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Generalised geometric Brownian motion: Theory and applications to option pricing

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  • Viktor Stojkoski
  • Trifce Sandev
  • Lasko Basnarkov
  • Ljupco Kocarev
  • Ralf Metzler

Abstract

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 develop a generalisation of GBM where the introduction of a memory kernel critically determines the behavior of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and obtain the corresponding probability density functions by 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) 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.

Suggested Citation

  • Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    • Handle: RePEc:arx:papers:2011.00312
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    References listed on IDEAS

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    9. Jolakoski, Petar & Pal, Arnab & Sandev, Trifce & Kocarev, Ljupco & Metzler, Ralf & Stojkoski, Viktor, 2023. "A first passage under resetting approach to income dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    10. Rajeev Rajaram & Nathan Ritchey, 2023. "Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy," Mathematics, MDPI, vol. 11(15), pages 1-16, July.
    11. Petar Jolakoski & Arnab Pal & Trifce Sandev & Ljupco Kocarev & Ralf Metzler & Viktor Stojkoski, 2022. "The fate of the American dream: A first passage under resetting approach to income dynamics," Papers 2212.13176, arXiv.org.
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    14. Ma, Pengcheng & Najafi, Alireza & Gomez-Aguilar, J.F., 2024. "Sub mixed fractional Brownian motion and its application to finance," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).

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