Kelly Criterion Position Sizing in Volatile Crypto Markets

Abstract

Position sizing is often the most neglected aspect of trading system design, yet it determines 70-80% of long-term returns. This paper presents our implementation of fractional Kelly Criterion with dynamic volatility and confidence scaling, demonstrating superior risk-adjusted performance in crypto market backtests.

The Kelly Criterion

The Kelly Criterion, developed by John L. Kelly Jr. at Bell Labs in 1956, provides a mathematically optimal bet sizing formula:

$f^* = \frac{pb - q}{b}$

Where:

• $f^*$ = optimal fraction of capital to bet
• $p$ = probability of winning
• $q$ = probability of losing $(1 - p)$
• $b$ = ratio of net profit on a win to net loss on a loss

For symmetric payoffs (win/loss ratio of 1):

$f^* = p - q = 2p - 1$

Example Calculation

Given:

• Win rate: 55% ($p = 0.55$, $q = 0.45$)
• Win/loss ratio: 1:1 ($b = 1$)

Full Kelly:

$f^* = \frac{0.55 \times 1 - 0.45}{1} = 0.10 = 10\%$

Why Full Kelly Is Dangerous

While mathematically optimal for maximizing long-term growth, full Kelly has severe practical problems:

1. Volatility of Returns

Full Kelly produces extremely volatile equity curves. A 10-trade losing streak (which happens approximately once every 2,000 trades at 55% win rate) can draw down 65% of capital.

The probability of $n$ consecutive losses is:

$P(n) = q^n = (1 - p)^n$

For $p = 0.55$ and $n = 10$:

$P(10) = 0.45^{10} \approx 0.00034 = 0.034\%$

2. Parameter Uncertainty

The formula assumes perfect knowledge of win rate and payoff ratio. In reality, these are estimated from historical data and change over time.

3. Correlation Effects

Kelly assumes independent bets. In trading, positions are often correlated, effectively increasing risk.

4. Psychological Tolerance

Few traders can stomach the drawdowns that full Kelly produces without abandoning their system.

Our Implementation: Fractional Kelly

We use fractional Kelly (0.25-0.5) with dynamic adjustments:

python
def calculate_position_size(
    account_value: float,
    win_rate: float,
    signal_confidence: float,
    current_volatility: float,
    baseline_volatility: float,
    current_drawdown: float,
    correlation_factor: float
) -> float:
    """
    Calculate position size using fractional Kelly with adjustments.
    """
    # Base Kelly calculation
    kelly = (2 * win_rate - 1)
    
    # Apply fractional Kelly (0.25 by default)
    base_size = account_value * kelly * 0.25
    
    # Volatility scaling: reduce in high vol, increase in low vol
    vol_multiplier = baseline_volatility / current_volatility
    vol_multiplier = min(max(vol_multiplier, 0.5), 1.5)  # Cap at 0.5x-1.5x
    
    # Confidence scaling: linear with signal confidence
    confidence_multiplier = signal_confidence
    
    # Drawdown throttling
    if current_drawdown > 0.15:
        drawdown_multiplier = 0.5
    elif current_drawdown > 0.10:
        drawdown_multiplier = 0.75
    else:
        drawdown_multiplier = 1.0
    
    # Correlation adjustment
    correlation_multiplier = 1.0 - (correlation_factor * 0.3)
    
    # Final position size
    position_size = (
        base_size *
        vol_multiplier *
        confidence_multiplier *
        drawdown_multiplier *
        correlation_multiplier
    )
    
    # Apply absolute limits
    max_position = account_value * 0.10  # Never exceed 10%
    min_position = account_value * 0.01  # Minimum meaningful position
    
    return max(min(position_size, max_position), min_position)

Adjustments Explained

Volatility Scaling

In high volatility periods, we reduce position size proportionally:

Confidence Scaling

Signal confidence (from our ML models) directly scales position:

• 90%+ confidence: Full calculated position
• 70-90%: 70-90% of calculated position
• 50-70%: 50-70% of calculated position
• < 50%: No trade

Drawdown Throttling

When underwater, we systematically reduce exposure:

• 0-10% drawdown: No adjustment
• 10-15% drawdown: 25% reduction
• 15-20% drawdown: 50% reduction

• 20%: Flatten all positions

Correlation Adjustment

High correlation between positions indicates concentrated risk:

• Correlation < 0.3: No adjustment
• Correlation 0.3-0.5: 5-15% reduction
• Correlation 0.5-0.7: 15-21% reduction
• Correlation > 0.7: 21-30% reduction

Backtesting Results

We tested three approaches over 3 years of crypto market data:

Key observations:

• Full Kelly maximizes returns but with unacceptable drawdowns
• Half Kelly is a common compromise
• Our approach sacrifices some return for dramatically better risk-adjusted performance

Real-World Considerations

Transaction Costs

Position sizing must account for costs:

python
# If expected edge is small, reduce position size
expected_return = win_rate * avg_win - (1 - win_rate) * avg_loss
cost_adjusted_return = expected_return - total_transaction_cost

if cost_adjusted_return < min_expected_return:
    position_size = 0  # Skip trade

Liquidity Constraints

Large positions require scaling:

python
max_daily_volume_participation = 0.01  # 1% of daily volume
liquidity_constrained_size = daily_volume * max_daily_volume_participation
position_size = min(position_size, liquidity_constrained_size)

Margin Requirements

For leveraged positions:

python
max_leverage = 3.0  # Platform limit
available_margin = account_value / max_leverage
position_size = min(position_size, available_margin)

Conclusion

Position sizing is not glamorous, but it's essential. Our fractional Kelly implementation with dynamic adjustments provides:

1. Mathematical foundation: Based on proven optimal growth theory
2. Practical adjustments: Accounts for real-world constraints
3. Risk management: Built-in drawdown throttling and correlation awareness
4. Adaptability: Responds to changing market conditions

The result is a position sizing system that maximizes risk-adjusted returns while maintaining the drawdown profile necessary for long-term survival.

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For more on our risk management approach, see our blog post on risk-first trading.