Filtered Historical Simulation (GJR-GARCH)

EWMA updates variance from the squared shock, so it is blind to sign: a 3% fall and a 3% rise are identical news about tomorrow. Equity markets disagree, consistently and across every market that has been studied. Volatility rises far more after a decline than after a rally of the same magnitude, an asymmetry usually called the leverage effect after the original explanation that a falling equity value raises a firm's debt-to-equity ratio and hence the riskiness of its equity.

The GJR-GARCH(1,1) model of Glosten, Jagannathan and Runkle (1993) captures it with one extra parameter that switches on only for negative shocks. For a drift-removed log return :

A positive shock contributes ; a negative shock of identical size contributes . Fitted is typically positive and often comparable in magnitude to on equity data.

The surrounding pipeline is identical to the EWMA variant: log space, per-asset sample-log-mean demeaning, standardization, IID whole-row residual resampling, forward variance recursion, and the empirical cumulant drift normalization that keeps the conditional expected gross return equal to the declared daily target.

GARCH-family fits fail in ways that are easy to miss and expensive to trust: the optimizer stops at a boundary, persistence lands at essentially one so the model has no mean reversion, or the fitted variance process is wildly off the sample variance. Every fit must therefore pass five checks before it is admitted. All thresholds are versioned engineering constants, revised only through validation evidence, not universal statistical truths.

The Ljung-Box check deserves emphasis, because it is the one that validates the method's core premise. Filtered historical simulation is only legitimate if the standardized residuals are close enough to IID to be resampled independently. Testing the squared residuals for leftover autocorrelation is a direct test of exactly that assumption, and a failure means the process is not entitled to resample them.

Admission is decided per asset, not per portfolio. A holding whose GJR fit fails any check falls back to EWMA for that holding alone; the rest of the portfolio keeps its GJR fits. Because EWMA is stored as the integrated special case of the same GJR form, with , , , , the mixed portfolio simulates through one uniform vectorized recursion with no special-casing.

The result reports which holdings fell back and the specific reason for each. A run where every asset silently degraded to EWMA would otherwise be indistinguishable from a successful GJR run, which is precisely the kind of invisible degradation the engine is built to avoid.

The process requires 1,260 aligned observations, roughly five trading years, against 756 for EWMA. GJR has four parameters per asset to EWMA's effective one, and asymmetry in particular is estimated from the negative-shock subsample, so it needs more data before the estimate means anything.