A prototype model of stock exchange

– A prototype model of stock market is introduced and studied numerically. In this self-organized system, we consider only the interaction among traders without external influences. Agents trade according to their own strategy, to accumulate their assets by speculating on the price’s fluctuations which are produced by themselves. The model reproduced rather realistic price histories whose statistical properties are also similar to those observed in real markets. In the modern market of stocks, currencies, and commodities, trading patterns are becoming more and more global. Market-moving information is being transmitted quickly to all the participants (at least in principle). However, not all the participants interpret the information the same way and react at the same time delay. In fact, every participant has a certain fixed framework facing external events. It is well known that the global market is far from being at equilibrium [1], the collective behavior of the market can occasionally have violent bursts (rallies or crashes) and these violent events follow some empirically well-established scaling laws. These are currently the subject of intensive studies [2]-[6]. It is not settled yet whether these fluctuations are due to external factors or to the inherent interaction among market’s players. From a physicist’s point of view, the market is an excellent example of self-organized systems: each agent decides according to his own perception of the events. In the simplest framework, these events consist in the price fluctuations, the only available information. Each participant’s action will in turn influence the price. In a true economy there are external driving factors, such as politics, natural disasters, human psychology, etc. Another systematic effect is due to the periodicity of human life (days, weeks, months and years) which also influences the dynamics of prices. From the theoretical side, it is interesting to understand whether the statistical properties of prices depend directly on the external driving factors or whether they are self-generated by the system itself. In the present work, sticking to a physicist’s point of view, we shall address this issue by investigating this system in the absence of external factors. Thus in our market all the participants are speculators: they trade with the sole aim to increase their capital. We shall see that a very rich and complex statistics of price fluctuations emerges from such a c © Les Editions de Physique 480 EUROPHYSICS LETTERS closed system of traders which speculate on the price fluctuations they produce themselves. In spite of the simplicity of our model and of the strategies of the single participants, and the outright exclusion of economic external factors, we shall find a market which behaves surprisingly realistically. These results suggest that a stock market can be considered as a selforganized critical system: The system reaches dynamically an equilibrium state characterized by fluctuations of any size, without the need of any parameter fine tuning or external driving. Let us define our model more precisely. Each player is initially given the same amount of capital in two forms: cash Mi,t=0 and stock Si,t=0. At any time t the capital of player i is given by Ci,t = Mi,t + ptSi,t, where pt is the current price of the stock. There is only one stock in this model, e.g., a foreign currency. All trading consists of switching back and forth between cash and this stock. Each player has a strategy that makes recommendation for buying or selling a certain amount of stock for the next time step. This depends solely on the information available, i.e. the past price history. All the players have equal access to the price history. The actions taken by each player are bounded by his belongings. Player i can invest only a fraction of his stocks which, at any time t, is given by his strategy: At time t, the general form of the strategy of player i is Xi,t = Fi[ pt, pt−1, . . .], where Xi,tSi,t is the amount of stock player i decides to buy (Xi,t > 0) or to sell (Xi,t < 0). Our model draws inspiration from Brian Arthur’s model of Bar Attendance [7], where each bar hopper can formulate his own prediction, based on the past observation. This shows that, in a game of interacting strategies, the measure of a strategy’s efficacy can be given only a posteriori. Any strategy is, a priori, as good as any other. Therefore, in our case, initially the strategies are randomly chosen. Then, at each time step, the agent with the smallest capital is eliminated and replaced by one with a new (random) strategy. This refreshing rule keeps the population of the traders dynamic and it is a simple application of Darwinism to economy. Since, the “space of strategies” is enormous, finding some “local maximum” of the fitness is nearly impossible. Moreover, it is unrealistic to assume that the action of player i is independent of his belongings Mi,t and Si,t. For these two reasons we i) parameterized the functions Fi in terms of indicators Ik{p} and ii) we introduced a restoring effect which tends to balance the ratio Si,t/Mi,t to the current price pt. For the indicators we choose moving time averages of combinations of time derivatives of log pt (e.g., I1 = 〈∂t log pt〉, I2 = 〈∂ t log pt〉, I3 = 〈[∂t log pt] 〉, etc). The time averages were done over a time period of typically 10–100 time steps [8]. Note that considering time differences of log p and not of pt, makes the indicators, and hence the strategies, to depend on relative fluctuations of pt and not on his absolute value. The strategies were then parameterized by ` numbers ηi,k: