IDEAS home Printed from https://ideas.repec.org/p/pra/mprapa/90549.html
   My bibliography  Save this paper

Modeling Returns of Stock Indexes through Fractional Brownian Motion Combined with Jump Processes and Modulated by Markov Chains

Author

Listed:
  • Carpinteyro, Martha
  • Venegas-Martínez, Francisco
  • Martínez-García, Miguel Ángel

Abstract

We develop a mathematical model useful to describe the stochastic dynamics and return distribution of the stock indexes of world’s main economies (USA, Eurozone, UK and Japan) and of the main emerging markets (China, Brazil, and México) incorporating risk factors as: idiosyncratic volatility, market volatility, and regime-switching volatility. It is assumed that the returns of the stock indexes are driven by fractional Brownian motions combined with Poisson processes and modulated by Markov chains. To do that, we calibrate Jump-GARCH models and estimate Markov regime-switching stochastic volatility models. The proposed models properly describe the stochastic dynamics of the returns of the stock indexes under study during 1994-2017. The main empirical finding is that the USA stock market stays in high volatility most of the time and presents more jumps than other indexes, and that Brazil stock market has the biggest intensity of jumps during 1994-2017. The outcome supports the hypothesis of long-term memory of stock markets. Resumen: Esta investigación desarrolla un modelo matemático útil para describir la dinámica estocástica y la distribución de los rendimientos de los índices bursátiles de las principales economías del mundo (EE.UU. Zona Euro, Reino Unido y Japón) y de los mayores mercados emergentes (China, Brasil y México) mediante la incorporación de factores de riesgo como: volatilidad idiosincrática, volatilidad del mercado y volatilidad de cambio de régimen. Se supone que los rendimientos de los índices bursátiles son conducidos por movimientos fraccionales brownianos combinados con procesos de Poisson y modulados por cadenas de Markov. Para lograr este objetivo se calibran modelos Jump-GARCH y se estiman modelos de volatilidad estocástica de cambio de régimen markoviano. Los modelos propuestos describen adecuadamente la dinámica estocástica de los rendimientos de los índices bursátiles bajo estudio durante 1994-2017. Los principales hallazgos empíricos reflejan que el mercado de capitales de EE.UU. se mantiene en alta volatilidad la mayor parte del tiempo y presenta más saltos que los otros índices y el mercado accionario de Brasil presenta grandes saltos con mayor intensidad. El resultado sostiene la hipótesis de memoria de largo pazo en el mercado de capitales.

Suggested Citation

  • Carpinteyro, Martha & Venegas-Martínez, Francisco & Martínez-García, Miguel Ángel, 2018. "Modeling Returns of Stock Indexes through Fractional Brownian Motion Combined with Jump Processes and Modulated by Markov Chains," MPRA Paper 90549, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:90549
    as

    Download full text from publisher

    File URL: https://mpra.ub.uni-muenchen.de/90549/1/MPRA_paper_90549.pdf
    File Function: original version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Jamdee, Sutthisit & Los, Cornelis A., 2007. "Long memory options: LM evidence and simulations," Research in International Business and Finance, Elsevier, vol. 21(2), pages 260-280, June.
    2. Harvey, Campbell R. & Siddique, Akhtar, 1999. "Autoregressive Conditional Skewness," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 34(4), pages 465-487, December.
    3. Carr, Peter & Wu, Liuren, 2007. "Stochastic skew in currency options," Journal of Financial Economics, Elsevier, vol. 86(1), pages 213-247, October.
    4. Byeong-Je An & Andrew Ang & Turan G. Bali & Nusret Cakici, 2014. "The Joint Cross Section of Stocks and Options," Journal of Finance, American Finance Association, vol. 69(5), pages 2279-2337, October.
    5. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 74(1), pages 3-30, September.
    6. Andrew Ang & Robert J. Hodrick & Yuhang Xing & Xiaoyan Zhang, 2006. "The Cross‐Section of Volatility and Expected Returns," Journal of Finance, American Finance Association, vol. 61(1), pages 259-299, February.
    7. Christian Bender & Tommi Sottinen & Esko Valkeila, 2010. "Fractional processes as models in stochastic finance," Papers 1004.3106, arXiv.org.
    8. Huyěn Pham & Nizar Touzi, 1996. "Equilibrium State Prices In A Stochastic Volatility Model1," Mathematical Finance, Wiley Blackwell, vol. 6(2), pages 215-236, April.
    9. Bates, David S., 2008. "The market for crash risk," Journal of Economic Dynamics and Control, Elsevier, vol. 32(7), pages 2291-2321, July.
    10. Peter Christoffersen & Steven Heston & Kris Jacobs, 2009. "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well," Management Science, INFORMS, vol. 55(12), pages 1914-1932, December.
    11. Pedro Santa-Clara & Shu Yan, 2010. "Crashes, Volatility, and the Equity Premium: Lessons from S&P 500 Options," The Review of Economics and Statistics, MIT Press, vol. 92(2), pages 435-451, May.
    12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    13. Hamilton, James D & Gang, Lin, 1996. "Stock Market Volatility and the Business Cycle," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(5), pages 573-593, Sept.-Oct.
    14. Martijn Cremers & Michael Halling & David Weinbaum, 2015. "Aggregate Jump and Volatility Risk in the Cross-Section of Stock Returns," Journal of Finance, American Finance Association, vol. 70(2), pages 577-614, April.
    15. Johnson, Timothy C, 2002. "Volatility, Momentum, and Time-Varying Skewness in Foreign Exchange Returns," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 390-411, July.
    16. Cajueiro, Daniel O. & Tabak, Benjamin M., 2005. "Possible causes of long-range dependence in the Brazilian stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 345(3), pages 635-645.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Carpinteyro, Martha & Venegas Martínez, Francisco & Martínez García, Miguel Ángel, 2019. "Modelado de rendimientos de índices bursátiles mediante movimiento fraccional browniano combinado con procesos de saltos y modulado por cadenas de Markov / Modeling Returns of Stock Indexes through Fr," Estocástica: finanzas y riesgo, Departamento de Administración de la Universidad Autónoma Metropolitana Unidad Azcapotzalco, vol. 9(2), pages 163-180, julio-dic.
    2. Christoffersen, Peter & Heston, Steven & Jacobs, Kris, 2010. "Option Anomalies and the Pricing Kernel," Working Papers 11-17, University of Pennsylvania, Wharton School, Weiss Center.
    3. Du Du & Dan Luo, 2019. "The Pricing of Jump Propagation: Evidence from Spot and Options Markets," Management Science, INFORMS, vol. 67(5), pages 2360-2387, May.
    4. Peter Christoffersen & Mathieu Fournier & Kris Jacobs, 2018. "The Factor Structure in Equity Options," The Review of Financial Studies, Society for Financial Studies, vol. 31(2), pages 595-637.
    5. Jozef Barunik & Michael Ellington, 2020. "Dynamic Network Risk," Papers 2006.04639, arXiv.org, revised Jul 2020.
    6. Chung, Kee H. & Wang, Junbo & Wu, Chunchi, 2019. "Volatility and the cross-section of corporate bond returns," Journal of Financial Economics, Elsevier, vol. 133(2), pages 397-417.
    7. Ruan, Xinfeng, 2020. "Volatility-of-volatility and the cross-section of option returns," Journal of Financial Markets, Elsevier, vol. 48(C).
    8. Christoffersen, Peter & Jacobs, Kris & Ornthanalai, Chayawat, 2012. "Dynamic jump intensities and risk premiums: Evidence from S&P500 returns and options," Journal of Financial Economics, Elsevier, vol. 106(3), pages 447-472.
    9. Horatio Cuesdeanu & Jens Carsten Jackwerth, 2018. "The pricing kernel puzzle: survey and outlook," Annals of Finance, Springer, vol. 14(3), pages 289-329, August.
    10. Christoffersen, Peter & Jacobs, Kris & Chang, Bo Young, 2013. "Forecasting with Option-Implied Information," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 581-656, Elsevier.
    11. Lee, Kiryoung & Jeon, Yoontae & Nam, Eun-Young, 2021. "Chinese Economic Policy Uncertainty and the Cross-Section of U.S. Asset Returns," International Review of Economics & Finance, Elsevier, vol. 76(C), pages 1063-1077.
    12. Peter Christoffersen & Kris Jacobs & Chayawat Ornthanalai, 2012. "GARCH Option Valuation: Theory and Evidence," CREATES Research Papers 2012-50, Department of Economics and Business Economics, Aarhus University.
    13. Chen, Chin-Ho, 2019. "Downside jump risk and the levels of futures-cash basis," Pacific-Basin Finance Journal, Elsevier, vol. 57(C).
    14. Ballotta, Laura & Rayée, Grégory, 2022. "Smiles & smirks: Volatility and leverage by jumps," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1145-1161.
    15. Chen, Xi & Wang, Junbo & Wu, Chunchi, 2022. "Jump and volatility risk in the cross-section of corporate bond returns," Journal of Financial Markets, Elsevier, vol. 60(C).
    16. Chernov, Mikhail & Graveline, Jeremy & Zviadadze, Irina, 2018. "Crash Risk in Currency Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 53(1), pages 137-170, February.
    17. Bingxin Li, 2020. "Option-implied filtering: evidence from the GARCH option pricing model," Review of Quantitative Finance and Accounting, Springer, vol. 54(3), pages 1037-1057, April.
    18. Tim Bollerslev, 2008. "Glossary to ARCH (GARCH)," CREATES Research Papers 2008-49, Department of Economics and Business Economics, Aarhus University.
    19. Agarwal, Vikas & Arisoy, Y. Eser & Naik, Narayan Y., 2017. "Volatility of aggregate volatility and hedge fund returns," Journal of Financial Economics, Elsevier, vol. 125(3), pages 491-510.
    20. Ruan, Xinfeng & Zhang, Jin E., 2018. "Risk-neutral moments in the crude oil market," Energy Economics, Elsevier, vol. 72(C), pages 583-600.

    More about this item

    Keywords

    Stock index return; fractional Brownian motion; Markov regime-switching; jump processes.;
    All these keywords.

    JEL classification:

    • G15 - Financial Economics - - General Financial Markets - - - International Financial Markets

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:90549. See general information about how to correct material in RePEc.

    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 CitEc recognized a bibliographic reference but did not link an item in RePEc 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 RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Joachim Winter (email available below). General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.