Exact and Asymptotic Solutions of the Call Auction Problem

Abstract

The call auction is a widely used trading mechanism, especially during the opening and closing periods of¯nancial 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 F, 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'°ows.

Clearing price distributions in call auctions

“…The unrestricted freedom in the choices for F A , F B and the distribution of (N A , N B ) enables use of empirical fits for these distributions from previous intervals. It is also possible to make definite, default choices for these quantities: for instance, choosing independent Poisson distributions for N A and N B would correspond to the assumption of Poisson order flow, which is omnipresent in the literature (see, among many others, [24,9,1,8,20] for examples in context of continuous double auctions and [18,19] for examples in the standard call auction).…”

“…In this subsection we discuss some possibilities for the distribution of (N A , N B ) (the so-called order flow distribution) and their consequences for clearing price distributions. The common assumption in the (early) literature is what is called Poisson order flow : for continuous double auctions [24,9,1,8,20] and call auctions [18,19], Poisson order flow follows from assumed independent Poisson processes for the arrival of buy and sell orders. Here, we would take,…”

“…The distribution of transacted volume is derived, together with the clearing price expectation. Technically, this paper is related to the work of [19], who gives the full solution of Mendelson's call auction model, deriving distributions for transacted volume, and lower/upper clearing prices, as well as asymptotic distributions in very liquid call auctions.…”

“…R (·) = α F A (·) + (1 − α)F B (·),we rewrite equation(19) as follows,1 τ (X) √ N (F R (X) − F R (x E )) − D(X) w.…”

“…The exception (recall, these five stocks have been selected randomly from the Eurostoxx 50 index) is the Anheuser Busch Inbev NV stock, with a KS-statistic that indicates evidence (p = 0.0198) to reject the null-hypothesis and visual inspection through the QQ-plot reveals underestimation of the up-side tail. One could wonder how the market clearing model performs if the extreme beta distribution for order flow imbalance is replaced by Poisson order flow (this essentially corresponds to the call auction model of[19]). To investigate this option, we performed exactly the same analysis using the estimated clearing price distribution (as in definition 6.1) to obtain a sample of quantiles, but now with Poisson order flow: (N A , N B ) ∼ Pois(50) × Pois(50).…”