Call Auctions with Contingent Orders

We introduce a new mechanism for call auctions which are widely used in stock exchanges. By incorporating contingent claims (buy stock A, if selling stock B) into the price discovery process, we achieve a higher liquidity in call auctions and lower volatility after opening call auctions. We also show that present call auctions and the proposed mechanism have similar incentive properties.


Introduction
Intraday price volatility of stocks generally tends to be higher at the opening interval than other trading time periods during the day. Among others, Lockwood and Linn (1990) support this claim using the data of Dow Jones Industrial Average between . Most stock exchanges around the globe implement call auctions at the opening and closing of trading sessions in order to make an e¤ective system for price discovery, reduce the opening volatility, and avoid closing price manipulations.
In this manuscript, we introduce a new call auction mechanism with an innovative algorithm. This new design lets prospective investors to place contingent buy orders and determines the auction price by using these contingent orders, which will help to solve the problems mentioned above. Our mechanism is also theoretically shown to increase the liquidity level in auctions. We also consider the incentive properties of present call auctions and the proposed new mechanism and …nd that they are similar.
Investors regularly update their portfolio decisions with incoming news received overnight, and they may need to change their holdings at the forthcoming trading day. But if they want to realize this change at the opening session in order to bene…t from the concentration of liquidity at the call auction, in some cases these traders may not have su¢ cient sources to execute their buy orders before completing their sell orders. One can argue that the actual delivery and payment (settlement) procedure is not instantaneous in many stock exchanges; therefore traders can do this exchange without using contingent orders. However, …nancial intermediaries are exposed to settlement risk, and the risk management considerations increase the transaction costs of these traders. On the other hand, contingent orders reduce transaction costs and provide the opportunity of simultaneously buying and selling di¤erent stocks at the opening or closing call auction.
Other potential groups of investors who may use contingent orders are "pairs traders." Pairs trading is a well-known and commonly used trading strategy, which is formed by, at …rst, …nding two stocks that are highly correlated with each other depending on historical records, then, buying one and selling the other one whenever the spread between them increases signi…cantly. Gatev et al. (2006) show empirically that this simple strategy brings up to 11% average annual excess return. They also note that the pro…tability of this strategy, used largely by institutional investors, depends on the traders'transaction costs. It has also been shown that call auctions are preferable to continuous auctions for these investors, as call auctions exhibit lower transaction costs and less market impact (see Snell andTonks 2003 andEconomides andSchwartz 1995). Also, the reference price property of closing auction prices attracts institutional investors to be more active and aggressive at the closing auction (Pagano and Schwartz 2003). An arbitrageur, with a pairs trading strategy, can avoid the risk of "buying a lower priced stock without selling a higher priced one"by using contingent orders in our proposed auction market.

Call Auctions
Di¤erent stock exchanges use very similar types of uniform price auction mechanism for their call auction sessions. We demonstrate the general idea of these auctions below.

Current Call Auction Mechanism
Investors can only enter the call auction with limit orders, which consist of the code of the individual stock, price, and quantity for the order. In the call auction (CA) buyers and sellers announce their values and quantities (v; q). Then the call auction price (CAP) is chosen to be one of the prices that maximize the total sales (total sale with a price p is de…ned as the minimum of "total quantities demanded by buyers with values v p" and "total quantities supplied by sellers with values v p"). In case there are multiple maximizers of the total sale, the price closest to the last closing/sale price is chosen. For CAP equal to p; there could be excess demand or supply. In case of an excess demand (supply), then all matching sell (buy) orders will be executed, and the lowest buy (highest sell) o¤ers may be partially executed. In case of multiplicity of lowest buy (highest sell) o¤ers, these o¤ers will be executed on a pro rata basis.

Call Auction Mechanism with Contingent Orders
Consider an investor who holds stock A and would like to buy stock B only if she sells stock A. Now, suppose we allow investors to announce their orders as "I would like to buy Stock B (at most price x), if I can sell Stock A (q units of the stock for a price at least y)." After all regular limit orders and contingent orders are collected until the call auction, the Call Auction with Contingent Orders (CACO) mechanism is run by the following algorithm in order to determine the auction prices (CACOP) of di¤erent stocks: Stage 1: All regular limit orders and selling limit orders of the contingent orders will be entered and the price vector p 1 will be determined according to CA mechanism described above. If any of the contingent selling limit orders is executed with p 1 ; corresponding buying limit orders of these contingent orders 1 are entered into the auction book and the algorithm moves to Stage 2.
. . Stage k: CA mechanism determines the new price vector p k with the updated orders in the auction book. If any of the contingent selling limit orders is executed with p k ; corresponding buying limit orders of these contingent orders are entered into the auction book and the algorithm moves to Stage k + 1. Otherwise the algorithm ends with the resulting CACOP vector p k : One very important feature of this mechanism is that at each stage k, only new buy orders can be added to the auction book. Hence, matched sellers of stages 1; :::; k 1 will also be matched in stage k and p k is nondecreasing in k. This feature guarantees that there will be no loops in the algorithm and it will end in …nitely many steps.
Below, we establish that with CACO there are no remaining matching buy and sell orders at the CACOP that can be processed into the continuous trading session for the opening call auction. Moreover, we easily show that trading volume of CACO is greater than the volume with CA.
Proposition 1 There are no remaining matching buy and sell orders at the CACO. Moreover, the trading volume of CACO is greater than or equal to the volume with CA.
Proof. The …rst claim follows by de…nition. Since the algorithm moves to a new stage whenever there are matching buy and sell orders, there cannot be any matching buy and sell limit orders when the algorithm ends. The second claim follows by noting that the volume traded in the …rst stage of the CACO algorithm is equal to the volume traded in CA. In the following stages (if there are any), the volume traded has to increase. Therefore, the claim follows.
Remark 1 Since there are no remaining matching buy and sell orders at the CACO, ceteris paribus, the volatility after opening CACO will be lower than that of opening CA.

Incentives Properties of Call Auctions
In a CA, it is not a weakly dominant strategy to announce the true types (values and quantities). Consider the following example.
The above example shows that investors may have incentive to misreport their type. However, the incentive to misreport her type arises only when that investor becomes a price setter after the deviation: Proposition 2 In a CA, given any type announcement of other investors, an investor is never better o¤ (compared to truthful announcement) by announcing another type, unless with that announcement she becomes a price setter (that is, CAP is equal to her announced value).
Proof. Consider a buyer b i whose true value is v b i : If by a truthful announcement she is not a successful buyer, then CAP has to be greater than v b i : In that case, the only way she can be a successful buyer is to increase her type, which makes the CAP even greater and she obtains a negative utility. Therefore, under this case there is no pro…table deviation. If by a truthful announcement she is a successful buyer, then CAP has to be smaller than v b i : For a deviation to be pro…table, b i has to be a successful buyer after the deviation. If CAP after the deviation is not equal to the new value announcement of b i ; that means that CAP remained the same. This is because CAP is una¤ected by the values announced of the successful buyers, who are not price setters. Therefore, the only way a lie can be bene…cial is when after the lie, the investor becomes a price setter (as in the above example). Analogous arguments can be made for the sellers.
A similar version of the above incentive result continues to hold in a CACO.
Proposition 3 In a CACO, given any type announcement of other investors, an investor is never better o¤ (compared to truthful announcement) by announcing another type, unless with the best deviation she becomes a price setter (that is, CACOP is equal to her announced value).
Proof. For non-contingent investors the arguments are the same as above. Since all the investors care about is the …nal price that the algorithm produces, for the buyers or the sellers of a stock, the only way to bene…t is by becoming a price setter. Now consider a contingent investor who would like to buy stock A only if she can sell stock B. She can lie in two dimensions, value for A and value for B. If she is not a successful seller of Stock B by announcing truthfully-in contrast with CA-she may still want to lie and become a successful seller. This may be the case if her value for Stock A is su¢ ciently high so that she can o¤set some loss in selling Stock B by the pro…t in buying Stock A. Nevertheless, in the best deviation this investor will be becoming a price setter for Stock B. Her incentive in the announcement of values for A is the same as a regular buyer. Hence the result follows. Propositions 2 and 3 establish that although CA and CACO are not truthful, the incentives from lying could be present only for "price setters." Therefore, we do not expect the behavior of the traders to di¤er signi…cantly under CA and CACO. Thus, comparing CA and CACO for the same "inputs"is a fair argument, and conclusions of Proposition 1 and Remark 1 are justi…able in this environment.