Bivariate Copulas

This tutorial introduces Bivariate Copulas estimators that are the building blocks of VineCopula.

Introduction

Bivariate copulas are mathematical functions that allow us to construct a joint distribution by combining the individual marginal distributions of two variables with a separate model for their dependence structure. This approach enables the marginal behavior of each variable to be modeled independently, while the copula captures how these variables move together.

There are two primary families of copulas used in finance:

1. Elliptical Copulas:

   • Gaussian Copula: Based on the multivariate normal distribution with a correlation parameter \(\rho \in [-1, 1]\). It is symmetric but does not exhibit tail dependence.

   • Student’s t Copula: Derived from the multivariate Student’s t-distribution with \(\rho \in [-1, 1]\) and degrees of freedom \(\nu > 2\). It captures symmetric tail dependence, making it more suitable for modeling extreme co-movements.

2. Archimedean Copulas:

   • Gumbel Copula: Characterized by a parameter \(\theta \in [1, \infty)\) and models upper tail dependence.

   • Clayton Copula: Uses a parameter \(\theta \in (0, \infty)\) and capture lower tail dependence.

   • Joe Copula: Defined with \(\theta \in [1, \infty)\) and models upper tail dependence.

Copula	Family	Parameters	Tail Dependence	Symmetry
Gaussian	Elliptical	\(\rho \in [-1, 1]\)	None	Symmetric
Student’s t	Elliptical	\(\rho \in [-1, 1]\); \(\nu > 2\)	Both upper and lower tail dependence	Symmetric
Gumbel	Archimedean	\(\theta \in [1, \infty)\)	Upper tail dependence	Asymmetric
Clayton	Archimedean	\(\theta \in (0, \infty)\)	Strong lower tail dependence	Asymmetric
Joe	Archimedean	\(\theta \in [1, \infty)\)	Strong upper tail dependence	Asymmetric

Rotation of Archimedean Copulas

Standard Archimedean copulas are inherently designed to capture dependence in one specific tail:

• Gumbel and Joe: Naturally capture upper tail dependence.

• Clayton: Naturally captures lower tail dependence.

However, financial data may exhibit tail behavior opposite to a copula’s inherent design, or even negative dependence. Rotation is a transformation that adjusts the copula to model the opposite tail. In effect, rotation swaps the roles of the upper and lower tails, enabling the model to capture tail dependence where it is most relevant.

Available rotations include 0° (unrotated), 90°, 180°, and 270°. During the fitting process, both the copula parameters and the optimal rotation are estimated.

Data

We load the S&P 500 dataset and select Bank of America (BAC) and JPMorgan (JPM) stocks starting from 1990-01-02 up to 2022-12-28:

import numpy as np
from plotly.io import show

from skfolio.datasets import load_sp500_dataset
from skfolio.distribution import (
    ClaytonCopula,
    Gaussian,
    GaussianCopula,
    GumbelCopula,
    JoeCopula,
    JohnsonSU,
    StudentT,
    StudentTCopula,
    select_bivariate_copula,
    select_univariate_dist,
)
from skfolio.preprocessing import prices_to_returns

prices = load_sp500_dataset()
prices = prices[["BAC", "JPM"]]
X = prices_to_returns(prices)
print(X.tail())

                 BAC       JPM
Date
2022-12-21  0.015223  0.011248
2022-12-22 -0.008848 -0.011355
2022-12-23  0.002443  0.004749
2022-12-27  0.001875  0.003504
2022-12-28  0.007360  0.005463

Marginal Distribution

First, we fit the marginal distributions for each asset independently. We use the utility function select_univariate_dist to select the optimal univariate distribution based on information criterion (BIC or AIC).

candidates = [Gaussian(), StudentT(), JohnsonSU()]
X1, X2 = X[["BAC"]], X[["JPM"]]

bac_dist = select_univariate_dist(X=X1, distribution_candidates=candidates)
print(f"BAC: {bac_dist.fitted_repr}")

jpm_dist = select_univariate_dist(X=X2, distribution_candidates=candidates)
print(f"JPM: {jpm_dist.fitted_repr}")

BAC: StudentT(loc=0.00033, scale=0.013, df=2.5)
JPM: JohnsonSU(a=-0.041, b=1.1, loc=-0.00032, scale=0.015)

Let’s plot the PDF of the fitted distribution versus the historical data:

jpm_dist.plot_pdf(X2)

Let’s analyse the Q-Q plot:

jpm_dist.qq_plot(X2)

Let’s explore the difference versus the Gaussian distribution:

gaussian = Gaussian()
gaussian.fit(X2)
gaussian.plot_pdf(X2)

gaussian.qq_plot(X2)

The Q-Q plot and zooming in on the tail of the Johnson Su distribution shows that its fat tails are well captured, which is not the case for the Gaussian distribution.

Uniform Marginals

Before working with copulas, we transform the asset returns into uniform marginals using their univariate CDFs:

X = np.hstack([bac_dist.cdf(X1), jpm_dist.cdf(X2)])

Bivariate Copulas

We use the utility function select_bivariate_copula to select the optimal bivariate copula based on information criterion (BIC or AIC).

candidates = [
    GaussianCopula(),
    StudentTCopula(),
    ClaytonCopula(),
    GumbelCopula(),
    JoeCopula(),
]
copula = select_bivariate_copula(X, copula_candidates=candidates)
print(copula.fitted_repr)
print(f"AIC: {copula.aic(X):,.2f}")

StudentTCopula(rho=0.748, dof=2.65)
AIC: -7,525.40

Let’s plot the 2D probability density function (PDF) of the Student-t copula. The x-axis and y-axis represent the uniform variates (u and v) obtained by applying the marginal cumulative distribution functions (CDFs) to the returns of BAC and JPM, respectively.

fig = copula.plot_pdf_2d()
fig.update_layout(height=700)

Each contour line connects points of equal copula density, effectively delineating regions with the same level of joint dependence between the two assets. Areas where the contours are closely spaced indicate a rapid change in density, reflecting regions with higher concentrations of joint probability.

The Student-t copula captures tail dependence: extreme positive or negative returns in one asset tend to occur simultaneously with extreme returns in the other. This is visualized as pronounced “bulges” in the tail regions of the contour plot.

Note: The copula density is defined on the scale of the transformed uniform marginals. To recover the joint density of the original asset returns, the copula density must be multiplied by the marginal probability density functions, which take on lower values in the tails.

Now, let’s plot the 3D PDF:

fig = copula.plot_pdf_3d()
fig.update_layout(scene_camera=dict(eye=dict(x=-1.2, y=1.4, z=0.8)))
fig

Tail Dependencies

For a bivariate random vector \((X,Y)\) with marginal distribution functions \(F_X\) and \(F_Y\), the tail dependence coefficients quantify the probability that one variable is extreme given that the other is extreme. They are defined as follows:

Upper Tail Dependence Coefficient:

\[\lambda_U = \lim_{u \to 1^-} P\left( Y > F_Y^{-1}(u) \mid X > F_X^{-1}(u) \right)\]

Lower Tail Dependence Coefficient:

\[\lambda_L = \lim_{u \to 0^+} P\left( Y \le F_Y^{-1}(u) \mid X \le F_X^{-1}(u) \right)\]

A positive \(\lambda_U\) (or \(\lambda_L\)) indicates that extreme high (or low) values of \(X\) and \(Y\) occur together more frequently than if the variables were independent.

Let’s print the Student’s t Copula tail dependence. The model indicates a tail dependence coefficient of approximately 51%, suggesting a relatively high likelihood that extreme returns (both negative and positive) occur simultaneously for the assets. Since the Student’s t copula is symmetric, the tail dependence is the same for both the lower and upper tails.

print(f"Lower Tail Dependence: {copula.lower_tail_dependence:.2%}")
print(f"Upper Tail Dependence: {copula.upper_tail_dependence:.2%}")

Lower Tail Dependence: 51.21%
Upper Tail Dependence: 51.21%

Tail Concentration

Tail concentration refers to how much probability mass is concentrated at the tails of a distribution. Let’s plot the tail concentration of the copula model versus the historical data:

copula.plot_tail_concentration(X)

Comparison with the Gaussian Copula

Let’s explore the difference versus the Gaussian Copula:

copula = GaussianCopula()
copula.fit(X)
print(copula.fitted_repr)
print(f"Rho: {copula.rho_:0.2f}")
print(f"AIC: {copula.aic(X):,.2f}")
print(f"Lower Tail Dependence: {copula.lower_tail_dependence:.2%}")
print(f"Upper Tail Dependence: {copula.upper_tail_dependence:.2%}")

GaussianCopula(rho=0.748)
Rho: 0.75
AIC: -6,346.65
Lower Tail Dependence: 0.00%
Upper Tail Dependence: 0.00%

Let’s plot the 2D PDF:

fig = copula.plot_pdf_2d()
fig.update_layout(height=700)

Let’s plot the tail concentration of the copula model versus the historical data:

copula.plot_tail_concentration(X)

As expected, the tail concentration plot shows that the Gaussian Copula cannot capture the tail dependencies of the historical data.

Comparison with the Joe Copula

Let’s now compare with the Joe Copula:

copula = JoeCopula()
copula.fit(X)
print(copula.fitted_repr)
print(f"Rotation: {copula.rotation_}")
print(f"Rho: {copula.theta_:0.2f}")
print(f"AIC: {copula.aic(X):,.2f}")
print(f"Lower Tail Dependence: {copula.lower_tail_dependence:.2%}")
print(f"Upper Tail Dependence: {copula.upper_tail_dependence:.2%}")

JoeCopula(theta=3.18, rot=180°)
Rotation: 180°
Rho: 3.18
AIC: -4,921.77
Lower Tail Dependence: 75.61%
Upper Tail Dependence: 0.00%

Let’s plot the 2D PDF:

fig = copula.plot_pdf_2d()
fig.update_layout(height=700)
show(fig)

Let’s plot the tail concentration of the copula model versus the historical data:

copula.plot_tail_concentration(X)

The Joe Copula, when rotated at 180° (as in this example), exhibits strong lower tail dependence. Although Archimedean copulas are typically not appropriate for stock returns, they can be a good fit for other asset classes such as agricultural commodities, derivatives, or CDSs.

Conclusion

Bivariate copulas are used to model complex dependencies between financial assets. Elliptical copulas, such as the Gaussian and Student’s t, offer a straightforward approach with symmetric dependence, while Archimedean copulas (Gumbel, Clayton, Joe) provide specialized modeling of tail dependencies. The ability to rotate Archimedean copulas further enhances their flexibility, allowing for a more accurate representation of the observed tail behavior in financial data.