The NGARCH option pricing model with Variance Gamma distributed innovations

In this article, we present a discrete-time NGARCH option pricing model with Variance Gamma (VG) innovations ζn+1VG,(δ)=ζVG,(δ)(vn+1) as a special case of Lévy GARCH models. The NGARCH VG model is driven by non-standardized, time-varying VG innovations entering both the asset price and the NGARCH variance processes. This preserves the original VG distribution, in contrast to standardized Lévy GARCH models based on zero-mean, unit-variance innovations. The VG jump process and the NGARCH variance process evolve jointly, generating an endogenous interaction between jump dynamics and conditional volatility, in contrast to standardized Lévy GARCH models where volatility is generated solely by the GARCH process, which decouples the Lévy distribution from the volatility process. The Lévy component affects primarily higher order moments, while the volatility process is exogenous to the Lévy innovation distribution. The comparison between the standardized and non-standardized specifications from the point of view of structural dependence, likelihood function geometry, and martingale measure construction is provided. Statistical analysis under the physical measure P is presented. Model specific risk-neutral measures are derived by the correction of the conditional mean mn+1(m)=m(vn+1) and in the analytical form by Esscher transformation inducing a conditional time-varying skewness process θn+1∗=θ(vn+1). The non-standardized NGARCH VG model exhibits improved performance under both the physical P and the risk-neutral Q measures.