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SP90

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## Homework Statement

A simple dynamical model for the price P (in £) of shares in a single stock or commodity traded in a stock market describes the behaviour of all the traders in the market with the same simple rule. All traders buy or sell shares at each "tick" (or time step [itex]\Delta t[/itex]. To decide his/her trade, each trader makes a rough guess by thinking of a random number of shares [itex]\eta[/itex] (dimensionless number) that he/she will buy or sell, then phones [itex]f[/itex] trader friends to ask what their trades will be. The trader then takes an average of his/her rough guess with the trades of his/her friends to determine his/her trade. The average volume, that is, value of the stock traded in each time step is [itex]V[/itex] (in £ per tick [itex]\Delta t[/itex])

1. Describe Buckingham [itex]\Pi[/itex] Theorem, and use it to find the relevant parameters for the model.

2. What happens for very small and very large values of your parameters?

3. What is the role played by [itex]\frac{\eta}{f}[/itex]? What happens as [itex]\frac{\eta}{f} \rightarrow 0[/itex] and [itex]\frac{\eta}{f}[/itex] large?

## The Attempt at a Solution

1. (I'm fairly sure on this definition, not so on the rest of Q1 and Q2)

So Buckingham [itex]\Pi[/itex] Theorem states that if a system is described by [itex]F(Q_{1},...Q_{n})[/itex] where [itex]Q_{i}(D_{1},...,D_{m})[/itex] are relevant macroscopic variables which are functions of dimensions [itex]D_{i}[/itex] (e.g. time, mass etc), then [itex]F[/itex] is a function of dimensionless functions [itex]\pi_{j}(Q_{1},...Q_{n})[/itex].

There are [itex]j[/itex] such [itex]\pi_{i}[/itex] where [itex]j=n-m[/itex].

And [itex]F(\pi_{1},...\pi_{n})=C[/itex] is the general solution.

However, I'm not sure which macroscopic variables describe the system.

**Not sure of the following:**The dimensions are price (£) and time (T).

The macroscopic variables are the following (where [] indicates their dimension):

[itex]\eta_{a}=[1][/itex] - the average amount each trader will sell

[itex]P=[£][/itex] - the price of each share

[itex]\Delta t=[T][/itex] - the time of each tick

[itex]V=\frac{[£]}{[T]}[/itex] - the average volume

Then there are four variables and two dimensions. So that means there are two [itex] \pi_{ i} [/itex]

Obviously [itex] \pi_{1}= \eta_{a} [/itex].

The second is [itex] \pi_{2} = \frac{V \Delta t}{£} [/itex]

Then we get [itex] \eta_{a} = C (\frac{V \Delta t}{£})^{n} [/itex], that the average number of trades is some constant multiplied by [itex] (\frac{V \Delta t}{£})^{n}[/itex] to some index.

2. So effectively if the stock price is large, the amount of trades made is low and vice-versa. Also, more trades are likely to be made in a longer tick.

3. (Just to check I've understood this bit, I'm fairly sure it's right)

I know that [itex]\frac{\eta}{f}[/itex] is the disorderly factor. It represents the individual following their own path rather than the group When [itex]\frac{\eta}{f} \rightarrow 0[/itex] the system is very orderly - they tend to be enslaved to the crowd and do what everyone else does. When [itex]\frac{\eta}{f}[/itex] is large, however, there is chaotic behaviour, where everyone acts randomly and unpredictably. The individual term dominates that of the group and their behaviour becomes the driving factor.

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