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This will cause a sharp decrease in X resulting in a crash. This is similar to the market discontinuity described in reference [23]. This result is robust and does not depend on the assumed preference. Fluctuations in the investment proportions induce price fluctuations, and in turn, fluctua- tions in the return on the stock. One of the fluctuations is big enough to cause a discontinuous crash.

This happens first at around time Once the price is down, the dividends become significant again in their contribution to the returns on the stock, and the returns are high. After 15 periods, when the catastrophic return resulting from the crash is beyond the memory span of the investors, the stock price jumps back up.

This cycle repeats itself very regularly with a period of 30 trades 15 highs, 15 lows. Due to the investor-specific nature of the noise, not all investors behave in the same way in spite of the homogeneity of preference, memory span, and strategy. Hence, there will be trade in the market and investors will become diverse in respect to their wealth. How is the wealth distributed across investors throughout the run?

The answer to this question is shown in Figure 2. The horizontal axis depicts the wealth as a percentage of the average wealth for a given time period , and the vertical axis depicts the number of investors with that wealth. As can be seen, at time 1, the distribution is very narrow — almost all investors have the same wealth — the average wealth.

Some investors are "lucky" — they happen to buy stocks before the booms and sell before the crashes whereas the "unlucky" investors do the opposite. As would be expected, the wealth distribution in this case assumes the form of a normal distribution with a standard deviation that increases over time. How is the behavior of the market affected by the memory span of the investors? The dynamics of a market of homogeneous investors with logarithmic utility functions and memory span 5 is shown in Figure 3.

The dynamics is very similar to that depicted in Figure 1, except that the cycle is reduced to 10 time periods because of the shorter memory span 5 highs, 5 lows.

Mathematical Model Formulation

The cycles are shorter because of the shorter effective memory, however, the cyclic booms and crashes persist. What can we learn from Figures 1, 3 and 4? First, with perfectly rational homogeneous investors, we obtain very unrealistic results: the stock price increases continuously at a constant rate the straight line in Fig. Furthermore, as all investors are assumed to be identical in all respects, they all have identical demand functions and we have a no trade market. When we introduce investor-specific noise, there will be trade but we still obtain unrealistic results: booms and crashes with cyclical regularity occur, hence they are fully, or almost fully, predictable.

This is in complete contradiction to market efficiency. The introduction of exponentially decaying reoccurrence of returns probability leads to shorter cycles of booms and crashes, but does not make the dynamics more realistic.

Microscopic Simulation of Financial Markets - Semantic Scholar

Thus, the assumption of a representative investor leads from reasonable microfoundations to totally unreasonable macroeconomic results. The introduction of investor-specific noise does not help for further details on the effects of varying degrees of noise see [22]. We now turn to analyze the effects of other sources of heterogeneity.

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The first case of non-homogeneous investors studied was that of a market with two types of investors: investors with memory span 15 and investors with memory span 5. All investors were assumed to have logarithmic utility functions and are divided equally between these two groups. The results are shown in Figure 5. As can be seen, what we have here is clearly not a simple superposition of Figure 1 and Figure 3.

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The dynamics resulting from the interaction between these two investor types is very complex, perhaps even chaotic. We still see booms and crashes, however, they are much less predictable. Thus, heterogeneous memory spans induce more realistic results than those observed in Figures 1 and 3. It is interesting to note that around time , there is a transition from cycles of about 10 periods corresponding to a 5-period memory span to longer cycles of about 30 periods corresponding to a period memory span.

This can be partially understood from Figure 6. In Figure 6 we see that although the two investor groups start with the same wealth distribution, after a number of time periods they form two distinct wealth classes. The population with the larger memory span dominates most of the money m the market. This is probably the reason for the transition to longer cycles. We continued to make investors more heterogeneous and for each investor in the market, we chose a random memory span between 1 and 20 utility functions are still all logarithmic.

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This data correspond to the simulation presented in Figure 5. The general trend of transition to longer cycles persists but it occurs later and becomes more complex. We then studied a second form of heterogeneity — diversity of preference. All investors had memory span Figure 8 shows that the ordered cycles are still evident, with period 30, as expected. Thus, heterogeneity of preference alone does not lead to reasonable market behavior. Although it looks as if we have the dynamics of a homogeneous investor market, Figure 9 shows that there is a power struggle between the two investor groups.

It would appear that the more diverse the investors, the more complex and realistic the market. The rest of the investors we took as in Figure 10 with memory span and preference chosen randomly. The simulation results are presented in Figure We see that the "long runners" have a stabilizing effect on the market, in that the fluctuations are smaller compared with Figure Figures 12 and 13 show the underlying tug-of-war between the "trend followers" and the "long runners".

This is presented in Figure Summary In this paper we suggest microscopic models of the stock market. We employ microscopic sim- ulation to study the macroscopic market behavior resulting from different microscopic models. We focus on the difference between the macroeconomic behavior resulting from representative investor models and heterogeneous investor models.

We analyzed the following forms of investor heterogeneity: investor-specific noise, hetero- geneous preference, heterogeneous expectations and heterogeneous investment strategies. The main findings of this paper are: 1 With homogeneous investors and no investor-specific noise the stock price increases at a constant rate and is fully predictable.

There is no trade taking place in the market. Combinations of the different forms of heterogeneity produce the most realistic macroscopic results. The more diverse the investors, the more complex and realistic the resulting market behavior. Our results are in sharp contrast to the results implied by models that rely on the repre- sentative investor assumption.

Microscopic simulation allows us to relax this assumption and to study the stock market realistically. We believe that microscopic simulation holds great potential for new insights into the complex behavior of the stock market. Appendix A In this appendix we show that a moderate change in the investment proportion in the stock dramatically affects the stock price. For simplicity, let us assume only one investor with an unchanging number of stocks N.

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This is identical to a market with many homogeneous investors and no noise because, in such a market, all investors act identically. First, recall that in the homogeneous case there is no trade. Hence, if money is not withdrawn for consumption which is the assumption in our model , all the additional cash flow can be invested only in bonds, which are exogenous to our model. However, to keep the constant investment proportions in the bond and the stock, there will be a demand for the stock, its price will go up until in equilibrium the proportion of investment in the bond and in the stock is restored.

Thus, if we are able to calculate the growth rate of the money held in bonds, we can deduce the growth rate of the stock price, which must be the same rate, otherwise the proportions held do not remain constant. As no stocks are bought or sold, all income gains are invested in the bond. The first term represents the compounded interest on the bond. The second term represents dividends that are reinvested in the bond. Thus, in the long run, we obtain: [1] Varian H.

Louis pp. Finance 37 Finance 42 Finance 46 Binder, Ed. II, to appear. Today World Modern Phys. C, Our selective review outlines the main ingredients of some influential models of multiagent dynamics in financial markets like Levy, Levy, and Solomon Economics Letters, 45, and Lux and Marchesi International Journal of Theoretical and Applied Finance, 3, The introduction of kinetic equations permits to study the asymptotic behavior of the wealth and the price distributions and to characterize the regimes of lognormal behavior and the ones with power-law tails.

Unable to display preview. Download preview PDF. Skip to main content. Advertisement Hide. Microscopic and kinetic models in financial markets. Chapter First Online: 06 June This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Bouchaud, J.