Littérature scientifique sur le sujet « Calibration sets »

Zentrale Grenzwertsätze und Konfidenzmengen werden in zwei verschiedenen, nichtparametrischen, inversen Problemen ähnlicher Struktur untersucht, und zwar in der Kalibrierung eines exponentiellen Lévy-Modells und im Dekonvolutionsmodell. Im ersten Modell wird eine Geldanlage durch einen exponentiellen Lévy-Prozess dargestellt, Optionspreise werden beobachtet und das charakteristische Tripel des Lévy-Prozesses wird geschätzt. Wir zeigen, dass die Schätzer fast sicher wohldefiniert sind. Zu diesem Zweck beweisen wir eine obere Schranke für Trefferwahrscheinlichkeiten von gaußschen Zufallsfeldern und wenden diese auf einen Gauß-Prozess aus der Schätzmethode für Lévy-Modelle an. Wir beweisen gemeinsame asymptotische Normalität für die Schätzer von Volatilität, Drift und Intensität und für die punktweisen Schätzer der Sprungdichte. Basierend auf diesen Ergebnissen konstruieren wir Konfidenzintervalle und -mengen für die Schätzer. Wir zeigen, dass sich die Konfidenzintervalle in Simulationen gut verhalten, und wenden sie auf Optionsdaten des DAX an. Im Dekonvolutionsmodell beobachten wir unabhängige, identisch verteilte Zufallsvariablen mit additiven Fehlern und schätzen lineare Funktionale der Dichte der Zufallsvariablen. Wir betrachten Dekonvolutionsmodelle mit gewöhnlich glatten Fehlern. Bei diesen ist die Schlechtgestelltheit des Problems durch die polynomielle Abfallrate der charakteristischen Funktion der Fehler gegeben. Wir beweisen einen gleichmäßigen zentralen Grenzwertsatz für Schätzer von Translationsklassen linearer Funktionale, der die Schätzung der Verteilungsfunktion als Spezialfall enthält. Unsere Ergebnisse gelten in Situationen, in denen eine Wurzel-n-Rate erreicht werden kann, genauer gesagt gelten sie, wenn die Sobolev-Glattheit der Funktionale größer als die Schlechtgestelltheit des Problems ist.<br>Central limit theorems and confidence sets are studied in two different but related nonparametric inverse problems, namely in the calibration of an exponential Lévy model and in the deconvolution model. In the first set-up, an asset is modeled by an exponential of a Lévy process, option prices are observed and the characteristic triplet of the Lévy process is estimated. We show that the estimators are almost surely well-defined. To this end, we prove an upper bound for hitting probabilities of Gaussian random fields and apply this to a Gaussian process related to the estimation method for Lévy models. We prove joint asymptotic normality for estimators of the volatility, the drift, the intensity and for pointwise estimators of the jump density. Based on these results, we construct confidence intervals and sets for the estimators. We show that the confidence intervals perform well in simulations and apply them to option data of the German DAX index. In the deconvolution model, we observe independent, identically distributed random variables with additive errors and we estimate linear functionals of the density of the random variables. We consider deconvolution models with ordinary smooth errors. Then the ill-posedness of the problem is given by the polynomial decay rate with which the characteristic function of the errors decays. We prove a uniform central limit theorem for the estimators of translation classes of linear functionals, which includes the estimation of the distribution function as a special case. Our results hold in situations, for which a square-root-n-rate can be obtained, more precisely, if the Sobolev smoothness of the functionals is larger than the ill-posedness of the problem.

La vegetazione di ambiente montano è nota per essere sensibile alle variazioni climatiche. Il forte gradiente climatico altitudinale, che caratterizza le aree montane, dà luogo ad un marcato gradiente ecologico, in cui numerosi ecotoni sono presenti in una piccola area. Sequenze polliniche investigate all’interno o poco al di sopra dell’ecotono attuale della timberline sono archivi ideali per investigare le relazioni esistenti fra clima ed ecosistemi. Ricostruzioni quantitative delle condizioni climatiche del passato da record pollinici fossili richiedono la comprensione della rappresentazione pollinica attuale lungo gradienti climatici ed ecologici. Obbiettivi di questa ricerca di dottorato sono: lo sviluppo di transetti altitudinali di polline-vegetazione-clima attuale, l’elaborazione di modelli e la loro validazione per valutare le relazioni polline-clima grazie a gradienti altitudinali, la ricerca di nuovi archivi naturali ad alta risoluzione per ottenere dati proxy e ricostruire le variazioni paleoambientali e paleoclimatiche durante l’Olocene, l’applicazione di questi modelli ai dati pollinici stratigrafici ed infine il confronto dei risultati con diverse ricostruzioni su base proxy. E’ stata dimostrata l’importanza di transetti altitudinali locali con campioni pollinici moderni e temperature sito-specifiche come strumento per le ricostruzioni climatiche. Sono stati sviluppati due transetti altitudinali (la Valle di La Thuile e l’Alta Val Brembana) per ottenere consistenti correlazioni locali polline-clima, per trovare taxa pollinici sensibili utili per le ricostruzioni paleoclimatiche, per stimare gli effetti di parametri locali (lapse rate, clima, trasporto di polline verso l’alto e impatto antropico) e infine sono stati utilizzati come test sets per valutare i modelli polline-clima basati su calibration sets estratti dall’European Modern Pollen Database. Sono state investigate le relazioni rappresentazione pollinica attuale-vegetazione-clima lungo un gradiente altitudinale in Alta Val Brembana. Qui la rappresentazione pollinica attuale (trappole polliniche e campioni di muschio), vegetazione, quota e clima sono stati rilevati in 16 siti di campionamento situati lungo un gradiente altitudinale da 1240 m slm a 2390 m slm. I risultati della CCA mostrano una buona concordanza con studi precedenti, che hanno individuato la quota come il maggior gradiente nella variazione della rappresentazione pollinica attuale e della vegetazione in aree montane. Lo studio stratigrafico di proxies paleoecologici e sedimentari nella torbiera Armentarga hanno permesso di ricostruire la vegetazione e il clima negli ultimi 10 ka in un distretto oceanico di alta quota delle Alpi Italiane. Sono state ottenute e validate ricostruzioni quantitative di Tluglio e Pann applicando funzioni di trasferimento create da un vasto dataset di calibrazione polline-clima. Il record paleobotanico proveniente dalla torbiera Armentarga ha mostrato che questa composizione altitudinale della vegetazione è principalmente guidata da un aumento di precipitazioni nel medio-tardo Olocene, indipendente dalle sequenze millenarie di anomalie termiche già note da altri proxies alpini di alta quota (ad es. ghiacciai, timberline, chironomidi, speleotemi). Le variazioni delle precipitazioni annuali durante l’Olocene sono avvenute in tre step principali, partendo da un inizio Olocene moderatamente umido, caratterizzato dalla presenza precoce di foreste di Alnus viridis e seguito da un primo incremento delle precipitazioni a partire da 6.2 ka cal BP. Uno step prominente in avanti avvenne nella transizione medio-tardo Olocene, datata alla torbiera Armentarga tra 4.7 e 3.9 ka, che ha portato verso i valori attuali tipici di clima oceanici montani (Pann 1700-1850 mm) ed è stato probabilmente accompagnato da un incremento di precipitazioni nevose e runoff ed ha avuto un maggior impatto sulla depressione della timberline e l’espansione di praterie.<br>Montane vegetation is traditionally known to be particularly sensitive to climate changes. The strong elevational climatic gradient that characterises mountain areas results in a steep ecological slope, with several ecotones occurring in a small area. Pollen sequences investigated in or shortly above the modern timberline ecotone are ideal archives to analyze the relationships between climate and ecosystems. Quantitative reconstruction of past climate conditions from fossil pollen records requires understanding modern pollen representation along climatic and ecological gradients. The aims of this PhD research are the development of modern pollen-vegetation-climate elevational transects, model processing and validation for evaluating pollen-climate relationships thanks to elevational gradients, looking for new high-resolution natural archives to obtain proxy data and to reconstruct paleoenvironmental and palaeoclimatic changes during the Holocene, the application of these models to pollen-stratigraphical data and comparing the results with different proxy-based reconstructions. The importance of local elevational transects of modern pollen samples with site-specific temperature as a tool for paleoclimate reconstructions in the Alps was demonstrated. The two elevational transects (La Thuile Valley and Upper Brembana Valley) were developed to derive consistent local pollen-climate correlations, to find sensitive pollen taxa useful for paleoclimate reconstructions; to estimate the effects of local parameters (elevational lapse rate, climate, uphill pollen transport and human impact) and were used as test sets to evaluate pollen-climate models based on calibration sets extracted from the European Modern Pollen Database. Modern pollen assemblages-vegetation-climate relationships along an elevational gradient in the Upper Brembana Valley were investigated. Here modern pollen assemblages (pollen traps and moss samples), vegetation, elevation and climate have been collected at 16 sampling sites placed along an elevational gradient stretching from 1240 m asl to 2390 m asl. The results of CCA analysis demonstrated a general good agreement with previous studies, which identified elevation as the main gradient in the variation of modern pollen and vegetation assemblages in mountain areas. The stratigraphic study of paleoecological and sedimentary proxies in the Armentarga peat bog allowed to reconstruct the vegetation and climate history during the last 10 ka in a high-elevation, oceanic district of the Italian Alps. Quantitative reconstructions of Tjuly and Pann were obtained and validated by applying numerical transfer functions built on an extensive calibration pollen-climate dataset. The palaeobotanical record of the Armentarga peat bog has shown this elevational vegetation arrangement to be primarily driven by a Middle to late Holocene precipitation increase, substantially independent from the millennial sequence of thermal anomalies already known from other high-elevation Alpine proxies (i.e. glaciers, timberline, chironomids, speleothems). Changes in annual precipitation occurred in three main steps during the Holocene, starting with a moderately humid early Holocene marked by early occurrence of the Alnus viridis dwarf forests, and followed by a first step of precipitation increase starting at 6.2 ka cal BP. A prominent step forward occurred at the Middle to Late Holocene transition, dated between 4.7 and 3.9 ka at the Armentarga site, which led to present values typical for oceanic mountain climates (Pann 1700-1850 mm) and was probably accompanied by increased snowfall and runoff, and had a major impact on timberline depression and grassland expansion.

Pas de résumé<br>In this thesis, we study the problem of registering 2D image sets and 3D point clouds under threedifferent acquisition set-ups. The first set-up assumes that the image sets are captured using 2Dcameras that are fully calibrated and coupled, or rigidly attached, with a 3D sensor. In this context,the point cloud from the 3D sensor is registered directly to the asynchronously acquired 2D images.In the second set-up, the 2D cameras are internally calibrated but uncoupled from the 3D sensor,allowing them to move independently with respect to each other. The registration for this set-up isperformed using a Structure-from-Motion reconstruction emanating from images and planar patchesrepresenting the point cloud. The proposed registration method is globally optimal and robust tooutliers. It is based on the theory Sum-of-Squares polynomials and a Branch-and-Bound algorithm.The third set-up consists of uncoupled and uncalibrated 2D cameras. The image sets from thesecameras are registered to the point cloud in a globally optimal manner using a Branch-and-Prunealgorithm. Our method is based on a Linear Matrix Inequality framework that establishes directrelationships between 2D image measurements and 3D scene voxels

Cette thèse est consacrée à l'étude théorique et numérique de deux problèmes non linéaires au sens de McKean en finance. Nous abordons dans la première partie le problème de calibration d'un modèle à volatilité locale et stochastique pour tenir compte des prix d'options Européennes vanilles observés sur le marché. Ce problème se traduit par l'étude d'une équation différentielle stochastique (EDS) non linéaire au sens de McKean à cause de la présence dans le coefficient de diffusion d'une espérance conditionnelle du facteur de volatilité stochastique par rapport à la solution de l'EDS. Nous obtenons l'existence du processus dans le cas particulier où le facteur de volatilité stochastique est un processus de sauts ayant un nombre fini d'états. Nous obtenons de plus la convergence faible à l'ordre 1 de la discrétisation en temps de l'EDS non linéaire au sens de McKean pour des facteurs de volatilité stochastique généraux. Dans l'industrie, la calibration est effectuée efficacement à l'aide d'une régularisation de l'espérance conditionnelle par un estimateur à noyau de type Nadaraya-Watson, comme proposé par Guyon et Henry-Labordère dans [JGPHL]. Nous proposons également un schéma numérique demi-pas de temps et étudions le système de particules associé que nous comparons à l'algorithme proposé par [JGPHL]. Dans la deuxième partie de la thèse, nous nous intéressons à un problème de valorisation de contrat avec appels de marge, une problématique apparue avec l'application de nouvelles régulations depuis la crise financière de 2008. Ce problème peut être modélisé par une équation différentielle stochastique rétrograde (EDSR) anticipative avec dépendance en la loi de la solution dans le générateur. Nous montrons que cette équation est bien posée et proposons une approximation de sa solution à l'aide d'EDSR standards linéaires lorsque la durée de liquidation de l'option en cas de défaut est petite. Enfin, nous montrons que le calcul des solutions de ces EDSR standards peut être amélioré à l'aide de la méthode de Monte-Carlo multiniveaux introduite par Giles dans [G]<br>This thesis is dedicated to the theoretical and numerical study of two problems which are nonlinear in the sense of McKean in finance. In the first part, we study the calibration of a local and stochastic volatility model taking into account the prices of European vanilla options observed in the market. This problem can be rewritten as a stochastic differential equation (SDE) nonlinear in the sense of McKean, due to the presence in the diffusion coefficient of a conditional expectation of the stochastic volatility factor computed w.r.t. the solution to the SDE. We obtain existence in the particular case where the stochastic volatility factor is a jump process with a finite number of states. Moreover, we obtain weak convergence at order 1 for the Euler scheme discretizing in time the SDE nonlinear in the sense of McKean for general stochastic volatility factors. In the industry, Guyon and Henry Labordere proposed in [JGPHL] an efficient calibration procedure which consists in approximating the conditional expectation using a kernel estimator such as the Nadaraya-Watson one. We also introduce a numerical half-step scheme and study the the associated particle system that we compare with the algorithm presented in [JGPHL]. In the second part of the thesis, we tackle the pricing of derivatives with initial margin requirements, a recent problem that appeared along with new regulation since the 2008 financial crisis. This problem can be modelled by an anticipative backward stochastic differential equation (BSDE) with dependence in the law of the solution in the driver. We show that the equation is well posed and propose an approximation of its solution by standard linear BSDEs when the liquidation duration in case of default is small. Finally, we show that the computation of the solutions to the standard BSDEs can be improved thanks to the multilevel Monte Carlo technique introduced by Giles in [G]

This doctoral thesis is devoted to estimation and examination of default probabilities (PDs) within credit risk management and comprises three various studies. In the first study, we introduce a structural credit risk model based on stable non-Gaussian processes in order to overcome distributional drawbacks of the classical Merton model. Following the Moody’s KMV estimation methodology, we conduct an empirical comparison between the results obtained from the Merton model and the stable Paretian one. Our results suggest that PDs are generally underestimated by the Merton model and that the stable Lévy model is substantially more sensitive to the periods of financial crises. The second study is devoted to examination of the performance of static and multi-period credit-scoring models for determining PDs of financial institutions. Using an extensive sample of U.S. commercial banks provided by the FFIEC, we focus on evaluating the performance of the considered scoring techniques. We find that our models provide a high predictive accuracy in distinguishing between default and non-default financial institutions. Despite the difficulty of predicting defaults in the financial sector, the proposed models perform very well also in comparison to results on scoring techniques for the corporate sector. Finally, in the third study, we examine the relationship between distress risk and returns of U.S. renewable energy companies. Using the Expected Default Frequency (EDF) measure obtained from Moody’s KMV, we demonstrate that there is a positive cross-sectional relationship between returns and evidence for a distress risk premium in this sector. The positively priced distress premium is also confirmed by investigating returns corrected for common Fama and French and Carhart risk factors. We further show that raw and risk-adjusted returns of value-weighted portfolios that take a long position in the 20% most distressed stocks and a short position in the 20% safest stocks generally outperforms the S&amp;P 500 index throughout our sample period.

This thesis is devoted to the study of some applications of quantization to Financial Mathematics, especially to option pricing and calibration of financial data. Quantization is a technique that comes originally from numerical probability, and consists in approximating random variables and stochastic processes taking infinitely many values, with a discrete version of them, in order to simplify the quadrature algorithms for the computation of expected values. The purpose of this thesis is to show the great flexibility that quantization can have in the area of numerical probability and option pricing. In the literature, often there are ad hoc methods for a particular type of model or derivative, but no general framework seems to exist. Finite difference methods are heavily affected by the curse of dimensionality, while Monte Carlo methods need intense computational effort in order to have good precision, and are not designed for calibration purposes. Quantization can give an alternative methodology for a broad class of models and deriva- tives. The aim of the thesis is twofold: first, the extension of the literature about quantization to a broad class of models, namely local and stochastic volatility models, affine, pure jumps and polynomial processes, is an interesting theoretical exercise in itself. In fact, every time we deal with a different model we have to take in consideration the properties of the process and therefore the quantization algorithm must be adapted. Second, it is important to consider the computational results of the new types of quantization introduced. Indeed, the algorithms that we have developed turn out to be fast and numerically stable, and these aspects are very relevant, as we can overcome some of the issues present in literature for other types of approach. The first line of research deals with a technique called Recursive Marginal Quantization. Introduced in Pagès and Sagna (2015), this methodology exploits the conditional distribution of the Euler scheme of a one dimensional stochastic differential equation in order to construct a step-by-step approximation of the process. In this thesis we deal with the generalization of this technique to systems of stochastic differential equations, in particular to the case of stochastic volatility models. The Recursive Marginal Quantization of multidimensional stochastic process allows us to price European and path dependent options, in particular American options, and to perform calibration on financial data, giving then an alternative, and sometimes overcoming, to the usual Monte Carlo techniques. The second line of research takes a different perspective on quantization. Instead of using discretization schemes in order to compute the distribution of a stochastic process, we exploit the properties of the characteristic function and of the moment generating function for a broad class of processes. We consider the price process at maturity as a random variable, and we focus on the quantization of the stochastic variable, instead of focusing on the quantization of the whole stochastic process. This gives a faster and more precise technology for the pricing of options, and allows the quantization of a huge set of models for which the Recursive Marginal Quantization cannot be applied or is not numerically competitive.<br>Questa tesi si occupa dello studio delle applicazioni della quantizzazione alla Finanza Matematica, in particolare al prezzaggio di opzioni e alla calibrazione su dati finanziari. La quantizzazione è una tecnica che ha le sue origini dalla probabilità numerica, e consiste nell’approssimare variabili aleatorie e processi stocastici continui nello spazio delle realizzazioni con una versione discreta, allo scopo di semplificare gli algoritmi di quadratura per il calcolo di valori attesi. L’obiettivo di questa tesi è di mostrare la grande flessibilità che può avere la quantizzazione nell’ambiente della probabilità numerica e del prezzaggio di opzioni. Nella letteratura spesso esistono metodi ad hoc per ogni tipo di modello e di derivato, ma non sembra esserci una metodologia unica. I metodi alle differenze finite soffrono fortemente della curse of dimensionality, mentre i metodi Monte Carlo necessitano di un grande sforzo computazionale, e non sono pensati per esercizi di calibrazione. La quantizzazione può a volte risolvere problemi specifici di queste tecnologie, e presenta una metodologia alternativa per una grande classe di modelli e di derivati. Lo scopo della tesi è duplice: in primo luogo, l’estensione della letteratura sulla quantizzazione ad un’ampia gamma di processi, cioè processi a volatilità locale e stocastica, affini, di puro salto e polinomiali, è di per se un interessante esercizio teorico. Infatti, per ogni tipo di processo dobbiamo considerare le sue proprietà specifiche, adattando quindi l’algoritmo di quantizzazione. Inoltre, è importante considerare i risultati computazioni dei nuovi tipi di quantizzazione introdotti, in quanto è fondamentale sviluppare algoritmi che siano veloci e stabili numericamente, allo scopo di superare le problematiche presenti nella letteratura per altri tipi di approcci. Il primo filone di ricerca si occupa di una tecnica chiamata Quantizzazione Marginale Ricorsiva. Introdotta in Pagès and Sagna (2015), questa metodologia sfrutta la distribuzione condizionale dello schema di Eulero di un’equazione differenziale stocastica unidimensionale per costruire un’approssimazione passo passo del processo. In questa tesi generalizziamo questa tecnica ai sistemi di equazioni differenziali stocastiche, in particolare al caso dei modelli a volatilità stocastica. La Quantizzazione Marginale Ricorsiva di processi stocastici multidimensionali permette il prezzaggio di opzioni Europee e di opzioni path dependent, in particolare le opzioni Americane, e di effettuare calibrazione su dati finanziari, dando quindi un’alternativa, e spesso superandole, alle tipiche tecniche di Monte Carlo. La seconda linea di ricerca tratta la quantizzazione da una prospettiva differente. Invece di usare schemi di discretizzazione per il calcolo della distribuzione di un processo stocastico, viene sfruttata le proprietà della funzione caratteristica e della funzione generatrice dei momenti di una vasta classe di processi. Consideriamo infatti il processo del prezzo a maturità come una variabile aleatoria, e ci focalizziamo sulla quantizzazione della variabile casuale, invece di considerare tutto il processo stocastico. Questo approccio porta a una tecnologia più veloce e precisa per il prezzaggio di opzioni, e permette la quantizzazione di un vasto insieme di modelli, che non potevano essere affrontati dalla Quantizzazione Marginale Ricorsiva.