To evaluate the effect of the arbitrageurs’ presence on the market, it would be ideal to have access to a direct measure of their activity. Unfortunately, to my knowledge, such a measure has not been presented to any academic study to date. However, a metric used in previous works (Foucault et al., 2017; Rösch, 2014) to calculate arbitrageurs’ activity—or lack thereof—is the presence of arbitrage opportunities, which is the approach I take here. In Subsection V.A, I show evidence that my measure actually captures arbitrageurs’ activity rather than fortuitous mispricings.
they had a minimum trading price below 1 CA Dollar (CAD), resulting in sub-penny quoting; and 8 because they traded less than 20 days in 2013. All variables are winsorized at the 2.5% level. Observations for the stock of Telus Corporation are dropped for the month of January since the company went through a conversion of non-voting shares into common shares, altering the normal trading for the shares and the cross-listed counterparts.
9This is to examine arbitrage in the stricter sense, i.e., by settling all trades instantaneously, rather than waiting for netting.
10Thomson Reuters’ closest competitor is Electronic Broking Services, which dominates the market for all other pairs (Chaboud et al., 2014). In the Appendix I compare the FX data from Thomson Reuters to the aggregate feed obtained from Olsen data, a company that consolidates the quotes of multiple FX dealers. I find that Thomson Reuters generally quotes a wider spread than the market as a whole. The bias resulting from employing Thomson Reuters, thus, would be to underestimate the presence of arbitrage opportunities, asAr bit sis decreasing in the bid-ask spread.
In order to correctly identify actual trading opportunities, I need to match bid and ask prices on the three markets that would be involved in an arbitrage transaction: the TSE, the US stock market, and the FX market. I do so at a one-second frequency, calculating, at the end of each second, the best bid and ask on the US stock market, using TAQ data, the corresponding quotes on the Canadian market, using prices prevailing on the TSE, and the best bid and ask quotes for an FX transaction.
I define the arbitrage opportunity for stock pairi, on dayt, at secondsas AOit s =max
"
BidC A
AskF X
−AskU S,BidU S− AskC A
BidF X
,0
#
(1) where BidU S, AskU S, BidC A, AskC A, BidF X, and AskF X are defined intuitively and the FX rate is expressed as CA per US dollar. The first term in the max operator corresponds to the strategy of selling a share in Canada by hitting the standing bid-price and covering the position by lifting a share in the US at the ask-price, while the second term corresponds to the opposite strategy. The 0 term constrains the arbitrage trade to have a non-negative profit.11
For example, on January 2, 2013, at 3:40:53 PM, Toronto Dominion was quoted at a bid and ask of USD 84.47 and 84.48, respectively, in the US, and at a bid and ask of CAD 83.26 and 83.27, respectively, in Canada. Contemporaneously, 1 US dollar could buy 0.9854 CA dollars (BidF X) and 0.9855 CA dollars could buy 1 US dollar (AskF X). The arbitrage opportunity was, therefore, maxf83.26
0.9855 −84.48,84.47− 0.985483.27,0g
= max [0.005,−0.034,0] = 0.005, i.e., the arbitrageur could have locked in a riskless profit of half a cent by selling the stock in Canada for CAD 83.26, buying USD 0.985583.26 = 84.485 with the sale proceeds, and covering her position by buying the stock in the US for USD 84.48. Figure 2 shows the arbitrage opportunities for Toronto Dominion on January 2, 2013, between 3:40:00 PM and 3:40:59. Panel A shows the arbitrage opportunities shaded in grey, calculated as per Equation 1. To highlight the importance of taking the FX liquidity cost into consideration, Panel B shows the arbitrage opportunities calculated as per Equation 1, but where corresponding FX midquote is employed instead of BidF X and AskF X. Clearly, the liquidity of the FX is an important determinant of arbitrage profitability, and ignoring this aspect will lead to an overestimation of the frequency of arbitrage opportunities.
Insert Figure 2 here.
In the remainder of the paper, I focus on a daily measure of arbitrage opportunities, namely their frequency Ar bit:
Ar bit = 1 S
S
X
1
I[AOit s > 0] (2)
11The existence of make- and take-fees would require the arbitrage opportunity to be larger than the sum of the fees on the two markets. I account for this extra cost by repeating our analysis and only countingAOit s inAr bit if it is larger than 0.006, which is a conservative estimate consistent with (twice) the fee structure reported in Foucault, Kadan, and Kandel (2013) and Malinova and Park (2015). The results are reported in the Appendix.
whereAOit swas defined in Equation 1,S is the total amount of trading seconds on dayt, andI is the indicator function. A daily observation of Ar bit = 0.03 should be interpreted as showing that arbitrage opportunities arose 3% of the time for stocki, on dayt. Alternative arbitrage measures, which will be used to verify the robustness of the main results in the paper in Section VI, are defined as follow:
M ax Rel Ar bit =max
s
AOit s 1
2BidU S+ 12AskU S
Rel Ar bit = 1 S
S
X
1
AOit s 1
2BidU S+ 12AskU S
where M ax Rel Ar bit measures the maximum return to an arbitrage opportunity for stock i on dayt, Rel Ar bit measures the size of the arbitrage opportunity relative to the capital required to take advantage of it, i.e., the price of the stock. Capturing a different dimension of the arbitrage opportunities, I calculate their average durationDur ationit. Finally, I obtain a more conservative estimate of Ar bit, requiring the arbitrage opportunity to yield at least a return of 1 basis point, Ar b1bpit = 1SPS
1I f
AOit s > 0.0001BidU S+2AskU Sg .
As a final alternative measure of arbitrageurs’ activity, I calculate the half-life of a pricing shock, i.e., how long it takes for the co-integrated price system to absorb half of a 1$ pricing shock.
I define the half-life as H alf Lif eit = log(1log(0.5)+αU S−αC A). In this definition, the αs are adjustment parameters to the (1,−1) co-integrating vector in an error correction model estimated at a daily frequency using second-level midquotesPit sU S,Pit sC A,U SD(where the latter is measured in USD). The half-life measures how long it takes for the system to absorb half of the shock to either price. The average half-life in the sample is 49 seconds, meaning that, on average, following a USD 1 change in one of the prices, the difference in prices diminishes to USD 0.5 within 49 seconds.12
Panel A of Table I reports descriptive statistics for the arbitrage measures. The median cross-listed stocks pair is mispriced 1.3% of the time, corresponding to about 5 minutes a day. However the arbitrage opportunities are generally small, as only two minutes a day the arbitrage is larger than 1 basis point. In about 8% of stock-days, there are no arbitrage opportunities, while 50% of the stock-days feature mispricing between 0.3% and 3.4% of the time, corresponding to 1 and 13 minutes, of which only between 20 seconds and 8 minutes feature a sizeable arbitrage.
Insert Table I here.
For the median stock, the largest daily arbitrage opportunity is 5 bps, with an interquartile spread of 7.8 bps. The arbitrage opportunities are, in general, small, with 95% of the stock-days having a maximum arbitrage below 27 bps. The arbitrage opportunities are short-lived, with half
12I estimate
∆Pit sU S
∆Pit sU S
!
= γU Sit γitC A
! + αU Sit
αC Ait
!
Pi,t,s−1U S −Pi,t,s−1C A,U SD +it s
whereαC Ait >0> αU Sit . Allowing for short term dynamics has a negligible effect onH alf Lif eit.
of them lasting less than 5 seconds. On the vast majority of days, the average arbitrage opportunity lasts less than one and a half minute.