Estimation of the Value at Risk Using the Stochastic Approach of Taylor Formula

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. As shown by Sklar (1959), see Nelsen [2], a multivariate cumulated distribution function (cdf) F � (F1, . . . , Fn) of a random continuous vector X � (X1, . . . , Xn) is canonically linked to a unit uniform cdf CF which can be written as for all (u1, . . . , un) ∈ [0, 1] ,


Introduction
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.
As shown by Sklar (1959), see Nelsen [2], a multivariate cumulated distribution function (cdf ) F � (F 1 , . . . , F n ) of a random continuous vector X � (X 1 , . . . , X n ) is canonically linked to a unit uniform cdf C F which can be written as for all (u 1 , . . . , u n ) ∈ [0, 1] n , where F − 1 i (u) � inf x ∈ R, F i (x) ≥ u , for 1 ≤ i ≤ n, is the quantile function of F i . Joe and Xu [1] and Nelsen [2] provided the key standard references for copulas analysis. ey provided detailed and readable introductions to copulas and their statistical and mathematical foundations. In the same vein, Bouyé et 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 value-at-risk (VaR) is also intrinsically linked in its expression to this function, making it possible to bridge the copula function with the VaR. 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 F t , and the univariate component of the vector VaR α (X) is, for all portfolio X, where F − 1 t being here the right continuous inverse of F t . As stated in the summary, the purpose of this paper is to propose another form of the VaR approximation. We went around on the estimates that were proposed, given that there are several methods of estimating the copula. 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. us, the global log likelihood can be expressed as follows: log D n C t F 1 x 1 , θ 1 ; . . . , F n x n , θ n ; α is the vector of the parameters containing the parameter θ i of each marginal F i and the parameter α of the copula. At first, the parameters of each marginal law are estimated independently by the method of maximum likelihood θ i � arg max T i�1 ln f i (x i , θ i ). en, we estimate α taking into account the estimates obtained, so we have is estimate of the copula is a good approximation. It is widely used in numerical simulations of the copula. We could have used this approximation of the copula. Another idea is to use the limited development formula to get an estimate. Our approach would be better and much more precise. It is in this logic we thought to use the Taylor-Young formula. We will later use a method of limited development theory (Taylor's formula) in this paper to provide an estimate of the value at risk. In calculus, Taylor's theorem gives an approximation of a ktime differentiable function around a given point by a k-th order Taylor polynomial. For analytic functions, the Taylor polynomials at a given point are finite-order truncations of its Taylor series, which completely determines the function in some neighborhood of the point. It can be thought as the extension of linear approximation to higher order polynomials, and in the case of k equals to 2, it is often referred to as a quadratic approximation.
Modeling the risk of multivariate portfolio, it is also on the one hand to choose the appropriate mathematical tools for interpreting the data of a portfolio and, on the other hand, to provide an interface to the financial world. But between these two stages a rigorous and precise work should be carried out.
In this paper, we have given an approximation of the VaR using the Taylor-Young formula. So, we first tested at order 2 before and, subsequently, extended this approximation to higher orders.

Preliminaries
e works of Clauss [7], Coles [8], Dean [9], and Nelsen [2] have inspired us a lot on the theory of copulas. Before trying to model our different relationships on VaR, the works of Artzner et al. [10], Lee and Prékopa [11], and Yves Bernadin Loyara et al. [12] make us understand that VaR and its derived measures are measured under consistent risk. e use of copulas in finance has been well illustrated in the works of Böcker and Klüppelberg [13], Cherubini et al. [4], Marius Hofert [14], and Lee and Prékopa [11]. All of these authors have allowed us to understand and develop the relationships that appear in the following sections.
In this section, we have an overview of the key statements (definitions, propositions, and theorems) which will be useful thereafter. us, the conditional approach of Sklar's theorem [15] will play a central role as far the concept of scalar product which allows us to highlight a link between the conditional copula and the notion of norms in the metric spaces.

Scalar Product and Copulas Applications on the VaR.
Using the above relation (1) (positiveness of the volume of any hyper-rectangle of R n ), Yves Bernadin Loyara and Barro [16] provided the following result by extending a proposition of Patton [15] both to space-varying case and to higher dimensional framework.
where f w is the spatial density of the law of W t . Moreover, the following properties are satisfied: Let us consider a linear portfolio consisting of n different financial instruments (risks and actions) X � (X 1 , . . . , X n ). Furthermore, let p 0 � (p 0,1 , . . . , p 0,n ); the initial value of the portfolio is given by V 0 � n i�1 x i p 0,i for a realization x � (x 1 , . . . , x n ) of X. At the next date at time t, the uncertain profit and loss functions of the portfolio are given by Particularly, from the integral probability transforms, one can associate with F t a parametric copula C t as, for all International Journal of Mathematics and Mathematical Sciences C t u 1 , . . . , u n � P F 1 X 1 ≤ u 1 ; . . . ; F n X n ≤ u n , (9) where u i � F i (x i ), for all i ∈ 1, . . . , n { }. Let us consider the particular case where X denotes a portfolio of risks X i of potential losses in independent lines of business for an assurance company and suppose that P t � (p 1,t e z t,1 ; . . . ; p n,t e z n,1 ). In the same way, Loyara et al. [16] proposed the following result which extended the conditional copula to VaR. Theorem 1. For a realization x � (x 1 , . . . , x n ) of X, at the next date at time t, the uncertain profit and loss functions of the portfolio are given by Then, where P t � (p 1,t e z t,1 ; . . . ; p n,t e z n,1 ) and is a value at risk of X and ‖·‖ Euclidean norm. Even in stochastic analysis, the Taylor-Young formula plays a key role in the estimation of the number of functions.

Taylor-Young Formula in Standardized Vector Spaces.
e exact content of "Taylor's theorem" is not universally agreed upon. Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial. e statement of the most basic version of Taylor's theorem is as follows: let k ≥ 1 be an integer and let the function f: R ⟶ R be k times differentiable at the point a ∈ R. en, there exists a function h k : R ⟶ R such that with D k x f(a) � z k f(a)/zx k and lim x⟶a h k (x) � 0. Let E and F be two normalized vector spaces. If a function f: E ⟶ F is n times differentiable into a point a ∈ E, then it admits at this point a development limited to the order n, given by where h k denotes the k-tuple (h, . . . , h) ∈ E k . In the next section, we will use the relation (15) to have a limited development of the time-varying conditional copula function. en, using the relation (11), we will express an approximation of the value at risk.

Main Results
In this subsection, we develop the second-order relation established by Yves Bernadin Loyara and Barro [] [16], and we took as a basic notion, probability metric of Rachev et al. [17]. We also use the Taylor-Young relation to obtain an estimate of the value-at-risk norm.

Our Working Assumptions
(H 1 ) e space of work is a Euclidean one [12]: so, we can use the characteristic elements of metric spaces and scalar product to clarify the magnitude of the dependence between the VaR and the conditional timevarying copula. (H 2 ) In all the following, ϕ denotes a bilinear application mapping L([0, 1] 2 ) such that for all (u, v) and (H 3 ) e copula differentiating formula (1) shows that the density function of the copula is equal to the ratio of the joint density h of H to the product of marginal densities h i such that for all (u 1 , . . . , u n ) ∈ [0, 1] n , c u 1 , . . . , u n � z n C u 1 , . . . , u n zu 1 , . . . , zu m More preciously, the consequence of the relation (17) is that the copula satisfies the Taylor formula.

Value-At-Risk Modeling with Time-Varying Conditional
Copulas. Let us consider a linear portfolio consisting of 2 different financial instruments (risks and actions) X � (X 1 , X 2 ) and let p t � (p 1,t , p 2,t ) at a given date measured at given time t. Furthermore, let p 0 � (p 0,1 ; p 0,2 ), and the initial value of the portfolio is given by V 0 � x 1 p 0,1 + x 2 p 0,2 for a realization x � (x 1 , x 2 ) of X. At the next date at time t, the uncertain profit and loss functions of the portfolio are given by International Journal of Mathematics and Mathematical Sciences 3 where Z t � (z t,1 , z t,2 ) is the log price vector such that z t,i � log p t,i with i ∈ 1, 2 { }. Particularly, from the integral probability transforms, we can associate with F t a parametric copula C t such that, for all (u 1 , u 2 ) ∈ [0, 1] 2 and u i � F i (x i ), ∀i ∈ 1, 2 { }, X 2 ), at the next date at time t, the uncertain profit and loss functions of the portfolio are given by relation (18); then is a value at risk of X and ‖·‖ Euclidean norm.
Proof. Consider the following relation: e relation (3) allows us to write that If we consider relation (18), we obtain C t u 1 , u 2 � VaR u 1 X 1 p t,1 e z t,1 − p t,1 + VaR u 2 X 2 p t,2 e z t,2 − p t,2 . (23) Consider P t � (p 1,t e z t,1 , p 2,t e z t,2 ) and VaR u (X) � (VaR u 1 (X 1 ), VaR u 2 (X)). en, e application defines on E a norm, called Euclidean norm and noted ‖ ‖. ∀(x, y) ∈ E 2 , we have the following: Cauchy-Schwartz inequality: Cauchy-Schwartz equality (in the case where (x, y) do not form a free family): Value at risk is intrinsically linked to the portfolio and therefore to the initial amount and the amount at a given time t. We will suppose linked vector VaR u (X) and vector p t − P t .
Based on the relation (27), we obtain en, □

Estimation of the VaR via Taylor-Young Formula of Order 2.
We introduce the linear application to avoid having the aspect of the development formula limited in our relation. Let us consider ϕ ∈ L([0, 1] 2 ) such that In short, ϕ is a bilinear application. Under the key assumption (we are in a Euclidean space [] [1]), let us consider here Z t � (z t,1 ; z t,2 ) is the log price vector such that z t,i � log p t,i with i ∈ 1, 2 { }. Particularly, from the integral probability transforms, we can associate with F t a parametric copula C t such that, for all (u 1 , Proposition 3. Let C t be a copula of order 2 and I � [0, 1] 2 ⊂ U a subset of R 2 (u 1 , u 2 ) ∈ I, and for any h ∈ [0, 1], where ΔC t (u 1 , u 2 ) and D i C t (u 1 , u 2 ), i � 1,2, denote, respectively, the Laplacian and the partial derivatives of the copula C t .

International Journal of Mathematics and Mathematical Sciences
If we use the relation (11), then the following equation is obtained: Under the key assumption, it follows Moreover, by pulling the value at risk, the following equation is obtained: Let us take another variant of the previous relation (15). For a function C t : [0, 1] p ⟶ [0, 1] twice differentiable and a ∈ [0, 1] p , we have where ∇C t is a gradient of C t and H(a) is the Hessian matrix at point a. By considering our third key assumption (H 3 ), the following characterization of the VaR (47) can be obtained. □ Corollary 1. Let copula C t : [0, 1] p ⟶ [0, 1] be twice differentiable and u ∈ [0, 1] p , and we have where ∇f is a gradient of f and H(a) Hessian matrix evaluated in a, with the condition u + h ∈ [0, 1] p . en, Proof. For the proof action of eorem 1, we will skip the relation (48), i.e.,