This thesis focuses on measuring extreme risks in insurance business. We mainly use extreme value theory to develop asymptotics for risk measures. We also study the characterization of upper comonotonicity for multiple extreme risks. Firstly, we conduct asymptotics for the Haezendonck--Goovaerts (HG) risk measure of extreme risks at high confidence levels, which serves as an alternative way to statistical simulations. We split the study of this problem into two steps. In the first step, we concentrate on the HG risk measure with a power Young function, which yields certain explicitness. Then we derive asymptotics for a risk variable with a distribution function that belongs to one of the three max-domains of attraction separately. We extend our asymptotic study to the HG risk measure with a general Young function in the second step. We study this problem using different approaches and overcome a lot of technical difficulties. The risk variable is assumed to follow a distribution function that belongs to the max-domain of attraction of the generalized extreme value distribution and we show a unified proof for all three max-domains of attraction. Secondly, we study the first- and second-order asymptotics for the tail distortion risk measure of extreme risks. Similarly as in the first part, we develop the first-order asymptotics for the tail distortion risk measure of a risk variable that follows a distribution function belonging to the max-domain of attraction of the generalized extreme value distribution. In order to improve the accuracy of the first-order asymptotics, we further develop the second-order asymptotics for the tail distortion risk measure. Numerical examples are carried out to show the accuracy of both asymptotics and the great improvements of the second-order asymptotics. Lastly, we characterize the upper comonotonicity via tail convex order. For any given marginal distributions, a maximal random vector with respect to tail convex order is proved to be upper comonotonic under suitable conditions. As an application, we consider the computation of the HG risk measure of the sum of upper comonotonic random variables with exponential marginal distributions. The methodology developed in this thesis is expected to work with the same efficiency for generalized quantiles (such as expectile, Lp-quantiles, ML-quantiles and Orlicz quantiles), quantile based risk measures or risk measures which focus on the tail areas, and also work well on capital allocation problems.

This thesis applies entropy as a model independent measure to address research questions concerning the dynamics of various financial time series. The thesis consists of three main studies as presented in chapters 3, 4 and 5. Chapters 3 and 4 apply an entropy measure to conduct a bivariate analysis of drawdowns and drawups in foreign exchange rates. Chapter 5 investigates the dynamics of investment strategies of hedge funds using entropy of realised volatility in a conditioning model. In all three studies, methods from information theory are applied in novel ways to financial time series. As Information Theory and its central concept of entropy are not widely used in the economic sciences, a methodology chapter was therefore included in chapter 2 that gives an overview on the theoretical background and statistical features of the entropy measures used in the three main studies. In the first two studies the focus is on mutual information and transfer entropy. Both measures are used to identify dependencies between two exchange rates. The chosen measures generalise, in a well defined manner, correlation and Granger causality. A different entropy measure, the approximate entropy, is used in the third study to analyse the serial structure of S&P realised volatility. The study of drawdowns and drawups has so far been concentrated on their uni- variate characteristics. Encoding the drawdown information of a time series into a time series of discrete values, Chapter 3 uses entropy measures to analyse the correlation and cross correlations of drawdowns and drawups. The method to encode the drawdown information is explained and applied to daily and hourly EUR/USD and GBP/USD exchange rates from 2001 to 2012. For the daily series, we find evidence of dependence among the largest draws (i.e. 5% and 95% quantiles), but it is not as strong as the correlation between the daily returns of the same pair of FX rates. There is also dependence between lead/lagged values of these draws. Similar and stronger findings were found among the hourly data. We further use transfer entropy to examine the spill over and lead-lag information flow between drawup/drawdown of the two exchange rates. Such information flow is indeed detectable in both daily and hourly data. The amount of information transferred is considerably higher for the hourly than the daily data. Both daily and hourly series show clear evidence of information flowing from EUR/USD to GBP/USD and, slightly stronger, in the reverse direction. Robustness tests, using effective transfer entropy, show that the information measured is not due to noise. Chapter 4 uses state space models of volatility to investigate volatility spill overs between exchange rates. Our use of entropy related measures in the investigation of dependencies of two state space series is novel. A set of five daily exchange rates from emerging and developed economies against the dollar over the period 1999 to 2012 is used. We find that among the currency pairs, the co-movement of EUR/USD and CHF/USD volatility states show the strongest observed relationship. With the use of transfer entropy, we find evidence for information flows between the volatility state series of AUD, CAD and BRL.Chapter 5 uses the entropy of S&P realised volatility in detecting changes of volatility regime in order to re-examine the theme of market volatility timing of hedge funds. A one-factor model is used, conditioned on information about the entropy of market volatility, to measure the dynamic of hedge funds equity exposure. On a cross section of around 2500 hedge funds with a focus on the US equity markets we find that, over the period from 2000 to 2014, hedge funds adjust their exposure dynamically in response to changes in volatility regime. This adds to the literature on the volatility timing behaviour of hedge fund manager, but using entropy as a model independent measure of volatility regime. Finally, chapter 6 summarises and concludes with some suggestions for future research.

Cette thèse vise à mieux comprendre les décisions individuelles des acteurs des marchés de l'assurance et de la réassurance face à différents types d'information, la couverture des risques extrêmes, et les arrangements contractuels entre assureurs et réassureurs. Le contexte général de l'assurance des risques extrêmes est marqué par une intensification et un accroissement des catastrophes naturelles et industrielles depuis une vingtaine d'années dans le monde, ce qui rend délicat une évaluation correcte des caractéristiques des événements et questionne l'assurabilité de ces risques du fait de la présence d'incertitudes. La thèse est constituée de deux parties utilisant une large palette d'outils quantitatifs. La première partie se concentre sur l'analyse des effets de l'information sur les décisions d'assurance, en particulier sur les primes que les agents sont prêts à payer ou accepter pour transférer ou supporter des risques spécifiques. L'information disponible peut donner lieu à des situations de risque ou d'ambiguïté, en distinguant deux types d'ambiguïté: l'imprécision, situation dans laquelle l'information est consensuelle mais imprécise, et le conflit, situation où il y a désaccord entre experts. Cette partie s'appuie sur des enquêtes distribuées auprès d'un échantillon représentatif de la population en tant qu'acheteurs potentiels d'assurance (chapitre 2) ou auprès de professionnels de l'assurance (chapitre 3) afin d'étudier les comportements d'assurance du côté de l'offre et de la demande. Cette partie présente aussi une approche expérimentale de ces comportements avec rémunération des participants (chapitre 4). La seconde partie étudie le marché de la réassurance. Après avoir présenté les spécificités et mécanismes de ce marché (chapitre 5), l'offre de réassurance est analysée à travers des comportements d'enchères de réassureurs (chapitre 6). A partir d'une base de données exclusive, cette partie détermine les facteurs explicatifs des primes de réassurance qui dépendent des caractéristiques des risques, mais également de celles des traités, des réassureurs et du marché. / This thesis aims to better understand the individual decisions of insurance and reinsurance market actors facing different information types, the coverage of extreme risks, and the contractual arrangements between insurers and reinsurers. The insurance context of extreme risks is characterized by the intensification and the increased number of natural and industrial disasters for the last twenty years in the world, which makes it difficult to properly evaluate the event characteristics and questions the insurability of these risks due to the presence of uncertainties. The thesis is composed of two parts using a wide range of quantitative tools. The first part focuses on the analysis of information effects on insurance decisions, especially the premium that actors are willing to pay or to accept in order to transfer or to take specific risks. The available information can lead to situations of risk or ambiguity, distinguishing two ambiguity types: imprecision in which the information is consensual but imprecise, and conflict where there is a disagreement between experts. This part is based on surveys distributed to a representative sample of the population as potential insurance buyers (chapter 2) or to insurance professionals (chapter 3) in order to study demand and supply behaviors. This part also presents an experimental approach in which participants are paid according to their performance (chapter 4). The second part examines the reinsurance market. After presenting the characteristics and mechanisms of the market (chapter 5), reinsurance supply is examined through auction behaviors of reinsurers (chapter 6). 8ased on a proprietary data-base, this part identifies the underlying factors of reinsurance premiums which depend on the risk characteristics, but also on the treaties, the reinsurers and the market.