Discussing the article: "Bivariate Copulae in MQL5 (Part 2): Implementing Archimedean copulae in MQL5"

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Check out the new article: Bivariate Copulae in MQL5 (Part 2): Implementing Archimedean copulae in MQL5.

In the second installment of the series, we discuss the properties of bivariate Archimedean copulae and their implementation in MQL5. We also explore applying copulae to the development of a simple pairs trading strategy.

A bivariate Archimedean copula is a specific type of copula C(u,v) used in statistics to model the dependence structure between two random variables with uniform marginal distributions. Its defining property is its generative form, which can be expressed using a single, continuous, strictly decreasing, and convex function called the generator function ϕ.

The generator ϕ must satisfy ϕ(1)=0. This structure gives Archimedean copulae a high degree of symmetry (C(u,v)=C(v,u)) and allows for a wide range of dependence structures to be modeled simply by choosing different generator functions. Archimedean copulae are fundamentally more flexible than common copulae, but that flexibility is usually reduced for practical use. Theoretically, there are an infinite number of choices for the generator function. Every slight change in the function creates a new, unique copula. In practice, one usually chooses a parametric family governed by a unique generator function, defined by a single scalar input variable. This makes estimation and modeling much easier. Therefore, most common Archimedean families are characterized by a single parameter embedded within the generator function, which governs the strength of the dependence.

The generator function defines an Archimedean copula and captures the entire dependence structure between the random variables. In simple terms, its role is to translate the marginal probabilities into a dependence structure that can be easily combined using simple addition. The function transforms the marginal variable. Because the generator is strictly decreasing, low probabilities get mapped to large numbers, and high probabilities get mapped to 0. This is an inversion of the probability scale. The specific shape and parameters of the generator function entirely determine the resulting copula family and, therefore, the exact way the two variables depend on each other. We begin our exploration of Archimedean copulae with the bivariate Frank copula in the next section.

Author: Francis Dube