Assume the market operates in the continuous time t , i.e. all trading occurs in a continuous flow, so we shall not need to distinguish between each individual order. We will only consider the price (or index ) of the market p(t) as a continuous function of time. At any time t , the joint market force f (t) acts upon the market and drives the price p(t) up or down in a continuous flux of movement. Note that the market force f(t) may include all kinds of supply or demand, greed or fear, rational decisions or emotional reactions, etc.

But it must realize its impact to the market price through all kinds of orders and transactions. We shall not concern ourselves with all the order details unlike the agent-based models such as one proposed by Farmer (1998). However, we shall follow some part of Farmer’s path in model building but our development is rather going into a direction quite different from his agent-based model.

Let p(t) be the price (or index) at time , the relative return Rt (t)is defined as

where t is the time scale. In general, it is more common to use the log-return rt (t) defined as

For small changes in p(t) , the log-return rt (t) and the relative return Rt (t) are approximately equal.

Suppose at time t the price p(t) is changed to immediately after the force f (t) acts upon the market, and this process can be expressed as

where is the abstract mass of the market, which reflects how much the price will change relative to the magnitude of the force, and which may depend on the capitalization of the market and the past trading history.

The price change should satisfy the following basic conditions

1) The price is always positive but finite

2) is an increasing function of the force , meaning that the price impact is in the direction of the joint force and (maybe nonlinearly) proportional to force magnitude

3) If there is no force at all there is no market impact, i.e.

4) is additive in the force

5) is a decreasing function of the mass , meaning that the price impact is inversely (maybe nonlinearly) proportional to the mass

6) The price return due to the impact of the force is completely determined by and m

δ is called the market impact function, and it must be an increasing function of f and a decreasing function of m according to equations (5) and (8). Applying equation (9) into equation (7) gives a fundamental equation of the market impact function

This equation has a simple but remarkable solution satisfying equations (5) and (8)

Therefore, the basic price impact function is derived as

The basic log-return dynamics function is thus

Or we should remember the time,

We see that this model is similar to Newton’s Second Law in physics. However, The abstract mass m is not a constant, but a slowly varying quantity in time relative to the variation of force f (t). It can be understood as a scale factor that normalizes the order size and be considered as the liquidity. However, we prefer to consider it as an abstract quantity whose variation may be related to the clustered volatility.

Considering the high autocorrelation of the clustered volatility, m may be considered constant for a limited time period. However, we must be aware that m may vary slowly. A varying mass may be compared to the Einstein’s relativity theory in physics. Although professional traders are aware of the relativity of price in mass psychology, a modeling of price relativity requires further research.

The log-price at the current time t can be obtained through the integral of the log-return from the starting time t =0 to the current time t

We mainly consider two types of forces: dynamic swings and physical cycles. For generality, assume there are L levels of dynamic swing forces fl, l=1,2.., L, and K levels of physical cycle forces gk=1,2.., K. The joint force f(t) is the sum of multilevel dynamic swing forces and multilevel physical cycle forces

Substituting equation (16) into equation (15) gives

where

are respectively the contribution to the current log-price ln p(t) by the evolutionary impact of the dynamic swing force fl(t) and the physical cycle force gk(t).

Let

equation (17) can be rewritten as

Apparently, F(t), ?(t) are the joint dynamic swing component and the joint physical cycle component in the log-price.

Prof. Heping Pan

Next: Multilevel Fractal Swings In Log-Periodic Power Laws

Summary: Index