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Box
–
Cox realized asymmetric stochastic volatility model
Didit B. Nugrohoa and Takayuki Morimotob
aDepartment of Mathematics, Satya Wacana Christian University bDepartment of Mathematical Sciences, Kwansei Gakuin University
Abstract
This study proposes a class of non-linear realized stochastic volatility model by applying the Box–Cox (BC) transformation, instead of the logarithmic transformation, to the realized estimator. The proposed models are fitted to daily returns and realized kernel of six stocks: SP500, FTSE100, Nikkei225, Nasdaq100, DAX, and DJIA using an Markov Chain Monte Carlo (MCMC) Bayesian method, in which the Hamiltonian Monte Carlo (HMC) algorithm updates BC parameter and the Riemann manifold HMC (RMHMC) algorithm updates latent variables and other parameters that are unable to be sampled directly. Empirical studies provide evidence against log and raw versions of realized stochastic volatility model.
Keywords Box–Cox transformation, HMC, MCMC, realized stochastic volatility
1.
Introduction
Volatility of financial asset returns, defined as the standard deviation of log returns, has played a major role in many financial applications such as option pricing, portfolio allocation, and value-at-risk models. In the context of SV model, very closely related studies of joint models has been proposed by Dobrev & Szerszen (2010), Koopman & M. Scharth (2013), and Takahashi et al. (2009). Their model is known as the realized stochastic volatility (RSV) model.
Recently, the Takahashi et al. (2009)'s model has been extended by Takahashi et al. (2014) by applying a general non-linear bias correction in the realized variance (RV) measure. This study extends the asymmetric version of Takahashi et al. (2014)'s models, known as the realized asymmetric SV (R-ASV) model, by applying the BC transformation to the realized measure. Here we focus on the realized kernel (RK) measure that has some robustness to market microstructure noise. The realized kernel data behave very well and better than any available realized variance statistic Goncalves and Meddahi (2011).
Adding an innovation to obtain an RSV model substantially increase the difficulty in parameter estimation. Nugroho & Morimoto (2015) provided some comparison of estimation results between HMC-based samplers and multi-move Metropolis--Hastings sampler provided by Takahashi et al. (2019) and Takahashi et al. (2014), where the RMHMC sampler give the best performance in terms of inefficiency factor. Therefore, this study employs HMC-based samplers methods to estimate the parameters and latent variables in our proposed model when draws cannot be directly sampled.
2.
Box-Cox R-ASV model
measure of variance. The logarithmic transformation of RV is known to have better finite sample properties than RV. Goncalves and Meddahi (2011) provided a short overview of this transformation and suggested that the log transformation can be improved upon by choosing values of BC parameter other than zero (corresponding to the log transformation). Motivated by this result, we apply the BC transformation to the RV measure to obtain a class of R-ASV model called the BCR-ASV model that takes the following form:
ü = exp -›
Notice that this transformation contains the log transformation for RV (when G = 0) and the raw statistic (when G = 1) as special cases.
As in previous studies, in order to adopt a Gaussian nonlinear state space form where the error terms in both return and volatility equations are uncorrelated, we introduce a
3.
MCMC simulation in the BCR-ASV model
Let = )ü +/# , ½ = )ü +/# , _= )ℎ +/# , & = -G, ¼, , …. , and & =
m, †, Æ,T . By Bayes' theorem, the joint posterior distribution of &,_ given R and V is
& ,& ,_| , ½ = & ,& × |_ × ½|& ,_ × _|& , , (3) where & ,& is a joint prior distribution on parameter & ,& .
The MCMC method simulates a new value for each parameter in turn from its conditional posterior distribution assuming that the current values for the other parameters are true values. We employ the MCMC scheme as follows:
0) Initialize & ,& ,_ .
For steps (1), (3), and (4), we implement the same sampling scheme proposed in Nugroho & Morimoto (2014, 2015). In the following, we study the sampling scheme in step (2) only.
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which is not of standard form, and therefore we cannot sample G from it directly. In this case, we implement the HMC algorithm for sampling G because the metric tensor required to implement the RMHMC sampling scheme cannot be explicitly derived from the above distribution.
4.
Empirical results on real data sets
In this section, the BCR-ASV model and MCMC algorithm discussed in the previous section are applied to the daily returns and RK data for six different international stock indices: DAX, DJIA, FTSE100, Nasdaq100, Nikkei225, and SP500. These data sets were downloaded from Oxford-Man Institute ``realised library" and range from January 2000 to December 2013. Analyses are presented for the sampling efficiency, key parameters that build the extension models.
We run the MCMC algorithm for 15,000 iterations but discard the first 5,000 draws. From the resulting 10,000 draws which have passed the Geweke (1992)'s convergence test for each parameter, we calculate the posterior mean, standard deviation (SD), and 95% highest posterior density (HPD) interval. Summary of the posterior simulation results for G are presented in Table 1.
Table 1. Summary of the posterior sample of parameter G in the BCR-ASV model for six data sets from January 2000 to December 2013.
Statistic DAX DJIA FTSE100 Nasdaq100 Nikkei225 SP500
Mean –0.0406 0.1226 0.1572 0.0420 0.0503 0.1149
SD 0.0104 0.0099 0.0096 0.0096 0.0128 0.0094
LB –0.0609 0.1022 0.1385 0.0242 0.0241 0.0968
UB –0.0202 0.1409 0.1765 0.0617 0.0741 0.1340
Note: LB and UB denote lower and upper bounds of 95% HPD interval, respectively.
The posterior means of G are positive for DJIA, FTSE100, Nasdaq100, Nikkei225, and SP500 and negative for DAX. In all cases, the 95% HPD interval of G exclude 0 (the log transformation) or 1 (the raw version). Although not reported, we find that even 99% HPD interval of G exclude 0 or 1, providing significant evidence to transform the RV series using the BC transformation against both the logarithmic transformation and no transformation cases.
Figure 1. Marginal densities of the log RV and BC-RV for six different international stock indices.
5.
Conclusions and extensions
This study presented an extension to the R-ASV models, in which the Box-Cox transformation is applied to realized measure. The HMC and RMHMC algorithms were proposed to implement the MCMC method for updating several parameters and the latent variables in the proposed model, taking advantage of updating the entire latent volatility at once. The model and sampling algorithms were applied to the daily returns and realized kernels of six international stock indices. The estimation results showed very strong evidence supporting the Box-Cox transformation of realized measure. The model can be extended by assuming a non-Gaussian density of return errors.
References
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SWUP Takahashi, M., Omori, Y., & Watanabe, T. (2014). Volatility and quantile forecasts by realized
