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Theory"],"abstract":"<jats:p>This article studies the weak convergence of discrete-time generalized autoregressive conditional heteroskedasticity (GARCH) and stochastic volatility (SV) option pricing models that allow for fat tails, multi-component volatilities, and non-monotonic pricing kernels. We introduce a general affine framework that enables the simultaneous derivation of new diffusion limits for Gaussian and inverse Gaussian GARCH models based on distributional invariant parametric convergence rates. When restricted to one-component specifications, our limits yield non-degenerate bivariate diffusions, generalizing the existing results in the affine GARCH literature. By using alternative parametric convergence rates, we further provide a comprehensive classification of all possible limits for two classes of affine and non-affine discrete-time SV models. Specifically, we show that the canonical affine classes of models popularized in discrete and continuous time are not analogous to one another. Our theoretical results are supported by a series of numerical experiments that investigate the impact of parametric scaling assumptions and model features on the convergence of European option prices. 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