Дисертації з теми "Diffusions affines"

Cette thèse traite l'inférence statistique de quelques processus de diffusion affine dans \( \R^m_+ \times \R^n \), avec m,n\in\N. Cette sous-classe de diffusions, notée par \textit{AD}(m,n), est appliquée à la tarification des options sur obligations et des actions, ce qui est illustré pour les modèles de Vasicek, Cox-Ingersoll-Ross (CIR) et Heston. Dans cette thèse, nous considérons deux différents modèles: le premier lorsque \( m = 1 \) et \( n \in \mathbb{N} \) et le deuxième lorsque \( m = 2 \) et \( n = 1 \). Pour le modèle \mathit{AD}(1, n), nous introduisons, au Chapitre 2, un résultat de classification où nous distinguons trois cas différents : sous-critique, critique et surcritique. Ensuite, nous étudions la stationnarité et l'ergodicité de sa solution sous certaines hypothèses sur les paramètres du drift. Pour le problème d'estimation paramétrique, nous utilisons deux méthodes différentes : l'estimation par maximum de vraisemblance (MLE) et l'estimation des moindres carrés conditionnels (CLSE). Au Chapitre 2, nous présentons l'estimateur obtenu par la méthode MLE basée sur des observations en temps continu et nous étudions sa consistance et son comportement asymptotique dans des cas ergodiques et non-ergodiques particuliers. Au Chapitre 3, nous présentons l'estimateur obtenu par la méthode CLSE basée sur des observations en temps continu puis discret avec haute fréquence et horizon infini et nous étudions sa consistance et son comportement asymptotique dans des cas ergodiques et non-ergodiques particuliers. Il est à noter ici que nous obtenons les mêmes résultats asymptotiques que dans le cas continu sous des hypothèses supplémentaires sur le pas de discrétisation \( \Delta_N \). Au Chapitre 4, nous étudions le modèle \mathit{AD}(2, 1), également appelé modèle de double Heston. Dans un premier temps, nous introduisons sa classification suivant les cas sous-critique, critique et surcritique. Dans un second temps, nous établissons les théorèmes de stationnarité et d'ergodicité y associés. Dans la partie statistique de ce chapitre, nous étudions les estimateurs par la méthode MLE et la méthode CLSE du modèle de double Heston en se basant sur des observations en temps continu dans le cas ergodique et nous introduisons les théorèmes de consistance et de normalité asymptotique pour chaque estimateur obtenu
This thesis deals with statistical inference of some particular affine diffusion processes in the state space \R_+^m\times\R^n, where m,n\in\N. Such subclass of diffusions, denoted by \mathit{AD}(m,n), is applied to the pricing of bond and stock options, which is illustrated for the Vasicek, Cox-Ingersoll-Ross (CIR) and Heston models. In this thesis, we consider two different cases : the first one is when m=1 and n\in\N and the second one is when m=2 and n=1. For the \mathit{AD}(1,n) model, we introduce, in Chapter 2, a classification result where we distinguish three different cases : subcritical, critical and supercritical. Then, we study the stationarity and the ergodicity of its solution under some assumptions on the drift parameters. For the parameter estimation problem, we use two different methods: the maximum likelihood estimation (MLE) and the conditional least squares estimation (CLSE). In Chapter 2, we present the estimator obtained by the MLE method based on continuous time observations and we study its consistency and its asymptotic behavior in ergodic and particular non-ergodic cases. In Chapter 3, we present the estimator obtained by the CLSE method based on continuous then discrete time observations with high frequency and infinite horizon and we study its consistency and its asymptotic behavior in ergodic and particular non-ergodic cases. It is worth to note here that we obtain the same asymptotic results in both discrete and continuous sets under additional assumptions on the discretization step \Delta_N. In Chapter 4, we study the \mathit{AD}(2,1) model, called also double Heston model, we introduce first its classification with respect to subcritical, critical and supercritical case and we establish the relative stationarity and ergodicity theorems. In the statistical part of this chapter, we study the MLE and the CLSE of the ergodic double Heston model based on continuous time observations and we introduce its consistency and asymtotic normality theorems for each estimation method

The central theme of this thesis is the study of stochastic processes in the infinite dimensional setup of (non-negative) measures. We introduce a class of measure-valued processes, which – in analogy to their finite dimensional counterparts – will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e. a representation of the conditional marginal moments via a system of finite dimensional linear PDEs. Furthermore, we characterize the corresponding infinitesimal generators obtaining a representation analogous to polynomial diffusions on R^m_+, in cases where their domain is large enough. In general, the infinite dimensional setting allows for richer specifications strictly beyond this representation. As a special case, we recover measure-valued affine diffusions, sometimes also called Dawson-Watanabe superprocesses. The polynomial framework is especially attractive from a mathematical finance point of view. Indeed, it allows to transfer some of the most famous finite dimensional models, such as the Black-Scholes one, to an infinite dimensional measure-valued setting. We outline the applicability of our approach to energy markets term structure modeling by introducing a framework allowing to employ (non-negative) measure-valued processes to consider electricity and gas futures. Interpreting the process' spatial structure as time to maturity, we show how the Heath-Jarrow-Morton (HJM) approach can be translated to such framework, thus guaranteeing arbitrage free modeling in infinite dimensions. We derive an analogue to the HJM-drift condition, then considering existence of (non-negative) measure-valued diffusions satisfying this condition in a Markovian setting. To analyze mathematically convenient classes of models, we also consider measure-valued polynomial and affine diffusions allowing for tractable pricing procedures via the moment formula and Fourier approaches.

This thesis examines the empirical performance of option pricing models in the continuous- time affine jump-diffusion (AID) class. In models of this class, the underlying returns are governed by stochastic volatility diffusions and/or jumps and the dynamics of the whole system has affine dependence on the state variables. The thesis consists of three essays. The first essay calibrates a wide range of AID option pricing models to S&P 500 index options. The aim is to empirically identify how best to structure two types of risk components- stochastic volatility and jumps - within the framework of multi-factor AID specifications. Our specification analysis shows that the specifications with more-than-two diffusions perform well and that a three-factor specification should be preferred, in which jump intensities are allowed to depend on an independent diffusion process. Having identified the well-performing pricing model specifications, the second essay examines how such a model can be used to forecast realized volatility using only option prices as an input. To do so, the dynamics of volatility implied by the model are used to construct a forecasting equation in which the spot volatilities extracted from observed option prices act as the key predictors. The analysis indicates that the option-based multi-factor forecasting model outperforms other popular models in forecasting realized volatility of S&P 500 Index returns over most of the short-term horizons considered. The final essay investigates if a two-factor AJD model can fit option pricing patterns generated by a single-factor long memory volatility model. Our simulation experiments show that this model does well in this respect. Remarkably, however, at the fitted parameter values it does not generate the volatility auto-correlation patterns that are characteristic of long-memory volatility models.

The object of study in this thesis is the most general affine term structure model characterized by Duffie and Kan (1996), which nests, as special cases, many of the interest rate models previously formulated in the literature. The purpose of the dissertation is two-fold: to derive fast and accurate pricing solutions for the general term structure framework under analysis, which enable the effective use of model’ specifications yet unexplored due to their analytical intractability, and, to implement a simple and robust model’ estimation methodology that enhances the model' fit to the market interest rates covariance surface. Concerning the first (theoretical) goal, analytical exact pricing solutions, for several interest rate derivatives, are first derived under a (simpler and) nested Gaussian affine specification Then, and as the main contribution of the present dissertation, such Gaussian formulae are transformed into first order approximate closed-form pricing solutions for the most general stochastic volatility model’ formulation. These approximate solutions arc shown to be both extremely fast to implement and accurate, which make them an effective alternative to the existing numerical pricing methods available. Related to second thesis’ (empirical) goal, and in order to enable the model’ estimation from a panel-data of interest rate contingent claims’ prices, a general equilibrium model’ specification is derived under non-severc preferences’ assumptions and in the context of a monetary economy. The corresponding state-space model’ specification is estimated through a non-linear Kalman filter and using a panel-data of not only swap rates (as it is usual in the Finance literature) but also (for the first time) of caps and European swaptions prices It is shown that although the model' fit to the level of the yield curve is extremely good, short-term caps and swaptions are systematically mispriced. Finally, a time-inhomogeneous HJM formulation is proposed, and the model’ fit to the market interest rates covariance matrix is substantially improved.

This thesis is dedicated to two problems arising from geometric control theory, regarding control-affine systems $\dot q= f_0(q)+\sum_{j=1}^m u_j f_j(q)$, where $f_0$ is called the drift. In the first part we extend the concept of complexity of non-admissible trajectories, well understood for sub-Riemannian systems, to this more general case, and find asymptotic estimates. In order to do this, we also prove a result in the same spirit as the Ball-Box theorem for sub-Riemannian systems, in the context of control-affine systems equipped with the L1 cost. Then, in the second part of the thesis, we consider a family of 2-dimensional driftless control systems. For these, we study how the set where the control vector fields become collinear affects the diffusion dynamics. More precisely, we study whether solutions to the heat and Schrödinger equations associated with this Laplace-Beltrami operator are able to cross this singularity, and how its the presence affects the spectral properties of the operator, in particular under a magnetic Aharonov–Bohm-type perturbation.

Cette thèse est consacrée au problème d'évaluation de produits exotiques dans un modèle de diffusion a sauts de type affine-quadratique. Les formules d'évaluation sont obtenues de façon explicite en utilisant la caractérisation du modèle affine qui ramène le calcul de la transforme de Laplace d'une variable aléatoire en la détermination de fonctions satisfaisant des équations de Riccati. Nous considérons ensuite le variance swap présentant le produit financier, les produits dérivés de ce contrat et les méthodes d'évaluation. Nous étudions en détail les options sur variance afin d'obtenir un modèle permettant d'évaluer et de couvrir les produits sur variance. Nous cherchons la dynamique d'un variance swap pour en déduire la dynamique des prix de produits dérivés. Nous portons une attention particulière aux modèles affine-quadratiques pour lesquels, dans certains cas particulier, nous obtenons des formules fermées. La dernière partie de la thèse est consacrée au modèles hybrides pour calculer les prix de produits actions-taux et actions-crédits
This thesis is concerned with the pricing of exotic options within an affine quadratic jump diffusion model. In this case the computational difficulties can be reduced to solving a system of Riccati equations a number of times and performing a numerical integration using the resulting values via the FFT technique. We then present the variance swap contract and explain the reasons why it became a traded underlying. Since the variance swap contract is just a forward on the annualised realised variance we choose to infer its dynamic from the dynamic of the stock price. We therefore make the variance swap the new underlying and diffuse it over time in order to price options on the quadratic variation and more generally derivatives on the volatility. The properties of the affine-quadratic model allow us in some special cases to recover closed-form solutions. To conclude we extend the approach to the hybrid markets and consider the equity-rate and equity-credit products

This thesis examines the empirical performance of four Affine Jump Diffusion models in pricing and hedging S&P 500 Index options: the Black Scholes (BS) model, Heston???s Stochastic Volatility (SV) model, a Stochastic Volatility Price Jump (SVJ) model and a Stochastic Volatility Price-Volatility Jump (SVJJ) model. The SVJJ model structure allows for simultaneous jumps in price and volatility processes, with correlated jump size distributions. To the best of our knowledge this is the first empirical study to test the hedging performance of the SVJJ model. As part of our research we derive the SVJJ model minimum variance hedge ratio. We find the SVJ model displays the best price prediction. The SV model lacks the structural complexity to eliminate Black Scholes pricing biases, whereas our results indicate the SVJJ model suffers from overfitting. Despite significant evidence from in and out-of-sample pricing that the SV and SVJ models were better specified than the BS model, this did not result in an improvement in dynamic hedging performance. Overall the BS delta hedge and SV minimum variance hedge produced the lowest errors, although their performance across moneyness-maturity categories differed greatly. The SVJ model???s results were surprisingly poor given its superior performance in out-of-sample pricing. We attribute the inadequate performance of the jump models to the lower hedging ratios these models provided, which may be a result of the negative expected jump sizes.

This thesis presents the modeling of credit risk by using structural approach. Three fundamental questions of credit risk literature are analyzed throughout the research: modeling single firm credit risk, modeling portfolio credit risk and credit risk pricing. First we analyze these questions under the assumptions that firm value follows a geometric Brownian motion and the interest rates are constant. We discuss the weaknesses of the geometric brownian motion assumption in explaining empirical properties of real data. Then we propose a new extended model in which asset value, volatility and interest rates follow affine jump diffusion processes. In our extended model volatility is stochastic, asset value and volatility has correlated jumps and interest rates are stochastic and have jumps. Finally, we analyze the modeling of single firm credit risk and credit risk pricing by using our extended model and show how our model can be used as a solution for the problems we encounter with simple models.

This thesis considers the price and characteristics of an exotic option called the Volatility-Cap-Target-Level(VCTL) option. The payoff function is a simple European option style but the underlying value is a dynamic portfolio which is comprised of two components: A risky asset and a non-risky asset. The non-risky asset is a bond and the risky asset can be a fund or an index related to any asset category such as equities, commodities, real estate, etc. The main purpose of using a dynamic portfolio is to keep the realized volatility of the portfolio under control and preferably below a certain maximum level, denoted as the Volatility-Cap-Target-Level (VCTL). This is attained by a variable allocation between the risky asset and the non-risky asset during the maturity of the VCTL-option. The allocation is reviewed and if necessary adjusted every 15th day. Adjustment depends entirely upon the realized historical volatility of the risky asset. Moreover, it is assumed that the risky asset is governed by a certain group of stochastic differential equations called affine jump-diffusion models. All models will be calibrated using out-of-the money European call options based on the Deutsche-Aktien-Index(DAX). The numerical implementation of the portfolio diffusions and the use of Monte Carlo methods will result in different VCTL-option prices. Thus, to price a nonstandard product and to comply with good risk management, it is advocated that the financial institution use several research models such as the SVSJ- and the Seppmodel in addition to the Black-Scholes model. Keywords: Exotic option, basket option, risk management, greeks, affine jumpdiffusions, the Black-Scholes model, the Heston model, Bates model with lognormal jumps, the Bates model with log-asymmetric double exponential jumps, the Stochastic-Volatility-Simultaneous-Jumps(SVSJ)-model, the Sepp-model.

Esta Tese explora o Processo de Cox quando sua intensidade pertence a uma família de difusões afim. A forma da funçâo densidade de Probabilidade do Processo de Cox é obtida quando a intensidade é descrita por uma difusão fim d-dimensional arbitrária. Analisa-se também o acoplamento e convergência para o Processo de Cox com intensidade afim. Para ilustrar assume-se que a intensidade do Processo é governada por uma difusão de Feller e resultados mais detalhados são obtidos. Adicionalmente, os parâmetros da intensidade do Processo são estimados por meio do Filtro de Kalman conjugado com o estimador de Quase-Máxima Verossimilhança.
This Thesis deals with the Cox Process when its intensity belongs to a family of affine diffusions. The form of the probability density function of the Cox process is obtained when the density is described by an arbitrary d-dimensional affine diffusion. Coupling and convergence results are also addressed for a general Cox process with affine intensity. We adopted the Feller diffusion for driving the underlying intensity of the Cox Process to illustrate our results. Additionally the parameters of the underlying intensity processes are estimated by means of the Kalman Filter in conjunction with Quasi-Maximum Likelihood estimation.

Les équations différentielles stochastiques sont des outils des mathématiques très utilisés, que ce soit en finance, en physique ou encore en biologie ; ces modèles peuvent être très efficaces pour modéliser de nombreux phénomènes. Afin de mieux comprendre ces équations différentielles stochastiques, on s'intéresse dans cette thèse aux solutions de certaines d'entre elles, appelées processus de Bernstein ou processus de Schrödinger, dont la construction fait apparaître des propriétés liées à l'équation de la chaleur. Deux catégories de résultats sont présentés ici. Des résultats purement liés à l'équation de la chaleur et complètement indépendants du contexte probabiliste, comme par exemple le calcul explicite des flots associés à l'équation de la chaleur pour trois types de potentiels, ou encore la structure de l'algèbre de Lie des symétries de ces équations. D'autres résultats sont liés aux processus stochastiques, on donne ici une paramétrisation des modèles affines de taux d’intérêt à un paramètre (modèles utilisés en finance) par des processus de Bernstein ainsi une condition nécessaire à la paramétrisation des modèles affines en dimension par des processus de Bernstein
Stochastic differential equations are a powerfull tool of mathematics. Applications range from finance or physics to biology. Those models can be very efficient to modelise numerous phenomenons where uncertainties are involved. In order to have a better understanding of those stochastic differential equations, this work studies the solutions of a subclass, called Bernstein (or Schrödinger) processes. Those processes are linked to the heat equation by construction. Two types of results are presented here. Some are about the heat equation and totally independant from any probabilistic context. For example, we compute the flows associated with the heat equation for three different potential and we study the structure of the Lie algebra of symmetries for those equations. Other results are presented: we show how it is possible to parametrize one factor affine models with Bernstein processes. We also give a necessary condition for the parametrization of -factor affine models with Berntein processes

A "self-exciting" market is one in which the probability of observing a crash increases in response to the occurrence of a crash. It essentially describes cases where the initial crash serves to weaken the system to some extent, making subsequent crashes more likely. This thesis investigates if equity markets possess this property. A self-exciting extension of the well-known jump-based Bates (1996) model is used as the workhorse model for this thesis, and a particle-filtering algorithm is used to facilitate estimation by means of maximum likelihood. The estimation method is developed so that option prices are easily included in the dataset, leading to higher quality estimates. Equilibrium arguments are used to price the risks associated with the time-varying crash probability, and in turn to motivate a risk-neutral system for use in option pricing. The option pricing function for the model is obtained via the application of widely-used Fourier techniques. An application to S&P500 index returns and a panel of S&P500 index option prices reveals evidence of self excitation.

In life insurance contracts, actuaries generally value premiums using deterministic mortality rates and interest rates. They have ignored them stochastically in most of the studies. However it is known that neither interest rates nor mortality rates are constant. It is also known that companies may encounter insolvency problems such as ruin, so the ruin probability need to be added to the valuation of the life insurance contracts process. Insurance companies should model their surplus processes to price some types of life insurance contracts and to see risk position. In this study, mortality rates and surplus processes are modeled and financial strength of companies are utilized when pricing life insurance contracts.

This dissertation is dedicated to study the design and utilization of financial contracts and pricing mechanisms for managing the demand/price risks in electricity markets and the price risks in carbon emission markets from different perspectives. We address the issues pertaining to the efficient computational algorithms for pricing complex financial options which include many structured energy financial contracts and the design of economic mechanisms for managing the risks associated with increasing penetration of renewable energy resources and with trading emission allowance permits in the restructured electric power industry. To address the computational challenges arising from pricing exotic energy derivatives designed for various hedging purposes in electricity markets, we develop a generic computational framework based on a fast transform method, which attains asymptotically optimal computational complexity and exponential convergence. For the purpose of absorbing the variability and uncertainties of renewable energy resources in a smart grid, we propose an incentive-based contract design for thermostatically controlled loads (TCLs) to encourage end users' participation as a source of DR. Finally, we propose a market-based approach to mitigate the emission permit price risks faced by generation companies in a cap-and-trade system. Through a stylized economic model, we illustrate that the trading of properly designed financial options on emission permits reduces permit price volatility and the total emission reduction cost.

Cette thèse propose de coupler deux outils préexistant pour la modélisation mathématique en mécanique : l’homogénéisation périodique et la réduction de modèle, afin de modéliser la corrosion des structures de béton armé exposées à la pollution atmosphérique et au sel marin. Cette dégradation est en effet difficile à simuler numériquement, eu égard la forte hétérogénéité des matériaux concernés, et la variabilité de leur microstructure. L’homogénéisation périodique fournit un modèle multi-échelle permettant de s’affranchir de la première de ces deux difficultés. Néanmoins, elle repose sur l’existence d’un volume élémentaire représentatif (VER) de la microstructure du matériau poreux modélisé. Afin de prendre en compte la variabilité de cette dernière, on est amenés à résoudre en temps réduit les équations issues du modèle multi-échelle pour un grand nombre VER. Ceci motive l’utilisation de la méthode POD de réduction de modèle. Cette thèse propose de recourir à des transformations géométriques pour transporter ces équations sur la phase fluide d’un VER de référence. La méthode POD ne peut, en effet, pas être utilisée directement sur un domaine spatial variable (ici le réseau de pores du matériau). Dans un deuxième temps, on adapte ce nouvel outil à l’équation de Poisson-Boltzmann, fortement non linéaire, qui régit la diffusion ionique à l’échelle de la longueur de Debye. Enfin, on combine ces nouvelles méthodes à des techniques existant en réduction de modèle (MPS, interpolation ITSGM), pour tenir compte du couplage micro-macroscopique entre les équations issues de l’homogénéisation périodique
In this thesis, we manage to combine two existing tools in mechanics: periodic homogenization, and reduced-order modelling, to modelize corrosion of reinforced concrete structures. Indeed, chloride and carbonate diffusion take place their pores and eventually oxydate their steel skeleton. The simulation of this degradation is difficult to afford because of both the material heterogenenity, and its microstructure variability. Periodic homogenization provides a multiscale model which takes care of the first of these issues. Nevertheless, it assumes the existence of a representative elementary volume (REV) of the material at the microscopical scale. I order to afford the microstructure variability, we must solve the equations which arise from periodic homogenization in a reduced time. This motivates the use of model order reduction, and especially the POD. In this work we design geometrical transformations that transport the original homogenization equations on the fluid domain of a unique REV. Indeed, the POD method can’t be directly performed on a variable geometrical space like the material pore network. Secondly, we adapt model order reduction to the Poisson-Boltzmann equation, which is strongly nonlinear, and which rules ionic electro diffusion at the Debye length scale. Finally, we combine these new methods to other existing tools in model order reduction (ITSGM interpolatin, MPS method), in order to couple the micro- and macroscopic components of periodic homogenization

Yes
This paper compares several widely-used and recently-developed methods to extract risk-neutral densities (RND) from option prices in terms of estimation accuracy. It shows that positive convolution approximation method consistently yields the most accurate RND estimates, and is insensitive to the discreteness of option prices. RND methods are less likely to produce accurate RND estimates when the underlying process incorporates jumps and when estimations are performed on sparse data, especially for short time-to-maturities, though sensitivity to the discreteness of the data differs across different methods.
The full-text of this article will be released for public view at the end of the publisher embargo on 9 Sep 2021.

In this Thesis, we address the modelling of stochastic mortality, a key issue for life insurance, pension funds, public policy and fiscal planning. Indeed, the prospective increase of longevity can be an advantage for individuals, but it represents also a relevant social achievement. The stability and consistency of social welfare systems are put in danger worldwide due to the combined phenomenon of the progressive increase in life expectancy, along with the reduction of birth-rates in industrialized Countries. This phenomenon needs to be interpreted in the context of the connected world in which we live, where the multiple networks arising from the globalization, the Internet communication and the global economic development propagate any event in a very short time, making risks more complex. Due to their very nature, insurance and reinsurance deal with several risks on their balance sheet and, when determining the total risk of a portfolio, they need to establish the rules for aggregating the various risks that compose it. The introduction of market-consistent accounting and risk-based solvency requirements has called for the integration of mortality risk analysis into stochastic valuation models; moreover mortality-linked securities have attracted the interest of capital market investors, who in turn demand transparent tools to price demographic and financial risks in an integrated fashion. Accordingly, a coherent mathematical framework for studying the changes in financial and demographic conditions over time, is suitable. The class of the affine processes has been used in a wide range of applications in financial and actuarial sciences, thanks to its computational tractability and flexibility. For instance, affine processes have been extensively used in modelling the term structure of interest rates, that underpin extensive literatures on the pricing of bonds and interest-rate derivatives and are also at the basis of many of the pricing systems used by the financial industry. Affine models for the force of mortality have been developed in the literature under the assumption of both dependence and independence between mortality and interest rate dynamics. The core of this Thesis are the affine models and their properties for modelling the evolution of mortality. We propose and discuss two contributions: (i) we fit and compare past mortality trends among different Countries under the mathematical framework of the Feller process; (ii) we design a multiplicative affine model for the future evolution of mortality, by combining two components: the forecast provided by any existing mortality model, representing the deterministic baseline, and an affine driving process that stochastically affects the baseline over the forecasting time horizon. The so structured model not only is affine, thus fitting well our targets, but, when assessing its forecasting performance, it proves to be parsimonious and to provide a more accurate forecast with respect to the baseline. Within such a model, the affine driving factor is tasked with describing the dynamics over time of a measure of the fitting error of the existing mortality model providing the baseline and it is stochastically described by a Cox-Ingersoll-Ross process. For our numerical application, we choose, as the existing mortality model giving the baseline, the Cairns-Blake-Dowd (or M5) model, that is combined with the CIR process describing the stochastic factor affecting the baseline in a multiplicative way. The resulting model is called mCBD. Using the Italian females mortality data, for fixed ages, and implementing the backtesting procedure, over both a static time horizon and fixed-length windows rolling one-year ahead through time, we empirically test the performance of the CBD and the mCBD models in forecasting death rates. On the basis of average measures of forecasting errors and information criteria, we demonstrate that the mCBD model is a parsimonious model providing better results in terms of predictive accuracy than the CBD model and showing a stronger potential to gain accuracy in the long-run when a rolling windows analysis (dynamic approach) is performed. To conclude, in the Thesis, we explore and test the properties and capabilities of some affine models in fitting and forecasting mortality data both by themselves and as dynamic driving processes multiplying a deterministic baseline. Combining models and mixing techniques prove to give satisfactory results and show a concrete potential to bring the research forward. Our future research is thus oriented to use approaches that combine Monte Carlo simulations and benefit from the synergy between different techniques.