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Option pricing in subdiffusive Bachelier model


  • Marcin Magdziarz
  • Sebastian Orzel
  • Aleksander Weron


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.

Suggested Citation

  • Marcin Magdziarz & Sebastian Orzel & Aleksander Weron, 2011. "Option pricing in subdiffusive Bachelier model," HSC Research Reports HSC/11/05, Hugo Steinhaus Center, Wroclaw University of Technology.
  • Handle: RePEc:wuu:wpaper:hsc1105

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    Cited by:

    1. Sebastian, Orzeł & Agnieszka, Wyłomańska, 2010. "Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times," MPRA Paper 28593, University Library of Munich, Germany.
    2. Lv, Longjin & Xiao, Jianbin & Fan, Liangzhong & Ren, Fuyao, 2016. "Correlated continuous time random walk and option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 447(C), pages 100-107.

    More about this item


    Subdiffusion; Fractional Fokker-Planck equation; Bachelier model; Option pricing; Infinitely divisible distribution; Tempered stable distribution;

    JEL classification:

    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing


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