異質與分群訊息在金融市場的交易行為及績效分析 - 政大學術集成

全文

(1)國立政治大學社會科學院經濟學系碩士論文 Department of Economics College of Social Sciences National Chengchi University Master Thesis. 指導教授：莊委桐博士 Advisor: Wei- Torng Juang, Ph. D.. 立. 政治大. ‧ 國. 學. ‧. 異質與分群訊息在金融市場的交易行為及績效分析 Nat. n. er. io. sit. y. Analysis on Heterogeneous and Subgroup a l Information i v n Ch e n g cMarkets in Financial hi U. 研究生：楊祐宗撰 Name: Yu- Tsung Yang. 中華民國九十九年一月 January, 2010.

(2) 謝辭本篇論文得以順利完成, 最應該感謝指導教授莊委桐博士。從研究題目、研究方法、研究結果等, 自始至終給予我莫大的鼓勵與幫助。教授學術涵養之深厚、邏輯思考能力之靈活、處理問題態度之積極, 凡此種種我皆望塵莫及。短短兩年時光中, 我瞭解了學問的浩瀚以及研究的精深。更重要的是從教授身邊所學, 能令我終身受用的研究過程, 這是我一輩子的寶藏。教授總在百忙中給予我建議與方向, 讓我能朝正確的方向摸索邁進。真的非常感謝教授您的指導, 是我一生受用無窮的回憶。. 立. 政治大. 另外當然也要感謝口試委員袁國芝教授與葉俊顯教授珍貴的建議與幫助,. ‧ 國. 學. 沒有他們從另一種角度的切入討論, 本篇論文不可能如此周延完善。也感謝教導我的政治大學, 讓我在此成長與茁壯。. ‧. 感謝在背後支持我的朋友、同學與家人。如果沒有桂方的幫助就不會有本篇. y. Nat. io. sit. 論文的誕生。也感謝昶翔與維傑, 他們總是熱烈且耐心的跟我討論論文的細節。. n. al. er. 特別是維傑, 竟然還認真的幫我架構了一段程式的內容, 雖然最後無法派上用場,. Ch. i Un. v. 但是他的熱血使我感動。也謝謝莊委桐教授指導的同儕賽局小組成員們, 組長柏. engchi. 鈞總是給人意想不到的思考方向, 奎后常常帶來精闢而深入的討論, 最後一位加入的成員修懷則伴隨著許多細部的補強。還有感謝花了許久時間一直嘗試找出我論文缺陷與漏洞並想方法予以補強的柏園、之元與壬德。最後當然還要感謝這段期間一直給予我支持鼓勵與幫助的朋友、同學與親人們。只能說需要感謝的人真的是太多了。於此獻上我最真誠的感謝, 給教授、老師、朋友、同學以及親人們, 謝謝你們一路走來的照顧。楊祐宗謹誌於政治大學經濟研究所民國九十九年一月. i.

(3) 摘要金融市場中存在著許多預測機構, 他們各自召集信眾並且不時地釋放訊息給其會員好讓他們能在交易中獲利。每筆訊息皆代表著各機構對此資產價值的預測, 會員則依此訊息至市場上尋找機會交易。他們交易前會先理性地觀察市場過往的波動。如果市場走勢所預測的訊息與自己的訊息一致, 那交易者交易時大概不會有所顧慮。然而當市場趨勢與自己的訊息不一致時, 交易者勢必會陷入兩難。仔細地衡量斟酌兩股力量的輕重後, 進而選擇他覺得對的決定。如果交易者放棄. 政治大如果某一機構的會員人數龐大, 則他們勢必會影響市場價格的波動。不知情立. 自己的訊息而追隨前人交易的腳步, 那我們可定義這是一種群聚的行為。. ‧ 國. 學. 的交易者在看到價格趨勢如此時, 可能會放棄自己的訊息轉而追隨過往交易者的選擇。然而此種交易伴隨著風險, 因為不知道正確的訊息為何, 當價格已經達. ‧. 到機構所預測的目標時, 知情的會員便開始反向操作, 而不知情的交易者可能會. sit. y. Nat. 持續地採取此一交易策略。於是當資產真正價值揭露時, 不知情交易者便可能因. n. al. 情勢的變化時, 便可能面臨損失的風險。. Ch. engchi. er. io. 此被套牢。跟隨大眾的決策相對保險, 但是當追隨的人沒有額外的訊息無法查覺. i Un. v. 我們建構了一個存在兩種類型交易者的市場, 一方是沒有參加機構的一般交易者, 另一方是同時參加某一機構的會員交易者。透過私有訊息與公開歷史交易預測的權衡, 交易者必須想辦法在這一次機會的交易中獲利。而我們想找出是否對任何交易者而言, 參加預測機構是有利可圖的。想當然爾, 市場中會員交易者的多寡對於機構預測目標價位的達成頗為重要, 因為影響力的大小間接決定了市場雙方的利潤。當然每位交易者對訊息的信心也有所不同, 這些因素都會影響雙方的利潤。而本篇論文即是嘗試找出在哪種條件之下, 參與機構交易者的交易績效會比沒有參與機構交易者的績效為佳。. ii.

(4) Abstract. Traders with their own heterogeneous hidden information are coming to the market to trade in order to maximize their expected profits. They will observe the trends of prices and compare it to their private signals and then make the right decisions. The trends might not consistent with the private signals.. If the traders choose to abandon. his own signals and follow the actions made by predecessors, we called the action. 政治大 In this paper, we set a mechanism to harmonize with these two powers. 立. “Herds.”. Also we. ‧ 國. 學. put the traders into two subgroups, and one of the groups will send another signal to its members. For simplicity, we use a sequential trading model to see the trade. ‧. patterns. Since we use the closing price to measure traders’ profits, traders in the. Nat. sit. y. market need to presume what the closing price will be. Then we calculate the profits. n. al. er. io. of each group and find out their performance.. i Un. v. We want to see under what kind of conditions, the performance of one group will. Ch. e n gIf cweh ican. be better than that of another group.. find the conditions of better. performance, it is worth for the traders to join that group.. Keywords: subgroups; heterogeneous information; herds; performance.. iii.

(5) Table of Contents. 謝辭............................................................................................................. i 摘要............................................................................................................ ii Abstract ..................................................................................................... iii Table of Contents ...................................................................................... iv. 治政大 List of Figures and Tables ........................................................................ vi 立 ‧ 國. 學. 1. Introduction ..........................................................................................1 2. Literature Review ................................................................................4. ‧. 3. The Basic Model ..................................................................................5. y. Nat. er. io. sit. 4. The Equilibrium Decision Rule .........................................................14 4.1. The concepts of profits a ...................................................................14. n. iv l C n h e............................................................... 4.2. The process of simulation 15 ngchi U 4.3. The numerical example ..................................................................18 4.3.1. The one round game .................................................................18 4.3.2. The two rounds game ...............................................................19 4.4. The explication of the diagrams .....................................................20 5. Discussion of Results .........................................................................25 5.1. The basic cases ...............................................................................25 5.1.1. Case I (sc ~ N (120, 1), T = 100, w22 = 0.5) .............................25. iv.

(6) 5.1.2. Case II (sc ~ N (150, 0), T = 100, w22 = 0.5) ............................28 5.1.3. A brief summary .......................................................................30 5.2. The symmetric cases .......................................................................31 5.2.1. Case III (sc ~ N (80, 0), T = 100, w22 = 0.5).............................31 5.2.2. Case IV (sc ~ N (50, 0), T = 100, w22 = 0.5) ............................32 5.2.3. Case V (sc ~ N (67, 0), T = 100, w22 = 0.5)..............................32 5.3. The longer periods’ cases ...............................................................33. 政治大. 5.3.1. Case VI (sc ~ N (120, 1), T = 200, w22 = 0.5) ..........................33. 立. 5.3.2. Case VII (sc ~ N (150, 0), T = 200, w22 = 0.5) .........................33. ‧ 國. 學. 5.4. The extended cases .........................................................................34. ‧. 5.4.1. Case VIII (sc ~ N (150, 0), T = 100, w22 = 0.9)........................34. sit. y. Nat. 5.4.2. Case IX (sc ~ N (150, 0), T = 100, w22 = 1) .............................35. er. io. 5.4.3. Case X (sc ~ N (150, 0), T = 100, w22 = 0.1)............................35. n. al 5.5. Summary ......................................................................................... 36 iv 6.. n U i e h ngc Extension............................................................................................ 38. Ch. 7. Conclusion .........................................................................................39 8. Appendix ............................................................................................40 References .................................................................................................57. v.

(7) List of Figures and Tables. Figure A 1: The stock price pattern of completely herding. .....................40. Table A 1: The results of proposition 1. ....................................................40 Table A 2: The relationship between actions and profits. .........................41 Table A 3: The one round example. ..........................................................42. 政治大 Table A 5: The price patterns 立 of case I......................................................43 Table A 4: The two rounds example. ........................................................42. ‧ 國. 學. Table A 6: The relative performance of case I. .........................................45 Table A 7: The absolute performance of case I.........................................46. ‧. Table A 8: The price patterns of case II. ...................................................47. sit. y. Nat. Table A 9: The relative performance of case II.........................................48. er. io. Table A 10: The absolute performance of case II. ....................................49. n. a l of case IV. .................................................. Table A 11: The performance 50 iv Un. C. h ecase i Table A 12: The performance of 51 n gVII. c h................................................ Table A 13: The performance of case VIII. ..............................................52 Table A 14: The relative performance of case X. .....................................53 Table A 15: The absolute performance of case X. ....................................54 Table A 16: The three-dimensional figures of the relative performance. .55. vi.

(8) 1.. Introduction. In the financial market, there are so many investment forecasting institutions which are providing various forecasting information for its members only. Sometimes it is difficult to get unless you join the institutions. Once you have paid the fee and become one of their members, the manager of the institution will give you their studies, the information about the assets. With this information in mind, you come. 政治大. to the market and try to find a best chance to trade. However, what the information. 立. points may not like the current trends in the market.. The traders will consider these. ‧ 國. 學. two indices carefully under this circumstance and then make the right decisions under rationality.. As you can imagine, there are a lot of traders with their private. ‧. information on the market, some are the members of institutions and others are not.. y. Nat. io. sit. In this paper, we try to find out that under what kind of conditions, the members will. n. al. er. earn more than the traders who do not join the institutions in the financial market.. Ch. i Un. v. Banerjee [1] showed that individuals make a onetime decision under incomplete. engchi. and asymmetric information sequentially.. Glosten and Milgrom [10] set a trading. market with heterogeneous informed traders.. Following the spirits of them, we. construct a sequential trading line market which is trading by the traders with heterogeneous information in this paper. For simplicity, there is only one institution on the market and traders are separating into two groups.. One group is consisted of traders who join the institution in order. to get extra information, and the other group is consisted of traders who need to collect information by their own. When traders come to the market with their own information, after the observations of the past actions, they can make their decisions.. 1.

(9) Bikhchandani, Hirshleifer, and Welch [19] found that individuals rationally ignore their own information and imitate their predecessors, a phenomenon of herding.. In. this model through our mechanism, the herding will not persist which is equal to the result of Cipriani and Guarino [17] because of the adjustment of asset price. Rational herds, defined by Bikhchandani and Sharma [20], occur when sequential individuals make identical decisions and they do not give their private information up necessarily.. Celen and Kariv [4] gave a clear definition between informational. cascade and herd behavior.. In this paper, traders observe the past actions and will. compare the history trend to their private information, and then the herds may occur.. 治政 If the institution is trying to influence the price to the大 value they set, it is not enough 立 to reach the target price by their own power of its members. The institution needs to ‧ 國. 學. guarantee that the non-members will be infected. Since the members have extra. ‧. information, if they can influence the decisions made by non-members through price,. sit. y. Nat. they can have better profits.. al. We believe that the price connection between two days is not. iv n C Traders bring their private and observe the public history, h e n information gchi U n. independent.. io. of Brown motion.. er. The mechanism of asset price in this paper is not using the recently popular method. and then make their decisions.. In some sense, the private information reveals the. value of the asset, and it is like the fundamental analysis; in another sense, the public information affects the private information and it is like the technical analysis. Traders will use both of them as their judgments when they are trading.. A lot of. papers ignore this point. The aim of this paper is trying to find out if there is any benefit for traders to join the institutions. Hence we need to compare the performance of the members and the non-members. We set a lot of parameters to see that what conditions make the performance of the members better through simulation. 2.

(10) In chapter 3, we introduce the set up of this paper.. In chapter 4, we are starting to. describe the calculation of performance and the method of simulation. two examples to explain our simulation.. In chapter 5, we list ten different cases to. contrast the performance with these two groups.. 立. Finally, we make our conclusions.. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. Also we use. engchi. 3. i Un. v.

(11) 2.. Literature Review. Following Glosten and Milgrom [10], we ignored the bid and ask prices to make the herds apparently.. Traditionally, there was always an asset value v, and the. signals would be randomly drawn from two states.. If you had drawn a good signal,. you could guess that the value of the asset would be high, and then you probably would call a buy order.. On the contrary, if you had drawn a bad signal, you would. 政治大 and Holt [11] set another mechanism 立. put a sell order. Scharfstein and Stein [7] used this assumption to observe the herd behaviors.. Anderson. ‧ 國. Alevy, Haigh, and List [9] were. 學. information cascades in the laboratory.. to observe the. experimented the information cascades in the laboratory.. The asset value was. ‧. simply becoming dichotomy and could not sophisticated deal with.. Nat. sit. y. Park and Sabourian [3] assumed that there are two types of traders, informed and. n. al. er. io. noise. A lot of papers assumed the comparative traders trade for liquidity, but in fact. i Un. v. most traders come to the market for profits. They are not just trading for liquidity, if. Ch. engchi. we had made this assumption, we would underestimate the ability of noise traders. In recent paper, Amador and Weill [13] assumed there are a public and a noisy signal and they wanted to see the speed of learning through observing other’s actions. We made the assumption further: there are two private signals and one public signal. Traders will act dependent on the weighted averages method of these three signals. Also the aim of analysis would be different, what we measured and interested is the relative performance between these two subgroups.. 4.

(12) 3.. The Basic Model. We mainly refer to the sequential trading model by Glosten and Milgrom (1985). Hereinafter we called it the GM model. In this paper, there is only one asset traded by a sequence of traders who interact with a market maker.. In our analysis, the risky asset may be thought as a stock.. Time is represented by a countable set of trading dates indexed by t = 1, 2, 3…. 立. I. The traders. 政治大. ‧ 國. 學. There are N traders with risk-neutral utility on the market and they all want to. ‧. maximize their own payoffs. Each of them is waiting to be selected to trade in the. Nat. y. Each time t, a trader will be randomly selected to trade the stock. sit. sequential model.. n. al. er. io. with the market maker among the total population. Supposed there are totally T. i Un. v. periods, following this process, we could find the total T traders to trade in this sequential model.. Ch. engchi. To make this model reasonable, we need to assume N > T.. If. T/N is sufficiently small, the chance for a trade would be precious. We need to add another property among the traders. Observing the traders on the realistic financial world, there are various kinds of traders who come to trade with various hidden information.. We do not know where their information comes from. and what exactly the information is. previous traders made.. What we might know are the actions the. Hence we could guess what the traders’ information is. basing on the actions we observe.. Before a trader making his decision, he will. observe the decisions made by previous traders and then compare it with his own. 5.

(13) information in order to make the right decision.. To simplify our model, we supposed. there are two types of traders (type I and type II). Type I group consists of general traders. signal (sg) from the public source.. Each general trader receives one general. The signals represent the estimated asset value.. Every trader gets one signal to help him evaluate the asset value and then, based on it, to make trading decision.. One could get the signal through the public source like. newspaper, magazine, television programs, etc. drawn from the same distribution.. We assumed each general signal is. Although each trader obtains one general signal,. the realized signal may be different due to traders’ heterogeneity in preference, ability,. 治政 etc. Hereinafter we denoted it as n = 1, 2, 3… N . 大立 Type II is the club traders who have another extra signal (s ). 1. 1. c. Imagine that the. ‧ 國. 學. club traders join an institution, such as a stock club, an investment bank, a legal. ‧. person, etc. To avoid the signals to be too disorder to affect our main analysis, we. y. sit. io. n. al. er. the same signal.. Nat. assumed that the entire club traders are in the same institution and they will receive. Ch II. The institution manager. engchi. i Un. v. Since the club traders could get an extra signal from the institution manager, of course they need to pay for it. Supposed that before the starting of this game, the traders will decide whether to join the institution or not.. If the traders want to join. the institution, the participants need to pay a fee (F) for getting the extra signal. Maybe the signal the manager gave was the report made by him, his collection from the public information, or even a number he randomized selected. We do not know how he got the signal.. All we concern is that the members do have faith on the. signal the manager gave them.. To simplify our work, we assumed that the 6.

(14) participants were given before the trading procedure began.. That is to say, the club. traders are going to be fixed at some amount through the whole game. The manager knows the signal he gave has the critical power among his members. If he wants to make profit, he has to make sure his members will have positive profits after they accepted the suggestion made by him.. If the member fee is the only. revenue the manager could get, he will want to maximize his total profit, πm = n*F – cm, where n is the amount of his members and cm is his cost. From the perspective of the members, if they paid a fee and still lost money when they had a chance to trade, then they will choose to leave the institution. Hence we. 治政 simply supposed that the members will stay when they 大did earn some profits, even 立 though the profits were getting close to zero. To make sure the profits will be ‧ 國. 學. positive for his members, the manager needs to affect the general traders’ decisions.. sit. y. Nat. io. er. III. The model structure. al. n We assumed that the. ‧. This is the only way to let his members earn positive profits.. iv n C tradersh cannot communicate with engchi U. each other.. If. communication does happen, there will be no longer any hidden information, the signals will be revealed and the herds will not happen.. In such case, our work will. be vain. Also the club members would not permit to know each other. Observing the decisions and traders could deduce the signals the decision makers carrying-on.. If every trader could trace the former traders’ hidden signals, then when. the club traders have an advantage among the proportion of the whole population, the club traders will have an invisible power and influence every decision the general traders are going to make. The manager knows this point. Hence he will have no incentive to give his members the inconsistent signals. 7. Because if he gave his.

(15) members more than two signals, the past actions will confuse the general traders and the herds may not occur. Now you could find the most different part of these two type traders is that the type II traders have one different signal.. Since the club traders are on the same market,. they could also get the general signals.. Hereinafter we denoted the number of the. club traders as n2 = 1, 2, 3 … N2. Hence the whole population are N = N1 + N2, and the proportion of the club traders is β ［Pr(N2/N) = β ≥ 0］, and the proportion of the general traders is 1－β ［Pr(N1/N) = 1－β ≥ 0］, where β ∈ 0,1 , and the number of traders should be positive integer.. 治政 The structure of the model is a common knowledge大 to all traders. We could then 立 move to how this game works. After traders received their signals, they stayed on ‧ 國. 學. the market until the time they had been selected. Once they have a shot to trade,. ‧. with their signals in mind, they have to make a decision. Before the decisions had. io. be discussed later.. The mechanism of this part would. er. price will affect the ideas of traders definitely.1. sit. y. Nat. been made, they could observe how the past price moved. The trend of the stock. al. n. iv n C Facing the only one risky asset,hwhose true value e n g c h i Uis v ∈ R, the traders need to. decide their actions to maximize their expected profits. When the game ends, the market would be closed, and we could get a closing price. Comparing to the traders’ transaction costs, we may count the profit for each trader. When on one trader’s turn, he will decide to buy or to sell the asset, or he could decide not to trade.. However,. since T/N is sufficiently small, if the trader chooses to hold his trade, the probability he will be selected again would be zero, which means if he did not act in this time, his profit would definitely be zero. 1. After the elimination, the actions would be. Think about if the trend goes up, but your signals tell you the value should be lower.. confused, and then you need to choose one side to be believed.. 8. You might be.

(16) rationalized. The action space A = {buy, sell}. For simplicity, only one unit of asset would be allowed to trade each time. We ignored how the transaction money the traders have or come from; we just want to see the profits after their trades. Now we move to the decision process of the traders. As mention before, every trader could observe the history price and compare it with his own signals.. Both. these two elements help their decisions.. IV. The private signals. 立. 政治大. The signals represented the estimation of the stock price.. Because traders got. ‧ 國. 學. their general signal by their own source, we assumed that the general signal would be. ‧. i .i .d .. drawn from a normal distribution, sg. N (Pg, σg). That is to say, trader will have. Nat. sit. y. his own general signal.. i Un. v. N (Pc, σc). The club manager drew one number and sent it. been drawn from sc to all his members.. al. n. i .i .d .. er. io. As to the club traders, the manager will give them one same club signal, which had. Ch. engchi. One important assumption is that the club signal is not. necessarily correlated to the intrinsic value of the asset. Members have to believe what they had received from the manager. When the club traders acted united, they could have power to influence the general traders and then the herds will happen.. V. The public signals. To show this part of the history prices, we used the average price of all past prices. If you are familiar with the technical analysis, you might use the moving average lines. 9.

(17) to help your decisions. the past prices out. prices.. However, the moving average lines are cutting one part of. It is a branch of waves of the history prices, not the whole past. In the real financial world, it will work out.. In our simple one side. transaction model, the entire history prices contain implicative information. if we ignored some prices, the results will be unconvinced.. Hence. Because the trends are. using the past prices to estimate the future prices, the first few transactions would be a lot of influence through the whole game. At any given point of time t, there were a series of the history prices, ht = {p0, p1, p2, p3… pt-1}. t 1. Observing the history prices, the trader made whole the past prices. average ( Pt ) / t . t 0. 政治大 Since the trader needed to guess what the final T price is, he had 立. ‧ 國. 學. to find the trend of the stock price. The trend line we used in this paper is setting the average price as the previous price and then collocating with the opening price p0 to. ‧. conjecture the final price pT.. Mathematically, let ht be the final price a trader. Nat. t 1. [( Pt ) / t ]  p0. n. a l ht = Ch. t 0. t 1. er. io. sit. y. conjectures at time t,. * T  p0. engchi U. v ni. (1). Every trader could get a conjecture value of the closing price through this step. To prevent the variation of the conjecture would be too large to accurately reflect our results, thus we need to add one more restriction.. If the variation of the. conjecture is going to be too large, then we add the lower and the upper bounds. The lower or the upper bounds would be started when the conjecture is exceeding a range of the real previous price, pt-1.. In this paper, we assumed that range is fifty percent.. Supposed the trend gives us an expected closing stock price, Pˆ T. When the trend goes up, and it does exceed the fifty percent range, which means 1.5*pt-1, then we will use the 1.5*pt-1 to substitute the Pˆ T; and vice versa. You could prove by yourself. 10.

(18) that when the trend goes up, and the trend would never touch the lower bound. We defined it as kt, the evolution of history.. The information kt can also be. regarded as the projected future trend in the market, and it is a public circulating conjecture.. It will make the faith of optimism become more optimistic, and the faith. of pessimism become more pessimistic.. VI. The base function. After the introduction of the history forecasting part, we would like to show you. 治政 how the traders combine with the trend line and the signals. 大立 For general traders, they will mix their signals and history, because both parts ‧ 國. 學. estimate a closing stock price..  w1 * sg  (1  w1 )* kt. y. I t. (2). sit. Nat. base. ‧. formation. The decision-making bases on the following. n. al. and the kt is the history estimation at any given time t.. Ch. engchi U. er. io. Where the w1 is the weight for the general traders to mix their signals, w1 ∈ [0, 1], Inv the real economy world, i n. the weight w1 would be different person by person. For simplicity in this paper, we assumed all the general traders have the same weight. Moving to the club traders, the base formation is. base. II t.  w21 * sg  w22 * sc  w23 * kt. (3). Where the w21, w22, and w23 are the weights for the club traders to mix their signals, w21 ∈ [0, 1], w22 ∈ [0, 1], w23 ∈ [0, 1], and w21  w22  w23  1 .. Definition 1 If 1-w1= w23= 1, then there is no one caring about his private information, every trader on this market follows the herds, a completely herds.. 11.

(19) If the traders put more weight on their private signals, the herds probably would not easily succeed.. In the appendix figure A1, we use a diagram to show this result, and. the detailed description will be introduced in the next chapter.. Definition 2 If β= 0, there is no any club traders on this market.. The general. traders use their own special prowess to make the profits. We called the style of such game a general-trade game.. If β≠ 0, but the entire club traders do not believe in the signal the manager gave,. 治政 which means w = 0, then we could get the same result 大 like a general-trade game. 立 Since you need to pay an entry fee to get the extra signal, and then you do not trust the 22. ‧ 國. 學. signal, it sounds illogical. Hence we eliminated this situation.. ‧. sit. y. Nat. Proposition 1 If w1= w21= 1, every traders on the market believe their own signals,. io. er. the herds would never exist.. al. n. iv n C Every trader depends on their own to make a decision, automatically, the h esignals ngchi U herds will not happen.. Because nobody sees what the actions others had done, and. then the imitation would not happen. We also showed this result in the appendix table A1. We haven’t introduced how the two mathematical formulas work. At any given time t, a trader will face the previous price pt-1, if the estimative base is higher than or equal to pt-1, then he will buy the asset one unit; on the contrary, if the estimative base is lower than pt-1, then he will sell the asset one unit.. Hence the actions could. represent by at = {1(buying the asset), -1(selling the asset)}. When the traders made their actions, the price would change.. If the trader decides to buy the asset, then the 12.

(20) stock price will go up for one unit; otherwise, if he decides to sell the asset, then the stock price will go down for one unit. After the price had changed, the time goes to the next period, t+1. Next trader will repeat the steps we just mentioned, and the successors will follow the same steps again and again until the game ends.. VII. The price schedule. Now we could handle the price schedule t 1. Pt  Pt 1  at  P0   at. (4) 政治大 For simplicity, we ignored the bid and ask price. The history of this game is 立 t 1. ‧ 國. 學. public information, Ht = {H0, H1, H2, H3… Ht-1} = {(a1, p1), (a2, p2), (a3, p3)… (at-1, pt-1)}, and the initial state is H0 = {(a0, p0)} = {(0, p0)}. Above are the basic. ‧. assumption of our model in this paper, we may find an important proposition here.. sit. y. Nat. n. al. er. io. Proposition 2 Throughout the assumption we made, the former traders will have a better influence than the latter.. Ch. engchi. i Un. v. Since the latter traders need to calculate the entire previous prices, the decisions made by the predecessors would be influenced.. 13.

(21) 4.. The Equilibrium Decision Rule. In this chapter we are going to show how to count the profits and how does the model work by simulation.. 4.1. The concepts of profits. 政治大. We would like to introduce the strategies, γ = { s }sS , where  s is the mixed. 立. probability of action, ∀ a ∈ A,  s ,a ≥ 0,. . aA. 學. ‧ 國. strategy of all traders if they observe s; and  s  ( s ,buy , s ,sell ) , where  s ,a is the.  s ,a  1 .. ‧. Nat. sit. y. Definition 3 At any trading time t, the equilibrium in the market consists of a. n. al. er. io. trading strategy correspondence  s* ( P t ) : R2 , for every trader s ∈ S, and a price Pt * , such that. Ch. en. hi. i Un. v. gcR2 , s  S ,  * (Pt* t )  [ s* (P t )]sS  s* ( P t )  argmaxEt[ s ( P)], P   ( A). Traders are trying to maximize their expected profits. The expected profit for a trader, when he gets the signals s, the price schedule P, and plays the strategy γ ∈ ∆(A) is Et[ s ( P)]   s , sell [ P  Et (base S , H )]   s ,buy [ Et (base S , H )  P]. (5). Because we only allowed one action for one trade, there is only one strategy would be executed and the probability would be one under the time the decision had been. 14.

(22) made. To make it clear, we rewrote it below. After the final period the last trader had acted, the market would be closed, and every trader got the closing price to calculate their profits. The profit functions for the two type traders are tI  at *( PT  Pt ). (6). tII  at *( PT  Pt )  F. (7). We assumed traders need to transact under current price, pt. prevents the influential power of the last trader.. This assumption. If traders transact with previous. 政治大. price pt-1, the last trader will gain under every circumstance. Using the backward. 立. induction, this game will be a total different one.. ‧ 國. 學. If a trader’s action is 1, there will be two situations after the market closing; the first situation is that when the closing price pT is higher than the current price pt, the. ‧. trader will gain, and the difference between this two prices is his positive profit; the. y. Nat. io. sit. second situation is that when the closing price pT is lower than the current price pt, the. n. al. er. trader will lose, and the difference between this two prices is his negative profit.. Ch. i Un. v. As to the part of selling, which means at is -1, and there is no need to go into details, you could deduce by yourself.. engchi. In the appendix table A2, we used a series of simple. diagrams to show the relationship between actions and profits.. 4.2. The process of simulation. After the preliminary realization of our model, we would like to interpret the main contribution in this paper. The sequential trading line model is randomly drawing traders into the game to trade.. If the game had run for only one time, the profits we wanted to calculate must 15.

(23) be un-precisely. There are a lot of traders on the market waiting to be selected, if we re-run this game again, the traders, the sequential order of traders, the signals, the conjectures, the actions, and the prices, even the profits will be a totally different one. Every run for this game will get a diametrically opposed result, the probability for having the same result will almost be zero, and hence we have to run this game over and over until we have found its properties. Simulation provides a good way for simplifying the complicated works. Throughout the simulations, it does help us confirm the ideas we assumed in this paper, thus we put concentration not on the mathematical proofs but on the simulating results.. 立. 政治大. A run of this game we called it as one round, we ran this game for n rounds.2. The. ‧ 國. 學. procedure of this game is a random process, and the results of every round are. ‧. independent and hence could not ignore any one of them. We then have to make the. sit. y. Nat. average of every round to display our results.. io. al. er. In order to simulate the model with a convenient way, we substituted the discrete. n. uniform distribution for the hyper geometric distribution.. Ch. n engchi U. iv. To prevent the traders. selecting repeatedly, we set the population is ten times total periods, which means N = 10*T.. In the following simulations, we basically assumed that the T would be 100,. and the N would be 1,000. The initial price p0 would be 100, and we assumed that the general signals would not deviate too far from the opening price, sg ~ N (100, 10).. It implied that there. would not have any material changes. This assumption helps us focus on the signal the manager gave. After calculating the conjectures of the history trend lines, we will give them the. 2. We use the program MATLAB.. 16.

(24) exogenous w1, w21, and w22 to find out the base functions. The traders will make their actions and then the price will change. When the game ended, the profit for each trader will be counted.. We set the club. fee be zero in the following model. At the same time, the profit of the manager will be omitted.. The analysis of the market maker’s profit would also be omitted.. We have the actions, the price schedule, and the profits now. Then we would like to recommend the concepts of performances. It is meaningless to calculate the profit for each trader. Since the trading order is a random process, it will be different during each round.. It forces us to calculate the 治政 profit of each type. If we do know how many traders 大 of each type are in the 立 sequential line, then we could calculate the average individual profit for each type.. ‧ 國. 學. Using the vertical aggregation and dividing by total rounds, we could get the average. ‧. market price and the average profits.. If. sit. y. Nat. Then we move to the tremendous works, the average performance of each type.. io. er. we use the average individual profit of type II to subtract the average individual profit. al. of type I, we could get the absolute performance. The other measure way, relative. n. iv n C complicated. could get it by h e nWe gchi U. performance, is much. using the absolute. performance to divide by the average individual profit of type I. However, if the average individual profit of type I is negative, we need to add a minus sign to the relative performance as our final relative performance, while if the average individual profit of type I is zero, and the relative performance would be equal to the average individual profit of type II. After the completely known of this game, we know that we could control the n, T, p0, Pg, σg, Pc, σc, N, beta, w1, w21, w22, the upper bound, and the lower bound to master this game.. 17.

(25) 4.3. The numerical example. We use two simple examples to show the process of simulations.. 4.3.1. The one round game. For simplicity and clarity, we assumed there were only four periods, the opening price was 100, both the general signals and the club signal were drawing from N (100,. 政治大 the range of the bounds were 50 %. 立. 10), the N was 10, the beta was 0.5, the w1 was 0.5, the w21 and the w22 were 1/3, and. ‧ 國. 學. In the beginning of this game, the manager sent the club signal, 111.9092. first period, the market selected a type II trader to trade.. At the. His general signal was. ‧. 111.9004, his history was 100, and thus his base was 107.9336. When facing the. Nat. sit. y. previous price p0, the base was larger than the previous price, and then his action was. n. al. er. io. 1. The price went to 101. The second trader was coming from type I with his. i Un. v. general signal 99.6237 and the history 102. The base for him was 100.8118 which. Ch. engchi. were smaller than the previous price 101. Thus he chose to sell the asset, and the price backed to 100. The third trader also came from type I, his general signal was 103.2729, the history was 100.6667, and hence the base was 101.9698 which were larger than the previous price 100. He bought one unit and the price went to 101. The last trader was a type II trader, his club signal was also 111.9092, and the general signal was 106.8278.. With the 100.6667 history, his base was 104.7741 which were. larger than the previous price 101, thus his action was 1 and the closing price was 102. The game ended. According to the final price 102, using the profit functions, we could find out what. 18.

(26) the profit for each trader was. Here we sequentially wrote down the traders’ profits, 1, -2, 1, 0. As you might know, the first and the last traders were belonging to type II.. Hence the group of type II won total 1 dollar, on the contrary the group of type I. won -1 dollar. Because there were both two traders for each type in the market, the average individual profit of type I was -0.5, while this part for type II was 0.5. Having the above information, the absolute performance for this one round game was 1, and the relative performance was 2. We concluded that type II traders performed well this time.. In the appendix table A3, some diagrams were displayed to show this. result.. 立. 政治大. 4.3.2. The two rounds game. ‧ 國. 學 ‧. Following the set up above, we are going to see how the two rounds example works.. al. er. io. sit. y. Nat. We presented it by a table.. n. Table 1: The numerical example of two rounds game.. C hThe first round. U n i engchi. v. The sc was 105.689. Period. Type. sg. History. Base. Action. Price. Profit. 1. II. 97.4435. 100. 101.0442. 1. 101. -1. 2. I. 96.2253. 102. 99.1127. -1. 100. 0. 3. II. 97.0411. 100.6667 101.1323. 1. 101. -1. 4. I. 85.2487. 100.6667. -1. 100. 0. 19. 92.9577.

(27) Table 1 (continued) The second round. The sc was 97.66. Period. Type. sg. History. Base. Action. Price. Profit. 1. II. 101.1844. 100. 99.6148. -1. 99. -1. 2. I. 103.1481. 98. 100.574. 1. 100. 0. 3. II. 114.4351. 99.3333. 103.8095. 1. 101. -1. 4. II. 96.4903. 100. 98.0501. -1. 100. 0. 政治大 In the first round, the average individual profit of type I was 0; while the average 立. ‧ 國. In the second round, the average individual profit. 學. individual profit of type II was -1.. of type I was 0; while the average individual profit of type II was -2/3.. Hence the. ‧. average individual profit of type I in the two rounds was 0, and the average individual. y. Having the above information, the. sit. Nat. profit of type II in the two rounds was -5/6.. io. al. v ni. We concluded that type I traders performed well averagely.. n. was also -5/6.. er. absolute performance for this two rounds game was -5/6, and the relative performance. Ch. engchi U. In the. appendix table A4, some diagrams were displayed to show the results.. 4.4. The explication of the diagrams. In the more complicated case, we moved to the case of one hundred rounds. We assumed the periods were one hundred periods, the N was 1000, and the other variables remained unchanged. We listed some values to help your realization, the average club signal was 100.2611, the average general signal was 100.1335, the average individual profit of. 20.

(28) type I was -2.2644, and the average individual profit of type II was 1.7818. Hence the absolute performance was 4.0462, and the relative performance was 1.7869. We used a series of eight diagrams to record these results, and we changed the order to make a clear explanation.. Table 2: The explanation of the eight figures. The figures. The explanation Figure V represented the stock. 立. price in the final round. The 治政大 vertical axis denoted the stock price and the horizontal axis. ‧ 國. 學. denoted the time.. Since we. ‧. set the price would only. not like the pattern of the real stock price.. n. al. er. io. sit. y. Nat. changed for one unit, it might. Ch. engchi. iv n UFigure. I. represented. the. average market price of one hundred rounds.. Using the. vertical aggregation and then making. them. average,. we. could get the average market price.. You could see that. although it went down at the beginning, it eventually went. 21.

(29) up because the average signals were higher than the opening price. Figure. VI. represented. the. profits of type I in the final round.. The stems reflected. the traders’ profits at the given time.. 立. If the profits were zero,. probably traders at that time 治政大 had zero profit or type I traders did not make any trade, which. ‧ 國. 學. implied that at that time the. You could see that. the traders who had positive profits were lesser than the. n. al. er. io. sit. y. Nat. traders.. ‧. trade was making by type II. Ch. engchi. iv n Utraders. who. had. negative. profits. Figure. II. represented. the. profits of type I on average. We summed them by vertical aggregation and made them average.. You could see that. the performances for type I traders were not good.. 22.

(30) Figure VII represented the profits of type II in the final round. You could see that the traders. who. had. positive. profits were more than the traders. who. had. negative. profits. Figure. 立. III. represented. the. profits of type II on average. 政治大 You could see that the. ‧ 國. 學. performances. for. type. II. traders were much better.. ‧. n. er. io. sit. y. Nat. al. Figure VIII represented the. Ch. engchi. i Un. v. profits of every trader in the market in the final round. could. be. got. by. It. vertical. aggregating the profits of type I and type II.. We used this. figure. show. to. the. performances of the entire traders in the market.. 23.

(31) Figure. IV. represented. the. profits of every trader on average.. We could see that. the traders had negative profits averagely.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 24. i Un. v.

(32) 5. Discussion of Results. We will give different conditions to see under what conditions the performances of type II are better than that of type I.. 5.1. The basic cases. 政治大. 5.1.1. Case I (sc ~ N (120, 1), T = 100, w22 = 0.5). 立. ‧ 國. 學. For the first case, we assumed the club manager drew the club signal from N (120, 1). We varied the proportion of the club traders (β) from 0 to 1, and the weights (w1,. ‧. w21) from 0 to 0.5 to see under what conditions that the club traders performed well.. y. Nat. io. sit. In the following cases, we all adopted the set up below: the rounds n was 100; the. n. al. er. periods T was 100; the opening price p0 was 100; the general signal sg was drawing. Ch. i Un. v. from N (100, 10); the population N was 1,000; the weight on the club signal w22 was 0.5; and the bounds were 50 %.. engchi. Since the club signals were not deviated too far from the general signals, the variations of the conjectures between different traders would not be too large. appendix table A5, we listed some price patterns under different conditions.. In the The. traders both completely focused on the history on the top rows. When we moved down, both traders put more weight on their general signals, and at the bottom the general traders put half weight on their general signals and history, while the club traders completely ignored the history. When both traders completely ignored their private signals, the completely herds. 25.

(33) occurred although the club traders still put half weight on the club signal.. The price. reached its height earlier when beta increased, and this explained the club power had increased in the market when the proportion of the club traders increased. The price smoothly climbed up, and smoothly went down after the price touched its height.. If. we looked down, the traders put more concentrations on their general signals, and the price became fluctuant. We found that given any w1 and w21, when beta was larger than or equal to 0.3, the stock price went up over the value 120, the phenomenon of overshooting.. The price. quickly climbed up to the club signal 120, we could think it as an exogenous random. 治政 shock. This phenomenon also interpreted that if type 大II traders wanted to have an 立 influence on the price, the club manager had to ensure that the proportion of type II in ‧ 國. 學. this market would be over 0.2. When type II traders were large enough, through. ‧. every trade they made, they could influence the behaviors of the general traders.. sit. y. Nat. The general traders did not exactly know the club signal, what they knew was only a. io. al. er. range. When the price was over the club signal, type II traders might put a sell order,. n. but type I traders might still call a buy order, and then it would give the club traders a. Ch chance to snipe at the general traders.. iv n the performance of type II would better e Hence ngchi U. than the performance of type I. Also you could find out that when both traders put half weight on the general signals, the phenomenon of overshooting would not exist.. Traders seldom followed. the herds, and then the club traders would lose their united influential power. The signals were too noise when there were little club traders in the market (β ≤ 0.2).. If. the proportion of the club traders became more and the signals would be clearer, the general traders would precisely guess what the club signal was, the price height of the overshooting might decline. After introducing the price patterns, we are going to find out the better conditions 26.

(34) for type II traders.. In the appendix table A6, there were some diagrams to show the. relative performance. From the top left corner, both traders put half weight on their general signals.. If we moved to the right, the general traders focused more. concentrations on the history, while if we moved down, the club traders focused more concentrations on the history.. If we moved to the lower right corner, both traders put. more weight on the history. Starting from the figure w1= 0.5, w21= 0.5, if you took a horizontal glimpse, the height of relative performance would move to the right.. Since the general traders no. longer insisted on their private signals, the herds would happen easily.. In the case of 治政 w = 0 and w = 0.5, the club traders made their biggest 大profits when they had forty 立 percents of population in the market. Why the numbers of type II traders could not 21. 學. be any higher?. ‧ 國. 1. Because type II traders were more over the range in the market, the. sit. y. Nat. general traders.. ‧. club signal would be distinct, and then the club traders would not easily cheat on the. io. history.. er. Then we took a vertical glimpse, the club traders put more concentrations on the. al. However this time the relative performance fluctuated largely, the. n. iv n C performance of club traders evenhbecame negative.U But if the absolute value of engchi. difference between w1 and w21 became small, the situations would reverse to the situations that the club traders performed well. Since the club traders put more concentrations on the history, this time they would ignore the objectivity and jump into the trap. Hence we might conclude that if w1 equals to w21, the club traders perform little well; if w1 is smaller than w21, the club traders perform very well; while if w1 is larger than w21, the club traders perform very bad. In the appendix table A7, the corresponding absolute performance, you could find that the club traders had positive absolute performance for the most time. Comparing the relative and absolute performance, there were some inconsistent 27.

(35) heights.. Hence we could conclude that the inconsistent heights came from the tiny. average individual profit of type I.. 5.1.2. Case II (sc ~ N (150, 0), T = 100, w22 = 0.5). We would like to loosen the variations of the conjectures between different traders in the second case. Hence we set the club signal drawing from N (150, 0); the manager sent the value 150 of club signal through the entire game. To save the space of pages, we only listed some representative diagrams to show. unless both traders ignored their general signals.. 學. ‧ 國. 治政 the results. As you could see in the appendix table A8, 大the price went into the sky at 立 the beginning, unlike case I, the price height seldom touched the club signal 150 Since the club signal was deviated. ‧. too far from the opening price and the general signals, there would exist overshooting. y. sit. It was the reason why the price seldom touched the value 150.. io. er. market.. Nat. unless both traders focused on the history and type II traders were majorities in the. al. In the diagrams whenever the price reached its height and then it might went down,. n. iv n C it was because that the general signals around the opening h e n were gchi U traders would put a sell order after the price height. related to the assumption we made.. price and some. Also the price patterns were. Because there was a restriction on the volumes. of trades and the price was made by average.. If we took a look among one of the. simulation, the price might be more realistic. After introducing the price patterns, we are going to find out the better conditions for type II traders.. In the appendix table A9, there were some diagrams to show the. relative performance.. Unlike the relative performance of case I, the relative. performance of case II transformed a lot. The relative performance had only one peak for the most conditions in case I, but there were many peaks in case II. 28.

(36) Comparing it to the relative performance in the appendix table A6, we found that the basic results did not change too much.. In the north-east of the. northwest-southeast side diagonal, the place where the club traders put more concentrations on their general signals than the general traders did, the club traders always performed well. On the northwest-southeast side diagonal, the club traders performed ordinary this time.. In the south-west of the northwest-southeast side. diagonal, the place where the club traders put more concentrations on the history than the general traders did, the club traders performed badly. Comparing the relative performance to the absolute performance in the appendix. 治政 table A10, the reason for the peaks should attribute大 to the tiny average individual 立 profit of type I. If we took a horizontal glimpse, the absolute performance would ‧ 國. 學. decline with the increasing beta given any w21. The general traders would precisely. sit. y. Nat. signal.. ‧. guess what the club signal was when beta increased because of the deviated club. io. er. If we took a vertical glimpse, type II traders had negative profits when beta was. al. smaller than 0.4, while if we added more club traders into the market, the profits of. n. iv n C type II traders would become positive. could attribute the reason to the herding h e nWe gchi U power.. Although the general traders focused on their general signals than the club. traders did, the club traders did have the influential power when they were predominant on the population.. As you could see, in the south-west of the. northwest-southeast side diagonal, the profits when beta was one were larger than the profits when beta was smaller than one.. Even in the north-east of the. northwest-southeast side diagonal, the rank of the profits when beta was one was in the middle.. 29.

(37) Proposition 3 If the club signal does not deviate too far from the general signals, the relative performance of type II will be better when beta is smaller than 0.6.. Proposition 4 If the club signal does deviate too far from the general signals, the relative performance of type II will be better when beta is in the range of 0.2 to 0.6 under w21 ≥ w1. The relative performance of type II will be better when beta is 0.6 under w21 < w1.. If there are full of the club traders in the market, their profits will. reach its maximum under w21 ≤ w1.. the sixty percents trades in the market.. 學. ‧ 國. 治政 According to these propositions, if the manager 大 wanted his members to earn 立 positive expected profits under the deviated club signal, his members had to occupy On the contrary, when the club manager sent. ‧. a signal which was closing to the general signals, if there were some club traders in. n. sit er. io. 5.1.3. A brief summarya. y. Nat. the market and then the club traders would earn the biggest relative performance.. iv l C n hengchi U. Averagely, the club traders in case I performed well when we compared both absolute performances in case I and case II. There was little difference between the general signals 100 and the club signal 120, hence the club traders would be easily filtered through the general traders in case I and of course the club traders would make positive profits. Since the difference between the general signals 100 and the club signal 150 was too large, the club traders could not infiltrate into the general traders through their little members in the market.. No wonder the club traders. performed badly. According to the results of both cases, we could generalize a proposition below. 30.

(38) Proposition 5 If both type traders put the same weight on their own general signals, the relative performance of type II will little better than the relative performance of type I; else if the general traders put the weight on their general signals more than the club traders do, the relative performance of type II will worse than the relative performance of type I; else if the club traders put more weight on their general signals than the general traders do, the relative performance of type II will better than the relative performance of type I.. This tells us if you want to make a big profit when you already joined an institution,. 治政 you have not to completely infect by the herds. When 大you have nothing in mind, it 立 is not a bad idea to follow the herds. However you may lose profit on average ‧ 國. 學. because you do not realize what information is in their mind.. ‧ er. io. sit. y. Nat. 5.2. The symmetric cases. n. a l 0), T = 100, w22 = 0.5)i v 5.2.1. Case III (sc ~ N (80, Ch. n engchi U. In the above cases, we all set the club signal drawing from the price which was higher than the general signals.. From now on we move to the cases the club signals. are lower than the general signals.. First we drew the club signal from N (80, 0), and. other set up remained unchanged. Comparing the relative and the absolute performance of case I and case III, we could find out that the relative performance patterns and the performance heights were quite equal after excluding the situations of sampling error.. Although the variances. of the club signal were changing from 1 to 0, the difference was too small to see the. 31.

(39) fluctuations. Therefore we could not find the significant difference between these two cases. They were symmetric basically. We omitted the diagrams of this part to save our space.. 5.2.2. Case IV (sc ~ N (50, 0), T = 100, w22 = 0.5). Moving to the club signal case of N (50, 0), there were some fluctuations in the performance. Comparing the performance in case IV to the performance in case II, there were eight diagrams which did not like the corresponding diagrams of both. 治政 cases. The major difference between these two series大 diagrams was the performance 立 value when beta was equal to 0.1 or 0.2. We listed them in the appendix table A11. ‧ 國. 學. If we took a look on the absolute performance, we could find out that the absolute. ‧. performance decreased when beta increased in case II, while the absolute performance. sit. y. Nat. increased first and then decreased when beta increased. The difference diagrams. io. al. If the general traders ignored their general signals, the absolute. iv n C became hconcave. The U e n g c h i club traders. n. followed the history.. er. were on the upper right corner, the condition when the general traders completely. performance of type II. would have their. maximized profits when beta was 0.6 through the influential power. This result was one of herding.. 5.2.3. Case V (sc ~ N (67, 0), T = 100, w22 = 0.5). There were no difference between case I and case III, while there were some differences between case II and case IV. The relationship between them is like an arithmetic progression, then we would like to know if there might have the possibility that it presents as a geometric progression. The relationship between 150 and 100 is 32.

(40) like the relationship between 100 and 50 in the arithmetic progression, and the relationship between 150 and 100 is like the relationship between 100 and x in the geometric progression.. The x is approximately equal to 66.6667.. We drew the club signal from N (67, 0) to see if this club signal would satisfy our hypothesis. Although they were alike in some diagrams of relative performance, but if we compared it to the relative performance of case II and case IV, there were still unlike in a lot of diagrams.. Hence we concluded that the signals were adapted to the. arithmetic progression. The club signal 80 would be symmetric to the club signal. 治政 120, while the club signal 50 would be symmetric to大 the club signal 150. 立 had the symmetric cases and we omitted the diagrams to save our space.. Here we. ‧ 國. 學 ‧. 5.3. The longer periods’ cases. n. er. io. al. sit. y. Nat. 5.3.1. Case VI (sc ~ N (120, 1), T = 200, w22 = 0.5). Ch. engchi. i Un. v. If the periods become longer, the variations of the conjectures will be larger, we could imagine that the fluctuations will be exacerbated. We set the periods T turning into 200 and the population N became 2,000. The relative performance did not change when the periods became longer. The variations of the conjectures were not enough to rewrite the conclusion.. 5.3.2. Case VII (sc ~ N (150, 0), T = 200, w22 = 0.5). Under the case of N (150, 0), the results will be totally different.. 33. We listed some.

(41) diagrams in the appendix table A12. Under the longer periods, comparing the performance to the same weight performance when T was 100, we could find that the absolute performance were better in every level of betas. The absolute and the relative performance heights would move to the right when beta increased.. When both traders focused more. concentrations on the history and the conjectures became large because of the longer periods, the influential power of type II would increase.. Hence if the periods. became longer, the influential power of type II in the market became large, the general traders would be led to a worse situation.. 政治大. 立. Proposition 6 When the periods become longer in this game, the influential power. ‧ 國. 學. of the club traders becomes large, hence the general traders will be infected and their. ‧. profits will become lesser.. sit. y. Nat. n. al. er. io. 5.4. The extended cases. i Un. Ch. v. 5.4.1. Case VIII (sc ~ N (150, 0),eTn=g100, c h wi 22 = 0.9). What would happen when the club traders put more weight on the club signal? We listed two cases for this subject, one was w22= 0.9 and the other was w22= 1. In the appendix table A13, we listed some diagrams for case VIII.. In this case, the. members ignored the objectivity and became more confidence in the club signal. The relative performance patterns of case VIII became smoother than that of case II. When we looked at the absolute performance, we could find that there were positive profits for the club traders when beta was larger than 0.4. Unlike the previous cases,. 34.

(42) this time they had to gather more group power to influence the behavior of the general traders’, we would find out that the absolute and the relative performance were higher when beta was also higher. This implied the power of group.. 5.4.2. Case IX (sc ~ N (150, 0), T = 100, w22 = 1). The club traders only followed the instruction of the institution this time. The relative and the absolute performance patterns were quite alike comparing to the performance of case VIII. The only difference was that the value of the performance. 治政 was higher in case IX given any beta under different weights. 大 We would get a higher 立 value of performance, if we completely trusted the club signal and acted based on it.. Nat. sit. y. ‧. ‧ 國. 學. 5.4.3. Case X (sc ~ N (150, 0), T = 100, w22 = 0.1). io. al. We moved to the case when w22 was 0.1.. n. signal?. er. What would happen when the club traders did not pay much attention to the club. Ch. n engchi U. The v i. basic idea of the. performance was not changing. The club traders would have better performance when w21 ≥ w1.. We discussed the absolute performance in the appendix table A15 first. Comparing it to the table A10, in the north-east of the northwest-southeast side diagonal, the absolute performance of case X was greater than that of case II given any beta. While in the south-west of the northwest-southeast side diagonal, the absolute performance of case X was lesser than that of case II given any beta. We already knew that the club traders would perform better than the general traders under w21 > w1, and the club traders would perform worse than the general traders under w21 < w1. However this time the club absolute performance was better under w22= 0.1 35.

(43) than that under w22= 0.5 on the upper right corner, while the club absolute performance was worse under w22= 0.1 than that under w22= 0.5 on the bottom left corner. Starting from the case II of w22= 0.5, if we moved to the case VIII of w22= 0.9, the club absolute performance would overall rise; while if we moved to the case X of w22= 0.1, then there would have two effects.. The first effect was that when they. already performed well under w21 > w1, they would perform even better and the second effect was that when they already performed badly under w21 < w1, they would perform even worse.. 治政 If the club traders did not pay much attention to the大 club signal, which implied that 立 they ignored the subgroup heterogeneous information, the club absolute performance ‧ 國. 學. would accelerate the fluctuations because of the little influential club signal power.. ‧. Hence if we denoted the horizontal axis as the value of w22 from 0 to 1, and denoted. sit. y. Nat. the vertical axis as the value of the average absolute performance, then we could have. io. er. one U-shape curve under w21 > w1, and an up-warp sloping curve under w21 < w1.. al. In the appendix table A14, the relative performance was listed for your reference.. n. iv n C The peaks would be different in case comparing toU the peaks in case II under w21 > h eX n gchi w1, and the club traders had the negative performance when w1= 0.6 and the relative performance increased when beta increased.. 5.5. Summary. We summarized our work by the three-dimensional diagrams.. Fixed the w1 as the. x axis, the w21 as the y axis, we wanted to see under what conditions the club traders could get the biggest relative performance given any beta.. 36.

(44) As you could see the figure 1 below, if sc ~ N (150, 0), T= 100, w1= 0.4, w21= 0.5 and w22= 0.5, the club traders would get the biggest performance when beta was 0.1. When the group power was lower in the market, the club traders should not follow the herds.. It could prevent the club traders give up their heterogeneous information.. Figure 1: The 3-D of beta = 0.1 given sc ~ N (150, 0).. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. We sequentially wrote down the best conditions for the club traders when beta moved from 0.1 to 0.9: (0.4, 0.5), (0.4, 0.5), (0.2, 0.5), (0.1, 0.4), (0.1, 0.4), (0.2, 0.5), (0.2, 0.5), (0.1, 0.4), (0.1, 0.4). When beta was lower, the general traders should keep more focus on their general signals in order to prevent following the noise signals.. When beta increased, in order to let the general traders be infected, the. general traders should put more weight on the history. In the appendix table A16, we listed the remaining diagrams for your reference. 37.

(45) 6.. Extension. We simply assumed that there were only two type subgroups which were competing with each other.. In the real world, there is more than one institution.. The various institutions are sending their own signals to their members and trying to make their members the biggest expected profits.. If we assumed there were more. than two subgroups in the market, though the signals would be noised, the. 政治大 In this paper, in order to see the phenomenon of herds, we monotonously set that 立. competition between these subgroups would definitely be fascinating.. ‧ 國. 學. the general signals were around the opening price. We could further change the relative location of general signals.. If general signals were around the value of 120. ‧. and club signal was around the value of 150, the price height might raise.. However. Nat. sit. y. if we assumed that general signals were around the value of 80 and club signal was. n. al. signals would let the decisions become complicated.. Ch. engchi. er. io. around the value of 150, this analysis would be quite interesting. The contrary. i Un. v. If the time became endogenous, the manager could send the club signal in a multi-stage. He could send the signal earlier to his members who pay highest, and send the signal later to his members who pay little.. Throughout the continued. subgroup power of entering into this market, the influential herding power would increase. We could forward assume that the manager could send a more precisely Bayesian signal with the changing prices after the market opened.. The cloud. computing is very popular in recent days. Through a super computer, the New York Stock Exchange could send the signals to its major clients three seconds earlier than its ordinary clients.. This spirits would like the derivative works of this paper.. 38.

(46) 7. Conclusion. There are a lot of people on the financial market, and the reasons for their trades are various. But you might agree that most traders are seeking profits.. Throughout. their own private social connections, they came to the market and waited a better chance to trade with their hidden information.. Since the market will vary from. minute to minute, the traders will be infected with the phenomenon.. 政治大 confused between their own private signals and the status quo. 立. Although they. came to the market with the signals in mind, after observing the trends, they might be Hence some of them. ‧ 國. signals.. 學. will abandon their signals and follow the herds; others will still persist in their private For the traders, there are always two powers pulling and dragging, despite. ‧. their final choices and their final profits, they must make the decision under. sit. y. Nat. rationality.. n. al. er. io. We use a sequential trading line model and simply restrict their action space. The. i Un. v. entire traders need to guess what the final price of the single asset will be. We set. Ch. engchi. there are two type traders with different signals, and both of them will be infected with their own private signals and the public history trend line.. Based on the. structure, they have to make their actions to maximize their expected profits.. In this. paper, we provide a simulation way to observe the stock price patterns and the performance of both type traders. Through the transaction functions we made, in order to make the biggest performance, the club traders must not pay attentions on the history prices, or they would be infected by the herds. Hence we conclude the traders should not easily follow the herds.. With the heterogeneous hidden. information in mind, if the traders choose to follow the herds, it seems insane.. 39.

(47) 8. Appendix. Figure A 1: The stock price pattern of completely herding.. 立. 政治大. ‧. ‧ 國. 學. Figure A1 represents the stock price of completely herds, we may see under some. y. Nat. n. al. er. io. sit. conditions in the following figures that the completely herds do occur.. i Un. v. Table A 1: The results of proposition 1.. Ch. engchi. A one hundred rounds average. A one hunderd rounds average. A one hundred rounds average. price pattern.. profits of the general traders.. profits of the club traders.. In these series of figures, the club traders have half proportion among the population. We could see that the price pattern is irregular, and both traders have negative profits on average. Hence if you want to trade without consulting the history, you might have a great chance to get a negative profit. 40.

(48) Table A 2: The relationship between actions and profits.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. In these figures, the actions on the top correspond to the profits at the bottom. You could see the actions which decided the pattern of the price; while the final price which decided the profits of the traders. Using the concepts of the profits, and you will realize the relationship between the actions and the profits.. 41.

(49) Table A 3: The one round example.. The market price.. The profits of type I.. The profits of type II.. The profits of all traders.. Table A 4: The two rounds example.. 政治大. The average profits of. The average profits of. The average profits of all. type I.. type II.. traders.. Nat. n. al. er. io. sit. y. ‧. price.. 學. The average market. ‧ 國. 立. Ch. engchi. i Un. v. The market price in the. The profits of type I in. The profits of type II in. The profits of all traders. first round.. the first round.. the first round.. in the first round.. The market price in the. The profits of type I in. The profits of type II in. The profits of all traders. second round.. the second round.. the second round.. in the second round.. 42.

(50) Table A 5: The price patterns of case I. Sc ~ N (120, 1), T = 100, w1 = 0, w21 = 0, w22 = 0.5. Beta = 0. Beta = 0.1. Beta = 0.2. Beta = 0.3. Beta = 0.4. Beta = 0.5. Beta = 0.6. Beta = 0.7. 政治大. 學. Beta = 0.8. Beta = 0.9. Beta = 1. ‧. ‧ 國. 立. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Sc ~ N (120, 1), T = 100, w1 = 0.3, w21 = 0.3, w22 = 0.5. Beta = 0. Beta = 0.1. Beta = 0.2. 43. Beta = 0.3.

(51) Table A5 (continued) Beta = 0.4. Beta = 0.5. Beta = 0.6. Beta = 0.8. Beta = 0.9. Beta = 1. Beta = 0.7. 政治大. 立. 學. Beta = 0. Beta = 0.1. Beta = 0.2. Beta = 0.3. ‧. ‧ 國. Sc ~ N (120, 1), T = 100, w1 = 0.5, w21 = 0.5, w22 = 0.5.. n. er. io. sit. y. Nat. al. Ch. Beta = 0.4. Beta = 0.5. Beta = 0.8. Beta = 0.9. engchi. i Un. v. Beta = 0.6. Beta = 1. 44. Beta = 0.7.

(52) Table A 6: The relative performance of case I. Sc ~ N (120, 1), T = 100, w22 = 0.5.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 45. i Un. v.

(53) Table A 7: The absolute performance of case I. Sc ~ N (120, 1), T = 100, w22 = 0.5.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 46. i Un. v.

(54) Table A 8: The price patterns of case II. Sc ~ N (150, 0), T = 100, w1 = 0.1, w21 = 0.1, w22 = 0.5. Beta = 0.1. Beta = 0.5. Beta = 0.9. Sc ~ N (150, 0), T = 100, w1 = 0.3, w21 = 0.3, w22 = 0.5. Beta = 0.1. 立. Beta = 0.5 政治大. Beta = 0.9. ‧. ‧ 國. 學 y. Nat. Beta = 0.5. n. al. Ch. engchi. 47. er. io. Beta = 0.1. sit. Sc ~ N (150, 0), T = 100, w1 = 0.5, w21 = 0.5, w22 = 0.5.. i Un. v. Beta = 0.9.