High-Frequency Jump Analysis of the Bitcoin Market
aa r X i v : . [ q -f i n . S T ] J un High-Frequency Jump Analysis of the Bitcoin Market
Olivier Scaillet ∗ Adrien Treccani † Christopher Trevisan ‡§ June 27, 2017first draft: April 2017
Abstract
We use the database leak of Mt. Gox exchange to analyze the dynamics of the price ofbitcoin from June 2011 to November 2013. This gives us a rare opportunity to study anemerging retail-focused, highly speculative and unregulated market with trader identifiersat a tick transaction level. Jumps are frequent events and they cluster in time. The orderflow imbalance and the preponderance of aggressive traders, as well as a widening of thebid-ask spread predict them. Jumps have short-term positive impact on market activity andilliquidity and induce a persistent change in the price.
JEL classification: C58, G12, G14.Keywords: Jumps, Liquidity, High-frequency data, Bitcoin. ∗ University of Geneva and Swiss Finance Institute, 40 Bd du Pont d’Arve, 1211 Geneva, Switzerland. Voice:+41 22 379 88 16. Fax: +41 22 379 81 04. Email: [email protected]. Corresponding author. † University of Geneva and Swiss Finance Institute, 40 Bd du Pont d’Arve, 1211 Geneva, Switzerland. Voice:+41 22 379 81 66. Fax: +41 22 379 81 04. Email: [email protected]. ‡ Ecole Polytechnique F´ed´erale de Lausanne and Swiss Finance Institute, EPFL CDM-DIR, 1015 Lausanne,Switzerland. Voice: +41 21 693 01 28. Email: christopher.trevisan@epfl.ch. § Acknowledgements: We thank participants at the ”Market Microstructure and High Frequency Data” con-ference 2017 in Chicago for helpful comments. Introduction
Bitcoin, a distributed digital currency, was created in 2009 and is the most popular cryptocur-rency with a multi-billion dollar capitalization since 2013. It is the first such currency to gainrelatively widespread adoption. The technology provides an infrastructure for maintaining apublic accounting ledger and for processing transactions with no central authority. Unliketraditional currencies, which rely on central banks, bitcoin relies on a decentralized computernetwork to validate transactions and grow money supply (see Yermack (2015) and Yermack(2017) for further background on the bitcoin and its technology). Each bitcoin is effectivelya (divisible) unit which is transferred between pseudonymous addresses through this network.Its promising potential and scarcity have driven the market price of bitcoin to parity with theU.S. dollar in February 2011 and above $1,000 in November 2013. It is estimated that by theend of our period of study in 2013, bitcoin had approximately one million users worldwide witha three-digit annual growth. Mt. Gox was the largest exchange platform to provide bitcointrading for U.S. dollar until it went bankrupt early 2014 as a result of the theft of client fundsby hackers. An important part of Mt. Gox internal database leaked, revealing a full history oftrades on the period April 2011–November 2013. This data set gives us a rare opportunity toobserve the emergence of a retail-focused, highly-speculative and unregulated market at a tickfrequency with trader identifiers at the transaction level.Bitcoin has experienced numerous episodes of extreme volatility and apparent discontinuitiesin the price process. On one hand, the absence of solid history and exhaustive legal frameworkmake bitcoin a very speculative investment. Because it does not rely on the stabilizing policyof a central bank, the reaction to new information, whether fundamental or speculative, resultsin high volatility relative to established currencies. On the other hand, the relative illiquidityof the market with no official market makers makes it fundamentally fragile to large trading The Japanese courts are holding pre-trial hearings, and the claim process enters its fourth year. Japanesepolice have found part of the missing bitcoins, and the 24,000 or so claimants are waiting for a final settlement.
Let us first briefly introduce the bitcoin. Bitcoin is a novel form of electronic money that isbased on a decentralised network of participating computers. It has no physical counterpart; it ismerely arbitrary (divisible) units that exist on this network. There is no central bank and thereare no interest rates. The system has a pre-programmed money supply that grows at a decreasingrate until reaching a fixed limit. This semi-fixed supply exacerbates volatility and deflationarypressure. Each user of bitcoin can generate an address (like an email address or account number)through which to make and receive transactions, making bitcoin pseudonymous. The crucialaspect that makes bitcoin work is that it solves the double-spending problem without relying ona central authority. In other words, it is possible to send a bitcoin securely, without then beingable to spend that bitcoin again, without someone else being able to forge a transaction, andalso without your being able to claim that bitcoin back (i.e., a chargeback). These transactionsget recorded in a decentralised ledger (known as the blockchain), which is maintained by anetwork of computers (called ’miners’). Miners maintain consensus in the blockchain throughsolving difficult mathematical problems, and are rewarded with bitcoins and optional (voluntary)transaction fees. The additional rewarded bitcoins are the mechanism that increases the bitcoinmoney supply.For our empirical study, we use transaction-level data with trader identifiers for the Mt. Gox5itcoin exchange. We conduct our analysis over the uninterrupted period from June 26, 2011to November 29, 2013. Mt. Gox was the leading bitcoin trading platform during that periodand processed the majority of trading orders.We extract the data from the Mt. Gox database leak of March 2014, following Mt. Goxsuspension of its operation and bankruptcy filing. This data set is available on the BitTorrentnetwork and includes a history of all executed trades. The data is organized as a series ofcomma-separated files with each row listing a time stamp, a trade ID, a user ID, a transactiontype (buy or sell), the currency of the fiat leg, the fiat and bitcoin amounts, and the fiat andbitcoin transaction fees. A subset of the trades additionally reveals the country and state ofresidence of the user. We ignore these last pieces of information as they are only available for alimited number of trades. A heuristic analysis of trade IDs reveals that they correspond to theconcatenation of a POSIX timestamp and a microsecond timestamp. We parse the timestampsaccordingly to define the execution time of each trade with a microsecond precision.The respective legs of the trades are split across multiple lines. We initiate the cleaningprocedure by aggregating trade entries according to their trade IDs. We filter out trades whosebid leg or ask leg are missing, and remove all duplicates. We also remove from the sampletrades for which the same user identifier appears on both legs. Those trades are either due to abug in the order book matching algorithm, or are simple data errors. Finally, we only considerU.S. dollar-denominated trades and filter out trades whose fiat amount is smaller than $0.10 toavoid numerical errors in the computation of the price. We define the tick-time price series asthe ratio of the bitcoin amount over the fiat amount for the chronological trades series, roundedto the third decimal.We confirm the authenticity of the remaining data by comparing them to the data setpublished by Mt. Gox in 2013 and its subsequent updates. However, the comparison alsoreveals two problems related to multi-currency trades. First, 92,174 trades have a systematic On August 27, 2011, Mt. Gox implemented a form of order book aggregation across currencies, with theexchange acting as intermediary. For exemple, a market buy order in USD could match a limit sell order in and remove tradeswhose exchange price lies outside of the high-low interval with a 20% margin. We also discard‘bounceback’ outliers as defined in A¨ıt-Sahalia, Mykland, and Zhang (2011). The resulting setof trades is used for our analysis of the bitcoin market.The data set only includes information on executed trades. It lacks limit orders, and conse-quently provides no explicit information on the bid-ask spread across time or the depth of theorder book. The published data set provides an additional field specifying whether orders areinitiated by the buyer or the seller, that is, if they are aggressive bids or aggressive asks. Thisrecording is important for our analysis of the potential determinants of jump occurence. Wedefine the best bid series as the price series of aggressive ask orders, and the best ask series asthe price series of aggressive bid orders. In the rare occurrences where the best bid price getshigher than the best ask price, we update the best ask to the value of the bid price; reciprocally,we update the best bid price if the best ask price crosses it.We construct calendar-time price series by computing the median of the tick-time priceswithin each interval of 5 minutes. In the case where no trade occurs, we propagate the pricefrom the previous period. We build the calendar-time volume series by summing the respective EUR, triggering a pair of trades between the users and Mt. Gox. The two legs share the same trade ID, whichallows us to identify them easily. The published data set further distinguishes the primary and non-primary legsof a multi-currency trade. The primary leg is the one where Mt. Gox is selling bitcoins in exchange for fiat. Allmissing trades are non-primary legs. Log BTC/USD Log volume [USD]
Volume Moving average
Figure 1: BTC/USD exchange rate and volume
The figures display respectively the bitcoin price in dollar terms and the trading volume at a dailyfrequency on Mt Gox exchange platform from June 2011 to November 2013. volumes within each interval, and the trades number series by taking the number of trades oneach period.The final data set contains 6.4 million transactions involving 90,382 unique traders. Thetransactions amount to a total volume of $2.1 billion, or on average $2.4 million per day. Figure 1shows the time series of the price and volume on a logarithmic scale during the period. Theprice of bitcoin increases from $16 on June 26, 2011 to an all-time high of $1,207 on November29, 2013. Volume increases significantly during the period as well, and the linear correlationbetween price and volume exceeds 70%. The price of bitcoin has experienced several booms andbusts. The clearest example is the crash of April 10, 2013 which saw the bitcoin value drop by61% in only hours for no obvious reason, after doubling over the previous week. No stabilizingmechanisms mitigate those large swings. There are no central banks, no market makers, andno circuit breakers in the bitcoin market. 8
Methodology
Many pricing models rely on the assumption that the dynamics of the underlying asset fol-low a continuous trajectory. For instance, Black and Scholes (1973) propose a diffusion modelwith constant volatility and Heston (1993) augments it with a second factor to allow for het-eroskedasticity. The empirical literature challenges continuous models (see, e.g., A¨ıt-Sahalia,2002; Carr and Wu, 2003). The probability of large moves disappears asymptotically as thehorizon shrinks, which does not provide consistent short-term skewness and kurtosis.There are mainly two approaches to overcome this limitation. First, we can introduce ajump component in the price process (e.g., Merton, 1976; Bates, 1996). Jumps are discontinuousprice changes occurring instantaneously, no matter the frequency of observations. Alternatively,we can consider models with highly dynamic volatility, such as the two-factor stochastic volatilitymodel of Chernov, Gallant, Ghysels, and Tauchen (2003) and Huang and Tauchen (2005). Theprobability of sudden moves asymptotically still vanishes, yet those models allow for bursts ofvolatility leading to significant changes on short-term horizons.Identifying whether a price process is continuous or has jumps is important because of theimplications for financial management such as pricing, hedging and risk assessment. For deepout-of-the-money call options, there may be relatively low probability that the stock price ex-ceeds the strike price prior to expiration if we exclude the possibility of jumps. However, thepresence of jumps in the price dynamics significantly increases this probability, and hence, makesthe option more valuable. The converse holds for deep in-the-money call options. This phe-nomenon is exacerbated with short-maturity options. Barndorff-Nielsen and Shephard (2006),A¨ıt-Sahalia and Jacod (2009), Mancini (2001), Lee and Mykland (2008) develop statistical toolsto test for the presence of jumps. Their modeling approach assumes that the data is not con-taminated by microstructure noise, preventing a high-frequency analysis. Christensen et al.(2014) show that it is crucial to test for jumps at a high frequency to avoid misclassification of Another alternative would be to consider L´evy jumps of infinite activity (see, e.g., A¨ıt-Sahalia, 2004). , F t , P ), where Ω is the set ofevents of the bitcoin market, {F t : t ∈ [0 , T ] } is the right-continuous information filtration formarket participants, and P is the physical measure. We denote the log-price P and model itsdynamics on a given day as d P t = σ d W t + aY t d J t , where W t is a Brownian motion, J t is a jump counting process, Y t is the size of the jump, σ isthe volatility assumed to be constant on a one-day period, and a is 0 under the null hypothesisof no jump and 1 otherwise. The log-price P stands for the unobservable, fundamental price in an ideal market. Thebitcoin market is relatively illiquid and is subject to multiple frictions such as trading fees.Consequently, the observed price is contaminated by noise. We define the observed price ˜ P as˜ P t i = P t i + U t i , where t i is the time of observation , i = 1 , ..., n, with n being the number of observations perday. Here U denotes the market microstructure noise with mean 0 and variance q . Figure 2shows the autocorrelation function at a tick frequency of the observed log-returns on June 10,2013. The significant dependence in the first lags suggests that the microstructure noise hasserial dependence. We therefore allow U to have a ( k − k = 4. We omit the drift term in our log-price model as it has no impact in the jump detection test asymptotically,as explained in Mykland and Zhang (2009). We assume that Assumption A of Lee and Mykland (2012) about the density of the sampling grid holds. We observe a similar pattern of significantly negative 1–3 lag coefficients throughout the sample. ag S a m p l e A u t o c o rr e l a t i on -0.4-0.200.20.40.60.81 Figure 2: Autocorrelation of log BTC/USD returns for June 10, 2013
The figure displays the autocorrelogram of the bitcoin price series on June 10, 2013. Dashedhorizontal lines show the -confidence levels. The autocorrelation is significant up to order . We define the block size as M = ⌊ C ( n/k ) / ⌋ , where ⌊ x ⌋ denotes the integer part of the num-ber x , and follow the recommendations of Lee and Mykland (2012), Section 5.4, for specifyingthe parameter C . We compute the averaged log-price over the block size M asˆ P t j = 1 M ⌊ j/k ⌋ + M − X i = ⌊ j/k ⌋ ˜ P t ik , and test for the presence of jumps between t j and t j + kM using the asymptotically normalstatistic L defined as L ( t j ) = ˆ P t j + km − ˆ P t j , for j = 0 , kM, kM, . . . The asymptotic variance of the test statistic is given by V = lim n →∞ V n = . σ T + 2 q where the limit holds in probability. We estimate the volatility ˆ σ using the consistent estimatorof Podolskij and Vetter (2009), which is robust to the presence of noise and jumps. We use11roposition 1 of Lee and Mykland (2012) to estimate the noise variance ˆ q , that is,ˆ q = 12( n − k ) n − k X m =1 ( ˜ P t m − ˜ P t m + k ) . Our estimate of the asymptotic variance is therefore ˆ V n = . ˆ σ T + 2ˆ q .Lee and Mykland (2012) show the convergence in distribution of the test statistics B − n √ M √ V n max j |L ( t j ) | − A n ! −→ ξ, for j = 0 , kM, kM, . . . , where ξ follows a standard Gumbel distribution with cumulative dis-tribution function P ( ξ ≤ x ) = exp ( − e − x ), and the constants are as follows A n = (cid:18) (cid:22) nkM (cid:23)(cid:19) / − log( π ) + log (cid:0) log (cid:0)(cid:4) nkM (cid:5)(cid:1)(cid:1) (cid:0) (cid:0)(cid:4) nkM (cid:5)(cid:1)(cid:1) / ,B n = 1 (cid:0) (cid:0)(cid:4) nkM (cid:5)(cid:1)(cid:1) / . We test the presence of jumps on a given day by identifying a divergence of the test statisticfrom the Gumbel distribution. As emphasized in Bajgrowicz et al. (2016), it is crucial to accountfor multiple testing when applying a statistical test more than once. Indeed, if the rejectionthreshold is fixed, the proportion of rejections converges to the size of the test under the nullhypothesis because of type I errors, preventing any statistical inference. The FDR ensuresthat at most a certain expected fraction of the rejected null hypotheses correspond to spuriousdetections. The FDR approach results in a threshold for the p -value that is inherently adaptiveto the data. It is higher when there are few true jumps, i.e., the signal is sparse, and lowerwhen there are many jumps, i.e., the signal is dense. Setting the FDR target parameter to0 is equivalent to a strict control of the family-wise error rate. It is very conservative asit asymptotically admits no spurious detection due to multiple testing. We prefer a FDRtarget level of 10%, which results in a more liberal threshold than with family-wise error rate12ontrol. The power of the test is therefore improved, at the cost of accepting that up to 10%of detected jump days may be spurious. We refer to Barras, Scaillet, and Wermers (2010) andBajgrowicz and Scaillet (2012) for further discussion, background, and applications of the FDRmethodology in finance (see also Harvey, Liu, and Zhu (2016) for multiple testing issues in factormodeling). In this section, we study the dynamics of jump arrivals on the bitcoin market. We aim to assessthe presence of jumps and their distributional properties. We qualify market conditions favoringthe apparition of discontinuities and show that jumps have a positive impact on market activityand illiquidity.
We apply the high-frequency jump detection test of Lee and Mykland (2012) with FDR controlat a 10% level and find 124 jump days in the period June 2011 to November 2013, or approxi-mately one jump date per week. Table 1 reports the summary statistics for the jumps detectedfrom 5-min intervals and Figure 3 shows the histogram of jump sizes. In 70 cases, the jumphas a positive size, and in 54 cases, a negative size. This contrasts with the common idea thatjumps depict mainly price crashes. The average size of a positive jump is 4.7%, and that of anegative jump is − . p -values ofthe jump test statistics, as well as the 1% confidence threshold and the FDR threshold. We seethat a fixed level of 1% is too permissive and leads to many spurious detections. Interestingly,the thresholding only discards 35% of rejections, where Bajgrowicz et al. (2016) marked up to95% as spurious detections on Dow Jones stocks. This is due to the adaptiveness of the FDRcontrol, which is less strict where there are many true jumps in the data.13 able 1: Summary statistics of jumps The table shows summary statistics for the jump detections from June 26, 2011 to November29, 2013. The first column considers all jumps. The second and last columns consider positiveand negative jumps, respectively.
All jumps Positive jumps Negative jumpsN 124 70 54Mean 0.82% 4.65% -4.14%Mean (abs.) 4.43% 4.65% 4.14%Med (abs.) 3.51% 3.47% 3.52%Max 32.13% 32.13% -0.76%Min -12.20% 1.24% -12.20%Std dev. 5.69% 4.37% 2.43%Skewness 1.33 4.05 -1.09Kurtosis 9.26 24.27 3.94
Jump size -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R e l a t i v e nu m be r o f ob s e r v a t i on s Figure 3: Histogram of the size of jumps
The figure shows the distribution of jump sizes for the detections from June 26, 2011 toNovember 29, 2013. In 70 cases, the jump exhibits a positive size. The average size of a positivejump is 4.7%, and that of a negative jump is 4.1%. The largest discontinuity is a positive jump of32%.
A widely-used assumption is that jump arrival times follow a simple Poisson process, orequivalently that durations between successive jumps are independent and exponentially dis-tributed. We study the dynamics of jump arrivals to assess whether this assumption is consistentwith empirical data. Figure 5 shows the number of jump detections per quarter on the wholedata set. It suggests that the frequency of days with jumps varies across time. Because our testonly indicates whether at least one jump occurred on a given date but does not give the exactnumber of jumps within that day, we cannot test the null hypothesis of exponential inter-jump14 ul 2011 Nov 2011 Mar 2012 Jul 2012 Nov 2012 Mar 2013 Jul 2013 Nov 2013 p - v a l ue TestsFalse discoveriesTrue jumps
Figure 4: p -values of detection statistics The figure displays the p -values of Lee and Mykland (2012) statistics, for every day from June26, 2011 to November 29, 2013. The solid line indicates the 1% confidence level and the dashedline indicates the FDR threshold. The 1%-level is too permissive and leads to many spuriousdetections due to multiple testing. Table 2: Runs test
The table shows the results of runs tests applied to jump detections from June 26, 2011 to Novem-ber 29, 2013, as well as on three sub-periods of equal length.
Period p -value Jumps DaysJun 26, 2011 – Apr 16, 2012 0.01 67 296Apr 17, 2012 – Feb 6, 2013 0.09 21 296Feb 7, 2013 – Nov 29, 2013 0.95 36 296Entire sample < umps/day Figure 5: Number of jumps per day across time
The figure displays the number of jump detections across time, grouped by quarters.
The dynamics of jumps on the bitcoin market contrast with previous literature on high-frequency jump analysis. Bajgrowicz et al. (2016) and Christensen et al. (2014) identify a smallnumber of jumps on large markets such as Dow Jones constituents, market-wide U.S. equityindices and foreign currencies. Bajgrowicz et al. (2016) do not identify clustering in the fewremaining jumps. We investigate the hypothesis that the relative illiquidity of the bitcoinmarket coupled with abnormal market activity is key to understanding sudden moves.
Figure 6 shows an example of a 5% positive jump that occurred on June 10, 2013. The high-lighted region emphasizes the time interval with the maximum absolute value of L ( t j ) duringthat day. As illustrated in Panel (c), the jump occurs after an apparent increase in the tradingvolume and the order flow imbalance. Panel (d) also reveals multiple spikes in the bid-ask spreadas well as a general widening of the spread shortly before the discontinuity. In this section, weinvestigate the conjecture that the relative illiquidity of the bitcoin market coupled with abnor-mal market activity is key to understanding sudden moves. Specifically, we hypothesize thatjumps are the result of liquidity drying up in certain market conditions, in conjunction with a16 a) Price (b) Bids and asks(c) Volume (d) Spread Figure 6: Jump event of June 10, 2013
The panel illustrates the jump detection of June 10, 2013. Panel (a) displays the price series (solid)and the pre-averaged price (dashed). The dark region shows the jump detection period. Panel(b) emphasizes the bid and the ask prices across time. Panel (c) shows the directional volume.Positive (negative) bars count the cumulative volume initiated by aggressive buyers (sellers). Darkbars represent the order flow. Panel (d) shows the evolution of the bid-ask spread. regime change in the order flow. We consider a regular time series at a 5-minute frequency on the whole sample. For each5-minute period i , we set Y t i = 1 if a jump was identified during the period i and 0 otherwise,and compute the following statistics using the tick data: • M S i is the bid-ask spread, calculated as the median of the ratio of the bid-ask differenceto the mid-price. We use the bid-ask spread factor as a proxy for market illiquidity. • OF i is the absolute order flow imbalance, defined as the absolute value of the difference For a study on the importance of the order flow on price discovery, see, e.g., Evans (2002), Evans and Lyons(2002), Green (2004) and Brandt and Kavajecz (2004). Our results are robust to the choice of frequency. We get similar estimates at a 10-minute and 20-minutefrequency, but the significance of estimates decreases strongly at 20-minute. We also try alternative measures ofthe spread such as the maximum spread on the period or the average spread with no qualitative change. We notethat because the jump test of Lee and Mykland (2012) only reveals the largest jump of the day, we might haveseveral time indices i for which Y t i is incorrectly set to 0 in the regression. OF i thusindicates excessive buying pressure in the market. • W R i is the ‘whale’ index calculated as the ratio of the number of unique passive tradersto the total number of unique traders during the period. The ratio is large when fewaggressive traders are responsible for most of the transactions. • P i is the median observed price. • RV i is the realized variance of the latent price during the period, given by the noise-robustestimator of Podolskij and Vetter (2009). • N V i is the variance of the microstructure noise, estimated as in Lee and Mykland (2012).The order flow imbalance OF i and the whale ratio W R i quantify two different aspects of thetrading pressure that were not directly observable by market participants. The former measuresexcess directional volume, irrespective of the number of traders responsible for the divergence.For the latter, we take advantage of the richness of our data set that allows us to track theactivity of each individually identified trader. The whale index thus gives us a measure of theimbalance between liquidity providers and liquidity takers: a large estimate indicates that fewtraders are responsible for most of the liquidity taking.We apply a binary probit model to assess the predictive power of these statistics on theprobability a jump in the next period and verify our hypothesis. Formally, P [ J i +1 | M S i , OF i , W R i , P i , RV i , N V i ] = Φ (cid:0) β + β ,i + β ,i + β MS M S i + β OF OF i + β W R
W R i + β P P i + β RV RV i + β NV N V i (cid:1) , (1)where Φ is the Gaussian cumulative distribution function and 1 t : t ,i = 1 if t ≤ i ≤ t , zero The term ‘whale’ is frequently used to describe the big money bitcoin players that show their hand in thebitcoin market. The large players being referred to are institutions such as hedge funds and bitcoin investmentfunds. able 3: Jump predictability The table displays estimates of the probit regression model in Equation 1. On Panel A, we compute statistics for periods of 5 minutes. On PanelB, we compute statistics for periods of 10 minutes. First four columns show estimates for the model including fixed effects; last four columns do notinclude fixed effects. The ‘Marg. prob.’ columns shows the marginal probability change induced by a one-standard deviation change in the values ofthe covariates from their respective sample averages.
With fixed effects Without fixed effectsCoefficient Est. Std error p -value Marg. prob. Est. Std error p -value Marg. prob.Panel A: 5-minute periodsIntercept -3.61 0.13 < < < < < < < < < < R < < < < < < R therwise. We add fixed effects for the same sub-periods as in Section 4.1 to control for thechanging market conditions associated with the rapid development of the market for bitcoin.Table 3 exhibits the parameter estimates and their respective significance levels. The adjustedpseudo- R = 0 .
07 confirms the predictive power of the regression, and the unreported likelihoodratio test rejects the constant model at the 0 .
1% level.The estimates for β MS and β OF are both positive and significant, showing the strong impactof market illiquidity and order flow on jump risk. This confirms the results of Jiang, Lo, and Verdelhan(2011), who find that illiquidity factors and order flow imbalance play a positive role in the oc-currence of jumps in the U.S. Treasury market. The estimate of β W R is significantly positive aswell, indicating that it is not only an imbalance in volume that increases jump risk, but also anasymmetry in the number of aggressive traders relative to their passive counterparts. For β P , itis significantly negative, supporting the intuition that jumps have less probability of occurringas the bitcoin market develops and its size increases. Microstructure noise variance plays anegative role in the occurrence of jumps. We can explain the negative sign by the probit modelcapturing the dominant effect that very large values (or at least above the time series average)of microstructure noise variance are not being followed by a jump most of the time. When themicrostructure noise variance is large, the market participants do not get a clear signal of thefundamental value of the asset and do not seem to adjust their expectations in an abrupt way.Yet, in contrast to Jiang et al. (2011), realized variance has no significant impact on jump risk.Setting aside the obvious differences between the markets for U.S. Treasuries and bitcoin, webelieve that the divergence is explained by our use of robust-to-noise estimators and multipletesting adjustments for jump detection on 5-min intervals. The positive impact of the realizedvariance in their empirical results from jump detection on 5-min intervals for many consecutivedays could be a consequence of spurious detections.Panel B of Table 3 reports the estimation of the same model for periods of 10 minutes. Theresults are consistent with the estimation with 5-minute periods, albeit less categorical, with20 slightly lower adjusted pseudo- R and the coefficient for microstructure noise variance losingsignificance, which again highlights the importance of considering high-frequency data for suchan analysis. Our findings thus indicate that jumps are systematically associated with marketconditions characterized by a low level of liquidity and the presence of few large and activedirectional traders. We perform a post-jump analysis of the market dynamics. On Figure 7, we plot the averagedynamics of the whale index, the bid-ask spread, the noise variance and the absolute order flowaround jumps. The graphs show that these measures are affected before and after a jump. Thewhale ratio surges right before a jump, as shown already in Section 4.2, but quickly reverts to itsprevious level. The bid-ask spread and the microstructure noise variance gradually increase andpeak right around the jump, followed by a slow reversion. The order flow imbalance massivelyincreases before the occurrence but falls to below-average levels right after that. This figureillustrates the intuition of the previous section about the influence of market forces on pricediscontinuities: aggressive traders placing massive orders, in conjunction with market illiquidityare a significant signal for the occurrence of jumps.The figure emphasizes the market reaction and dynamics after the jumps. We aim to deter-mine if market conditions are affected and how persistent the possible subsequent changes are.We consider the same set of statistics as in the model of Equation (1), and include additionallythe trading volume and the number of traders. For each jump, we compute the statistics onfour consecutive spans of 15 minutes following the detection period. We compare the statisticsto a reference period preceding respective jumps by one hour. We define the test statistics asthe log-ratio of the post-jump measure over the reference measure for each period. We run aStudent t -test to assess changes in the means. Table 4 gathers the results of t -tests, grouped bytheir respective spans. We find that all measures are exacerbated in the 15 minutes immediately21 ime -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 W ha l e s ( nu m . o f t r ade s ) (a) Whales Time -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 M ed . s p r ead (b) Spread Time -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 N o i s e v a r i an c e × -5 (c) Noise variance Time -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 A b s . o r de r f l o w (d) Order flow imbalance Figure 7: Market factors around a jump
The panel illustrates four statistics averaged across all jump detections for different periods aroundjump times. Panel (a) displays the ratio of passive traders over the total number of traders. Ahigh value indicates that few traders are responsible for most liquidity-taking. We observe a clearspike right before the jump occurrence. Panel (b) shows the median bid-ask spread (normalizedby the price). We observe a slow widening of the spread punctuated with a large increase beforethe jump, and a slow reversion to normal levels afterwards. Panel (c) shows the microstructurenoise variance, with a significant spike right before and right after a jump detection. The level ofthe microstructure noise is higher on average after the jump than before. Panel (d) displays theabsolute order flow imbalance, which rises sharply before a jump and quickly reverts to normallevels afterwards. following a jump. The trading volume and the absolute order flow imbalance are abnormallyhigh. At the same time, the number of active traders, and the proportion of aggressive tradersare significantly larger. Liquidity proxies including the bid-ask spread and the microstructurenoise variance see an increase too, as well as the realized variance. However, the impact of jumpsdampens after 30 minutes already. After 45 minutes, all measures revert to anterior levels exceptthe market price: a positive jump generally induces a persistent lower price—and reciprocally, anegative jump induces a higher price. Figure 8 illustrates this feature by showing the (rescaled)average price around positive and negative jumps, respectively. Jumps tend to occur in episodes22 ime -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 M ed . p r i c e (a) Negative jumps Time -30 -25 -20 -15 -10 -5 5 10 15 20 25 30 M ed . p r i c e (b) Positive jumps Figure 8: Jump impact
The panel illustrates the average price levels before and after the occurrence of jumps for a setof 5-minute tranches. Prices are normalized with respect to the price minutes before the jumpto be comparable. Each bar corresponds to the median normalized price on the 5-minute periodconsidered. Panel (a) only considers negative jumps and Panel (b) only considers positive jumps. of massive price trends and act in an opposite direction to allow for an abrupt and quick pricecorrection. This type of correction is not observed on other markets with stability and liquidityproviding mechanisms. The presence of jumps in the dynamics of asset prices remains a debated question in the empiricalliterature. While many jumps may be detected in low-frequency data, recent studies basedinstead on high-frequency data have shown that most are in fact misidentified bursts of volatilityin continuous price paths. True jumps in large-cap stock prices appear to be rare which preventssystematic studies of their properties.In this paper, we have been able to conduct such a study for the bitcoin-to-U.S. dollar(BTC/USD) exchange rate using transaction-level data obtained from Mt. Gox exchange , theleading platform during the sample period of June 2011 to November 2013. We contribute tothe literature in several ways. First, in contrast to large-cap stock markets, we find that jumpsare frequent: out of the 888 sample days, we identify 124 jump days, or on average one jumpday per week. In contrast to the intuition that relates jumps to crash events, most jumps are in23 able 4: Jump impact
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