Abstract
We investigate the behaviour of the maximum likelihood estimator (MLE) for stochastic volatility jump-diffusion models commonly used in financial risk management. A simulation study shows the practical conditions under which the MLE behaves according to theory. In an extensive empirical study based on nine indices and more than 6000 individual stocks, we nonetheless find that the MLE is unable to replicate key higher moments. We then introduce a moment-targeted MLE – robust to model misspecification – and revisit both simulation and empirical studies. We find it performs better than the MLE, improving the management of financial risk.
Funding source: Simon Fraser University
Funding source: Natural Science and Engineering Research Council of Canada
Acknowledgment
The authors would like to thank Louis Arsenault-Mahjoubi and Geneviève Gauthier for their helpful suggestions and comments. Bégin wishes to acknowledge the financial support of the Natural Science and Engineering Research Council of Canada (NSERC) and Simon Fraser University. Boudreault also thanks the financial support of NSERC. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Research funding: This work was supported by Simon Fraser University and Natural Science and Engineering Research Council of Canada.
References
Aït-Sahalia, Y., C. Li, and C. X. Li. 2021. “Closed-form Implied Volatility Surfaces for Stochastic Volatility Models with Jumps.” Journal of Econometrics 222 (1): 364–92. https://doi.org/10.1016/j.jeconom.2020.07.006.Search in Google Scholar
Amaya, D., J.-F. Bégin, and G. Gauthier. 2022. “The Informational Content of High-Frequency Option Prices.” Management Science 68 (3): 2166–201. https://doi.org/10.1287/mnsc.2020.3949.Search in Google Scholar
Andersen, T. G., and B. E. Sørensen. 1996. “GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study.” Journal of Business & Economic Statistics 14 (3): 328–52, https://doi.org/10.1080/07350015.1996.10524660.Search in Google Scholar
Artzner, P., F. Delbaen, J. Eber, and D. Heath. 1997. “Thinking Coherently.” Risk 10: 68–71.Search in Google Scholar
Bakshi, G., C. Cao, and Z. Chen. 1997. “Empirical Performance of Alternative Option Pricing Models.” The Journal of Finance 52 (5): 2003–49. https://doi.org/10.1111/j.1540-6261.1997.tb02749.x.Search in Google Scholar
Bardgett, C., E. Gourier, and M. Leippold. 2019. “Inferring Volatility Dynamics and Risk Premia from the S&P 500 and Vix Markets.” Journal of Financial Economics 131 (3): 593–618. https://doi.org/10.1016/j.jfineco.2018.09.008.Search in Google Scholar
Bartolucci, F., and G. De Luca. 2001. “Maximum Likelihood Estimation of a Latent Variable Time-Series Model.” Applied Stochastic Models in Business and Industry 17 (1): 5–17. https://doi.org/10.1002/asmb.426.Search in Google Scholar
Bates, D. S. 1996. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” Review of Financial Studies 9 (1): 69–107. https://doi.org/10.1093/rfs/9.1.69.Search in Google Scholar
Bates, D. S. 2000. “Post-’87 Crash Fears in the S&P 500 Futures Option Market.” Journal of Econometrics 94 (1): 181–238, https://doi.org/10.1016/s0304-4076(99)00021-4.Search in Google Scholar
Bates, D. S. 2006. “Maximum Likelihood Estimation of Latent Affine Processes.” Review of Financial Studies 19 (3): 909–65. https://doi.org/10.1093/rfs/hhj022.Search in Google Scholar
Bégin, J.-F., and M. Boudreault. 2020. “Likelihood Evaluation of Jump-Diffusion Models Using Deterministic Nonlinear Filters.” Journal of Computational & Graphical Statistics 30: 452–66. https://doi.org/10.1080/10618600.2020.1840995.Search in Google Scholar
Bégin, J.-F., D. Amaya, G. Gauthier, and M.-E. Malette. 2020. “On the Estimation of Jump-Diffusion Models Using Intraday Data: A Filtering-Based Approach.” SIAM Journal on Financial Mathematics 11 (4): 1168–208. https://doi.org/10.1137/19m1266915.Search in Google Scholar
Brandt, M. W., and P. Santa-Clara. 2002. “Simulated Likelihood Estimation of Diffusions with an Application to Exchange Rate Dynamics in Incomplete Markets.” Journal of Financial Economics 63 (2): 161–210. https://doi.org/10.1016/s0304-405x(01)00093-9.Search in Google Scholar
Christoffersen, P. F. 1998. “Evaluating Interval Forecasts.” International Economic Review 39 (4): 841–62. https://doi.org/10.2307/2527341.Search in Google Scholar
Christoffersen, P., K. Jacobs, and K. Mimouni. 2010. “Volatility Dynamics for the S&P 500: Evidence from Realized Volatility, Daily Returns, and Option Prices.” Review of Financial Studies 23 (8): 3141–89. https://doi.org/10.1093/rfs/hhq032.Search in Google Scholar
Christoffersen, P., K. Jacobs, and C. Ornthanalai. 2012. “Dynamic Jump Intensities and Risk Premiums: Evidence from S&P 500 Returns and Options.” Journal of Financial Economics 106 (3): 447–72. https://doi.org/10.1016/j.jfineco.2012.05.017.Search in Google Scholar
Clements, A., S. Hurn, and S. White. 2006. “Estimating Stochastic Volatility Models Using a Discrete Non-linear Filter.” Working Paper.Search in Google Scholar
Cont, R. 2001. “Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues.” Quantitative Finance 1 (2): 223–36. https://doi.org/10.1088/1469-7688/1/2/304.Search in Google Scholar
Danielsson, J. 1994. “Stochastic Volatility in Asset Prices Estimation with Simulated Maximum Likelihood.” Journal of Econometrics 64 (1–2): 375–400. https://doi.org/10.1016/0304-4076(94)90070-1.Search in Google Scholar
Das, S. R., and R. K. Sundaram. 1999. “Of Smiles and Smirks: A Term Structure Perspective.” Journal of Financial and Quantitative Analysis 34 (2): 211–39. https://doi.org/10.2307/2676279.Search in Google Scholar
Duffie, D., J. Pan, and K. Singleton. 2000. “Transform Analysis and Asset Pricing for Affine Jump-Diffusions.” Econometrica 68 (6): 1343–76. https://doi.org/10.1111/1468-0262.00164.Search in Google Scholar
Engle, R., and J. Mezrich. 1996. “GARCH for Groups.” Risk 9 (8): 36–40.Search in Google Scholar
Eraker, B. 2001. “MCMC Analysis of Diffusion Models with Application to Finance.” Journal of Business & Economic Statistics 19 (2): 177–91. https://doi.org/10.1198/073500101316970403.Search in Google Scholar
Eraker, B., M. Johannes, and N. Polson. 2003. “The Impact of Jumps in Volatility and Returns.” The Journal of Finance 58 (3): 1269–300. https://doi.org/10.1111/1540-6261.00566.Search in Google Scholar
Francq, C., L. Horváth, and J.-M. Zakoïan. 2011. “Merits and Drawbacks of Variance Targeting in GARCH Models.” Journal of Financial Econometrics 9 (4): 619–56. https://doi.org/10.1093/jjfinec/nbr004.Search in Google Scholar
Fridman, M., and L. Harris. 1998. “A Maximum Likelihood Approach for Non-Gaussian Stochastic Volatility Models.” Journal of Business & Economic Statistics 16 (3): 284–91, https://doi.org/10.2307/1392504.Search in Google Scholar
Gill, P. E., W. Murray, and M. H. Wright. 1981. Practical Optimization. New York: Academic Press.Search in Google Scholar
Gordon, N. J., D. J. Salmond, and A. F. Smith. 1993. “Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation.” In IEEE Proceedings F (Radar and Signal Processing), 107–13.10.1049/ip-f-2.1993.0015Search in Google Scholar
Harvey, A. C., and N. Shephard. 1996. “Estimation of an Asymmetric Stochastic Volatility Model for Asset Returns.” Journal of Business & Economic Statistics 14 (4): 429–34. https://doi.org/10.1080/07350015.1996.10524672.Search in Google Scholar
Harvey, A., E. Ruiz, and N. Shephard. 1994. “Multivariate Stochastic Variance Models.” The Review of Economic Studies 61 (2): 247–64. https://doi.org/10.2307/2297980.Search in Google Scholar
Heston, S. 1993. “A Closed-form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” Review of Financial Studies 6 (2): 327. https://doi.org/10.1093/rfs/6.2.327.Search in Google Scholar
Hull, J. C. 2018. Options, Futures, and Other Derivatives, 10 ed. New York: Pearson.Search in Google Scholar
Hurn, A. S., K. A. Lindsay, and A. J. McClelland. 2015. “Estimating the Parameters of Stochastic Volatility Models Using Option Price Data.” Journal of Business & Economic Statistics 33 (4): 579–94. https://doi.org/10.1080/07350015.2014.981634.Search in Google Scholar
Jacquier, E., N. G. Polson, and P. E. Rossi. 1994. “Bayesian Analysis of Stochastic Volatility Models.” Journal of Business & Economic Statistics 12 (4): 371–89. https://doi.org/10.2307/1392199.Search in Google Scholar
Johannes, M., R. Kumar, and N. G. Polson. 1999. “State Dependent Jump Models: How Do Us Equity Indices Jump.” Working Paper.Search in Google Scholar
Johannes, M. S., N. G. Polson, and J. R. Stroud. 2009. “Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices.” Review of Financial Studies 22 (7): 2759–99. https://doi.org/10.1093/rfs/hhn110.Search in Google Scholar
Julier, S., and J. Uhlmann. 1997. “A New Extension of the Kalman Filter to Nonlinear Systems.” In SPIE Proceedings Series, 182–93. Society of Photo-Optical Instrumentation Engineers.10.1117/12.280797Search in Google Scholar
Kitagawa, G. 1987. “Non-Gaussian State-Space Modeling of Nonstationary Time Series.” Journal of the American Statistical Association 82 (400): 1032–41. https://doi.org/10.2307/2289375.Search in Google Scholar
Kristensen, D., and O. Linton. 2004. “Consistent Standard Errors for Target Variance Approach to GARCH Estimation.” Econometric Theory 20 (5): 990–3. https://doi.org/10.1017/s0266466604225099.Search in Google Scholar
Kupiec, P. H. 1995. “Techniques for Verifying the Accuracy of Risk Measurement Models.” Technical Report. Division of Research and Statistics, Division of Monetary Affairs, Federal Reserve Board.10.3905/jod.1995.407942Search in Google Scholar
Langrock, R., I. L. MacDonald, and W. Zucchini. 2012. “Some Nonstandard Stochastic Volatility Models and Their Estimation Using Structured Hidden Markov Models.” Journal of Empirical Finance 19 (1): 147–61. https://doi.org/10.1016/j.jempfin.2011.09.003.Search in Google Scholar
Li, J. 2013. “An Unscented Kalman Smoother for Volatility Extraction: Evidence from Stock Prices and Options.” Computational Statistics & Data Analysis 58: 15–26. https://doi.org/10.1016/j.csda.2011.06.001.Search in Google Scholar
Lord, R., R. Koekkoek, and D. Van Dijk. 2010. “A Comparison of Biased Simulation Schemes for Stochastic Volatility Models.” Quantitative Finance 10 (2): 177–94. https://doi.org/10.1080/14697680802392496.Search in Google Scholar
Maheu, J. M., and T. H. McCurdy. 2004. “News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns.” The Journal of Finance 59 (2): 755–93. https://doi.org/10.1111/j.1540-6261.2004.00648.x.Search in Google Scholar
Melino, A., and S. M. Turnbull. 1990. “Pricing Foreign Currency Options with Stochastic Volatility.” Journal of Econometrics 45 (1–2): 239–65. https://doi.org/10.1016/0304-4076(90)90100-8.Search in Google Scholar
Merton, R. C. 1976. “Option Pricing when Underlying Stock Returns Are Discontinuous.” Journal of Financial Economics 3 (1): 125–44. https://doi.org/10.1016/0304-405x(76)90022-2.Search in Google Scholar
Nelder, J. A., and R. Mead. 1965. “A Simplex Method for Function Minimization.” The Computer Journal 7 (4): 308–13. https://doi.org/10.1093/comjnl/7.4.308.Search in Google Scholar
Nelson, D. B. 1988. “The Time Series Behavior of Stock Market Volatility and Returns.” PhD thesis. Cambridge: Massachusetts Institute of Technology.Search in Google Scholar
Pan, J. 2002. “The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study.” Journal of Financial Economics 63 (1): 3–50. https://doi.org/10.1016/s0304-405x(01)00088-5.Search in Google Scholar
Pitt, M. K., S. Malik, and A. Doucet. 2014. “Simulated Likelihood Inference for Stochastic Volatility Models Using Continuous Particle Filtering.” Annals of the Institute of Statistical Mathematics 66 (3): 527–52. https://doi.org/10.1007/s10463-014-0456-y.Search in Google Scholar
Powell, M. J. 1968. “A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations.” Technical Report. Harwell: Atomic Energy Research Establishment.Search in Google Scholar
Sewell, M. 2011. “Characterization of Financial Time Series.” Working Paper.Search in Google Scholar
Shephard, N. 1993. “Fitting Nonlinear Time-Series Models with Applications to Stochastic Variance Models.” Journal of Applied Econometrics 8 (S1): S135–52. https://doi.org/10.1002/jae.3950080509.Search in Google Scholar
Taylor, S. J. 1986. Modelling Financial Time Series. New York: Wiley.Search in Google Scholar
Todorov, V., and G. Tauchen. 2011. “Volatility Jumps.” Journal of Business & Economic Statistics 29 (3): 356–71. https://doi.org/10.1198/jbes.2010.08342.Search in Google Scholar
Watanabe, T. 1999. “A Non-linear Filtering Approach to Stochastic Volatility Models with an Application to Daily Stock Returns.” Journal of Applied Econometrics 14 (2): 101–21. https://doi.org/10.1002/(sici)1099-1255(199903/04)14:2<101::aid-jae499>3.3.co;2-110.1002/(SICI)1099-1255(199903/04)14:2<101::AID-JAE499>3.3.CO;2-1Search in Google Scholar
Zhang, X., and P. W. Glynn. 2018. “Affine Jump-Diffusions: Stochastic Stability and Limit Theorems.” Working Paper.Search in Google Scholar
Zhang, Y., and S. Nadarajah. 2018. “A Review of Backtesting for Value at Risk.” Communications in Statistics – Theory and Methods 47 (15): 3616–39. https://doi.org/10.1080/03610926.2017.1361984.Search in Google Scholar
Supplementary Material
This article contains supplementary material (https://doi.org/10.1515/snde-2023-0028).
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Articles in the same Issue
- Frontmatter
- Research Articles
- Multiscale SUR Estimation of Systematic Risk
- A Simulation and Empirical Study of the Maximum Likelihood Estimator for Stochastic Volatility Jump-Diffusion Models
- Core Inflation Rate for China and the ASEAN-10 Countries: Smoothed Signal for Score-Driven Local Level Plus Scale Models
- Diversified Reward-Risk Parity in Portfolio Construction
- Time-Varying Parameter Four-Equation DSGE Model
- Does State Dependence Matter in Relation to Oil Price Shocks on Global Economic Conditions?