Market-consistent actuarial valuation of insurance liabilities is important approach not only for regulatory framework Solvency II but also generally for financial and actuarial modeling in insurance companies. It is the reason why we will focus on derivation of basic theory for valuation of cash flow from insurance liabilities by real world probability measure with deflators and risk neutral measure with bank account numeraire (also called equivalent martingale measure). We will show on illustrative examples ekvivalence of both approaches. Further, we will focus on spot rate modeling using discrete time Vasicek model. We use discrete time Vasicek model in Valuation Portfolio theory, where we are trying to replicate insurance liabilities by financial instruments. In theory and also example we use important assumption about independent decoupling of financial events and insurance technical events for theirs modeling.

In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

This thesis aims at contributing to the study of the valuation of insurance liabilities and the management of the assets backing these liabilities. It consists of four parts, each devoted to a specific topic. In the first part, we study the pricing of a classical single premium life insurance contract with profit, in terms of a guaranteed rate on the premium and a participation rate on the (terminal) financial surplus. We argue that, given the asset allocation of the insurer, these technical parameters should be determined by taking explicitly into account the risk management policy of the insurance company, in terms of a risk measure such as the value-at-risk or the conditional value-at-risk. We then design a methodology that allows us to fix both parameters in such a way that the contract is fairly priced and simultaneously exhibits a risk consistent with the risk management policy. In the second part, we focus on the management of the surrender option embedded in most life insurance contracts. In Chapter 2, we argue that we should model the surrender time as a random time not adapted to the filtration generated by the financial assets prices, instead of assuming that the surrender time is an optimal stopping time as it is usual in the actuarial literature. We then study the valuation of insurance contracts with a surrender option in such a model. We here follow the financial literature on the default risk and in particular, the reduced-form models. In Chapter 3 and 4, we study the hedging strategies of such insurance contracts. In Chapter 3, we study their risk-minimizing strategies and in Chapter 4, we focus on their ``locally risk-minimizing' strategies. As a by-product, we study the impact of a progressive enlargement of filtration on the so-called ``minimal martingale measure'. The third part is devoted to the systematic mortality risk. Due to its systematic nature, this risk cannot be diversified through increasing the size of the portfolio. It is thus also important to study the hedging strategies an insurer should follow to mitigate its exposure to this risk. In Chapter 5, we study the risk-minimizing strategies for a life insurance contract when no mortality-linked financial assets are traded on the financial market. We here extend Dahl and Moller’s results and show that the risk-minimizing strategy of a life insurance contract is given by a weighted average of risk-minimizing strategies of purely financial claims, where the weights are given by the (stochastic) survival probabilities. In Chapter 6, we first study the application of the HJM methodology to the modelling of a longevity bonds market and describe a coherent theoretical setting in which we can properly define the longevity bond prices. Then, we study the risk-minimizing strategies for pure endowments and annuities portfolios when these longevity bonds are traded. Finally, the fourth part deals with the design of ALM strategies for a non-life insurance portfolio. In particular, this chapter aims at studying the risk-minimizing strategies for a non life insurance company when inflation risk and interest rate risk are taken into account. We derive the general form of these strategies when the cumulative payments of the insurer are described by an arbitrary increasing process adapted to the natural filtration of a general marked point process and when the inflation and the term structure of interest rates are simultaneously described by the HJM model of Jarrow and Yildirim. We then systematically apply this result to four specific models of insurance claims. We first study two ``collective' models. We then study two ``individual' models where the claims are notified at a random time and settled through time.

Dans cette thèse nous considérons le problème de la maximisation d’utilité et de la formation des prix d’indifférence pour les modèles semimartingales exponentiels dépendant d’un facteur aléatoire ξ. L’enjeu est de résoudre le problème des prix d’indifférence en utilisant le grossissement de l’espace et de la filtration. Nous réduisons le problème de maximisation dans la filtration élargie au problème conditionnel, sachant {ξ = v}, que nous résolvons en utilisant une approche duale. Pour HARA-utilités nous introduisons les informations telles que les entropies relatives et les intégrales de type Hellinger, ainsi que les processus d’information correspondants, enfin d’exprimer, via ces processus, l’utilité maximal. En particulier, nous étudions les modèles de Lévy exponentiels, où les processus d’information sont déterministes ce que simplifie considèrablement les calculs des prix d’indiffrence. Enfin, nous appliquons les rèsultats au modèle du mouvement brownien géométrique et au modèle de diffusion-sauts qui inclut le mouvement brownien et les processus de Poisson. Dans les cas d’utilité logarithmique, de puissance et exponentielle, nous fournissons les formules explicites des informations, et puis, en utilisant les méthodes numériques, nous résolvons les équations pour obtenir les prix d’indifférence en cas de vente d’une option européenne. / This thesis explores the utility maximisation problem and indifference pricing for exponential semimartingale models depending on a random factor ξ. The main idea to solve indifference pricing problem consists in the enlargement of the space and filtration. We reduce the maximization problem on the enlarged filtration to the conditional one, given {ξ = v}, which we solve using dual approach. For HARA-utilities we introduce the information quantities such that the relative entropies, Hellinger type integrals, and the corresponding information processes, and we express the maximal utility via these processes. As a particular case, we study exponential Levy models, where the information processes are deterministic and this fact simplify very much indifference price calculus. Finally, we apply the results to Geometric Brownian motion model and jump-diffusion model which incorporates Brownian motion and Poisson process. In the cases of logarithmic, power and exponential utilities, we provide the explicit formulae of information quantities and using the numerical methods we solve the equations for the seller’s and buyer’s indifference prices of European put option.

Cette thèse porte sur les questions d'évaluation et de couverture des options dans un modèle exponentiel-Lévy avec changements de régime. Un tel modèle est construit sur un processus additif markovien un peu comme le modèle de Black- Scholes est basé sur un mouvement Brownien. Du fait de l'existence de plusieurs sources d'aléa, nous sommes en présence d'un marché incomplet et ce fait rend inopérant les développements théoriques initiés par Black et Scholes et Merton dans le cadre d'un marché complet. Nous montrons dans cette thèse que l'utilisation de certains résultats de la théorie des processus additifs markoviens permet d'apporter des solutions aux problèmes d'évaluation et de couverture des options. Notamment, nous arrivons à caracté- riser la mesure martingale qui minimise l'entropie relative à la mesure de probabilit é historique ; aussi nous dérivons explicitement sous certaines conditions, le portefeuille optimal qui permet à un agent de minimiser localement le risque quadratique associé. Par ailleurs, dans une perspective plus pratique nous caract érisons le prix d'une option Européenne comme l'unique solution de viscosité d'un système d'équations intégro-di érentielles non-linéaires. Il s'agit là d'un premier pas pour la construction des schémas numériques pour approcher ledit prix. / This thesis focuses on the pricing and hedging problems of financial derivatives in a Markov-modulated exponential-Lévy model. Such model is built on a Markov additive process as much as the Black-Scholes model is based on Brownian motion. Since there exist many sources of randomness, we are dealing with an incomplete market and this makes inoperative techniques initiated by Black, Scholes and Merton in the context of a complete market. We show that, by using some results of the theory of Markov additive processes it is possible to provide solutions to the previous problems. In particular, we characterize the martingale measure which minimizes the relative entropy with respect to the physical probability measure. Also under some conditions, we derive explicitly the optimal portfolio which allows an agent to minimize the local quadratic risk associated. Furthermore, in a more practical perspective we characterize the price of a European type option as the unique viscosity solution of a system of nonlinear integro-di erential equations. This is a rst step towards the construction of e ective numerical schemes to approximate options price.

Le but de cette thèse est l'étude de certains aspects d'un marché financier comportant plusieurs actifs risqués et des options écrites sur ces actifs. Dans un premier essai, nous proposons une expression de la distribution implicite du prix d'un actif sous-jacent en fonction du smile de volatilité associé aux options écrites sur cet actif. L'expression obtenue pour la densité implicite prend la forme d'une densité log-normale plus deux termes d'ajustement. La mise en œuvre de ce résultat est ensuite illustrée à travers deux applications pratiques. Dans le deuxième essai, nous obtenons deux caractérisations de l'absence d'opportunité d'arbitrage en termes de fonctions copules. Chacune de ces caractérisations conduit à une méthode de détection des situations d'arbitrage. La première méthode proposée repose sur une propriété particulière des copules de Bernstein. La seconde méthode est valable dans le cas bivarié et tire profit de résultats sur les bornes de Fréchet-Hoeffding en présence d'information additionnelle sur la dépendance. Les résultats de l'utilisation de ces méthodes sur des données empiriques sont présentés. Enfin, dans le troisième essai, nous proposons une approche pour couvrir avec des options sur spread l'exposition au risque de dépendance d'un portefeuille d'options écrites sur deux actifs. L'approche proposée repose sur l'utilisation de deux modèles paramétriques de dépendance que nous introduisons: les copules Power Frank (PF) et Power Student's t (PST). Le fonctionnement et les résultats de l'approche proposée sont illustrés dans une étude numérique. / This thesis is dedicated to the study of a market with several risky assets and options written on these assets. In a first essay, we express the implied distribution of an underlying asset price as a function of its options implied volatility smile. For the density, the obtained expression has the form of a log-normal density plus two adjustment terms. We then explain how to use these results and develop practical applications. In a first application we value a portfolio of digital options and in another application we fit a parametric distribution. In the second essay, we propose a twofold characterization of the absence of arbitrage opportunity in terms of copula functions. We then propose two detection methods. The first method relies on a particular property of Bernstein copulas. The second method, valid only in the case of a market with two risky assets, is based upon results on improved Fréchet-Hoeffding bounds in presence of additional information about the dependence. We also present results obtained with the proposed methods applied to empirical data. Finally, in the third essay, we develop an approach to hedge, with spread options, an exposure to dependence risk for a portfolio comprising two-asset options. The approach we propose is based on two parametric models of dependence that we introduce. These dependence models are copulas functions named Power Frank (PF) and Power Student's t (PST). The results obtained with the proposed approach are detailed in a numerical study.

運用傳統再保險契約移轉風險受限於承保能量的逐年波動，尤其自90年代起，全球巨災頻繁，保險人損失巨幅增加，承保能量急遽萎縮，基於巨災市場之資金需求，再保險轉向資本市場，預期將巨災風險移轉至投資人，促成保險衍生性金融商品之創新，本研究針對佔有顯著交易量的巨災風險債券進行分析，基於Cummins和Geman (1995)所建構巨災累積損失模型，引用Duffie 與Singleton (1999)於違約債券的計價模式，將折現利率表示為短期利率加上事故發生率及預期損失比例之乘積，並將債券期間延長至多年期，以符合市場承保的需求，應用市場無套利假設及平賭測度計價的方法計算合理的市場價值，巨災損失過程將分成損失發展期與損失確定期，以卜瓦松過程表示巨災發生頻率，並利用台灣巨災經驗資料建立合適之損失幅度模型，最後以蒙地卡羅方法針對三種不同型態的巨災風險債券試算合理價值，並具體結論所得的數值結果與後續之研究建議。 / Using traditional reinsurance treaties to transfer insurance risks are restrained due to the volatility of the underwriting capacity annually. Catastrophe risks have substantially increased since the early 1990s and have directly resulted significant claim losses for the insurers. Hence the insurers are pursuing the financial capacities from the capital market. Transferring the catastrophe risks to the investor have stimulated the financial innovation for the insurance industry. In this study, pricing issues for the heavily traded catastrophe risk bonds (CAT-bond) are investigated. The aggregated catastrophe loss model in Cummins and Geman (1995) are adopted. While the financial techniques in valuing the defaultable bonds in Duffie and Singleton (1999) are employed to determine the fair prices incorporating the claim hazard rates and the loss severity. The duration of the CAT-bonds is extended from single year to multiple years in order to meet the demand from the reinsurance market. Non- arbitrage theory and martingale measures are employed to determine their fair market values. The contract term of the CAT-bonds is divided into the loss period and the development period. The frequency of the catastrophe risk is modeled through the Poisson process. Taiwan catastrophe loss experiences are examined to build the plausible loss severity model. Three distant types of CAT-bonds are analyzed through Monte Carlo method for illustrations. This paper concludes with remarks regarding some pricing issues of CAT-bonds.

本論文提出利用機率平賭性質由選擇權市場價格還原風險中立機率測度的雙目標規劃模型。假設對應同一標的資產且不同履約價的選擇權均為歐式選擇權，到期時標的資產的狀態為離散點且個數有限。若市場不存在套利機會時，建構出最小化離差總和及最大化平滑的雙目標規劃模型。將此雙目標規劃模型利用權重法轉換成單一目標之非線性模型，即可還原風險中立機率測度，並利用此風險中立機率測度評價選擇權的公平價格。最後，我們以台指選擇權(TXO)為例，驗證此模型的評價能力。 / This thesis proposes a biobjective nonlinear programming model to derive risk-neutral probability distribution of underlying asset. The method are used to choose probabilities that minimize the deviation between the observed price and the theoretical price as well as maximize the smoothness of the resulting probabilities. A weighting method is used to covert the model into a single objective model. Given a non-arbitrage observed option price, a risk-neutral probability distribution consistent with the observed option can be recovered by the model. This risk-neutral probability is then utilized to evaluate the fair price of options. Finally, an empirical study applying to Taiwan’s market is given to verify the pricing ability of this model.

Cette thèse porte sur les questions d'évaluation et de couverture des options dans un modèle exponentiel-Lévy avec changements de régime. Un tel modèle est construit sur un processus additif markovien un peu comme le modèle de Black- Scholes est basé sur un mouvement Brownien. Du fait de l'existence de plusieurs sources d'aléa, nous sommes en présence d'un marché incomplet et ce fait rend inopérant les développements théoriques initiés par Black et Scholes et Merton dans le cadre d'un marché complet. Nous montrons dans cette thèse que l'utilisation de certains résultats de la théorie des processus additifs markoviens permet d'apporter des solutions aux problèmes d'évaluation et de couverture des options. Notamment, nous arrivons à caracté- riser la mesure martingale qui minimise l'entropie relative à la mesure de probabilit é historique ; aussi nous dérivons explicitement sous certaines conditions, le portefeuille optimal qui permet à un agent de minimiser localement le risque quadratique associé. Par ailleurs, dans une perspective plus pratique nous caract érisons le prix d'une option Européenne comme l'unique solution de viscosité d'un système d'équations intégro-di érentielles non-linéaires. Il s'agit là d'un premier pas pour la construction des schémas numériques pour approcher ledit prix. / This thesis focuses on the pricing and hedging problems of financial derivatives in a Markov-modulated exponential-Lévy model. Such model is built on a Markov additive process as much as the Black-Scholes model is based on Brownian motion. Since there exist many sources of randomness, we are dealing with an incomplete market and this makes inoperative techniques initiated by Black, Scholes and Merton in the context of a complete market. We show that, by using some results of the theory of Markov additive processes it is possible to provide solutions to the previous problems. In particular, we characterize the martingale measure which minimizes the relative entropy with respect to the physical probability measure. Also under some conditions, we derive explicitly the optimal portfolio which allows an agent to minimize the local quadratic risk associated. Furthermore, in a more practical perspective we characterize the price of a European type option as the unique viscosity solution of a system of nonlinear integro-di erential equations. This is a rst step towards the construction of e ective numerical schemes to approximate options price.