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  - 
TY  - JOUR
A1  - Basnarkov, Lasko
A1  - Tomovski, Igor
A1  - Sandev, Trifce
A1  - Kocarev, Ljupčo
T1  - Non-Markovian SIR epidemic spreading model of COVID-19
JF  - Chaos, solitons & fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science
N2  - We introduce non-Markovian SIR epidemic spreading model inspired by the characteristics of the COVID-19, by considering discrete-and continuous-time versions. The distributions of infection intensity and recovery period may take an arbitrary form. By taking corresponding choice of these functions, it is shown that the model reduces to the classical Markovian case. The epidemic threshold is analytically determined for arbitrary functions of infectivity and recovery and verified numerically. The relevance of the model is shown by modeling the first wave of the epidemic in Italy, Spain and the UK, in the spring, 2020.
KW  - Epidemic spreading models
KW  - Non-Markovian processes
KW  - COVID-19
KW  - SIR model
Y1  - 2022
U6  - https://doi.org/10.1016/j.chaos.2022.112286
SN  - 0960-0779
SN  - 1873-2887
VL  - 160
PB  - Elsevier
CY  - Oxford [u.a.]
ER  -