TY  - JOUR
A1  - Stojkoski, Viktor
A1  - Sandev, Trifce
A1  - Basnarkov, Lasko
A1  - Kocarev, Ljupco
A1  - Metzler, Ralf
T1  - Generalised geometric Brownian motion
BT  - theory and applications to option pricing
JF  - Entropy
N2  - 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.
KW  - geometric Brownian motion
KW  - Fokker– Planck equation
KW  - Black– Scholes model
KW  - option pricing
Y1  - 2020
U6  - https://doi.org/10.3390/e22121432
SN  - 1099-4300
VL  - 22
IS  - 12
PB  - MDPI
CY  - Basel
ER  -