A closed-form solution for the stochastic volatility model with applications on international stock markets

Abstract

This paper proposes a closed-form estimator for the stochastic volatility (SV) model. Compared to the usual maximum likelihood estimation (MLE), which is difficult to perform without appropriate approximations, the proposed method can be easily implemented and does not require the use of any numerical optimizer or starting values for iterations. Moreover, closed-form estimates can be supplied as initial values to MLE, for instance, conducted with a novel Laplace approximation. Denoted by MLE-C, this method consistently outperforms other estimators including the Markov chain Monte Carlo (MCMC). This is confirmed with simulation studies consisting of various combinations of true parameters and sample sizes. Our empirical data include daily returns of S&P 500, Nikkei 225 and DAX 100 over 2011–2020. The SV model estimated by MLE-C almost uniformly beats the popular GARCH counterparty, based on both the in-sample fit and out-of-sample forecasting criteria. Value-at-Risk analyses further demonstrate the capability of the SV model to accurately describe the tail behaviors of negative returns.

Keywords:

• Stochastic volatility
• closed-form solution
• maximum likelihood estimation
• value-at-risk analysis

Acknowledgements

We are grateful to the Macquarie University for their support. The author particularly thanks the Editor-in-Chief (Zhe George Zhang), Associate Editor and one anonymous referee for providing valuable and insightful comments on earlier drafts. The usual disclaimer applies.

Computational packages

We use the software R to perform the computations of all models. GARCH model is implemented by the fGARCH package. SV model with MLE and MCMC are run by the stochvol and stochvolTMB packages, respectively, with modifications.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Note that as discussed in Section 2.2, σ_h may be different from 1. We have examined the cases σ_h set to 0.5 and 2, and those changes will lead to robust results.

2 For the estimated $\widehat{\varphi }$ using closed-form solution, we follow D. Kristensen and Linton (2006) and let $\widehat{\varphi }$ equal to the simple average of ${\widehat{\rho }}_y(k)/{\widehat{\rho }}_y(k-1)$ (defined in Remark 4) with $k=1,2,$ and 3. When $\left|\widehat{\varphi }\right|\ge 1,$ and we let $ϵ=0.001.$

3 In addition to MLE-R, we also considered alternatives such as MLE based on starting values produced by a global optimizer via differential evolution, as discussed in Mullen et al. (2011). It is shown that although the overall accuracy of this approach slightly beats that of MLE-R, MLE-C is still the best performing method. Nevertheless, the computational time of MLE-C is much shorter than that of MLE with starting values supplied by differential evolution. Specific results are available upon request.