In a financial derivative trading situation, there are two outcomes to a bet (win/lose), but I don't necessarily lose my entire stake if I lose the bet, because I can buy my way out of the bet, taking a net loss. Nonetheless, I have to bet the entire stake in order to play. However, if I win the bet, I receive my stake plus a fraction as a gain.
For example, let's say:
- my bankroll is \$100
- if I win, I will gain 5% of my stake
- if I lose, I will lose 30% of my stake
- I expect to win 91% of the time
What I want to determine is how much of my bankroll to put at stake.
According the Wikipedia article, I should be using this formula:
$$ f*=(\frac pa)-(\frac qb) $$
where:
- p = probability of winning
- q = probability of losing ($\equiv 1-p$)
- a = amount of loss
- b = amount of gain
- f* = percentage of bankroll to bet
However, when I do the math:
$$ f*=(\frac{.91}{.3})-(\frac{.09}{.05})={3.0\overline3}-{1.8}={1.2\overline3} $$
That's obviously very wrong. I'm expecting a number where $0 \leq f* \leq 1$.
A colleague provided a function that he claims is a Kelly formula, and it gives values that I would expect, but I don't understand how he got the formula he uses.
$$ k*=p-\frac q{(\frac ba)} $$
The math works closer to what I would expect:
$$ k*={0.91}-\frac {0.09}{(\frac {.05}{.3})}={0.91}-\frac {0.09}{.1\overline6}={0.91}-{.54}={.37} $$
So I would put 37% of my bankroll at stake, which seems about right.
What I don't understand is how my colleague's math works to the Kelly Criterion, or how I misapplied the formula from Wikipedia.