Modeling the Variance Risk Premium of Equity Indices: The Role of Dependence and Contagion
Abstract
The variance risk premium (VRP) refers to the premium demanded for holding assets whose variance is exposed to stochastic shocks. This paper identifies a new modeling framework for equity indices and presents for the first time explicit analytical formulas for their VRP in a multivariate stochastic volatility setting, which includes multivariate non-Gaussian Ornstein--Uhlenbeck processes and Wishart processes. Moreover, we propose to incorporate contagion within the equity index via a multivariate Hawkes process and find that the resulting dynamics of the VRP represent a convincing alternative to the models studied in the literature up to date. We show that our new model can explain the key stylized facts of both equity indices and individual assets and their corresponding VRP, while some popular (multivariate) stochastic volatility models may fail.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Y. Ait-Sahalia, J. Cacho-Diaz, and R. J. Laeven, Modeling Financial Contagion Using Mutually Exciting Jump Processes, Tech. report, National Bureau of Economic Research, 2010.
2.
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Stud. Adv. Math. 93, Cambridge University Press, Cambridge, UK, 2004.
3.
E. Bacry, S. Delattre, M. Hoffmann, and J. F. Muzy, Modelling microstructure noise with mutually exciting point processes, Quant. Finance, 13 (2013), pp. 65--77.
4.
O. E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein--Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. Ser. B Stat. Methodol., 63 (2001), pp. 167--241.
5.
O. E. Barndorff-Nielsen and N. Shephard, Econometric analysis of realized volatility and its use in estimating stochastic volatility models, J. R. Stat. Soc. Ser. B Stat. Methodol., 64 (2002), pp. 253--280.
6.
O. E. Barndorff-Nielsen and R. Stelzer, Positive-definite matrix processes of finite variation, Probab. Math. Statist., 27 (2007), pp. 3--43.
7.
O. E. Barndorff-Nielsen and A. E. D. Veraart, Stochastic volatility of volatility and variance risk premia, J. Financial Econometrics, 11 (2013), pp. 1--46.
8.
T. Bollerslev, Modelling the coherence in short-run nominal exchange rates: A multivariate generalised ARCH model, Rev. Econom. Statist., 72 (1990), pp. 498--505.
9.
T. Bollerslev, J. Marrone, L. Xu, and H. Zhou, Stock return predictability and variance risk premia: Statistical inference and international evidence, J. Financial Quantitative Anal., 49 (2014), pp. 633--661.
10.
T. Bollerslev, G. Tauchen, and H. Zhou, Expected stock returns and variance risk premia, Rev. Financial Studies, 22 (2009), pp. 4463--4492.
11.
O. Bondarenko, Variance trading and market price of variance risk, J. Econometrics, 180 (2014), pp. 81--97.
12.
P. Carr and L. Wu, Variance risk premiums, Rev. Financial Studies, 22 (2009), pp. 1311--1341.
13.
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2003.
14.
J. Da Fonseca, M. Grasselli, and C. Tebaldi, A multifactor volatility Heston model, Quant. Finance, 8 (2008), pp. 591--604.
15.
I. Drechsler and A. Yaron, What's vol got to do with it, Rev. Financial Studies, 24 (2011), pp. 1--45.
16.
J. Driessen, P. Maenhout, and G. Vilkov, The price of correlation risk: Evidence from equity options, J. Finance, 64 (2009), pp. 1377--1406.
17.
D. Dufresne, The Integrated Square-Root Process, working paper, University of Melbourne, 2001.
18.
R. Engle, Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models, J. Bus. Econom. Statist., 20 (2002), pp. 339--350.
19.
E. Errais, K. Giesecke, and L. R. Goldberg, Affine point processes and portfolio credit risk, SIAM J. Financial Math., 1 (2010), pp. 642--665.
20.
C. Gouriéroux, Continuous time Wishart process for stochastic risk, Econometric Rev., 25 (2006), pp. 177--217.
21.
A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58 (1971), pp. 83--90.
22.
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.
23.
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Corundlehren Math. Wiss. 288, Springer-Verlag, Berlin, 1987.
24.
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New York, 1991.
25.
J. Muhle-Karbe, O. Pfaffel, and R. Stelzer, Option pricing in multivariate stochastic volatility models of ou type, SIAM J. Financial Math., 3 (2012), pp. 66--94.
26.
E. Nicolato and E. Venardos, Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type, Math. Finance, 13 (2003), pp. 445--466.
27.
P. Protter, Stochastic Integration and Differential Equations: Version 2.1, Stoch. Model. Appl. Probab. 21, Springer, New York, 2005.
28.
D. Serre, Matrices: Theory and Applications, Grad. Texts in Math. 216, Springer, New York, 2010.
29.
A. Sokol and R. N. Hansen, Exponential martingales and changes of measure for counting processes, Stoch. Anal. Appl., 33 (2015), pp. 823--843.
30.
V. Todorov, Variance risk-premium dynamics: The role of jumps, Rev. Financial Studies, 23 (2010), pp. 345--383.
Information & Authors
Information
Published In
SIAM Journal on Financial Mathematics
Pages: 382 - 417
DOI: 10.1137/15M1011822
ISSN (online): 1945-497X
Copyright
© 2016, Society for Industrial and Applied Mathematics.
History
Submitted: 9 March 2015
Accepted: 11 April 2016
Published online: 14 June 2016
Keywords
MSC codes
Authors
Funding Information
European Commissionhttp://dx.doi.org/10.13039/501100000780: PCIGII-GA-2012-321707
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- The dynamics of price jumps in the stock market: an empirical study on Europe and U.S.The European Journal of Finance, Vol. 28, No. 7 | 19 March 2020
- Geometric ergodicity of the multivariate COGARCH(1,1) processStochastics, Vol. 93, No. 7 | 19 November 2020
- A Gamma Ornstein–Uhlenbeck model driven by a Hawkes processMathematics and Financial Economics, Vol. 15, No. 4 | 24 March 2021
- Exact simulation of gamma-driven Ornstein–Uhlenbeck processes with finite and infinite activity jumpsJournal of the Operational Research Society, Vol. 72, No. 2 | 20 December 2019
- Geometric ergodicity of affine processes on conesStochastic Processes and their Applications, Vol. 130, No. 7 | 1 Jul 2020
- A Gamma Ornstein-Uhlenbeck Model Driven by a Hawkes ProcessSSRN Electronic Journal, Vol. 6 | 1 Jan 2019
- Stochastic Modelling of Energy Spot Prices by LSS ProcessesAmbit Stochastics | 2 November 2018
- The Dynamics of Price Jumps in the Stock Market: An Empirical Study on Europe and U.S.SSRN Electronic Journal, Vol. 74 | 1 Jan 2017
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].