Frank

systematica.models.arbitrage_index.frank

Frank

Frank(
    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.

The Frank copula is a widely used copula in statistics and finance for modeling dependencies between random variables. The Frank copula captures symmetric dependence and is particularly useful when there is no tail dependence.

1. Symmetric Dependency : The Frank copula is suitable for datasets where dependency between variables is symmetric and without tail dependence.

2. Modeling Joint Distributions : Used in scenarios where the relationship between random variables is non-linear but consistent across their range.

3. Risk Management and Finance : It can model dependencies between financial assets, insurance claims, or other risk variables with moderate dependencies.

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.

Method generated by attrs for class Frank.

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	Frank dependence (theta) must not be 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 density.

Raises:

Type	Description
AssertionError	Frank dependence (theta) must not be 0.

arbitrage

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

Compute the h-function (partial derivative) for the bivariate Frank 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	Frank dependence (theta) must not be 0.

partial_derivative

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

Compute the h-function (partial derivative) for the bivariate Frank 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	Frank dependence (theta) must not be 0.

score

score(
    self,
    u: numpy.ndarray,
    v: numpy.ndarray,
    best_fit: bool = False,
) ‑> 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.
best_fit	bool	False	Apply best fitted rotation. Defaults to False.

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	Frank dependence (theta) must not be 0.