High-low range in GARCH models of stock return volatility

ABSTRACT

We suggest a simple and general way to improve the GARCH volatility models using the intraday range between the highest and the lowest price to proxy volatility. We illustrate the method by modifying a GARCH(1,1) model to a range-GARCH(1,1) model. Our empirical analysis conducted on stocks, stock indices and simulated data shows that the range-GARCH(1,1) model performs significantly better than the standard GARCH(1,1) model both in terms of in-sample fit and out-of-sample forecasting ability.

KEYWORDS:

• Volatility
• high
• low
• range
• GARCH

JEL CLASSIFICATION:

• C22
• G17

Acknowledgements

I would like to thank Jonas Andersson, Milan Bašta, Ray Chou, Stein-Erik Fleten, Ove Hetland, Einar Cathrinus Kjenstad, Svein Olav Krakstad, Lukáš Lafférs, Gudbrand Lien and Kjell Nyborg for helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ For most of the assets, mean daily return is much smaller than its standard deviation and therefore can be considered equal to zero. In this article, we assume that it is indeed zero. This assumption not only makes further analysis simpler, but it actually helps to estimate volatility more precisely. In the words of Poon and Granger (2003): ‘The statistical properties of sample mean make it a very inaccurate estimate of the true mean, especially for small samples, taking deviations around zero instead of the sample mean typically increases volatility forecast accuracy’.

² Even though the GARCH(1,1) is a very simple model, it still works surprisingly well in comparison with much more complex volatility models (Hansen and Lunde 2005).

³ By rolling window forecasting with window size 100 we mean that we use the first 100 observations to forecast volatility on the 101, then we use observations 2–101 to forecast volatility for day 102 and so on.

⁴ In our data, the DM test statistic never lies above 95-percentile.

⁵ However, there is on difference worth mentioning. The Parkinson volatility estimator estimates the volatility only for the open-to-close period. If we estimate RGARCH model on close-to-close returns, we must be careful with interpretation of the $\alpha$-coefficient in the RGARCH model. As long as opening jumps are present, the Parkinson volatility estimator underestimates volatility of daily returns,

(25) $E\left(\widehat{{\sigma }_P^2}\right)<E(r^2)={\sigma }^2.$(25)

As a result, the estimated coefficient $\alpha$ will be larger to balance this bias in $\widehat{{\sigma }_P^2}.$ This intuition can explain one seemingly surprising result. The RGARCH model estimated on the close-to-close data typically yield coefficients $\alpha$ and $\beta$ such that $\alpha +\beta >1,$ even though estimation of the standard GARCH(1,1) model yields coefficients $\alpha$ and $\beta$ such that $\alpha +\beta <1.$ However, as we just explained, these $\alpha$-coefficients are not directly comparable in presence of opening jumps. We illustrate this on a simple example. If we specify GARCH(1,1) in the following form:

${\sigma }_t^2=\omega +\alpha \frac{r_{t-1}^2}{2}+\beta {\sigma }_{t-1}^2,$

then the estimated coefficient $\alpha$ will be exactly twice as large as when we estimate Equation (5). Therefore, if the RGARCH model is estimated on the close-to-close returns, the coefficient $\alpha$ does not have the same interpretation as in standard GARCH models. Even though we expect $\alpha$ to increase and $\beta$ to decrease, if we use close-to-close returns, we must focus on the coefficient $\beta$ only. The coefficient $\beta$ will change only because a less noisy volatility proxy is used, whereas change in coefficient $\alpha$ is caused by both high precision and bias of the Parkinson volatility estimator.

⁶ Components of stock indices change over time. These stocks were DJI components on 1 January 2009.

⁷ Since historical data for KFT (component of DJI) are not available for the complete period, we use its competitor CAG instead.

⁸ A comparison of the forecasted volatility with squared returns will penalize the volatility forecast whenever the squared return and volatility forecast differ, even if the volatility forecast was perfect. Moreover, when we have two models and one of them forecasts volatility to be ${\sigma }^2={0.1}^2$ on the day when the stock return is $r=1$ and the second model forecasts volatility to be ${\sigma }^2=3^2$ on the day when stock return is $r=\sqrt{10},$ then MSE (RMSE) will favour the first model $({\left({0.1}^2-1^2\right)}^2<{\left(10-3^2\right)}^2),$ even though the probability of the return $r=1$ being drawn from the distribution $N(0,{0.1}^2)$ is more than 10⁴°-times smaller than probability of the return $r=\sqrt{10}$ being drawn from the distribution $N(0,3^2).$

⁹ Heber et al. (2009).

¹⁰ We use 100,000 discrete steps for the approximation of the continuous Brownian motion.