Advanced Mathematical Methods for Finance

Abstract

Over the last decade, it has been shown that the concept of comonotonicity is a helpful tool for solving several research and practical problems in the domain of finance and insurance. In this paper, we give an extensive bibliographic overview -without claiming to be complete -of the developments of the theory of comonotonicity and its applications, with an emphasis on the achievements over the last five years. These applications range from pricing and hedging of derivatives over risk management to life insurance. ComonotonicityOver the last two decades, researchers in economics, financial mathematics and actuarial science have introduced results related to the concept of comonotonicity in their respective fields of interest. In this paper, we give an overview of the relevant literature in these research fields, with the main emphasis on the development of the theory and its applications in finance and insurance over the last five years. Although it is our intention to give an extensive bibliographic overview, due to the high number of papers on applications of comonotonicity, it is impossible to present here an exhaustive overview of the recent literature. Further, we restrict this paper to a description of how and where comonotonicity comes in and refer to the relevant papers for a detailed mathematical description. In order to make this paper self-contained, we also provide a short overview of the basic definitions and initial main results of comonotonicity theory, hereby referring to part of the older literature on this topic.The concept of comonotonicity is closely related to the following well-known result, which is usually attributed to both Hoeffding (1940) and Fréchet (1951): For any n-dimensional random vector X ≡ (X 1 , X 2 , . . . , X n ) with multivariate cumulative distribution function (cdf) F X and marginal univariate cdf's F X 1 , F X 2 , . . . , F Xn and for any x ≡ (x 1 , x 2 , . . . , x n ) ∈ R n it holds that F X (x) ≤ min (F X 1 (x 1 ) , F X 2 (x 2 ) , . . . , F Xn (x n )) .(1)In the sequel, the notation R n (F X 1 , F X 2 , . . . , F Xn ) will be used to denote the class of all random vectors Y ≡ (Y 1 , Y 2 , . . . , Y n ) with marginals F Y i equal to the respective marginals F X i . The set R n (F X 1 , F X 2 , . . . , F Xn ) is called the Fréchet class related to the random vector X.The upper bound in (1) is reachable in the Fréchet class R n (F X 1 , F X 2 , . . . , F Xn ) in the sense that it is the cdf of an n-dimensional random vector with marginals given by F X i , i = 1, 2, . . . , n. * Department of Mathematics, ECARES, Université Libre de Bruxelles, CP 210, 1050 Brussels, Belgium † Faculty of Business and Economics, Katholieke Universiteit, 3000 Leuven, Belgium. ‡ Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, 9000 Gent, Belgium 1 In order to prove the reachability property, consider a random variable U , uniformly distributed on the unit interval (0, 1). Then one has that F −1 X 1 (U ), F −1 X 2 (U ), . . . , F −1 Xn (U ) ∈ ...