Clayton

systematica.models.arbitrage_index.clayton

Clayton

Clayton(
    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 Clayton Copula Estimation.

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

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

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

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

It is defined by:

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

where $\theta > 0$ is the dependence parameter. As $\theta -> 0$, the Clayton copula converges to the independence copula. Larger values of $\theta$ result in stronger lower-tail dependence.

The Clayton copula is widely used in statistics and quantitative finance for modeling dependency structures between random variables, particularly those exhibiting lower-tail dependence .

The conditional probability formulas derived from the Clayton copula describe the dependency between two variables $X$ and $Y$ (or more generally, $U$ and $V$, representing uniform marginals) conditional on one variable.

1. Tail Dependence : Clayton copula is ideal for modeling lower-tail dependence, capturing the likelihood of extreme low values in $X$ and $Y$ occurring simultaneously.

2. Risk Management : In financial risk, conditional probabilities from the Clayton copula are used to assess joint default probabilities and systemic risk.

3. Dependency Analysis : Quantifies the strength and nature of dependency between random variables.

Rotations are needed for Archimedean copulas (e.g., Joe, Gumbel, Clayton) 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 Clayton.

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
tp.Array1d	Log probability density.

Raises:

Type	Description
AssertionError	Clayton dependence (theta) must be > -1 and not 0.

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	Clayton dependence (theta) must be > -1 and not 0.

arbitrage

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

Compute the h-function (partial derivative) for the bivariate Clayton 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	Clayton dependence (theta) must be > -1 and not 0.

partial_derivative

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

Compute the h-function (partial derivative) for the bivariate Clayton 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	Clayton dependence (theta) must be > -1 and not 0.

score

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

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

For Clayton, the PDF is given by:

$$c(u,v) = (\theta+1) \Bigl(u^{- \theta} + v^{- \theta} - 1\Bigr)^{ -\frac{1}{\theta}-2} (u, v)^{-\theta-1}$$

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	Clayton dependence (theta) must be > -1 and not 0.