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Characterising trader manipulation in a limit-order driven market
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Mathematics and Computers in Simulation 93 (2013) 43–52
Original article
Characterising trader manipulation in a limit-order driven market R.M. Withanawasam a,∗ , P.A. Whigham a , T.F. Crack b b
a Department of Information Science, Commerce Building, University of Otago, Dunedin 9054, New Zealand Department of Accountancy and Finance, Commerce Building, University of Otago, Dunedin 9054, New Zealand
Received 22 May 2012; received in revised form 21 August 2012; accepted 25 September 2012 Available online 8 October 2012
Abstract Use of trading strategies to mislead other market participants, commonly termed trade-based market manipulation, has been identified as a major problem faced by present day stock markets. Although some mathematical models of trade-based market manipulation have been previously developed, this work presents a framework for manipulation in the context of a realistic computational model of a limit-order market. The Maslov limit order market model is extended to introduce manipulators and technical traders. We show that “pump and dump” manipulation is not possible with traditional Maslov (liquidity) traders. The presence of technical traders, however, makes profitable manipulation possible. When exploiting the behaviour of technical traders, manipulators can wait some time after their buying phase before selling, in order to profit. Moreover, if technical traders believe that there is an information asymmetry between buy and sell actions, the manipulator effort required to perform a “pump and dump” is comparatively low, and a manipulator can generate profits even by selling immediately after raising the price. © 2012 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Limit order markets; Market manipulation; Market micro-structure models; Bayesian learning
1. Introduction A stock market is a mechanism to trade company stocks among market participants at an agreed price. These market participants compete with each other with the expectation of maximising their trading profits. The imbalance between supply and demand of these participants determines the price direction. Market participants are characterised in terms of the information that they utilise in making a trading decision. Common trader categories are liquidity traders (uninformed), technical traders (information seekers), and informed traders. Motivations of liquidity trader actions are exogenous to the market conditions. Informed traders have private information and they utilise that information to take advantage of others. Technical traders try to generate information using past trading data and are influenced by their perception of informed trading. Manipulators generate false information in order to mislead other market participants. A common form of market structure is the use of a double-auction or limit-order mechanism for determining the bid and ask prices for a stock. This approach is characterised by a limit-order book that stores current bid and ask prices, and a last traded price representing the last sale. Market orders are submitted as buy/sell at the current market ∗
Corresponding author. Tel.: +64 3479 8097. E-mail addresses: [email protected] (R.M. Withanawasam), [email protected] (P.A. Whigham), [email protected] (T.F. Crack). 0378-4754/$36.00 © 2012 IMACS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.matcom.2012.09.012
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28 27 24
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Price
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Date Fig. 1. The price graph of the “pump and dump” manipulation recorded in Universal stock (ticker symbol UVV) in the US market between April 1999 and July 1999.
price. Maslov [16], Rosu [20], and Krause [13] have developed computational models that incorporate the mechanics and trader behaviours of limit order markets. The limit order market model developed by Maslov [16] simplifies the concept of a market place for a particular stock by constructing a limit-order book manipulated by an infinite pool of uninformed traders. In each time step, a new trader is drawn from this pool as a buyer (probability qb ) or a seller (qs ), and either a limit-order (probability qlo ) or a market order (1 − qlo ) is submitted. If a limit-order is submitted, a value is drawn from a distribution Δ as an offset from the last traded price p(t) (trader computes buy/sell limit order price with a negative/positive Δ from p(t)). Each buy or sell order involves a single unit of a stock. Liquidity traders do not consider the state of the current limit-order book, such as the spread, or any past patterns of the last traded price. All limit orders are assumed to be “good till cancelled”. Withanawasam et al. [21] introduced an alternative pricing method (denoted by M*), which computes limit buy/sell order prices with a negative/positive offset (Δ) from best ask/bid price of the Maslov order book (i.e., with respect to the contra side best prices). If the contra side is empty, the limit prices are computed with respect to the last traded price. The M* model does not produce the unrealistic strong cone-shaped patterns in price that can be observed with the Maslov model and is therefore used in experiments with this work [21]. Use of trading strategies with the intention of misleading other market participants (i.e., generating false information) is called “market manipulation”. One such manipulation scenario is commonly termed “pump and dump”, where manipulators pretend to be informed and mislead the market. They buy at successively higher prices, giving the appearance of activity at a higher price than the actual market value, and then sell or dump shares at an inflated price. Fig. 1 illustrates the price behaviour of a “pump and dump” manipulation recorded for “Universal” stock (ticker symbol UVV) between April 1999 and July 1999 in the US (U.S. Securities and Exchange Commission Litigation Release No. 16621/July 6, 2000). Allen and Gale [2] categorised manipulations under three main types, namely action based (e.g., not bidding in an auction), information based (e.g., spreading false rumours and news) and trade based (e.g., performing a series of trades to mislead the market). They also discussed the possibilities of trade-based manipulation and showed that a manipulator can pretend to be informed and mislead a market. Investors always have a problem distinguishing informed traders from manipulators. This problem leads to a manipulator being able to pretend to be informed and mislead the market. Van Bommel [7] showed that a potentially informed trader can pretend to be informed and manipulate the market. The author also analysed situations where a trader could spread rumors in order to distort the true value of stocks. Allen and Gorton [3] showed that asymmetry between the information associated with buying and selling implies that market participants assume buy transactions contain more information than sales. Hence, “a manipulator can repeatedly buy
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Technical trader’s learned demand increases the price further Manipulators buy at a higher rate, so the price starts to go up All the traders are started performing buy/sell actions with equal probability
Price collapses
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Price
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The manipulator sells and profits from the market
tS
Ignition Momentum Call−off Period Period Period tC tM tI
Time
Fig. 2. A graphical representation of the behaviour of the “pump and dump” model.
stocks, causing a relatively large effect on prices, and then sell with relatively little effect” [1, p. 1919]. Aggarwal and Wu [1] presented common characteristics of manipulated stocks. Mei et al. [17] also analysed the properties of data on “pump and dump” cases prosecuted by the U.S. Securities and Exchange Commission from January 1980 to December 2002. Some other notable papers in this area are Berle [6], and Kyle and Viswanathan [14]. Most of the existing manipulation detection methods involve classification of datasets. Statistical methods such as discriminant analysis, logit regression and machine learning methods such as support vector machines, and artificial neural networks have been used in manipulation analysis [18]. Although some mathematical models of market manipulation have been previously developed [1,8], this paper presents a framework for manipulation in the context of a realistic computational model of a limit-order market. The Maslov model is extended by introducing technical traders and manipulators. In the past literature, the Maslov model has not been considered in the context of analysing stock manipulations. Uninformed traders in the original Maslov model do not consider the behaviour of other market participants when taking trading decisions. Because of this, manipulation is not possible in the original Maslov model. We can, however, introduce technical traders who base their trading on the behaviour of informed traders that are believed to be trading in the market. These technical traders extract information from past trading and use that information when making their buy/sell decisions [11]. The manipulator can then introduce wrong information to the market to mislead these technical traders in an attempt to make illegal profits. For our purposes, technical traders utilise a Bayesian learning technique to generate information from informed persons that are believed to be trading in the market. This Bayesian model is useful because it allows us to model a belief structure of a trader type and also allows us to control and tune these beliefs explicitly in order to analyse their learning processes and its effects on the order book. A probability based belief tree and observed trading actions (i.e., orders and trades) are employed in this Bayesian learning technique. Past work of Glosten and Milgrom [10], Almgren and Lorenz [4], Hautsch and Hess [12], and Pastor and Veronesi [19] used Bayesian learning in modelling stock markets. Technical traders can also use alternative methods such as the use of historical prices in predicting future market behaviour in order to generate future trading signals [11]. The graphical representation of the model is depicted in Fig. 2. Fig. 2 is a stylised version of Fig. 1. The manipulator uses the price behaviour in response to supply/demand along with the behaviour of technical traders when attempting to inflate or deflate the price and mislead the market. In doing this, the manipulator continues to buy with a higher probability for a certain period of time. This results in an increase of the buying probability of technical traders and influences them to buy more. This artificial demand created by the manipulator inflates the price. After allowing the technical traders to raise the price further, the manipulator repeatedly sells his stocks as smoothly as possible, without being noticed by other traders, and profits from the market. The supply created by the manipulator changes the buying probability of technical traders and the price collapses.
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The market manipulation model is presented in Section 2. Results, implications of results, and future work are discussed in Section 3. 2. Methodology The “pump and dump” manipulation scenario is modelled using manipulators who mislead technical traders by introducing misleading information to the market. We extended the M* model by adding technical traders, and manipulators. Throughout the simulation, liquidity traders, technical traders, and manipulators are considered to be in a pool and are selected for trading with probabilities pL , pT , and pM , respectively. The simulation (i.e., tT time steps) is divided into two periods: a non-manipulation period and a manipulation period. In the manipulation period, manipulators use the “pump and dump” strategy to introduce the manipulation. In the nonmanipulation period the manipulator’s strategy is similar to the strategy of uninformed traders. The manipulation period starts after tS time steps. All these traders are assumed to have initial stocks and money. The final profit is computed as the difference between initial and final wealth of the trader. In computing the wealth, the stocks at hand are liquidated with respect to the last traded price at that moment. In nullifying the effect of market parameters such as price, profits of manipulators and technical traders are computed with respect to the average liquidity trader profit. Technical traders use a Bayesian learning model to generate information from past informed trading. Using this Bayesian learning model, each technical trader models his probability of price going up at time t as a common πt (starting πt is π0 ). The observed trading actions (orders/trades) are used in revising πt . In other words, the posterior probability of πt given the observed trading action at time t becomes the probability of price going up at time t + 1. A Bayes probability tree based on the trader’s beliefs (Fig. 3) is used in computing the posterior probabilities. When forming this Bayesian belief model, a technical trader takes the following conditions into consideration: whether the trader involved in the observed order/trade is likely to be an informed trader or a liquidity trader, whether this trader is acting as a buyer or a seller, and whether he is placing a limit order or a market order. All these conditions are characterised by conditional probabilities as shown in Fig. 3. Based on this Bayesian learning process, the prior probability of price going up in the future can be stated as: πt =
π0 (A + E)w (B + F )x (C + G)y (D + H)z π0 (A + E) (B + F ) (C + G)y (D + H)z + (1 − π0 )(I + M)w (J + N)x (K + O)y (L + P)z w
x
(1)
where A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, and P are probability multiples of each branch in the Bayesian probability tree given in Fig. 3, and w, x, y, and z denote the number of observed limit buy, market buy, limit sell, and market sell trader actions, respectively. When using the generated information, the technical trader considers the probability of price going up as his probability of buying (i.e., qbT = πt ) (i.e., the technical trader assumes that the signal of price going up is favourable for buying). As a result, buy and sell actions in the market influence the qbT to go up and go down, respectively. This model allows technical traders to carry forward information in past trading in order to generate information when making their decisions to buy or sell. When attempting to inflate or deflate the price and mislead the market, a manipulator uses the price behaviour in response to supply/demand, along with the behaviour of technical traders. The manipulation period is divided into three periods which we term “ignition period”, “momentum period” and “call-off period”. In the ignition period (i.e., tI time steps), the manipulator buys with a probability qbI . The momentum period (i.e., tM time steps) is used by the manipulators to allow the technical traders to raise the price further. Finally, in the call-off period, the manipulator sells his stocks for tC period of time with probability qsC and exits from the market. The time-line is depicted in Fig. 2. The M* model is initialised with p(0) = 1000, qb = qlo = 0.5, π0 = 0.5 and simulated for tT = 26,000 time steps (initially the model is run for 1000 time steps to populate the order book before we start recording the properties). The price offset (Δ) is drawn uniformly from a discrete set of random numbers from 1 to 4. Profits of each trader type are averaged over 100 simulations. In the Bayesian belief model, technical traders assume that when the price is to go up, informed traders are definitely buying (μ1 = 1), and they definitely sell (μ3 = 0) when the price is to go down. It is also assumed that the probabilities of submitting limit/market orders are equal (θ 1 = θ 2 = θ 3 = θ 4 = θ 5 = θ 6 = θ 7 = θ 8 = 0.5). In simulating the “pump and dump” manipulation, the manipulator parameters are set as: tS = 1000, tI = tC = 10, 000,
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Fig. 3. Bayesian diagram to compute posterior probabilities.
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1000 900 800 700
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M* prices M* prices with 20% technical traders M* prices with 50% technical traders
0
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Time Fig. 4. Price behaviour when technical traders are introduced to the M* model. (Note that data were re-sampled at every 50th point for clarity.)
tM = 5000, qbI = 0.9, and qbC = 0.1. This means that the manipulator buys with 0.9 probability for 10,000 steps, waits 5000 steps, and then sells with 0.9 probability for 10,000 steps. The information content possessed by a technical trader depends on the past records that he/she could see when computing πt . For simplicity, we assume that all traders arrive in the market at the same time, and therefore they have the same set of information (i.e., common πt ). The asymmetry between buying and selling is introduced to the technical trader belief model through the percentage of liquidity buying (i.e., μ2 ). The role of supply and demand in determining price direction, how technical traders extract information from trading, and how manipulators use these principles to alter market information are presented through the model properties. 3. Results and discussion Fig. 4 shows price behaviour of the M* model with technical traders. Here, in the technical trader belief model, technical traders expect 0.1% of informed trading and equal percentage of liquidity buying and selling (i.e., μ2 = 0.5). The percentage of technical trading increases the positive correlation in prices. This observation indicates that technical trading adds a persistence behaviour to the M* prices. Positively correlated/persistence price behaviour often observed in real stock prices (i.e., Hurst exponent is greater than 0.5) is a consequence of trader reactions to market information [5]. In Fig. 5, between the time steps 10,000 and 20,000, the manipulator performs buy/sell actions with a higher probability (0.9) to induce a supply and demand imbalance to drive the price up/down. Fig. 5 also shows how the momentum of technical trading amplifies the effect of this price change. Due to this momentum effect, the price continues to go up/down even after the manipulator stops his continuous buying/selling at the 20,000 time step. This momentum can be exploited by the manipulator in his “pump and dump” strategy. Table 1 summarises our results and compares manipulator profit with only liquidity traders and manipulator profit with technical traders in simulations with/without the momentum period and with/without information asymmetry. In a simulation with only technical and liquidity traders (pL = pT = 0.5, λ = 0.1, μ2 = 0.5, and tT = 26,000), it is observed that technical traders are making higher profits (i.e., average relative profit of 91,887 with standard error 10,815)1 than liquidity traders. This confirms our technical traders’ ability to predict the market using past information and use this ability to generate profits. The standard deviations of these trader profits are higher due to the highly random variation of simulated prices.
1 These results are averaged over 100 simulation runs. Note that a lower standard error value can be obtained by using a higher number of simulation runs [15].
1000 1200 1400 1600 1800
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Price drive up without technical traders, pL = 0.8, pT = 0, pM = 0.2 Price drive up with technical traders, pL = 0.4, pT = 0.4, pM = 0.2 Price drive down without technical traders, pL = 0.8, pT = 0, pM = 0.2 Price drive down with technical traders, pL = 0.4, pT = 0.4, pM = 0.2
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Fig. 5. Price driving up/down with/without technical trading. (Note that data were re-sampled at every 50th point for clarity.) Table 1 Mean, standard deviation, and standard error values of technical trader and manipulator profits with respect to the liquidity trader profit averaged over 100 simulation runs. pL , pT , pM denote the liquidity, technical, and manipulator trader selection probabilities, respectively. tM denotes the length of the momentum period and μ2 represents the assumed percentage of liquidity buying in the Bayesian belief model. Simulation
Technical trader profit
Manipulator profit
pL
pT
pM
tM
μ2
Mean
Std dev
Std error
Mean
Std dev
Std error
0.5 0.5
0.5 0
0 0.5
NA 5000
0.5 NA
91,887 NA
108,157 NA
10,815 NA
NA −15,430
NA 46,115
NA 4611
0.4 0.3 0.2
0.3 0.4 0.5
0.3 0.3 0.3
0 0 0
0.5 0.5 0.5
4700 8862 5048
16,518 17,163 17,047
1651 1716 1704
14,332 37,529 49,020
17,294 15,598 16,483
1729 1559 1648
0.5 0.4 0.3 0.2 0.2
0.2 0.3 0.4 0.5 0.5
0.3 0.3 0.3 0.3 0.3
5000 5000 5000 5000 7500
0.5 0.5 0.5 0.5 0.5
6300 14,966 22,441 11,530 89,972
22,985 32,703 40,937 39,159 133,382
2298 3270 4093 3915 13,338
7963 30,278 56,707 71,508 60,829
28,410 38,231 45,394 43,478 59,295
2841 3823 4539 4347 5929
0.6 0.6 0.5 0.4
0.1 0.2 0.3 0.4
0.3 0.3 0.3 0.3
0 0 0 0
0.45 0.45 0.45 0.45
−16,371 −20,664 −19,408 −15,631
11,064 16,348 21,610 27,700
1106 1634 2161 2770
3526 24,747 42,620 63,812
16,252 20,519 23,506 21,433
1625 2051 2350 2143
0.35
0.35
0.3
5000
0.45
−60,510
84,010
8401
197,692
54,409
5440
Manipulator profit with only liquidity trading is very low (i.e., −15,430). This indicates that “pump and dump” type of manipulation is not possible with liquidity type traders. If there is no momentum period and the overall manipulation behaviour is symmetric (i.e., ignition period is almost identical to the call-off period (i.e., tI = tC and qbI = qsC ) and there is no information asymmetry considered between buying and selling (i.e., μ2 = 0.5)), the manipulator profit with technical trading is also comparatively low. These results indicate that with symmetric behaviour, the manipulators are not able to make any significant profits by selling immediately after raising the price either with liquidity traders or technical traders. These results indicate that “pump and dump” type of manipulation is not possible when the market is only composed of liquidity traders. This is because liquidity traders cannot introduce either momentum or information asymmetry to stock prices. When there are technical traders, however, the manipulator can exploit technical trader behaviour by waiting some time between the ignition period and call-off period (e.g., momentum period, tM = 5000) in order to make a higher profit. Here, the technical trader percentage increases the manipulator profit. When the manipulator momentum period increases (i.e., tM = 7500), the manipulator profit decreases. This is because the perception the
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2e+05 0e+00 −2e+05
Trader Profit
4e+05
50
Manipulator Technical Trader
0.40
0.45
0.50
µ2
0.55
0.60
Fig. 6. Last traded price graphs for different liquidity buying percentages (μ2 ), assumed informed trader percentage λ = 0.1. (Note that data were re-sampled at every 50th point for clarity.)
1200
1400
µ2 = 0.4 µ2 = 0.5 µ2 = 0.6
Ignition period
1000
Last Traded Price
1600
manipulator generated among technical traders may die out and the advantage he has produced can be lost. Hence, there is a memory effect for technical traders. Moreover, if technical traders believe in an information asymmetry between buy and sell transactions (i.e., if they believe a buy contains more information than a sale), manipulation is possible/profitable even without this momentum period. In addition, the percentage of technical trading increases the manipulator profit. This information asymmetry is introduced to the model by varying the percentage of liquidity buying in the Bayesian belief model (i.e., μ2 < 0.5). Allen and Gorton [3] also used the percentage of liquidity buying/selling in order to introduce information asymmetry and show the possibility of manipulation in the Glosten and Milgrom [10] model. In this approach, technical traders assumes that μ2 < 0.5 and respond more to buy transactions than sell transactions. As a result, the price change in the manipulators’ ignition period becomes higher than the price change in the call-off period. Price graphs in Fig. 6 illustrate how a manipulator uses this information asymmetry to buy with a higher effect on prices and sell with little effect. Note that Fig. 6 (i.e., simulated pump-and-dump prices) has the same characteristics as Fig. 1 (i.e., real pump-and-dump prices). Moreover, Fig. 7 illustrates how the manipulator profit in comparison with the technical trader profit, varies from positive to negative with the assumed percentage of liquidity buying (μ2 ) in the technical trader belief model. Based on these results, we confirm the Allen and Gorton findings related to the role of information asymmetry in manipulation [3]. Technical trading makes manipulation possible and the manipulator profit increases with the percentage of technical traders (Fig. 8). When analysing the manipulator profits, we set manipulator selection probability pM = 0.3, length of
0
5000
Momentum period
10000
15000
Call−off period
20000
25000
Time Fig. 7. Manipulator profit behaviour for different liquidity buying percentages (μ2 ), assumed informed trader percentage λ = 0.1.
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50000 −100000
0
Trader Profit
150000
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Manipulator Technical Trader
0.0
0.1
0.2
0.3
0.4
0.5
pT Fig. 8. Manipulator profits with the percentage of technical trading (i.e., when changing pT while pM = 0.3, μ2 = 0.45, and tM = 5000).
the momentum period tM = 5000, and liquidity buying percentage μ2 = 0.45 in the Bayesian belief model. This result confirms the findings of Aggrawal and Wu [1] (i.e., manipulation is more profitable when there are more information seekers in the market). Based on these results, using a computational model of a limit order market, we confirm that “pump and dump” manipulation is not profitable with only liquidity trading. Rather, manipulators require the presence of technical traders to profit. In addition, increasing the percentage of technical trading makes manipulation more profitable. When exploiting the behaviour of technical traders, a manipulator can wait some time after the ignition phase to make a profit. However, if technical traders believe that there is an information asymmetry in the market, the manipulator effort required to perform “pump and dump” is low and they can even make profits by selling immediately after raising the price. As future work, this manipulation model could be extended to introduce and simplify the dynamics of other manipulation mechanisms such as “cyclic trading”, “marking the close”, “wash sales”, and “painting the tape” [9]. In the future, these market manipulation models will be considered in terms of analysing market manipulation behaviour and finding manipulation detection mechanisms. 4. Conclusion A computational model was developed to characterise price manipulation in a limit order driven market. The Maslov [16] limit order market model was extended to introduce technical traders and manipulators to be used as a general framework for modelling manipulation strategies. We showed that a manipulator can pretend to be informed to mislead the technical traders in performing a “pump and dump” strategy in a limit order market. Concepts such as the role of supply and demand in determining price direction, how technical traders extract information from trading, and how manipulators use these principles to alter information, have been presented via the model properties. We showed that “pump and dump” manipulation is not possible with solely liquidity type of trading. Rather, manipulators require the presence of technical traders to profit. When exploiting the behaviour of technical traders, a manipulator should wait some time after accumulating stocks to generate a profit. If technical traders believe that there is an information asymmetry in the market, however, then manipulators can even generate profits by selling immediately after buying stocks. Acknowledgement This work has been partially funded by a University of Otago Postgraduate Scholarship. References [1] R.K. Aggarwal, G. Wu, Stock market manipulations, Journal of Business 79 (2006) 1315–1336. [2] F. Allen, D. Gale, Stock-price manipulation, Review of Financial Studies 5 (1992) 503–529.
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[3] F. Allen, G. Gorton, Stock price manipulation, market microstructure and asymmetric information, European Economic Review 36 (1992) 624–630. [4] R. Almgren, J. Lorenz, Bayesian adaptive trading with a daily cycle, Journal of Trading 1 (2006) 38–46. [5] J. Alvarez-Ramirez, J. Alvarez, E. Rodriguez, G. Fernandez-Anaya, Time-varying Hurst exponent for US stock markets, Physica A 387 (2008) 6159–6169. [6] A.A. Berle, Stock market manipulation, Columbia Law Review 38 (1938) 393–407. [7] J.V. Bommel, Rumors, Journal of Finance 58 (2003) 1499–1520. [8] A. Chakraborty, B. Yilmaz, Informed manipulation, Journal of Economic Theory 114 (2004) 132–152. [9] D. Cumming, S. Johan, Global market surveillance, American Law and Economics Review 1 (2008) 1–10. [10] L.R. Glosten, P.R. Milgrom, Bid, ask and transaction prices in a specialist market with heterogeneously informed traders, Journal of Financial Economics 14 (1985) 71–100. [11] L. Harris, Trading and Exchanges: Market Microstructure for Practitioners, Oxford University Press, 2002. [12] N. Hautsch, D. Hess, Bayesian learning in financial markets: testing for the relevance of information precision in price discovery, Journal of Financial and Quantitative Analysis 42 (2007) 189. [13] A. Krause, Fat tails and multi-scaling in a simple model of limit order markets, Physica A 368 (2006) 183–190. [14] A.S. Kyle, S. Viswanathan, How to define illegal price manipulation, American Economic Review 98 (2008) 274–279. [15] D.V. Lindley, A statistical paradox, Biometrika 44 (1957) 187–192. [16] S. Maslov, Simple model of a limit order-driven market, Physica A 278 (2000) 571–578. [17] J. Mei, G. Wu, C. Zhou, Behavior based manipulation: theory and prosecution evidence, unpublished manuscript, New York University. [18] H. Ogut, M.M. Doganay, R. Aktas, Detecting stock-price manipulation in an emerging market: the case of Turkey, Expert Systems with Applications 36 (2009) 11944–11949. [19] L. Pastor, P. Veronesi, Learning in financial markets, Review of Financial Economics 11 (2009) 374. [20] I. Rosu, A dynamic model of the limit order book, Review of Financial Studies 22 (2009) 4601–4641. [21] R.M. Withanawasam, P.A. Whigham, T.F. Crack, I.M. Premachandra, Empirical investigation of the Maslov limit order market model, discussion paper (2010/04), Department of Information Science, University of Otago.
Mathematics and Computers in Simulation 93 (2013) 43–52
Original article
Characterising trader manipulation in a limit-order driven market R.M. Withanawasam a,∗ , P.A. Whigham a , T.F. Crack b b
a Department of Information Science, Commerce Building, University of Otago, Dunedin 9054, New Zealand Department of Accountancy and Finance, Commerce Building, University of Otago, Dunedin 9054, New Zealand
Received 22 May 2012; received in revised form 21 August 2012; accepted 25 September 2012 Available online 8 October 2012
Abstract Use of trading strategies to mislead other market participants, commonly termed trade-based market manipulation, has been identified as a major problem faced by present day stock markets. Although some mathematical models of trade-based market manipulation have been previously developed, this work presents a framework for manipulation in the context of a realistic computational model of a limit-order market. The Maslov limit order market model is extended to introduce manipulators and technical traders. We show that “pump and dump” manipulation is not possible with traditional Maslov (liquidity) traders. The presence of technical traders, however, makes profitable manipulation possible. When exploiting the behaviour of technical traders, manipulators can wait some time after their buying phase before selling, in order to profit. Moreover, if technical traders believe that there is an information asymmetry between buy and sell actions, the manipulator effort required to perform a “pump and dump” is comparatively low, and a manipulator can generate profits even by selling immediately after raising the price. © 2012 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Limit order markets; Market manipulation; Market micro-structure models; Bayesian learning
1. Introduction A stock market is a mechanism to trade company stocks among market participants at an agreed price. These market participants compete with each other with the expectation of maximising their trading profits. The imbalance between supply and demand of these participants determines the price direction. Market participants are characterised in terms of the information that they utilise in making a trading decision. Common trader categories are liquidity traders (uninformed), technical traders (information seekers), and informed traders. Motivations of liquidity trader actions are exogenous to the market conditions. Informed traders have private information and they utilise that information to take advantage of others. Technical traders try to generate information using past trading data and are influenced by their perception of informed trading. Manipulators generate false information in order to mislead other market participants. A common form of market structure is the use of a double-auction or limit-order mechanism for determining the bid and ask prices for a stock. This approach is characterised by a limit-order book that stores current bid and ask prices, and a last traded price representing the last sale. Market orders are submitted as buy/sell at the current market ∗
Corresponding author. Tel.: +64 3479 8097. E-mail addresses: [email protected] (R.M. Withanawasam), [email protected] (P.A. Whigham), [email protected] (T.F. Crack). 0378-4754/$36.00 © 2012 IMACS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.matcom.2012.09.012
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price. Maslov [16], Rosu [20], and Krause [13] have developed computational models that incorporate the mechanics and trader behaviours of limit order markets. The limit order market model developed by Maslov [16] simplifies the concept of a market place for a particular stock by constructing a limit-order book manipulated by an infinite pool of uninformed traders. In each time step, a new trader is drawn from this pool as a buyer (probability qb ) or a seller (qs ), and either a limit-order (probability qlo ) or a market order (1 − qlo ) is submitted. If a limit-order is submitted, a value is drawn from a distribution Δ as an offset from the last traded price p(t) (trader computes buy/sell limit order price with a negative/positive Δ from p(t)). Each buy or sell order involves a single unit of a stock. Liquidity traders do not consider the state of the current limit-order book, such as the spread, or any past patterns of the last traded price. All limit orders are assumed to be “good till cancelled”. Withanawasam et al. [21] introduced an alternative pricing method (denoted by M*), which computes limit buy/sell order prices with a negative/positive offset (Δ) from best ask/bid price of the Maslov order book (i.e., with respect to the contra side best prices). If the contra side is empty, the limit prices are computed with respect to the last traded price. The M* model does not produce the unrealistic strong cone-shaped patterns in price that can be observed with the Maslov model and is therefore used in experiments with this work [21]. Use of trading strategies with the intention of misleading other market participants (i.e., generating false information) is called “market manipulation”. One such manipulation scenario is commonly termed “pump and dump”, where manipulators pretend to be informed and mislead the market. They buy at successively higher prices, giving the appearance of activity at a higher price than the actual market value, and then sell or dump shares at an inflated price. Fig. 1 illustrates the price behaviour of a “pump and dump” manipulation recorded for “Universal” stock (ticker symbol UVV) between April 1999 and July 1999 in the US (U.S. Securities and Exchange Commission Litigation Release No. 16621/July 6, 2000). Allen and Gale [2] categorised manipulations under three main types, namely action based (e.g., not bidding in an auction), information based (e.g., spreading false rumours and news) and trade based (e.g., performing a series of trades to mislead the market). They also discussed the possibilities of trade-based manipulation and showed that a manipulator can pretend to be informed and mislead a market. Investors always have a problem distinguishing informed traders from manipulators. This problem leads to a manipulator being able to pretend to be informed and mislead the market. Van Bommel [7] showed that a potentially informed trader can pretend to be informed and manipulate the market. The author also analysed situations where a trader could spread rumors in order to distort the true value of stocks. Allen and Gorton [3] showed that asymmetry between the information associated with buying and selling implies that market participants assume buy transactions contain more information than sales. Hence, “a manipulator can repeatedly buy
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Technical trader’s learned demand increases the price further Manipulators buy at a higher rate, so the price starts to go up All the traders are started performing buy/sell actions with equal probability
Price collapses
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The manipulator sells and profits from the market
tS
Ignition Momentum Call−off Period Period Period tC tM tI
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Fig. 2. A graphical representation of the behaviour of the “pump and dump” model.
stocks, causing a relatively large effect on prices, and then sell with relatively little effect” [1, p. 1919]. Aggarwal and Wu [1] presented common characteristics of manipulated stocks. Mei et al. [17] also analysed the properties of data on “pump and dump” cases prosecuted by the U.S. Securities and Exchange Commission from January 1980 to December 2002. Some other notable papers in this area are Berle [6], and Kyle and Viswanathan [14]. Most of the existing manipulation detection methods involve classification of datasets. Statistical methods such as discriminant analysis, logit regression and machine learning methods such as support vector machines, and artificial neural networks have been used in manipulation analysis [18]. Although some mathematical models of market manipulation have been previously developed [1,8], this paper presents a framework for manipulation in the context of a realistic computational model of a limit-order market. The Maslov model is extended by introducing technical traders and manipulators. In the past literature, the Maslov model has not been considered in the context of analysing stock manipulations. Uninformed traders in the original Maslov model do not consider the behaviour of other market participants when taking trading decisions. Because of this, manipulation is not possible in the original Maslov model. We can, however, introduce technical traders who base their trading on the behaviour of informed traders that are believed to be trading in the market. These technical traders extract information from past trading and use that information when making their buy/sell decisions [11]. The manipulator can then introduce wrong information to the market to mislead these technical traders in an attempt to make illegal profits. For our purposes, technical traders utilise a Bayesian learning technique to generate information from informed persons that are believed to be trading in the market. This Bayesian model is useful because it allows us to model a belief structure of a trader type and also allows us to control and tune these beliefs explicitly in order to analyse their learning processes and its effects on the order book. A probability based belief tree and observed trading actions (i.e., orders and trades) are employed in this Bayesian learning technique. Past work of Glosten and Milgrom [10], Almgren and Lorenz [4], Hautsch and Hess [12], and Pastor and Veronesi [19] used Bayesian learning in modelling stock markets. Technical traders can also use alternative methods such as the use of historical prices in predicting future market behaviour in order to generate future trading signals [11]. The graphical representation of the model is depicted in Fig. 2. Fig. 2 is a stylised version of Fig. 1. The manipulator uses the price behaviour in response to supply/demand along with the behaviour of technical traders when attempting to inflate or deflate the price and mislead the market. In doing this, the manipulator continues to buy with a higher probability for a certain period of time. This results in an increase of the buying probability of technical traders and influences them to buy more. This artificial demand created by the manipulator inflates the price. After allowing the technical traders to raise the price further, the manipulator repeatedly sells his stocks as smoothly as possible, without being noticed by other traders, and profits from the market. The supply created by the manipulator changes the buying probability of technical traders and the price collapses.
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The market manipulation model is presented in Section 2. Results, implications of results, and future work are discussed in Section 3. 2. Methodology The “pump and dump” manipulation scenario is modelled using manipulators who mislead technical traders by introducing misleading information to the market. We extended the M* model by adding technical traders, and manipulators. Throughout the simulation, liquidity traders, technical traders, and manipulators are considered to be in a pool and are selected for trading with probabilities pL , pT , and pM , respectively. The simulation (i.e., tT time steps) is divided into two periods: a non-manipulation period and a manipulation period. In the manipulation period, manipulators use the “pump and dump” strategy to introduce the manipulation. In the nonmanipulation period the manipulator’s strategy is similar to the strategy of uninformed traders. The manipulation period starts after tS time steps. All these traders are assumed to have initial stocks and money. The final profit is computed as the difference between initial and final wealth of the trader. In computing the wealth, the stocks at hand are liquidated with respect to the last traded price at that moment. In nullifying the effect of market parameters such as price, profits of manipulators and technical traders are computed with respect to the average liquidity trader profit. Technical traders use a Bayesian learning model to generate information from past informed trading. Using this Bayesian learning model, each technical trader models his probability of price going up at time t as a common πt (starting πt is π0 ). The observed trading actions (orders/trades) are used in revising πt . In other words, the posterior probability of πt given the observed trading action at time t becomes the probability of price going up at time t + 1. A Bayes probability tree based on the trader’s beliefs (Fig. 3) is used in computing the posterior probabilities. When forming this Bayesian belief model, a technical trader takes the following conditions into consideration: whether the trader involved in the observed order/trade is likely to be an informed trader or a liquidity trader, whether this trader is acting as a buyer or a seller, and whether he is placing a limit order or a market order. All these conditions are characterised by conditional probabilities as shown in Fig. 3. Based on this Bayesian learning process, the prior probability of price going up in the future can be stated as: πt =
π0 (A + E)w (B + F )x (C + G)y (D + H)z π0 (A + E) (B + F ) (C + G)y (D + H)z + (1 − π0 )(I + M)w (J + N)x (K + O)y (L + P)z w
x
(1)
where A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, and P are probability multiples of each branch in the Bayesian probability tree given in Fig. 3, and w, x, y, and z denote the number of observed limit buy, market buy, limit sell, and market sell trader actions, respectively. When using the generated information, the technical trader considers the probability of price going up as his probability of buying (i.e., qbT = πt ) (i.e., the technical trader assumes that the signal of price going up is favourable for buying). As a result, buy and sell actions in the market influence the qbT to go up and go down, respectively. This model allows technical traders to carry forward information in past trading in order to generate information when making their decisions to buy or sell. When attempting to inflate or deflate the price and mislead the market, a manipulator uses the price behaviour in response to supply/demand, along with the behaviour of technical traders. The manipulation period is divided into three periods which we term “ignition period”, “momentum period” and “call-off period”. In the ignition period (i.e., tI time steps), the manipulator buys with a probability qbI . The momentum period (i.e., tM time steps) is used by the manipulators to allow the technical traders to raise the price further. Finally, in the call-off period, the manipulator sells his stocks for tC period of time with probability qsC and exits from the market. The time-line is depicted in Fig. 2. The M* model is initialised with p(0) = 1000, qb = qlo = 0.5, π0 = 0.5 and simulated for tT = 26,000 time steps (initially the model is run for 1000 time steps to populate the order book before we start recording the properties). The price offset (Δ) is drawn uniformly from a discrete set of random numbers from 1 to 4. Profits of each trader type are averaged over 100 simulations. In the Bayesian belief model, technical traders assume that when the price is to go up, informed traders are definitely buying (μ1 = 1), and they definitely sell (μ3 = 0) when the price is to go down. It is also assumed that the probabilities of submitting limit/market orders are equal (θ 1 = θ 2 = θ 3 = θ 4 = θ 5 = θ 6 = θ 7 = θ 8 = 0.5). In simulating the “pump and dump” manipulation, the manipulator parameters are set as: tS = 1000, tI = tC = 10, 000,
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Fig. 3. Bayesian diagram to compute posterior probabilities.
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Time Fig. 4. Price behaviour when technical traders are introduced to the M* model. (Note that data were re-sampled at every 50th point for clarity.)
tM = 5000, qbI = 0.9, and qbC = 0.1. This means that the manipulator buys with 0.9 probability for 10,000 steps, waits 5000 steps, and then sells with 0.9 probability for 10,000 steps. The information content possessed by a technical trader depends on the past records that he/she could see when computing πt . For simplicity, we assume that all traders arrive in the market at the same time, and therefore they have the same set of information (i.e., common πt ). The asymmetry between buying and selling is introduced to the technical trader belief model through the percentage of liquidity buying (i.e., μ2 ). The role of supply and demand in determining price direction, how technical traders extract information from trading, and how manipulators use these principles to alter market information are presented through the model properties. 3. Results and discussion Fig. 4 shows price behaviour of the M* model with technical traders. Here, in the technical trader belief model, technical traders expect 0.1% of informed trading and equal percentage of liquidity buying and selling (i.e., μ2 = 0.5). The percentage of technical trading increases the positive correlation in prices. This observation indicates that technical trading adds a persistence behaviour to the M* prices. Positively correlated/persistence price behaviour often observed in real stock prices (i.e., Hurst exponent is greater than 0.5) is a consequence of trader reactions to market information [5]. In Fig. 5, between the time steps 10,000 and 20,000, the manipulator performs buy/sell actions with a higher probability (0.9) to induce a supply and demand imbalance to drive the price up/down. Fig. 5 also shows how the momentum of technical trading amplifies the effect of this price change. Due to this momentum effect, the price continues to go up/down even after the manipulator stops his continuous buying/selling at the 20,000 time step. This momentum can be exploited by the manipulator in his “pump and dump” strategy. Table 1 summarises our results and compares manipulator profit with only liquidity traders and manipulator profit with technical traders in simulations with/without the momentum period and with/without information asymmetry. In a simulation with only technical and liquidity traders (pL = pT = 0.5, λ = 0.1, μ2 = 0.5, and tT = 26,000), it is observed that technical traders are making higher profits (i.e., average relative profit of 91,887 with standard error 10,815)1 than liquidity traders. This confirms our technical traders’ ability to predict the market using past information and use this ability to generate profits. The standard deviations of these trader profits are higher due to the highly random variation of simulated prices.
1 These results are averaged over 100 simulation runs. Note that a lower standard error value can be obtained by using a higher number of simulation runs [15].
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Price drive up without technical traders, pL = 0.8, pT = 0, pM = 0.2 Price drive up with technical traders, pL = 0.4, pT = 0.4, pM = 0.2 Price drive down without technical traders, pL = 0.8, pT = 0, pM = 0.2 Price drive down with technical traders, pL = 0.4, pT = 0.4, pM = 0.2
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Fig. 5. Price driving up/down with/without technical trading. (Note that data were re-sampled at every 50th point for clarity.) Table 1 Mean, standard deviation, and standard error values of technical trader and manipulator profits with respect to the liquidity trader profit averaged over 100 simulation runs. pL , pT , pM denote the liquidity, technical, and manipulator trader selection probabilities, respectively. tM denotes the length of the momentum period and μ2 represents the assumed percentage of liquidity buying in the Bayesian belief model. Simulation
Technical trader profit
Manipulator profit
pL
pT
pM
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μ2
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Std dev
Std error
Mean
Std dev
Std error
0.5 0.5
0.5 0
0 0.5
NA 5000
0.5 NA
91,887 NA
108,157 NA
10,815 NA
NA −15,430
NA 46,115
NA 4611
0.4 0.3 0.2
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0 0 0
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4700 8862 5048
16,518 17,163 17,047
1651 1716 1704
14,332 37,529 49,020
17,294 15,598 16,483
1729 1559 1648
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6300 14,966 22,441 11,530 89,972
22,985 32,703 40,937 39,159 133,382
2298 3270 4093 3915 13,338
7963 30,278 56,707 71,508 60,829
28,410 38,231 45,394 43,478 59,295
2841 3823 4539 4347 5929
0.6 0.6 0.5 0.4
0.1 0.2 0.3 0.4
0.3 0.3 0.3 0.3
0 0 0 0
0.45 0.45 0.45 0.45
−16,371 −20,664 −19,408 −15,631
11,064 16,348 21,610 27,700
1106 1634 2161 2770
3526 24,747 42,620 63,812
16,252 20,519 23,506 21,433
1625 2051 2350 2143
0.35
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84,010
8401
197,692
54,409
5440
Manipulator profit with only liquidity trading is very low (i.e., −15,430). This indicates that “pump and dump” type of manipulation is not possible with liquidity type traders. If there is no momentum period and the overall manipulation behaviour is symmetric (i.e., ignition period is almost identical to the call-off period (i.e., tI = tC and qbI = qsC ) and there is no information asymmetry considered between buying and selling (i.e., μ2 = 0.5)), the manipulator profit with technical trading is also comparatively low. These results indicate that with symmetric behaviour, the manipulators are not able to make any significant profits by selling immediately after raising the price either with liquidity traders or technical traders. These results indicate that “pump and dump” type of manipulation is not possible when the market is only composed of liquidity traders. This is because liquidity traders cannot introduce either momentum or information asymmetry to stock prices. When there are technical traders, however, the manipulator can exploit technical trader behaviour by waiting some time between the ignition period and call-off period (e.g., momentum period, tM = 5000) in order to make a higher profit. Here, the technical trader percentage increases the manipulator profit. When the manipulator momentum period increases (i.e., tM = 7500), the manipulator profit decreases. This is because the perception the
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2e+05 0e+00 −2e+05
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4e+05
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Fig. 6. Last traded price graphs for different liquidity buying percentages (μ2 ), assumed informed trader percentage λ = 0.1. (Note that data were re-sampled at every 50th point for clarity.)
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manipulator generated among technical traders may die out and the advantage he has produced can be lost. Hence, there is a memory effect for technical traders. Moreover, if technical traders believe in an information asymmetry between buy and sell transactions (i.e., if they believe a buy contains more information than a sale), manipulation is possible/profitable even without this momentum period. In addition, the percentage of technical trading increases the manipulator profit. This information asymmetry is introduced to the model by varying the percentage of liquidity buying in the Bayesian belief model (i.e., μ2 < 0.5). Allen and Gorton [3] also used the percentage of liquidity buying/selling in order to introduce information asymmetry and show the possibility of manipulation in the Glosten and Milgrom [10] model. In this approach, technical traders assumes that μ2 < 0.5 and respond more to buy transactions than sell transactions. As a result, the price change in the manipulators’ ignition period becomes higher than the price change in the call-off period. Price graphs in Fig. 6 illustrate how a manipulator uses this information asymmetry to buy with a higher effect on prices and sell with little effect. Note that Fig. 6 (i.e., simulated pump-and-dump prices) has the same characteristics as Fig. 1 (i.e., real pump-and-dump prices). Moreover, Fig. 7 illustrates how the manipulator profit in comparison with the technical trader profit, varies from positive to negative with the assumed percentage of liquidity buying (μ2 ) in the technical trader belief model. Based on these results, we confirm the Allen and Gorton findings related to the role of information asymmetry in manipulation [3]. Technical trading makes manipulation possible and the manipulator profit increases with the percentage of technical traders (Fig. 8). When analysing the manipulator profits, we set manipulator selection probability pM = 0.3, length of
0
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15000
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25000
Time Fig. 7. Manipulator profit behaviour for different liquidity buying percentages (μ2 ), assumed informed trader percentage λ = 0.1.
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Manipulator Technical Trader
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pT Fig. 8. Manipulator profits with the percentage of technical trading (i.e., when changing pT while pM = 0.3, μ2 = 0.45, and tM = 5000).
the momentum period tM = 5000, and liquidity buying percentage μ2 = 0.45 in the Bayesian belief model. This result confirms the findings of Aggrawal and Wu [1] (i.e., manipulation is more profitable when there are more information seekers in the market). Based on these results, using a computational model of a limit order market, we confirm that “pump and dump” manipulation is not profitable with only liquidity trading. Rather, manipulators require the presence of technical traders to profit. In addition, increasing the percentage of technical trading makes manipulation more profitable. When exploiting the behaviour of technical traders, a manipulator can wait some time after the ignition phase to make a profit. However, if technical traders believe that there is an information asymmetry in the market, the manipulator effort required to perform “pump and dump” is low and they can even make profits by selling immediately after raising the price. As future work, this manipulation model could be extended to introduce and simplify the dynamics of other manipulation mechanisms such as “cyclic trading”, “marking the close”, “wash sales”, and “painting the tape” [9]. In the future, these market manipulation models will be considered in terms of analysing market manipulation behaviour and finding manipulation detection mechanisms. 4. Conclusion A computational model was developed to characterise price manipulation in a limit order driven market. The Maslov [16] limit order market model was extended to introduce technical traders and manipulators to be used as a general framework for modelling manipulation strategies. We showed that a manipulator can pretend to be informed to mislead the technical traders in performing a “pump and dump” strategy in a limit order market. Concepts such as the role of supply and demand in determining price direction, how technical traders extract information from trading, and how manipulators use these principles to alter information, have been presented via the model properties. We showed that “pump and dump” manipulation is not possible with solely liquidity type of trading. Rather, manipulators require the presence of technical traders to profit. When exploiting the behaviour of technical traders, a manipulator should wait some time after accumulating stocks to generate a profit. If technical traders believe that there is an information asymmetry in the market, however, then manipulators can even generate profits by selling immediately after buying stocks. Acknowledgement This work has been partially funded by a University of Otago Postgraduate Scholarship. References [1] R.K. Aggarwal, G. Wu, Stock market manipulations, Journal of Business 79 (2006) 1315–1336. [2] F. Allen, D. Gale, Stock-price manipulation, Review of Financial Studies 5 (1992) 503–529.
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[3] F. Allen, G. Gorton, Stock price manipulation, market microstructure and asymmetric information, European Economic Review 36 (1992) 624–630. [4] R. Almgren, J. Lorenz, Bayesian adaptive trading with a daily cycle, Journal of Trading 1 (2006) 38–46. [5] J. Alvarez-Ramirez, J. Alvarez, E. Rodriguez, G. Fernandez-Anaya, Time-varying Hurst exponent for US stock markets, Physica A 387 (2008) 6159–6169. [6] A.A. Berle, Stock market manipulation, Columbia Law Review 38 (1938) 393–407. [7] J.V. Bommel, Rumors, Journal of Finance 58 (2003) 1499–1520. [8] A. Chakraborty, B. Yilmaz, Informed manipulation, Journal of Economic Theory 114 (2004) 132–152. [9] D. Cumming, S. Johan, Global market surveillance, American Law and Economics Review 1 (2008) 1–10. [10] L.R. Glosten, P.R. Milgrom, Bid, ask and transaction prices in a specialist market with heterogeneously informed traders, Journal of Financial Economics 14 (1985) 71–100. [11] L. Harris, Trading and Exchanges: Market Microstructure for Practitioners, Oxford University Press, 2002. [12] N. Hautsch, D. Hess, Bayesian learning in financial markets: testing for the relevance of information precision in price discovery, Journal of Financial and Quantitative Analysis 42 (2007) 189. [13] A. Krause, Fat tails and multi-scaling in a simple model of limit order markets, Physica A 368 (2006) 183–190. [14] A.S. Kyle, S. Viswanathan, How to define illegal price manipulation, American Economic Review 98 (2008) 274–279. [15] D.V. Lindley, A statistical paradox, Biometrika 44 (1957) 187–192. [16] S. Maslov, Simple model of a limit order-driven market, Physica A 278 (2000) 571–578. [17] J. Mei, G. Wu, C. Zhou, Behavior based manipulation: theory and prosecution evidence, unpublished manuscript, New York University. [18] H. Ogut, M.M. Doganay, R. Aktas, Detecting stock-price manipulation in an emerging market: the case of Turkey, Expert Systems with Applications 36 (2009) 11944–11949. [19] L. Pastor, P. Veronesi, Learning in financial markets, Review of Financial Economics 11 (2009) 374. [20] I. Rosu, A dynamic model of the limit order book, Review of Financial Studies 22 (2009) 4601–4641. [21] R.M. Withanawasam, P.A. Whigham, T.F. Crack, I.M. Premachandra, Empirical investigation of the Maslov limit order market model, discussion paper (2010/04), Department of Information Science, University of Otago.