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Kelly Criterion

The bet size that maximises long-run logarithmic growth.

Kelly gives the fraction of capital to wager that maximises geometric growth without risking ruin. Full Kelly is volatile, so practitioners use 'half-Kelly'. For continuous returns it reduces to mean over variance.

f* = (b·p − q) / b        (discrete bets)
f* = μ / σ²              (continuous returns)

# discrete: b = win/loss payoff ratio, p = win prob
kelly = (b*p - (1 - p)) / b
# continuous returns
f_star = returns.mean() / returns.var()