# Pair-Copula Constructions for Financial Applications: A Review

## Abstract

**:**

## 1. Introduction

## 2. The Pair-Copula Construction and the Regular Vine

- Tree ${T}_{1}$ has nodes ${N}_{1}=\{1,...,d\}$ and edges ${E}_{1}$.
- For $i=2,...,d-1$, the nodes in tree ${T}_{i}$ are the edges in tree ${T}_{i-1}$, i.e., ${N}_{i}={E}_{i-1}$.
- Proximity condition: if two edges in tree ${T}_{i}$ are to be joined as nodes in tree ${T}_{i+1}$ by an edge, they must share a common node in ${T}_{i}$.

**X**determined by the indices constituting $D\left(e\right)$. Then, Theorem 4.2 in [9] states that the joint density of $\mathit{X}$ can be written as:

#### 2.1. Simplifying Assumption

#### 2.2. Canonical Vines and D-Vines

#### 2.3. Serial Dependence

## 3. Inference

#### 3.1. Structure Selection

#### 3.2. Choosing Copula Families

#### 3.3. Parameter Estimation for a Given Structure and Copula Families

#### Time-Varying Models

#### 3.4. Pruning and Truncation

#### 3.4.1. Pruning

#### 3.4.2. Truncation

## 4. Model Validation

## 5. Financial Applications

#### 5.1. Market Risk

#### 5.2. Capital Asset Pricing

#### 5.3. Credit Risk

#### 5.4. Operational Risk

#### 5.5. Liquidity Risk

#### 5.6. Systemic Risk

#### 5.7. Portfolio Optimization

#### 5.8. Option Pricing

## 6. Conclusions

## Conflicts of Interest

## References

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**Figure 1.**A regular vine (R-vine) with 7 variables, 6 trees and 21 edges. Each edge may be may be associated with a pair-copula.

**Figure 2.**A D-vine with 5 variables, 4 trees and 10 edges. Each edge may be may be associated with a pair-copula.

**Figure 3.**A canonical vine with 5 variables, 4 trees and 10 edges. Each edge may be may be associated with a pair-copula.

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Aas, K. Pair-Copula Constructions for Financial Applications: A Review. *Econometrics* **2016**, *4*, 43.
https://doi.org/10.3390/econometrics4040043

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Aas K. Pair-Copula Constructions for Financial Applications: A Review. *Econometrics*. 2016; 4(4):43.
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Aas, Kjersti. 2016. "Pair-Copula Constructions for Financial Applications: A Review" *Econometrics* 4, no. 4: 43.
https://doi.org/10.3390/econometrics4040043