The Libor Market Model: from theory to calibration

This thesis is focused on the financial model for interest rates called the LIBOR Market Model. In the appendixes, we provide the necessary mathematical theory. In the inner chapters, firstly, we define the main interest rates and financial instruments concerning with the interest rate models, then, we set the LIBOR market model, demonstrate its existence, derive the dynamics of forward LIBOR rates and justify the pricing of caps according to the Black’s formula. Then, we also present the Swap Market Model, which models the forward swap rates instead of the LIBOR ones. Even this model is justified by a theoretical demonstration and the resulting formula to price the swaptions coincides with the Black’s one. However, the two models are not compatible from a theoretical point. Therefore, we derive various analytical approximating formulae to price the swaptions in the LIBOR market model and we explain how to perform a Monte Carlo simulation. Finally, we present the calibration of the LIBOR market model to the markets of both caps and swaptions, together with various examples of application to the historical correlation matrix and the cascade calibration of the forward volatilities to the matrix of implied swaption volatilities provided by the market.

Jarrow-Yildirim model for inflation: theory and applications

This thesis deals with inflation theory, focussing on the model of Jarrow & Yildirim, which is nowadays used when pricing inflation derivatives. After recalling main results about short and forward interest rate models, the dynamics of the main components of the market are derived. Then the most important inflation-indexed derivatives are explained (zero coupon swap, year-on-year, cap and floor), and their pricing proceeding is shown step by step. Calibration is explained and performed with a common method and an heuristic and non standard one. The model is enriched with credit risk, too, which allows to take into account the possibility of bankrupt of the counterparty of a contract. In this context, the general method of pricing is derived, with the introduction of defaultable zero-coupon bonds, and the Monte Carlo method is treated in detailed and used to price a concrete example of contract. Appendixes: A: martingale measures, Girsanov's theorem and the change of numeraire. B: some aspects of the theory of Stochastic Differential Equations; in particular, the solution for linear EDSs, and the Feynman-Kac Theorem, which shows the connection between EDSs and Partial Differential Equations. C: some useful results about normal distribution.

An introduction to scalar-tensor gravity: theory and experiment in the Brans-Dicke model

Le teorie della gravità scalare-tensore sono una classe di teorie alternative alla rel- atività generale in cui l’interazione gravitazionale è descritta sia dalla metrica, sia da un campo scalare. Ne costituisce un esempio caratteristico la teoria di Brans-Dicke, in- trodotta come estensione della relatività generale in modo da renderla conforme con il principio di Mach. Il presente lavoro di tesi è volto a presentare un’analisi di questa teoria nei suoi aspetti principali, studiandone i fondamenti teorici e il modello cosmologico derivante, sottolineandone inoltre i limiti e le criticità; in seguito vengono esposti i risultati degli esperimenti fino ad ora svolti per verificare fondamenti e previsioni del modello.

Adiabatic and local approximations for the kohn-sham potential in the hubbard model

We obtain the exact time-dependent Kohn-Sham potentials Vks for 1D Hubbard chains, driven by a d.c. external field, using the time-dependent electron density and current density obtained from exact many-body time-evolution. The exact Vxc is compared to the adiabatically-exact Vad-xc and the “instantaneous ground state” Vigs-xc. The effectiveness of these two approximations is analyzed. Approximations for the exchange-correlation potential Vxc and its gradient, based on the local density and on the local current density, are also considered and both physical quantities are observed to be far outside the reach of any possible local approximation. Insight into the respective roles of ground-state and excited-state correlation in the time-dependent system, as reflected in the potentials, is provided by the pair correlation function.

Analytical model for the prediction of wrinkling of rear pressure bulkheads

In Airbus GmbH (Hamburg) has been developed a new design of Rear Pressure Bulkhead (RPB) for the A320-family. The new model has been formed with vacuum forming technology. During this process the wrinkling phenomenon occurs. In this thesis is described an analytical model for prediction of wrinkling based on the energetic method of Timoshenko. Large deflection theory has been used for analyze two cases of study: a simply supported circular thin plate stamped by a spherical punch and a simply supported circular thin plate formed with vacuum forming technique. If the edges are free to displace radially, thin plates will develop radial wrinkles near the edge at a central deflection approximately equal to four plate thicknesses w0/ℎ≈4 if they’re stamped by a spherical punch and w0/ℎ≈3 if they’re formed with vacuum forming technique. Initially, there are four symmetrical wrinkles, but the number increases if the central deflection is increased. By using experimental results, the “Snaptrhough” phenomenon is described.

Model categories

The purpose of this thesis is to present some fundamental results about model categories, and to give some examples of categories that can be equipped with a model structure.