Price action in an efficient market

At any time in any market, we can choose an offset to the price d (think of it as either a possible target price for a trade or a possible stop loss price for a trade) and consider the probability distribution of the maximum move in the opposite direction before the price d is reached. By a simple translation, make the current price 0 to simplify and clarify the notation.

Then  if the efficient market hypothesis (EMH) was true (don't worry, it isn't ;) ), the probability distribution would have to be such that any choice of target to go with the stop d or any choice of stop to go with the target d would have to be unprofitable. Neglecting spreads and transaction costs, the inequalities in both directions imply that any such trade would have to break even.

This implies that for every offset in the opposite direction f the odds of the maximum extension of the price graph exceeding this point before reaching price d must be d : f . To put it another way, the probability of exceeding -f must be d / (d+f) = 1 / (1 + d/f).

This requirement provides us with a precise cumulative frequency for the probability of the extreme. To find the implied probability distribution, we simply differentiate w.r.t. f. This gives us the predicted distribution of extremes of moves (the sign is because the cumulative distribution is reversed).

E(f) = d / (d + f)^2 

With a stop d = 1, here are the cumulative frequency and probability density graphs of extensions in the opposite direction implied by the EMH. All others have to be the same other than a multiple of d.

Note of course these graphs can be interpreted as the probability of hitting a stop before a chosen target as the bid-ask spread is ignored.

Price action topology

Relative price action forward or backward from a time forms a continuous function f, with f(0) = 0.
Key to trade management by stops and targets are the extremes of this price action. Let E be the set of positive reals e s.t. f(e) does not lie in f( [0, e) ). An extreme of a continuous function f: [0, Inf) -> (-Inf, Inf) is an extreme e s.t. all extremes in some future neighbourhood of e are of the opposite sign.

The image under f of E consists of semi-open intervals of the forms (p_i, p_i+1] and [n_i+1, n_i), where {p_I} are positive extremes and {n_i} are negative extremes and the latter interval is in reverse temporal order (taking time as the domain of f).