Modelling Universal Order Book Dynamics in Bitcoin Market
Fabin Shi, Nathan Aden, Shengda Huang, Neil Johnson, Xiaoqian Sun, Jinhua Gao, Li Xu, Huawei Shen, Xueqi Cheng, Chaoming Song
MModelling Universal Order Book Dynamics in Bitcoin Market
Fabin Shi,
1, 2
Nathan Aden, Shengda Huang, Neil Johnson, Xiaoqian Sun, Jinhua Gao, Li Xu, Huawei Shen,
1, 2
Xueqi Cheng,
1, 2 and Chaoming Song ∗ CAS Key Laboratory of Network Data Science and Technology,Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China University of Chinese Academy of Sciences, Beijing 100049, China Department of Physics, University of Miami, Coral Gables, Florida 33142, USA Physics Department, George Washington University, Washington D.C. 20052
Abstract
Understanding the emergence of universal features such as the stylized facts in markets is a long-standing challenge that has drawn much attention from economists and physicists. Most existingmodels, such as stochastic volatility models, focus mainly on price changes, neglecting the complextrading dynamics. Recently, there are increasing studies on order books, thanks to the availabilityof large-scale trading datasets, aiming to understand the underlying mechanisms governing themarket dynamics. In this paper, we collect order-book datasets of Bitcoin platforms across threecountries over millions of users and billions of daily turnovers. We find a 1+1D field theory, governby a set of KPZ-like stochastic equations, predicts precisely the order book dynamics observed inempirical data. Despite the microscopic difference of markets, we argue the proposed effective fieldtheory captures the correct universality class of market dynamics. We also show that the modelagrees with the existing stochastic volatility models at the long-wavelength limit. ∗ e-mail: [email protected] a r X i v : . [ q -f i n . T R ] J a n nderstanding universal emergent properties in different markets is a long-standing chal-lenge for both economists and physicists. As early as the 1960s, Mandelbrot pointed outthat the distribution of logarithmic price return was heavy-tailed in the cotton market which,soon after, was found to hold true in numerous other markets . Since then many manystylized facts have been observed as common across a wide range of instruments, markets,and time periods . This raises a fundamental question: what are the general mechanismsin a financial market leading to these phenomena.Existing approach esof modeling price evolution as a stochastic process to capture thevolatility of a market, such as stochastic volatility (SV) models , have been met withsuccess when attempting to tease out numerous stylized facts such as the volatility clusteringand heavy-tailed price return distributions. However since these models do not includeaspects of the actual trading process, connections between these facts and human behaviorremain outside their scope. A natural extension is then to include the actions of tradersas the mechanism behind creating the price by incorporating all limit orders into bid/askorder books at a given time t and price x , and the matching price at the position x = 0 atwhich buyers and sellers agree to trade. Thanks to technological advances during the pastdecade there are an increasing number of datasets available about order-book dynamicswhich provide the microscopic details of trading dynamics . These details have been usedto construct several models that attempt to bridge the gap between human behavior andmarket dynamics . Bak et.al. considered orders as particles and models the movementof each particle along the price lattice using a random walk . Further work also took intoaccount fixed limit orders and market orders that trigger transactions . More recently,orders were modelled as Brownian colloidal particles bumbling along in a price-fluid .However the common approach in these models is the discretization of price which has thepotential to obfuscate the behavior/market connection with details about how orders aretransacted in the specific market analyzed. In our model we ignore some of these details bysmoothing out the limit order price axis into a continuous spacial dimension. Along with acontinuous time axis, we propose a 1+1D field theory to explain some of the stylized factsas resulting directly from the tendencies of traders in a limit order based market.2 . ANALYZING AND MODELING THE ORDER-BOOK DYNAMICS Despite there being several studies based on the order-book datasets for varies securities,these datasets are often limited by quantity, time span, and accessibility. The novelty ofBitcoin however lies in the decentralized nature of how transactions are executed. Tradesinvolving BTC are only recognized as valid once they have been communally mined into thepublicly available ledger, which is known as Blockchain. This intrinsic market data avail-ability has lead to extensive study since its inception in 2009. The first Bitcoin exchangesemerged in 2010 providing a uniquely public look into the mechanics of exchange tradingincluding orderbook dynamics. Some early analyses of this data focused primarily on stan-dard financial methods to compare Bitcoin to normal currencies . Later works exploredprice prediction and stability analysis . More recently, Bitcoin has entered the publicdiscourse by exploding in value throughout 2017 and then bursting soon after in early 2018thereafter continuing to rise and fall in diminishing motions, seemingly approaching a stablevalue. This long and varied public economic history makes it an ideal candidate on whichto test our model.We use three datasets collected through different online Bitcoin trading platforms: (i) OKCoin was the largest Bitcoin exchange platform in China, consisting of millions of usersand billions of turnovers per day until being shut down in 2017 due to government policy.We collected order-book data from OKCoin from Nov. 3rd, 2016 to Jul. 28th, 2017 (withan unfortunate gap from Jan. 4th, 2017 to Mar. 1st, 2017 due to machine failure). SinceOKCoin introduced an additional transaction fee on each order after Jan. 24, 2017, wedecided to split the data in two: Nov. 3rd, 2016 to Jan. 4th, 2017 (OKCoin1) and Mar.1st, 2017 to Jul. 28th, 2017 (OKCoin2). (ii)
We also collected data from BTC-e, one of thelargest Bitcoin trading platforms headquartered in Russia, from May 3rd, 2017 to Jul. 26th,2017. (iii)
And lastly from Coinbase, a US-based Bitcoin trading platform, from Jan. 23rd,2018 to Apr. 18th, 2018. The order-book datasets collected for each of these three domainsrecord the profiles of the bid (limit buy) and ask (limit sell) orders every few seconds duringthe stated observation period. We are unable to track the instantaneous change of eachorder. Nevertheless, for OKCoin1 we also collected the market order transaction data persecond by recording the total number of market orders which are higher/lower than the bestprice (bid/ask) and immediately match to one or more active orders upon arrival.3e introduce a 1 + 1 D continuous field (CF) model to explain the dynamics found in thebid/ask order volumes, n + ( x, t ) and n − ( x, t ). The spatial dimension x ≡ ± ln p ( t ) ∓ ln p x ≥ p x and the trading price p ( t ) with thetwo signs correspond to the bid/ask axes respectively for the notational convenience ofkeeping x positive. Fig. 1a demonstrates a typical bid/ask order-book profile over time.Figure 1c–f plots the distribution of the order volume change among bids, ∆ n + ( x, t ) ≡ n + ( x, t ) − n + ( x, t − ∆ t ), for a fixed x and n and various values of ∆ t , revealing a fat-tailednature for both positive and negative tails. Similar results observed among the ask side for∆ n − . Any change in the volume of orders away from the x = 0 boundary must come fromone of three possible order-book operations, i) order placement (OP), ii) order cancellation(OC), and iii) order modification (OM), as illustrated in Fig. 1b. We model these threeoperations as follows: (1) Order Placement : Traders place a new order on top of previous orders at some price x (cid:54) = 0. It suggests that in the continuous case we can model the change in order volume dueto order placement, notated as dn OP ± ( x, t ) as dn OP ± ( x, t ) = σ in ± ( x ) ξ ± ( x, t ) dt, (1)where ξ ± ( x, t ) is continuous set of random variables satisfying some one-sided stable dis-tribution. We find that we must allow the scale parameter σ in ± to depend on the position x . This general ingredient of order-book dynamics has been found in both the Paris StockExchange and the London Stock Exchange . (2) Order Cancellation : Traders cancel orders which they have placed previously. InFig. 1g–j, we have plotted the time averaged change of order volume at some fixed x , (cid:104) ∆ n + ( x, t ) (cid:105) t against the current order volume. Unlike the Order Placement (1) wherechanges are independent of n , we see a linear dependence consistent with an existing study from which we can intuit the form of the order cancellation term to be dn OC ± ( x, t ) = − σ out ± ( x ) n ± ( x, t ) ζ ± ( x, t ) dt. (2)The scale parameter σ out ± , similar to σ in ± , depends on the current position x and again ζ ± ( x, t )is a random variable satisfying the same stable distribution above. (3) Order Modification : Traders change the price of orders that they own. Empiricallythere exists a negative correlation between ∆ n + ( x, t ) at different positions in Fig. 1k–n4uggesting that the order modification operation can be modeled as a diffusion processalong the order-books. Therefore, the order modification term is dn OM ± ( x, t ) = ∂ ∂x D ± ( x ) n ± ( x, t ) dt, (3)where the diffusion rate D ± ( x ) depends on the position in general. It is possible that thenegative correlation we observed is due to a combination of order modification and thecorrelated behaviors of adding/removing orders, perhaps through different users. As aneffective field model such microscopic differences are effectively the same and all capturedby the diffusion term (see Supplementary Section S2 for a direct validation of (3) using anadditional dataset).Directly from the chain rule we obtain dn ± ( x, t ) dt = ∂n ± ( x, t ) ∂t ± ∂n ± ( x, t ) ∂x v ( t ) , (4)where v ( t ) ≡ d ln p ( t ) /dt is the velocity of logarithmic price. The total derivative would thenbe simply the sum of the effects of order operations determined above (1)–(3) leading to ourfirst stochastic differential equation ∂n ± ( x, t ) ∂t = ∂ D ± ( x ) n ± ( x, t ) ∂x ∓ v ( t ) ∂n ± ( x, t ) ∂x + σ in ± ( x ) ξ ± ( x, t ) − σ out ± ( x ) n ± ( x, t ) ζ ± ( x, t ) . (5)Unlike limit orders, when market orders are placed they are set to execute immediately atthe trading price – even before limit orders momentarily existing at the x = 0 boundary.Therefore the discrepancy in these orders placed in a short period of time, denoted J ( v ) ≡ ∆ n MO + ( v, t ) − ∆ n MO − ( v, t ) controls the flow of orders through the x = 0 boundary meaninga positive excess would indicate more buyers than sellers so the discrepancy would begindepleting the reservoir of ask limit orders and vice-versa. Applying the continuity equationgives the rate of change of the total volume in n ± ( x, t ) as ∂∂x ( D ± (0 , t ) n ± (0 , t )) ∓ v ( t ) n ± (0 , t )which must be conserved by the market orders leading to v ( t ) = 1 n ( t ) (cid:20) J ( v, t ) + ∂D − n − ∂x (0 , t ) − ∂D + n + ∂x (0 , t ) (cid:21) , (6)where n ( t ) = n + (0 , t ) + n − (0 , t ). Equations. (5)–(6) give a complete description of our CFmodel which exhibits the relationship between order placement, order cancellation, ordermodification, and price change.From here we describe two important aspects of the traders’ reactions to velocity of theprice, J ( v, t ). The first is the influence of trend-following. The intuition being that the5raders will try to follow the changing price e.g. that traders would prefer placing bid ordersas the price is increasing and ask orders as it is decreasing. In Fig. 2a, we observe exactlythis: J ( v, t ) is the linear response to v for small velocity but also saturates at high speeds.The work done by Kanazawa suggests that this curve approximately follows a hyperbolictangent. Thus we set J ∝ tanh( v/v ), fitting the empirical data well. The other is theinfluence of market activity. When the market is moving at high speeds in either direction,it seems to cause more activity among the traders. In Fig. 2b, the total change in marketorder volume over a small time-step ∆ n MO + + ∆ n MO − is observed to increase as the magnitudeof the velocity grows, verifying the existence of this influence. We chose a natural fit to thisdata using ∆ n MO + + ∆ n MO − ∝ − sech( v/v ). These two equations combine to describe thebehavior of market orders (Fig. 2c),∆ n MO ± = [ ± k tanh( v/v ) + k ∞ − k sech ( v/v )] v . (7)We also analyzed the rms change in the total number of limit orders over short periodof time and found that it too approximately follows equation (7) according to (Fig. 2d).It is then reasonable to believe that the traders’ reactions to the movement of the tradingprice at any x should mirror in form that of the reaction seen in market order activity.We propose that the limit order placement activity function is of the form σ in ( x, v ) =[ k in ( x ) tanh( v/v in ( x )) + k in ∞ ( x ) − k in ( x ) sech ( v/v in ( x ))] v in ( x ) where to avoid cluttering thenotation we have left off the ± subscripts. II. MODEL PREDICTIONS
To test the validity of our model, we conduct some simulations of the order-book dynam-ics (Supplementary Section S1) and compare the simulation results with empirical data inthe OKCoin1, OKCoin2, BTC-e, and Coinbase datasets. We first indirectly provide evidencesupporting the validity of the form of the three trader operations that we have included inthe model. A consideration of the diffusion-less ( D ± ( x ) = 0) and point process J = 0 limitsof our model consisting only of traders placing and canceling orders at random leads toa linear relationship between (cid:104) ∆ n ± ( x, t ) (cid:105) t and (cid:104) n ± ( x, t ) (cid:105) t which is verified in (Fig. 1g–j)since empirically the contributions of the diffusion term were small on time scales where thevelocity doesn’t change very much. We also see justification for the heavy tails of ξ ± ( x, t )6nd ζ ± ( x, t ) in the heavy tail observed on the distribution for ∆ n + (Fig. 1c–f). The finalrow of figures (Fig. 1k–n) show the classic signs of the negative rebounds on either side ofthe self-correlating spike common to diffusion processes with a more detailed analysis givenin the supplementary materials (Supplementary Section S2).Moreover, we measure the distribution of the absolute value of instantaneous price re-turn, which is important for understanding the market, quantifying risk, and optimizingportfolios . Because it is defined as the logarithmic ratio of the price before and after thesmallest discernible unit of time in the market (tic, τ ) the price return is equivalent to thevelocity times the tic | v ( t ) τ | . In Fig. 3a–d, we plot the pdf for the price return normalized toabsorb the effect of tic size which is of course irrelevant in our continuous model. We showthat the heavy tail of this distribution decays with an exponent of α ≈ − . As it will be shown, the methodin which our model predicts this exponent is very general suggesting that this mechanismis a sufficient explanation for this universality class independent of Bitcoin specific marketdetails.Simulations reveal the diffusion terms in equation 6 to be negligible in the influence onprice movement allowing v ( t + τ ) ≈ J ( v ( t )) /n ( t S ) where care has been taken in the writingthe correct time dependence. Thus we construct the infinitesimal for velocity as dv ≈ J ( v ) n − v where every quantity is now evaluated at the same time. We can construct the Fokker-Planckequation for the distribution of the returns by writing the itˆo SDE dv = µ ( v ) dt + σ ( v ) dW t .We then measure the drift and diffusion coefficients by finding the relationships (cid:104) ∆ v (cid:105) = − v so that µ ( v ) = dd t (cid:104) v (cid:105) ≈ τ (cid:104) ∆ v (cid:105) = − vτ and (cid:104) Var( v ) (cid:105) ≈ v = 0 so that k ≈ k ∞ and σ ( v ) = v n τ (cid:104) k tanh (cid:16) vv (cid:17) + (cid:16) k ∞ − k sech (cid:16) vv (cid:17)(cid:17)(cid:105) (Supplementary Section S3). We thenuse the Fokker-Planck equation, ∂∂t p ( v ) = − ∂∂v [ µ ( v ) p ( v )] + ∂ ∂v [ σ ( v )2 p ( v )] . (8)to solve for the stable solution p ( v ) ∝ σ ( v ) e (cid:82) µ ( v ) σ v ) dv . (9)which is the general form of the price return distribution. A summary of solutions to this7quation are given below p ( v ) ∝ exp (cid:18) − n v v ( k ∞ − k ) (cid:19) | v | → p ( v ) ∝ v − − k − n (cid:28)| v | (cid:28) v p ( v ) ∝ exp (cid:18) − n v v ( k ∞ + k ) (cid:19) v (cid:28)| v | In the regime where the power law dominates we find that n ≈ k in the OKCoin1dataset (Supplementary Section S3) which gives a power of − and KSTT . The CS modeldeals with specific trader behavior in allowing for the placement and cancellation of orderswhereas the KSTT model focuses on the traders’ reaction to changing price. However neitherproduce a price return distribution with the appropriate universal exponent (Fig. 4a) sincein both models the variance of the change in velocity is independent of velocity which,according to equation (9), implies that the distribution of price return follows a Gaussian.In addition to price return, we also verify some other useful quantities from the modelwith our data. The relation between second moment of the velocity (cid:104) v (cid:105) and market ordervolume n is calculated with an expectation with respect to the conditional distribution on n so we have (cid:104) v (cid:105) = (cid:90) p ( v | n ) v dv. (10)In our model, the theoretical value of (cid:104) v (cid:105) fits the empirical data well in Fig. 4b. (cid:104) v (cid:105) iscomposed of an exponential decay and/or a power-law decay with power-law exponent -2for different limits on n summarized in the supplementary section (Supplementary SectionS4). We again note the predictions from the CS and KSTT models are insufficient to fullyexplain this observation. In the CS model σ (∆ v ) ∝ n − as in our model however the fit ispoor and in the KSTT model the conditional probability p ( v | n ) is independent of n givingan approximately constant result.Another point of distinction for our model is the correlation between velocity and totalchange of order volume. In our model, the correlation. Fig. 4c shows that (cid:104) v, ∆ n (cid:105) decreasesfrom positive to negative for bid order and vice-versa for ask orders and both go to 0as x → ∞ in agreement with the empirical data. The previous works capture only theproperties of order-book dynamics in certain regimes. The KSTT model assumes all of theinvestors are high-frequency traders, which enlarges the influence of trend-following in the8egion far away from price leading to a correlation that does not taper to zero far awayfrom the trading price. In contrast, the CS model completely ignores traders’ reaction tothe changing price velocity, leading to the deviation from an empirical value near the price.Both of the previous works also neglect the influence of market activity therefore, (cid:104) ∆ n (cid:105) / is approximately equal at different velocities while the curve our model produces agrees withthe empirical data (Fig. 4d).To conclude, the above simulation results and analysis indicate that our model can pre-cisely capture and potentially explain the power law decay of the price return distributionfound to be universal across a wide range of markets. We also report the success of ourmodel in demonstrating some of the key features of order-book dynamics as an improve-ment over previous work. However one obvious limitation of our model is the lack of oftemporal correlation in the traders’ reactions. It is reported that the time series constructedby assigning the value +1 to incoming buy orders and − . Since the order placement ∆ n OP in our model follows astable distribution which leads to the absence of long memory in the order flow, we cannotpredict these results.Finally, we point out that our model can be used for price prediction provided qualitydata. Current methods of predicting price, such as ARMA and GARCH , are based onthe price series and while order book data can be over-valued in some financial analyses,our model would constitute the basis for a complementary approach to more conventionalmethods. Since our model is only concerned with the mesoscopic details of order booktrading, many of the complications of using order book data for price return predictionaren’t an issue such as so-called iceberg orders wherein a single market maker tries to sell alarge amount of a security secretly by not listing it all at once. As long as the orders followone-sided stable distributions and general trader reaction trends, our model is applicable. ACKNOWLEDGEMENTS
The authors thank Hao Zhou for helpful comments. X.S was supported by the NationalNatural Science Foundation of China under award numbers 61802370. X.C was supportedby the National Natural Science Foundation of China under award numbers 60873245. H.Swas supported by the K.C. Wong Education Foundation.9
UTHOR CONTRIBUTIONS
C.Song conducted the project. F.S. collected and curated datasets, and performed nu-meric simulation. C.S., F.S., N.A., S.H. developed the model and calculated analyticalresults. C.S., F.S., N.A., S.H., X.S., J.G., L.X., H.S., X.C., and N.J. analyzed data andcontributed to the writing of the manuscript.
COMPETING INTERESTS
The authors declare no competing interests.10
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Order Placement (OP) Order Cancellation (OC)Order Modification (OM) relative positionrelative position x=ln p ( t ) - ln p x x=ln p x - ln p ( t ) x=0p ( t ) n - ( x,t ) n + ( x,t ) -2 0 2 4 6 8 1010 -3 -2 -1 P ( n | n ) n n=1n=5n=10 CF model ( n=10 )stable distribution x=0.00005 -2 0 2 4 6 8 1010 -4 -3 -2 -1 x=0.001 P ( n | n ) n n=1n=5n=10 CF model ( n=10 )stable distribution -2 0 2 4 6 8 1010 -4 -3 -2 -1 x=0.0005 P ( n | n ) n n=1n=5n=10 CF model ( n=10 )stable distribution -2 0 2 4 6 8 1010 -3 -2 -1 x=0.0001 P ( n | n ) n n=1n=5n=10 CF model ( n=10 )stable distribution 〈 n 〉 n DataCF model x=0.00005 〈 n 〉 n DataCF model x=0.001 〈 n 〉 n DataCF model x=0.0005 〈 n 〉 n DataCF model x=0.0001 V a r ( n x , n y ) x y=0.0025y=0.0035 V a r ( n x , n y ) x y=0.025y=0.04 V a r ( n x , n y ) x y=0.02y=0.03 V a r ( n x , n y ) x y=0.0003y=0.003 Fig. 1.
Analysis and modelling the three types of order-book operations in differentBitcoin markets. a ) A typical ask/bid order-book profile. b ) The schematic description of threeorder-book operations. c–f ) The conditional distribution, P (∆ n | n ) for c ) OKCoin1, d ) OKCoin2, e BTC-e and f ) Coinbase. Dots denote measurements from data and lines are measurements fromsimulation. g–j ) The change of order volume ∆ n versus order volume n for g ) OKCoin1, h ) OK-Coin2, i BTC-e and j ) Coinbase. Dots denote measurements from data and lines are measurementsfrom simulation. k–n ) The correlation (cid:104) ∆ n x , ∆ n y (cid:105) (the correlation between the change of ordervolume at different positions) versus position x for k ) OKCoin1, l ) OKCoin2, m ) BTC-e and n )Coinbase. Dots denote measurements from data and lines are measurements from simulation. bc d -20 -10 0 10 20-40-2002040 OKCoin1 〈 n + M O - n - M O 〉 v -20 -10 0 10 2004080120160 OKCoin1 〈 n + M O + n - M O 〉 v -20 -10 0 10 200306090120 OKCoin1 〈 n M O 〉 v n -MO n +MO -2 -1 0 1 2 〈 n 〉 / v Bid order
OKCoin1 〈 n 〉 / Ask order
Fig. 2.
The traders’ reaction to velocity. a ) The discrepancy of market order ∆ n MO + − ∆ n MO − versus velocity v in OKCoin1. Dots denote measurements from data, whereas the curve is a guideto the eye, following ∆ n MO + − ∆ n MO − ∝ k tanh( v/v ). b ) The market order volume ∆ n MO + +∆ n MO − versus velocity v in OKCoin1. Dots denote measurements from data, whereas the curve is a guideto the eye, following ∆ n MO + + ∆ n MO − ∝ k ∞ − k ∞ sech( v/v ). c ) The market order ∆ n MO ± versusvelocity v in OKCoin1. Dots denote measurements from data, whereas the curve is a guide to theeye, following ∆ n MO ± ∝ [ ± k tanh( v/v ) + k ∞ − k sech ( v/v )] v . d ) The root mean square of ∆ n versus normalized v in OKCoin1. Dots denote measurements from data, whereas the line is a guideto the eye, following (cid:104) ∆ n (cid:105) / ∝ [ ± k (cid:48) ( x ) tanh (cid:16) v/v (cid:48) ( x ) (cid:17) + k (cid:48) ∞ ( x ) − k (cid:48) ( x ) sech ( v/v (cid:48) ( x ))] v (cid:48) ( x ). bc d -1 -5 -4 -3 -2 -1 OKCoin1 P r ob a b ilit y DataCF model
Absolute of price return α =-4 -1 -5 -4 -3 -2 -1 OKCoin2 P r ob a b ilit y DataCF model
Absolute of price return α =-4 -1 -5 -4 -3 -2 -1 BTC-e P r ob a b ilit y DataCF model
Absolute of price return α =-4 -1 -5 -4 -3 -2 -1 Coinbase P r ob a b ilit y DataCF model
Absolute of price return α =-4 Fig. 3.
The distribution of absolute value of the normalized price return in differentBitcoin market.
The probability distribution of absolute value of the normalized price returnfor a ) OKCoin1, b ) OKCoin2, c ) BTC-e and d ) Coinbase. Dots denote measurements from data,lines are measurements from simulation. The dot dash, shown as a guide to the eye, represents apower-law decay with exponent α = − bc d -1 -6 -4 -2 P r ob a b ilit y Absolute of price return
DataCF modelCS modelKSTT model
OKCoin1 -1 〈 v 〉 n DataCF modelCS modelKSTT modelEquation (10) n OKCoin1 〈 v , n 〉 x OKCoin1 -0.2-0.10.00.1
Ask order
DataCF modelCS modelKSTT model 〈 v , n 〉 Bid order -2 -1 0 1 20246 〈 n 〉 v DataCF modelCS modelKSTT modelOKCoin1
Fig. 4.
The properties of order-book dynamic in OKCoin1. a ) The probability distributionof the absolute value of normalized price return in OKCoin1. Dots denote measurements fromdata and lines are measurements from different models. b ) The variance of velocity (cid:104) v (cid:105) versusorder volume n in OKCoin1. Dots denote measurements from data and lines are measurementsfrom different models. c ) The correlation (cid:104) v, ∆ n (cid:105) (correlation between the change of order volume∆ n and velocity v ) versus position x in OKCoin1. Dots denote measurements from data andlines are measurements from different models. d ) The root mean square of the change of ordervolume (cid:104) ∆ n (cid:105) / versus velocity v in OKCoin1. Dots denote measurements from data and linesare measurements from different models.in OKCoin1. Dots denote measurements from data and linesare measurements from different models.