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Article

Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

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Faculty of Economics, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia
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Research Centre for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia
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Institute of Physics & Astronomy, University of Potsdam, D-14776 Potsdam-Golm, Germany
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Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia
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Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, P.O. Box 393, 1000 Skopje, Macedonia
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(12), 1432; https://doi.org/10.3390/e22121432
Received: 30 October 2020 / Revised: 11 December 2020 / Accepted: 16 December 2020 / Published: 18 December 2020
(This article belongs to the Special Issue New Trends in Random Walks)
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. View Full-Text
Keywords: geometric Brownian motion; Fokker–Planck equation; Black–Scholes model; option pricing geometric Brownian motion; Fokker–Planck equation; Black–Scholes model; option pricing
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MDPI and ACS Style

Stojkoski, V.; Sandev, T.; Basnarkov, L.; Kocarev, L.; Metzler, R. Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing. Entropy 2020, 22, 1432. https://doi.org/10.3390/e22121432

AMA Style

Stojkoski V, Sandev T, Basnarkov L, Kocarev L, Metzler R. Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing. Entropy. 2020; 22(12):1432. https://doi.org/10.3390/e22121432

Chicago/Turabian Style

Stojkoski, Viktor, Trifce Sandev, Lasko Basnarkov, Ljupco Kocarev, and Ralf Metzler. 2020. "Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing" Entropy 22, no. 12: 1432. https://doi.org/10.3390/e22121432

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