PS wrote:I am trying to consider the realistic model in which we have not only probabilities of Win/Lose and corresponding amounts W/L, but also fixed time horizon , our target (profit we want) on this horizon, and our bankroll (money we are willing to risk on this time horizon).
It's clear you're using mechanics of Kelly, which don't express what you're after. The mechanics of Optimal-f speak about initial wealth (W), Terminal Wealth Relative (TWR) and a Geometric Mean (G). Using trade frequency, you can derive number of trades expected in a fixed horizon (N), and then the expected TWR by taking W * G**N. I would use something like Monte Carlo simulation to derive the chance of actually hitting TWR in the future.
PS wrote:Please note, that in my case maximization of the expected value is acomplished on the set of optimal fixed-fraction betting strategies, where optimality was defined as a solution of the two-criteria optimization problem. In your case the optimization is accomplished on the set of all possible strategies.
Maybe it is useful to start enumerating the bet size strategies you are interested in. Fixed fraction is simply (Equity*Fraction)/(Equity at Risk), and there are numerous interpretaions and methods which derive a fraction.
PS wrote:It seems that maximization of geometric growth is not a practical criterion. Optimal-f does not take into account [...] our preferences (..., risk-reward ratio, etc.)
I agree (given the edits).
PS wrote:Optimal-f is greater (or equal ??) than Kelly's value.
In my experience, Optimal-f is always less than Kelly value.
MCT wrote:What you are attempting to do is fix in place whatâ€™s by nature a moving target. Remember, unlike a game of black jack the rules that generate these statistical properties are continually changing with time. And time, my friend, makes it difficult to determine whatâ€™s really optimal. You can't determine what your profit will be, but you can determine what your loss will be.
Agreed. That is where Ralph delves into how the probabilities of blackjack are nothing like trading, and why Kelly misses the point. If I remember correctly, when you do the mathematical redux, Optimal-f becomes Kelly for binomial distributions.
The other difficulty MCT mentions is that optimal changes with the series of trades, so that Optimal-f varies. This makes it highly likely that what you thought was optimal is actual trading too hot (or too cold ... but too hot's the one that bites back).