Filtered historical simulation

Sep 20, 2026, filed under risk, time series · part of VaR/ES risk engine

Plain historical simulation treats every day in the sample window as equally informative about tomorrow. That’s the part filtering fixes: yesterday’s volatility regime should count for more than a quiet stretch from six months ago.

Let $z_t = r_t / \sigma_t$ , where $\sigma_t$ is the GARCH(1,1) conditional volatility. Under a correctly specified model, $\{z_t\}$ is approximately $\text{i.i.d.}$ with unit variance.

Proof. Dividing $r_t$ by its fitted conditional standard deviation removes the time-varying scale by construction: $\mathrm{Var}(z_t) = \mathrm{Var}(r_t)/\sigma_t^2 = 1$ . What’s left is the standardized shape of the shock, which GARCH assumes is drawn independently each day. ◻

Filtered historical simulation rescales each historical standardized residual $z_t$ by today’s forecast $\sigma_{t+1}$ before reading VaR and ES off the resulting sample.

That’s the whole trick: keep the empirical shape of the tail from history, but let today’s volatility regime set its scale.