I focus on the trading strategies available to arbitrageurs, once they enter a market with exogenously given bid- and ask-prices. I then derive implications on the co-movement of liquidity, given the arbitrageurs’ trading behavior. The classic characterization of arbitrageurs trading as buying and selling mispriced securities suggests that arbitrageurs will only trade via market orders, hitting existing limit orders and subtracting liquidity from the market. Chaboud et al. (2014) show that, on the FX market, algorithmic traders increase informational efficiency, reducing arbitrage opportunities, whileprovidingliquidity, suggesting that arbitrageurs can profit from strategies that include liquidity provision. In light of these findings, in order to understand the effect that arbitrage trading has on liquidity co-movement, one cannot assume that arbitrageurs only trade with market
5Differentiating between arbitrageurs and market makers based on their access to the markets is common in the literature. Goldstein, Li, and Yang (2014) model arbitrageurs as traders having access to a larger set of assets, as compared to mutual funds and individuals, whose investment opportunity set is small. Similarly, in Foucault et al.
(2017), market makers are specialized in a single security while arbitrageurs trade across securities.
orders, but should instead consider all possible strategies available to them, including those that improve market liquidity.
I derive the strategies available to the arbitrageurs in the context of typical order-driven equity markets, with price-time priority order books, such as the New York Stock Exchange, the NASDAQ, and the Toronto Stock Exchange, i.e., the markets that are object of most of the empirical work in this study. Figure 1 exemplifies the scenarios arbitrageurs face when trying to profit from the mispricing between two securities, security 1 and security 2, linked by arbitrage and traded on markets 1 and 2. A1and A2stand for the ask prices, B1and B2stand for the bid prices in markets 1 and 2, respectively.
Insert Figure 1 here.
Panel A of Figure 1 provides an instance of crossed quotes. This setting represent the textbook example of arbitrage, since the arbitrageur can contemporaneously hit the order at the bid of market 2 and lift the order at the ask in market 1, purchasing the security for A1and selling it for B2, thus locking in a risk-less profit of B2− A1 > 0 dollars per unit traded. The arbitrageurs can repeat this trading strategy, causing the ask-price in market 1 to raise and the bid-price in market 2 to diminish, as long asB2is larger than A1. This strategy demands liquidity from both markets, and widens the bid-ask spreads in market 1 and 2.
Alternatively, to exploit the mispricing and her dominant role as cross-market trader, the arbitrageur could post a limit order to sell atA2or lower (to exploit price priority) and a limit order to buy at B1 or higher. In case one of the limit orders was to be hit, the arbitrageur could then close her position instantaneously by trading, at a profit, on the other market. For example, if her limit order to buy just above B1 on market 1 was hit, she could close her position by hitting the bid in market 2, hence selling the security for B2 > B1. As the first strategy leads to a risk-less, instantaneous, and certain profit, I expect the arbitrageur to first take advantage of the mispricing betweenB2andA1, thus turning the markets from displaying crossed quotes to overlapping quotes, as shown in Panel B.
In Panel B, no arbitrage opportunity is readily available to be profited on, since the bid- and ask-prices of the two markets do not cross. However, the arbitrageur is still able to profit from the privileged position she holds, i.e., being able to trade in both markets. The arbitrageur can offer to acquire the security on market 1 by posting a limit order to buy aboveB1. If the limit order to buy is hit, she acquires the security on market 1, and she can close her position by selling that same security at B2on market 2. She can repeat the strategy for as long as B1and B2differ. Similarly, she can post a limit order to sell at A2or lower and, when that is hit, close her position by buying the security for A1on market 1. She can repeat this strategy as long as A1andA2differ.
Finally, Panel C shows a case of included quotes, where the bid- and ask-prices of one market are included within the bid- and ask-prices of the other. In this situation, the arbitrageur can still
profit from her position as a cross-market trader by posting a limit order to sell atA1or lower (buy atB1or higher) and, if it is hit, close her position by lifting the order at A2(B2).
In all the cases discussed above, the trading strategies of the arbitrageur imply the convergence of the ask-prices, i.e., the convergence of A1to A2, and the convergence of the bid-prices, i.e., the convergence of B1 to B2. The expectation held in the existing literature that arbitrageurs make prices converge across markets, where each market is represented by a single price, carries through when considering a more realistic setting, where each market features a price to sell and a price to buy: Arbitrageurs make bid-(ask-) prices converge. As the bid and ask sides of market 1 and 2 converge when the arbitrageur is trading, it follows that so does the bid-ask spread. As arbitrageurs are active on the market, the bid-ask spreads between markets converge.
Given the trading strategies and the intuition above, I formalize the effect that arbitrageurs’
trading has on the correlation in bid-ask spreads, i.e., the commonality in liquidity between markets, by developing a simple framework of arbitrageurs’ trading behavior. Let us assume that a security with underlying valueαis traded in two order-driven markets. In both markets, the tick size equals 1 cent. The bid-price in market 1 isB1 =α+ε1whereε1takes the value of -0.01, 0, or 0.01 with equal probability13. The ask-price is given byA1 = B1+µ1whereµ1is equal to 0.01 or 0.02, with equal probability 12. SimilarlyB2= α+ε2andA2= B2+µ2whereε2and µ2are independent and identically distributed toε1 and µ1, respectively. In this framework, the bid-ask spreads quoted on the two markets are ex-ante uncorrelated, since ε1, µ1, ε2, and µ2 are uncorrelated. While this feature is certainly a simplification, I only aim to determine the marginal effect of including arbitrage trades to a setting without arbitrageurs. I assume that the order book consists of one share available at each price level at and below the bid-price and at and above the ask-price. I do this to focus on a single dimension of liquidity, summarized by the bid-ask spread, i.e., a round-trip cost measure.
The time-line of the events I am considering is as follows. The arbitrageur arrives on the market and observes A1, B1, A2, and B2. The arbitrageur can submit market orders, hence executing trades at the existing prices. Alternatively, she can submit limit orders above the bid-or below the ask-prices, to take advantage of price pribid-ority. After the arbitrageur’s submission, a trader enters the market, submits a market order and trades at one of the four available prices, with equal probability. Finally, the arbitrageur can submit a last market order to close her position.
After that, the arbitrageur cancels her remaining orders, if any, the quotes change and the game starts from the beginning, with a new set of quotes.
As a means of comparison, consider a similar setting without the presence of the arbitrageur.
Before the advent of the trader and her trade, the bid-ask spreads in the two markets are uncorrelated, by construction. When the trade takes place, one of the bid- or ask-sides is hit and moves either closer or further away from its counterpart on the other market. While the correlation between quotes is null, one of the two bid-ask spreads widens, while the other remains unchanged, thus causing the liquidity measure to be negatively correlated across markets.
When the arbitrageur is present, however, both quotes and liquidity measures are more positively correlated. First of all, let us consider the liquidity in the markets prior to the advent of the trader but following the arbitrageur’s orders submission. Consider the case in Panel A of Figure 1.6 In the case of crossed quotes, the arbitrageur subtracts liquidity from both markets, diminishing the higher bid and increasing the lower ask. Her trades widen both bid-ask spreads, and contribute to their positive correlation.
When the quotes are overlapping (as in Panel B of Figure 1) or included (as in Panel C of Figure 1), the arbitrageur’s strategy includes providing liquidity to the market that has the least of it: In this setting, the arbitrageur has to better the prices to obtain execution of her limit orders, as she only submits limit orders in the market(s) where µ = 0.02, hence bettering that market’s liquidity. If both markets have a wide bid-ask spread (µ = 0.02), she will provide liquidity to both, reducing the bid-ask spreads. If only one of the markets has a wide bid-ask spread, the arbitrageur will provide liquidity to that market alone. The arbitrageur’s actions thus contribute to the positive correlation between bid-ask spreads, by submitting limit orders and tightening the bid-ask spread on the market that has the largest one. It follows that, before the advent of the trader, the bid-ask spreads of the two markets are positively correlated, in the presence of the arbitrageur. Alternatively, considering liquidity separately for each side, the arbitrageur’s strategy implies raising the lower bid, thus providing liquidity to the market where a seller would have found a worse liquidity, and lowering the higher ask, hence providing liquidity to the market where a buyer would have found a worse liquidity. None of the arbitrageur’s strategies, when she provides liquidity, includes providing liquidity by raising the higher bid or lowering the lower ask.
After the trader’s arrival, the execution of her market order subtracts liquidity from one market, while holding the other market’s liquidity unchanged, thus contributing negatively to the commonality between bid-ask spreads. If the quotes were crossed to start with, the arbitrageur would have no influence on the liquidity in the markets after the trader’s trade. If the quotes were overlapping or included, however, the presence of the arbitrageur’s order can dampen the effect of the trader’s market order: If the trader’s order hits the arbitrageur’s standing limit order, the trader subtracts liquidity from, e.g., market 1. The arbitrageur, subsequently, closes her position by hitting an order in market 2, hence lowering the liquidity of market 2, contributing positively to the co-movement of the markets’ liquidities. Finally, if the trader’s order does not hit the arbitrageur’s order and the arbitrageur cancels her orders, and all but one of the final quotes revert to their original positions, i.e., the state before the actions of the arbitrageur. It follows that the overall effect of the presence of the arbitrageur, both before and after the trader’s arrival, contributes positively to the commonality in liquidity between the two markets.
I formalize my findings in the following proposition:
Proposition 1 Consider two securities linked by arbitrage. When arbitrageurs are present on the
6I assume that the arbitrageur prefers locking in a sure profit first, without the use of limit orders. Relaxing this assumption and assuming the arbitrageur would post limit orders does not affect the results.
market:
1) The bid prices of the two markets are more positively correlated than in the arbitrageurs’
absence. The same is true for ask prices, midquotes, and returns.
2) The bid-ask spreads of the two securities co-move more than in the arbitrageurs’ absence.
Calculations can be found in the Appendix. The predictions are robust to allowing multiple trades from exogenous traders to take place, and allowing trades to be correlated across markets, i.e., allowing trades to take place at both bids or both asks at the same time, as it would be the case it traders split their orders across markets. Trading volume, another possible liquidity measure, can also be trivially shown to co-move more in the presence of an arbitrageur.
To summarize, based on a simple framework of the arbitrageur’s trading behavior, I expect the liquidities of the markets for securities traded on different exchanges and connected by arbitrage to co-move more when arbitrageurs are present on the market.