Kelly criterion: reconciliate discrete and continuous case

Question

The Kelly criterion for the discrete case is: $$\tag{1}f=\frac{pW-(1-p)L}{LW}$$ where $f$ is the fraction of wealth to invest, $p$ the probability of winning, $(1-p)$ the probability of losing, $W$ the return if win, $-L$ the return if loss.

The continuous case is: $$\tag{2}f=\frac{E[r]}{V(r)}$$ where $E[r]$ is the expected return, and $V(r)$ is the variance of returns.

I am trying to derive the discrete case from the continuous case. So, the expected return $E[r]$ when the distribution of returns is discrete (it takes two values $W$ and $-L$) is actually equal to $pW-(1-p)L$, which is the numerator in the discrete case. So far so good. However, $V(r)$ should be equal to $pq(W-L)^2$, which is quite different from $LW$ in formula 1. Why is it so?

Answer 1

So imagine you specified your discrete outcomes by virtue of a classic "stop-loss" and "target" price level. You buy/sell and hold, until either critical level is met, creating your discrete set of outcomes.

Assuming: p is the probability of hitting the upside barrier before the downside equivalent U is the upside payoff D is the downside payoff L is your leverage, ie your Kelly bet.

Your expected log wealth (ELW) becomes ELW = p * ln(1+LU) - (1-p) * ln(1+LD) I want to maximise DPW with respect to L.

The derivative of ELW wrt L = pU/(1+LU) - (1-p)D/(1+LD) Equals zero at a maximum, so the two parts above must equate

Do the algebra, and you will end up with: L = (p*U - (1-p)*D) / (U * D)

Which is "edge over odds" in the traditional formulation. Why U*D represents "odds" is maybe not intuitive. Just divide everything above by D so U is an x:1 odds bet. Which gives you the classic Kelly discrete formula.

hope this explains to your intuition and satisfaction, DEM