Modelling bid-ask spread conditional distributions using hierarchical correlation reconstruction

TL;DR

This paper introduces hierarchical correlation reconstruction (HCR), a method for predicting the entire probability distribution of bid-ask spreads from accessible data, enabling uncertainty quantification and more accurate financial modeling.

Contribution

The paper presents a novel HCR approach that models conditional distributions using orthonormal polynomials and moments, providing an interpretable and computationally efficient prediction framework.

Findings

High accuracy in predicting bid-ask spread distributions across 22 DAX companies.

Large differences in behavior between companies necessitate individual models.

Method is computationally inexpensive, suitable for real-time applications.

Abstract

While we would like to predict exact values, available incomplete information is rarely sufficient - usually allowing only to predict conditional probability distributions. This article discusses hierarchical correlation reconstruction (HCR) methodology for such prediction on example of usually unavailable bid-ask spreads, predicted from more accessible data like closing price, volume, high/low price, returns. In HCR methodology we first normalize marginal distributions to nearly uniform like in copula theory. Then we model (joint) densities as linear combinations of orthonormal polynomials, getting its decomposition into (mixed) moments. Then here we model each moment (separately) of predicted variable as a linear combination of mixed moments of known variables using least squares linear regression - getting accurate description with interpretable coefficients describing linear relations between moments. Combining such predicted moments we get predicted density as a polynomial, for which we can e.g. calculate expected value, but also variance to evaluate uncertainty of such prediction, or we can use the entire distribution e.g. for more accurate further calculations or generating random values. There were performed 10-fold cross-validation log-likelihood tests for 22 DAX companies, leading to very accurate predictions, especially when using individual models for each company as there were found large differences between their behaviors. Additional advantage of the discussed methodology is being computationally inexpensive, finding and evaluation a model with hundreds of parameters and thousands of data points takes a second on a laptop.