Exact Kelly comes to 18.018%. The approximation overestimates. thought your approximation always underestimates?
It's just an approximation. In real trading it works better because there are more than two outcomes and the vast majority of trade returns have magnitudes well under 1.
I'm slightly confused why we're using an example with discrete outcomes in a thread about continous Kelly. My understanding of CK is if you have an expected annualised Sharpe Ratio (mu/sigma) of say 1.0 then Kelly optimal will be to target annualised risk of 100%. SR of 0.5 implies annualised risk of 50%. It's trivial to prove this by simulation (plot final log utility versus annualised risk for a given Sharpe Ratio), as I did in the graph here. Personally I'd use half-Kelly, and then on a very pessimistic estimate of Sharpe Ratio, but the CK formula is still correct.
Set your stop outside the noise. Never risk more than 2% of TLNW on any one trade/idea. That max loss is in relation to the entry price only. Do not continue calculating the 2% as you accumulate unrealized gains. Forget formulas. Keep it simple. Thank you for your time.
The CK formula is not correct. It is in fact an approximation of an approximation. The approximation I developed in the "Kelly for traders" thread listed below is exactly equivalent to k ~ mu/( sigma^2 + mu^2 ) If we assume that sigma >> mu, we get Thorp's approximation: CK == mu/sigma^2 , Let's revisit the practical example from the first post: There is a 89% chance that you win 13% of your trade ; there is a 10% chance that you lose 5% of your trade ; there is a 1% chance that you lose 100% of your trade. The OTF equation for this system is 0 == (.89)(.13)/(1+.13k) + (.10)(-.05)/(1-.05k) + (.01)(-1)/(1-k). The Kelly fraction is the smallest positive solution to the OTF equation : k == 0.89834 , obtained by numerical solution. My proprietary sizing formula yields k0 == 0.9000 The first approximation formula yields k1 == sum1/sum2 == mu/( sigma^2 + mu^2 ) = 3.981654 Finally, Thorp's approximation yields CK == mu/(sigma^2) == 6.646641 Thorp's approximation is a shortcut to money-management hell.
I don't understand where you get the numbers from, or indeed your 'CK' formula. The mean, mu, of this example bet is 11.1%. The sigma is 12.3%. This gives an optimal annualised risk target of 90%. The only CK formula I know of, as I showed empirically was correct in my post, is mu/sigma. I've never seen anyone quote mu/sigma^2. I don't know where you found this straw man. I accept that mu/sigma is an approximation, and that the empirical result might be wrong for strategies with extreme skew. The contrived example has a skew of -7. Even selling at the money straddles unconditionally has a real world skew of maybe -2. For this mu/sigma is still empirically very close to optimal. I don't think there are many strategies where you would know your skew is -7 ex-ante. Most likely if you had a strategy which backtested with a payoff profile like your example you wouldn't run it. Its a matter of judgement wether you are doing something for which there is likely be negative skew out of sample in live trading, or you will end up like LTCM. It goes without saying that someone with extreme negative skew, or the likely danger of it, shouldn't use full Kelly even if they are using a correct version of the CK formula - in practice the likely jump risk means thats a dangerous thing to do. As a catious person I would run a zero skew strategy at half Kelly max, and a -2 skew strategy at quarter Kelly. I'd never run something that I expected to have -7 skew.
Your numbers are wrong. And mu/sigma is the Sharpe ratio, which, besides being a crappy performance metric, has buggerall to do with position sizing.
I guess we shouldn't bother continuing the argument since my apparently wrong approach gives exactly the same result as yours, and I think we can both agree that sizing to mu/sigma^2 - whatever name you want to give it - is far too aggressive. (Though its hard to see how I've managed to calculate the mu or the sigma incorrectly since these are quite trivial formula on which there shouldn't be any disagreement....)