Copula modeling has taken the world of finance and insurance, and well beyond, by storm. Why is this? In this paper I review the early start of this development, discuss some important current research, mainly from an applications point of view, and comment on potential future developments. An alternative title of the paper would be “Demystifying the copula craze”. The paper also contains what I would like to call the copula must-reads. Keywords: copula, extreme value theory, Fréchet–Hoeffding bounds, quantitative risk management, Value–at–Risk 1

Properties of risk measures for extreme risks have become an important topic of research. In the present paper we discuss sub- and superadditivity of quantile based risk measures and show how multivariate extreme value theory yields the ideal modeling environment. Numerous examples and counter-examples highlight the applicability of the main results obtained.

The copula of a multivariate distribution is the distribution transformed so that one dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions and the resulting distribution we call the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima and empirical processes. We discuss implications for aggregation of risk and offer some examples.

We study a model for default contagion in intensity-based credit risk and its consequences for pricing portfolio credit derivatives. The model is specified through default intensities which are assumed to be constant between defaults, but which can jump at the times of defaults. The model is translated into a Markov jump process which represents the default status in the credit portfolio. This makes it possible to use matrix-analytic methods to derive computationally tractable closed-form expressions for single-name credit default swap spreads and k th-to-default swap spreads. We ”semi-calibrate” the model for portfolios (of up to 15 obligors) against market CDS spreads and compute the corresponding k th-to-default spreads. In a numerical study based on a synthetic portfolio of 15 telecom bonds we study a number of questions: how spreads depend on the amount of default interaction; how the values of the underlying market CDS-prices used for calibration influence k th-th-to default spreads; how a portfolio with inhomogeneous recovery rates compares with a portfolio which satisfies the standard assumption of identical recovery rates; and, finally, how well k th-th-to default spreads in a nonsymmetric portfolio can be approximated by spreads in a symmetric portfolio.

We describe several analytical and numerical procedures to obtain bounds on the distribution function of a sum of n dependent risks having fixed overlapping marginals. As an application, we produce bounds on quantile-based risk measures for portfolios of financial/actuarial interest.

We describe several analytical and numerical procedures to obtain bounds on the distribution function of a sum of n dependent risks having fixed overlapping marginals. As an application, we produce bounds on quantile-based risk measures for portfolios of financial/actuarial interest.

for their financial backing and for giving me the opportunity to keep a toe in the credit markets. I would like to thank all those at OCIAM who have provided me with the guidance and support necessary to complete this thesis. I am particularly grateful to my supervisor, William Shaw, for allowing me the freedom to pursue my own research interests and I would especially like to thank Christoph Reisinger for more helpful discussions than I can count, not to mention the opportunity to play with his code. I would also like to mention all my friends, in Oxford and elsewhere, who have contributed in so many ways to my life over the last three years and to the ultimate form of my research. Finally, I would like to thank my family for their continued support in all my endeavours, and in particular, my sister, for giving me the impetus I needed to return to student life and Multi-asset credit derivatives trade in huge volumes, yet no models exist that are capable of properly accounting for the spread behaviour of dependent

Abstract. Using a large set of daily US and Japanese stock returns, we test in detail the relevance of Student models, and of more general elliptical models, for describing the joint distribution of returns. We find that while Student copulas provide a good approximation for strongly correlated pairs of stocks, systematic discrepancies appear as the linear correlation between stocks decreases, that rule out all elliptical models. Intuitively, the failure of elliptical models can be traced to the inadequacy of the assumption of a single volatility mode for all stocks. We suggest several ideas of methodological interest to efficiently visualiseand compare different copulas. We identify the rescaled difference with the Gaussian copula and the central value of the copula as strongly discriminating observables. We insist on the need to shun away from formal choices of copulas with no financial interpretation. 1.

• Copulas (e.g., Renard and Lang, 2006; Mikosch, 2006, and ensuing discussions). • Regional Frequency Analysis (RFA)

An important problem in statistics is determining a joint probability distribution from its marginals. In 2D case, the marginal probability density functions f1(x) and f2(y) are related to their joint distribution f(x,y) via the horizontal and vertical line integrals. So, the problem of determining f(x,y) from f1(x) and f2(y) is an ill-posed inverse problem. In statistics the notion of copula is exactly introduced to obtain a solution to this problem. Interestingly, this is also a problem encountered in X ray tomography image reconstruction where f(x,y) is an image representing the distribution of the material density and f1(x) and f2(y) are the horizontal and vertical line integrals. In this paper, we try to link the notion of copula to X ray Computed Tomography (CT) and to see if we can use the methods used in each domain to the other one. 1