IDEAS home Printed from
MyIDEAS: Login to save this paper or follow this series

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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL:
File Function: Original version, 2012
Download Restriction: no

Paper provided by Hugo Steinhaus Center, Wroclaw University of Technology in its series HSC Research Reports with number HSC/12/04.

in new window

Length: 20 pages
Date of creation: 2012
Date of revision:
Publication status: Forthcoming in Acta Phys. Polon. B 43(5), 1093-1110.
Handle: RePEc:wuu:wpaper:hsc1204
Contact details of provider: Postal: Wybrzeze Wyspianskiego 27, 50-370 Wroclaw
Phone: +48-71-3203530
Fax: +48-71-3202654
Web page:

More information through EDIRC

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

as in new window
  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.
Full references (including those not matched with items on IDEAS)

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:wuu:wpaper:hsc1204. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Rafal Weron)

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.