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…
I. The traders
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 sequential model. Each time t, a trader will be randomly selected to trade the stock with the market maker among the total population. Supposed there are totally T periods, following this process, we could find the total T traders to trade in this sequential model. 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
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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. Each general trader receives one general signal (sg) from the public source. 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. We assumed each general signal is drawn from the same distribution. Although each trader obtains one general signal, the realized signal may be different due to traders’ heterogeneity in preference, ability, etc. Hereinafter we denoted it as n1 = 1, 2, 3… N1.
Type II is the club traders who have another extra signal (sc). Imagine that the 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 assumed that the entire club traders are in the same institution and they will receive the same signal.
II. The institution manager
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
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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 simply supposed that the members will stay when they did earn some profits, even though the profits were getting close to zero. To make sure the profits will be positive for his members, the manager needs to affect the general traders’ decisions.
This is the only way to let his members earn positive profits.
III. The model structure
We assumed that the traders cannot communicate with 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. Because if he gave his
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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.
The structure of the model is a common knowledge to all traders. We could then move to how this game works. After traders received their signals, they stayed on 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. After the elimination, the actions would be
1 Think about if the trend goes up, but your signals tell you the value should be lower. You might be confused, and then you need to choose one side to be believed.
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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
The signals represented the estimation of the stock price. Because traders got their general signal by their own source, we assumed that the general signal would be drawn from a normal distribution, sg
. . . i i d
N (Pg, σg). That is to say, trader will have his own general signal.
As to the club traders, the manager will give them one same club signal, which had been drawn from sc 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
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the past prices out. It is a branch of waves of the history prices, not the whole past prices. In the real financial world, it will work out. In our simple one side transaction model, the entire history prices contain implicative information. Hence if we ignored some prices, the results will be unconvinced. 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, 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 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, ˆPT. 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 ˆPT; and vice versa. You could prove by yourself
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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 how the traders combine with the trend line and the signals.
For general traders, they will mix their signals and history, because both parts 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
21* 22* 23* information, every trader on this market follows the herds, a completely herds.
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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, which means w22= 0, then we could get the same result like a general-trade game.
Since you need to pay an entry fee to get the extra signal, and then you do not trust the signal, it sounds illogical. Hence we eliminated this situation.
Proposition 1 If w1= w21= 1, every traders on the market believe their own signals, the herds would never exist.
Every trader depends on their own signals to make a decision, automatically, the 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
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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
1 assumption of our model in this paper, we may find an important proposition here.
Proposition 2 Throughout the assumption we made, the former traders will have a better influence than the latter.
Since the latter traders need to calculate the entire previous prices, the decisions made by the predecessors would be influenced.