Faster Comparison of Stopping Times by Nested Conditional Monte Carlo

We show that deliberately introducing a nested simulation stage can lead to significant variance reductions when comparing two stopping times by Monte Carlo. We derive the optimal number of nested simulations and prove that the algorithm is remarkably robust to misspecifications of this number. The method is applied to several problems related to Bermudan/American options. In these applications, our method allows to substantially increase the efficiency of other variance reduction techniques, namely, Quasi-Control Variates and Multilevel Monte Carlo

Computation of Multivariate Barrier Crossing Probability, and Its Applications in Finance

In this thesis, we consider computational methods of finding exit probabilities for a class of multivariate stochastic processes. While there is an abundance of results for one-dimensional processes, for multivariate processes one has to rely on approximations or simulation methods. We adopt a Large Deviations approach in order to estimate barrier crossing probabilities of a multivariate Brownian Bridge. We use this approach in conjunction with numerical techniques to propose an efficient method of obtaining barrier crossing probabilities of a multivariate Brownian motion. Using numerical examples, we demonstrate that our method works better than other existing methods. We present applications of the proposed method in addressing problems in finance such as estimating default probabilities of several credit risky entities and pricing credit default swaps. We also extend our computational method to efficiently estimate a barrier crossing probability of a sum of Geometric Brownian motions. This allows us to perform a portfolio selection by maximizing a path-dependent utility function

Asymptotic analysis of dependent risks and extremes in insurance and finance

In this thesis, we are interested in the asymptotic analysis of extremes and risks. The heavy-tailed distribution function is used to model the extreme risks, which is widely applied in insurance and is gradually penetrating in finance as well. We also use various tools such as copula, to model dependence structures, and extreme value theorem, to model rare events. We focus on modelling and analysing of extreme risks as well as demonstrate how the derived results that can be used in practice. We start from a discrete-time risk model. More concretely, consider a discrete-time annuity-immediate risk model in which the insurer is allowed to invest its wealth into a risk-free or a risky portfolio under a certain regulation. Then the insurer is said to be exposed to a stochastic economic environment that contains two kinds of risk, the insurance risk and financial risk. The former is traditional liability risk caused by insurance loss while the later is the asset risk resulting from investment. Within each period, the insurance risk is denoted by a real-valued random variable $X$, and the financial risk $Y$ as a positive random variable fulfils some constraints. We are interested in the ruin probability and the tail behaviour of maximum of the stochastic present values of aggregate net loss with Sarmanov or Farlie-Gumbel-Morgenstern (FGM) dependent insurance and financial risks. We derive asymptotic formulas for the finite-ruin probability with lighted-tailed or moderately heavy-tailed insurance risk for both risk-free investment and risky investment. As an extension, we improve the result for extreme risks arising from a rare event, combining simulation with asymptotics, to compute the ruin probability more efficiently. Next, we consider a similar risk model but a special case that insurance and financial risks following the least risky FGM dependence structure with heavy-tailed distribution. We follow the study of Chen (2011) that the finite-time ruin probability in a discrete-time risk model in which insurance and financial risks form a sequence of independent and identically distributed random pairs following a common bivariate FGM distribution function with parameter $-1\leq \theta \leq 1$ governing the strength of dependence. For the subexponential case, when $-1<\theta \leq 1$, a general asymptotic formula for the finite-time ruin probability was derived. However, the derivation there is not valid for $\theta =-1$. In this thesis, we complete the study by extending Chen's work to $\theta =-1$ that the insurance risk and financial risk are negatively dependent. We refer this situation as the least risky FGM dependent insurance risk and financial risk. The new formulas for $\theta = −1$ look very different from, but are intrinsically consistent with, the existing one for $-1<\theta\leq 1$, and they offer a quantitative understanding on how significantly the asymptotic ruin probability decreases when $\theta$ switches from its normal range to its negative extremum. Finally, we study a continuous-time risk model. Specifically, we consider a renewal risk model with a constant premium and a constant force of interest rate, where the claim sizes and inter-arrival times follow certain dependence structures via some restriction on their copula function. The infinite-time absolute ruin probabilities are studied instead of the traditional infinite-time ruin probability with light-tailed or moderately heavy-tailed claim-size. Under the assumption that the distribution of the claim-size belongs to the intersection of the convolution-equivalent class and the rapid-varying tailed class, or a larger intersection class of O-subexponential distribution, the generalized exponential class and the rapid-varying tailed class, the infinite-time absolute ruin probabilities are derived

Modèles de dépendance avec copule Archimédienne : fondements basés sur la construction par mélange, méthodes de calcul et applications

Le domaine de l’assurance est basé sur la loi des grands nombres, un théorème stipulant que les caractéristiques statistiques d’un échantillon aléatoire suffisamment grand convergent vers les caractéristiques de la population complète. Les compagnies d’assurance se basent sur ce principe afin d’évaluer le risque associé aux évènements assurés. Cependant, l’introduction d’une relation de dépendance entre les éléments de l’échantillon aléatoire peut changer drastiquement le profil de risque d’un échantillon par rapport à la population entière. Il est donc crucial de considérer l’effet de la dépendance lorsqu’on agrège des risques d’assurance, d’où l’intérêt porté à la modélisation de la dépendance en science actuarielle. Dans ce mémoire, on s’intéresse à la modélisation de la dépendance à l’intérieur d’un portefeuille de risques dans le cas où une variable aléatoire (v.a.) mélange introduit de la dépendance entre les différents risques. Après avoir introduit l’utilisation des mélanges exponentiels dans la modélisation du risque en actuariat, on démontre comment cette construction par mélange nous permet de définir les copules Archimédiennes, un outil puissant pour la modélisation de la dépendance. Dans un premier temps, on démontre comment il est possible d’approximer une copule Archimédienne construite par mélange continu par une copule construite par mélange discret. Puis, nous dérivons des expressions explicites pour certaines mesures d’intérêt du risque agrégé. Nous développons une méthode de calcul analytique pour évaluer la distribution d’une somme de risques aléatoires d’un portefeuille sujet à une telle structure de dépendance. On applique enfin ces résultats à des problèmes d’agrégation, d’allocation du capital et de théorie de la ruine. Finalement, une extension est faite aux copules Archimédiennes hiérarchiques, une généralisation de la dépendance par mélange commun où il existe de la dépendance entre les risques à plus d’un niveau.The law of large numbers, which states that statistical characteristics of a random sample will converge to the characteristics of the whole population, is the foundation of the insurance industry. Insurance companies rely on this principle to evaluate the risk of insured events. However, when we introduce dependencies between each component of the random sample, it may drastically affect the overall risk profile of the sample in comparison to the whole population. This is why it is essential to consider the effect of dependency when aggregating insurance risks from which stems the interest given to dependence modeling in actuarial science. In this thesis, we study dependence modeling in a portfolio of risks for which a mixture random variable (rv) introduces dependency. After introducing the use of exponential mixtures in actuarial risk modeling, we show how this mixture construction can define Archimedean copulas, a powerful tool for dependence modeling. First, we demonstrate how an Archimedean copula constructed via a continuous mixture can be approximated with a copula constructed by discrete mixture. Then, we derive explicit expressions for a few quantities related to the aggregated risk. The common mixture representation of Archimedean copulas is then at the basis of a computational strategy proposed to compute the distribution of the sum of risks in a general setup. Such results are then used to investigate risk models with respect to aggregation, capital allocation and ruin problems. Finally, we discuss an extension to nested Archimedean copulas, a general case of dependency via common mixture including different levels of dependency.Résumé en espagno

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstracts

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstract