Il filtro di Kalman: analisi di serie storiche e applicazione ai dati finanziari

Historia magistra vitae, scriveva Cicerone nel De Oratore; il passato deve insegnare a comprendere meglio il futuro. Un concetto che a primo acchito può sembrare confinato nell'ambito della filosofia e della letteratura, ma che ha invece applicazioni matematiche e fisiche di estrema importanza. Esistono delle tecniche che permettono, conoscendo il passato, di effettuare delle migliori stime del futuro? Esistono dei metodi che permettono, conoscendo il presente, di aggiornare le stime effettuate nel passato? Nel presente elaborato viene illustrato come argomento centrale il filtro di Kalman, un algoritmo ricorsivo che, dato un set di misure di una certa grandezza fino al tempo t, permette di calcolare il valore atteso di tale grandezza al tempo t+1, oltre alla varianza della relativa distribuzione prevista; permette poi, una volta effettuata la t+1-esima misura, di aggiornare di conseguenza valore atteso e varianza della distribuzione dei valori della grandezza in esame. Si è quindi applicato questo algoritmo, testandone l'efficacia, prima a dei casi fisici, quali il moto rettilineo uniforme, il moto uniformemente accelerato, l'approssimazione delle leggi orarie del moto e l'oscillatore armonico; poi, introducendo la teoria di Kendall conosciuta come ipotesi di random walk e costruendo un modello di asset pricing basato sui processi di Wiener, si è applicato il filtro di Kalman a delle serie storiche di rendimenti di strumenti di borsa per osservare se questi si muovessero effettivamente secondo un modello di random walk e per prevedere il valore al tempo finale dei titoli.

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.

Pricing in stochastic-local volatility models with default

In recent years is becoming increasingly important to handle credit risk. Credit risk is the risk associated with the possibility of bankruptcy. More precisely, if a derivative provides for a payment at cert time T but before that time the counterparty defaults, at maturity the payment cannot be effectively performed, so the owner of the contract loses it entirely or a part of it. It means that the payoff of the derivative, and consequently its price, depends on the underlying of the basic derivative and on the risk of bankruptcy of the counterparty. To value and to hedge credit risk in a consistent way, one needs to develop a quantitative model. We have studied analytical approximation formulas and numerical methods such as Monte Carlo method in order to calculate the price of a bond. We have illustrated how to obtain fast and accurate pricing approximations by expanding the drift and diffusion as a Taylor series and we have compared the second and third order approximation of the Bond and Call price with an accurate Monte Carlo simulation. We have analysed JDCEV model with constant or stochastic interest rate. We have provided numerical examples that illustrate the effectiveness and versatility of our methods. We have used Wolfram Mathematica and Matlab.