, which hint to the localization of the field (joint work with Erwin Bolthausen and Noemi Kurt).

-Abstract Gnoatto
We present a flexible approach for the valuation of interest rate derivatives based on Affine Processes. We extend the methodology proposed in Keller-Ressel et al. (2009) by changing the choice of the state space. We provide semi-closed-form solutions for the pricing of caps and floors. We then show that it is possible to price swaptions in a multifactor setting with a good degree of analytical tractability. This is done via the Edgeworth expansion approach developed in Collin-Dufresne and Goldstein (2002). A numerical exercise illustrates the flexibility of the Wishart Libor model in describing the movements of the implied volatility surface. (Joint work with José Da Fonseca and Martino Grasselli).

-Abstract Montes
While explicit formulae for pricing generic interest rate derivatives under a continuous-time Markov chain (CTMC) can be obtained, as shown in [4], these expressions however entail a matrix exponential that must be numerically computed. As an alternative, we investigate instead a combined analytic and simulations-based approach to pricing. Based on [3], we obtain an explicit formula, different from the matrix exponential as in [4]. It is based on probabilistic considerations related to Markov chains as in [2] and involves the computation of a finite, although random, number of powers of matrices rather than matrix exponentials. Furthermore, for deterministic payoffs, the randomness is expressed in terms only of the jump-count distribution. This facilitates Monte Carlo simulations because only the jump-counts and final jump-time enters the pricing formula. Using a recursively defined "Prototype Product", our method can be applied to derive bond prices as well as interest derivatives such as swaps caps and floors. In this way we show how one can obtain a pricing model that is suitable for a computational approach and that avoids a pure analytic or pure simulations-based solution. Finally, some numerical results are provided.

Rif. int. P. Dai Pra