165 research outputs found
Some Applications of Markov Additive Processes as Models in Insurance and Financial Mathematics
Cette thÚse est principalement constituée de trois articles traitant des processus markoviens additifs, des processus de Lévy et d'applications en finance et en assurance.
Le premier chapitre est une introduction aux processus markoviens additifs (PMA), et une présentation du problÚme de ruine et de notions fondamentales des mathématiques financiÚres. Le deuxiÚme chapitre est essentiellement l'article "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" écrit en collaboration avec Manuel Morales et publié dans la revue European Actuarial Journal. Cet article étudie le problÚme de ruine pour un processus de risque markovien additif. Une identification de systÚmes de Lévy est obtenue et utilisée pour donner une expression de l'espérance de la fonction de pénalité actualisée lorsque le PMA est un processus de Lévy avec changement de régimes. Celle-ci est une généralisation des résultats existant dans la littérature pour les processus de risque de Lévy et les processus de risque markoviens additifs avec sauts "phase-type".
Le troisiĂšme chapitre contient l'article "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" qui est soumis pour publication. Cet article prĂ©sente une extension de l'espĂ©rance de la fonction de pĂ©nalitĂ© actualisĂ©e pour un processus subordinateur de risque perturbĂ© par un mouvement brownien. Cette extension contient une sĂ©rie de fonctions escomptĂ©e Ă©spĂ©rĂ©e des minima successives dus aux sauts du processus de risque aprĂšs la ruine. Celle-ci a des applications importantes en gestion de risque et est utilisĂ©e pour dĂ©terminer la valeur espĂ©rĂ©e du capital d'injection actualisĂ©. Finallement, le quatriĂšme chapitre contient l'article "The Minimal entropy martingale measure (MEMM) for a Markov-modulated exponential LĂ©vy model" Ă©crit en collaboration avec Romuald HervĂ© Momeya et publiĂ© dans la revue Asia-Pacific Financial Market. Cet article prĂ©sente de nouveaux rĂ©sultats en lien avec le problĂšme de l'incomplĂ©tude dans un marchĂ© financier oĂč le processus de prix de l'actif risquĂ© est dĂ©crit par un modĂšle exponentiel markovien additif. Ces rĂ©sultats consistent Ă charactĂ©riser la mesure martingale satisfaisant le critĂšre de l'entropie. Cette mesure est utilisĂ©e pour calculer le prix d'une option, ainsi que des portefeuilles de couverture dans un modĂšle exponentiel de LĂ©vy avec changement de rĂ©gimes.This thesis consists mainly of three papers concerned with Markov additive processes, LĂ©vy processes and applications on finance and insurance.
The first chapter is an introduction to Markov additive processes (MAP) and a presentation of the ruin problem and basic topics of Mathematical Finance. The second chapter contains the paper "Lévy Systems and the Time Value of Ruin for Markov Additive Processes" written with Manuel Morales and that is published in the European Actuarial Journal. This paper studies the ruin problem for a Markov additive risk process. An expression of the expected discounted penalty function is obtained via identification of the Lévy systems. The third chapter contains the paper "On a Generalization of the Expected Discounted Penalty Function to Include Deficits at and Beyond Ruin" that is submitted for publication. This paper presents an extension of the expected discounted penalty function in a setting involving aggregate claims modelled by a subordinator, and Brownian perturbation. This extension involves a sequence of expected discounted functions of successive minima reached by a jump of the risk process after ruin. It has important applications in risk management and in particular, it is used to compute the expected discounted value of capital injection. Finally, the fourth chapter contains the paper "The Minimal Entropy Martingale Measure (MEMM) for a Markov-Modulated Exponential" written with Romuald Hérvé Momeya and that is published in the journal Asia Pacific Financial Market. It presents new results related to the incompleteness problem in a financial market, where the risky asset is driven by Markov additive exponential model. These results characterize the martingale measure satisfying the entropy criterion. This measure is used to compute the price of the option and the portfolio of hedging in an exponential Markov-modulated Lévy model
Innovations in Quantitative Risk Management
Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science
Innovations in Quantitative Risk Management
Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science
Estimation of Default Probabilities with Support Vector Machines
Predicting default probabilities is important for firms and banks to operate successfully and to estimate their specific risks. There are many reasons to use nonlinear techniques for predicting bankruptcy from financial ratios. Here we propose the so called Support Vector Machine (SVM) to estimate default probabilities of German firms. Our analysis is based on the Creditreform database. The results reveal that the most important eight predictors related to bankruptcy for these German firms belong to the ratios of activity, profitability, liquidity, leverage and the percentage of incremental inventories. Based on the performance measures, the SVM tool can predict a firms default risk and identify the insolvent firm more accurately than the benchmark logit model. The sensitivity investigation and a corresponding visualization tool reveal that the classifying ability of SVM appears to be superior over a wide range of the SVM parameters. Based on the nonparametric Nadaraya-Watson estimator, the expected returns predicted by the SVM for regression have a significant positive linear relationship with the risk scores obtained for classification. This evidence is stronger than empirical results for the CAPM based on a linear regression and confirms that higher risks need to be compensated by higher potential returns.Support Vector Machine, Bankruptcy, Default Probabilities Prediction, Expected Profitability, CAPM.
A Dynamic Programming Algorithm for the Valuation of Guaranteed Minimum Withdrawal Benefits in Variable Annuities
In this paper we present a dynamic programming algorithm for pricing variable annuities
with Guaranteed Minimum Withdrawal Benefits (GMWB) under a general LĂ©vy processes
framework. The GMWB gives the policyholder the right to make periodical withdrawals
from her policy account even when the value of this account is exhausted. Typically, the
total amount guaranteed for withdrawals coincides with her initial investment, providing
then a protection against downside market risk. At each withdrawal date, the policyholder
has to decide whether, and how much, to withdraw, or to surrender the contract. We
show how different levels of rationality in the policyholderâs withdrawal behaviour can be modelled. We perform a sensitivity analysis comparing the numerical results obtained for
different contractual and market parameters, policyholder behaviours, and different types
of LĂ©vy processes
On Some Stochastic Optimal Control Problems in Actuarial Mathematics
The event of ruin (bankruptcy) has long been a core concept of risk management interest in the literature of actuarial science. There are two major research lines. The first one focuses on distributional studies of some crucial ruin-related variables such as the deficit at ruin or the time to ruin. The second one focuses on dynamically controlling the probability that ruin occurs by imposing controls such as investment, reinsurance, or dividend payouts. The content of the thesis will be in line with the second research direction, but under a relaxed definition of ruin, for the reason that ruin is often too harsh a criteria to be implemented in practice.
Relaxation of the concept of ruin through the consideration of "exotic ruin" features, including for instance, ruin under discrete observations, Parisian ruin setup, two-sided exit framework, and drawdown setup, received considerable attention in recent years. While there has been a rich literature on the distributional studies of those new features in insurance surplus processes, comparably less contributions have been made to dynamically controlling the corresponding risk. The thesis proposes to analytically study stochastic control problems related to some "exotic ruin" features in the broad area of insurance and finance.
In particular, in Chapter 3, we study an optimal investment problem by minimizing the probability that a significant drawdown occurs. In Chapter 4, we take this analysis one step further by proposing a general drawdown-based penalty structure, which include for example, the probability of drawdown considered in Chapter 3 as a special case. Subsequently, we apply it in an optimal investment problem of maximizing a fund manager's expected cumulative income. Moreover, in Chapter 5 we study an optimal investment-reinsurance problem in a two-sided exit framework. All problems mentioned above are considered in a random time horizon. Although the random time horizon is mainly determined by the nature of the problem, we point out that under suitable assumptions, a random time horizon is analytically more tractable in comparison to its finite deterministic counterpart.
For each problem considered in Chapters 3--5, we will adopt the dynamic programming principle (DPP) to derive a partial differential equation (PDE), commonly referred to as a Hamilton-Jacobi-Bellman (HJB) equation in the literature, and subsequently show that the value function of each problem is equivalent to a strong solution to the associated HJB equation via a verification argument. The remaining problem is then to solve the HJB equations explicitly. We will develop a new decomposition method in Chapter 3, which decomposes a nonlinear second-order ordinary differential equation (ODE) into two solvable nonlinear first-order ODEs. In Chapters 4 and 5, we use the Legendre transform to build respectively one-to-one correspondence between the original problem and its dual problem, with the latter being a linear free boundary problem that can be solved in explicit forms. It is worth mentioning that additional difficulties arise in the drawdown related problems of Chapters 3 and 4 for the reason that the underlying problems involve the maximum process as an additional dimension. We overcome this difficulty by utilizing a dimension reduction technique.
Chapter 6 will be devoted to the study of an optimal investment-reinsurance problem of maximizing the expected mean-variance utility function, which is a typical time-inconsistent problem in the sense that DPP fails. The problem is then formulated as a non-cooperative game, and a subgame perfect Nash equilibrium is subsequently solved. The thesis is finally ended with some concluding remarks and some future research directions in Chapter 7
On the design of customized risk measures in insurance, the problem of capital allocation and the theory of fluctuations for LĂ©vy processes
Dans cette thĂšse, nous Ă©tudions quelques problĂšmes fondamentaux en mathĂ©matiques financiĂšres et actuarielles, ainsi que leurs applications. Cette thĂšse est constituĂ©e de trois contributions portant principalement sur la thĂ©orie de la mesure de risques, le problĂšme de lâallocation du capital et la thĂ©orie des fluctuations. Dans le chapitre 2, nous construisons de nouvelles mesures de risque cohĂ©rentes et Ă©tudions lâallocation de capital dans le cadre de la thĂ©orie des risques collectifs. Pour ce faire, nous introduisons la famille des "mesures de risque entropique cumulatifs" (Cumulative Entropic Risk Measures). Le chapitre 3 Ă©tudie le problĂšme du portefeuille optimal pour le Entropic Value at Risk dans le cas oĂč les rendements sont modĂ©lisĂ©s par un processus de diffusion Ă sauts (Jump-Diffusion). Dans le chapitre 4, nous gĂ©nĂ©ralisons la notion de "statistiques naturelles de risque" (natural risk statistics) au cadre multivariĂ©. Cette extension non-triviale produit des mesures de risque multivariĂ©es construites Ă partir des donnĂ©es financiĂ©res et de donnĂ©es dâassurance. Le chapitre 5 introduit les concepts de "drawdown" et de la "vitesse dâĂ©puisement" (speed of depletion) dans la thĂ©orie de la ruine. Nous Ă©tudions ces concepts pour des modeles de risque dĂ©crits par une famille de processus de LĂ©vy spectrallement nĂ©gatifs.The aim of this thesis is to study fundamental problems in financial and insurance mathematics particularly the problem of measuring risk and its application within financial and insurance frameworks. The main contributions of this thesis can be classified in three main axes: the theory of risk measures, the problem of capital allocation and the theory of fluctuation. In Chapter 2, we design new coherent risk measures and study the associated capital allocation in the context of collective risk theory. We introduce the family of Cumulative Entropic Risk Measures. In Chapter 3, we study the optimal portfolio problem for the Entropic Value at Risk coherent risk measure for particular return models which are based on relevant cases of Jump-Diffusion models. In Chapter 4, we extending the notion of natural risk statistics to the multivariate setting. This non-trivial extension will endow us with multivariate data-based risk measures that are bound to have applications in finance and insurance. In Chapter 5, we introduce the concepts of drawdown and speed of depletion to the ruin theory literature and study them for the class of spectrally negative LĂ©vy risk processes
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