Definition
The Kelly criterion (or Kelly formula) calculates the bet size that maximises the expected logarithm of wealth — i.e., the size that produces the highest long-run compounded growth rate. It was developed by John Kelly Jr. at Bell Labs in 1956 and has been widely adopted in both gambling and algorithmic trading.
The formula
f* = (bp − q) / b
Where: f* = fraction of bankroll to bet, b = net odds (profit per $1 wagered), p = probability of winning, q = probability of losing (1 − p).
Kelly in arbitrage
In pure arbitrage, the win probability is theoretically 100% (both legs fill, contracts settle correctly). The formula simplifies: the position size is constrained primarily by available liquidity, not by probability. In practice, Arbitrage Agent caps position size at the available order book depth at the quoted price to avoid price impact.
Fractional Kelly
Many practitioners use half-Kelly or quarter-Kelly to reduce variance. Full Kelly maximises long-run growth but produces large swings in portfolio value. Using a fraction (e.g., 0.5×Kelly) reduces volatility at the cost of slightly lower long-run returns. This is the standard approach in production arbitrage systems.
Why it matters for arbitrage
Without a position-sizing rule, there's a temptation to bet maximum on every opportunity. This concentrates risk: a string of circuit breaker triggers or unusual resolution events can cause large drawdowns. Kelly-based sizing distributes capital across opportunities and limits the damage of any single trade going wrong.