Zhang and Zhang modify Duan's (1995) locally risk-neutral valuation relationship (LRNVR) for GARCH option-pricing models. In their modified relationship (mLRNVR), the conditional variances differ across the physical and risk-neutral measures — the variance process is more persistent under the risk-neutral measure — so the model captures the variance risk premium. Empirically, GARCH option-pricing models under the mLRNVR price the SPX one-month variance-swap rate (the CBOE VIX) accurately, and the authors recommend the mLRNVR for pricing options with GARCH models.
"In our mLRNVR, the conditional variances under two measures are designed to be different and the variance process is more persistent in the risk-neutral measure than in the physical one, so that one is able to capture the variance risk premium."
A focused, practical fix to a known defect: Duan's LRNVR is elegant but forces the physical and risk-neutral variance dynamics to coincide, which mechanically kills the variance risk premium and leaves GARCH models under-pricing the VIX. Making the risk-neutral variance more persistent is a minimal, interpretable change that restores the premium and matches the one-month variance-swap rate. The natural caveat is scope: the paper validates against the one-month VIX rather than the full option surface or the VIX term structure, so how the single persistence wedge fares across maturities and strikes is the obvious next question. It complements the continuous-time Heston and characteristic-function FFT machinery catalogued elsewhere with the discrete-time, directly-estimable GARCH route.