Let us leave traders’ decision-making processes and turn to the adjustment of the stock-market price. We assume the existence of a market maker, such as a specialist in the New York stock exchange. The role of the market maker is to give an execution price to incoming orders and to execute transactions. The market maker announces a price at the beginning of each trading period. Traders then determine their excess demand, based on the announced price and on their expected prices. When the market maker observes either excess demand or excess supply, he applies the so-called short-side rule to the demands and supplies, taking aggregate transactions for the stock to be equal to the minimum of total supply and demand. Thus traders on the short side of the market will realize their desired transactions. At the beginning of the next trading period, he announces a new price. If the excess demand in period t is positive (negative), the market maker raises (reduces) the price for the following period t + 1. The process then is repeated. Let κ and ξ be the fractions of chartists and of noise traders in the total number of traders, respectively. Then the process of price adjustment can be written as
The term used to describe an investor who makes decisions regarding buy and sell trades without the use of fundamental data. These investors generally have poor timing, follow trends, and over-react to good and bad news. Let us consider the noise traders’ decision making. They are assumed to base decisions on noise in the sense of a large number of small events. The behavior of a noise trader can be formalized as maximizing the quadratic utility function
where xtn and ytn represent the noise trader’s excess demand for stock and for money at time t, respectively. The noise εt is assumed to be an IID random variable. In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d. or iid or IID) if each random variable has the same probability distribution as the others and all are mutually independent. The excess demand function for stock is given as
where γ denotes the strength of the reaction to noisy information. In short, noise traders try to buy stock if they believe the noise to be good news (εt > 0). Inversely, if they believe the noise to be bad news (εt < 0), they try to sell it.
where xtc and ytc represent the chartist’s excess demand for stock and for money at period t, and pt+1c denotes the price expected by him. The chartist’s excess demand function for the stock is given by
where the parameter µ(0 < µ < 1) is a so-called error correction coefficient. Chartists’ decisions are based on observation of the past price-data. This type of trader, who simply extrapolates patterns of past prices, is a common stylized example, currently in popular use in heterogeneous agent models. It follows that chartists try to buy stock when they anticipate a rising price for the next period, and, in contrast, try to sell stock when they expect a falling price.