Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for the dynamics of an asset due to the fat-tailed properties found in empirical distributions of the asset returns. To capture this phenomenon three approaches have been proposed: i) stochastic volatility, where the amplitude of the noise itself is a stochastic process; ii) utilizing fat-tailed distributions for the noise; and iii) generalizations of GBM based on subdiffusion. Here, we turn our attention to the last approach and develop a generalization of GBM where the introduction of a memory kernel critically determines the behavior of the stochastic process. We utilize the resulting generalized GBM to examine the empirical performance of a selected group of memory kernels in pricing European call options. Our results indicate that the performance of kernel ultimately depends on the maturity of the option and its moneyness. This yields novel inights to both finance theory and stochastic processes in general.