S is a price process which follows Geometric Brownian motion with no drift: dS=S*vol*dW, vol=const., W is a Wiener process.

Define the following ratio: R=E[Max(f(S)-S(T),0)]/E[f(S)], where S(T) is the stock price at maturity time T, and f={S(t) for some 0<=t<=T, picked according to some unknown rule}. In other words, the functional 'f' takes the entire stock price path from 0 to T as input, and its output is some stock price S(t) along the path (0<=t<=T), according to some unknown rule. The questions are:

1) What is the functional 'f' that maximizes the ratio R above? For example, if I choose 'f' so that it picks the maximum stock price along the path, f=Max(S(t), 0<=t<=T), then the numerator is clearly maximized, but so is the denominator. Conjecture: the maximum functional f=Max(S(t), 0<=t<=T) maximizes the ratio R - true or false?

2) Even if we can't find 'f' explicitly, can we find an upper bound for R which is better than the trivial case 1?