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Stock-price dynamics: thermodynamic limit

Many meaning can be given to the system we have just considered. In contrast, the application we now examine is quite specific. In fact, we look at a stock market with N agents trading in a single asset, whose log price at time t is x(t). The agents buy, sell, or don't trade, and we shall denote by jn (t),n = 1,...,N, the demand for stock at time t. The strategies from which the agents may choose are: to be a bull, +1, to be neutral, 0, to be a bear, — 1. The aggregate excess demand for the asset at time t is then A(t) = jn (t), and we suppose that the price return

is proportional to A(t), that is,

* ^ N

where A (t) = У jn (t). у is the excess demand needed to move the

percentage return by one unit. For the sake of simplicity we take у = 1. We aim at evaluating the distribution of returns. This means that we must determine the joint distribution of {jn (t )}n N . Finally we suppose that the agents' changes of strategy are governed by a probability which is exchangeable and invariant in such a way that the main theorem holds.

At a given time, which for the sake of simplicity we do not make explicit, let N = (N+, N-, N0), N+ + N-+ N0 = N, be the state of the system, that is, N+ is the number of bulls, N that of bears, and N0 that of neutral agents. Hence, A = N + — N- holds. Moreover, we suppose that p+,p and p0 are the initial probabilities of the three strategies. At each step one (or more) agent(s) may change strategy. For positive X the probability that an agent chooses a strategy grows with the number of agents following the considered strategy. Considering a negative value of X, the probability that an agent chooses a strategy diminishes with the number of agents following the considered strategy. For |X| ^ the probability of his choice does not depend on the number of agents following one or the other strategy.

We have seen that in general the equilibrium distribution is (30). In the case we are considering this becomes

Now we suppose that the thermodynamic this limit holds, that is *P 0 N ^ = x = const. The limit we are considering implies

that the total number of agents N, and X p0, both grow without limit, with the proviso that this does not change the mean number of bulls and bears. This can be interpreted by saying that among the infinite number of neutral agents there are always agents that may become active while among bulls and bears there are agents that may become neutral.

If the thermodynamic limit holds, (35) factorizes. This means that p(N) tend (in distribution) to p(N+)p(N_) given these distributions negative binomial, that is

If this is the case, the moments of the equilibrium distribution of excess demand are functions of X and y. In fact, considering that

hence

Due to the stochastic independence of N+ and N_, we have

The last formula shows that the kurtosis of the negative binomial is large for small (Ap + + Ap _). A lot can be said about this. We limit ourselves to two remarks. First we note that the value of the kurtosis characterizes the "herd behavior" of the agents. Second, we have derived three equations that link the moments of the probability distribution of the excess demand with the three parameters of this model. As a consequence we have the possibility to estimate these parameters using the corresponding observed values. The estimated values may be inductively used, that is to say, they may be used to calculate the probabilities of future trends of the market. We note that this does not contradict the postulate according to which the behavior of the market is unpredictable. The unpredictability of the market does not prevent us from looking for probabilities, just as the unpredictability of how a tossed coin falls does not prevent us from determining the probabilities of heads and tails. As a matter of fact, searching for probabilities of uncertain events is exactly what people have been doing since the emergence of probability in the Renaissance, and what Keynes with his Treatise on Probability recommended to us at the beginning of the twentieth century.

 
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