یک مدل تصادفی سیستم های دینامیکی از بازار سهام سبک خاص

We propose a random dynamical systems model for a stylized equity market. The model generalises previous deterministic models for price formation in equity markets. We provide analytic results (existence of fixed points, existence of invariant measures) as well as numerical results indicating the dynamical richness of this simple model. The model can be used to assess the effects of uncertainty on the fundamentals on stock price dynamics.

The rôle of agents on the formation of prices in the stock market is a subject that has fascinated economists for a long time. There is a vast related literature on the subject to which thinkers of the caliber of John Maynard Keynes [1] or Fischer Black [2] have contributed. In recent years, models for the formation of prices using the theory of complex dynamical systems have been proposed. Richard Day [3], [4] and [5] has proposed a dynamical systems model for an equity market. Eventhough the model was very simple (a one-dimensional map) it was designed in a very clever way so as to reflect well-established stylized facts for the stock market structure. As a result, the model could reproduce very well qualitative features of stock market prices such as transition between bull and bear regimes, etc. The model was purely deterministic. The basic features of the model were the following. Each stock was supposed to somehow reflect the value of the fundamental it was representing. The stock market was assumed to consist of two types of investors, αα-investors and ββ-investors and a mediator through which transactions were made. Each type of agent followed a different policy towards investment decisions. Type αα-investors were those using sophisticated estimates of long run investment value. Such investors are those trying to buy when prices are below investment value (i.e., when chances of capital gain seem to be high) and trying to sell in the opposite case. Type ββ-investors cannot afford to be as sophisticated as αα-investors. Such investors make investments decisions based on current investment value. In some sense, ββ-investors are the noise traders (to follow the terminology of Fisher Black) entering the market when prices are high and exiting the market in the opposite case. The third type of agent is the market maker (or mediator) who sets the price in response to excess demand or supply. The prices are supposed to be formed by Walrasian tattonement. This simple dynamical model was found to present complicated behaviour giving rise to irregular oscillations of the prices (chaos). Furthermore, despite its simplicity it reproduced well-qualitative features of the prices, and could be used to understand the relative importance of the various types of investors in the market. As a result, it was greeted with great interest by the community (see the discussion following [5]). The model of Day was a purely deterministic model. Eventhough there have been followups of Day's research (see e.g. Ref. [6] and references therein), all of them, at least to the best of our knowledge, have kept the deterministic nature of Day's original model. It is the aim of the present model to include randomness in Day's model. The randomness is introduced through the current investment value which is used by the ββ-investors in their investment decisions. The model can then be formulated as a random dynamical system. Through the use of recent advances of the theory of random dynamical systems, a qualitative and quantitative study of the evolution of prices in the stock market as a result of the interaction of different types of agents is possible. We see that the salient features and findings of the Day model (such as the existence of fixed points or the existence of invariant measures) are robust in the presence of random effects. Furthermore, the existence of unstable dynamics in the stock market, leading possibly to erratic oscillations, is shown through the calculation of Lyapunov exponents. The model allows us to monitor the effect of uncertainty of the fundamental to the stock prices and assess the effect of the underlying market dynamics on the propagation of uncertainty from the level of the fundamental to the level of the stock. The model, despite its simplicity, contributes to the interesting and important field of market microstructure, an active area of financial mathematics, the study of which is necessary for the understanding of the structure and function of financial markets. The paper is organized as follows. In Section 2, we present the general features of our model and provide two special forms, a piecewise linear form and a piecewise monotone form. In Section 3, we present several qualitative and analytic results on the models, that is the existence of fixed points, the existence and approximation of invariant measures and a method for the calculation of Liapunov exponents, based on the existence of the invariant measure. In Section 4, we present our numerical results on the proposed models, which illustrate the dynamical richness of the random dynamical system governing the market dynamics. Finally, in Section 5, we summarize our findings and propose topics for further research.

One of the aims of the present paper was to examine the robustness of the qualitative features of the original model of Day and coworkers [3], [4] and [5] under the effect of uncertainty on the value of the fundamental. Assuming that the fundamental follows a stochastic process, Day's original model takes the form of a random dynamical system with additive or multiplicative noise. Through the study of this random dynamical system, we found the following persistent qualitative features between the deterministic and the random model: 1. Existence of random fixed points that are random processes invariant under the dynamics. 2. Existence of an invariant measure. 3. Existence of positive Liapunov exponents. We provide an approximation scheme for the density of the invariant measure using finite dimension approximations of the Frobenius–Perron operator as well as a scheme for the calculation of dynamical quantities of interest such as Liapunov exponents or the dispersion of the stock prices. Other examples can be estimates of quantities related to the viability of the market such as the maximum transaction fees Hmax:=Ep(α(p)+β(p))Hmax:=Ep(α(p)+β(p)). Two specific models are treated in detail: a piecewise linear model and a piecewise monotone case. Both models are stochastic generalizations of their corresponding deterministic counterparts. The morphology of the invariant measure allows us to visualize features of the dynamics that exhibit a very rich behaviour. For example we see that for certain parameter values the invariant measure presents a single mode, or it becomes bimodal or even multimodal. We observe features that are generated by noise and thus are absent in the original deterministic model, such as for instance the intermittent behaviour of the stock prices between bull and bear zones which are caused by the uncertainty of the fundamental. Another interesting feature of the model is that it allows us to assess the effect of market dynamics on the transmission of uncertainty from the level of the fundamentals to the level of the stock prices. One can imagine that the market mechanism (the tattônement process) acts as nonlinear filter on the uncertainty at the level of the fundamental values. We can quantify the above argument through the examination of the Liapunov exponent and the variance of the prices as a function of the variance of the fundamental. In particular, we find that regular market dynamics tend to amplify the effects of the uncertainty of the fundamentals on the stock prices, whereas, on the contrary, chaotic (unstable) market dynamics tend to suppress it.