The stock-market is usually modelled as a Brownian motion or a random walk as shown by Mandlebrot amongst others. Now Brownian motion is an odd type of motion because the speed at which particles travel depends on the time between you making measurements of where they are.
On a molecular level Brownian motion is caused by lots of collisions of much smaller objects on a larger object. As time increases these average out and so changes of direction cancel as the object is pummelled from all directions. So as I measure the distance travelled over longer times it increases proportional to the square root of the time since I started measurement. If I measure it very frequently then I see rapid changes over short distances but over longer times the rate of change of position is smaller and so the speed decreases.
This square root relationship applies to all the random walks where changes of direction are possible - so share prices go up and down depending on the buffeting of individual transactions. If I look at the movements over a day they will be slower because of averaging than movements over an hour. This is this square root relationship. The problem with high frequency trading is that the time-scales are shrunk to seconds and so the speed of change is much higher than these longer sampling averaged times because of the NOISE at this level. That is the point high frequency trading depends on the noise that gives rapid price changes. There is no sense of investment or any rational relationship to the economy this is pure speculation and their is no investment involved.
Mathematically and politically this is a foolish and reckless process that makes no sense. On average you will gain nothing. Some days you will be well ahead and others well behind because that is how a random process works. If you are consistently ahead then you must have information that tips the random process in your favour like card counting. This can only be done if you have insider information that others do not and is suggestive of insider trading, or at least an uneven playing field. There is no justifiable reason for the high frequency trading that has become a major part of the markets.
The EU wanted to introduce a transaction tax that would have stopped this practice but the UK was opposed as the equity markets are supposedly important for the UK economy (more of the inequity of equities another time). Such a tax would kill these transactions as they would no longer be profitable and end one of the hedge funds little tricks. This would put another nail in their coffin if short selling is also banned. It is time there was some sense in the markets and a transaction tax would be a good step forward.
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