IDEAS home Printed from https://ideas.repec.org/p/fip/fedgfe/2009-40.html
   My bibliography  Save this paper

Bayesian analysis of stochastic volatility models with Lévy jumps: application to risk analysis

Author

Abstract

In this paper I analyze a broad class of continuous-time jump diffusion models of asset returns. In the models, stochastic volatility can arise either from a diffusion part, or a jump part, or both. The jump component includes either compound Poisson or Lvy alpha-stable jumps. To be able to estimate the models with latent Lvy alpha-stable jumps, I construct a new Markov chain Monte Carlo algorithm. I estimate all model specifications with S&P500 daily returns. I find that models with Levy alpha-stable jumps perform well in capturing return characteristics if diffusion is a source of stochastic volatility. Models with stochastic volatility from jumps and models with Poisson jumps cannot represent excess kurtosis and tails of return distribution. In density forecast and VaR analysis, the model with Levy alpha-stable jumps and joint stochastic volatility performs the best among all other specifications, since both diffusion and infinite activity jump part provide information about latent volatility.

Suggested Citation

  • Pawel J. Szerszen, 2009. "Bayesian analysis of stochastic volatility models with Lévy jumps: application to risk analysis," Finance and Economics Discussion Series 2009-40, Board of Governors of the Federal Reserve System (U.S.).
    • Handle: RePEc:fip:fedgfe:2009-40
    as

    Download full text from publisher

    File URL: http://www.federalreserve.gov/pubs/feds/2009/200940/200940abs.html
    Download Restriction: no
    File URL: http://www.federalreserve.gov/pubs/feds/2009/200940/200940pap.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 2002. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 69-87, January.
    3. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    4. Gallant, A. Ronald & Tauchen, George, 1996. "Which Moments to Match?," Econometric Theory, Cambridge University Press, vol. 12(4), pages 657-681, October.
    5. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    6. Christoffersen, Peter F, 1998. "Evaluating Interval Forecasts," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 841-862, November.
    7. Jones, Christopher S., 2003. "The dynamics of stochastic volatility: evidence from underlying and options markets," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 181-224.
    8. repec:bla:jfinan:v:59:y:2004:i:3:p:1405-1440 is not listed on IDEAS
    9. Chib, Siddhartha & Greenberg, Edward, 1996. "Markov Chain Monte Carlo Simulation Methods in Econometrics," Econometric Theory, Cambridge University Press, vol. 12(3), pages 409-431, August.
    10. repec:bla:jfinan:v:59:y:2004:i:3:p:1367-1404 is not listed on IDEAS
    11. Haitao Li & Martin T. Wells & Cindy L. Yu, 2008. "A Bayesian Analysis of Return Dynamics with Lévy Jumps," The Review of Financial Studies, Society for Financial Studies, vol. 21(5), pages 2345-2378, September.
    12. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    13. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    14. W. R. Gilks & P. Wild, 1992. "Adaptive Rejection Sampling for Gibbs Sampling," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 41(2), pages 337-348, June.
    15. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    16. 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, December.
    17. Torben G. Andersen & Luca Benzoni & Jesper Lund, 2002. "An Empirical Investigation of Continuous‐Time Equity Return Models," Journal of Finance, American Finance Association, vol. 57(3), pages 1239-1284, June.
    18. Das, Sanjiv Ranjan & Sundaram, Rangarajan K., 1999. "Of Smiles and Smirks: A Term Structure Perspective," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 34(2), pages 211-239, June.
    19. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    20. 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.
    21. Diebold, Francis X & Gunther, Todd A & Tay, Anthony S, 1998. "Evaluating Density Forecasts with Applications to Financial Risk Management," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 863-883, November.
    22. Durham, Garland B., 2006. "Monte Carlo methods for estimating, smoothing, and filtering one- and two-factor stochastic volatility models," Journal of Econometrics, Elsevier, vol. 133(1), pages 273-305, July.
    23. Bjørn Eraker & Michael Johannes & Nicholas Polson, 2003. "The Impact of Jumps in Volatility and Returns," Journal of Finance, American Finance Association, vol. 58(3), pages 1269-1300, June.
    24. Chernov, Mikhail & Ghysels, Eric, 2000. "A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation," Journal of Financial Economics, Elsevier, vol. 56(3), pages 407-458, June.
    25. Yacine Ait-Sahalia, 2003. "Disentangling Volatility from Jumps," NBER Working Papers 9915, National Bureau of Economic Research, Inc.
    26. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    27. Jacquier, Eric & Polson, Nicholas G. & Rossi, P.E.Peter E., 2004. "Bayesian analysis of stochastic volatility models with fat-tails and correlated errors," Journal of Econometrics, Elsevier, vol. 122(1), pages 185-212, September.
    28. Jingzhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time- Changed Levy Processes," Finance 0401002, University Library of Munich, Germany.
    29. Chib, Siddhartha & Nardari, Federico & Shephard, Neil, 2002. "Markov chain Monte Carlo methods for stochastic volatility models," Journal of Econometrics, Elsevier, vol. 108(2), pages 281-316, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Qi Wang & Jos'e E. Figueroa-L'opez & Todd Kuffner, 2019. "Bayesian Inference on Volatility in the Presence of Infinite Jump Activity and Microstructure Noise," Papers 1909.04853, arXiv.org.
    2. Kostrzewski, Maciej & Kostrzewska, Jadwiga, 2019. "Probabilistic electricity price forecasting with Bayesian stochastic volatility models," Energy Economics, Elsevier, vol. 80(C), pages 610-620.
    3. Kaeck, Andreas & Rodrigues, Paulo & Seeger, Norman J., 2018. "Model Complexity and Out-of-Sample Performance: Evidence from S&P 500 Index Returns," Journal of Economic Dynamics and Control, Elsevier, vol. 90(C), pages 1-29.

    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. Carverhill, Andrew & Luo, Dan, 2023. "A Bayesian analysis of time-varying jump risk in S&P 500 returns and options," Journal of Financial Markets, Elsevier, vol. 64(C).
    2. Jingzhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time- Changed Levy Processes," Finance 0401002, University Library of Munich, Germany.
    3. Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Center for Research in Economics and Statistics.
    4. Carr, Peter & Wu, Liuren, 2007. "Stochastic skew in currency options," Journal of Financial Economics, Elsevier, vol. 86(1), pages 213-247, October.
    5. Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," PIER Working Paper Archive 05-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    6. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2006. "Volatility and Correlation Forecasting," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 15, pages 777-878, Elsevier.
    7. Christoffersen, Peter & Jacobs, Kris & Ornthanalai, Chayawat & Wang, Yintian, 2008. "Option valuation with long-run and short-run volatility components," Journal of Financial Economics, Elsevier, vol. 90(3), pages 272-297, December.
    8. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    9. Peter Carr & Liuren Wu, 2004. "Variance Risk Premia," Finance 0409015, University Library of Munich, Germany.
    10. Peter Christoffersen & Kris Jacobs & Chayawat Ornthanalai, 2009. "Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options," CIRANO Working Papers 2009s-34, CIRANO.
    11. Bates, David S., 2012. "U.S. stock market crash risk, 1926–2010," Journal of Financial Economics, Elsevier, vol. 105(2), pages 229-259.
    12. Maciej Kostrzewski & Jadwiga Kostrzewska, 2021. "The Impact of Forecasting Jumps on Forecasting Electricity Prices," Energies, MDPI, vol. 14(2), pages 1-17, January.
    13. Kozarski, R., 2013. "Pricing and hedging in the VIX derivative market," Other publications TiSEM 221fefe0-241e-4914-b6bd-c, Tilburg University, School of Economics and Management.
    14. Jin Zhang & Yi Xiang, 2008. "The implied volatility smirk," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 263-284.
    15. Li, Junye & Favero, Carlo & Ortu, Fulvio, 2012. "A spectral estimation of tempered stable stochastic volatility models and option pricing," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3645-3658.
    16. Gonçalo Faria & João Correia-da-Silva, 2014. "A closed-form solution for options with ambiguity about stochastic volatility," Review of Derivatives Research, Springer, vol. 17(2), pages 125-159, July.
    17. 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.
    18. F. Cacace & A. Germani & M. Papi, 2019. "On parameter estimation of Heston’s stochastic volatility model: a polynomial filtering method," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 503-525, December.
    19. Andreas Kaeck & Carol Alexander, 2013. "Stochastic Volatility Jump†Diffusions for European Equity Index Dynamics," European Financial Management, European Financial Management Association, vol. 19(3), pages 470-496, June.
    20. Kaeck, Andreas, 2013. "Asymmetry in the jump-size distribution of the S&P 500: Evidence from equity and option markets," Journal of Economic Dynamics and Control, Elsevier, vol. 37(9), pages 1872-1888.

    More about this item

    Keywords

    Stocks; Rate of return;

    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:fip:fedgfe:2009-40. 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: Ryan Wolfslayer ; Keisha Fournillier (email available below). General contact details of provider: https://edirc.repec.org/data/frbgvus.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.