Pairs Trading with Kalman Filters: Dynamic Hedge Ratios

Why a static OLS beta falls short: estimating a time-varying hedge ratio with a state-space model and building a cointegration strategy.

The state-space model#

We model the hedge ratio as a latent state:

$y_t = \beta_t x_t + \epsilon_t, \qquad \beta_t = \beta_{t-1} + \eta_t$

where $\epsilon_t \sim N(0, R)$ is observation noise and $\eta_t \sim N(0, Q)$ is state noise.

Python implementation#

import numpy as np

def sharpe(r, rf=0.0, ann=252):
    ex = r - rf
    return np.sqrt(ann) * ex.mean() / ex.std()

# Kalman-filtered dynamic hedge ratio
for t in range(1, n):
    P_p = P[t-1] + Q
    K = P_p * x[t] / (x[t]**2 * P_p + R)
    b[t] = b[t-1] + K * (y[t] - x[t]*b[t-1])
    P[t] = (1 - K * x[t]) * P_p

Full article in progress — backtest results and parameter selection to follow.