See here: http://www.bjmath.com/bjmath/thorp/paper.htm, equation 7.4. See here: http://quantivity.wordpress.com/2011/02/21/why-log-returns/ Some of my strategies have Kelly-optimal leverage of 180.
There's a huge difference between log(r) and log(1+r). And the Kelly equation is the result of first taking the logarithm of the gain before taking the differential. It looks like you're conflating a derivation with a final result. And do you trust this number? And how is this leverage applied? To the bankroll or to the trade size?
Both can be used, depending on how you calculate the return: LN(1 + (price2 - price1) / price1) = LN(price2 / price1) = LN(price2) - LN(price1)
Not at all. Here's what you posted. Nowhere in the link you provided is ln(r) or log(r) mentioned. What is talked about is log(1+r), which is vastly different from log(r), and which I already said is one step in the derivation of the Kelly equation. The derivation of the Kelly equation is straightforward. There's nothing ambiguous about it. I am now convinced that the CK formula is just an approximation, and not a very good one. To be sure, all of the Kelly formulae are approximations for all but the simplest scenarios (binary outcomes, like coin flips). But some approximations are better than others, and I see nothing to recommend the CK formula, especially if it is giving results like a leverage of 180. :eek:
A Tale of Two Horses II At long last, so have I. I tried different ways but got nowhere with my unit-bet analysis. So I went back to the relationship established by Splawndarts and, with the aid of an on-line equation solver, I got the following results: F == 5.72% and S == 8.43% Surprisingly, at least to me, both F and S are larger than their respective single-bet Kelly fractions. Specifically, F is 4.4% larger than k14 and S is 2% larger than k16. This implies some sort of synergy is created by combining mutually exclusive bets. Any thoughts?