Exact and asymptotic solutions of the call auction problem

TL;DR

This paper provides exact and asymptotic analytical solutions for the distributions of traded volume and clearing prices in a call auction model with Poisson order arrivals and random prices, including explicit formulas and limit behaviors.

Contribution

It introduces explicit formulas for the distributions of key auction metrics and derives their asymptotic limits, advancing the theoretical understanding of call auction dynamics.

Findings

Traded volume and clearing price bounds are asymptotically normal.

The price range of clearing prices is asymptotically exponential.

Explicit distribution parameters are derived from order flow characteristics.

Abstract

The call auction is a widely used trading mechanism, especially during the opening and closing periods of financial markets. In this paper, we study a standard call auction problem where orders are submitted according to Poisson processes, with random prices distributed according to a general distribution, and may be cancelled at any time. We compute the analytical expressions of the distributions of the traded volume, of the lower and upper bounds of the clearing prices, and of the price range of these possible clearing prices of the call auction. Using results from the theory of order statistics and a theorem on the limit of sequences of random variables with independent random indices, we derive the weak limits of all these distributions. In this setting, traded volume and bounds of the clearing prices are found to be asymptotically normal, while the clearing price range is asymptotically exponential. All the parameters of these distributions are explicitly derived as functions of the parameters of the incoming orders' flows.