Jumps, Cojumps, and Eciency in the Foreign Exchange Market

Transcription

1 Jumps, Cojumps, and Eciency in the Foreign Exchange Market Abstract Failing to control for intraweek patterns in volatility substantially biases the source of jumps and cojumps to be pre-scheduled macroeconomic news. Instead, jumps and cojumps occur when the market becomes increasingly one-sided culminating in large shocks to market state. Jumps and cojumps are preceded by pre-jump drifts with increasing cumulative abnormal returns, increasing quote volume, increasing illiquidity, increasing order ow, and decreasing information content of trade. Jump and cojump arrivals also display serial and cross-serial correlation, rejecting jump arrival independence. Finally, triangular arbitrage violations during jump events are studied and jump discontinuity serves as an important limit to arbitrage. JEL Classication: G12, G14, G15 Keywords: Jumps, Cojumps, Foreign exchange market, Market eciency

2 Jumps are rare discontinuous events in asset prices that have important implications for risk management (Boes, Drost, and Werker (2007), Christoersen, Jacobs, and Ornthanalai (2012), Eraker (2004), Jiang, Lo, and Verdelhan (2011), Johannes (2004), Kim, Oh, and Brooks (1994), Maheu and McCurdy (2004), and Merton (1976)) optimal portfolio allocation (Das and Uppal (2004), Jin and Zhang (2012), Liu and Pan (2003), and Liu, Longsta, and Pan (2003)), and the risk premium (Jiang and Yao (2014), Maheu, McCurdy, and Zhao (2013), and Yan (2011)). Despite the advances that this literature has made on the implications of jumps for portfolio theory, there have been relatively few empirical studies looking at the nature of jump events, the sources of jumps, and their eects on market eciency. This paper lls this gap in the literature by looking at market state and eciency surrounding jump and cojump events in the spot foreign exchange market, where a cojump event of order n is dened in this paper as n exchange rates experiencing jump events simultaneously. The foreign exchange market is ideal for studying the nature of jump events due to its twenty-four hour high-frequency trading environment which allows markets to freely trade with minimal frictions. This greatly reduces the potential of erroneous jump identication that is exacerbated by greater trading frictions (Aït-Sahalia, Mykland, and Zhang (2005), Zhang, Mykland, and Aït-Sahalia (2005), and Andersen, Bollerslev, and Dobrev (2007)). This paper identies jump events at the ve-minute frequency using the intraday jump identication methodology of Andersen, et al. (2007) and Lee and Mykland (2008). Intraweek patterns in volatility are controlled for by using the Boudt, Croux, and Laurent (2011) weighted standard deviation (WSD) estimator, which is a robust estimator of patterns in the diusion term of returns in the presence of jumps. Since previous literature has shown that foreign exchange rates follow intraweek patterns in volatility (Andersen and Bollerslev (1998), Baillie and Bollerslev (1991), and Harvey and Huang (1991)), controlling for these patterns is important so as to avoid falsely identifying a normal increase in volatility as a jump event. First, the propensity of jumps to arrive with pre-scheduled macroeconomic news announcements is examined after controlling for intraweek volatility patterns. A much larger high-frequency tickby-tick dataset of fourteen exchange rates is used in this paper than in previous studies with a news dataset that contains nearly 15,000 pre-scheduled macroeconomic news events from eleven economic regions during the January 2007 to December 2010 period. 8,551 news release times of the total news 1

3 sample are unique news release times. Evidence from other nancial markets has shown that in the moments surrounding pre-scheduled macroeconomic news releases, 90 percent of two-year treasury note jumps occur (Jiang, et al. (2011)) and many stocks experience signicant increases in jump intensity (Lee (2012)). Specically in the foreign exchange market, Andersen, Bollerslev, Diebold, and Vega (2003) show that exchange rates jump on key pre-scheduled macroeconomic new releases and Chatrath, Miao, Ramchander, and Villupurum (2014) show that news events generate 9-15 percent of jump arrivals. This paper nds that failing to control for intraweek volatility patterns erroneously identies a large number of jumps to occur with pre-scheduled macroeconomic news releases. After controlling for intraweek volatility patterns, identied jump events largely occur in the absence of pre-scheduled macroeconomic news releases. The conditional probability of a jump event in an individual exchange rate, conditional on a news event occurring ranges from a low of 0.40 percent to a high of 1.01 percent. Conditioning on specic nations' news releases, the probability of a jump event occurring in a specic exchange rate ranges from 0.00 percent to percent. Although news events generate jumps with low probability, the mean news jump magnitude is twice as large as the mean magnitude of no-news jumps. This lack of explanatory power that prescheduled macroeconomic news releases have for predicting exchange rate jump arrivals motivates the need to identify other causes for their occurrences. Event studies are used in this paper to test how Amihud (2002) illiquidity, quote volume, order ow, Hasbrouck (1991) information content of trade, return variance, and cumulative abnormal returns (CARs) vary surrounding jump and cojump events. Jumps and cojumps are associated with a pre-jump drift in CARs, greater quote volume, greater illiquidity, greater order ow in the direction of jump sign, and lower Hasbrouck (1991) information content of trades. News jumps and no-news jumps display diering characteristics. Leading up to a news jump, relative to no-news jumps, there is no pre-jump drift in CARs, liquidity is higher, quote volume is lower, and return variance is lower. Following a news jump, relative to no-news jumps, there is a post-jump drift in CARs in the direction of jump sign, liquidity is higher, quote volume is higher, informed trade is lower, and order ow in the direction of jump sign is higher. These diering patterns in state variables following news-jumps and no-news jumps are in agreement with the post-news patterns proposed by Tetlock (2010) when information asymmetries are resolved by public news releases. 2

4 The post-event drift following news-jumps on public information and the mild reversal following a no-news jump indicates that traders underreact to value-relevant public information and overreact to other price shocks. Similar post-news dynamics have recently been found in the stock market by Vega (2006) and Savor (2012). Next, cojump events in the foreign exchange market are examined. The presence of cojump events indicate that jump events can be systemic in nature. The unconditional probability of a cojump event is 0.28 percent and the probability of a trading day experiencing at least one cojump is 43 percent. The conditional number of cojump events, conditioned on a cojump day having occurred, is Similarly to the full jump sample, cojumps largely occur in the absence of pre-scheduled macroeconomic news announcements. Ex-ante, the probabilities of a news event generating a cojump event range from zero percent for a CJ(11) to 1.13 percent for a CJ(2). Expost, the probability of a pre-scheduled news event having ocurred conditioning on a cojump event having occurred ranges from a low of zero percent for a CJ (11)) to a high of percent for a CJ (8). Ex-post conditional news event probabilities, however, are generally increasing in cojump order. Illiquidity, quote volume, and order ow in the direction of jump sign are increasing in cojump order, while the information content of trade is decreasing in cojump order. This indicates that cojump events are the result of broad discount rate shocks in exchange rates. While jump events within exchange rates are rare events, jump events across the exchange rate space are substantially less rare. 99 percent of jump interarrival times in individual exchange rates are less than hours and 99 percent of jump interarrival times across the exchange rate space are less than hours. 99 percent of cojump interarrival times are less than hours. Jump events and cojump events also display clustering in excess of what should be observed in the commonly used constant jump intensity Poisson model of jumps. Under the null hypothesis of a constant jump intensity Poisson process, the cumulative 10 percent of jump interarrival times in individual exchange rates should be less than 4.75 hours. In fact, the observed jump cumulative distribution has a cumulative 10 percent jump interarrival time of less than 0.92 hours. Similar jump clustering is found across the exchange rate space as well as in cojump events. Under the null hypothesis of constant intensity Poisson arrivals, the cumulative 10 percent of interarrival times for any jump in the exchange rate space and of interarrival times for cojumps are 1.33 hours and 3

5 1.58 hours, respectively. However, in the observed distributions, the cumulative 10 percent of the interarrival times are less than 0.17 hours and 0.25 hours, respectively. These jump intensity results suggest that jump intensities should not only be modeled as autoregressive processes, as in Oldeld, Rogalski, and Jarrow (1977), Chan and Maheu (2002), and Maheu and McCurdy (2004), but also as correlated ones as proposed by Chan (2003). Since jump events only create market incompleteness if they are purely discontinuous in nature, the quote discontinuity of identied jumps is tested for. Mean maximum quote discontinuity during a jump event represents 25 percent of mean jump magnitude. However, quote discontinuity varies during news jumps and no-news jumps. Mean quote discontinuity, as a fraction of mean jump magnitude, is 18 percent for no-news jumps and it is 50 percent for news jumps. Cojumps also exhibit weak evidence of increased quote discontinuity compared to the full jump sample with the average maximum quote discontinuity of cojumps being 50 percent greater than that of the full jump sample. Finally, the eects that jump and cojump events have on triangular arbitrage prots are tested for. Liu, Longsta, and Pan (2003) model jump discontinuity as an illiquidity cost that forms a limit to arbitrage. Jump discontinuity is found to be a signicant limit to triangular arbitrage in this paper. Mean magnitudes of triangular arbitrage violations are 47 percent larger during jump events than normal times. Within the jump event subsample, there is weak evidence that news related jump events and cojump events further exacerbate triangular arbitrage violations. For those with signicant dierences, arbitrage prots during news jump events are 57 percent greater than during no-news jump events and arbitrage prots during cojump events are 37 percent greater than in the full jump event sample. Further tests reveal, however, that it is not the jump event per-se that leads to a limit to arbitrage, but rather the quote discontinuity and illiquidity associated with jump events that lead to a limit to arbitrage. In fact, identied jump events that are primarily driven by an increased demand to trade are associated with smaller arbitrage prots. The remainder of the paper is organized as follows. Section I presents the dataset. Section II presents results from jump event studies. Section III presents results from cojump event studies. Section IV examines quote discontinuity during jump events and triangular arbitrage violations during jump events. Section V contains concluding remarks. 4

6 I Data Tick-by-tick quote data for exchange rates is obtained from Gain Capital 1 for the following exchange rates: AUD/JPY, AUD/USD, EUR/GBP, EUR/USD, GBP/JPY, GBP/USD, USD/CHF, USD/JPY, USD/CAD, NZD/USD, CHF/JPY, EUR/AUD, EUR/CHF, and NZD/JPY covering the sample period of January 1, 2007 to December 31, Exchange rates are quoted such that they represent the price of one unit of the rst currency in units of the second currency. Therefore, the second currency is viewed as the domestic currency and the rst one is viewed as the foreign currency. Each observation contains an exchange rate identier, date, time-stamp (to the second), bid rate, and ask rate. Bid and ask rates are rm meaning that if an order arrives, then the dealer must transact at the posted rates. Since there is negligible foreign exchange activity during the weekend period, observations from 17:00 ET on Friday to 22:00 ET on Sunday are dropped from the sample. Observations on days containing a bank holiday in the United States, United Kingdom, or Japan are also dropped from the sample. [ Insert Figure 1 about here ] Data on pre-scheduled news events is obtained from FXStreet.com 2. News observations contain the time of release, country of origin, and name of economic indicator or event. If the news release is an economic indicator, then there are further elds stating the actual value, the consensus estimated value, and the economic indicator's previously recorded value. In addition to economic indicators, pre-scheduled publicized speeches of key economic personnel are included since these events often allude to future economic policy that the market pays close attention to. In total, there are 14,764 news observations from eleven nations for the January 1, 2007 to December 31, 2010 period. Table A1 of Appendix A presents summary statistics on the sample of scheduled macroeconomic news releases. Panel A displays the number of news observations by country of origin. The largest number of pre-scheduled news releases originates in the United States at 3,763, representing percent of the sample, while the fewest number originates in France at 6, representing 0.04 percent of the sample. In many cases multiple economic indicators are released by a country simultaneously. Panel

7 B presents statistics on the number of unique news release times. In total, there are 8,551 unique news release times. Panel C tabulates the number of observations from each economic indicator. No single indicator makes up more than 2.55 percent of the news sample. Figure 1 plots the intraday distribution of news releases. Approximately 17.5 percent of news releases occur at 08:30 ET, representing the largest volume. Notable news release volumes also cluster at 10:00 ET, 04:00-05:00 ET, and 19:00-21:30 ET. The 04:00-05:00 ET and 19:00-21:30 ET periods are concurrent with the openings of European and Asian trading centers, respectively. Between 11:00 ET and 18:00 ET is largely absent of scheduled news releases. Since many of the news releases occur concurrently with other releases by the same country, for the remainder of the paper only unique pre-scheduled news release times by each country are used. II Jump and Cojump Events A Jump Events The jump detection methodology of Andersen, et al. (2007) and Lee and Mykland (2008) is used in this paper to detect intraday jumps at the 5-minute frequency. Using a 5-minute frequency is a good compromise between retaining the information content of high-frequency data and avoiding microstructure biases that sampling more frequently would result in (Aït-Sahalia, Mykland, and Zhang (2005), Zhang, Mykland, and Aït-Sahalia (2005), and Andersen, Bollerslev, and Dobrev (2007)). A jump is identied if a 5-minute period's standardized absolute return is in excess of the 99.9 percent critical level of the standard normal distribution. Under the null hypothesis of the jump detection test, there should be a 0.1 percent unconditional probability of a jump occurring. Intraweek patterns in volatility, the diusive component of exchange rate returns, is controlled for by using the weighted standard deviation (WSD) estimator of Boudt, et al. (2011). They show that the WSD estimator is a robust measure of the diusive component of returns in the presence of jumps. Appendix B outlines the jump detection methodology and the WSD estimator is described in Appendix C. All jump and cojump statistics presented are stochastic volatility robust estimates unless otherwise noted. 6

8 A.1 Jump Statistics Table I presents jump statistics. Exchange rates experience from 331 to 656 jumps over the sample period. Unconditional jump probabilities range from 0.13 percent for the CHF/JPY pair to 0.25 percent for the EUR/CHF pair. Mean absolute jump sizes are presented in the eighth column. Mean jump magnitudes range from a low of 0.14 percent for the EUR/CHF to a high of 0.44 percent for the NZD/JPY. Jumps are distributed symmetrically as indicated by the jump return means being close to zero. Column ten presents the unconditional probability that at least one jump will occur in a day. Exchange rates experience jumps on 271 to 455 days, presented in column ve, or to percent of trading days. The CHF/JPY has the fewest number of jump days at 271 and the EUR/CHF has the greatest number of jump days at 455. The expected number of intraday jumps, conditional on at least one jump having occurred on a given day, are presented in the second to last column. Expected intraday jump frequencies, given at least one jump occurs, range from for the EUR/AUD to for the EUR/CHF currency pair. [ Insert Table I about here ] Figure 2 plots the intraday distribution of identied jump events without controlling for intraweek volatility patterns in the top panel and controlling for intraweek volatility patterns using the WSD estimator in the bottom panel. Clearly, failing to control for intraweek volatility patterns severely biases the identication of jump events to occur surrounding the release of pre-scheduled macroeconomic news. Once intraweek volatility patterns are controlled for, the intraday distribution of identied jumps appears to be more uniformly distributed. There are a slightly greater number of jumps occurring from 21:00 ET on day (t 1) to 02:00 ET on day t and slightly fewer jumps occurring from 10:00 ET to 16:00 ET. An above average frequency of jumps, 1.2 percent, continues to be observed at 08:30 ET even after adjusting for the intraweek patterns in volatility. Figure 3 plots the time series of the fraction of daily realized variance attributable to intraday jump variance, averaged across exchange rates. Intraweek patterns in volatility are not controlled for in the top panel and the patterns in intraweek volatility are controlled for in the bottom panel with the WSD estimator. Failing to control for intraweek patterns in volatility also biases the magnitude of identied jumps upward. Generally, jumps make up from 4-70 percent of daily realized variance. 7

9 This fraction tends to be fairly stable over the sample period. [ Insert Figure 2 and Figure 3 about here ] A number of previous studies have shown that jumps largely tend to occur with pre-scheduled news events without explicitly controlling for intraweek patterns in volatility, including Jiang, et al. (2011) in the market for treasury securities, Lee (2012) and Maheu and McCurdy (2004) in the stock market, and Andersen, et al. (2003) in the foreign exchange market. Table II presents the probabilities of a jump occurring in response to a pre-scheduled news announcement in each exchange rate, controlling for the bias in jump identication that stochastic volatility can create. These probabilities are the ex-ante probabilities that news events will generate jumps. Generally, contrary to previous ndings in the literature, the probability of a jump occurring in an individual exchange rate in response to a news event is less than one percent in Table II. These small probabilities indicate that pre-scheduled news events largely do not generate jumps. Notable exceptions are the impacts that U.K. and Australian news have on GBP and AUD exchange rates, respectively. The probabilities of jump events, conditioned on a U.K. news event, in the EUR/GBP exchange rate and GBP/USD exchange rates are 3.14 percent and 3.41 percent, respectively. Conditioned on an Australian news event occurring, the probabilities of a jump in the AUD/JPY, AUD/USD, and EUR/AUD are 5.95 percent, 7.57 percent, and 6.59 percent, respectively. [Insert Table II and Table III about here] Andersen, et al. (2003) shows that pre-scheduled macroeconomic news may not be a good predictor of jump events since many news releases provide redundant information based on the time of the month in which they are released. Table III presents conditional probabilities of how likely a news event occurred conditional on a jump event having occurred. These probabilities reveal ex-post how important news events were in generating jumps, in contrast to Table II which shows ex-ante how well news events predict jumps. U.S. news is revealed to be the most likely news to generate a jump with conditional probabilities ranging from 1.68 percent to 4.61 percent. Australian news is revealed to be the most important news for Australian exchange rates with the conditional probabilities that a news event occurred, given that a jump occurred in the AUD/USD 8

10 and EUR/AUD, being percent and percent, respectively. The nal column of Table III shows that even conditioning on a jump event having occurred, news is unlikely to be the identied source of the jump ex-post. News jumps, jumps generated from a news event occurring in any nation in the same ve-minute period as a jump event, as a fraction of the full jump sample range from a low of 6.86 percent for the EUR/USD exchange rate to a high of percent for the NZD/JPY exchange rate. Together, Tables II and III motivate the need to identify other sources for the occurrences of jumps in exchange rates. A.2 Jump Event Studies This section presents event study tests of market state surrounding jump events where the event is the occurrence of an exchange rate jump. The previous twelve 5-minute periods and the following twelve 5-minute periods are included for a total event window size of twenty-ve 5-minute periods. Market state variables examined in this paper include cumulative abnormal returns (CAR) signed in the direction of jump sign, Amihud (2002) price impact illiquidity (P I), quote volume (QV ), return variance (V AR), Hasbrouck's (1991) contribution of informed trade to return variance ( S2XW ), and order ow (OF ) signed in the direction of jump sign. These state variables are chosen since the literature has shown that they have important eects on market eciency and price discovery. OF and S2XW are included since Evans and Lyons (2008) and Love and Payne (2008) show that order ow in the foreign exchange market contributes signicantly to price discovery. Menkveld, et al. (2012), Pasquariello and Vega (2009) and Locke and Onayev (2007) also nd that order ow plays an important role for price discovery in the U.S. Treasury bond market and S&P 500 futures prices, respectively. QV and P I are included to test how trading activity and illiquidity vary around jump events, since Mancini, Ranaldo, and Wrampelmeyer (2013) shows that illiquidity shocks can be signicant in the foreign exchange market. Formal denitions of P I, S2XW, and OF are given in Appendix D. When calculating abnormal returns, a constant-mean return model is used. Abnormal returns for exchange rate i are dened to be AR i,t+j = r i,t+j E T [r i ] (1) 9

11 where r i,t+j is the 5-minute log mid-quote revision and E T [ ] denotes the expected value, conditioned on the full sample of data. is the frequency of discretely observed intraweek returns ( =1/2016 in this paper), t denotes the week, and j = {1, 2,..., 1 }. Since P I, QV, V AR, S2XW, and OF display intraweek patterns, each is standardized to have a mean of zero and variance of one for each intraweek period x i,t+j = (x i,t+j E T [x i,j ]) VT [x i,j ] (2) where x {P I, QV, V AR, S2XW, OF } and V T [ ] is the sample variance operator. [ Insert Table IV about here ] Jump event study results are presented in Table IV. Serving as the control sample, standardized state variables are approximately equal to zero at all event window dates when there is not a jump event. At the jump date, P I increases to , QV increases to , V AR increases to , S2XW decreases to , and OF increases to All state variables except for P I display pre-jump drifts. QV increases monotonically from the start of the event window to the event date, increasing from to V AR increases from at the beginning of the event window to in the period prior to the event date. From the start of the event window to the period prior to the event date, OF increases from to S2XW has a mild pre-jump drift, decreasing from to Both QV and V AR mean-revert to normal levels slowly following a jump and continue to remain at heightened levels at the end of the event window. In the moments following a jump event, P I decreases to and remains lower than normal for the remainder of the event window, indicating that there is improved liquidity following a jump. OF and S2XW are and in the period following the event date and both also remain at lower than normal levels for the remainder of the event window, providing further evidence that order ow tends to originate from uninformed liquidity providers following jumps. There is mild evidence of a pre-jump drift in CARs in the direction of jump sign, indicating that jump events may be anticipated. CARs increase from 0.00 percent to 0.07 percent immediately prior to a jump event. While this may be a small amount in other nancial markets, this is an 10

12 economically meaningful amount in the foreign exchange market where heavy leverage is often used. Evidence that jumps represent permanent innovations to traders' information sets is provided by the absence of post-jump reversals in CARs. Mean CARs mildly decline from 0.33 percent at the event date to 0.31 percent at the end of the event window. [ Insert Table V about here ] A pre-jump drift in CARs can occur for information-reasons, such as private information about an upcoming news announcement, as well as for information-unrelated reasons, such as market onesidedness. The patterns in state variables are expected to dier between news jumps and no-news jumps. Pre-scheduled macroeconomic news releases resolve asymmetric information in exchange rates. Tetlock (2010) shows that following when asymmetric information is resolved, returns are serially correlated, high volume news is a better predictor of returns than low volume news, trading volume and return variance are higher, and lastly that illiquidity and informed trade decreases. Vega (2006) and Savor (2012) provide empirical evidence that when public news has less impact on resolving asymmetric information and when large price changes are not information-related, the less information-related price changes display no drift to a modest return reversal. Based on this extant evidence, expectations are that following a news jump, relative to a no-news jump, there will be a larger post-jump drift in CARs, QV will be higher, OF will be higher, V AR will be higher, P I will be lower, and S2XW will be lower. Comparisons of news jumps and jumps that occur in the absence of news, no-news jumps, are presented in Table V. A news jump is dened as a jump that occurs in the same 5-minute period as the release of a pre-scheduled macroeconomic news event from any country in the sample. Jumps are conditioned on news originating from any country in the sample since Table II and Table III show that jumps in exchange rates occur in association with news, regardless of nation of origin. News jump magnitudes are greater than no-news jump magnitudes by 17 pips (1 pip = ), or percent greater. There is no evidence of a pre-jump drift in CARs prior to news jumps rejecting the hypothesis that traders possess private information about pre-scheduled macroeconomic news. However, the pre-jump drift continues to be observable for no-news jumps. CARs following news jumps increase from 0.42 percent at the event date to 0.47 percent at the end of the event window 11

13 indicating that news jumps do not fully adjust to innovations in the public information set. This post-event drift is consistent with information being impounded into prices indirectly through order ow (Evans and Lyons (2008) and Love and Payne (2008)) with dealers that are Bayesian updaters as in Glosten and Milgrom (1985). In the period before a no-news jump, the mean CAR is 8 pips. A slight negative post-jump drift is observed following no-news jumps providing evidence that the returns to liquidity provision following a no-news jump is about 0.03 percent for a one hour holding period. P I levels at the event date for news jumps and no-news jumps are approximately the same; however, pre-jump P I for news jumps is lower than pre-jump P I for no-news jumps. Whereas P I is in the ve minutes prior to a news jump, P I is in the ve minute period prior to no-news jumps. QV, however, is larger during news related jumps than no news jumps, indicating a greater shock to the demand to trade during news jumps. QV is during news jumps and it is during no-news jump events. QV, however, is lower prior to news jumps than no-news jumps indicating that traders take a wait-and-see approach prior to pre-scheduled news. News jumps also have a smaller informed trade contribution than no-news jumps. Whereas S2XW is at the news jump event date, it is at the no-news jump event date consistent with the release of pre-scheduled macroeconomic news resolving asymmetric information. Following a news jump, P I and S2XW are lower than in the no-news jump case. P I increases from to in the post news jump event window and S2XW increases from to in the post news jump event window. The changes in P I and S2XW in the post no-news jump event window are respectively to and to That the information content of trades is lower following news jumps than following no-news jumps is in contrast to the ndings by Green (2004) that the information content of trades in government bonds is greater following the release of macroeconomic news announcements. QV and OF are larger following news jumps than during no-news jumps. QV decreases from (1.6125) to (0.1616) for news jumps (no-news jumps) and OF increases from ( ) to ( ) for news jumps (no-news jumps). In contrast to expectations, V AR is lower following a news jump than it is following a no-news jump. 12

14 B Cojump Events This section tests for cojumps in the foreign exchange market. Cojumps have been shown to have important implications for optimal portfolio allocation (Das and Uppal (2004) and Jin and Zhang (2012)) as well as for the optimal use of leverage (Das and Uppal (2004). A cojump event is dened in this paper as two or more exchange rates simultaneously experiencing jump events CJ t+j (n) = 1 ( 14 ) 1 (κ q,t+j 0) = n q=1 (3) where 1 (x) is the indicator function which is equal to one if x is true and equal to zero otherwise. κ q,t+j is the identied jump return whose precise denition is given in Appendix B. Unconditional cojump probabilities are given by P{CJ (n)} = ( T 1) T i=1 ( 14 ) 1 (κ q,i 0) = n q=1 (4) P{CJ (n)} denotes the unconditional probability of n exchange rates experiencing jumps simultaneously. B.1 Cojump Statistics Cojump statistics are presented in Table VI. Cojump order is indicated in the rst column. The highest cojump order identied during the sample period is a CJ(11). Unconditionally, the probability of a cojump order occurring decreases from 0.28 percent for a CJ(2) to less than 0.01 percent for a CJ(11). The number of cojump days for a given cojump order decreases from 487 days for a CJ(2) to 1 day for a CJ(11) with associated probabilities of percent and 0.09 percent. The expected number of cojumps conditioned on a cojump day occurring range from 1 for a CJ(11) to 1.48 for a CJ(2). [ Insert Table VI about here ] News cojump probabilities are presented in the nal two columns. The ex-ante probability of news generating a cojump of order n, presented in the second to last column, decreases from

15 percent for a CJ(2) to less than 0.01 percent for a CJ(11). These low probabilities indicate that systemic jump events generally are not generated by pre-scheduled macroeconomic news events. Ex-post probabilities of a news event having occurred conditioned on a jump having occurred, presented in the nal column, range from a low of zero percent for a CJ(11) to a high of percent for a CJ(8). Generally, however, the ex-post probability of a pre-scheduled macroeconomic news event having occurred conditioned on a jump event having occurred is increasing in cojump order. These probabilities, however, are still low. Table VI provides evidence that jump risk can be systemic in nature and that pre-scheduled macroeconomic news releases generally are not the source of cojump events. [Insert Figure 4 about here] Figure 4 plots the intraday distribution of CJ( 2) events. Intraweek volatility patterns are not controlled for in the top panel and the WSD estimator is used in the bottom panel to control for these patterns. Failing to control for volatility patterns results in too many cojump events being identied coinciding with pre-scheduled macroeconomic news releases. Once the volatility patterns are controlled for, the distribution of cojump events is more uniformly distributed. Above average probabilities of cojump arrival continue to be observed at 04:30 ET, 08:30 ET, 19:30 ET, and 21:30 ET, coinciding with dense pre-scheduled macroeconomic news release times, however. B.2 Cojump Event Studies Table VII presents event studies of CJ( 2) events. The average CAR leading up to a cojump event is 0.07 percent indicating that cojump events may also be anticipated by the market. Following a cojump event, mean CARs reverse by 0.03 percentage points. Mean jump magnitudes are approximately the same in the cojump subsample as in the full jump sample. P I, QV, and V AR across exchange rates involved in a cojump event are also similar to those of the full jump sample at , , and , respectively. Pre-cojump and post-cojump drifts in the state variables are also similar to as in the full jump sample. [ Insert Table VII about here ] 14

16 In order to study the heterogeneity in market state for varying cojump orders, Figure 5 plots event studies of market state by cojump order. P I, QV, S2XW, and OF are presented in the top-left panel, top-right panel, bottom-left panel, and bottom-right panel, respectively. None of the cojump orders display a pre-cojump drift in P I; however, P I is positively related to the order of exchange rate cojump. P I during a cojump event increases from for a CJ(2) to for a CJ(8). This pattern could arise from common illiquidity shocks as in Mancini, et al. (20130) or from illiquidity contagion as in Cespa and Foucault (2014). All cojump orders experience signicant improvements in liquidity in the ve-minute period immediately following a cojump. P I in the 5- minute period immediately following a cojump ranges from a low of for a CJ(4) to a high of for a CJ(7). [ Insert Figure 5 about here ] Cojump order is also increasing in QV. QV increases from for a CJ(2) to for a CJ(8). The pre-cojump drift in QV is observed in all cojump orders. Following a cojump event, QV decreases, although remains at above average levels through the end of the event window. QV gradually decreases and decreases more slowly for cojumps of higher order than for those cojumps of lower order. One hour following a cojump event, QV for a CJ(8) is , whereas it is for a CJ(2). The information content of trades and cojump order are negatively related. S2XW is for a CJ(2) and decreases to for a CJ(8). Cojumps of higher order are less information driven. The bottom-right panel shows that cojump order is positively related to OF, signed in the direction of jump sign, at the event date. OF at the event date increases from for a CJ(2) to for a CJ(2). The increase is not monotonic, however. OF reaches a high of for a CJ(7). In the 5-minute period following a cojump event, order ow is contrarian and the level of contrarianism is increasing in cojump order. In the 5-minute period following a cojump, OF decreases from for a CJ(2) to for a CJ(8). These results indicate that systemic jump events are followed by greater liquidity provision and less informed trade in the following hour. 15

17 III Jump and Cojump Clustering and Determinants A Jump and Cojump Clustering Extant asset pricing literature generally assumes that jumps follow independent Poisson processes with constant jump intensity parameters. Under this assumption, the interarrival time of jumps is exponentially distributed and clustering jump events are not permitted. In contrast to this, more recently, Maheu and McCurdy (2004) show that jump intensity should be modeled as an autoregressive process in the stock market to model jump clustering. Figure 6 plots the distribution of intra-exchange rate jump interarrival times, the distribution of inter-exchange rate jump interarrival times, and the distribution of cojump arrival times. Intraexchange rate jump arrival times are plotted in the top-left panel. The exponential distribution is also plotted to provide evidence of how well the constant jump intensity Poisson process models the empirical distribution. Maximum likelihood is used to estimate the intensity parameter of the exponential distribution. For the full sample of exchange rates, the intensity parameter of intra-exchange rate jump arrivals is estimated to be Jump clustering is apparent from the abnormally large frequency of jumps that occur within one hour of each other. This is consistent with the notion that jump intensities in the foreign exchange market should be modeled as autoregressive processes. For interarrival times in excess of two hours, the Poisson distribution appears to model jump arrivals well. [ Insert Figure 6 about here ] The top-right panel of Figure 6 plots the distribution of jump interarrival times across the exchange rate space. Whereas 99 percent of intra-exchange rate jump interarrival times are less than hours, 99 percent of jump interarrival times are less than hours across the exchange rate space. Jump events may be rare in individual exchange rates, but they are substantially more common in the space of exchange rates. The exponential distribution associated with the interarrival times of jumps in the exchange rate space is also plotted and the intensity parameter is estimated to be Signicant jump clustering is observable in the exchange rate space as well. There is a greater density of interarrival times that are shorter than would be present in a compound Poisson 16

18 process with constant intensity parameter. Cojump interarrival times are plotted in the bottom-left panel of Figure 6 along with the distribution of interarrival times implied by the exponential distribution. The intensity parameter of a cojump arrival is estimated to be Similarly to the other series, cojumps display signicant jump clustering. In summary, Figure 6 presents evidence that jump clustering is an important consideration for risk-management. Jump intensities in the foreign exchange market should not only be modeled as autoregressive processes, but as correlated processes as well. B Jump and Cojump Determinants This section formally tests for the impact that market state has on the probabilities of jump and cojump events occurring. The following Probit model is estimated κ i,t+j = β 0 + β 1 CAR (12) i,t+(j 1) + β 2NEW S t+j + β 3 P I i,t+(j 1) + β 4 BV i,t+(j 1) (5) + β 5 QV i,t+(j 1) + β 6 S2XW i,t+(j 1) + β 7 OF i,t+(j 1) 4 h + β 7+q CAR (12) i,t+(j 1) x i,t+(j 1) + β 11+q 1 ( κ i,t+(j q) 0 ) + q=1 g β 11+hi +z1 ( CJ t+(j z) (n) > 1 ( κ i,t+(j z) 0 )) + ε i,t+j z=1 where x {P I, OF, S2XW, NEW S}. One-period lagged state variables are used so that they are contained in traders' information sets. CAR (12) i,t+(j 1) q=1 is the one-hour cumulative abnormal return from time t + (j 12) to time t + (j 1), NEW S t+j is a dummy variable equal to one if a pre-scheduled macroeconomic news event occurs at time t + j and is equal to zero otherwise, and BV i,t+j is the trading day's realized bipower variation for exchange rate i. P I, QV, S2XW, and OF are dened as before. CAR and OF are not signed in the direction of jump sign for these tests. Interaction terms with CAR are included to test if the observed pre-jump drift in CARs is information-related or information-unrelated. The last two terms in eqn. (5) are lagged own jump dummy variables and lagged cojump dummy variables. These are included to capture the explanatory power of jump-clustering within-exchange rates and jump clustering across the 17

19 exchange rate space for predicting exchange rate jumps. Lag lengths are chosen optimally using the Campbell and Perron (1991) methodology. First, the own jump lag length is set at 12 and iterated down until the largest lag length is signicant at the one percent level and then, keeping the optimally found own jump lag length xed, the cojump lag length is set at 12 and iterated down until the largest lag length is signicant at the one percent level. Coecient estimates from eqn. (5) can be interpreted as the increase in z-score of jump probability given a one unit increase in explanatory variable. [ Insert Table VIII about here ] Table VIII presents the results from estimating eqn. (5). Column headers denote subsamples of positive and negative jumps. Columns two and three use the full jump sample. N EW S enters signicantly increasing the probability of a jump occurring. Pre-scheduled macroeconomic news events increase the z-score of jump probability by for negative jumps and by for positive jumps. P I, QV, S2XW, and OF also signicantly increase the probability of a jump event. Jump probability z-score is increased by 0.032, 0.080, 0.011, and for one standard deviation increases in P I, QV, S2XW, and OF, respectively, for positive jumps. Market state eects on negative jump probabilities are similar. CAR signicantly increases the probably of a jump occurring, and in the same direction, as the CAR. A unit increase in CAR increases the jump probability z-score by for positive jumps. The signicant coecients on CAR can arise for information-related and information-unrelated reasons. An information hypothesis is that a subset of traders posses privately leaked information regarding a news event that occurs. An information-unrelated hypothesis is that the market becomes increasingly one-sided until a liquidity shock causes a jump or cojump event (Mancini, et al. (2013), and Cespa and Foucault (2014). The CAR interaction terms attempt to distinguish between these two hypotheses. Looking at positive jumps events, the CAR interaction terms with OF and S2XW signicantly increase the probability of a positive jump. CAR interaction terms with P I and N EW S decrease the probability of a positive jump event. These results indicate that exchange rates partially adjust to incoming pre-scheduled macroeconomic news prior to the release time and that jumps occur with greater probability when the market becomes more one-sided with informed 18

20 trade about order ow. Results are qualitatively similar for negative jumps. Signicant evidence of autocorrelation in jump intensity as well as cross-exchange rate jump dependence is provided in the optimally chosen lag lengths for lagged own jumps and lagged cojumps denoted by JLAGS and CJLAGS, respectively. Estimated JLAGS and CJLAGS coecients are not presented to conserve space. In the case of positive jumps, exchange rate i's jump history up to 25 minutes in the past continues to have explanatory power for a jump in exchange rate i in the current period. Jumps anywhere in the exchange rate space have a more long-lived dependence structure. A jump event anywhere in the sample of exchange rates (excluding a jump in exchange rate i if one occurred) for up to 45 minutes in the past continue to have explanatory power for predicting a jump event in the current period for exchange rate i. The dependence structure for negative jumps is slightly longer-lived with optimal JLAGS and CJLAGS being 5 and 11, respectively. Columns four and ve of Table VIII restrict jump events to only news jumps. CAR enters weakly signicantly for positive news jumps only. P I and S2XW are no longer signicant. QV continues to enter signicantly and with coecient values that are relatively unchanged from the full jump sample. OF is not signicant for negative news jumps, but enters with a negative sign for positive news jumps. None of the CAR interaction terms enter signicantly which is consistent with the lack of a pre-jump drift in CARs for news jumps. The dependence structure between current jumps and past within-exchange rate jumps and jumps anywhere in the exchange rate space are dramatically dierent for news jumps. Now, the optimal JLAGS and CJLAGS for positive jumps is one and one, respectively, for positive news jumps. None of the lag lengths are signicant in the case of negative news jumps. These news jump results indicate that the occurrence of news jumps is largely unconditional on market state and the history of jump events. Probit results for jumps that are part of a cojump event are presented in the nal two columns of Table VIII. NEW S, P I, QV, S2XW, CAR in the direction of jump sign, and OF in the direction of jump sign increase the probability of a cojump occurring. The signs and magnitudes of these market state variables are of similar sign and magnitude to those in the full jump sample with the exception of NEW S. NEW S has a larger eect on cojump probability than it has in the full jump sample. N EW S increases the z-score of cojump probability by for negative jumps during 19

21 cojump events and by for positive jumps during cojump events. The dependence structure with lagged cojumps for jumps that are part of a cojump event are longer-lived than those of the full jump sample. CJLAGS is optimally chosen to be 12 for positive and negative jumps that are part of a cojump event. [ Insert Table IX about here ] Exchange rates are partitioned into a USD exchange rate sample and a cross exchange rate sample in Table IX. Eqn. (5) is estimated on each sample to test how jump determinants dier for USD rates and cross-rates. CAR is more important in determining USD exchange rate jump probabilities than it is in determining jump probability in cross rates. In the USD and cross-rate sample, the coecients on CAR are consistent with its pre-jump drift. NEW S and QV are slightly more likely to generate a jump in cross-rates than in USD exchange rates. S2XW and OF in the direction of the jump are slightly more likely to generate a jump in USD exchange rates than they are in cross-rates. P I is more likely to generate a jump in the USD sample than in the cross-rate sample. As in Table VII, the interaction terms of CAR with OF and S2XW increase the probability of jump events. This eect is stronger in the USD sample than in the cross-rate sample. NEW S interacted with CAR decreases the probability of a jump event when CAR is in the direction of jump sign by a larger amount in the USD sample indicating that USD rates partially-adjust to future pre-scheduled macroeconomic news by a larger amount than cross rates do. JLAGS and CJLAGS indicate that the dependence structure of jump arrivals on past jump arrivals diers for USD rates and cross rates. The dependence structure is longer-lived for cross rates than for USD exchange rates. IV Jump Events and Market Eciency A Quote Discontinuity Discontinuous jumps form a market incompleteness that cannot be hedged. Panel A of Table X presents magnitudes of quote discontinuity during jump events as measured by the mean maximum quote revision during the jump event, dened as max{ ln (p τ /p τ 1 ) }, where here τ indicates quote 20

22 time. The mean maximum quote revision across exchange rates is 2.07 pips (1 pip equals ) in normal times and it is 6.87 pips during jump events. Columns ve to ten restrict the sample to include only jump event observations and tests how quote discontinuity changes with news jump events and cojump events. News jumps are more discontinuous than no-news jumps. Whereas the mean maximum quote revision across exchange rates is 4.83 pips for no-news jumps, it is pips for news jumps. For all exchange rates, the mean maximum quote discontinuity is signicantly greater for news jumps than for no-news jumps. Quote discontinuity dierences between cojump events and jump events are smaller. Mean maximum quote discontinuity across exchange rates is 7.58 pips for cojumps, 2.63 pips greater than for jumps. Further, cojump quote discontinuity is only statistically signicantly greater than jump quote discontinuity in eight out of the fourteen exchange rates. [ Insert Table X about here ] Panel B presents the mean fraction of jump return that is discontinuous for each of the exchange rates. All jumps are considered in the second column, the sample is restricted to no-news jumps only in the third column, and the sample is restricted to news jumps only in the nal column. Mean quote discontinuity ranges from percent to percent of total jump return. The CHF/JPY has the least discontinuous identied jumps and the EUR/AUD has the most discontinuous identied jumps. News jumps are generally more than twice as discontinuous as no-news jumps. No-news jump discontinuity as a fraction of jump size ranges from a low of percent for the AUD/JPY rate to a high of percent for the EUR/CHF rate. News jump discontinuity as a fraction of jump size ranges from a low of percent for the CHF/JPY rate to a high of percent for the EUR/GBP rate. B Triangular Arbitrage and Jump Events Since identied jumps have a large discontinuous component, this discontinuity serves as an illiquidity cost that limits arbitrage as in the model of Liu, et al. (2003). Table XI presents triangular arbitrage prots during normal times, during all jump events, during news jump events, and during cojump events. The sample is further partitioned into two halves: one containing the 21