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Anomalous dynamics of Black–Scholes model time-changed by inverse subordinators


  • Marcin Magdziarz
  • Janusz Gajda


In this paper we consider a generalization of one of the earliest models of an asset price, namely the Black–Scholes model, which captures the subdiffusive nature of an asset price dynamics. We introduce the geometric Brownian motion time-changed by infinitely divisible inverse subordinators, to reflect underlying anomalous diffusion mechanism. In the proposed model the waiting times (periods when the asset price stays motionless) are modeled by general class of infinitely divisible distributions. We find the corresponding Fractional Fokker–Planck equation governing the probability density function of the introduced process. We prove that considered model is arbitrage-free, construct corresponding martingale measure and show that the model is incomplete. We also find formulas for values of European call and put option prices in subdiffusive Black–Scholes model and show how one can approximate them based on Monte Carlo methods. We present some Monte Carlo simulations for the particular case of tempered alpha-stable distribution of waiting times. We compare obtained results with the classical and subdiffusive alpha-stable Black–Scholes prices.

Suggested Citation

  • Marcin Magdziarz & Janusz Gajda, 2012. "Anomalous dynamics of Black–Scholes model time-changed by inverse subordinators," HSC Research Reports HSC/12/04, Hugo Steinhaus Center, Wroclaw University of Technology.
  • Handle: RePEc:wuu:wpaper:hsc1204

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    File Function: Original version, 2012
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    References listed on IDEAS

    1. Aleksander Janicki & Aleksander Weron, 1994. "Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook9401.
    2. Joanna Janczura & Rafal Weron, 2012. "Inference for Markov-regime switching models of electricity spot prices," HSC Research Reports HSC/12/01, Hugo Steinhaus Center, Wroclaw University of Technology.
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    Cited by:

    1. Foad Shokrollahi, 2017. "The evaluation of geometric Asian power options under time changed mixed fractional Brownian motion," Papers 1712.05254,
    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


    Black-Scholes model; alpha-stable distribution; time-changed Brownian motion; fractional Fokker–Planck equation; martingale measure;

    JEL classification:

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


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