The one-factor Gaussian model is well-known not to fit simultaneously the prices of the different tranches of a collateralized debt obligation (CDO), leading to the implied correlation smile. Recently, other one-factor models based on different distributions have been proposed. Moosbrucker [12] used a one-factor Variance Gamma model, Kalemanova et al. [7] and Guégan and Houdain [6] worked with a NIG factor model and Baxter [3] introduced the BVG model. These models bring more flexibility into the dependence structure and allow tail dependence. We unify these approaches, describe a generic one-factor Lévy model and work out the large homogeneous portfolio (LHP) approximation. Then, we discuss several examples and calibrate a battery of models to market data.
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MSC:primary 62G20 secondary 60F05
Keywords:Weak limit theorems Functions of regular variation Domain of attraction of a stable law Sample coefficient of variation Sample dispersion Student's t-statistic Extreme value theory The coefficient of variation and the dispersion are two examples of widely used measures of variation. We show that their applicability in practice heavily depends on the existence of sufficiently many moments of the underlying distribution. In particular, we offer a set of results that illustrate the behavior of these measures of variation when such a moment condition is not satisfied. Our analysis is based on an auxiliary statistic that is interesting in its own right. Let (X i ) i Ն 1 be a sequence of positive independent and identically distributed random variables with distribution function F and define for n ∈ N T n := X 2 1 + X 2 2 + · · · + X 2 n (X 1 + X 2 + · · · + X n ) 2 .Mainly using the theory of functions of regular variation, we derive weak limit theorems for the properly normalized random quantity T n , given that 1 − F is regularly varying. Following a distributional approach based on T n , we then analyze asymptotic properties of the sample coefficient of variation. As a second illustration of the same method, we then turn to the sample dispersion. We also include asymptotic properties of the first moments of these quantities. Finally, we give a distributional result on Student's t-statistic which is closely related to T n . The main message of this paper is to show that the unconscientious use of some measures of variation can lead to wrong conclusions.
Abstract. Let {X 1 , X 2 , . . .} be a sequence of independent and identically distributed positive random variables of Pareto-type and let {N (t); t 0} be a counting process independent of the X i 's. For any fixed t 0, define:if N (t) 1 and T N (t) := 0 otherwise. We derive limits in distribution for T N (t) under some convergence conditions on the counting process. This is even achieved when both the numerator and the denominator defining T N (t) exhibit an erratic behavior (EX 1 = ∞) or when only the numerator has an erratic behavior (EX 1 < ∞ and EX 2 1 = ∞). Armed with these results, we obtain asymptotic properties of two popular risk measures, namely the sample coefficient of variation and the sample dispersion.
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