Time series

Mean-reversion half-life

Turns an estimated reversion speed into a holding period, and therefore into turnover and costs.

t1/2=ln⁡2λ,Var∞=σ22θt_{1/2} = \frac{\ln 2}{\lambda}, \qquad \mathrm{Var}_\infty = \frac{\sigma^2}{2\theta}

Where

λ\lambda
Reversion coefficient from Δst=−λst−1+εt\Delta s_t = -\lambda s_{t-1} + \varepsilon_t.
θ\theta
The same speed, written for an Ornstein–Uhlenbeck process.

Assumptions

  • The spread is genuinely stationary — test the residual before believing the half-life.

Sanity check. A half-life longer than your capital can wait is a signal you cannot actually trade.

Where this is taught