Gumbel

systematica.models.arbitrage_index.gumbel

Gumbel

Gumbel(
    kendall_tau: float,
    rotation: str | systematica.models.arbitrage_index.utils.BaseCopulaRotation = BaseCopulaRotation.R0,
    start: float = 0.001,
    stop: float = 0.999,
    num: int = 100,
)

Bivariate Gumbel Copula Estimation.

The Gumbel copula is an Archimedean copula characterized by strong upper tail dependence and little to no lower tail dependence.

In its unrotated form, it is used for modeling extreme co-movements in the upper tail (i.e. simultaneous extreme gains).

Rotations allow the copula to be adapted for different types of tail dependence:

• A 180 rotation captures extreme co-movements in the lower tail (i.e. simultaneous extreme losses).
• A 90 rotation captures scenarios where one variable exhibits extreme losses while the other shows extreme gains.
• A 270 rotation captures the opposite scenario, where one variable experiences extreme gains while the other suffers extreme losses.

Gumbel copula generally exhibits weaker upper tail dependence than the Joe copula.

It is defined by:

$$C_{\theta}(u, v) = \exp\Bigl(-\Bigl[(-\ln u)^{\theta}+(-\ln v)^{\theta}\Bigr]^{1/\theta}\Bigr)$$

where $\theta \ge 1$ is the dependence parameter. When $\theta = 1$, the Gumbel copula reduces to the independence copula. Larger values of $\theta$ result in stronger upper-tail dependence.

The Gumbel copula is a widely used copula in statistics and quantitative finance, particularly for modeling upper-tail dependence , which captures the likelihood of extreme high values occurring simultaneously in two or more random variables. The conditional probability formulas
derived from the Gumbel copula describe the relationship between two dependent random variables $U$ and $V$ (uniform marginals).

1. Upper-Tail Dependence : The Gumbel copula is ideal for scenarios where variables exhibit strong upper-tail dependence, such as extreme co-movements in financial markets.

2. Risk Management : Used for modeling dependencies in extreme events like catastrophic losses in insurance or simultaneous stock crashes.

3. Joint Probabilities : Helps compute joint or conditional probabilities for events involving highly correlated variables.

Rotations are needed for Archimedean copulas (e.g., Joe, Gumbel, Gumbel) because their parameters only model positive dependence, and they exhibit asymmetric tail behavior. To model negative dependence, one uses rotations to “flip” the copula’s tail dependence.

References:

• “An Introduction to Copulas (2nd ed.)”, Nelsen (2006)
• “Multivariate Models and Dependence Concepts”, Joe, Chapman & Hall (1997)
• “Quantitative Risk Management: Concepts, Techniques and Tools”, McNeil, Frey & Embrechts (2005)
• “The t Copula and Related Copulas”, Demarta & McNeil (2005)
• “Copula Methods in Finance”, Cherubini, Luciano & Vecchiato (2004)

Method generated by attrs for class Gumbel.

Ancestors

• systematica.models.arbitrage_index.base.BaseCopula
• abc.ABC

Instance variables

• lower_tail_dependence: float: Theoretical lower tail dependence coefficient.

• upper_tail_dependence: float: Theoretical upper tail dependence coefficient.

Methods

density

density(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> numpy.ndarray

Calculate log probability density of the bivariate copula:

$$P(U=u, V=v)$$

Parameters:

Name	Type	Default	Description
u	tp.Array1d	--	First uniform marginal.
v	tp.Array1d	--	Second uniform marginal.

Returns:

Type	Description
float	Log probability density.

Raises:

Type	Description
AssertionError	Gumbel dependence (theta) must be >= 1.

cumulative_density

cumulative_density(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> numpy.ndarray

Calculate cumulative density of the bivariate copula:

$$P(U<=u, V<=v)$$

Parameters:

Name	Type	Default	Description
u	tp.Array1d	--	First uniform marginal.
v	tp.Array1d	--	Second uniform marginal.

Returns:

Type	Description
tp.Array1d	Cumulative probability density.

Raises:

Type	Description
AssertionError	Gumbel dependence (theta) must be >= 1.

arbitrage

arbitrage(
    self,
    ui: float,
    vi: float,
) ‑> Tuple[float, float]

Compute the h-function (partial derivative) for the bivariate Gumbel copula, a.k.a. the mispricing index, for every time step in the trading period using the estimated copula.

Parameters:

Name	Type	Default	Description
ui	float	--	The first component of the bivariate input.
vi	float	--	The second component of the bivariate input.

Returns:

Type	Description
tp.Tuple[float, float]:	The mispricing indices for the copula, a.k.a. arbitrage.

Raises:

Type	Description
AssertionError	Gumbel dependence (theta) must be >= 1.

partial_derivative

partial_derivative(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> numpy.ndarray

Compute the h-function (partial derivative) for the bivariate Gumbel copula, a.k.a. the mispricing index, for every time step in the trading period using the estimated copula.

Parameters:

Name	Type	Default	Description
u	tp.Array1d	--	The first component of the bivariate input.
v	tp.Array1d	--	The second component of the bivariate input.

Returns:

Type	Description
tp.Array2d	The mispricing indices for the copula, a.k.a. arbitrage.

Raises:

Type	Description
AssertionError	u and v must have the same shape.
AssertionError	Gumbel dependence (theta) must be >= 1.

score

score(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
) ‑> numpy.ndarray

Compute the log-likelihood score of each sample (log-pdf) under the model.

u and v are bivariate inputs (u, v) where each row represents a bivariate observation. Both u and v must be in the interval [0, 1], having been transformed to uniform marginals.

Parameters:

Name	Type	Default	Description
u	tp.Array1d	--	The first component of the bivariate input.
v	tp.Array1d	--	The second component of the bivariate input.

Returns:

Type	Description
tp.Array1d	The log-likelihood score of each sample under the fitted copula.

Raises:

Type	Description
AssertionError	u and v must have the same shape.
AssertionError	Gumbel dependence (theta) must be >= 1.