Half-Life for Position Sizing
Mean reversion strategies profit from price deviations returning to a long-term average. The half-life of mean reversion quantifies this tendency. It measures the time for a price deviation to dissipate by 50%. A shorter half-life indicates stronger mean reversion. A longer half-life suggests weaker mean reversion or a trend.
Calculate half-life using the Ornstein-Uhlenbeck (OU) process. The OU process models mean-reverting asset prices. Its formula is $dr_t = \lambda(\mu - r_t)dt + \sigma dW_t$. Here, $r_t$ is the price, $\mu$ is the mean, $\lambda$ is the speed of reversion, $\sigma$ is the volatility, and $dW_t$ is a Wiener process. The half-life $T_{1/2}$ relates to $\lambda$ by $T_{1/2} = \frac{\ln(2)}{\lambda}$. Estimate $\lambda$ from historical price data.
Use half-life to size positions. A shorter half-life implies a quicker return to the mean. This allows for larger position sizes. The market corrects deviations faster. A longer half-life suggests a slower return. This necessitates smaller position sizes. The market takes longer to correct.
Consider a pairs trading strategy involving two co-integrated stocks, XOM and CVX. We form a spread, $S_t = \text{ln(XOM)}_t - \beta \text{ln(CVX)}_t$. We estimate $\beta$ using ordino least squares regression over a 252-day lookback window. We then model the spread's mean reversion.
Suppose our analysis of the XOM/CVX spread from January 1, 2023, to December 31, 2023, yields a $\lambda$ of 0.15 per day. The half-life is $T_{1/2} = \frac{\ln(2)}{0.15} \approx 4.62$ days. This indicates a relatively fast mean reversion. A deviation from the mean should halve within approximately 4.62 trading days.
Now, consider another pair, MSFT and GOOGL, over the same period. Suppose this pair yields a $\lambda$ of 0.03 per day. The half-life is $T_{1/2} = \frac{\ln(2)}{0.03} \approx 23.1$ days. This pair mean-reverts much slower.
For the XOM/CVX pair with a 4.62-day half-life, a trader might allocate 2% of their capital per trade. For the MSFT/GOOGL pair with a 23.1-day half-life, the same trader might reduce allocation to 0.5% of capital. The faster mean-reverting asset justifies a larger allocation. The slower mean-reverting asset requires a smaller allocation to manage risk over a longer reversion period.
Half-Life for Stop-Loss Placement
Half-life also informs stop-loss placement. A stop-loss should trigger when the mean-reverting assumption breaks down. A price deviation persisting beyond its expected half-life suggests this breakdown.
A common approach sets stop-losses based on standard deviations from the mean. For a mean-reverting process, deviations are expected to return. However, extreme deviations or prolonged deviations signal a regime change. Half-life provides a time-based threshold.
If a trade opens when the spread is 2 standard deviations from its mean, we expect it to revert. If the spread continues to move against the position for a duration significantly longer than its half-life, the mean reversion hypothesis weakens. This prolonged deviation could signal a trend formation or a structural change in the relationship.
For the XOM/CVX pair with a 4.62-day half-life, if a trade is initiated at a 2-standard-deviation extreme, and after 10 trading days (more than twice the half-life) the spread has moved further against the position (e.g., to 3 standard deviations), consider exiting. The probability of reversion decreases. The cost of holding increases.
For the MSFT/GOOGL pair with a 23.1-day half-life, a 10-day period is less significant. A deviation persisting for 10 days is still within the expected range of its slower reversion. A stop-loss might trigger after 40-50 days (roughly twice the half-life) if the spread continues to move adversely.
Define stop-loss levels as multiples of the standard deviation of the spread. Simultaneously, define a time-based stop. If the spread reaches a pre-defined standard deviation threshold or if it exceeds a time threshold (e.g., 2 times the half-life) without reverting, close the position.
Combining Half-Life with Risk Management
Integrate half-life into a comprehensive risk management framework. Position sizing and stop-loss placement are essential. Half-life provides a dynamic input.
Consider a statistical arbitrage strategy on a basket of 20 mean-reverting pairs. Each pair has a unique half-life. A portfolio manager allocates capital based on these individual half-lives. Faster mean-reverting pairs receive larger capital allocations. Slower pairs receive smaller allocations.
For example, a total capital of $10 million. For a pair with a 5-day half-life, allocate $500,000. For a pair with a 20-day half-life, allocate $125,000. This ensures capital deploys efficiently across assets with varying reversion speeds.
Set stop-losses for each pair individually. A pair with a 5-day half-life might have a time-based stop at 10 days. A pair with a 20-day half-life might have a time-based stop at 40 days. This prevents "dead money" from tying up capital in non-reverting trades.
Simultaneously, implement hard dollar-loss stops. If a trade loses 1% of allocated capital, close it, regardless of half-life or standard deviation. This provides an absolute risk ceiling.
This multi-faceted approach combines quantitative insights from half-life with traditional risk controls. It optimizes capital deployment and minimizes exposure to failed mean reversion trades. Practical application requires continuous recalculation of half-life. Market dynamics change. Recalculate $\lambda$ and $T_{1/2}$ regularly (e.g., daily or weekly) using a rolling lookback window. This adapts the strategy to evolving market conditions.