We consider an interacting particle system for the stock price fluctuation. The change of the stock price with a feedback by the price considering the herding behavior (majority orienting behavior) of traders, gives the van der Pol equation as a deterministic approximation. Considering the investment position of each trader, we introduce the delayed van der Pol equation. The history of investment positions, for example sell or buy, of each trader for a stock makes a memory effect, which is modeled by using the time retardation. The delayed van der Pol equation model seems to be natural and explains typical phenomena, for example triangle pattern, volatility jumps, price jumps and price trends, known for the time series of a stock price.
Using the Ising model, the analysis of the market has been discussed in Refs. [1] and [2]; besides, modeling the financial market by a certain form of Ising structure of the interactions of agents seems to be successfully achieved in several studies [3] and [4]. We think that most traders are influenced by rumors, excessively or under excessively react to the information, and like the subjective desirability more than the objective probability [5] and [6]. The minority traders of a market, who are diffident to their investment position in many cases, are going to follow the decision of the majority. Because they tend to think that the majority of the traders have more accurate information than themselves. A majority orienting model [7] is introduced, which is composed of three elements: the mutation of dealers, the majority rule and the feedback by the price, as basic elements for the change of a stock price in a real market. This model is a ternary interaction model of a finite particle, which makes excursions that are similar to the Ising model [8], assuming a mutation to the other type for each particle. The van der Pol equation is obtained as a deterministic approximation, which seems to explain the oscillation of a stock price.
Traders make use of information from the history of a stock price in order to gain profits by dealing stocks. We develop the majority orienting model taking into account of the feedback rule considering the history of buying then selling of traders and introduce a delayed van der Pol equation. Our present model makes the majority orienting model more realistic and helps to understand the dynamics for the change of the stock price. Our model seems to give an explanation for the typical phenomena, known for the chart analysis of the time series of a stock price, which is commonly used by traders; for example, triangular pattern volatility jumps, price trends and price jumps, those are perturbed by random noise in a real market.
We have introduced the delayed van der Pol equation by considering the effect by a trader's holding position in the van der Pol equation obtained by the majority orienting rule. In this model, the position of each trader makes a memory effect. The system depends on the history of price changes, because each trader calculates his profit and loss with reference to the book value of his holding position. The orbit of the van der Pol equation is asymptotically independent on the initial value at t=0. Because of that the delayed van der Pol equation seems to be strongly dependent on the value on the time interval [-u,0], and can generate triangle patterns, price jumps, volatility jumps and price trends which are frequently observed in time series of changes of stock prices. Our present study will show that the delayed van der Pol equation driven by random noise will give a good model for the change of stock price.
In this model, we assume that the linearity of the feedback power of the price to reference price is proportional to the difference of the price between the book value and the current price. The Prospect theory by Kahneman and Tversky, which is a theory about decision making of the human being in an atmosphere of uncertainty, says that human uses the “value function”, as a measure of decision making instead of the linear function, whose conceptual figure of the value function is shown in Fig. 5[15]. By using the non-linear function, the delayed van der Pol equation shows more intricate orbit, which is an interesting next subjects.The Prospect theory is one section of the “Behavioral Finance”, which is a new finance theory conditioning on the action of the investor who is not necessarily rational under such uncertain environment [5] and [6]. It includes the above-mentioned “information cascade” in many cases. In “Efficient Markets Hypothesis”, which is one of the important hypotheses of the conventional finance theory, each market player is assumed to be “enough rational”. If market players are enough rational, the useful information to estimate the future price will not remain in the past price transaction, because they use up the information in order to make profits. Therefore, if the Efficient Markets Hypothesis is true, then the price transition will become completely random noise, and the deterministic dynamics cannot be estimated from the analysis of the past price transaction. However, this assumption is unrealistic as shown by some irrational behaviors in Prospect theory. We think that the market players are “moderate” rational. The news which destabilizes a market occurs at random, we have to use a statistical model. However, we think that a dynamics is applicable in the behaviors of them after the news, which will be estimated with modeling the behavior of market players.
