Kelly Criterion

Kelly Criterion calculates the optimal betting fraction given the expected outcome and probability for both negative and positive scenarios. Optimal means the highest expected geometric return. The proof follows below.

This is intuitive as an exponential function with an exponent less than 1 is concave. Thus the graph's marginal growth is decreasing. Also, note that geometric mean is approximately arithmetic mean minus one-half of the variance which can be derived by 2nd approximation of the Maclaurin series.

 

Note that when Sharpe ratio is calculated with arithmetic mean, any betting size will lead to the same Sharpe ratio. This is one reason why it makes more sense to target a specific volatility than a specific return in calculating the betting size (or the leverage ratio).