Market Analysis of Foreign Exchange Rate Movements based on Conditional Probability

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Quantitative Finance

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Random walk or a run. Market microstructure analysis of foreign exchange

rate movements based on conditional probability

Yuko Hashimoto^a; Takatoshi Ito^b; Takaaki Ohnishi^c; Misako Takayasu^d; Hideki Takayasu^e; Tsutomu Watanabe^f

a Statistics Department, International Monetary Fund, N.W. Washington, DC 20431, USA ^b Faculty of Economics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan ^c The Canon Institute for Global Studies, Bunkyo-ku, Tokyo 113-0033, Japan ^d Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Midori-ku, Yokohama 226-8502, Japan ^e Sony Computer Science Laboratories, 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan ^f Institute of Economic Research, Hitotsubashi University, Kunitachi-city, Tokyo 186-8603, Japan

First published on: 13 December 2010

To cite this Article Hashimoto, Yuko , Ito, Takatoshi , Ohnishi, Takaaki , Takayasu, Misako , Takayasu, Hideki and Watanabe, Tsutomu(2010) 'Random walk or a run. Market microstructure analysis of foreign exchange rate movements based on conditional probability', Quantitative Finance,, First published on: 13 December 2010 (iFirst)

To link to this Article: DOI: 10.1080/14697681003792237 URL: http://dx.doi.org/10.1080/14697681003792237

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Quantitative Finance, 2010, 1–13, iFirst

Random walk or a run. Market microstructure

analysis of foreign exchange rate movements based

on conditional probability

YUKO HASHIMOTOy, TAKATOSHI ITO*z, TAKAAKI OHNISHIx,

MISAKO TAKAYASU{, HIDEKI TAKAYASUyy and TSUTOMU WATANABEzz

yStatistics Department, International Monetary Fund, 700 19th Street, N.W. Washington, DC 20431, USA zFaculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan xThe Canon Institute for Global Studies, 11F, Shin-Marunouchi Bldg., 1-5-1 Marunouchi, Chiyoda-Ku, Tokyo 100-6511, Japan and Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,

Tokyo 113-0033, Japan

{Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, 4259-G3-52 Nagatsuta-cho, Midori-ku,

Yokohama 226-8502, Japan

yySony Computer Science Laboratories, 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan zzInstitute of Economic Research, Hitotsubashi University, 2-1 Naka, Kunitachi-city, Tokyo 186-8603, Japan

(Received 10 March 2009; in final form 15 March 2010)

Using tick-by-tick data for the dollar–yen and euro–dollar exchange rates recorded on the actual transaction platform, a ‘run’—continuous increases or decreases in deal prices for the past several ticks—does have some predictable information on the direction of the next price movement. Deal price movements, that are consistent with order flows, tend to continue a run once it is started. Indeed, conditional probabilities of a run continuing in the same direction after several consecutive observations exceed 0.5. However, quote prices do not show such a run tendency. Hence, a random walk hypothesis is refuted in a simple test of a run using tick- by-tick data. In addition, a longer continuous increase of the price tends to be followed by a larger reversal. The findings suggest that those market participants who have access to real- time, tick-by-tick transaction data may have an advantage in predicting exchange rate movements. The findings reported here also lend support to the momentum trading strategy.

Keywords: Exchange rates; International finance; International economics; Financial time series

1. Introduction

The foreign exchange market remains sleepless around the clock.xx Someone is trading somewhere all the time—24 hours a day, 7 days a week, 365 days a year. Analysing the behavior of the exchange rate has become a popular sport of international finance researchers, while global financial institutions are spending millions of dollars to build real- time computer trading systems (program trading).

High-frequency, reliable data are the key to finding robust results for good research for academics or prof- itable schemes for businesses.

Program trading typically uses an algorithm to find a

‘trend’, which is a one-directional movement of the price (foreign exchange rate) for several minutes to several hours. The computer program produces buy or sell orders depending on the detected trend. Finding a trend is usually based on a continuous increase or decrease (a run)

*Corresponding author. Email: tito@e.u-tokyo.ac.jp

xxThe foreign exchange markets in this paper are meant to be only the spot interbank transactions of Yen/Dollar and Dollar/Euro. In these markets, electronic broking, such as EBS, dominates the market shares and the globally linked broking system executes the transactions, continuously, 24 hours, 365 days, although trading on weekends and banking holidays is, of course, thin.

Quantitative Finance

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of the price in the past several minutes to several hours. This type of trading is known as a momentum trading strategy (as opposed to a contrarian strategy). What is attempted in this paper is to construct a very naı¨ve strategy, try to detect a trend by a run, and make a bet on the next move. A program trading strategy is much more sophisticated than the naı¨ve strategy of just detecting a run and making a bet. Therefore, even if the conditional probability is estimated to be different from 0.5, then it is just an example of such trading, and any profit opportunity detected would be an underestimate. On the other hand, all exercises would not be proof of profitability, because exercises are conducted in an in-sample manner. However, the point is to examine whether the conditional probability of an increase (or a decrease) could be different from 0.5—the prediction of the random walk hypothesis.

In order to mimic a program trading strategy, the data set has to be exactly the same as the dealers. Recently, the actual transactions data have become available. The EBS data set is a historical record of firm quotes and transaction prices recorded in one-second slices. The information is almost the same as what the dealers had. One of the advantages of this paper is the use of tick-by- tick (a one-second slice, to be precise) data of the EBS, which covers a large share of the world-wide spot exchange rate transactions. The data from the transaction platform is much more reliable and desirable, as shown by Ito and Hashimoto (2006).

The most popular data set among researchers, the FXFX screen of Reuters indicative quotes, is not appro- priate for this purpose because quotes are input by dealers to deliver the condition of the market. However, this is information only, without any commitment to trade. The reliability of indicative quotes in capturing the whole picture of market reality is much less than firm quotes. Firm quotes (ready to trade) and transaction price data are simply not available on the FXFX screen.y

Many academic researchers and officials believe that exchange rate changes follow a random walk. The standard benchmark in the literature is that a random walk hypothesis is hard to refute, in that the best predictor of the future price is the current price and past information is of no value in prediction.z Most papers are written using daily, hourly, or minute-by- minute data frequency from the FXFX screen of Reuters.

However, the literature does not prove the non-existence of any profit opportunity in the real world, since the FXFX screen does not represent actual trading possibil- ities and the tick-by-tick conditional trading strategy is not examined properly in the existing literature.

The test conducted in this paper is a test using the concept of a ‘run’—a continuous increase or decrease in deal or quote price (exchange rate) changes.x The probability of the next exchange rate change being positive has to be one-half, if the increase or decrease is a random event, irrespective of the history of exchange rate changes. However, momentum traders tend to perceive that the probability of the next change being positive is higher than one-half. Therefore, the test of the random walk is the conditional probability of the next change after a run of the same sign in exchange rate changes.{ Second, the size of the next change is examined. In particular, it is examined whether the step size of the next change may become different in the case of a run.

There are three kinds of innovations in this paper. First, our data set is much better than any other data set used in the literature, because it is a record of an actual trading platform. Second, the frequency is a one-second slice—almost tick-by-tick. Third, a very simple test is devised so that a test is not subject to model uncertainty or a structural break. It is possible to test whether the expected percentage change (direction times step size) can be estimated in a regression model, using both price and transaction volume information, to test a ‘no profit condition’, as in Ito and Hashimoto (2007). However, the regression model relies on the assumption of stable structural parameters.

Major findings with respect to price behavior, for both the dollar–yen and euro–dollar, are as follows. First, quote prices are less likely to continue rising or falling even if a run exists. For quote prices, the conditional probability of a run (movement in the same direction) is less than 0.5 for most cases. Second, deal prices are more likely to continue falling or rising. For deal prices done on the bid (ask) side, the conditional probability that the price continues falling (rising) is greater than 0.5. Third, the step size of the price change in a run is, in general, constant. The size of the price fall is larger than that of the price increase in the case of bid quote prices. On the contrary, the size of the price increase is larger than that of the price fall in the case of ask quote prices. ySee Goodhart and O’Hara (1997, p. 78) for detailed arguments on the advantage of using actual trading data, and Goodhart et al. (1996) and Goodhart and Payne (1996) for early attempts to make use of transactions data.

zThe seminal papers are those of Meese and Rogoff (1983a, b), and a survey of the nominal exchange rate behavior is found in Frankel and Rose (1995). Other papers failing to reject random walk properties include Rossi (2006). Papers failing to find a better forecasting model than the random walk hypothesis include Baghestani and van Wincoop (2007) and Baghestani (2009), both of which use quarterly data with a long time series. Engel (1994) found a Markov-switching model to be no better than a random walk model in forecasting. Wright (2008) and Carriero et al. (2009) use Bayesian Sims methodology to explore models that produce out- of-sample forecasts better than a random walk prediction.

xAn anonymous referee pointed out the possibility that what we call a ‘run’ may be the result of a large order executed in pieces in a short period of time. Since the data we purchased do not reveal the names of the institutions involved in particular deals, we cannot answer positively or negatively to the referee’s inquiry. However, a trader in a financial institution informed us that the reason for breaking up a large customer order into small pieces is to make no price impact by executing them by spacing timings appropriately. Therefore, if the run is a result of such behavior, then the trader is failing in the purpose of breaking up a large order.

{Bouchaud et al. (2004) and Lillo and Farmer (2004) examined price movements in the stock market and found that the conditional probability of the sign of the next order is greater than 0.5.

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Fourth, in the case of deal prices, the size of the price increase is larger than that of the price fall for bid-side deal prices (and vice versa for ask-side deal prices). Finally, the size of the correction after a run is found to depend on the number of continuous price increases (decreases) and on the cumulative price change in a run. The sample covers the period from January 1998 to October 2003 for the yen and from January 1999 to October 2003 for the euro. Saturdays, Sundays, Mondays and days when interventions were conducted by the monetary authorities (Bank of Japan, Federal Reserve Bank, and ECB) were omitted from the data.

The remainder of the paper is organized as follows. Section 2 describes the data. Section 3 shows the patterns of transactions (quotes) in the foreign exchange market based on the conditional probability. In section 4, the size of the price change at each transaction (quote revision) is examined conditional on a run of price increase/decrease. In section 5, the probability of a price increase/decrease is calculated conditional on the bid–ask spread. Section 6 concludes the paper.

2. Data

An increasing majority of spot interbank transactions of major currencies are now carried out through electronic broking systems. The EBS, whose data we use in the analysis below, has a strong market share (in absolute terms and in comparison with other electronic broking systems such as Reuter D-3000) in the yen/dollar rate and the euro/dollar rate.y The EBS is a provider of trading technology, and the quotes and transactions are shown continuously, 24 hours a day. The EBS screen shows the

‘firm bid’ and ‘firm ask’, the bid and offer that are committed to trade if someone on the other side is willing to trade at that price.z

A ‘firm ask’ (‘firm bid’) means that the institution that posts the quote is ready to sell (buy) the shown currency (e.g., the dollar in exchange for the quoted yen). The ask

quote is almost always higher than the bid quote.x When the deal is done at the ask side, it means that the firm ask (ready to sell) quote is ‘hit’ by a buyer. When the deal is done at the bid side, the firm bid quote is ‘hit’ by a seller. Therefore, the ask deal is a buyer-initiated deal, and the bid deal is a seller-initiated deal, according to the description of Berger et al. (2005).{ If ask (bid) deals are continuously hit, then the ask (bid) deal prices tend to move up (down), because of the buy (sell) pressure.

The data used in the analysis includes information on the price of the USD–JPY from January 1998 to October 2003 and the EUR–USD from January 1999 to October 2003. It contains information of best bid, best ask, bid-side deal prices, and ask-side deal prices. Every data point is on the one-second time slice. Bid and ask quote prices are recorded at the end of the time slice. Any bid/ask quote price movements within a second are not recorded. When there are multiple trades within one second, the ‘lowest given price’ and the ‘highest paid price’ will be shown. The highest paid is the deal price done on the ask side and the lowest given is the price done on the bid side within one second.

As part of facilitating an orderly market, EBS requires any newly linked institution to secure a sufficient number of other banks that are willing to open credit lines with the newcomer. A smaller or regional bank may have fewer trading relationships, thus not as many credit relationships. Then the best bid and ask for that institution may be different from the best bid and ask of the market. A smaller or regional bank may post more aggressive prices (higher bids or lower asks) because they will have relatively fewer credit relationships, implying that they will see fewer dealable prices generally.

This trend means that ‘hot potatoes’ (Lyons 1997) are less important, and a cool supercomputer is increasingly important. In other words, dealers’ tactics to transform order flows from the corporate sector into the interbank market may be less influential than before, and the dealers’ behavior in posting firm bids and asks through the electronic broking system is more influential than before.?

yReuters have significant market share in exchanges related to Sterling, the Canadian dollar, and Australian dollars.

zFor general references on the microstructure of the foreign exchange market, see Lyons (1995, 2001) and Goodhart and O’Hara (1997).

xHowever, in the EBS data set, the reversal (bid higher than ask) can happen, when the two parties do not have credit lines to each other, and there is no third party that has credit lines to the two quote posting parties. The EBS system facilitates, as part of the dealing rules, each institution to control bilateral credit lines. Namely, each EBS-linked institution sets credit lines (including zero) against all other potential counter-parties. Therefore, an institution faces the restriction of bid, offer, or deal from other institutions. When bid and offer rates are posted for the system, they are not necessarily available to all participants of the EBS system. The EBS- registered trader’s screen shows the best bid and best offer of the market and best bid and best offer for that particular institution. In normal times, the best bid of the market is lower than the best offer of the market. Otherwise, an institution that has positive credit lines with both institutions on the bid and ask sides will be able to make profits by arbitrage.

{The buyer-initiated trades (the seller-initiated trades) used by Berger et al. (2005) correspond to the number of deals on the ask side (the number of deals on the bid side) in our paper. The order flow, the net excess of buyer-initiated trades in Berger et al., corresponds to the netdeal in our paper. Berger et al. had access to the data of actual transaction volumes—proprietary data of EBS—while we use the number of seconds in which at least one deal was done. The number of deals, rather than the signed (actual) volume, is a good enough proxy for the volume of transactions. In fact, the actual transaction volume is not revealed to participants other than the parties involved, so that they would not be able to be used to predict price movements in real time.

?Our interviews (in November 2003) with banks with substantial foreign exchange trading in London revealed that they had reduced the degree of discretion of dealers and shifted proprietary trading to the specialized section. Computer models have replaced dealers’ instincts.

Random walk or a run 3

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3. Patterns of quote and deal prices 3.1. Run

In this section, the directional patterns of quote and deal price changes will be examined, calculating the conditional probabilities of the sign of the changes. In the foreign exchange market, quote prices fluctuate as limit orders are newly entered, hit or withdrawn, and deal prices fluctuate as deals are done at different prices. By taking the first difference of prices, pt–pt–1, a series of binary choices consisting of signs of price changes is generated. For example,^.^.^. þ þ þ þ þ þ þ þ þ

þ þ þ ^.^.^.is a new series.y

Many theoretical works derive or assume that the exchange rate follows a random walk process. The random walk hypothesis implies that the probability of a price increase, irrespective of history, in the next change should be 0.5 (given that the size of the increase/decrease in prices is symmetric). Thus, the probability that the price is revised higher for n successive trade (that is, the sign of the price change is positive for n successive quotes) is 0.5ⁿ. Similarly, the probability that the price is revised lower for n successive trade is 0.5ⁿ.

The probability, calculated from the new data series, that the next price revision is positive (negative) conditional on the n successive positive (negative) price revisions is as follows:

Pnðþj þ þ,^.^.^., þ ðn plusesÞÞ

¼ ^{Nðþj þ þ,}^.^.^.^{, þ , Þ}

Nðþj þ þ,^.^.^., þ , Þ þ Nðj þ þ,^.^.^., þ , Þ^, ^ð1Þ where Pn(|) is the calculated conditional probability of continuing a run, given N successive price change in the same direction; N(|) is the number of successive price changes; ‘þ’ indicates positive price revisions (price going up), and ‘’ indicates negative price revisions (price going down). The ‘run’—the number of successive changes in the same direction—is defined as those after the change in direction N þ 1 times earlier.z

If Pn(þ| þþ^.^.^.þ,) is greater than 0.5 and statistically different from the null of (0.5)ⁿ, then the process is said to have ‘momentum’. Given the history of a run—positive (negative) changes n times in a row—it is more likely that the price will move up (down) again in the next change. If it is less than 0.5, then the process is said to have ‘mean reversion’ nature, that is a sign reversal in the next change is more likely, given the history of a run.

In the data, the conditional probability of a run may be different from the theoretical probability with the assumption of random walk. Table 1 shows the number

of samples of a run. For example, N ¼ 2 (þ) shows the number of runs that have two consecutive increases in the prices. Five kinds of prices are considered in this sample: (1) bid quote; (2) ask quote; (3) mid-price quote; (4) bid- side deal price; and (5) ask-side deal price.

In the following, we calculate the conditional probability of equation (1) for the five different kinds of prices: bid quote, ask quote, mid-price of bid and ask quotes, bid-side deal price, and ask-side deal price. The probability is calculated for up to 13 successive (either positive or negative) changes. In calculating probability, Saturdays, Sundays and Mondays are excluded from the data set.x

3.2. Results

Figures 1–3 show the conditional probability of price changes in USD/JPY, and figures 4–6 show the conditional probability for EUR/USD. For the ‘price’, the mid- quote price, the bid-side deal price, and the ask-side deal price will be used. In each figure, the horizontal axis shows the numbers of successive positive/negative price revisions. The probability at n ¼ 0 can provide an unconditional test of the random walk hypothesis. The conditional probability at n ¼ j, for example, indicates the probability of a price increase/decrease in the next price change after j successive price hikes (down).

3.3. Conditional probability, USD/JPY

Figure 3 shows the calculated conditional probability of the mid-quote price (that is, the average of the bid and ask quote prices) for the currency pair USD/JPY.

The conditional probability of the mid-price for USD/JPY is shown in figure 1.{ No significance is detected between the probabilities of positive price revisions and negative revisions. For n ¼ 0, the conditional probability is almost exactly 0.5, i.e. consistent with the random walk hypothesis. However, for n between 1 and 7, the conditional probability is significantly below 0.5, indicating mean reversion, but as n becomes larger, price revisions become closer to unpredictable (P ¼ 0.5). The conditional probability becomes significantly larger than 0.5 for a run of more than 10 successive positive price revisions (white square). Negative quote changes are mean reversion in a small number (n56) run, but become a momentum after a large enough number run.

Figures 2 and 3 show the conditional probability of the bid-side deal and ask-side deal prices of USD/JPY, respectively. It is clear from these figures that the patterns yThe methodology employed in this paper follows the up-down analysis of figure 6 of Mizuno et al. (2003), where they used CQG quote data for a test of trend existence.

z Therefore, there is no ‘double counting’ of a run. That is, a run of four successive changes (þþþþ) in the same direction contains two successive changes and three successive changes as a subset of the four successive changes; they are not counted in the N ¼ 2 or N ¼ 3 definitions.

xMondays are excluded because many national holidays fall on Mondays. The patterns of price changes on Monday are slightly different from other business days.

{Calculations have been conducted for bid quote and ask quote separately, but are not shown here; see NBER working paper No. 14160, July 2008.

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Table 1. Summary of a run.

N ¼ 1 (all) N ¼ 2 N ¼ 3 N ¼ 4 N ¼ 5 N ¼ 6 N ¼ 7 N ¼ 8 N ¼ 9 N ¼ 10 N ¼ 11 N ¼ 12

Number of samples of a run with þ or USD/JPY, from January 1998 to October 2003

Quote bid þ 4,820,958 1,648,403 569,926 211,046 82,612 33,827 14,411 6486 3070 1469 712 367

4,442,196 1,269,642 382,374 126,624 44,893 16,890 6677 2775 1166 516 237 105

Quote ask þ 4,563,975 1,275,940 364,736 113,306 37,615 13,315 4991 1953 793 326 143 50

4,957,516 1,669,481 569,214 209,820 82,457 34,124 14,782 6658 3152 1559 802 166

Quote mid þ 8,015,204 3,182,504 1,234,233 508,938 218,742 97,330 44,664 21,086 10,224 5086 2595 1339

8,006,548 3,173,848 1,235,730 515,430 226,005 103,228 48,520 23,515 11,717 6034 3187 1688

Ask-side deal þ 1,694,588 942,219 527,967 293,037 159,842 85,665 45,239 23,521 12,075 6149 3137 1582

1,298,931 546,562 249,941 115,173 52,812 24,132 11,018 5027 2278 1037 464 209

Bid-side deal þ 1,242,564 519,298 234,787 106,710 48,202 21,554 9637 4295 1903 842 391 195

1,654,012 930,746 528,965 297,994 165,346 90,096 48,344 25,515 13,357 6926 3585 1873

Number of samples of a run with þ or EUR/USD, from January 1999 to October 2003

Quote bid þ 4,016,461 1,243,324 412,360 151,179 59,367 24,715 10,729 4818 2269 1116 559 290

3,874,695 1,101,558 351,711 124,955 47,409 18,987 7965 3431 1491 680 320 152

Quote ask þ 3,960,877 1,107,694 340,999 116,400 42,782 16,736 6925 2976 1324 599 281 149

4,108,672 1,255,490 411,685 150,236 58,990 24,535 10,788 4863 2261 1059 509 257

Quote mid þ 6,455,384 2,542,775 1,006,191 434,786 195,691 92,044 44,395 21,993 11,248 5913 3201 1788

6,427,969 2,515,360 995,392 433,340 197,693 93,880 45,516 22,567 11,404 5848 3024 1546

Ask-side deal þ 1,616,044 845,883 449,193 235,488 121,269 61,310 30,574 15,071 7376 3633 1773 846

1,376,109 605,948 288,292 137,957 65,417 30,597 14,202 6544 3017 1381 649 313

Bid-side deal þ 1,350,884 593,361 280,617 132,924 62,322 28,902 13,136 5920 2675 1184 536 250

1,599,561 842,038 450,945 240,016 125,793 64,663 32,762 16,432 8097 3962 1962 976

Randomwalkorarun5

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of the conditional probability for deal prices are quite different from those of quote prices in important ways.

The probability of having a run of ask-side deal price increases (white square in figure 3) is greater than 0.5, and so is a run of bid-side deal price decreases (black circle in figure 2). Once the price starts to rise on the ask side—that is, buyer-initiated deals—then the price increases tend to continue (P40.5). Similarly, once the price starts to decline on the bid side—that is, seller-initiated deals—then the price decreases tend to

continue (P40.5). These results are in contrast to the results implied by quote price movements, and consistent with the conventional real-world view that there are some moments in time when a momentum is formed. When buyers are eager to hit the ask quotes, and start to drive up prices, then this creates a momentum to push prices up further. Having deals is important in this process. Similarly, when sellers hit bid prices, driving prices down, with deals, this creates downward momentum. These movements support the view that the momentum

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 2 4 6 8 10 12

Probability

n Confidence interval

P(+|++...+) P(-|--...-)

Figure 1. Conditional probability of a run, mid-price, USD/JPY.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 2 4 6 8 10 12

Probability

P(+|++...+) P(-|--...-)

Figure 2. Conditional probability of a run, deal bid price, USD/JPY.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 2 4 6 8 10 12

Probability

P(+|++...+) P(-|--...-)

Figure 3. Conditional probability of a run, deal ask price, USD/JPY.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 2 4 6 8 10 12

Probability

P(+|++...+) P(-|--...-)

Figure 4. Conditional probability of a run, mid-price, EUR/USD.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 2 4 6 8 10 12

Probability

P(+|++...+) P(-|--...-)

Figure 5. Conditional probability of a run, deal bid price, EUR/USD.

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0 2 4 6 8 10 12

Probability

P(+|++...+) P(-|--...-)

Figure 6. Conditional probability of a run, deal ask price, EUR/USD.

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strategy is a winning strategy while the run continues. The run tends to continue for two to nine ticks.

The positive price change for bid-side deal prices (white square in figure 2) does not show the tendency of a run, but mean reversion, and so does the negative price change for ask-side deal prices (black circle in figure 3).

For deal bid prices, for 0 n 4, the conditional probability of a negative price revision is around 0.55–0.58. It then gradually declines to 0.53 for n ¼ 8. The probability of more than nine successive negative price revisions is not significantly different from 0.5. In contrast, the probability of a positive price revision is 0.42–0.43 for n ¼ 1 and 2, and remains stable around P ¼ 0.45 for up to n ¼ 9. The probability of more than 10 successive positive price revisions is not significantly different from 0.5. Once the bid-side deal prices (seller- initiated deals) start to decline, then a run of the same sign tends to continue.

The conditional probability of the ask deal price has exactly the opposite pattern to that of the bid deal price. The conditional probability of a positive price revision is around 0.56–0.58 and higher than that for negative price changes. In contrast to the deal bid price, the case of more than seven successive positive price revisions disappears.

3.4. Conditional probability, EUR/USD

Figure 4 shows the conditional probability of the mid- price quote for EUR/USD. The pattern of the probability is similar to that for USD/JPY. First, the conditional probability pattern shows that the quote price change exhibits no unconditional expected change (P0¼ 0.5), while it shows mean reversion for 15n56. Second, the conditional probability becomes neutral (P ¼ 0.5) for n47. Third, the conditional probability of 10 or more successive positive price revisions (white square) becomes significantly greater than 0.5. The last feature is in contrast to USD/JPY, where negative price revisions become a momentum after n410.

Figures 5 and 6 show the conditional probability of bid- side and ask-side deal prices, respectively. Similar to USD/JPY, a run tends to continue (P40.5) for a negative run of the bid-side deal price, and for a positive run of ask-side deal prices. Again, buying pressure (buyer- initiated deals) generates a positive run, and selling pressure (seller-initiated deals) generates a negative run. As n becomes larger, the conditional probability con- verges towards 0.5.

A positive price change for bid-side deal prices does not exhibit the tendency of a run, but rather mean reversion, and so does the negative price change for ask-side deal prices.

A similar analysis was conducted for EUR/JPY, but the patterns are the same, and the results are less robust (wider standard error), so are not presented here.

In summary, several salient features are detected regarding the conditional probability of a run. First, USD/JPY and EUR/USD show similar patters with regard to patterns of the conditional probability for a run

for five different kinds of prices. Second, the patterns for quote price changes are quite different from the patterns for deal price changes. Quote prices tend to show mean reversion (i.e. the conditional probability of a run is less than 0.5), regardless of a positive run or a negative run, either for bid or ask quotes. For deal prices, buyer- initiated deal prices (ask-side deal prices) tend to have a continuous positive run (i.e. the conditional probability of a run of positive changes to continue exceeds 0.5), and seller-initiated deal prices (bid-side deal prices) tend to have a tendency of a continuous negative run. The contrast is striking, and this shows that any study based on quote prices will be misleading in describing how transactions are proceeding in the market. Once a

‘momentum’, measured by deal prices fueled by buying pressure or selling pressure, is formed the momentum tends to continue.

4. Size of the price change in continuous quotes/deals 4.1. Acceleration or deceleration?

In this section, the size of the price change is examined. When a run is detected (i.e. an ask-side positive run or a bid-side negative run), is it more likely that the increase or decrease step becomes larger (acceleration in momentum)? This question is interesting for the following reasons.

As momentum trading works for several ticks (ask-side deals and bid-side deals), that is the conditional probability is greater than 0.5, those who detect this momentum may want to join the bandwagon. It may be conjectured that those one-side movements may gather force until the momentum stops and may possibly be followed by a sharp reversal. The process of momentum and an eventual stop may be a reflection of a (rational) ‘bubble’ phenomenon, if a stop means a correction of a significant degree, following an acceleration of momentum. A rational bubble requires a process whereby a greater degree of correction has to be compensated by a larger step towards the end of the bubble process. This kind of pattern is known as a stochastic bubble.

In contrast, a run may stop without having a reversal in the price. The run results in finding a new equilibrium level. Figure 7 (pattern 1) shows a typical bubble process if the increase is followed by a significant drop. Figure 8 (pattern 2) shows a process that is a convergence to a new equilibrium after digesting fundamental news. In order to distinguish the processes, we first examine whether the increase or decrease step would become larger or smaller as the run continues.

In the following, we analyse the size of the price change in successive quotes/deals,

size of price change ¼ priceðt þ nÞ priceðtÞ: ð2Þ We calculate the size of the price change for the price increase phase only (or price decrease only) for n successive quotes/deals. Four types of prices are used

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for calculation: bid, ask, mid-price, deals done on the bid side and deals done on the ask side.y

The results are shown in figures 9–11 for USD/JPY and figures 12–14 for EUR/USD. In each figure, the vertical axis shows the cumulative price increment (or decrease) and the horizontal axis shows the successive price increase (or decrease). The white circles show the price path of a continuous price increase and black circles show the price

path of a continuous price decline. The symmetric dashed straight lines indicate the smallest cumulative change of n successive price increases (decreases). The minimum price increase/decrease is called the ‘pip’ and its size is 0.01 for USD/JPY and 0.0001 for EUR/USD.

Figure 9 shows the mid-price of bid and ask quotes for USD/JPY. Again, the price paths of the price increase/fall are linear up to n ¼ 12, and they are almost symmetric. Tick

Price

Figure 8. Pattern 2: Pattern of convergence. Tick Price

Figure 7. Pattern 1: Pattern of a bubble.

–0.003 –0.002 –0.001 0 0.001 0.002 0.003

0 2 4 6 8 10 12 14

Price(t+n) - Price(t) [Dollar]

n +

–

Figure 12. Size of the price change in a run, mid-price, EUR/USD.

–0.3 –0.2 –0.1 0 0.1 0.2 0.3

0 2 4 6 8 10 12 14

Price(t+n) - Price(t) [yen]

n +

–

Figure 11. Size of the price change in a run, deal ask price, USD/JPY.

–0.3 –0.2 –0.1 0 0.1 0.2 0.3

0 2 4 6 8 10 12 14

n +

–

Figure 10. Size of the price change in a run, deal bid price, USD/JPY.

–0.3 –0.2 –0.1 0 0.1 0.2 0.3

0 2 4 6 8 10 12 14

n +

–

Figure 9. Size of the price change in a run, mid-price, USD/JPY.

yAgain, we do not report the results for the bid-quote and ask-quote prices. Refer to NBER working paper No. 14160, July 2008.

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The decrease of the mid-price quotes after n ¼ 12 accelerates, while no such acceleration is detected in the rise of the mid-price quote.

Figures 10 and 11 show the rise and fall of the deal price run. The pattern is in contrast to that for the run of bid/ask quotes. The price increase path (white circle) for a deal bid is well above the straight line, indicating that the size of the continuous price increase is larger than the continuous price fall in the run of bid-side deals, while the size of the continuous price fall is larger than the size of the price increase in the run of ask-side deals. In figure 11, for n ¼ 12 or larger, it is found that the ask-side deal price fall accelerates: when the transaction price continues to fall more than 12 times, the selling price starts falling faster.

As argued above, a price increase with buying pressure tends to exhibit a positive run of the ask-side deal price and a price decrease with selling pressure tends to exhibit a negative run of the bid-side deal price. For these price changes, no acceleration is detected.

Figures 12–14 show the price paths for EUR/USD. Figure 12 shows the price increment/fall of bid quotes. Figure 12 shows the mid-price of bid and ask quotes of EUR/USD. Again, the price paths of a price increase/fall are linear and almost symmetric.

Figures 13 and 14 show the price path of a run of deal price changes. The size of the price increase is slightly

wider than the size of the price fall for deals on the bid side, and vice versa for deal prices on the ask side. In both cases, the price paths are almost linear and there is no evidence that the price soars once the deal price start to increase.

In summary, there is no difference in the patterns of the price paths between the two currency pairs. It is found that the price increases (decreases) tend to accelerate as the price continues to rise (fall). The size at each price change is mostly constant regardless of the price—bid, ask, mid-price, deals done on the bid side, or deals done on the ask side.

4.2. Size of the correction after a run

In this section, we test whether the size of an opposite movement at the termination of a run becomes larger as the number of runs becomes larger and whether the cumulative increase (decrease) during the run becomes disproportionately larger. The following specifications are adopted for examination:

Dpnþ1

¼ þ n þ "nþ1^: ð3Þ After a run of length n, the direction of the price movement reverses, shown in Dpnþ1. If ¼ 0, then there is no relation between the size of the correction and the length of the run. The run is more likely associated with a transition from an old equilibrium to a new equilibrium. On the other hand, if 40, then the longer the run, the larger the correction. This is suggestive evidence of a speculative bubble. This would suggest a scenario where the price deviates from a fundamental value and then crashes back to a fundamental value. Recall figures 12 and 13, respectively, for the two cases.

The second specification of the same test is to use the cumulative price change rather than the just the number of successive changes,

jDp_nþ1j ¼ þ ^X

n

j¼1

Dp_jþ "_nþ1, ð4Þ

where^Pⁿ_j¼1Dpjdenotes the cumulative price change of a run of length n. The interpretation of is the same as in equation (3). In addition, if ¼ 1, then all the cumulative change is wiped out in one change after the run.

Tables 2 and 3 summarize the regression results of equations (3) and (4), respectively. In each regression, is estimated for either a run with positive price changes only (increase), or a run with negative price changes (decrease), based on White-robust standard errors. ‘All’ shows the estimated coefficient using all of the runs.

As shown in table 1, is estimated as being significantly positive in all cases regardless of the currency pair or the definition of the price (bid quote, ask quote, bid-side deal or ask-side deal). This implies that the longer the run, the larger the correction. We also find that the size of the correction is larger after a run with a price increase than a run with a price fall for the ask and/or deal price on the ask side. On the other hand, the size of the correction is larger after a price decrease run than a price increase run

–0.003 –0.002 –0.001 0 0.001 0.002 0.003

0 2 4 6 8 10 12 14

n +

–

Figure 13. Size of the price change in a run, deal bid price, EUR/USD.

–0.003 –0.002 –0.001 0 0.001 0.002 0.003

0 2 4 6 8 10 12 14

n +

–

Figure 14. Size of the price change in a run, deal ask price, EUR/USD.

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Table 2. Size of the correction (length of a run). Bid

increase Decrease All

Ask

increase Decrease all

Deal bid

Deal ask

USDJPY

# (s.e.)

6.93E-04 8.67E-06

8.39E-04 1.06E-05

7.57E-04 6.70E-06

1.08E-03 1.16E-05

5.90E-04 8.44E-06

7.95E-04 6.87E-06

2.22E-04 1.22E-05

1.44E-04 1.22E-05

3.64E-04 9.01E-06

1.85E-04 1.04E-05

2.06E-04 1.07E-05

3.60E-04 7.69E-06

nob 3108100 3108100 6216201 3219839 3219839 6439678 687279 687279 1374558 712094 712095 1424189

R-sqrd 3.7E03 3.6E03 3.6E03 5.2E03 2.2E03 3.4E03 7.0E04 2.2E04 1.6E03 4.7E04 4.4E04 1.6E03

DW 1.52 1.49 1.18 1.43 1.53 1.15 1.83 1.90 1.84 1.86 1.87 1.81

EURUSD

# (s.e.) nob

4.76E-06 5.63E-08 3313249

3.23E-06 5.97E-08 3313250

4.04E-06 4.10E-08 6626499

4.53E-06 9.10E-08 3406970

4.13E-06 5.67E-08 3406970

4.31E-06 5.19E-08 6813940

9.15E-07 7.95E-08 915094

1.31E-06 8.15E-08 915095

1.86E-06 5.86 E-08 1830189

1.83E-06 1.08E-07 927990

5.78E-07 5.78E-08 927990

6.8E-06 6.80E-08 1855980

DW 1.71 1.65 1.43 1.56 1.67 1.32 1.91 1.92 1.89 1.90 1.89 1.86

Note:indicates the significance level at 1%. Standard errors are Heteroscedasticity-consistent.

10Y.Hashimotoetal.

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Table 3. Size of the correction (cumulative price changes). Bid

Ask

Deal bid

Deal ask

USDJPY

#(s.e.) 1.17E01 4.30E03

2.24E01 7.44E03

1.78E01 4.53E03

2.67E01 9.38E03

1.12E01 4.23E03

2.02E01 6.33E03

3.27E02 1.28E03

4.55E02 3.31E03

4.36E02 1.82E03

4.95E02 3.20E03

3.55E02 2.88E03

4.60E02 2.21E03

nob 3108100 3108100 6216201 3219839 3219839 6439678 687279 687279 1374558 712094 712095 1424189

DW 1.56 1.51 1.18 1.45 1.56 1.16 1.84 1.90 1.85 1.87 1.88 1.83

EURUSD

#(s.e.) 6.69E-02 3.19E-03

1.16E-01 6.80E-03

9.27E-02 3.89E-03

1.69E-01 1.58E-02

6.59E-02 3.47E-03

1.23E-01 9.36E-03

2.22E-02 9.36E-03

3.25E-02 2.56E-03

2.97E-02 1.19 E-03

4.18E-02 4.16E-03

2.13E-02 9.41E-04

3.43E-02 221 E-03

nob 3313249 3313250 6626499 3406970 3406970 6813940 915094 901855 1830189 927990 927990 1855980

R-sqrd 1.1E02 3.6E02 2.2E02 6.9E02 9.1E03 3.4E02 2.2E-03 2.6E03 2.8E03 5.2E03 2.4E03 4.4E03

DW 1.73 1.67 1.44 1.58 1.69 1.33 1.91 1.92 1.90 1.91 1.89 1.86

Note:indicates the significance level at 1%. Standard errors are Heteroscedasticity-consistent.

Randomwalkorarun11

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for the bid and/or deal price on the bid side. Note that a price increase here means yen depreciation vis-a`-vis the US dollar, and the dollar depreciates vis-a`-vis the euro.

The estimation results of equation (4) are shown in table 3. Again, is estimated as being significantly positive in all cases regardless of the currency pair or the definition of the price. That is, the larger the cumulative price increases (decreases), the larger the size of the price revision after the run. We also find similar patterns in the coefficient—the size of the correction is larger after a run of price increases than after a run of price falls for the ask and/or deal price on the ask side, and vice versa for the bid and/or deal price on the bid side.

The results of tables 2 and 3, taken together, can be interpreted as supportive evidence that a run is the result of momentum trading that needs correction at the end. We may call this a mini-bubble that bursts within minutes, unlike the housing bubble that could go on for years.

5. Concluding remarks

In this paper, the random walk hypothesis is refuted in a simple test of a run—continuous increases or decreases in quote prices and deal prices—using tick-by-tick exchange rate data. Several questions exploiting the tick-by-tick data have been examined in this paper. Will the changes in ask (bid) deal prices, as well as ask (bid) quote prices, be influenced by the previous transactions? If a deal price goes up from the previous deal, then will the next deal price be more likely to go up or not? How long do prices continue to increase (decrease)? How about the size of price changes? We have examined the patterns of quote activities and transaction activities in the foreign exchange market using a very rich data set.

The main findings are as follows. First, the conditional probability that the price is revised in the same direction is less than 0.5 for both bid and ask quotes. Therefore, the quote price moves in a mean-reverting manner. On the contrary, deal prices are more likely to continue rising or falling. For deal prices done on the bid side, the conditional probability that the price is revised lower exceeds 0.5. Similarly, for deal prices done on the ask side, the conditional probability that the price is revised higher exceeds 0.5. Second, we do not observe the size of the price increment/fall at each transaction (quote) widening or shrinking as the run continues. Again, regardless of currency pair, the size of the price fall is larger than that of the price increase for bid-quote prices, and vice versa for ask-quote prices. In contrast, the size of the price increase is larger than that of the price fall for deals done on the bid side, and vice versa for deals done on the ask side. Third, we estimate the size of the price correction after a continuous price increase (decrease). The absolute size of the price correction is larger when a run becomes longer. It is also found that the absolute size of the price correction is larger when the cumulative change in price in a run is larger.

What is found at a very high-frequency level in foreign exchange markets is that the exchange rate mostly follows a random walk, but there appears a so-called mini-bubble once the exchange rate starts rising (falling). This is evident in the conditional probability of a deal price run, and the size of the correction after the run.

Acknowledgements

The authors are grateful to EBS for their appreciation of the value of academic research and for providing a proprietary data set for academic purposes with few restrictions and for a modest fee. Also, we are grateful to EBS analysts in New York for guidance on the nature of the data. Research support by JSPS Grants-in-aid for Scientific Research (No. 15203008) is gratefully acknowl- edged. Comments at the Econometric Society European Meeting, Budapest, August 26–31, 2008 and Meetings of the Statistics Association, September 8, 2007 are gratefully appreciated.

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