Option pricing in subdiffusive Bachelier model

Marcin Magdziarz (), Sebastian Orzeł and Aleksander Weron

No HSC/11/05, HSC Research Reports from Hugo Steinhaus Center, Wroclaw University of Science and Technology

Abstract: The earliest model of stock prices based on Brownian diffusion is the Bachelier model. In this paper we propose an extension of the Bachelier model, which reflects the subdiffusive nature of the underlying asset dynamics. The subdiffusive property is manifested by the random (infinitely divisible) periods of time, during which the asset price does not change. We introduce a subdiffusive arithmetic Brownian motion as a model of stock prices with such characteristics. The structure of this process agrees with two-stage scenario underlying the anomalous diffusion mechanism, in which trapping random events are superimposed on the Langevin dynamics.We find the corresponding fractional Fokker-Planck equation governing the probability density function of the introduced process. We construct the corresponding martingale measure and show that the model is incomplete. We derive the formulas for European put and call option prices. We describe explicit algorithms and present some Monte-Carlo simulations for the particular cases of alpha-stable and tempered alpha-stable distributions of waiting times.

Keywords: Subdiffusion; Fractional Fokker-Planck equation; Bachelier model; Option pricing; Infinitely divisible distribution; Tempered stable distribution (search for similar items in EconPapers)
JEL-codes: C46 C51 G13 (search for similar items in EconPapers)
Pages: 17 pages
Date: 2011
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Citations: View citations in EconPapers (5)

Published in J. Stat. Phys., 145 (2011) 187-203.

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http://dx.doi.org/10.1007/s10955-011-0310-z Final printed version, 2011 (text/html)

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Persistent link: https://EconPapers.repec.org/RePEc:wuu:wpaper:hsc1105

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