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Models for Heavy-tailed Asset Returns

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  • Szymon Borak
  • Adam Misiorek
  • RafaÅ‚ Weron

Abstract

Many of the concepts in theoretical and empirical finance developed over the past decades – including the classical portfolio theory, the Black- Scholes-Merton option pricing model or the RiskMetrics variance-covariance approach to VaR – rest upon the assumption that asset returns follow a normal distribution. But this assumption is not justified by empirical data! Rather, the empirical observations exhibit excess kurtosis, more colloquially known as fat tails or heavy tails. This chapter is intended as a guide to heavy-tailed models. We first describe the historically oldest heavy-tailed model – the stable laws. Next, we briefly characterize their recent lighter-tailed generalizations, the socalled truncated and tempered stable distributions. Then we study the class of generalized hyperbolic laws, which – like tempered stable distributions – can be classified somewhere between infinite variance stable laws and the Gaussian distribution. Finally, we provide numerical examples.

Suggested Citation

  • Szymon Borak & Adam Misiorek & RafaÅ‚ Weron, 2010. "Models for Heavy-tailed Asset Returns," SFB 649 Discussion Papers SFB649DP2010-049, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2010-049
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    Cited by:

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    6. Franziska Schulze, 2010. "Spatial Dependencies in German Matching Functions," SFB 649 Discussion Papers SFB649DP2010-054, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    7. Takashi Isogai, 2014. "Benchmarking of Unconditional VaR and ES Calculation Methods: A Comparative Simulation Analysis with Truncated Stable Distribution," Bank of Japan Working Paper Series 14-E-1, Bank of Japan.
    8. Adam Misiorek & Rafal Weron, 2010. "Heavy-tailed distributions in VaR calculations," HSC Research Reports HSC/10/05, Hugo Steinhaus Center, Wroclaw University of Technology.
    9. 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.
    10. Wyłomańska, Agnieszka & Chechkin, Aleksei & Gajda, Janusz & Sokolov, Igor M., 2015. "Codifference as a practical tool to measure interdependence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 412-429.
    11. Szymon Borak & Adam Misiorek & Rafał Weron, 2010. "Models for Heavy-tailed Asset Returns," SFB 649 Discussion Papers SFB649DP2010-049, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    12. Enno Mammen & Christoph Rothe & Melanie Schienle, 2010. "Nonparametric Regression with Nonparametrically Generated Covariates," SFB 649 Discussion Papers SFB649DP2010-059, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    13. Kenneth Bruninx & Erik Delarue & William D'haeseleer, 2013. "Statistical description of the error on wind power forecasts via a Lévy α-stable distribution," RSCAS Working Papers 2013/50, European University Institute.
    14. Greg Hannsgen, 2011. "Infinite-variance, Alpha-stable Shocks in Monetary SVAR: Final Working Paper Version," Economics Working Paper Archive wp_682, Levy Economics Institute.
    15. Ralf Sabiwalsky, 2010. "Executive Compensation Regulation and the Dynamics of the Pay-Performance Sensitivity," SFB 649 Discussion Papers SFB649DP2010-051, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    16. Joanna Janczura & Sebastian Orzel & Agnieszka Wylomanska, 2011. "Subordinated alpha-stable Ornstein-Uhlenbeck process as a tool for financial data description," HSC Research Reports HSC/11/03, Hugo Steinhaus Center, Wroclaw University of Technology.
    17. Janusz Gajda, 2012. "Modeling of short term interest rate based on tempered fractional Langevin equation," HSC Research Reports HSC/12/03, Hugo Steinhaus Center, Wroclaw University of Technology.
    18. Vladimir Panov, 2010. "Estimation of the signal subspace without estimation of the inverse covariance matrix," SFB 649 Discussion Papers SFB649DP2010-050, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

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    More about this item

    Keywords

    Heavy-tailed distribution; Stable distribution; Tempered stable distribution; Generalized hyperbolic distribution; Asset return; Random number generation; Parameter estimation;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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