# Percolation Model of Stock Market Prices

It is well known that we often observe “clustering” (or herding) phenomena in the financial markets—the situation when agents in the market prefer to make the same decisions. This behavior is clear in terms of psychology, because people are used to behaving dependently with each other; we might be easily influenced by others in many aspects of our lives. This correlation comes from random clustering. In our model, there are clusters of agents, i. e. groups of traders in the market that prefer to act together, i. e. to buy or to sell securities simultaneously.

For market simulation, each occupied site is regarded as an agent, and clusters are groups of traders who randomly decide to buy or to sell together (Stauffer 2001). Argument a is a measure of a one-time step. Small value of a corresponds to a small interval, and a value near the maximum of 1/2 corresponds to a large-time interval. Argument has influences on an agent’s decisions: each cluster decided randomly to sleep with probability 1 — 2a or to be active with some probability. Argument pbuy is the probability that an active agent prefers to buy. Argument pseii = 1 — pbuy is the probability that an active agent prefers to sell. This parameter helps us to consider influence of past and present trends on the market. It’s important to note that in this model, we have an assumption that agents have only two possible activities—to buy or to sell—and its sum is a full group of events.

Thus, for every time step, we analyze the existing clusters and find the number ns of clusters containing s investors each. The distribution of ns closely follows the percolation threshold of the scaling law:

ns ~ S~Tf[(p — Pc)sa] (1)

with two critical a, r exponents, and a function f decaying exponentially in its tails. Then each cluster randomly decides to buy, sell or sleep with some defined probabilities pbuyt pseii, and (1 — 2a) probability of sleeping. The price change in the market in a one-time step, which is labeled as A(f), is proportional to the difference of demand and supply in this market:

A (t) = ^ss — ^ss, (2)

where the total demand is sum of all agents in all clusters that decide to buy, and total supply is the sum of all agents in all clusters that decide to sell (Stauffer 2001).

The important part of this research is the analysis of model behavior and price change at the critical moment of percolation threshold occurrence. It’s possible to explain this market mechanism: when p <pc, price rises and more people enter the

market. Therefore p rises until a big crash occurs at p = pc. In this moment the price falls sharply, agents suffer losses and leave the market. As a result p falls and the cycle starts again at low p. The market crash during the moment of percolation threshold occurrence means that the most agents have the same opinion about their strategy. It leads to mass selling or buying; such a situation causes a market crash or a market boom (Chang et al. 2002).

Thus the basic purpose of the percolation model is to analyze the percolation threshold, which characterizes the threshold probability of a market crash. The model studies A empirical distribution as a distribution of price change in the market.

For the modeling of percolation theory, we use the Monte-Carlo method, which was realized in statistical environment R.

Results of our modeling were processed in MS Excel. The steps for the single iteration of Monte Carlo simulation are presented in Fig. 1.

We study the A empirical distribution with different values of model parameters. We have discovered a strong interrelationship of statistical characteristics of the received distribution of size A from parameters pbuy, a (Figs. 2 and 3). It is possible to note various curves shapes of the received functions. Sharper excess of function is marked at the maximum difference between probability of purchase pbuy and probability of sale 1 — pbuy and measure of time interval a.

 Fig. 2 Empirical distributions of Д with different value of a

 Fig. 3 Empirical distributions of Д with different value of pbuy

If it’s necessary to receive authentic distribution of market price change, it is important to pick up values of the parameters defining current market trends and preferences of agents in this market.