Modeling log-volatility with zero returns: empirical evidence for asymmetric SV and log-GARCH models

Document Type : Research Article

Authors

1 Laboratory of mathematics, computer science and applications (LMCSA), FST Mohammedia, Hassan II university, 20650 Casablanca, Morocco

2 Department of Mathematics and Statistics, College of Engineering, Abu Dhabi University, 59911 Abu Dhabi, UAE

Abstract

In this work, we address the challenges posed by zero returns in both stochastic volatility (SV) and log-GARCH models in their asymmetric form. Building upon EM imputation for handling zero returns, we propose a unified approach that enhances parameter estimation robustness for both model classes. Specifically, we employ the Quasi-Maximum Likelihood (QML) estimation, incorporating the Kalman filter for both asymmetric SV and asymmetric log-GARCH models, to ensure robust parameter estimation even in the presence of zero returns. By comparing the performance of these models under our proposed framework, we provide new insights into their relative strengths in capturing the asymmetric volatility dynamics in the presence of zero returns. This contribution extends the existing literature by proposing a computational framework applicable to such models, based on a logarithmic specification of volatility.

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References

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and
Mathematical Tables, Vol. 55, Courier Corp., North Chelmsford, MA, 1965.
[2] B.D.O. Anderson, J.B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ, 1979.
[3] M. Asai, A new method to estimate stochastic volatility models: A LOG-GARCH approach, J.
Japan Statist. Soc. 28 (1998) 101–114.
[4] S. Bhatnagar, N. Hemachandra, V.K. Mishra, Stochastic approximation algorithms for constrained
optimization via simulation, ACM Trans. Model. Comput. Simul. 21 (2011) 1–22.
[5] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econometrics 31 (1986)
307–327.
[6] P. Chirico, Iterative QML estimation for asymmetric stochastic volatility models, Stat. Methods
Appl. (2024).
[7] P. Christoffersen, K. Jacobs, C. Ornthanalai, GARCH option valuation: Theory and evidence, J.
Derivatives (2013).
[8] C. Francq, O. Wintenberger, J.-M. Zakoian, GARCH models without positivity constraints: Expo
nential or log GARCH?, J. Econometrics 177 (2013) 34–46.
[9] C. Francq, G. Sucarrat, An equation-by-equation estimator of a multivariate log-GARCH-X model
of financial returns, J. Multivariate Anal. 153 (2017) 16–32.
[10] C. Francq, J.-M. Zakoian, GARCH Models: Structure, Statistical Inference and Financial Applica
tions, Wiley, Hoboken, NJ, 2019.
[11] J. Geweke, Modelling the persistence of conditional variances: A comment, Econometric Rev. 5
(1986) 57–61.
[12] E. Ghysels, A.C. Harvey, E. Renault, Stochastic volatility, in: G.S. Maddala, C.R. Rao (Eds.),
Handbook of Statistics, Vol. 14, Elsevier, Amsterdam, 1996, pp. 119–191.
[13] L.R. Glosten, R. Jagannathan, D.E. Runkle, On the relation between the expected value and the
volatility of the nominal excess return on stocks, J. Finance 48 (1993) 1779–1801.
[14] A.C. Harvey, Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge Univ.
Press, Cambridge, 1990.
[15] A.C. Harvey, E. Ruiz, N. Shephard, Multivariate stochastic variance models, Rev. Econom. Stud.
61 (1994) 247–264
[16] A.C. Harvey, N. Shephard, Estimation of an asymmetric stochastic volatility model for asset re
turns, J. Bus. Econom. Statist. 14 (1996) 429–434.
[17] A. Melino, S.M. Turnbull, Pricing foreign currency options with stochastic volatility, J. Economet
rics 45 (1990) 239–265.
[18] S.G. Pantula, Modeling the persistence of conditional variances: A comment, Econometric Rev. 5
(1986) 71–74.
[19] E. Ruiz, Quasi-maximum likelihood estimation of stochastic volatility models, J. Econometrics 63
(1994) 289–306.
[20] S. Kim, N. Shephard, S. Chib, Stochastic volatility: Likelihood inference and comparison with
ARCHmodels, Rev. Econom. Stud. 65 (1998) 361–393.
[21] A. Settar, M. Badaoui, On the treatment of zero returns in the estimation of log-GARCH model:
Empirical study, IOP Conf. Ser. Mater. Sci. Eng. 1099 (2021) 012029.
[22] A. Settar, N.I. Fatmi, M. Badaoui, Numerical estimation of log-GARCH model with relaxed mo
ment assumptions, J. Phys. Conf. Ser. 1743 (2021) 012036.
[23] J.C. Spall, Adaptive stochastic approximation by the simultaneous perturbation method, IEEE
Trans. Automat. Control 45 (2000) 1839–1853.
[24] G. Sucarrat, A. Escribano, The power log-GARCH model, Working Paper, Dept. of Economics,
Univ. Carlos III Madrid, 2010.
[25] G. Sucarrat, A. Escribano, Unbiased QML estimation of log-GARCH models in the presence of
zero returns, Working Paper, Dept. of Economics, Univ. Carlos III Madrid, 2013.
[26] G. Sucarrat, S. Grønneberg, A. Escribano, Estimation and inference in univariate and multivariate
log-GARCH-X models when the conditional density is unknown, Comput. Statist. Data Anal. 100
(2016) 582–594.
[27] G. Sucarrat, The log-GARCH model via ARMA representations, in: Financial Mathematics,
Volatility and Covariance Modelling, Routledge, London, 2019, pp. 336–359.
[28] S.J. Taylor, Modelling Financial Time Series, Wiley, Chichester, 1986.
[29] S.J. Taylor, Modeling stochastic volatility: A review and comparative study, Math. Finance 4 (1994)
183–204.
[30] N. Vijayan, L.A. Prashanth, Smoothed functional-based gradient algorithms for off-policy rein
forcement learning: A non-asymptotic viewpoint, Syst. Control Lett. 155 (2021) 104988.
[31] A. Wills, T.B. Sch¨on, B. Ninness, Estimating state-space models in innovations form using the
expectation maximisation algorithm, in: Proc. 49th IEEE Conf. Decis. Control (CDC), IEEE, 2010,
pp. 5524–5529
Journal of Mathematical Modeling
Volume 14, Issue 2
May 2026
Pages 787-801

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