Here is a brand new (to the public) Kelly formula. It was designed for traders, not for gamblers. Trading is not a casino game. It doesn't have fixed probabilities and fixed payouts. Kelly formulae designed for casino games do not travel well when imported into trading. All that said, this new formula is presented "as is". There will be no history, no derivation, no explanation, no hints now or later. Enjoy. Nk = ( s1*s2*s2 - s1*s1*s3 )/( s2*s2*s2 + s1*s1*s4 - 2*s1*s2*s3 ) , where Nk is the estimation of the true Kelly fraction produced by the new formula ; S1 == sum[ Ri ]_i=1toN ; S2 == sum[ Ri*Ri ]_i=1toN ; S3 == sum[ Ri*Ri*Ri ]_i=1toN ; S4 == sum[ Ri*Ri*Ri*Ri ]_i=1toN.
If your returns are gaussian, this should end up as annualised mean divided by annualised standard deviation (Sharpe Ratio), which is the formula I use (having first checked my returns are gaussian or close). It punishes negative skew - good - though I'd personally do it more aggressively (use half the Nk that comes out, rather than about 20% lower for a skew of -2.0 as the formula does here). This isn't supported by any formula or theory; its just my blind prejudice against heavily negative skewed strategies where you don't really know if the skew if -2 or -5 unless you have enough data history, and where theoretical results don't help you when you get a big gap downwards or close limit down two weeks in a row. But I'm not sure I'd punish kurtosis as this formula does (positive skewed strategies, with mild kurtosis, end up with a lower ratio than gaussian normal), but I guess its better to be conservative.
Remember "no history, no derivation, no explanation, no hints now or later." The first rule of Kelly Criterion is.... (You might try googling Taylor series. Just don't tell anyone I told you.)
The problem with Kelly is that it handles more than one instrument poorly. It also doesn't discuss either strategy correlation or asset correlation (correlation itself a very fleeting measure of co-movement). It is this misunderstanding that lead to LTCM implosion since they were using something close to full Kelly. The real formula being used by modern portfolio managers is probably far more sophisticated.
Warning on believing any formula as applied to markets: http://archive.wired.com/techbiz/it/magazine/17-03/wp_quant
LTCM was trading nowhere near their real Kelly ratio. They were crazy-overleveraged, with a debt-to-equity ratio of like 25 to 1. If they had been using true Kelly, even ignoring correlations they probably would have been salvageable. Consider the following scenario: There is a 90% chance that you win 25% of your bet ; There is a 9% chance that you lose 5% of your bet ; There is a 1% chance that you lose 100% of your bet. This simulates a highly favorable trading situation that is dangerously close to the precipice that William Eckhardt warned about. It doesn't take much overestimation to land you right in the minefield or blow-up zone. But the formula I posted handles it. The true Kelly fraction is 0.9436081. The formula returns 0.96329443. I can live with that.
I can't believe you're comparing a down-to-earth, practical formula like Kelly to one of the quants' wackadoo pricing models. The instant I saw "Gaussian", I knew I was looking at pie-in-the-sky. Kelly makes no assumptions about probability distribution.