# The volume and behaviour of crowds

## Volume is a comparatively neglected variable in academic finance - price and return usually attract far more research interest. An interesting recent exception to this rule, which examines the interaction of volume with behavioural finance, is "Market crowd trading conditioning, agreement price, and volume implications" by a group of Chinese researchers. Automated Trader discusses the paper with its lead author, Leilei Shi of the University of Science and Technology of China.

Automated Trader: What prompted you to explore the significance and influence of volume and its interaction with trader behaviour?

Leilei Shi: Well, it is relevant to our research background and the price-volume probability wave 1 equation I developed in an earlier paper, "Does security transaction volume-price behavior resemble a probability wave?". That earlier paper was in turn inspired by George Soros' book The Alchemy of Finance, which made me wonder whether it was possible to derive a wave equation for the stock market.

In the course of attempting to derive the equation, I discovered that liquidity levels directly drove the joint behaviour of volume and price, as well as how volume was distributed across a price range. Where price was volatile upward and downward on a daily basis, accumulated trading volume would show kurtosis 2 around a price mean value (for a practical illustration of this based on a stock price history, see Figure 1.)

##### Figure 1

From this analysis it was eventually possible to derive a price-volume probability wave equation and obtain a closed form solution 3 for the volume distribution.

Automated Trader: Can you explain the assumptions behind the price-volume probability wave equation?

The fundamental assumption is the existence of a trading liquidity utility expressed in terms of accumulated trading wealth. It is the rate of liquidity, similar to power or the rate of work in physics, which we term liquidity energy.

We divide the liquidity utility into two parts:

• The trading momentum utility, which is the momentum energy that drives the distribution of momentum

• The supply/demand imbalance utility, which is the potential energy that produces reversal forces in states of stationary equilibrium as they occur in the market.

We found the states of stationary equilibrium mentioned in the second point above to be metastable 4 in the stock market. Some of the time, traded prices are volatile upward and downward after small supply-demand imbalances but continually revert to a stationary equilibrium price, thus resulting in a mean return of zero or very close to zero. On other occasions, the equilibrium price jumps after a large imbalance, resulting in a significant price mean return. The equilibrium price is defined as the price at which accumulative trading volume exhibits kurtosis during a given time interval.

##### Figure 2: Three-term contingency in an open feedback loop in trading conditioning

Automated Trader: So how does the price-volume probability wave equation interact with behavioural analysis 5 in the most recent research paper?

As we study the relationship between incremental liquidity, any equilibrium price jump, and volume, we also have to explore the volume uncertainty associated with a jump. (There are both positive and negative correlations between volume and a jump.)

We therefore attempted to incorporate subjective thinking into our model. Traders and animals behave extremely alike when learning in a highly uncertain environment. In terms of behaviour analysis, information and events on market return represent discriminative stimuli 6 , a trading action represents an operant, and return represents reinforcement/punishment (see Figure 2).

Volume probability in the wave equation can be used to measure the market crowd's operant frequency. Therefore, we introduced the notion of trading conditioning in terms of classical and operant conditioning. In this way, we were able to incorporate descriptively accurate behaviour analysis into the wave equation.

Automated Trader: Is the presumption that momentum and reversal traders' response to a jump in the equilibrium price is the cause of their over trading?

Their response to a jump by volume increase/decrease is one of the causes of their over trading. However, if one makes the presumption you mention, you still do not account for panic selling and 'autoshaping trading' (a special case of the built-in tendency to approach gain and withdraw loss). Momentum and reversal effects may disappear when traders respond to a jump in a certain environment. Thus, it is probably a better and more complete explanation that it is a contingency of return reinforcement/punishment (which includes a variety of internal and external causes) that results in their over trading.

Automated Trader: Your research seems to be primarily based upon the Chinese stock market, which has a relatively high proportion of active retail traders compared to many Western markets. Therefore would your research be as valid for a market with a higher proportion of institutional traders?

Yes. The hypotheses relating to traders as represented in our research are:

• Internal and biological behaviours can be represented to a large extent by external and intrinsically observable behavioural patterns

• Simple behaviour in learning is one of the elements of the trader

• Traders interact among themselves rather than being independent

All these hypotheses are valid in both the China and West, as well as for both retail and institutional traders.

Essentially, yes. Some traders are rational and others seem to be rational but are in fact not, because of biases, limits, and asymmetrical information et cetera. It is impossible and unnecessary for us to clarify who are rational or non-rational.

In our study, we have no restriction on market participants whether rational or non-rational. Our model abides by a supply-demand law and a trading rule - 'price first and time first', that is, the price volatility path is optimised by the least price volatility principle. It holds true for both a rational trading market, such as the AAA bond market, and a non-rational one, such as a stock market when a bubble is bursting.

Automated Trader: As part of your research, have you derived a closed form solution for the price/volume imbalance needed to cause a jump in the equilibrium price?

In the formal sense no, but we probably have implicitly. To do so formally, there are a number of possibilities. One is that the analytical volume distribution eigenfunctions 7 we already have might also be used to describe the supply-demand imbalance and jump as well.

Automated Trader: Your latest paper highlights a number of interactions between your previous research and behavioural finance. Where do you see its practical application?

I think the relationship between variables such as volume distribution over a price range and the causation between liquidity, price, and volume could have various applications. For example, it could be used to build a quantitative supply and demand law in economics.

In addition, in our most recent research, we found that stationary equilibrium exists extensively on a daily basis (four hours per day in the case of China's stock market). Assuming the price wave return during price equilibrium periods can cover costs, it might be possible to develop a feasible automated trading strategy based upon the price-volume probability wave equation.

## Footnotes

1. A wave that characterises a particle's travel where the square of the wave's magnitude at any given point corresponds to the probability of finding the particle at that point.

2. "Peakedness"

3. "An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum." ( http://mathworld.wolfram.com/Closed-FormSolution.html )

4. An unstable and transient, but relatively long-lived state

6. A discriminative stimulus influences the occurrence of an operant response, where an operant is behavior that is initially spontaneous, but that is subsequently influenced by its initial and ongoing consequences

7. The solution of a differential equation that satisfies specified conditions