Journal
ENTROPY
Volume 22, Issue 12, Pages -Publisher
MDPI
DOI: 10.3390/e22121432
Keywords
geometric Brownian motion; Fokker– Planck equation; Black– Scholes model; option pricing
Categories
Funding
- German Science Foundation (DFG) [ME 1535/6-1]
- Alexander von Humboldt Foundation
- Alexander von Humboldt Polish Honorary Research Scholarship from the Foundation for Polish Science (Fundacja na rzecz Nauki Polskiej, FNP)
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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.
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