# Optimization and Diversification of Cryptocurrency Portfolios: A Composite Copula-Based Approach
## 2 Methodology
### Vine-Copula  
This subsection describes the characteristics necessary for a function to be a copula, as well as some of their properties. Before mentioning them, some preliminary definitions and results are useful. In fact, here is the idea behind the copula approach. Let \(X=(X_{1},\ldots,X_{n})\) be a vector of \(n\) random variables for which we want to construct the joint distribution function. Let assume that, \(X_{1},\ldots,X_{n}\) are random variables with the following marginal distributions \(F_{1},\ldots,F_{n}\), such as \(F_{i}(x_{i})=P(X_{i}\leq x_{i})=u_{i}\) (the probability that the measure of \(X_{i}\) be less than \(x_{i}\)). Then the joint distribution function is given by  
\[F(x_{1},\ldots,x_{n})=P(X_{1}\leq x_{1},\ldots,X_{n}\leq x_{n}). \tag{1}\]  
**Definition 2**. : _An \(n\) dimensional (\(n\geq 2\)) copula is a function \(C:I^{n}\longrightarrow I\) satisfying the following properties:_  
1. \(C\) _is non-decreasing that is_ \(C(0,\ldots,x_{i},\ldots,0)=0\)_, for all_ \(x_{i}\in I=[0,1]\)__
2. \(C\) _possess one dimensional uniform margins on_ \(C_{i}\)_, that is:_ \(C_{i}(x_{i})=C(1,\ldots,1,x_{i},1,\ldots,1)=x_{i}\) _for all_ \(x_{i}\in I\)_._ \(C_{i}\) _is an invariant non-decreasing transformation of the marginal._  
We can now show the link between copulae and random variables. Copulas are of interest in statistics thanks to the theorem proposed by [PERSON] [14]. **Theorem 1**. : _Assume \(F=(F_{1},\ldots,F_{n})\) is an \(n\) dimensional joint distribution function with marginal distribution function \(F_{i}\)\((i=1,\ldots,n)\). Then, there exists a copula \(C\) such that for all \(\boldsymbol{x}=(x_{1},\ldots,x_{n})\in I^{n}\)_  
\[F(\boldsymbol{x})=C(F_{1}(x_{1}),\ldots,F_{n}(x_{n}))\]  
_If \(F_{1},\ldots,F_{n}\) are continue, then \(C\) is unique. Otherwise \(C\) is non-unique on \(I^{n}\)._  
_In addition, if \(F_{1},\ldots,F_{n}\) are distribution function on \(I\) and if \(C\) is a copula, then the function \(F(\boldsymbol{x})=C(F_{1}(x_{1}),\ldots,F_{n}(x_{n}))\) is a joint distribution function on \(I^{n}\)._  
The canonical representation of the copula density function is given as  
\[c(u_{1},\ldots,u_{n})=\frac{\partial^{n}C(u_{1},\ldots,u_{n})}{\partial u_{1},\ldots,\partial u_{n}} \tag{2}\]To obtain the density of the \(n\)-dimension distribution \(F\), the following relation is used  
\[f(x_{1},\ldots,x_{n})=c(F_{1}(x_{1}),\ldots,F_{n}(x_{n}))\prod_{i=1}^{n}f_{i}(x_{ i}) \tag{3}\]  
where \(f_{i}\) is the density of the marginal distribution \(F_{i}\). Copula functions constitute an advantageous statistical tool for constructing and simulating multivariate distributions. The literature devoted to the copula approach provides us with different types of density functions which can be summarized in two categories of families: the family of elliptical copulas and that of Archimedean copulas. Tables 1 and 2 show different properties of popular copulas functions. In practical applications, modelling using copula with a large set of high-dimensional variables, such as in this paper (a set of eight cryptocurrencies), has some limitations. These limitations can be parameters restriction and the selection of the appropriate copula function. [PERSON] [15] proposed an alternative which is the use of vine-copulas method for high-dimensional data. Thus vine copulas, described by [PERSON] and [PERSON] [16; 17] are flexible graphical models to describe multivariate copulas constructed using a cascade of bivariate copulas or pair-copulas. In this paper we will use C (canonical) and R (regular) vine copula specification. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**No. ** & **Elliptical Distribution** & **Parameter Range** & **Kendall’s \(\tau\)** & **Tail Dependence** \\ \hline
1 & Gaussian & \(\rho\in(-1,1)\) & \(\frac{2}{\pi}\arctan(\rho)\) & 0 \\
2 & Student-t & \(\rho\in(-1,1)\), \(\
u>2\) & \(\frac{2}{\pi}\arctan(\rho)\) & \(2t_{\
u+1}(-\sqrt{\
u+1}\sqrt{\frac{1-\rho}{1+\rho}})\) & \\ \hline
**No.