Opening and Closing Auctions

A call auction does not match orders as they arrive. It collects them, and then it solves for the single price at which the most shares can trade. Run that rule by hand on a book of thirteen limit orders and two market orders and two things fall out that no description of auctions mentions. The price clears at 100.50, which is two ticks above every price that traded in the session, because the rule is solving for the marginal order and not for where trading has been. And every filled order receives that marginal price, so the market-on-open buyer pays 100.50 on all 2,600 shares where walking a continuous book containing the identical orders would have averaged 100.2654. The auction did not save him the walk. It charged him the top of the walk on every share.

Prerequisites: Lesson 41, which put these two auctions at the edges of the day and left their mechanism unopened, and lesson 2, whose order book is the thing an auction crosses all at once.

One auction, run by hand

The session before this auction traded between 100.00 and 100.30, and the last print was 100.20. Into the auction come six buy limits — 300 at 100.50, 400 at 100.40, 600 at 100.30, 900 at 100.20, 700 at 100.10, 500 at 100.00 — and seven sell limits: 200 at 100.00, 400 at 100.10, 500 at 100.20, 700 at 100.30, 600 at 100.40, 800 at 100.50, 900 at 100.60. Then two orders that name no price: a market buy for 2,600 shares and a market sell for 300. Below, the two accumulated curves at every price on the book.

Price	Demand at or above	Supply at or below	Shares that can trade	Leftover
100.00	6,000	500	500	+5,500
100.10	5,500	900	900	+4,600
100.20	4,800	1,400	1,400	+3,400
100.30	3,900	2,100	2,100	+1,800
100.40	3,300	2,700	2,700	+600
100.50	2,900	3,500	2,900	−600
100.60	2,600	4,400	2,600	−1,800

The fourth column peaks at 100.50 with 2,900 shares, and the next best is two hundred shares short of it, so the first line of the rule settles the answer on its own and the tie-breaks never run. The auction clears at 100.50. Six hundred shares of sell interest go unfilled, and they sit at the clearing price itself: everyone who offered below 100.50 is filled in full, and the 800-lot resting at 100.50 is filled for 200 of its 800. That is where a call auction always leaves its leftover — on the marginal order, never on the orders that were inside it.

Look at where that price sits. The session traded from 100.00 to 100.30. The auction cleared at 100.50, two ticks above the high, and the high is not even a term in the calculation. The reason is in the fifth row: at 100.40, demand of 3,300 still exceeded supply of 2,700, so more shares could trade if the price went higher, and the rule’s only instruction is to trade more shares. It kept climbing until supply caught demand, and it did that because a market buy for 2,600 shares is demand at 100.40, and at 100.50, and at every price above that up to infinity. That is what a market order in an auction is: a promise to be on the demand curve at every price the algorithm considers.

What one more seller was worth

Now change exactly one thing. Add a single participant offering 1,200 more shares at 100.30 and leave everything else alone. Supply at 100.30 rises from 2,100 to 3,300, six hundred short of the 3,900 of demand standing there, so the executable volume at 100.30 becomes 3,300 — and 3,300 beats the 2,900 available at 100.50. And this time the tie-breaks do run, which is worth watching because they are the part of the rule a reader will otherwise never exercise. Supply at 100.40 rises to 3,900 against demand of 3,300, so 100.40 matches 3,300 as well and two prices are tied on volume. The leftovers are +600 and −600, equal in size and opposite in sign, so the second tie-break does not separate them and the third one points in both directions at once. What settles it is the last resort: the last traded price was 100.20, and 100.30 is the nearer of the two. The auction clears at 100.30, inside the session range this time, with 600 shares of buy interest left over rather than sell.

Outcome	The book as it stands	With 1,200 more offered at 100.30
Clearing price	100.50	100.30
Shares matched	2,900	3,300
Leftover, and on which side	600 sell	600 buy
The 2,600-share market buy costs	261,300	260,780
The same shares walking a continuous ladder	260,690	260,590
The buyer’s share of the matched volume	90%	79%

The fourth row is what one additional seller was worth to the buyer: 520 of currency on 2,600 shares, delivered by somebody who never spoke to him and whose only act was to place a limit order two ticks lower than the price that would otherwise have cleared. That is the whole of what depth does, stated as a number.

The fifth row is the uncomfortable one. Walking a continuous ladder made of the identical sell limits, the buyer’s 2,600 shares would have filled at an average of 100.2654 — 200 at 100.00, 400 at 100.10, 500 at 100.20, 700 at 100.30, 600 at 100.40 and the last 200 at 100.50. Same orders, same quantity, 610 less. The auction reached exactly the same top price as the walk and then charged it on every share instead of only on the last two hundred. And that is not an accident of this book: the clearing price is the price that clears the last matched share, so it can never be below the highest price a walk through the same orders would have reached, and it is charged on the whole quantity rather than on the tail of it. A call auction is systematically better for the side that rested and worse for the side that was in a hurry, and the difference is a transfer from one to the other.

The last row explains how the buyer ended up setting his own price. His 2,600 shares were 90 per cent of everything that matched. There was no crowd for him to hide in; the demand curve at 100.40 and 100.50 was almost entirely his own order, which means the algorithm was solving for where he would run out of sellers. With the extra seller present that fell to 79 per cent, and the price he paid fell with it. If you are most of the volume in an auction, the clearing price is not something that happens to you. It is something you compute for yourself, in public, and then pay.

Which is what a market-on-open or market-on-close order actually agrees to, stated plainly: not to trade at the open, but to be counted as demand at every price the rule tests, and to accept whatever number comes back. The instrument that keeps a bound on that is the limit-on-open or limit-on-close, which drops out of the demand curve above the price you name and takes the risk of not trading instead of the risk of the price. Every venue offers both. The choice between them is the choice between an uncertain fill and an uncertain price, and this book shows what the second one can cost when the book behind it is thin.

What this does not settle

That this is a real auction book. It is thirteen limit orders and two market orders at ten-cent ticks, sized so the arithmetic can be checked by hand in a minute. A real opening cross has thousands of orders at a hundredth of a cent, and the curves are smooth rather than stepped. What is the same at any size is the rule and the two things it forces: the price is set by the marginal order, and everybody filled receives it.

That the walk comparison proves continuous trading is cheaper. It proves the mechanical part only, and it assumes the same orders exist in both worlds, which is exactly what is not true. A continuous book in the first seconds of a session is thinner than the auction book precisely because the auction concentrates everyone’s orders into one instant, which is lesson 41’s third mechanism doing its work. The arithmetic above bounds the difference the mechanism makes on a fixed book. It does not settle which venue gives you a better fill on a real morning.

That the limit-on-open is the free alternative. It is offered as the instrument that keeps a bound on the price, and it is, and this lesson never priced the other side of it. Replace the 2,600-share market buy with a 2,600-share limit at 100.30 and the auction clears at 100.30 matching only 2,100 shares, because the demand that was holding the price up was his. Seven hundred shares of higher-priced buy interest fill first, leaving 1,400 to be shared at the margin among 3,200 shares of interest, so his pro-rata fill is about 1,138. He saves 228 on the shares he receives and does not receive 1,462 of them, which is 56 per cent of what he came for. The uncertain fill is the cheaper risk here, and it is not a small one, and it leaves no record: an order that did not trade produces no fill to regret.

That every venue clears the same way. The volume-maximising first step is close to universal; the tie-breaks are not. Some venues consult the reference price earlier in the sequence, some publish an indicative price and take orders against it throughout the accumulation period, and many impose a collar — a band around the reference outside which the cross is not allowed to print, which triggers an extension or a delay instead. A collar would have stopped the 100.50 print in this lesson. Find your venue’s band before you assume the rule above ran to completion.

That the published imbalance tells you what will happen. Exchanges publish the leftover in the minutes before the cross, and it is genuinely useful information: it says a one-sided order exists and has to clear. What it is not is a signal addressed to you. It goes to every subscriber on the same feed in the same instant, and the adjustment starts then, so what is left to trade afterwards is a residual that this lesson has not measured and that most of what is written about it has not measured either. It is a strong reason not to be positioned against the cross, which costs nothing, well before it is a reason to be positioned with it.

Problems

1. Run the rule on a book you can see. Take any venue that publishes its auction order book or its indicative price and quantity during the accumulation period, write down the resting orders at five or six prices, and build the two accumulated columns yourself. Then compare the price your arithmetic gives with the price that actually printed. If they differ, you have found either a tie-break you did not apply or a collar you did not know about, and either one is worth more than the exercise.
2. Work out what fraction of the cross you are. Before sending a market-on-close order, divide its size by the auction volume the same instrument printed at the close on an ordinary day. Most people have never computed this number and are startled by it. If the answer is a fraction of a per cent you are a price taker in the ordinary sense. If it is anything like the 90 per cent in the table above, the clearing price is largely your own arithmetic and the venue is only doing the sum for you.
3. Find your instrument’s collar. Look up the price band your venue applies to the opening and closing cross — how far from the reference price the auction is permitted to print before it extends, delays or cancels. It is published, it is usually a percentage, and it is the only hard bound that exists on what a market-on-open order can cost you. Write it down next to the position size you use into a cross. How to Collect a Base Rate is how the count is kept honest.

Sources. Ananth Madhavan, “Trading Mechanisms in Securities Markets” (The Journal of Finance, 1992), for the formal comparison of a call auction against continuous trading — the paper works out when batching produces a better price and when it fails to clear at all, which is the question this lesson answers with one book. Bruno Biais, Pierre Hillion and Chester Spatt, “Price Discovery and Learning during the Preopening Period in the Paris Bourse” (Journal of Political Economy, 1999), for what happens in the accumulation period itself: indicative prices early in the window carry little information and firm up sharply as the call approaches, which is the empirical version of the caution about imbalance data above. Eric Budish, Peter Cramton and John Shim, “The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response” (The Quarterly Journal of Economics, 2015), for the argument that the single-price cross is not a quirk of the open and the close but a design that removes the advantage of being first, and for what it would mean to run the whole day that way.

An auction is a discontinuity you can read in advance: it happens at a known time, by a known rule, and the venue publishes what is building up inside it. The calendar is full of other events with the first half of that property and not the second — an earnings release, a rate decision, a monthly data print, all of them scheduled to the minute and none of them announcing their content. Lesson 43 is about that asymmetry: what a known time and an unknown outcome do to a position, why the usual advice to trade the reaction rather than the announcement is arithmetic rather than temperament, and how to price the choice between holding through a scheduled event and standing aside.

Educational only. Trading involves substantial risk of loss. Not financial advice. Past performance does not guarantee future results.