Kelly Criterion

The Kelly Criterion is a formula for sizing bets or investments to maximize long‑term wealth growth. Developed by John L. Kelly Jr. in 1956, it was originally applied to gambling and later used by some investors to guide position sizing.

Core idea

The criterion computes the optimal fraction of your capital to allocate to a single trade or bet based on:
* W — the probability the trade will be profitable (win probability)
* R — the win/loss ratio (average win size divided by average loss size)

The result is a percentage of total capital to place on that single opportunity.

Formula

K% = W − [(1 − W) / R]

where:
* K% = fraction of capital to risk on the trade
* W = historical win probability of the trading system (a decimal)
* R = historical average win / average loss

If K% is negative, the formula indicates you should not take the bet.

How to estimate inputs

• Win probability (W): commonly estimated from recent historical trades (for example, counting favorable outcomes in the last 50–60 trades).
• Win/loss ratio (R): compute the average size of winning trades divided by the average size of losing trades over the same sample.

Enter these estimates into the formula to get the suggested position size.

Practical considerations and limitations

• Diversification: The Kelly Criterion determines optimal sizing for a single bet; it does not incorporate portfolio diversification. Relying solely on Kelly sizing for concentrated bets can create excessive risk.
• Estimation error: Inputs (W and R) are estimates and often noisy. Small errors can meaningfully change the suggested fraction.
• Personal constraints: Liquidity needs, risk tolerance, leverage limits, and other personal or regulatory constraints can make the pure Kelly allocation impractical. Some investors prefer conservative adjustments to the Kelly recommendation.
• Alternative frameworks: Expected utility theory and other decision frameworks may lead to different choices when personal utilities and constraints are considered.

Relation to option pricing

Both the Kelly Criterion and the Black‑Scholes model are mathematical tools that rely on probabilistic inputs. Black‑Scholes estimates theoretical option prices given model assumptions; Kelly focuses on optimal bet sizing given probabilities and payoff ratios. They address different questions—pricing versus capital allocation—but both depend on estimates of underlying uncertainties.

Key takeaways

• The Kelly Criterion provides a principled way to convert estimates of win probability and payoff ratio into a suggested fraction of capital to risk.
• It can help grow wealth over the long run when inputs are accurate, but it should be applied carefully because of estimation risk, lack of diversification consideration, and personal constraints.
• Use the Kelly result as one input among others (risk limits, diversification, utility preferences) rather than as a standalone rule.