@article{StojkoskiSandevBasnarkovetal.2020,
  author    = {Stojkoski, Viktor and Sandev, Trifce and Basnarkov, Lasko and Kocarev, Ljupco and Metzler, Ralf},
  title     = {Generalised geometric Brownian motion},
  series = {Entropy},
  volume    = {22},
  journal   = {Entropy},
  number    = {12},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {1099-4300},
  doi       = {10.3390/e22121432},
  pages     = {34},
  year      = {2020},
  abstract  = {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 investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions 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) in order 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.},
  language  = {en}
}