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Developing and
Implementing
Risk Management
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Fortis Group
Case Study



American Depository
Receipts in The
Global Equity Model



Analyzing the
Performance of
Crossing Networks
The Market
Impact Model™ — Part 4



The BARRA Brainteaser
 for Fall 1998
Solution to the Summer
1998 Brainteaser

Analyzing the Performance of Crossing Networks

by Nicolo G. Torre and Geoff A. Latham¹

In other articles we have drawn attention to the importance of controlling transaction costs. Electronic Crossing Networks (ECNs), such as the POSIT system², represent one approach to this problem. Participants in an ECN submit lists of orders they wish to do. The lists are matched up by a computer and matched orders are executed at the midpoint of the bid-ask spread. The system may support additional control by participants over the matching process, for instance the ability to specify minimum fill sizes. However, in this article we will focus on the bare bones of the matching process.

ECNs provide users with two key advantages. First, participants have the opportunity to search for a trade counterparty without revealing their trading intentions to the world at large. Second, trades are executed at the midpoint of the bid-ask spread and thus may be cheaper than trades executed in the wider marketplace, which could incur halfspread and incremental market impact costs. The principal disadvantage of ECNs is that trades only occur when a counterparty happens to arrive. Thus, participants must expect that many of their orders will go unmatched. These orders either will need to wait in the ECN until the arrival of a counterparty, or will need to be worked through some other trading mechanism. Thus, analyzing the determinants of the match rate is essential for understanding the performance of an ECN.

If we consider a match involving just two participants and assume that all orders are to either buy or sell 100 shares, we can derive analytic expressions for the results of the match. Suppose that the stock universe contains N names, and that the first participant submits a list of n orders and the second participant submits a list of p orders. The order lists are assumed to contain stock selections in which no stock is favored over another, and buy and sell orders occur with equal probability—that is, the lists are unbiased. Without loss of generality, we may suppose that pn. If we let p_j (p, n, N) denote the probability that exactly j names on the two lists match (i.e., the names are on both lists, once as a buy and once as a sell), then:

where

FIGURE 1 plots p_j (100,100, 1000). Clearly the most likely outcome is for 4 or 5 matches, but there is a tail of additional potential matches.

FIGURE 1: Probability of crosses

This situation can be summarized by giving the moments of the distribution, namely:

Here m, s² and k denote the mean, variance, and Fisher skewness, respectively. In our case:

m (100,100, 1000) = 5,
s (100,100, 1000) = 2.2,
and k (100,100, 1000) = 0.38.

To gain some insight into the multi-participant matching situation, let us suppose that first m participants with order sizes n₁, n₂, ... n_m match up among themselves, and then another participant submits p orders to match against the residual list of the other participants. The residual list will contain n names, which we may suppose to represent a reasonably large fraction of the total universe N. Let a = n/N. Then, assuming p is only a small fraction of N, we have, to a good order of approximation:

Empirically a = 0.2 gives a reasonable approximation of the results generated by POSIT. For this value we find:

Thus on a list of 100 names, one might expect 10 ± 3 crosses, with a skewness of 0.27 in the outcome.

Since the results of crossing depend fundamentally on the value of p, a natural approach to improving the performance of a crossing network is to allow substitutions. Thus, if for each stock on the list one also supplies an alternate, the number of opportunities for a cross is effectively doubled and the performance of the crossing network correspondingly improved. This concept of supporting substitutions is carried to its logical end in the latest release of POSIT, known as POSIT 4. This system allows the crossing list to be dynamically adjusted based on the order inventory currently present in the system as well as on the user's risk and return preferences. Allowing for dynamic substitutions can improve the matching rate above what can be achieved with static substitutions. Indeed, the crossing rate for dynamic substitutions will depend as much on the participant's flexibility in specifying substitutes as on the combinatorial properties which govern the static case. As crossing networks become increasingly powerful and flexible, they seem destined to play an ever more central role in getting the trade done.

¹ Communications Division, Defense Science and Technology Organization, Australian Department of Defense.

² POSIT is a joint venture of BARRA and ITG.