Generalised geometric Brownian motion
- 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 dependsClassical 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.…
Author details: | Viktor StojkoskiORCiD, Trifce SandevORCiDGND, Lasko BasnarkovORCiD, Ljupco KocarevGND, Ralf MetzlerORCiDGND |
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DOI: | https://doi.org/10.3390/e22121432 |
ISSN: | 1099-4300 |
Pubmed ID: | https://pubmed.ncbi.nlm.nih.gov/33353060 |
Title of parent work (English): | Entropy |
Subtitle (English): | theory and applications to option pricing |
Publisher: | MDPI |
Place of publishing: | Basel |
Publication type: | Article |
Language: | English |
Date of first publication: | 2020/12/18 |
Publication year: | 2020 |
Release date: | 2023/07/06 |
Tag: | Black– Scholes model; Fokker– Planck equation; geometric Brownian motion; option pricing |
Volume: | 22 |
Issue: | 12 |
Article number: | 1432 |
Number of pages: | 34 |
Funding institution: | German Science Foundation (DFG)German Research Foundation (DFG) [ME; 1535/6-1]; Alexander von Humboldt FoundationAlexander von Humboldt; Foundation; Alexander von Humboldt Polish Honorary Research Scholarship; from the Foundation for Polish Science (Fundacja na rzecz Nauki; Polskiej, FNP) |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |
DDC classification: | 5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik |
Peer review: | Referiert |
Publishing method: | Open Access / Gold Open-Access |
DOAJ gelistet | |
License (German): | CC-BY - Namensnennung 4.0 International |