The law of one price, arbitrage opportunities and price convergence: Evidence from cross-listed stocks

J. of Multi. Fin. Manag. 31 (2015) 126–145

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Journal of Multinational Financial Management journal homepage: www.elsevier.com/locate/econbase

The law of one price, arbitrage opportunities and price convergence: Evidence from cross-listed stocks Imen Ghadhab a,∗, Slaheddine Hellara b a b

Faculty of Science Economics and Management of Tunis (FSEG Tunis), University of Tunis El Manar, Tunisia Higher Institute of Management of Tunis (ISG Tunis), Bardo 2000 Tunis, Tunisia

a r t i c l e

i n f o

Article history: Received 25 November 2014 Received in revised form 14 April 2015 Accepted 6 May 2015 Available online 18 May 2015 JEL classification: F32 G15

a b s t r a c t The purpose of this study is to analyze price deviations, arbitrage opportunities and price convergence for cross-listed stock. Using a unique and comprehensive sample of dual-listed firms as well as firms with multiple foreign listings, we show that markets of cross-listed stocks are not efficient. We also show that the dynamic of price adjustment is correctly modeled by a multivariate STAR model for which the transition between regimes is affected by both transaction costs and cross-listing. © 2015 Elsevier B.V. All rights reserved.

Keywords: Cross-listing Arbitrage Law of one price STAR model

1. Introduction With the enhanced globalization of financial markets, an increasing number of companies choose to cross-list their shares in overseas market. Since prices of cross-listed stocks are the prices of the same security, a higher degree of financial integration is expected to improve market efficiency and ensure

∗ Corresponding author. Tel.: +216 21114447. E-mail addresses: [email protected] (I. Ghadhab), [email protected] (S. Hellara). http://dx.doi.org/10.1016/j.mulfin.2015.05.002 1042-444X/© 2015 Elsevier B.V. All rights reserved.

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price equality. Even if prices diverge, such discrepancy must be eliminated by arbitrage activities that bring prices toward equilibrium. In this paper we examine the following three questions involving the efficiency of market for crosslisted stocks. The first is whether price discrepancy exists. The second question concerns the existing of arbitrage opportunities. Finally, we address the issue of the dynamic of price convergence. Earlier empirical evidence on whether or not markets for cross-listed stocks are indeed fully efficient was mixed. Rosenthal (1983), Kato et al. (1991), Park and Tavakkol (1994) and Miller and Morey (1996) show that price deviations are not significant and arbitrage opportunities are infrequent. More recent studies like Suarez (2005), Gagnon and Karolyi (2010), Alsayed and McGroarty (2012) and Ansotegui et al. (2013) show that prices of cross-listed stocks may significantly deviate and that investors may make arbitrage profit by trading on these price differences. Those earlier studies focus on stocks listed on two countries or dual-listed stocks and are concentrated on the cross-listing on the US. Therefore, those previous results provide a non-conclusive answer to the question of whether markets for cross-listed stocks are efficient given that earlier studies do not consider further possible foreign listing for the same security. The purpose of this study is to contribute to the literature on the dynamic of price behavior of crosslisted stocks by providing the first empirical analysis of price deviations, arbitrage opportunities and price convergence for firms with multiple foreign listing. Furthermore, our new context allows us to provide a new empirical analysis on the effect of cross-listing (or multiple foreign listing) on the dynamic of price convergence. We study for the first time the non-linear effect of simultaneously the transaction costs and the cross-listing on stock prices based to both the transaction cost hypothesis and the limit to arbitrage hypothesis initiated by Lyons (2001). In addition to dual-listed stocks, our sample includes stocks with multiple foreign listing. We focus on highly traded European and Canadian firms cross-listed on major American and European exchanges. Using a high frequency intraday data for a shorter period of trading overlap, we show that markets for cross-listed stocks are not completely integrated and not fully efficient. In fact, price discrepancies exist and provide the opportunity to make a significant arbitrage profit. Our results also show that larger price deviations occur at the beginning and the ending of the trading overlap which is consistent with the finding of Werner and Kleidon (1996) and Miller and Morey (1996). To the best of our knowledge, the STAR model was only applied by Koumkwa and Susmel (2008) and Mehanaoui et al. (2012) to the price difference of cross-listed stocks. In contrast to these studies, I allow for a multivariate STAR model in which the transition function depend on two transition variables, in order to study the non-linear effect of multiple exogenous variables. The choice of these transition variables is motivated by the transaction cost hypothesis and the limit to arbitrage hypothesis. The linearity tests show that the determined exogenous variables are valid transition variables. We then estimate a STAR model with linear combination of these transition variables and show that this model is an adequate specification of price difference for stocks with multiple foreign listing. The estimation of STAR model supports the idea of smooth transition between regimes. This paper is organized as follows. Section 2 presents data and price deviation construction. Section 3 tests the presence of arbitrage opportunities. Section 4 analyses price convergence and estimates STAR models. Finally, Section 5 concludes. 2. Data and price discrepancy construction 2.1. Data description The aim of this paper is to study price deviations for cross-listed stocks that trade on U.S. and European exchanges. Our exercise is based on quotes prices from “Tickdatamarket” for the period from August 7, 2013 to October 31, 2013.1 This dataset is collected directory from exchanges and includes the best bid and ask quotes along with the time to the nearest even second. We also obtain

1 Due to the lack of available data on high frequency quote prices, and in order to have the same period for all firms considered in our paper, our sample period start from August 7 rather than from the first of August.

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Table 1 Sample description. Firm

Symbol Local and foreign listing

Aegon

AGN

Amsterdam NYSE

Bank of Ireland

BIR

Irish London NYSE

BAMA Brookfield Asset Management European Aeronautic EAD Defense and Space GSK Glaxosmithkline ING Groep

Royal Dutch Shell A

StMicroelectronics

Telefonica Yamana Gold Average

INGA

RDSA

STM

TEF YRI

Toronto NYSE Amsterdam Frankfort London NYSE Amsterdam Brussels NYSE London Amsterdam NYSE Milan NYSE Paris Madrid London Toronto NYSE

Cross-listing typea

Shares traded (in millions)

Number of trades

Average price

Average bid-ask spread (in basis point)

3937 57

267,930 206,136

5.69 7.64

8.17 98.34

4282 2950 45

18,800 52,794 153,236

0.22 0.22 12.14

10 10.43 300

Direct ADR

38 45 2689 10 460 138 14280 10 171

194,757 310,689 634,406 38,862 502,704 740,377 982,541 1593 506,486

38.5 37.11 46.49 46.48 16.19 51.24 8.72 8.66 11.72

100 200 133.8 270.4 73.7 134.13 11.4 278.8 107.4

Direct ADR

186 4601 136

268,353 560,304 621,398

2063.24 24.33 65.43

Direct

1564 71 1763 2671 436

316,958 238,104 275,828 667,338 1002

6.5 8.69 6.5 11.49 11.47

10 113.2 21.44 58.65 11431

Direct

176 520

481,423 1,822,075

10.87 10.47

101.64 105.9

1718

411,004

ADR

Direct ADR Direct Direct ADR

ADR Direct

8000 100 98

a

A company may list their shares in foreign markets either directory or by issuing Depositary Receipts (DR) which represent a specified number of company’s shares. American Depositary Receipts (ADR) are Depositary Receipts traded in US markets.

quote data for exchange rates from Dukascopy, Switzerland. Each quote contain a bid and an ask price. To identify arbitrage opportunities, we use trading cost estimates published by Elkins and McSherry at the end of the year 2012, as well as the ADR conversion fee which is set to 5 cent from bank of New York Mellon website. Unlike previous studies (e.g. Miller and Morey, 1996; Suarez, 2005; Alsayed and McGroarty, 2012; Ansotegui et al., 2013), we focus not only to cross-listed shares in the U.S. markets but we include firms with European foreign listing as well as firms with multiple foreign listing. Our sample consists of 8 European and 2 Canadian stocks cross-listed in U.S. market (NYSE) and European exchanges (including Euronext Amsterdam, Euronext Brussels, Euronext Paris, Frankfort stock exchange and London stock exchange), with a total of 14 foreign listing. We examine the prices over the overlap period when all markets are open (i.e. 14:30–16:30 GMT). Table 1 presents summary statistics of the 10 cross-listed stocks analyzed in this paper. Information are about firm’s name, ticker symbol, market listing, listing type, number of shares traded, number of trades, average price and average bid-ask spread. We conduct data cleaning approach inspired from Schultz and Shive (2010), which consists of filtering out the following observations:

- Bid price greater than the corresponding ask price. - Ask price more than four time the corresponding bid price. - Bid–bid or ask–ask returns exceeding 25%.

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2.2. Price discrepancy construction Similar to Suarez (2005), we make our analysis by using quote data rather that trade data. In this way, we can incorporate the bid-ask spread which is an important transaction cost that can impede arbitrage activities. We can also have a large number of observations as quote data are much more abundant than trade data and are updated even if there is no trading. We rely on intraday quote to measure our price deviation by focusing on a short period during which trading hours overlap. To form our price series, we opt for matching quotes from all markets at every 1 min interval and the final quote in each minute is chosen to represent all the quotes during that minute. Hence, we obtain synchronous price series for each stock with evenly sampled observations. To conduct our analysis of price deviation, we also use the appropriate exchange rate at the exact time of the price comparison. Exchange rate series are sampled at every minute during trading overlap. We use U.S. dollar as a common currency. When company is cross-listed on the U.S. market via ADRs (American Depositary Receipts), stock prices are multiplied by the appropriate ADR ratio. As our sample includes dual and multiple listing stocks, some companies, those with multiple foreign listing, has multiple price series (more than 2 price series). To construct our price deviation, let us denote the foreign bid and ask prices as BE and AE respectively. Furthermore, we denote the local bid and ask prices as BL and AL respectively. Finally, we denote the bid and ask prices of the exchange rate as BFX and AFX respectively. As Suarez (2005) and Alsayed and McGroarty (2012), we will incorporate the bid-ask spread into the definition of the price discrepancy and define it as follows:

DL/Ei =

⎧ B ∗ A − AEi ∗ BFX si AEi ∗ BFX < BL ∗ AFX ⎪ ⎨ L FX ⎪ ⎩

BEi ∗ AFX − AL ∗ BFX

si BEi ∗ AFX > AL ∗ BFX

0

otherwise

(1)

where L and Ei refer to the local and foreign market respectively. And:

DEi /E

j

i= / j

=

⎧ B ∗ A − AEj ∗ BFX si BEi ∗ AFX > AEj ∗ BFX ⎪ ⎨ Ei FX ⎪ ⎩

BEj ∗ AFX − AEi ∗ BFX

si BEj ∗ AFX > AEi ∗ BFX

0

otherwise

(2)

The first two cases of Eqs. (1) and (2) are situations where price deviations may be profitable. In other words, for an arbitrageur, the ask price of the stock to be bought is lower than the bid price of the stock to be sold. The last case of Eqs. (1) and (2) is a situation where D = 0. This means that there is an overlap between bid-ask spreads of stocks. In this case, price discrepancy may not be strictly zero, but it is unprofitable. In other words, markets are efficient and prices are very close to each other. In the case of Eq. (2), price deviation is only computed for firms with multiple foreign listing. This condition is not tested by earlier studies and so, our paper gives more conclusive results about the validity of the law of one price for cross-listed stocks. Table 2 depicts descriptive statistics of price discrepancy series. Price deviation is enclosed between (−2.0525, 4.0167) amount in U.S. dollars. Price deviations DL/Ei and DEi /Ej are both important. This means that price of cross-listed stock may deviate, not only from it local counterpart, but also from price on remaining foreign exchanges. Overall, we conclude that large price deviation exist for across-listed securities averaging approximately −7.1% across local and foreign markets and −2.2% across foreign markets when firms seek multiple overseas listing. Table 2 also shows that price deviation may be profitable in 41%, 39% and 20% of cases for respectively Foreign–Foreign, Local–U.S. and Local–European pairs. Fig. 1 depicts graphically price deviation series for four sample companies: Aegon, Bank of Ireland, ING Groep and StMicroelectronics. We illustrate price deviations with the presence of peaks throughout overlapping trading hours. However, there are no apparent patterns as to when these peaks occur within the overlap. Werner and Kleidon (1996) have found U-shaped patterns in intraday volatility and bid-ask spread for U.S. cross-listed British stocks. Quote price volatility is higher during the

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Table 2 Price deviation. Symbol

DL/E Mean

Min

Max

SD

AGN BAMA BIR EAD GSK INGA RDSA STM TEF YRI All sample

−0.0037 −0.0175 −0.0474 −0.02 −0.0174 −0.0126 −0.0186 0.01 −0.5744 −0.01 −0.071

−0.538 −0.332 −0.5964 −2.0525 −1.44 −0.6063 −1.199 −0.3468 −1.315 −0.3191 −2.0525

1.0318 0.29 1.3677 1.8965 1.2175 1.4 4.0167 1.3694 0.015 0.299 4.0167

0.05 0.05 0.1 0.09 0.22 0.06 0.18 0.11 0.1 0.03 0.1

% D>0

DEi /Ej Mean

Min

Max

SD

DL/US

−0.0416

−0.5718

0.7619

0.1

37 31 29

−0.024 −0.0144 −0.007

−0.5993 −0.9764 −0.35

0.5165 3.99 0.7622

0.07 0.1 0.06

46 36 46 52

−0.022

−0.9764

3.99

0.08

27 39

DL/Ei;i =/ US

DEi /Ej

6 4.8

30

50 46 18 0.007

48 42 50

20

41

morning and afternoon than at midday. In other words, they find that British cross-listed firms traded in the U.S. show a greater volatility and bid-ask spread at the opening of the U.S. market and the close of U.K. market. Miller and Morey (1996) show that volatility, bid-ask spread and arbitrage are related and find a similar U-pattern in the price differences. The authors suggest that an increase in price difference is caused by an increase in volatility and bid-ask spread. Miller and Morey (1996) analyze intraday pattern of price deviation for one British cross-listed stock traded in U.S. during the period 1.2

1.6 1.2

0.8

0.8 0.4 0.4 0.0 0.0 -0.4

-0.4

-0.8

-0.8 2500

5000

7500

10000

12500

15000

10000

Aegon

20000

30000

40000

Bank of Ireland

1.6

1.6 1.2

1.2

0.8 0.8 0.4 0.4 0.0 0.0

-0.4

-0.4

-0.8 10000

20000

30000

StMicroelectronics

40000

10000

20000

30000

40000

ING Groep

Fig. 1. Price deviation series. This figure depicts graphically price deviation series throughout overlapping trading hours for four sample companies: Aegon, Bank of Ireland, ING Groep and StMicroelectronics.

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Table 3 Price deviation per period of trading overlap. Symbol

AGN BAMA BIR EAD GSK INGA RDSA STM TEF YRI

Middle of trading

Opening of trading

Mean

Min

Max

−0.006 −0.017 −0.04 −0.03 −0.01 −0.01 −0.01 −0.01 −0.57 −0.01

−0.17 0.17 −0.33 0.29 −0.56 0.59 −1.02 1.89 −1.37 1.15 −0.3 0.2 −0.64 0.57 −0.34 0.33 −0.91 −0.31 −0.31 0.29

Closing of trading

SD

Mean

Min

Max

SD

Mean

Min

Max

SD

0.02 0.04 0.08 0.07 0.22 0.06 0.15 0.05 0.1 0.03

−0.006 −0.028 −0.03 −0.03 −0.01 −0.004 −0.02 −0.01 −0.57 −0.01

−0.19 −0.33 −0.59 −1.93 −1.44 −0.6 −0.58 −0.35 −0.86 −0.31

0.19 0.27 1.36 1.67 1.21 0.5 0.52 0.25 −0.33 0.29

0.02 0.06 0.15 0.12 0.21 0.06 0.15 0.04 0.1 0.04

0.014 −0.014 −0.04 −0.03 −0.01 −0.01 −0.01 0.09 −0.57 −0.01

−0.53 −0.22 −0.55 −2.05 −1.2 −0.39 −1.19 −0.34 −1.31 −0.11

1.03 0.17 0.23 0.1 0.98 1.39 4.01 1.36 0.01 0.09

0.11 0.04 0.08 0.12 0.19 0.09 0.19 0.21 0.11 0.01

of March 27, 1995–May 31, 1995. They find that mean price differences are higher in the first and last half-hours of the overlap. We extend the work of Miller and Morey (1996) to multiple foreign listing setting and analyze intraday price deviation pattern for different periods of the trading overlap. More particularly, we study price deviation in the first and last 15 min of the overlap. Table 3 depicts descriptive statistics for price discrepancy for each period of time overlap. For a substantial number of firms, larger deviations occur on the first and the last 15 min of trading overlap. In other words, price discrepancies are more frequents at the opening of U.S. markets or at the closing of European markets. Fig. 2 illustrate graphically price deviation series per period of trading overlap for each listing pair, stock–stock or stock–ADR, for the following companies: Aegon, Bank of Ireland, ING Groep and StMicroelectronics. Peaks in price deviations can occur either at the beginning or at the ending of trading overlap period. For instance, price deviation for the Irish company “Bank of Ireland” increases between 14:30 and 14:45 GMT during the earlier morning NYSE trading. This means that markets for cross-listed securities are not integrated since the opening and the closing of listing market exhibit exceptional levels of price deviation. As show in Fig. 2, these apparent patterns are caused by the trading of the security in the NYSE. In other words, when U.S. market open, the level of price deviation increases. More particularly, for U.S.–Irish pair and London–Irish pair. This pattern may be related to information not previously incorporated into London or Irish prices, which may lead to a higher level of volatility in the U.S. market. As suggested by Foster and Viswanathan (1993) and Holden and Subrahmanyam (1992), high volatility may be caused by investors with long live private information who trade aggressively at the opening of markets. Furthermore, Chan et al. (1996), Harris and Raviv (1993), Kim and Verrecchia (1991) and Wang (1994) show that high volatility may be caused by differences in how agents interpret public information. In our case, high level of price deviation for U.S.–Irish pair at the beginning of trading overlap may be explained by the fact that U.S. investors wait until NYSE open to trade on their private information or to reinterpret public information released during Irish trading hours. This result may also indicate that trades in U.S. market are more informative. We record the same price deviation pattern for London–Irish pair but not for U.S.–London pair. This result may be explained as follow. The fact that the open of New York market does not exhibit exceptional level of price deviation between U.S. and British exchanges show that both markets are, at some degree, integrated. So any new information incorporated in U.S. prices is followed by an immediate reaction of London prices to NYSE prices. This leads consequently to a high level of price deviation for London–Irish pair. Fig. 2 also show that price deviations may exhibit high levels at the end of trading overlap, i.e. 15 min before European market close. This pattern may be caused by new information, or is likely to result from the practice of prearranged so-called protected trades. These trades involve a dealer who at some point during trading agrees to give a customer the better of the prevailing for a specific (subsequent) period of the day. The trade is typically booked at close. Its prices will tend to differ from that of previously recorded regular trades (Werner and Kleidon, 1996). Hence, the large price deviations for the last period of trading overlap. Our conclusions from Table 3 and Fig. 2 are completed by the results presented in Table 4 which reports differences of means tests for each period of trading overlap.

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1.2

0.8

0.4

0.0

-0.4

-0.8 2500

5000

7500

10000

12500 15000

Aegon

1.6

1.6

Bank of Ireland

1.2

1.2

0.8

0.8

0.4

0.4

0.0

0.0

-0.4

-0.4

-0.8

-0.8

.8

Bank of Ireland

Bank of Ireland

.6 .4 .2 .0

2500

5000

7500

10000

12500

15000

-.2 -.4 -.6 2500

5000

Irish-London 1.6

7500

10000

12500

15000

2500

5000

Irish-NYSE 1.2

ING Groep

7500

10000

12500

15000

12500

15000

12500

15000

London-NYSE .6

ING Groep

1.2

ING Groep

.4

0.8 0.8

.2 0.4

0.4 .0

0.0 0.0

-.2

-0.4 -0.8

-0.4 2500

5000

7500

10000

12500

15000

-.4 2500

5000

Amsterdam-Bruxelles

1.6

7500

10000

12500

15000

2500

5000

Amsterdam-NYSE

1.6

StMicroelectronics

1.2

1.2

0.8

0.8

0.4

0.4

0.0

0.0

7500

10000

NYSE-Bruxelles

.8

StMicroelectronics

StMicroelectronics

.6 .4 .2 .0

-0.4

-.2

-0.4 2500

5000

7500

10000

Milan-Paris

12500

15000

-.4 2500

5000

7500

10000

Milan-NYSE

12500

15000

2500

5000

7500

10000

NYSE-Paris

Fig. 2. Price deviation series per period of trading overlap. This figure illustrates graphically price deviation series per period of trading overlap for each listing pair, stock–stock or stock–ADR, for the following companies: Aegon, Bank of Ireland, ING Groep and StMicroelectronics. Price deviations are illustrated at the first 15 min, at the middle and at the last 15 min of the trading overlap.

According to Miller and Morey (1996), the U-shape pattern exists if we find significantly differences in means between the first to the second and second to last period of overlap. We should also find no significant differences between the first and the last period of trading overlap. Table 4 shows that we can reject the null hypothesis of equality of means for the opening to the middle overlap and for middle to the ending of overlap in several cases. However, we can accept the U-shape hypothesis for only one case related to “EAD” stock. If prices of cross-listed securities deviate, are these deviations profitable. In the next section, we analyze arbitrage opportunities among sample stocks.

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Table 4 Differences of means test. Symbol

Opening–middle

Middle–closing

AGN BAMA

0.64 11.07***

17.63*** 1.02

8.37*** 8.3***

BIR Irish–London Irish–NYSE London–NYSE

19.24*** 8.31*** 0.95

3.51*** 0.93*** 0.769

8.08*** 4.74*** 1.27

3.17*** 0.36

5.49*** 0.15

1.16 0.42

INGA Amsterdam–Brussels Amsterdam–NYSE Brussels–NYSE

0.09 0.21 21.38***

4.35*** 6.29*** 0.55

2.74*** 3.78*** 16.46***

RDSA London–Amsterdam London–NYSE Amsterdam–NYSE

0.55 0.68 1.09

1.81* 0.8 4.99***

1.7* 0.1 3.26***

STM Milan–Paris Milan–NYSE NYSE–Paris TEF YRI

2.39*** 0.83 0.71 0.02 1.44

63.9*** 54.07*** 2.09*** 0.42 0.005

28.08*** 26.31*** 1.97** 0.3 1.03

EAD GSK

* ** ***

Opening–closing

Denotes significance at 10%. Denotes significance at 5%. Denotes significance at 1%.

3. Arbitrage for international cross-listed stocks To identify arbitrage opportunities, previous studies focused on price deviations between the local and one foreign market, especially U.S. market. However, results of the previous section of this paper show that price discrepancy may also exist across foreign listing markets. To analyze arbitrage opportunities in markets for cross-listed stocks, it is crucial to understand the process of arbitrage operation. When a firm is listed directory in foreign market, an arbitrageur simply buys the underpriced stock and sells the overpriced one. Whereas, when a firm seeks a foreign listing via the issue of ADRs, arbitrage operation is conducted through the contact of the depositary bank for ADR creation and cancellation. When an ADR is trading at a discount to the underlying share, an arbitrageur can buy an ADR, convert it into the underlying share and then sell them in the underlying market. If the ADR is selling at a premium, an arbitrageur can bought the underlying share and then request a custodian bank to issue the corresponding shares. We follow the methodology outlined in Suarez (2005) to our dataset and identify arbitrage opportunities whenever the price deviation, D, is strictly above a constant measure of transaction costs “C”. Transaction costs include the 5 cent ADR subscription/cancellation fee, and exchange trading cost published by Elkins and McSherry at the end of 2012. These include explicit costs (commissions, taxes and fees) as well as implicit costs (market impact costs computed by comparing the trade price to a VWAP benchmark price on the day of the trade). Table 5 presents all trading costs used in our arbitrage analysis.2 To test for arbitrage opportunity, we consider 3 different thresholds of C: C, 2C and 3C. Table 6 shows the percentage of observations in which price discrepancy exceeds the considered thresholds. On average, 42.6% of price deviations were identified as profitable arbitrage opportunities.3 2

Gagnon and Karolyi (2010) and Ansotegui et al. (2013) use the same source for trading costs. Our approach to test arbitrage opportunities, the direct arbitrage, is a standard approach used by Suarez (2005) and Ansotegui et al. (2013), but not a unique approach. Pairs trading (i.e. establishing a long/short position on price discrepancy, 3

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Transaction costs (in bp)a

Canada France buy France sell Germany Belgium Ireland buy Ireland sell Italia Netherland Spain U.K. buy U.K. sell U.S. ADRs conversion fee (in USD)

25.42 42.58 23.65 20.53 21.44 119.63 19.37 20.77 20.6 21.41 70.68 20.29 17.64 0.05

a

bp: Basis point.

Table 6 Arbitrage opportunities. Symbol AGN BAMA BIR Irish–London Irish–NYSE London–NYSE EAD GSK

Na

%>C

% > 2C

% > 3C

5690 4659

4.1 86

3.6 72.6

2.02 61.6

899 4488 4554 736 6977

21.4 51.8 47.9 57.7 71

21 24.2 19.21 45 47.3

21 10.8 7 43 29.2

INGA Amsterdam–Brussels Amsterdam–NYSE Brussels–NYSE

7644 5441 7350

100 5.5 31.4

100 2.9 2.6

100 2.7 0.37

RDSA London–Amsterdam London–NYSE Amsterdam–NYSE

7005 7035 6388

98 71 41.8

96 46.3 13.8

94 29 5.8

2701 7821 7528 1 4066 90983 22129 68854

45.4 17.3 12.1 100 75.4 42.6 57.3 47.9

36.2 10.9 2.73 0 56 34.4 42.9 33.5

34 9.7 2.1 0 42.2 29.5 31.9 28.5

STM Milan–Paris Milan–NYSE NYSE–Paris TEF YRI All Dual listed firms Firms with multiple foreign listings a

N is the number of observation when D > 0.

This percentage is about 57.3% and 47.9% for respectively dual-listed firms and firms with multiple foreign listing. Table 6 shows also that arbitrage opportunities decrease with the size of threshold. Table 7 presents descriptive statistics of arbitrage profit for three different sub-categories: category 1 represent arbitrage profit when D is between C and 2C; category 2 when D is between 2C and 3 C and category 3 is when D is above 3C.

and liquidating the position when prices converge) is also possible for stock-ADR pairs (Alsayed and McGroarty, 2012). The objective of our paper is to test the presence of arbitrage opportunities for cross-listed stocks. While our approach is not unique, it is appropriate for the research objective, and it allows us to show that price discrepancy may be profitable for arbitrageurs.

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Table 7 Arbitrage profits. Symbol

Profit 1(USD) C < D < 2C Mean

Profit 2 (USD) 2C < D < 3C

Total

Profit 3(USD) D > 3C

Mean

Total

AGN BAMA

0.032 0.002

1.02 1.34

0.08 0.006

7 3.3

BIR Irish–London Irish–NYSE London–NYSE EAD GSK

0 0.023 0.024 0.004 0.026

0 28 31.4 0.33 43.7

0 0.08 0.08 0.005 0.08

INGA Amsterdam–Brussels Amsterdam–NYSE Brussels–NYSE

0 0.016 0.012

0 2.2 26.2

0 0.07 0.08

RDSA London–Amsterdam London–NYSE Amsterdam–NYSE

0.002 0.03 0.023

0.3 47.5 40.2

STM Milan–Paris Milan–NYSE NYSE–Paris TEF YRI

0.002 0.01 0.01 0.006 0.002

All

0.012

236.6

Dual listed firms Firms with multiple foreign listings

0.012 0.012

48.1 188.6

0.5 5 7.3 0.006 1.65

Mean

Total

0.4 0.04

43 105.7

0 46.3 45.8 0.08 101.4

0.39 0.23 0.16 0.3 0.26

72.6 112 60 95.8 519.8

0 0.5 13.3

0.05 0.23 0.2

377 33.1 5.4

0.006 0.084 0.076

0.75 102 38.8

0.15 0.21 0.21

940.5 420.6 77.3

0.007 0.08 0.075 – 0.006

0.42 7.5 3.2 – 3.5

0.35 0.38 0.19 – 0.03

320.8 283 31 – 58.2

0.05

373.85

0.22

3555.8

0.029 0.053

115.28 258.57

0.17 0.23

822.5 2733.2

As we can see from Table 7, profits for all categories are positive for all sample stocks. Profits of category 3 are largest and an investor can earn an average profit of approximately 0.22 USD per share traded. Table 7 also shows that trading on multiple foreign listing’s stocks is more profitable than the trading on dual-listed stocks. More precisely, for categories 2 and 3, average arbitrage profit is respectively about 0.053 and 0.23 for firms with multiple foreign listing compared to 0.029 and 0.17 for dual-listed stocks. Although arbitrage opportunities exist and a trader may make a profit, they are few, if any, of cases in which limits to arbitrage cannot exist. These limits can impede arbitrage activities and so prevent prices to converge to parity and, consequently, price discrepancy may persist. However, as we discuss later, we suggest that these limits are not significant in our case. The first limit to arbitrage is the fundamental risk. It arises in case of not identical securities. Schultz and Shive (2010) analyze the mispricing of dual-class shares with highly fundamental risk since these assets present different voting rights. In our case, cross-listed stocks are identical securities in all respects (see Pulatkonak and Sofianos, 1999), and fundamental risk is then highly minimized. The second limit to arbitrage is liquidity risk. Marshall et al. (2013) find that liquidity decrease prior to arbitrage opportunities. Chung and Hrazdil (2010) find that higher liquidity facilitates arbitrage activities and consequently enhance market efficiency. Our sample comprises 10 of heavily traded stocks, with an average of 1718 millions traded shares4 (225,341 shares traded per minute). Finally, another limit to arbitrage which must be pointed out is order execution speed. Marshall et al. (2013) pointed out that order execution speed has increased over time. Bacidore et al. (2003) find

4

See Table 1.

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that order execution time is averaging 22.5 s for NYSE in 1999. More recently, Garvey and Wu (2009) show that execution time has a mean of 12 s for the period 1999–2003, and Hendershott and Moulton (2011) find a smaller execution time in 2006 of less than one second. Angel et al. (2010) pointed out that, European exchanges become more and more competitive in terms of execution speed, bid-ask spread and commissions. Overall, we suggest that, while limits to arbitrage may exist in our sample, they are small, and price discrepancy found in our paper may be exploited by arbitrageurs in practice. 4. Dynamic of price convergence for cross-listed firms At this point, we have shown that arbitrage opportunities exist in the markets for cross-listed stocks. Naturally, arbitrage activities tend to restore equality when stock prices deviate. In this section, we explore the dynamic of price convergence between different pairs of cross-listed stocks. More particularly, we seek to test the non-linearity adjustment and to specify an appropriate model for the dynamic of price convergence for cross-listed stocks. 4.1. Methodology In this sub-section we describe the model and the procedure of estimating STAR models. 4.1.1. The model Earlier literature on cross-listed stock has shown that price adjustment is non-linear due to transaction costs. This allows the presence of two regimes: an arbitrage and a non-arbitrage regime. The transition between regimes is more likely to be smooth as a result of non-synchronous adjustment by heterogeneous agents. We examine the dynamic of price convergence by employing the smooth transition autoregressive (STAR) model for price deviations, which can be written as:



yt =

ϕ10 +

p 





[1 − F(st , ˛, )] +

ϕ1i yt−i

i=1

ϕ20 +

p 



ϕ2i yt−i

F(st , ˛, ) + ε

(3)

i=1

where • • • • • •

yt : The price deviation between listing pairs of the cross-listed stock. F(.): The transition function bounded between 0 and 1. : The speed of transition between regimes,  > 0. st : The transition variable(s). ˛: The threshold parameter. ε: Error term.

STAR model is well suitable for our purpose since it allows for a smooth transition between regimes.5 The smooth transition function F(.) determines the degree of convergence. The speed of transition between regimes, , depends on the state of specified transition variables, st . The Logistic STAR (LSTAR) model uses the logistic function as the transition function: F(st , ˛, ) = (1 + exp[−(st − ˛)])−1 ;   0

(4)

When st → + ∞ ; F → 1. This corresponds to an upper regime given by:



yt =

ϕ20 +

p 



ϕ2i yt−i

+ε

(5)

i=1

5

Threshold autoregressive (TAR) model is another popular non linear model in which the change between regimes is abrupt.

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137

When st → − ∞ ; F → 0. This corresponds to a lower regime given by:



yt =

ϕ10 +

p 



ϕ1i yt−i

+ε

(6)

i=1

When  → ∞ , the transition function becomes a step function, so that the smooth transition model becomes a threshold autoregressive (TAR) model in which the transition between regimes is brutal. The exponential STAR (ESTAR) model uses the exponential function as the transition function: F(st , ˛, ) = (1 − exp[−(st − ˛)2 ]);   0

(7)

When st → ± ∞ ; F → 1. yt follow a simple linear model but with different speed of reversion. 4.1.2. Estimation procedure of a STAR model The estimation procedure of a STAR model is done in the following steps: • Identification of the transition variables. • Linearity tests and models estimation. 4.1.2.1. Identification of the transition variables. Earlier literature on the dynamic of price convergence of cross-listed stocks argued that price adjustment is non-linear. To the best of our knowledge, STAR models have only been applied to the price differences of cross-listed stocks by Koumkwa and Susmel (2008) and Mehanaoui et al. (2012) who use the lagged price difference, yt−d , as a transition variable according to the transaction cost hypothesis. As mentioned in both the empirical and theoretical literature, in the presence of transaction cost, price deviation follow a non-linear process in which the speed of convergence is related to the extent of past deviations. Unlike previous studies examining price convergence of dual-listed stocks, we are interested in addition to firms with multiple foreign listing by defining a new transition variable based both on the transaction cost hypothesis and the limits to arbitrage hypothesis developed by Lyons (2001). The latter proposes a non-linear model based on limits to speculation (or arbitrage), which explains that deviations from uncovered interest parity occur and persist because no one is willing to trade on these deviations since other investment opportunities have a higher profit. According to the transaction cost hypothesis, we define the first transition variable as the past price deviation, yt−d , where d > 0 the delay parameter that determines the lagged time between a shock and the response by the process. According to the limit to arbitrage hypothesis, we define a second transition variable zt−d as follows: When a firm is cross-listed in multiple foreign markets, limits to arbitrage hypothesis predicts that price deviation that allow for the highest profit attract arbitrage activities. In this case, profitable deviation will be eliminated and reversion to parity will occur. Conversely, when price deviations appear low, arbitrage opportunities is left unexploited and thereby price differences persist. In this case, the law of one price is violated and markets are not efficient. For example, when a firm is cross-listed on two foreign markets E1 and E2 , there will be three prices for the same security pl,t , pE1 ,t and pE2 ,t , where pl,t is the local price. In which case, three price differences may occur: y1,t = pl,t − pE1 ,t , y2,t = pl,t − pE2 ,t et y3,t = pE1 ,t − pE2 ,t . The first price deviation y1,t is preferred by a trader if:



 y1,t − Max y2,t , y3,t  0 condition (1)

The second price deviation y1,t is preferred by a trader if:



 y2,t − Max y1,t , y3,t  0 condition (2)

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The third price deviation y1,t is preferred by a trader if:



 y3,t − Max y1,t , y2,t  0 condition (3)

When one of these three conditions is satisfied, large price difference attracts arbitrage trades leading prices to the parity. Otherwise, price deviations persist and the law of one price is violated. The transition variable zt−d is defined by one of the above conditions. 4.1.2.2. Linearity tests and models estimation. To identify the appropriate transition variable, we test for linearity in the framework of Eq. (3) based on a Taylor approximation of the STAR model following Luukkonen et al. (1988). In a first step, we consider a uni-variate transition variable defined as follows: st = yt−d or st = zt−d A first, second, third and fourth order Taylor approximation of Eq. (3) around  = 0 gives: yt = ˇ00 +

p 



ˇ0j yt−j + ˇ1j yt−j st−d + t

(8)

j=1

yt = ˇ00 +

p 



2 ˇ0j yt−j + ˇ1j yt−j st−d + ˇ2j yt−j st−d + t

(9)

j=1

yt = ˇ00 +

p 



2 3 ˇ0j yt−j + ˇ1j yt−j st−d + ˇ2j yt−j st−d + ˇ3j st−d + t

(10)

j=1

yt = ˇ00 +

p 



2 3 4 ˇ0j yt−j + ˇ1j yt−j st−d + ˇ2j yt−j st−d + ˇ3j st−d + ˇ4j st−d + t

(11)

j=1

To test for linearity against LSTAR (ESTAR) model, we test the null hypothesis H0 : ˇ1j = 0 in Eq. (8) or H0 : ˇ1j = ˇ2j = ˇ3j = 0 in Eq. (10), (H0 : ˇ1j = ˇ2j = 0 in Eq. (9) or H0 : ˇ1j = ˇ2j = ˇ3j = ˇ4j = 0 in Eq. (11)), for j = 1, . . ., p. Rejection of linearity implies that st is a valid transition variable. STAR model estimated by earlier studies was typically univariate. The transition function depends on a single transition variable yt−d . In contrast to these studies, our paper allows for a multivariate transition function with multiple exogenous transition variables in order to estimate the non-linear effect of two variables simultaneously. The first variable yt−d is related to the non-linear effect of transaction cost and the second transition variable zt−d is related to the non-linear effect of cross-listing (or multiple foreign listing) on stock prices. Terasvirta (1994) recommends estimating the STAR model with the transition variable for which rejection of linearity is the strongest. This is a mean for selecting the delay “d” for the transition variables yt−d and zt−d . When the firm have multiple foreign listings, the rejection of linearity may occur for the two transition variables yt−d and zt−d . Within the case of multivariate transition function, Lof (2012) proposes to test the non-linearity effect of each transition variable and estimate simultaneously their weights in the transition function. We use a similar approach to Lof (2012) and estimate a linear combination of transition variables yt−d and zt−d for which rejection of linearity is the strongest, as follow: Xt = ˇ1 yt−d1 + ˇ2 zt−d2

(12)

Xt = ˇxt

(13)

Or

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139

where ˇ1 + ˇ2 = 1 and 0 < ˇ1 , ˇ2 < 1. ˇ is determined by estimating a Taylor approximation of order 1 with the multivariate transition variable Xt , for  = 0, lets: yt = ˇ00 +

p 



ˇ0j yt−j + ˇ1j yt−j ˇxt



+ t

(14)

j=1

Or yt = ˇ00 +

p 

p  

ˇ0j yt−j +

j=1



j1 yt−j yt−d + j2 yt−j zt−d + t

(15)

j=1

where ji = ˇi ∗ ˇ1j , pour i = 1, 2 et j = 1, . . ., p

(16)

The following restrictions: ˇ1 + ˇ2 = 1 and 0 < ˇ1 , ˇ2 < 1 Are used to determine ˇ1 and ˇ2 parameter. From (16), we have: p 

ji = ˇi ∗

j=1

p 

ˇ1j ,

i = 1, 2.

(17)

j=1

Under the condition 0 < ˇi , for i = 1, 2, we will have:

p  j=1 ji , for i = 1, 2. ˇi = p

(18)

ˇ j=1 1j

On the other hand: p  j=1

j1 +

p 

j2 = ˇ1 ∗

p 

j=1

j=1

ˇ1j + ˇ2 ∗

p 



p  

ˇ1j = ˇ1 + ˇ2 ∗

j=1

j=1

ˇ1j =

p 

ˇ1j

(19)

j=1

Under the condition ˇi < 1, (18) and (19) give:

p  j=1 ji p , for i = 1, 2. ˇi = p  + j=1 j2 j=1 j1

(20)

When ˇi parameters are determined, we perform a linearity test against a STAR model for the following transition variable: Xt = ˇ1 yt−d1 + ˇ2 zt−d2 . To conduct the linearity test against a LSTAR or ESTAR model, we proceed as follow. First, we check for the stationarity of price deviation series using the ADF test. If yt is stationary, we then use the information criteria AIC to select the appropriate lag order p that ensures no error correlation. Second, we perform a linearity test for the transition variable yt−d and zt−d for d = {1,2,3,4,5,6}.6 We choose the delay parameter for which rejection of linearity is the highest. For firms with multiple foreign listing,

6 The choice of the values of the delay parameter is similar to Koumkwa and Susmel (2008). We also perform the linearity test for several values of d > 6 and find that the rejection of linearity is much smaller.

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the linearity test is conducted on both yt−d and zt−d . If these variables are valid transition variables, then we re-conduct the linearity test on their linear combination Xt to select thereafter the appropriate STAR model. When the transition variables are well specified, STAR model is estimated by non-linear least square. 4.2. Empirical results In this sub-section we present empirical results related to linearity tests and model estimation. 4.2.1. Linearity test Table 8 presents empirical results related to ADF test, selected lag p and the linearity test. Results of the linearity test show the test statistic F and the corresponding p-value PF . The test statistic is asymptotically F (n, T − k − n − 1) distributed under the null hypothesis, where T is the number of observations, k is the number of unrestricted parameters and n is the number of restricted parameters. Rejection of linearity is the highest when p-value is the lowest or F is the highest. Table 8 shows that price deviation series are stationary in all cases and that our selected variables yt−d and zt−d are valid transition variables. In most cases, the delay parameter d takes the value 1 that is previous minute’s deviation from price parity. For dual-listed stocks, our results are consistent with transaction cost hypothesis and are in line with Koumkwa and Susmel (2008). For firms with multiple foreign listing, our results are also consistent with transaction cost hypothesis and we find in addition new empirical evidence that cross-listing (or multiple foreign listing) has a non-linear effect on the dynamic of price adjustment consistent with the limit to arbitrage hypothesis. Furthermore, results in Table 8 show that, for the pair “BIR, London-NYSE”, “INGA, Brussels-NYSE” and “RDSA, London-NYSE”, rejection of linearity is highest for the transition variable zt−d than yt−d . Columns 12 and 13 of Table 8 show high rejection of linearity for the transition variable Xt . For the majority of the firms, the LSTAR model is selected over the ESTAR model. 4.2.2. Estimation of STAR models In this section, we present the main results of the estimation of the STAR models by non-linear least square. STAR model are essentially governed by the transition parameters ˛ and . Terasvirta (1994) argue that convergence may not be reached in the estimation of STAR model since the joint estimation of ˛ and  is difficult. As proposed by Terasvirta (1994), we standardize the exponent of F by dividing it by  2 (y) the sample variance of yt . Estimation results of the STAR model are presented in Table 9 which reports the order of lags p selected using AIC, and transition parameters estimates. We also report the residuals diagnostic statistics which show the presence of heteroskedasticity and autocorrelation in several cases, thereby motivating the use of a robust covariance matrix. Despite the great difficulty of estimating the STAR model (LSTAR or ESTAR), convergence has been achieved for all listing pairs. Table 9 shows that transition parameters are highly significant in the majority of cases. The estimates of the smoothness parameters  are fairly small for most listing pairs which imply gradual transition between regimes. The only significant exception is the pair “RDSA, London-Amsterdam” for which  reaches a relatively high value suggesting a more abrupt type of switching.7 The estimation results of the STAR model show that ˛ is negative in several cases. This suggests that movement in the arbitrage regime occurs when price deviations are slightly higher than transaction costs or slightly more profitable than alternative opportunities. In other words, price deviations do not provide the strictly higher arbitrage profit. Our results show therefore that a STAR model (LSTAR or ESTAR) seems adequate for the price spread series in that transaction costs and cross-listing have a non-linear effect on price adjustments.

7 For the pairs “BIR, Irish–London”, “INGA, Brussels–NYSE” and “RDSA, Amsterdam–NYSE”, the smoothness parameters  reach also high values but the parameter is not significant.

Table 8 Linearity test. ADF test tˆ a

BIR Irish–London Irish–NYSE London–NYSE INGA Amsterdam–Bruxelles Amsterdam–NYSE Bruxelles–NYSE

−33*** −13.8*** −12.9*** −3.14*** −9.5*** −7.4***

dy

25 7 32 11 5 15

1 1 3 1 1 2

8 8 9

Fyt−d

L/Ec

PFy

dz

Fzt−d

PFz

ˇ1

ˇ2

FXt

PFX

31.6 17.9 738.7 5.8 36.1 20.3

<0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001

– – – – – –

– – – – – –

– – – – – –

– – – – – –

– – – – – –

– – – – – –

– – – – – –

L L L L L E

1 1 3

414.1 527.6 642.1

<0.0001 <0.0001 <0.0001

1 4 2

28.2 5.1 803.2

<0.0001 <0.0001 <0.0001

0.35 0.37 0.23

0.65 0.63 0.77

97.4 58.7 870.8

<0.0001 <0.0001 <0.0001

L L L

28 13 9

2 1 1

64.9 126.5 31.1

<0.0001 <0.0001 <0.0001

1 2 2

0.0087 <0.0001 <0.0001

0.69 0.31 0.25

0.31 0.69 0.75

15.5 100.9 29.1

<0.0001 <0.0001 <0.0001

L L L

t−d

3.05 34.3 46.8

t−d

t

RDSA London–Amsterdam London–NYSE Amsterdam–NYSE

−12.7*** −12.2*** −14.9***

6 21 9

1 1 1

716.9 135 1568.1

<0.0001 <0.0001 <0.0001

1 1 1

441.7 633.4 120.4

<0.0001 <0.0001 <0.0001

0.49 0.39 0.62

0.51 0.61 0.38

225.1 332.2 873.5

<0.0001 <0.0001 <0.0001

L L L

STM Milan–Paris Milan–NYSE NYSE-Paris

−25*** −15.1*** −8.9***

16 21 11

5 1 1

320.4 237.5 284

<0.0001 <0.0001 <0.0001

1 1 2

93.4 21.5 104.2

<0.0001 <0.0001 <0.0001

0.66 0.2 0.42

0.34 0.8 0.58

184.8 47.3 23.8

<0.0001 <0.0001 <0.0001

L L L

I. Ghadhab, S. Hellara / J. of Multi. Fin. Manag. 31 (2015) 126–145

Dual listed firms −10.8*** AGN −30.9*** BAMA −29.9*** EAD −22.1*** GSK −66.3*** TEF −23.9*** YRI Firms with multiple foreign listings

pb

a

tˆ is the test statistic of the ADF test distributed under the null hypothesis of non-stationarity, and Instead of the significance level of F statistic, we use the p-value which allows us to correctly choose the appropriate transition variable for which the rejection of linearity is the highest (for different values of d). c L and E refer respectively to LSTAR and ESTAR. b

***

Denotes the rejection of the null hypothesis at 1% level.

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Table 9 STAR models estimation. p

˛



ARCH(6)

AR(1–6)

Dual listed firms AGN

25

−0.002 (−4.2)***

0.02 (4.27)***

12.6 (0.0003)

0.99

BAMA

7

−0.017 (−14.8)***

0.002 (14.8)***

388.9 (<0.0001)

0.76

EAD

32

−1.01 (−26.9)***

0.13 (4.26)***

362.1 (<0.0001)

0.88

GSK

11

−0.084 (−2.29)**

1.01 (1.66)*

0.44 (0.5)

0.99

TEF

5

−0.52 (−28.7)***

0.009 (31.6)***

58 (<0.0001)

0.009

YRI

15

−0.006 (−0.97)

0.058 (3.68)***

1552 (<0.0001)

0.72

BIR Irish–London

8 8

London–NYSE

9

4.47 (0.58) 0.094 (0.64) 0.145 (2.56)***

1138.4 (<0.0001) 232.5 (<0.0001) 370.5 (<0.0001)

0.86

Irish–NYSE

0.28 (94.9)*** 0.92 (1.03) 0.2 (3.06)*** 0.037 (4.56)*** −0.015 (−2.34)*** −0.21 (−37)***

0.215 (4.38)*** 0.045 (5.03)*** 6.82 (0.21)

0.17 (0.67) 1.22 (0.26) 440 (<0.0001)

0.51

0.36 (120.2)*** −0.69 (−15.1)*** 0.21 (72.5)***

5.9 (3.18)*** 0.84 (2.47)*** 6.61 (0.37)

0.33 (0.56) 3.58 (0.058) 0.91 (0.33)

0.017

0.125 (7.1)*** 0.027 (5.11)*** 0.097 (1.39)

0.117 (3.7)*** 0.14 (2.5)*** 0.039 (3.83)***

175.5 (<0.0001) 157.4 (<0.0001) 16.5 (<0.0001)

0.3

Firms with multiple foreign listings

INGA Amsterdam–Brussels

28

Amsterdam–NYSE

13

Brussels-NYSE

9

RDSA London–Amsterdam

6

London–NYSE

21

Amsterdam–NYSE

9

STM Milan–Paris

16

Milan–NYSE

21

NYSE–Paris

11

0.065 0.86

0.0001 0.48

0.9 0.0008

0.22 0.0004

Robust standard errors are in parentheses below the corresponding parameter estimates. ARCH(6) reports the p-values associated with the standard test for heteroscedasticity. The last column report the p-values for the null hypothesis of no autocorrelation in the errors of order 1–6. * Denote significance at 10%. ** Denote significance at 5%. *** Denote significance at 1%.

To explicitly illustrate these dynamics of price convergence, panel A (panel B) of Fig. 3 shows the estimation transition functions for selected pairs plotted against the transition variable (against time). These figures visually support the non-linear adjustment of stock price and the choice of a STAR model as an appropriate specification for price deviation. In fact, panel A of Fig. 3 show an exponential

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143

Fig. 3. Transition functions illustration. This figure illustrates graphically the transition function plotted against the transition variable (Panel A) and against time (Panel B) for selected listing pairs for the following companies: Aegon, Telefonica, Yamana Gold, Bank of Ireland and StMicroelectronics.

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distribution (S-shaped) of observations around equilibrium which confirm the choice of ESTAR (LSTAR) model. Panel B of Fig. 3 show that arbitrage regime is attained several times during the sample period (the transition function is very close to unity) which correspond to a period when price deviations are profitable and the law of one price is more likely to hold. According to the limit to arbitrage hypothesis, this corresponds to period when price deviation is preferred among the alternative trading strategies. However, panel B of Fig. 3 show that for some pairs, like “BKIR, London-NYSE” and “Yamana gold”, there are few times of full arbitrage indicating that price deviations occur and persist. This may reflect limits to arbitrage for our sample. 5. Conclusion In this paper, we study the law of one price, arbitrage and price convergence of cross-listed stocks. We contribute to the pre-existing studies by considering a new sample stocks including firms with multiple foreign listing and so we provide a more completely study and better conclusive results about the efficiency of markets for cross-listed stocks. Our main results are as follow. First, the law of one price is violated and larger price deviations occur at the beginning and the ending of trading overlap which coincide with the opening of North American exchanges and the closing of European markets. Second, arbitrage opportunities exist and a trader may make a profit on dual-listed stocks as well as stocks with multiple foreign listing. We suggest that, while limits to arbitrage may exist in our case, they are minor, and arbitrage opportunities detected in our sample may be a reality in practice. Furthermore, the novelty of our sample stock provides a new opportunity to test the non-linear effect of cross-listing (or multiple foreign listing) on price adjustment. Based on transaction cost hypothesis and limit to arbitrage hypothesis, we show that the dynamic of price adjustments are non-linear and are correctly modeled by STAR models. To our knowledge, this the first evidence of the importance of simultaneously cross-listing and transaction cost in determining whether price adjustments are characterized by an arbitrage regime corresponding to the law of one price being valid and a non-arbitrage regime for which price deviations occur and persist. References Alsayed, H., McGroarty, F., 2012. Arbitrage and the law of one price in the market for American depository receipts. J. Int. Financ. Mark. Inst. Money 22, 1258–1276. Angel, J., Harris, L., Spatt, C.S., 2010. Equity Trading in the 21st Century, Available at http://papers.ssrn.com/sol3/papers. cfm?abstract id=1584026 Ansotegui, C., Bassiouny, A., Tooma, E., 2013. The proof is in the pudding: arbitrage is possible in limited emerging markets. J. Int. Financ. Mark. Inst. Money 23, 342–357. 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