Kelly Criterion: Mathematically Optimal Position Sizing for LMEX Traders

Most retail traders size positions wrong. Either they bet far too small (and underperform their actual strategy) or far too large (and blow up during normal drawdowns). The Kelly Criterion is the mathematical answer to "how much should I bet" — and the answer is almost always smaller than your gut wants.

This article walks through what Kelly actually is, how to compute it for a trading strategy, why everyone uses some fraction of Kelly rather than full Kelly, and how to implement Kelly-based sizing in a Python trading bot.

The Kelly formula

Kelly's original formulation, from John Kelly's 1956 paper, applied to betting with binary outcomes:

f* = (bp - q) / b

Where:

For continuous trading strategies, the formula generalises. The most useful form:

f* = (mean return) / (variance of returns)

This is the fraction of your account you should risk per trade to maximise long-term growth rate (geometric returns).

For a strategy with 60% win rate, average win 2%, average loss 1%:

That's full Kelly. And it's terrifying for most traders to actually run.

Why full Kelly is too aggressive

Full Kelly maximises geometric growth rate over the long run. It's mathematically optimal. But the cost is dramatic interim drawdowns.

A strategy run at full Kelly can drop 50% from peak and still be on track for its long-term optimal growth. Most traders cannot psychologically tolerate 50% drawdowns. They quit, switch strategies, or reduce sizing at the worst possible moment.

The other problem: Kelly assumes you know your strategy's true edge precisely. In reality, your win rate and average returns are estimates from limited historical data. If your estimates are off by even a small amount, full Kelly drastically over-leverages.

A strategy estimated at 60% win rate that's actually 55% in live trading produces dramatically more risk than intended. Full Kelly amplifies estimation error into actual losses.

Fractional Kelly

The standard solution: use a fraction of full Kelly. Common choices:

Most professional traders use somewhere between quarter and half Kelly. Full Kelly is essentially never used in practice except by academic researchers running simulations.

The math: if full Kelly gives expected growth rate G, then half Kelly gives 0.75 × G with much lower variance. Quarter Kelly gives 0.5 × G with much lower variance still. The tradeoff between growth and risk is non-linear, but the math favours stepping back from full Kelly significantly.

A Python implementation

Calculating Kelly from a trade history:

import numpy as np
import pandas as pd

def calculate_kelly(trades_df, fraction=0.5):
    """
    Calculate Kelly fraction from historical trades.
    
    trades_df should have a 'return' column with per-trade returns as decimals
    (e.g., 0.02 for 2% gain, -0.01 for 1% loss).
    """
    returns = trades_df['return']
    mean_return = returns.mean()
    variance = returns.var()
    
    if variance == 0:
        return 0
    
    full_kelly = mean_return / variance
    return full_kelly * fraction

def position_size_kelly(account_balance, kelly_fraction, stop_distance_pct):
    """
    Calculate position size in dollars based on Kelly and stop distance.
    
    stop_distance_pct is the maximum loss as a fraction (e.g., 0.02 for 2% stop).
    """
    risk_per_trade = account_balance * kelly_fraction * stop_distance_pct
    return risk_per_trade / stop_distance_pct

# Example usage
trades = pd.DataFrame({'return': [...]})  # your historical returns
kelly = calculate_kelly(trades, fraction=0.5)  # half-Kelly
print(f"Half-Kelly position size: {kelly*100:.1f}% of account per trade")

The implementation is simple. The hard part is having a good trade history to compute from. With fewer than 100 trades of data, Kelly estimates are unreliable. With 500+ trades across multiple market regimes, the estimates are more meaningful.

Re-computing as the strategy runs

Kelly isn't a one-time calculation. As your strategy runs and more trades accumulate, the optimal Kelly fraction shifts. Standard practice:

1. Calculate Kelly from initial backtest

2. Use half or quarter Kelly for live trading

3. Every 50 live trades, recalculate Kelly using only live data

4. Adjust position sizing based on the latest estimate

If live Kelly is significantly lower than backtest Kelly, the strategy's edge has likely degraded. Time to reassess.

If live Kelly is higher than backtest Kelly, either you got lucky early or the strategy is unexpectedly strong. Wait for more data before increasing sizes.

Why most people get this wrong

Two common Kelly mistakes:

**Confusing average return with edge.** A strategy with high average returns and high volatility might have a worse Kelly fraction than one with lower average returns and lower volatility. Kelly cares about return / variance, not return alone.

**Using stops as the only sizing input.** "I'll risk 2% per trade" is a position sizing rule that ignores edge. If your strategy has a 51% edge, risking 2% per trade is far too aggressive. If your strategy has a 65% edge, risking 2% leaves money on the table. Calibrate to actual strategy performance.

Frequently Asked Questions

Q: What if my strategy has variable trade sizes or holding periods?

Kelly still applies, but you need to be careful about defining your returns. Per-trade returns work if trades are roughly comparable in size and duration. For more variable strategies, use per-time-period returns (daily, weekly) instead.

Q: How does Kelly interact with diversification?

A portfolio of uncorrelated strategies each run at quarter Kelly produces a smoother equity curve than any single strategy at quarter Kelly. The Kelly fraction for each strategy in a portfolio is more complex math (involves the covariance matrix), but the rule of thumb is: each strategy gets a smaller individual allocation when running multiple in parallel.

Q: Does Kelly work for short trades and longer trades equally?

It works for both, but the formulation differs slightly. For directional trades with clear entry/exit points, the standard formula applies. For long-term holds (weeks to months), use the continuous-time formulation: f* = (μ − r) / σ², where μ is expected return, r is risk-free rate, and σ² is variance.

Q: How do I size when I don't have enough trade history?

Start with conservative defaults (1-2% risk per trade) until you have at least 50 trades. Then begin computing Kelly from your live data. Don't trust backtest Kelly numbers — they're almost always too high due to overfitting and survivorship effects.

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