Realized volatility measures

The standard realized volatility (SRV) over a certain time horizon T withN observations within T is defined according to the sample standard deviation as the square root of the

sample variance:¹²

where r_t is the logarithmic return at time t and ¯r is the mean return within the time horizon T (see, for example, Poon and Granger (2003)). Despite being a very popular volatility estimator, the SRV has several shortcomings that motivate the use of other realized measures.

Marquering and Verbeek (2004) argue that the true variance will be underestimated if returns are positively correlated. They use a realized measure that has a correction term, assuming that the daily return series within a month is appropriately described by an autoregressive process of order one. I use the square root of this variance estimator for the volatility analysis, calling itrealized volatility with autocorrelation correction (RVAC).

This measure is defined by¹³

RVAC(T) =

where ˆφ_T is the autocorrelation coefficient from an AR(1) model fitted to the N returns within the time horizon T.

If the sample variance is an unbiased estimator of the true variance, the SRV and RVAC are biased estimators of the true standard deviation due to Jensen’s inequality and a correction factor that depends on the return distribution would be necessary to eliminate the bias (Fleming (1998)). However, Fleming (1998) shows that the impact of the correction is very small. Therefore, no bias-corrected SRV will be considered in the following. Instead, I use another approach that is a direct and unbiased volatility estimator and thus serves as an appropriate extension of the robustness analysis. At the same time, it has a further

12This formula is usually written withoutN in the numerator at the beginning and thus estimates the average standard deviation for the returns at the given data frequency. To obtain the standard deviation for the whole time horizonT (one month here) the square root of time rule is applied by multiplying by

√N. This is also the case in some of the following formulas, which are usually defined without√ N.

13See Marquering and Verbeek (2004) for the underlying variance estimator. Given this definition, the RVAC is not identical to the SRV for an autocorrelation coefficient of zero becauseN−1 is in the divisor of the SRV and notN.

advantage compared to the SRV. Another drawback of the SRV estimator is its sensitivity to outliers caused by the squaring of returns. To circumvent this problem, Ederington and Guan (2006) propose taking the average absolute deviation of returns from the mean return.

Since they also define volatility as the standard deviation of returns, they make a further adjustment, assuming the returns to be normally distributed. Under this assumption, the approach further delivers an unbiased volatility estimator. The realized volatility based on the adjusted absolute deviation (RVAAD) is defined as

RVAAD(T) = 1

As discussed in Section 2.3, the mean absolute deviation seems to be a more intuitive measure of volatility. Therefore, I also want to consider the mean absolute deviation, refusing the adjustment necessary for the standard deviation. The disadvantage of measures based on this volatility definition compared to the standard deviation is the lack of general scaling rules. However, under the assumption of log-returns being not only independent and identically distributed but also normally distributed, the scaling with the square root of time rule leads to an appropriate estimator for the annualized mean absolute deviation.

Therefore, the square root of time rule is also applied here and allows for results that can be compared to the other measures. Therealized absolute deviation (RAD) is thus given by

A method similar to the RVAAD was introduced by Schwert (1990), whose volatility definition is again the standard deviation. Instead of adjusting the average absolute deviations from the mean, the author uses the residuals ˆu_t from a regression that regressed the daily log-returns within a month on 22 lagged returns, which cover approximately one month, and a dummy variableD representing the day of the week to capture differences

in mean returns:¹⁴ Therealized volatility based on adjusted residuals (RVAR) is than adjusted like the RVAAD:

RVAR(T) = 1

The last specification of realized volatilities that I use differs from the others in the amount of information used. Indeed, like all realized volatility estimators, it uses only price information from the relevant time horizon, but like many other estimators¹⁵—and in contrast to the SRV estimator—it takes also additional price information into account.

I use a relatively new estimator, which includes daily opening, closing, high, and low price information. Therealized volatility developed by Yang and Zhang (2000) (RVYZ) is given by¹⁶

Moreover, ¯o and ¯care the average o and c, respectively, within the time horizon T and k= ^α−1

α+^N+1_N−1. I follow the suggestion of Yang and Zhang (2000) and set α= 1.34.

14The only difference is that I use only five dummy variables for the day of the week because no trading on Saturday takes place in my sample.

15See, for example, Garman and Klass (1980), Parkinson (1980) and Rogers and Satchell (1991).

16Yang and Zhang (2000) suggest a method for an unbiased variance estimation. To be in line with the other approaches, I use the square root of their estimator as a volatility estimator. Again, this leads to the problem of a biased volatility estimator, since the variance estimator is unbiased.