Kelly Criterion

The Kelly Criterion is a formula developed by mathematician John L. Kelly Jr. in 1956 at Bell Labs, originally for telephone signal noise problems and later applied to gambling and investing by figures including Ed Thorp. It calculates the optimal fraction of a bankroll to stake on each bet to maximise long-run growth rate.

The formula: f* = (b × p − q) / b

Where: f* = fraction of bankroll to stake; b = net odds received on the bet (profit if you win, as a fraction of stake); p = probability of winning; q = probability of losing (1 − p).

Kelly's insight was that betting more than the optimal fraction reduces long-run growth rate — sometimes dramatically. Betting the full Kelly fraction maximises geometric growth but produces large drawdowns. Most practitioners use "half Kelly" (0.5 × f*) to reduce volatility.

When applied to binary options, the Kelly Criterion produces a critical, counterintuitive result. At a typical 80% payout rate, b = 0.80 (you win $80 per $100 staked). Let's calculate f* for various win rates:

At p = 55% win rate: f* = (0.80 × 0.55 − 0.45) / 0.80 = (0.44 − 0.45) / 0.80 = −0.0125 → Negative! This means Kelly says: do not bet. A 55% win rate at 80% payout is below the mathematical edge threshold.

At p = 58% win rate: f* = (0.80 × 0.58 − 0.42) / 0.80 = (0.464 − 0.42) / 0.80 = 0.044 / 0.80 = 5.5% of bankroll per trade.

At p = 65% win rate: f* = (0.80 × 0.65 − 0.35) / 0.80 = (0.52 − 0.35) / 0.80 = 0.17 / 0.80 = 21% of bankroll per trade.

The practical lesson: at realistic win rates of 55–60%, Kelly recommends risking 0–7% of bankroll per trade, not the 10–20% many retail binary traders use. Using half-Kelly is even more conservative: 0–3.5% per trade. This is why professional position sizing is far smaller than most retail traders instinctively select.