arbitragelab.codependence.optimal_transport

Implementations of Optimal Copula Transport dependence measure proposed by Marti et al.: https://arxiv.org/abs/1610.09659 And implemented in the blog post by Marti: https://gmarti.gitlab.io/qfin/2020/06/25/copula-optimal-transport-dependence.html

Module Contents

Functions

optimal_transport_dependence(→ float)

Calculates optimal copula transport dependence between the empirical copula of the two vectors and a target copula.
optimal_transport_dependence(x: numpy.array, y: numpy.array, target_dependence: str = 'comonotonicity', gaussian_corr: float = 0.7, var_threshold: float = 0.2) → float

Calculates optimal copula transport dependence between the empirical copula of the two vectors and a target copula.

This implementation is based on the blog post by Marti: https://gmarti.gitlab.io/qfin/2020/06/25/copula-optimal-transport-dependence.html

The target and forget copulas are being used to reference where between them does the empirical copula stand in the space of copulas. The forget copula used is the copula associated with independent random variables. The target copula is defined by the target_dependence parameter.

Currently, these target_dependence copulas are supported:

  • comonotonicity - a comonotone copula.

  • countermonotonicity - a countermonotone copula.

  • gaussian - a Gaussian copula with a custom correlation coefficient.

  • positive_negative - a copula of both positive and negative correlation.

  • different_variations - a copula with some elements having extreme variations, while those of others are relatively small, and conversely.

  • small_variations - a copula with elements being positively correlated for small variations but uncorrelated otherwise.

  • v-shape - a copula that is seen with vol index vs. returns index: when returns of the index are extreme, vol is usually high, when returns small in absolute value, vol usually low.

Parameters:

  • x – (np.array) X vector.

  • y – (np.array) Y vector.

  • target_dependence – (str) Type of target dependence to use when measuring distance. (comonotonicity by default)

  • gaussian_corr – (float) Correlation coefficient to use when creating gaussian and small_variations copulas. [from 0 to 1] (0.7 by default)

  • var_threshold – (float) Variation threshold to use for coefficient to use in small_variations. Sets the relative area of correlation in a copula. [from 0 to 1] (0.2 by default)

Returns:

(float) Optimal copula transport dependence.