Time-Varying Hedge Ratios

Say you hedge one share of stock A by shorting 1.4 shares of stock B, a ratio you got from regressing a year of daily prices. That 1.4 was true, on average, over that year. It is not necessarily true today. Businesses change, index weights get rebalanced, liquidity shifts — and a hedge ratio computed once and frozen slowly turns into a source of risk rather than a way of removing it.

Why a fixed ratio decays

A static ordinary least squares (OLS) hedge ratio uses every data point in the estimation window with equal weight, so a relationship that shifted three months ago is still being averaged against the older data pulling the estimate the wrong way. The fix is to let the ratio move: either by re-running the regression on a rolling window that slides forward each day, dropping old data as new data arrives, or by using a Kalman filter, which updates the ratio after every new observation, weighted by how uncertain the current estimate is. A rolling window reacts in discrete jumps as old data falls out; a Kalman filter reacts continuously and gives more weight to surprising new data than to noise.

Worked example: a rolling window catching a shift

A trader hedges stock A with stock B using a 60-day rolling OLS beta. In January the trailing regression gives a hedge ratio of 0.80 shares of B per share of A. By April, B's business mix has shifted and the relationship loosens; the 60-day rolling beta, recomputed each day, drifts to 1.15 by late April as the window fills with post-shift data. A desk still using the January 0.80 ratio would be short too little B for every share of A — on a 10,000-share position in A, that is a hedge shortfall of 3,500 shares of B, fully exposed to any move specific to B rather than the shared factor. The rolling estimate catches the drift within weeks; the frozen one never does.

What this means in practice

Rolling windows are simple and transparent but throw away information at the edge of the window in discrete jumps, which can whipsaw a hedge ratio around news events. Kalman filters are smoother and use every observation, but need a model of how fast the ratio is allowed to move (the process noise), and a badly tuned filter either lags real shifts or chases noise. Neither method removes the underlying risk that the relationship itself might break down entirely rather than just drift — that is a separate, sharper problem.