222 S. Muzzioli
Ta b le 2 .OLS regressionsIntercept ln(σc) ln(σp) ln(σh) Ad j R^2 DW χ^2 a χ^2 b Hausman Test
− 0. 01 1.05*** 0.77 1.73 13.139 0.10021
(0.915) (0.000) (0.00)
− 0. 018 1.047*** 0.76 1.77 13.139 0.25128
(0.853) (0.000) (0.00)
− 0. 29 0.82 0.65 2.12 7.517
(0.008) (0.000) (0.02)
− 0. 02 0.938*** 0.10+++ 0.76 1.87 1.288 0.47115
(0.850) (0.000) (0.400) (0.53)
− 0. 01 0.9631*** 0.082+++ 0.77 1.80 1.158 0.95521
(0.915) (0.000) (0.489) (0.56)
0.0006 0.372 0.6861*** 0.77 1.74 2.04 0.14977
(0.994) (0.244) (0.033) (0.35)aNote: The numbers in brackets are thep-values. Theχ 2 acolumn reports the statistic of a
χ^2 test for the joint null hypothesisα=0andβ=1 in the following univariate regressions
ln(σr)=α+βln(σi)whereσr= realized volatility andσi= volatility forecast,i=h,c,p.The
χ^2 bcolumn reports the statistic of aχ^2 test for the joint null hypothesisγ=0andβ=1inthe
following regressions: ln(σr)=α+βln(σi)+γln(σh),ln(σr)=α+βln(σp)+γln(σc),
whereσr= realized volatility,σi= volatility forecast,i=c,pandσh= historical volatility.
The superscripts , , indicate that the slope coefficient is not significantly different from
one at the 10%, 5% and 1% critical level respectively. The superscripts+++,++,+indicate
that the slope coefficient is not significantly different from zero at the 10%, 5% and 1% critical
level respectively. The last column reports the Hausman [11] specification test statistic (one
degree of freedom), where the 5% critical level is equal to 3.841.
features, which in our case do not require the dividend payment estimation. Another
possible explanation stems from the characteristics of the data set used. In particular
in our case put IV was on average lower than call IV, while in [3] the opposite is true.
As IV usually overpredicts realised volatility, if a choice has to be made between call
and put IV, a rule of thumb can be to choose the lowest of the two.
5 Conclusions
In this paper we have investigated the relation between IV, historical volatility and
realised volatility in the DAX index options market. Since IV varies across option type
(call versus put), we have run a horse race of different IV estimates: call IV, put IV. Two
hypotheses have been tested: unbiasedness and efficiency of the different volatility
forecasts. Our results suggest that both IV forecasts contain more information about
future realised volatility than historical volatility. In particular, they are unbiased
(after a constant adjustment) and efficient forecasts of realised volatility in that they
subsume all the information contained in historical volatility. In our sample both IV
forecasts obtain almost the same performance, with put IV marginally better than call