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Subordinated α-stable Ornstein–Uhlenbeck process as a tool for financial data description

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  • Janczura, Joanna
  • Orzeł, Sebastian
  • Wyłomańska, Agnieszka

Abstract

The classical financial models are based on the standard Brownian diffusion-type processes. However, in the exhibition of some real market data (like interest or exchange rates) we observe characteristic periods of constant values. Moreover, in the case of financial data, the assumption of normality is often unsatisfied. In such cases the popular Vasiček model, that is a mathematical system describing the evolution of interest rates based on the Ornstein–Uhlenbeck process, seems not to be applicable. Therefore, we propose an alternative approach based on a combination of the popular Ornstein–Uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior. The probability density function of the proposed process can be described by a Fokker–Planck type equation and therefore it can be examined as an extension of the basic Ornstein–Uhlenbeck model. In this paper, we propose the parameters’ estimation method and calibrate the subordinated Vasiček model to the interest rate data.

Suggested Citation

  • Janczura, Joanna & Orzeł, Sebastian & Wyłomańska, Agnieszka, 2011. "Subordinated α-stable Ornstein–Uhlenbeck process as a tool for financial data description," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4379-4387.
    • Handle: RePEc:eee:phsmap:v:390:y:2011:i:23:p:4379-4387
    DOI: 10.1016/j.physa.2011.07.007
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    References listed on IDEAS

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    1. Pavel Cizek & Wolfgang Karl Härdle & Rafal Weron, 2005. "Statistical Tools for Finance and Insurance," HSC Books, Hugo Steinhaus Center, Wroclaw University of Science and Technology, number hsbook0501, December.
    2. Burnecki, Krzysztof & Janczura, Joanna & Weron, Rafal, 2010. "Building Loss Models," MPRA Paper 25492, University Library of Munich, Germany.
    3. Borak, Szymon & Misiorek, Adam & Weron, Rafał, 2010. "Models for heavy-tailed asset returns," SFB 649 Discussion Papers 2010-049, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    4. repec:hum:wpaper:sfb649dp2010-049 is not listed on IDEAS
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    Cited by:

    1. Bielak, Łukasz & Grzesiek, Aleksandra & Janczura, Joanna & Wyłomańska, Agnieszka, 2021. "Market risk factors analysis for an international mining company. Multi-dimensional, heavy-tailed-based modelling," Resources Policy, Elsevier, vol. 74(C).
    2. Gu, Hui & Liang, Jin-Rong & Zhang, Yun-Xiu, 2012. "Time-changed geometric fractional Brownian motion and option pricing with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(15), pages 3971-3977.
    3. He, Yue & Kawai, Reiichiro, 2022. "Super- and subdiffusive positions in fractional Klein–Kramers equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 588(C).
    4. Sikora, Grzegorz & Michalak, Anna & Bielak, Łukasz & Miśta, Paweł & Wyłomańska, Agnieszka, 2019. "Stochastic modeling of currency exchange rates with novel validation techniques," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1202-1215.
    5. Szarek, Dawid & Bielak, Łukasz & Wyłomańska, Agnieszka, 2020. "Long-term prediction of the metals’ prices using non-Gaussian time-inhomogeneous stochastic process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).
    6. Foad Shokrollahi, 2016. "Subdiffusive fractional Brownian motion regime for pricing currency options under transaction costs," Papers 1612.06665, arXiv.org, revised Aug 2017.

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