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Modelling and Estimation for Bivariate Financial Returns

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  • Thomas Fung
  • Eugene Seneta

Abstract

Maximum likelihood estimates are obtained for long data sets of bivariate financial returns using mixing representation of the bivariate (skew) Variance Gamma (VG) and two (skew) t distributions. By analysing simulated and real data, issues such as asymptotic lower tail dependence and competitiveness of the three models are illustrated. A brief review of the properties of the models is included. The present paper is a companion to papers in this journal by Demarta & McNeil and Finlay & Seneta. Des estimateurs maximum de vraisemblance sont obtenus pour de longues séries bivariées de rendements financiers modélisées à partir d'un mélange (asymétrique) de type Variance‐Gamma et de deux mélanges (asymétriques) de type Student. L'analyse de données simulées et réelles permet d'illustrer quelques‐uns des aspects asymptotiques de ces trois modèles, tels que les dépendances asymptotiques des extrêmes dans la queue gauche, et leurs performances. Un bref compte‐rendu des propriétés de ces modèles est également inclus. Le présent travail accompagne et complète les articles de Demarta et McNeil (2005) et de Finlay et Seneta (2008) parus dans la même revue.

Suggested Citation

  • Thomas Fung & Eugene Seneta, 2010. "Modelling and Estimation for Bivariate Financial Returns," International Statistical Review, International Statistical Institute, vol. 78(1), pages 117-133, April.
    • Handle: RePEc:bla:istatr:v:78:y:2010:i:1:p:117-133
    DOI: 10.1111/j.1751-5823.2010.00106.x
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    References listed on IDEAS

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    1. Richard Finlay & Eugene Seneta, 2008. "Stationary‐Increment Variance‐Gamma and t Models: Simulation and Parameter Estimation," International Statistical Review, International Statistical Institute, vol. 76(2), pages 167-186, August.
    2. Kjersti Aas & Ingrid Hobaek Haff, 2006. "The Generalized Hyperbolic Skew Student's t-Distribution," Journal of Financial Econometrics, Oxford University Press, vol. 4(2), pages 275-309.
    3. Fung, Thomas & Seneta, Eugene, 2010. "Tail dependence for two skew t distributions," Statistics & Probability Letters, Elsevier, vol. 80(9-10), pages 784-791, May.
    4. Elisa Luciano & Wim Schoutens, 2006. "A multivariate jump-driven financial asset model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 385-402.
    5. Ines Fortin & Christoph Kuzmics, 2002. "Tail‐dependence in stock‐return pairs," Intelligent Systems in Accounting, Finance and Management, John Wiley & Sons, Ltd., vol. 11(2), pages 89-107, April.
    6. Thomas Fung & Eugene Seneta, 2010. "Tail dependence and skew distributions," Quantitative Finance, Taylor & Francis Journals, vol. 11(3), pages 327-333.
    7. Rafael Schmidt, 2002. "Tail dependence for elliptically contoured distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 55(2), pages 301-327, May.
    8. Banachewicz, Konrad & van der Vaart, Aad, 2008. "Tail dependence of skewed grouped t-distributions," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2388-2399, October.
    9. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    10. Branco, Márcia D. & Dey, Dipak K., 2001. "A General Class of Multivariate Skew-Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 99-113, October.
    11. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
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    Cited by:

    1. Zhang, Qingzhao & Li, Deyuan & Wang, Hansheng, 2013. "A note on tail dependence regression," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 163-172.
    2. Boris Buchmann & Benjamin Kaehler & Ross Maller & Alexander Szimayer, 2015. "Multivariate Subordination using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing," Papers 1502.03901, arXiv.org, revised Oct 2016.
    3. Fung, Thomas & Wang, Joanna J.J. & Seneta, Eugene, 2013. "Contaminated Variance–Mean mixing model," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 258-267.
    4. Buchmann, Boris & Kaehler, Benjamin & Maller, Ross & Szimayer, Alexander, 2017. "Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2208-2242.
    5. Roman V. Ivanov, 2023. "On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model," Risks, MDPI, vol. 11(6), pages 1-23, June.
    6. Vilca, Filidor & Balakrishnan, N. & Zeller, Camila Borelli, 2014. "Multivariate Skew-Normal Generalized Hyperbolic distribution and its properties," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 73-85.
    7. Fung, Thomas & Seneta, Eugene, 2011. "The bivariate normal copula function is regularly varying," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1670-1676, November.
    8. Liseo, Brunero & Parisi, Antonio, 2013. "Bayesian inference for the multivariate skew-normal model: A population Monte Carlo approach," Computational Statistics & Data Analysis, Elsevier, vol. 63(C), pages 125-138.
    9. Boris Buchmann & Kevin W. Lu & Dilip B. Madan, 2018. "Calibration for Weak Variance-Alpha-Gamma Processes," Papers 1801.08852, arXiv.org, revised Jul 2018.
    10. Thomas Fung & Eugene Seneta, 2023. "On Familywise Error Rate Cutoffs under Pairwise Exchangeability," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-13, June.
    11. Gardes, Laurent & Girard, Stéphane, 2015. "Nonparametric estimation of the conditional tail copula," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 1-16.
    12. Thomas Fung & Joanna J.J. Wang & Eugene Seneta, 2014. "The Deviance Information Criterion in Comparison of Normal Mixing Models," International Statistical Review, International Statistical Institute, vol. 82(3), pages 411-421, December.
    13. Olcay Arslan, 2015. "Variance-mean mixture of the multivariate skew normal distribution," Statistical Papers, Springer, vol. 56(2), pages 353-378, May.
    14. Boris Buchmann & Kevin W. Lu & Dilip B. Madan, 2019. "Calibration for Weak Variance-Alpha-Gamma Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1151-1164, December.

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