Pyramid scheme in stock market: a kind of financial market simulation
aa r X i v : . [ q -f i n . T R ] F e b Pyramid scheme in stock market: a kind of financialmarket simulation
Yong Shi , , , Bo Li , , ∗ and Guangle Du , School of Economics and Management,University of Chinese Academy of Sciences, Beijing 100190, China Research Center on Fictitious Economy and Data Science , Chinese Academy of Sciences, Beijing 100190, China Key Laboratory of Big Data Mining and Knowledge Management,Chinese Academy of Sciences, Beijing 100190, China Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang 325001, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Abstract
Artificial stock market simulation based on agent is an important means tostudy financial market. Based on the assumption that the investors are composedof a main fund, small trend and contrarian investors characterized by four param-eters, we simulate and research a kind of financial phenomenon with the character-istics of pyramid schemes. Our simulation results and theoretical analysis revealthe relationships between the rate of return of the main fund and the proportion ofthe trend investors in all small investors, the small investors’ parameters of takingprofit and stopping loss, the order size of the main fund and the strategies adoptedby the main fund. Our work are helpful to explain the financial phenomenon withthe characteristics of pyramid schemes in financial markets, design trading rulesfor regulators and develop trading strategies for investors. ∗ Email:[email protected] Introduction
Simulation of artificial financial market based on multi-agent models is an importantmethod to search for the dynamic laws in financial market [1–4], study the statisticalproperties of financial time series [5–8], and evaluate the possible effects of the financialregulations and/or rules [9, 10]. The multi-agent models for researching the dynamicsin financial market usually assume that the agents are heterogeneous, then study thebehaviors of the agents such as herd behavior or contrarian behavior [11–13], and theinfluences of the factors such as event shocks on market dynamics [1–4]. The universalstatistical properties in financial time series, autocorrelation and scaling exists in returns,volatility clustering, etc., generically called stylized facts [7, 8, 14], can be reproducedby artificial financial market simulation [5, 6]. Artificial market studies for evaluatingthe effects of the financial regulations and/or rules can give insights to the discussionon whether price variation limits and short selling regulation can prevent bubbles andcrushes or not, and estimate the effects of tick size, usage rate of dark pools, rules forinvestment diversification, order matching systems’ speed on financial exchanges [9, 10].In addition to the research objectives above, we can also roughly divide artificialfinancial market models from the characteristics of theirselves into the agent modelsdeveloped from minority games [15–18], the agent models based on existing physicalmodels [19–21], the agent models based on order book and so on [6, 22]. Recently, somestudies have combined deep learning with artificial financial markets, Ref. [23] considersthe case that there exists a class of agents who utilize deep learning to make decisions,and Ref. [24] involves taking advantage of deep reinforcement learning in agent basedfinancial market simulation to learn a robust investment strategy with an attractiverisk-return profile.In this work, we propose an agent model to simulate and research a kind of finan-cial phenomenon with the characteristics of the pyramid schemes [25–28], which haveorganizers who induce people to participate in the schemes with high returns, and theparticipants also recruit the next generation of participants until the schemes can notcontinue. In the secondary market of a certain security, there is often a financial phe-nomenon with the characteristics of the pyramid schemes: the main funds with a largeamount of money and securities may act as the organizers, who influence the price ofthe security through big orders to induce ordinary trend investors to follow the trendone after another, and profit through reverse operation finally.The financial phenomenon with the characteristics of pyramid schemes above is some-what similar to herd behavior in financial markets [11,29–31], they are all involved in thebehavior of following the trend, and there are some works about agent based models withherding or contrarian behaviors, for instances, Zhao et al. [11] and Liang et al. [12] deeply2tudy the mechanisms of herding behaviors and contrarian behaviors in adaptive sys-tems. But the difference between them is obvious, the core of the financial phenomenonwith the characteristics of pyramid schemes is that the main fund inducing the smallinvestors to follow the trend and making profits, while the herd behavior market empha-sizes the herd mentality of the ordinary investors in financial markets. In order to studythe financial phenomenon with the characteristics of pyramid schemes, we simulate thiskind of financial market by building an agent model with a special investor structure.While the difference between the two phenomena is obvious, the core of the financialphenomenon with the characteristics of pyramid schemes is the main fund inducing thesmall investors to follow the trend and making profits, while the herd behavior empha-sizes the herd mentality of the ordinary investors in financial markets. In order to studythe financial phenomenon with the characteristics of pyramid schemes, we simulate thiskind of financial market by building an agent model with a special investor structure. Inour proposed model, we assume there is only one main fund in the market, and the smallinvestors include homogeneous trend and contrarian investors described by 4 parametersas follows:
T ype , which represents the type of the strategy an investor used which isdepended on his or her specific trading directions relative to the market trend, trendor contrarian, and is different from many existing works on financial market simulationwhich often divide investors into fundamental investors, technical investors and otherinvestors [5,6,15,16,32]; r market , which determines when a investor decides to participatein trading according to the change of the security price in a certain trading period, dueto the confidentiality of investors’ strategies in the real market, we don’t specify themechanisms of the investors’ strategies as in Refs. [1, 6, 15, 16, 18], and only assume thatthe parameter r market of each investor is different and make some assumptions about theparameters’ overall distribution; r profit and r loss , which denote the take-profit point andstop-loss parameters set by investors. In the simulations, in order to fit the real marketbetter, we also set the activation probability p active for each small investor as in Ref. [6].Based on the artificial financial market established, we make use of computer simu-lation and theoretical analysis to reveal the relationships between the rate of return ofthe main fund and the proportion of the trend investors in all small investors, the smallinvestors’ parameters of taking profit and stopping loss, the order size of the main fundand the strategies adopted by the main fund. Our work clearly explains the financialphenomenon with the characteristics of pyramid schemes in financial markets, and itcan help regulators to design trading rules and assist investors in developing tradingstrategies.The rest of the paper is organized as follows: In the next section, we introduce ourproposed agent model. In Sec. 3 the simulation results and preliminary analysis are3isplayed, and Sec. 4 gives a further analysis. Some discussions and conclusions aregiven in Sec. 5. As the objective of our proposed model is to study the financial phenomenon withthe characteristics of pyramid schemes, we assume that there are three types of agentsin our models, the first one is the main fund, and the other two are small contrarianor trend investors who adopt contrarian strategies or trend strategies to trade. In thetrading process, the main fund buy or sell large orders to induce small trend investors,and then profit through reverse operation. The expected rate of return of the main fundlargely determines its behavior, which plays a leading role in the financial phenomenonwe research, thus our object is to research the relationships between the profit level ofthe main fund and the strategies of the main fund adopts, the order size of the mainfund and the structure of the small investors. The characteristics of the different agentsand specific trading rules are described in detail below.
The main fund in our model is the agent who has a large amount of money orthe traded security relative to other agents, and as a result, its trading behaviors candominate the change of the security price and it profits from this advantage. It shouldbe noted that the artificial financial market we simulate only trades one kind of security.In order to simplify the analysis process and better study the problems we are concernedabout, there is only one main fund in our model, and it does not have the ability tocompletely manipulate the market, that is to say, the main fund has no ability to swallowall the buy or sell orders from the small investors.The other two types of agents are small trend and contrarian investors, who employtrend strategies and contrarian strategies respectively in trading. Trend agents partici-pate in trading according to the trend of the security price, if they think there is a signof an upward trend in the security price, they go long the security, otherwise they goshort the security. Contrarian agents go short when the security price rises more than acertain extent, and go long when the security price falls by more than a certain range.We do not divide investors into fundamental investors and technical analysis investors asRefs. [5,6] do, where the fundamental investors and the technical analysis investors corre-spond to the investors using contrarian strategies and the investors using trend strategies4espectively. But in fact, fundamental investors will also follow the upward trend as thevaluation of the security increases, and technical analysis investors often adopt contrar-ian strategies. Therefore, we adopt a more reasonable way to distinguish investors basedon their investment strategies rather than on whether they adopt technical analysis orfundamental analysis.For the investors with trend strategies and contrarian strategies in our model, theinvestors with trend strategy trade at the market price and the investors with contrar-ian strategies trade at the limit price, we use parameter combinations (
T ype, r market ,r profit , r loss ) to characterize them, where T ype represents the types of strategies used byinvestors, trend or contrarian; r market refers to the trigger signals for trend investors ordetermines the prices of limit orders for contrarian investors, and the trend investors willparticipate in the trading when the security price’s range of rise and fall is more than r market over a certain period of time, while the contrarian investors place limit orders at(1 + r market ) P , where P is the initial price of the security in the period of time; r profit and r loss denote the take-profit and stop-loss parameters set by investors respectively.Generally, different parameters r profit and r loss can describe investors with differentbehavioral characteristics. For instance, for an investor with disposition effect [33], theabsolute value of stop-loss parameter is greater than that of take-profit parameter.In real financial markets, due to the confidentiality of investment strategies whichdetermine r market , we do not know investors’ specific investment strategies and when andhow they will participate in the trading. However, certain prices of securities may gathera large number of investors to participate in trading for some reasons, for instances, whensome common technical indicators such as moving averages (MAs) indicate buy signals,or the prices of securities reach key positions such integer positions, a large number ofinvestors will participate in trading, so we can make a hypothesis that the parameters r market are concentrated in the vicinity of some values and are normally distributed. Fig. 1shows the schematic diagram reflecting the structure of small investors, the parameters r market , r market , r market , r market and r market denote the rate of increase and decreaserelative to the initial price in the figure are different triggering signals for the trendstrategies and contrarian investors, and these parameters are normally distributed.Small investors in our model do not have the ability to affect the security price, theyare passive to participate in trading, only the main fund has the ability to influence thesecurity price. In addition, each small investor only sells or buys orders with size 1,but this does not affect the representativeness of our model, because in actual financialtransactions, an small investor who can buy and sell orders with the size greater than 1can be regarded as a collection of investors who can only buy and sell orders with size 1.5igure 1: The schematic diagram of the trend investors and contrarian investors in ourproposed model. In our model, the orders from the main fund and the small investors are divided intomarket orders which are active buy or sell orders, and limit orders which are passive.The matchmaking trading mode employed in our model is the fashion of time priorityand price priority, and we define the process from the issuance of a market order of themain fund, to the time time when there is no new transaction unless the main fund issuesa new market order as a trading period. In the following, we describe the procedures ofthe first trading period in a step-by-step way from the issuance of a market buy order ofthe main fund to the end of all transactions, and any other trading periods are similar.The procedures are:(i) The main fund issues a market buy order with the size N mf ( N mf is a positiveinteger), and it concludes transactions with the limit sell orders placed by N mf contrarianinvestors with r market >
0. After trading, the market price of the security will rise. Itshould be noted here that we set the activation probability p active for each contrarianinvestor, so even if its r market satisfies the trading conditions, it only has the probabilityof p active to trade. The same is true for trend investors.(ii) The contrarian investors who trades with the main fund in procedure (i) settake-profit and stop-loss orders according to their parameters r profit and r loss , and thetake-profit orders are limit orders while the stop-loss orders are market orders. In our6able 1: Parameters and numbers of trend investors and contrarian investors. Trend investors Contrarian investors
Mean ( r market ) σ ( r market ) Number p active Mean ( r market ) σ ( r market ) Number p active — — — — 0.1 0 20000 0.50.08 0.04 2000 ∗ ratio ∗ ratio ∗ ratio ∗ ratio ∗ ratio ∗ ratio setting, the small investors who have taken profit or stopped loss will no longer participatein trading in the same trading period.(iii) The rise of the security price caused by the buying of the main fund triggers thetrend traders whose parameters r market are no more than the rise to buy. The triggeredtrend traders will cause the price to rise again.(iv) The trend and contrarian investors trade in procedure (iii) set take-profit andstop-loss orders according to their parameters r profit and r loss . Same as contrarian in-vestors, the take-profit orders of the trend investors are limit orders and the stop-lossorders are market orders.(v) The rise caused by the triggered trend traders may trigger new trend traders tobuy. This process will continue until there are no trend investors are triggered, and thetrading period ends when the process stops.Tab. 1 shows parameters and numbers of trend investors and contrarian investors,where M ean ( r market ) and σ ( r market ) in each row are the average and standard deviationof the normal distribution satisfied by the parameter r market of the investors in the samerow, and the parameter ratio denotes the ratio of the trend investors to the contrarianinvestors. It is worth noting that there are 20000 contrarian investors with the parameter r market of 0.1 and -0.1 respectively, but no trend investors with the parameter r market of0.1 and -0.1. The purpose of our design is to be close to the real market, because whenthe price of a security deviates greatly from its due value, it will attract a large numberof contrarian investors to participate in the trading. We simulate the artificial financial markets with the characteristics of pyramid schemesand obtain the rate of returns of the main fund under different cases. Through the re-sults, we can preliminarily reveal the relationships between the rate of return of the mainfund and the proportion of the trend investors in all the small investors, the small in-7estors’ parameters of taking profit and stopping loss, the order size of the main fund andthe strategies adopted by the main fund. In our simulations, the main fund participatesin trading through the way of first buying and then selling, which is not fundamentallydifferent from the situation of selling first and then buying back, this is because the pa-rameters r market and numbers of small investors set in our agent model are symmetrical. In this subsection, we assume that the main fund only adopt one strategy, which is,buying a certain amount of securities through a single order in the first trading periodand selling all the securities also through one order in the next trading period. Inthe simulation, we also assume that all the small investors’ parameters of taking profitand stopping loss different, and we consider four different combinations of taking profitparameter r profit and stopping loss parameter r loss .The first case is the small investors don’t take profit and stop loss, we obtain therelationship between the order size of the main fund and its return, as shown in Fig. 2.Fig. 3, Fig. 4 and Fig. 5 shows shows the relationship between the order size of the mainfund and its rate of return under different parameter combinations of small investors tak-ing profit and stopping loss respectively. The parameter combinations of small investorstaking profit and stopping loss from Fig. 3 to Fig. 5 respectively are: r profit = | r loss | ,where r profit obeys the uniform distribution U (0 . , . r profit > | r loss | , where | r loss | obeys the uniform distribution U (0 . , .
08) and r profit obeys the uniform distribution U ( | r loss | , . r profit < | r loss | , where r profit obeys the uniform distribution U (0 . , . | r loss | obeys the uniform distribution U ( r profit , . ratio which are 0.1, 0.2, 0.4, 0.8 and 1.6 in these simulations.From these figures, we can see that when the order size of the main find reaches acritical value, its yield will suddenly increase, which is due to the distribution of r market of the small investors is sparse when the parameters r market are at larger values, andwhen the yield reaches the maximum, it will decrease with the increase of the order size.And we also find that the higher the proportion of trend investors, the smaller the ordersize required by the main fund to achieve the maximum yield. Besides above, we alsofind that different combinations of stopping loss and taking profit parameters of smallinvestors have an impact on the yield of the main fund.In our artificial financial market, the return of the main fund comes from the trendinvestors being triggered to push up the price, and the contrarian investors acceptingand buying the securities at a higher price in the next trading period. The reason whythe rate of return of the main fund increases first with the order size is that the largerorder size can trigger more trend investors which help the main fund to push the price8 2 U G H U V L ] H R I W K H P D L Q I X Q G 5 D W H R I U H W X U Q ' R Q W W D N H S U R I L W D Q G V W R S O R V V ratio ratio ratio ratio ratio Figure 2: The relationship between the order size of the main fund and its return, whenthe small investors don’t take profit and stop loss. 2 U G H U V L ] H R I W K H P D L Q I X Q G 5 D W H R I U H W X U Q r profit = |r loss | ratio ratio ratio ratio ratio Figure 3: The relationship between the order size of the main fund and its rate of return,when the parameters of taking profit and stopping loss of the small investors satisfy r profit = | r loss | , where r profit obeys the uniform distribution U (0 . , . r market parameterdistributed around 0.02, 0.04 and 0.08, while there are 20000 contrarian investors withparameters r market of 0.1, which can’t be swallowed by the main fund in our model, so9 2 U G H U V L ] H R I W K H P D L Q I X Q G 5 D W H R I U H W X U Q r profit > |r loss | ratio ratio ratio ratio ratio Figure 4: The relationship between the order size of the main fund and its rate of return,when the parameters of taking profit and stopping loss of the small investors satisfy r profit > | r loss | , where | r loss | obeys the uniform distribution U (0 . , .
08) and r profit obeys theuniform distribution U ( | r loss | , . 2 U G H U V L ] H R I W K H P D L Q I X Q G 5 D W H R I U H W X U Q r profit < |r loss | ratio ratio ratio ratio ratio Figure 5: The relationship between the order size of the main fund and its rate of return,when the parameters of taking profit and stopping loss of the small investors satisfy r profit < | r loss | , where r profit obeys the uniform distribution U (0 . , .
08) and | r loss | obeys theuniform distribution U ( r profit , . Next, we consider the cases that the main fund adopts the strategy of buying orselling a certain amount of securities in batches in multiple trading periods. Specifically,we assume that the main fund buy a certain amount of securities in D buy trading periodsand sell them in next D sell trading periods, where D buy ∈ [1 ,
5] and D sell ∈ [1 , D buy and D sell should not be taken as largervalues in our model. In addition, we also assume that in the process of buying or sellingsecurities, the main fund only places one market price order in each trading period, andthe sizes of the market price orders in different trading periods are the same. Thusthere are 5 × ratio is set as 0 .
4. When setting these twoparameters, we refer to the simulation results in the previous subsection. Same as theprevious subsection, we also assume that all the small investors’ parameters of takingprofit and stopping loss are different, and we consider four different situations.The first situation is that the small investors don’t take profit and stop loss, weobtain the relationship between the number of buying and selling periods of the mainfund and its rate of return, as shown in Fig. 6. Fig. 7, Fig. 8 and Fig. 9 shows therelationship between the number of buying and selling trading periods of the main fundand its rate of return under different parameter combinations of small investors takingprofit and stopping loss respectively. The parameter combinations of small investorstaking profit and stopping loss we consider also are r profit = | r loss | , where r profit obeysthe uniform distribution U (0 . , . r profit > | r loss | , where | r loss | obeys the uniformdistribution U (0 . , .
08) and r profit obeys the uniform distribution U ( | r loss | , . r profit < | r loss | , where r profit obeys the uniform distribution U (0 . , .
08) and | r loss | obeys theuniform distribution U ( r profit , . D buy í 5 D W H R I U H W X U Q ' R Q W W D N H S U R I L W D Q G V W R S O R V V D sell D sell D sell D sell D sell Figure 6: The relationship between the number of buying and selling periods of the mainfund and its rate of return when the small investors don’t take profit and stop loss. D buy 5 D W H R I U H W X U Q r profit = |r loss | D sell D sell D sell D sell D sell Figure 7: The relationship between the number of buying and selling periods of the mainfund and its rate of return when the parameters of taking profit and stopping loss ofthe small investors satisfy r profit = | r loss | , where r profit obeys the uniform distribution U (0 . , . D buy 5 D W H R I U H W X U Q r profit > |r loss | D sell D sell D sell D sell D sell Figure 8: The relationship between the number of buying and selling periods of the mainfund and its rate of return when the parameters of taking profit and stopping loss ofthe small investors satisfy r profit > | r loss | , where | r loss | obeys the uniform distribution U (0 . , .
08) and r profit obeys the uniform distribution U ( | r loss | , . D buy 5 D W H R I U H W X U Q r profit < |r loss | D sell D sell D sell D sell D sell Figure 9: The relationship between the number of buying and selling periods of the mainfund and its rate of return when the parameters of taking profit and stopping loss ofthe small investors satisfy r profit < | r loss | , where r profit obeys the uniform distribution U (0 . , .
08) and | r loss | obeys the uniform distribution U ( r profit , . To further understand the underlying mechanism of the proposed agent-based modeland avoid illusive results caused by the model setting, we conduct a theoretical analysison the model as follows. In the section, we analyze in detail the effects of the ordersize N mf of the main fund, the parameter ratio between the trend investors and thecontrarian investors, the stop-loss parameters r loss , take-profit parameters r profit and theactivation probability p active of the small investors on the yield of the main fund whenthe main fund adopt a single strategy. For the cases of that the main fund adopts thestrategies of buying or selling a certain equal amount of securities in batches in multipletrading periods, we analyze the impact of different numbers of buying or selling tradingperiods on the main fund yield. For the cases of the main fund adopting single strategy described in section 3.1, wefirstly arrange the parameters r market of the contrarian investors and trend investors intotwo series { R C − M , · · · , R C − , R C − , R C +1 , R C +2 , · · · , R C + N } and { R T − P , · · · , R T − , R T − ,R T +1 , R T +2 , · · · , R T + Q } respectively, where M , N , P and Q denote the numbers of thecontrarian investors and trend investors who have negative and positive r market , R C − i < R C + j > R T − k < R T + l > i , j , k and l are positive integers in [1 , M ], [1 , N ],[1 , P ] and [1 , Q ] respectively, R C − M ≤ · · · ≤ R C − ≤ R C − < R C +1 ≤ R C +2 ≤ · · · ≤ R C + N and R T − P ≤ · · · ≤ R T − ≤ R T − < R T +1 ≤ R T +2 ≤ · · · ≤ R T + Q . Since we set an activationprobability p active for each small investor, we assume that the sequences of parameters r market of the small investors activated in trading are { r C − m , · · · , r C − , r C − , r C +1 , r C +2 , · · · , r C + n } and { r T − p , · · · , r T − , r T − , r T +1 , r T +2 , · · · , r T + q } , where m , n , p and q denotethe numbers of the activated contrarian investors and trend investors with negative andpositive r market . According to the hypotheses of our model, after the transaction of themarket buy order with the scale of N mf issued by the main fund ( N mf is an integer andless than n ), the price of the security goes up to (1 + r C + N mf ) P where P is the initial priceof the security, and the cost of the main fund is given by M buy = N mf X i =1 (1 + r C + i ) P . (1)14he price rise may trigger some trend investors, and the price rise caused by these trendinvestors’ buying may trigger new trend investors. The numbers of the trend investorstriggered for the first time N t , the second time N t , ..., and the i -th time N it are givenby N t = q X i =1 I ( r T + i ≤ r C + N mf ) , N t = q X i>N t I ( r T + i ≤ r C + N mf + N t ) , · · · ,N it = q X i>N i − t I ( r T + i ≤ r C + N mf + N t + N t + ··· + N i − t ) , (2)where I ( x ) is the indicator function, when x is true I ( x ) is equal to 1, otherwise it is 0.We assume that this process lasts a total of x times, and define N Xt = P xi =1 N it , thenthe security price at the end of the first trading period is P (1) = (1 + r C + N mf + N Xt ) P . Itshould be noted that in our model setting, the contrarian investors are sufficient so thatthe expression N mf + N Xt < n is satisfied. In addition, in the above analysis, the smallinvestors don’t take profit and stop loss. In the second trading period, the main fund sellthe securities they have bought, and according to a similar analysis, the security priceat the end of the second trading period is P (2) = (1 + r C − N mf + N Yt ) P (1), where N Yt is thenumber of trend investors triggered in this trading period. The money M sell of the mainfund obtaining from selling the securities is given by M sell = N mf X i =1 (1 + r C − i ) P (1) , (3)then the rate of return of the main fund r mf is r mf = M sell M buy − r C + N mf + N Xt ) P N mf i =1 (1 + r C − i ) P N mf i =1 (1 + r C + i ) −
1= (1 + r C + N mf + N Xt ) 1 + P N mf i =1 r C − i /N mf P N mf i =1 r C + i /N mf − . (4)The above expression of r mf can explain our simulation results presented in Fig. 1.The left term (1 + r C + N mf + N Xt ) increases with the increase of N mf until it reaches themaximum (the reason for the existence the maximum is that there are enough con-trarian investors when the parameters r market are close to 0.1), and the right term(1 + P N mf i =1 r C − i /N mf ) / (1 + P N mf i =1 r C + i /N mf ) decreases with the increase of N mf , where P N mf i =1 r C − i /N mf and P N mf i =1 r C + i /N mf are actually the mean value of the sets { r C − i | i ∈ [1 , N mf ] ∩ N + } and { r C + i | i ∈ [1 , N mf ] ∩ N + } , and they reflect the increased average15ost of the main fund due to the price rise caused by its own selling and buying. Because { R C − M , · · · , R C − , R C − , R C +1 , R C +2 , · · · , R C + N } in our model is symmetric and the activa-tion probability of each small investor is equal, so the transaction situations of the mainfund buying and selling are almost the same, and it is easy to find that when r C + N mf + N Xt does not reach the maximum, the left term increases faster than the right term with theincrease of N mf , therefore, r mf increases with the increase of N mf and decreases when r C + N mf + N Xt reaches the maximum, which is consistent with the simulation results presentedin Fig. 2. In addition, the larger the ratio parameter ratio between the trend investorsand the contrarian investors, the larger the number N Xt of the triggered trend investors,and r mf will reach the maximum faster, which is also consistent with the simulationresults presented in Fig. 2. Here we must point out that the reason why the main fundgains profits is that the contrarian investors whose parameter r market is less than 0 acceptthe new benchmark price P (1) in the second trading period.Next, we analyze the effect of small investors’ taking profit and stopping loss pa-rameters on the yield of the main fund. In the first trading period, the contrarian in-vestors making deals with the main fund place the following stop-loss orders {− r loss (1 + r C + i ) P /P ( D − } and take-profit orders {− r return (1 + r C + i ) P /P ( D − } , where D isthe serial number of the trading period and P (0) = P . The take-profit orders will beadded to the new sequence { r C − m , · · · , r C − , r C − , r C + N mf +1 , r C + N mf +2 , · · · , r C + n } after trad-ing with the main fund, and the stop-loss orders will be added to the sequence { r T − p , · · · , r T − , r T − , r T +1 , r T +2 , · · · , r T + q } . We find that for the main fund, the smaller theabsolute values of the stop-loss parameters of the contrarian investors, the easier it isfor the contrarian investors to stop loss and help the main fund to push up the securityprice; the smaller the take-profit parameters of the contrarian investors are, the higherthe prices the contrarian investors buy back the securities from the main fund in thetrading periods when the main fund is selling. Therefore, the smaller the stop-loss andtake-profit parameters of reverse investors, the higher the yield of the main fund. Similaranalysis of the trend investors shows that the larger the absolute values of the takingprofit and stopping loss parameters, the higher the yield of the main fund.We design simulation experiments to verify our conclusion. We set the absolute valuesof stop-loss and take-profit parameters of the contrarian investors and trend investorsto random values obeying the uniform distribution U (0 . , .
04) respectively withoutsetting take-profit parameters and stop-loss parameters of the trend investors or con-trarian investors (no take-profit parameters and stop-loss parameters are equivalent tothe infinity of the absolute values of them). Comparing our simulation results in Fig. 10and Fig. 11 with the results in Fig. 2, we find that the returns in Fig. 10 are generallyhigher than those in Fig. 2 and the returns in Fig. 11 are generally lower than those16n Fig. 2, which are consistent with our analysis: the smaller the absolute values of thetake-profit and stop-loss parameters of the contrarian investors, the higher the yield ofthe main fund, while the situation of the trend investors is just the opposite. 2 U G H U V L ] H R I W K H P D L Q I X Q G 5 D W H R I U H W X U Q r profit = |r loss | D Q G r profit R E H \ V U(0.02, 0.04) I R U H D F K F R Q W U D U L D Q L Q Y H V W R U ratio ratio ratio ratio ratio Figure 10: The relationship between the order size of the main fund and its rate ofreturn, when the parameters of taking profit and stopping loss of the contrarian investorssatisfy r profit = | r loss | , where r profit obeys the uniform distribution U (0 . , . p active on r mf . In fact, the effect of lower p active is equivalent to that of larger N mf , sowithin a certain parameter range, the value of r mf increases with the decrease of p active and then decreases when r mf reaches the maximum. For the cases that the main fund adopts the strategies of buying or selling a certainequal amount of securities in batches in multiple trading periods described in section 3.2,we firstly assume N mf = n bmf D buy = n smf D sell , where n bmf and D buy are the number ofsecurities bought per trading period by the main fund and the total number of tradingperiods of buying, n smf and D sell are the corresponding quantities in the trading periodsof selling. In this subsection, we continue to use the notations defined in section 4.1,but note that r C + i , r C − i and N Xt are not the same in different trading periods, we denotethem as r C + i ( k ), r C − i ( k ) and N Xt ( k ) respectively, where k is the serial number of thetrading period of buying or selling. As the section section 4.1 has considered the effectof r profit and r loss on the main fund yield, we will not consider it in this subsection. The17 2 U G H U V L ] H R I W K H P D L Q I X Q G 5 D W H R I U H W X U Q r profit = |r loss | D Q G r profit R E H \ V U(0.02, 0.04) I R U H D F K W U H Q G L Q Y H V W R U ratio ratio ratio ratio ratio Figure 11: The relationship between the order size of the main fund and its rate ofreturn, when the parameters of taking profit and stopping loss of the trend investorssatisfy r profit = | r loss | , where r profit obeys the uniform distribution U (0 . , . M Dbuy is the sum of the cost M buy in each trading period, which is given by M Dbuy = D buy X j =1 M buy ( j ) = D buy X j =1 n bmf X i =1 [1 + r C + i ( j )] j Y k =1 [1 + f b ( k − P (5)where the function f b ( k ) = r C + n bmf + N Xt ( k ) ( k ), k is the serial number of the trading periodof buying and we define f b (0) = 0. The total money M sell of the main fund obtainingfrom selling the securities in multiple trading periods is given by M Dsell = D sell X j =1 M sell ( j ) = D sell X j =1 n smf X i =1 [1 + r C − i ( j )] j Y k =1 [1 + f s ( k − P s (6)where the function f s ( k ) = r C − n smf + N Yt ( k ) ( k ), k is the serial number of the trading periodof selling and we define f s (0) = 0, N Yt ( k ) is the corresponding quantity of N Xt ( k ) in thetrading periods of selling, P s is the price of the security at the end of the last trading18eriod of buying and P s = D buy Q l =1 [1 + f b ( l )] P . Then r mf = M Dsell M Dbuy − D buy Y l =1 [1 + f b ( l )] P D sell j =1 P n smf i =1 [1 + r C − i ( j )] j Q k =1 [1 + f s ( k − P D buy j =1 P n bmf i =1 [1 + r C + i ( j )] j Q k =1 [1 + f b ( k − − ≈ [1 + D buy X j =1 f b ( j )] 1 + P D sell j =1 [ P n smf i =1 r C − i ( j ) /n smf + P jk =1 f s ( k − /D sell P D buy j =1 [ P n bmf i =1 r C + i ( j ) /n bmf + P jk =1 f b ( k − /D buy − , (7)in the approximation, we only keep the first order quantity of r C + i or r C − i , becausein our simulations the absolute values of r C + i and r C − i are no more than 0.1, and theabsolute values of P D sell k =1 f s ( k −
1) and P D buy k =1 f b ( k −
1) are no more than 0.4. We define g b ( j )= P n bmf i =1 r C + i ( j ) /n bmf /D buy + P jk =1 f b ( k − /D buy , and the term P n bmf i =1 r C + i ( j ) /n bmf in g b ( j ) measures the increased average cost of the main fund due to the price rise causedby its own buying in the j -th trading period, which is less sensitive to the change of n bmf than the highest rise of the security represented by f b ( j ) in the same trading period.Then, we explain the change of the term P jk =1 f b ( k − /D buy with n bmf is not as sensitiveas that of f b ( j ) with n bmf as follows, which measures the increased cost due to the pricerise caused by the main fund buying before the j -th trading periods.The greater j is, the smaller the number of the trend and contrarian investors whoseparameters r market are greater than 0 because some of them have traded with the mainfund, and the contrarian investors play a leading role, this is because in our modelsettings, the number of them is much larger than that of the trend investors and thedistributions of trend investors and contrarian investors are the same except for theirnumbers. Therefore, with the increase of j , the main fund will be more easy to push upthe security price, that is, larger j leads greater increase of f b ( j ) when n bmf increases, so f b ( j ) is more sensitive to the change of n bmf with larger j , then f b ( j ) is more sensitive tothe change of n bmf than P jk =1 f b ( k − /D buy .Based on the above analysis, the term [1 + P D buy j =1 f b ( j )] plays a leading role inthe change of r mf when n bmf changes. As n bmf decreases, D buy will increase, and [1 + P D buy j =1 f b ( j )] will also increase according to the above analysis of f b ( j ). So in our modelsetting, the main fund increasing the number of times of buying will increase its yield.A similar analysis of { P D sell j =1 [ P n smf i =1 r C − i ( j ) /n smf + P jk =1 f s ( k − /D sell } shows thatdecreasing D sell will increase the yield of the main fund.19 Conclusion and discussion
We have established an agent model to study the financial phenomenon with thecharacteristics of pyramid schemes, and simulated this kind of artificial financial mar-ket under different investor structures. Through the simulation results and theoreticalanalysis, we obtain the relationships between the rate of return of the main fund andthe proportion of the small trend investors in all small investors, the small investors’parameters of taking profit and stopping loss, the order size of the main fund and thestrategies adopted by the main fund. Our main conclusions are: when the main fundadopts the strategy of buying a certain amount of securities through just one marketorder in the first trading period and selling them also through one market order in thesecond period, the higher the proportion of the small trend investors, the smaller theorder size of the main fund to obtain the maximum yield, and after that the yield ofthe main fund decreases slowly with the increase of the order size; when the main fundadopts the strategies of buying and selling securities in multiple trading periods, we findthat the less trading periods of selling and the more trading periods of buying, the higherthe yield.Our results are helpful to explain the financial phenomenon with the characteristicsof pyramid schemes in financial markets, design trading rules for regulators and developtrading strategies for investors. Specifically, for investors, they can use our model todevelop effective investment strategies after evaluating the structure of investors in themarket; for financial market regulators, through our agent model and introducing pa-rameters in real financial market, regulators can limit the maximum order size of thebig investors in a certain period of time, so as to avoid the inequality of investors withdifferent amount of money.
This research was supported by National Natural Science Foundation of China (No. 71932008,91546201).
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