Non-Gaussian Stochastic Volatility models: Laplace-variational Bayes inference
Published:
Dec 11, 2024
Volume:
22
Keywords:
Conditional volatility
Financial risks
Jumps
Laplace Approximations
Variational inference
Abstract
Stochastic volatility models are fundamental tools in finance for accurately estimating and managing risks due to their ability to accommodate a time-varying volatility structure. However, a relevant constraint within these models is the reliance on Gaussian processes to model the latent (log-)variance, which can limit their ability to effectively capture events such as sudden jumps or spikes in the latent volatility. To address this limitation, we propose in this short paper a non-Gaussian SV model utilizing an inference procedure that combines Laplace and Variational Bayes approximations. Our study showcases the advantages of this correction in modeling the conditional variance of Bitcoin's return series.
How to cite
João Pedro Nacinben, Márcio Laurini. Non-Gaussian Stochastic Volatility models: Laplace-variational Bayes inference. Brazilian Review of Finance, v. 22, n. 4, 2024. p. 13-25. DOI: 10.12660/rbfin.v22n4.2024.92572.
References
Andersen, T., & Sorensen, B. (1996). GMM estimation of a stochastic volatility model: A Monte Carlo study. *Journal of Business & Economic Statistics, 14*(3), 328–352. https://doi.org/10.1080/07350015.1996.10524660
Andersen, T., Sorensen, B., & Chung, H.-J. (1999). Efficient method of moments estimation of a stochastic volatility model: A Monte Carlo study. *Journal of Econometrics, 91*(1), 61–87. https://doi.org/10.1016/s0304-4076(98)00049-9
Barndorff-Nielsen, O. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. *Scandinavian Journal of Statistics, 24*(1), 1–13. https://doi.org/10.1111/1467-9469.00045
Cabral, R., Bolin, D., & Rue, H. (2023). Controlling the flexibility of non-Gaussian processes through shrinkage priors. *Bayesian Analysis, 18*(4), 1223–1246. https://doi.org/10.1214/22-ba1342
Cabral, R., Bolin, D., & Rue, H. (2024). Fitting latent non-Gaussian models using variational Bayes and Laplace approximations. *Journal of the American Statistical Association, 119*(548), 2983–2995. https://doi.org/10.1080/01621459.2023.2296704
Chaim, P., & Laurini, M. P. (2018). Volatility and return jumps in Bitcoin. *Economics Letters, 173*, 158–163. https://doi.org/10.1016/j.econlet.2018.10.011
Chaim, P., & Laurini, M. P. (2024). Bayesian inference for long memory stochastic volatility models. *Econometrics, 12*(4). https://doi.org/10.3390/econometrics12040035
de Zea Bermudez, P., Marín, J. M., Rue, H., & Veiga, H. (2021). Integrated nested Laplace approximations for threshold stochastic volatility models. *Econometrics and Statistics*. https://doi.org/10.1016/j.ecosta.2021.08.006
Gunawan, D., Kohn, R., & Nott, D. (2021). Variational Bayes approximation of factor stochastic volatility models. *International Journal of Forecasting, 37*(4), 1355–1375. https://doi.org/10.1016/j.ijforecast.2021.05.001
Harvey, A., Ruiz, E., & Shephard, N. (1994). Multivariate stochastic variance models. *The Review of Economic Studies, 61*(2), 247–264. https://doi.org/10.2307/2297980
Jordan, M. I., Ghahramani, Z., Jaakkola, T. S., & Saul, L. K. (1999). An introduction to variational methods for graphical models. *Machine Learning, 37*(2), 183–233. https://doi.org/10.1023/a:1007665907178
Kim, S., & Shephard, N. (1998). Stochastic volatility: likelihood inference and comparison with ARCH models. *The Review of Economic Studies, 65*(3), 361–393. https://doi.org/10.1111/1467-937x.00050
Martino, S., Aas, K., Lindqvist, O., Neef, L., & Rue, H. (2011). Estimating stochastic volatility models using integrated nested Laplace approximations. *European Journal of Finance, 17*(7), 487–503. https://doi.org/10.1080/1351847X.2010.495475
Nacinben, J. P. C. d. S. M., & Laurini, M. (2024). Multivariate stochastic volatility modeling via integrated nested Laplace approximations: A multifactor extension. *Econometrics, 12*(1). https://doi.org/10.3390/econometrics12010005
Polson, N., Rossi, P., & Jacquier, E. (1994). Bayesian analysis of stochastic volatility models. *Journal of Business & Economic Statistics, 20*(1), 69–87. https://doi.org/10.1198/073500102753410408
Rue, H., Martino, S., & Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. *Journal of the Royal Statistical Society, 71*(2), 319–392. https://doi.org/10.1111/j.1467-9868.2008.00700.x
Rue, H., Riebler, A., Sørbye, S. H., Illian, J. B., Simpson, D. P., & Lindgren, F. K. (2017). Bayesian computing with INLA: A review. *Annual Review of Statistics and its Application, 4*, 395–421. https://doi.org/10.1146/annurev-statistics-060116-054045
Sandmann, G., & Koopman, S. J. (1998). Estimation of stochastic volatility models via Monte Carlo maximum likelihood. *Journal of Econometrics, 87*(2), 271–301. https://doi.org/10.1016/s0304-4076(98)00016-5
Taylor, S. J. (1982). Financial returns modelled by the product of two stochastic processes – A study of daily sugar prices, 1961–79. In O. D. Anderson (Ed.), *Time Series Analysis: Theory and Practice, Volume 1* (pp. 203–226). Elsevier/North-Holland. https://doi.org/10.1093/oso/9780199257195.003.0003
Taylor, S. J. (1986). *Modelling Financial Time Series*. John Wiley & Sons.
Taylor, S. J. (1994). Modeling stochastic volatility: A review and comparative study. *Mathematical Finance, 4*(2), 183–204. https://doi.org/10.1111/j.1467-9965.1994.tb00057.x
Wang, Y., & Blei, D. M. (2019). Frequentist consistency of variational Bayes. *Journal of the American Statistical Association, 114*(527), 1147–1161. https://doi.org/10.1080/01621459.2018.1473776
Zhang, C., Bütepage, J., Kjellström, H., & Mandt, S. (2018). Advances in variational inference. *IEEE Transactions on Pattern Analysis and Machine Intelligence, 41*(8), 2008–2026. https://doi.org/10.1109/tpami.2018.2889774