A new generalized volatility proxy via the stochastic volatility model

ABSTRACT

This article proposes power transformation of absolute returns as a new proxy of latent volatility in the stochastic model. We generalize absolute returns as a proxy for volatility in that we place no restriction on the power of absolute returns. An empirical investigation on the bias, mean square error and relative bias is carried out for the proposed proxy. Simulation results show that the new estimator exhibiting negligible bias appears to be more efficient than the unbiased estimator with high variance.

KEYWORDS:

• Volatility
• stochastic volatility
• relative bias
• mean square error

JEL CLASSIFICATION:

• C32
• C52

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data

Supplemental data for this article can be accessed here.

Notes

¹ Kalman filter proposed by Kalman (1960) through a state space representation is a common tool for the SV model. Moreover, empirical studies of Andersen, Benzoni, and Lund (2002) and Eraker, Johannes, and Polson (2003) improve model fit of the SV model by incorporating price jumps. Recently, Harvey (2013) proposes the dynamic conditional score model which is close to SV specification, where the volatility is treated as an unobserved stochastic process.

² Note that Giles (2008) analysis is a special case in our model when we assume that k = 1.

³ The use of these ranges of the parameters is valid for the following reasons: (1) As discussed in Taylor (2005), the estimates of these parameters given by previous empirical studies are usually between the ranges. (2) This study chooses the same parameter ranges used in Giles (2008) for comparison with his study.

⁴ For simplicity, k takes such values, although in principle k can be extended to accommodate any values.

⁵ We find that the results shown in this study are not sensitive to other datasets constructed by different parameter values. The complete datasets used in this study and all of our empirical results are available in the Supplemental data.

⁶ We also find that for the typical parameter value ranges, as degrees of freedom increase the MSE of $|r_t{|}^k$ monotonically decreases.

⁷ We also obtain figures by employing different parameter values in Cases 1, 2 and 3. These figures are omitted to save space but the Supplementary data provides these figures.

⁸ Note: Additional information for this article can be found in the Supplementary data at http://facultypages.morris.umn.edu/ jongmink/research/OnlineAppendix.pdf