Here is the underlying thought experiment. Price starts at F. Trader starts flat, waits some time, not specified, and then if the price is greater than F, i.e. F+delta-F, for unspecified but positive delta-F, buys some amount at F+delta-F. The amount to buy is some constant g times delta-F.
If price subsequently corrects back to F, trader sells at F and returns to original flat position. Likewise, if price falls below F to F + delta-F for negative delta-F, trader sells at some F+delta-F, amount = g times delta-F, then if price corrects back to F, buys to go flat. The value of delta-F is not specified, and we have no knowledge of the underlying price process, except that an angel has a list of all the prices at which the trades took place, and works out the sum of the squared delta-Fs to give to us.
Assume the final position at the end of some time delta-t is flat.
It can be proved fairly easily (assuming I have set this up correctly) that the trader will make always make a loss whenever the price corrects back to F. Moreover if we have the value of the constant g, and the time taken for the series of trades, we can work out the average loss rate of the trader.
The question is then whether it makes sense delta-t to approach zero, but for the sum of squares to remain positive, so we can speak of a continuous time decay of the trader. Or does the thought experiment make sense only for a finite amount of time?
Alan will say it’s obvious, of course.