Abstract
The aim of this paper is to provide an approximation of the value-at-risk of the multivariate copula associated with financial loss and profit function. A higher dimensional extension of the Taylor–Young formula is used for this estimation in a Euclidean space. Moreover, a time-varying and conditional copula is used for the modeling of the VaR.
Highlights
A copula function is an instrument of probability theory that makes able to characterize joint dependence. e relationship between the marginal distributions of two or more random variables and the cumulate joint distribution is clarified by the associated copula independently of stochastic behavior of their marginal laws
Bouyeet al. [3], Cherubini et al [4], and Beirlant et al [5] dealt with applications of copulas to different levels of financial issues and derivative pricing. e relation (1) is justified since probability integral transformation returns every univariate variable X to the unit uniform random variable, U F(X)
E main limitation of the VaR lies in the fact that whatever the method used, the data injected into the calculation algorithm always come more or less from the market values found in the past, which are not necessarily a reflection of the evolutions
Summary
A copula function is an instrument of probability theory that makes able to characterize joint dependence. e relationship between the marginal distributions of two or more random variables and the cumulate joint distribution is clarified by the associated copula independently of stochastic behavior of their marginal laws.CF u1, . . . , un FtF−11 u1, . . . , F−n1 un, (1)where F−i 1(u) infx ∈ R, Fi(x) ≥ u, for 1 ≤ i ≤ n, is the quantile function of Fi. A copula function is an instrument of probability theory that makes able to characterize joint dependence. More generally in the multivariate study, for a random vector X satisfying the regularity conditions, one defines the multidimensional VaR at probability level α by. Where zL(α) is the boundary of the α − level set of Ft, and the univariate component of the vector VaRα(X) is, for all portfolio X, (3). One of the most famous methods is the marginal inference function (IFM), proposed by Carreau and Bengio [6] and Joe and Xu [1]. It consists in separating the estimation of the parameters of the marginal laws and those of the copula. It consists in separating the estimation of the parameters of the marginal laws and those of the copula. us, the global log likelihood can be expressed as follows: International Journal of Mathematics and Mathematical Sciences
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