  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
asset dynamics due to irregularities found when comparing its properties with
empirical distributions. As a solution, we develop a generalisation of GBM
where the introduction of a memory kernel critically determines the behavior of
the stochastic process. We find the general expressions for the moments,
log-moments, and the expectation of the periodic log returns, and obtain the
corresponding probability density functions by using the subordination
approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a
mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised
GBM (gGBM) to examine the empirical performance of a selected group of kernels
in the pricing of European call options. Our results indicate that the
performance of a kernel ultimately depends on the maturity of the option and
its moneyness.
