KEBAIER Ahmed

In this thesis, we examine several stochastic approximation methods for both the computation of financial risk measures and the pricing of derivatives.Since explicit formulas are rarely available for such quantities, the need for fast, efficient and reliable analytical approximations is of paramount importance to financial institutions.In the first part, we study several multilevel Monte Carlo approximation methods and apply them to two practical problems: the estimation of quantities involving nested expectations (such as initial margin) and the discretization of integrals appearing in rough models for the forward variance for VIX option pricing.In both cases, we analyze the asymptotic optimality properties of the corresponding multilevel estimators and numerically demonstrate their superiority over a classical Monte Carlo method.In the second part, motivated by the numerous examples from credit risk modeling, we propose a general metamodeling framework for large sums of weighted Bernoulli random variables, which are conditionally independent with respect to a common factor X. Our generic approach is based on the polynomial decomposition of the chaos of the common factor and on a Gaussian approximation. L2 error estimates are given when the factor X is associated with classical orthogonal polynomials.Finally, in the last part of this thesis, we focus on the short-time asymptotics of U.S. implied volatility and U.S. option prices in local volatility models. We also propose a law approximation of the VIX index in rough models for forward variance, expressed in terms of lognormal proxies, and derive expansion results for VIX options with explicit coefficients.

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In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We obtain Gaussian-type concentration. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

We consider a stable Cox--Ingersoll--Ross process driven by a standard Wiener process and a spectrally positive strictly stable L\'evy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. In all cases we prove strong consistency of the MLE in question, in the subcritical case asymptotic normality, and in the supercritical case asymptotic mixed normality are shown as well. In the critical case the description of the asymptotic behavior of the MLE in question remains open.

We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian L\'evy process of not necessarily bounded variation with a L\'evy measure concentrated on $(-1,\infty)$. We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. We also present some numerical illustrations to confirm our results.

We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian L\'evy process of not necessarily bounded variation with a L\'evy measure concentrated on $(-1,\infty)$. We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. We also present some numerical illustrations to confirm our results.

This paper focuses on the study of an original combination of the Multilevel Monte Carlo method introduced by Giles [10] and the popular importance sampling technique. To compute the optimal choice of the parameter involved in the importance sampling method, we rely on Robbins-Monro type stochastic algorithms. On the one hand, we extend our previous work [2] to the Multilevel Monte Carlo setting. On the other hand, we improve [2] by providing a new adaptive algorithm avoiding the discretization of any additional process. Furthermore, from a technical point of view, the use of the same stochastic algorithms as in [2] appears to be problematic. To overcome this issue, we employ an alternative version of stochastic algorithms with projection (see e.g.

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In this paper, we consider a one-dimensional jump-type Cox-Ingersoll-Ross process driven by a Brownian motion and a subordinator, whose growth rate is a unknown parameter. The L\'evy measure of the subordinator is finite or infinite. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Three cases are distinguished: subcritical, critical and supercritical. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To do so, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.

In this paper, we consider a one-dimensional Cox-Ingersoll-Ross (CIR) process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. To obtain these results, we use the Malliavin calculus techniques developed recently for CIR process by Al\`os et {\it al.} \cite{AE08} and Altmayer et {\it al.} \cite{AN14} together with the $L^p$-norm estimation for positive and negative moments of the CIR process obtained by Bossy et {\it al.} \cite{BD07} and Ben Alaya et {\it al.} \cite{BK12,BK13}. In this study, we require the same conditions of high frequency $\Delta_n\rightarrow 0$ and infinite horizon $n\Delta_n\rightarrow\infty$ as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier by Gobet \cite{G02}. However, in the non-ergodic cases, additional assumptions on the decreasing rate of $\Delta_n$ are required due to the fact that the square root diffusion coefficient of the CIR process is not regular enough. Indeed, we assume $n\Delta_n^{3}\to 0$ for the critical case and $\Delta_n^{2}e^{-b_0n\Delta_n}\to 0$ for the supercritical case.

We consider a jump-type Cox--Ingersoll--Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak consistency and asymptotic normality, and, under an additional moment assumption, strong consistency as well. In the supercritical case, we prove strong consistency and mixed normal (but non-normal) asymptotic behavior, while in the critical case, weak consistency and non-standard asymptotic behavior are described. We specialize our results to so-called basic affine jump-diffusions as well. Concerning the asymptotic behavior of the MLE in the supercritical case, we derive a stochastic representation of the limiting mixed normal distribution, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case, where a diffusion-type critical CIR process plays a role.

We consider a jump-type Cox–Ingersoll–Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak consistency and asymptotic normality, and, under an additional moment assumption, strong consistency as well. In the supercritical case, we prove strong consistency and mixed normal (but non-normal) asymptotic behavior, while in the critical case, weak consistency and non-standard asymptotic behavior are described. We specialize our results to so-called basic affine jump–diffusions as well. Concerning the asymptotic behavior of the MLE in the supercritical case, we derive a stochastic representation of the limiting mixed normal distribution, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case, where a diffusion-type critical CIR process plays a role.

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We obtain Gaussian-type concentration. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to compute the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.

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This thesis is devoted to the study of the strong convergence properties of the Ninomiya and Victoir scheme. The authors of this scheme propose to approximate the solution of a stochastic differential equation (SDE), denoted $X$, by solving $d+1$ ordinary differential equations (ODE) on each time step, where $d$ is the dimension of the Brownian motion. The aim of this study is to analyze the use of this scheme in a multi-step Monte-Carlo method. Indeed, the optimal complexity of this method is directed by the order of convergence towards $0$ of the variance between the schemes used on the coarse and on the fine grid. This order of convergence is itself related to the strong order of convergence between the two schemes. We then show in chapter $2$, that the strong order of the Ninomiya-Victor scheme, denoted $X^{NV,eta}$ and of time step $T/N$, is $1/2$. Recently, Giles and Szpruch proposed a multi-step Monte Carlo estimator realizing $Oleft(epsilon^{-2}right)$ complexity using a modified Milstein scheme. In the same spirit, we propose a modified Ninomiya-Victoir scheme that can be coupled at high order $1$ with the Giles and Szpruch scheme at the last level of a multi-step Monte Carlo method. This idea is inspired by Debrabant and Rossler. These authors suggest using a high low order scheme at the finest discretization level. Since the optimal number of discretization levels of a multi-step Monte Carlo method is directed by the low error of the scheme used on the fine grid of the last discretization level, this technique allows to accelerate the convergence of the multi-step Monte Carlo method by obtaining a high low order approximation. The use of the $1$ coupling with the Giles-Szpruch scheme allows us to keep a multi-step Monte-Carlo estimator realizing an optimal complexity $Oleft( epsilon^{-2} right)$ while taking advantage of the $2$ low order error of the Ninomiya-Victoir scheme. In the third chapter, we are interested in the renormalized error defined by $sqrt{N}left(X - X^{NV,eta}right)$. We show the stable law convergence to the solution of an affine SDE, whose source term is formed by the Lie brackets between the Brownian vector fields. Thus, when at least two Brownian vector fields do not commute, the limit is non-trivial. This ensures that the strong order $1/2$ is optimal. On the other hand, this result can be seen as a first step towards proving a central limit theorem for multi-step Monte-Carlo estimators. To do so, we need to analyze the stable law error of the scheme between two successive discretization levels. Ben Alaya and Kebaier proved such a result for the Euler scheme. When the Brownian vector fields commute, the limit process is zero. We show that in this particular case, the strong order is $1$. In chapter 4, we study the convergence to a stable law of the renormalized error $Nleft(X - X^{NV}right)$ where $X^{NV}$ is the Ninomiya-Victor scheme when the Brownian vector fields commute. We demonstrate the convergence of the renormalized error process to the solution of an affine SDE. When the dritf vector field does not commute with at least one of the Brownian vector fields, the strong convergence speed obtained previously is optimal.

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In the last decade, there has been a growing interest to use Wishart processes for modelling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that extend the recent findings of Gnoatto and Grasselli and are of independent interest.

In this thesis, we propose a progressive (forward) approach for the probabilistic representation of nonlinear and nonconservative Partial Differential Equations (PDEs), allowing to develop a particle-based algorithm to numerically estimate their solutions. The Nonlinear Stochastic Differential Equations of McKean type (NLSDE) studied in the literature constitute a microscopic formulation of a phenomenon modeled macroscopically by a conservative PDE. A solution of such a NLSDE is the data of a couple $(Y,u)$ where $Y$ is a solution of a stochastic differential equation (SDE) whose coefficients depend on $u$ and $t$ such that $u(t,cdot)$ is the density of $Y_t$. The main contribution of this thesis is to consider nonconservative PDEs, i.e. conservative PDEs perturbed by a nonlinear term of the form $Lambda(u,nabla u)u$. This implies that a pair $(Y,u)$ will be a solution of the associated probabilistic representation if $Y$ is still a stochastic process and the relation between $Y$ and the function $u$ will then be more complex. Given the law of $Y$, the existence and uniqueness of $u$ are proved by a fixed point argument via an original Feynmann-Kac formulation.

The objective of this thesis is to contribute to the improvement of the valuation of corporate bonds, in particular by trying to draw lessons from the recent economic and financial crisis. In order to achieve this objective, we propose an approach based on credit spreads. We start, in the first chapter, with an analysis of the main existing valuation models, which we reformulate from the point of view of spreads and simulate numerically. We show that, despite the attractive features of structural-type models, they have several shortcomings that can be misleading, especially in a crisis context. In the second and third chapters, we focus on empirical spreads, which we analyze during the subprime and eurozone crises. Through: (i) descriptive analysis, (ii) principal component analysis, and (iii) statistical regression analysis, we shed light on several factors affecting spread movements that are not captured by existing models. Among these factors, we show: (i) that the wave of bank bailouts during the crisis had a significant effect on credit spreads, and (ii) that the size of a firm also has an effect on its spreads. Based on these empirical results, in a fourth chapter we propose a contribution to the structural modeling of corporate bonds, which takes into account the possibility of firms to negotiate a bailout in case of distress. Using this model, we manage, on the one hand, to reproduce the empirical observations of lower spreads for higher bailout probabilities (as is the case for large banks), and on the other hand, to fill several gaps in existing models, such as simple bankruptcy mechanisms, or low credit spreads for short maturities.

This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008) 607-617] which is significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg-Feller type for the multilevel Monte Carlo method associated with the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [Ann. Probab. 26 (1998) 267-307], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. A complexity of the multilevel Monte Carlo algorithm is carried out.

In this thesis, we focus on the combination of variance reduction and complexity reduction methods of the Monte Carlo method. In a first part of this thesis, we consider a continuous diffusion model for which we build an adaptive algorithm by applying importance sampling to the Romberg Statistical method. We prove a Lindeberg Feller type central limit theorem for this algorithm. In this same framework and in the same spirit, we apply importance sampling to the Multilevel Monte Carlo method and we also prove a central theorem for the obtained adaptive algorithm. In the second part of this thesis, we develop the same type of algorithm for a non-continuous model, namely the Lévy processes. Similarly, we prove a central limit theorem of the Lindeberg Feller type. Numerical illustrations have been carried out for the different algorithms obtained in the two frameworks with jumps and without jumps.

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This paper deals with the problem of global parameter estimation in the Cox-Ingersoll-Ross (CIR) model. This model is frequently used in finance for example to model the evolution of short-term interest rates or as a dynamic of the volatility in the Heston model. We establish new asymptotic results on the maximum likelihood estimator (MLE) associated to the global estimation of the drift parameters of the CIR process. We obtain various and original limit theorems on our MLE, with different rates and different types of limit distributions. Our results are obtained for both cases : ergodic and nonergodic diffusion.

The efficiency of Monte Carlo simulations is significantly improved when implemented with variance reduction methods. Among these methods we focus on the popular importance sampling technique based on producing a parametric transformation through a shift parameter θ. The optimal choice of θ is approximated using Robbins-Monro procedures, provided that a non explosion condition is satisfied. Otherwise, one can use either a constrained Robbins-Monro algorithm (see e.g. Arouna [2] and Lelong [17]) or a more astute procedure based on an unconstrained approach recently introduced by Lemaire and Pagès in [18]. In this article, we develop a new algorithm based on a combination of the statistical Romberg method and the importance sampling technique. The statistical Romberg method introduced by Kebaier in [13] is known for reducing efficiently the complexity compared to the classical Monte Carlo one. In the setting of discritized diffusions, we prove the almost sure convergence of the constrained and unconstrained versions of the Robbins-Monro routine, towards the optimal shift θ^∗ that minimizes the variance associated to the statistical Romberg method. Then, we prove a central limit theorem for the new algorithm that we called adaptative statistical Romberg method. Finally, we illustrate by numerical simulation the efficiency of our method through applications in option pricing for the Heston model.

This Thesis is composed of two parts dealing respectively with the discretization of stochastic differential equations and the almost sure central limit theorem for martingales. The first part is composed of three chapters: The first chapter introduces the framework of the study and presents the results obtained. The second chapter is devoted to the study of a new convergence acceleration method, called the statistical Romberg method, for the computation of expectations of functions or functionals of a diffusion. The third chapter deals with the application of this method to density approximation by kernel methods. The second part of the thesis is composed of two chapters: the first chapter presents the recent literature concerning the almost sure central limit theorem and its extensions. The second chapter extends various TLCPS results to quasi-continuous left-handed martingales.

This Thesis is composed of two parts dealing respectively with the discretization of stochastic differential equations and the almost sure central limit theorem for martingales. The first part is composed of three chapters: The first chapter introduces the framework of the study and presents the results obtained. The second chapter is devoted to the study of a new convergence acceleration method, called the statistical Romberg method, for the computation of expectations of functions or functionals of a diffusion. This chapter is an extended version of an article to be published in the journal Annals of Applied Probability. The third chapter deals with the application of this method to density approximation by kernel methods. This chapter is based on a collaborative work with Arturo Kohatsu-Higa. The second part of the thesis is composed of two chapters: the first chapter presents the recent literature concerning the almost sure central limit theorem and its extensions. The second chapter, based on a collaborative work with Faouzi Chaâbane, extends various TLCPS results to quasi-continuous left-handed martingales.