Probabilistic Theory of Stock Exchanges - страница 35

In this paper we will also repeatedly provide various mathematical representations of this concept within the framework of probabilistic economics, complementing each other. For example, in the framework of our two-agent classical economics (negotiation model) let’s represent the S&D functions as follows:

In equations (1.6) and (1.7), we have defined at each time t the total buyer demand function, D^>0(t) and the total seller supply function, S^>0(t), as the multiplication of price and quantity quotations. For brevity, we shall hereafter refer to them simply as supply and demand functions, i.e., we shall omit the word «total» unless this could lead to confusion. These functions can easily be depicted in the time and S&D coordinate system, namely: [T, S&D], as shown in Fig. 1.4, which shows a diagram of the complete S&D functions. As expected, the S&D functions also intersect at the equilibrium point E_>1. More strictly, the equilibrium point is exactly the point in the diagram where the price and quantity quotations of the buyer and seller are equal. The fact that the S&D functions are also equal at this point is a simple consequence of their definition and the equality of prices and quantities in this point.

The last remark concerns the formula for estimating the market trade volume (Trade Volume, hereafter TV) in the market TV (t_>1^>E) between a buyer and a seller at those moments in time when they come to a mutual understanding and conclude a transaction at an equilibrium point. Clearly, one can simply multiply the equilibrium values of price and quantity in this classical market model to obtain the trade turnover, or the total volume of all transactions, which follows from the above formula. The dimension of trade volume is the product of the dimensions of price and quantity; in our example, it is $. The same is true for the dimensions of the functions S&D, namely D^>0(t) и S^>0(t). Based on Fig. 1.4 we can conclude that it is at the equilibrium point that the trade volume reaches its maximum value. This result, which is self-evident and trivial in this case, is, in our opinion, rather general and principled: using it, we can deduce an assumption that markets tend to reach the equilibrium where maximum sales in monetary terms are achieved. It is possible to formulate this statement differently – in the form of the following hypothesis: markets strive for the maximum trading volume that is reached in equilibrium conditions, which agrees with the principle of trade volume maximization.