# Kelly criterion: reconciliate discrete and continuous case

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?

## 1 Answer

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

• Thanks, but why the formula for the continuous case is different? Feb 23 at 7:02