Taking into account the time-varying, jump and leverage effect characteristics of asset price fluctuations, we first obtain the asset return rate model through the GJR-GARCH model (Glosten, Jagannathan and Rundle-generalized autoregressive conditional heteroskedasticity model) and introduce the infinite pure-jump Levy process into the asset return rate model to improve the model�s accuracy. Then, to be more consistent with reality and include more uncertainty factors, we integrate the more generalized parabolic fuzzy variable (which can cover the triangle and trapezoid fuzzy variable) to represent asset price volatility. Next, considering more general situations with fuzzy variables with mixed distributions, we apply fuzzy simulation technology to the least squares Monte Carlo algorithm to create fuzzy pricing numerical algorithms, that is the fuzzy least squares Monte Carlo algorithm. Finally, by using American options data from the Standard & Poor�s 100 index, we empirically test our fuzzy pricing model with different widely used infinite pure-jump Levy processes (the VG (variance gamma process), NIG (normal inverse Gaussian process) and CGMY (Carr-Geman-Madan-Yor process) under fuzzy and crisp environments. The results indicate that the fuzzy option pricing model is more reasonable; the fuzzy interval can cover the market prices of options and the prices that obtained by the crisp option pricing model, the fuzzy option pricing model is feasible one. © Springer Nature Switzerland AG 2019.