In study we presented the construction methodologies for the major volatility indices kept by stock exchanges, and also two more suggested in the literature.
We suggested three different methods to construct the VFTSE index for the UK stock market: the alternative interpolation scheme (AIS), which is a modified version of the VXO
43The filter is truncated for 500 observations.
44For the sample before July 1997 the filter is truncated for 500 observations, for the sample after July 1997 the filter is truncated for 250 observations.
construction methodology adapted for a market not as liquid as the US market; the out-of-the-money (OTM) index, which uses onlyOTM options data; and finally the model free (MF) volatility index which uses theVIXconstruction methodology.
We computed the implied option volatility for the American and European options both for trades and bid-ask quotes averages, and the results show that there is a clear smile effect in the
UK market, and that options implied volatilities are biased estimates of the realised volatility measure. As a consequence of this we computed the VFTSEindices using only options with a moneyness level (S/X) between 0.95 and 1.05. For the estimation of the implied volatilities, intraday data for the spot index level, as well as for the options, was used to ensure synchronous information.
We created volatility indices for the period from 14 June 1993 to 17 March 2000, using different construction methodologies and also considering different datasets: American, Eu-ropean, Complete (American and European) options, as well as trades and bid-ask averages.
Descriptive statistics for all the indices were computed and analysed. We show that the changes in the VFTSEOTM and VFTSEMF indices have a much higher level of autocorrelation than the
VFTSEAIS index.
To recommend the construction of theVFTSEusing one of the proposed methods we tested the information content of these volatility indices and also compared it to a historical high frequency realised volatility measure. Our results indicate that realised volatility is a better forecast of the future 22 trading day volatility than any of our volatility indices. As for the ranking of our volatility indices they all perform similarly well with the exception of the model free volatility index which is the worse of all measures. This is most probably due to the insufficient data on the FTSE-100index options market available to construct such index.
These results lead us to recommend the construction of theVFTSEusing the Alternative Interpolation Scheme. We also show that the use of bid-ask data along with the complete set of American and European data have the best statistical characteristics for a volatility index.
It also has the advantage of providing enough data for computing the VFTSEfor almost all days in the sample period considered.
We also present some stylised features of the volatility index, and show that the VFTSE
series is best described by a long memory process. TheVFTSEautocorrelations persist at high levels for a large number of lags. The series that results from using a long memory filter shows practically no signs of serial dependence.
As the estimate of the expected future volatility is of interest for academics and practi-tioners alike, the index here developed can spark much attention. It can be used to launch futures and options on the index which would be of great relevance for anyone interested in hedging volatility risk or even speculating on it.
It is of great interest to analyse the relation of theVFTSEand the stock market index since, in theUS market, it has been shown that they tend to move in different directions (Whaley, 1993, 2000; Fleming, Ostdiek and Whaley, 1995). We leave this to explore in future research.
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Table1:Summaryinformationaboutseveralvolatilityindices. IndexMarketUnderlying optionindexOptiontypeMaintainedbyastockmarket?Introduction dateHistoryMethodologyObservation VXOUSS&P100AmericanYes(CBOE)January19932January1986tothe presentDescribedinsection2.1.1Originallydesignatedwhenin- troducedasVIXandonlyon 22September2003itsname changedtoVXO VIXUSS&P500EuropeanYes(CBOE)22Septem- ber20032January1990tothe presentDescribedinsection2.1.2 OldVXNUSNASDAQ-100EuropeanYes(CBOE)January1998January1995to19 September2003ThesameastheVXOThisindexhasbeenreplaced on22September2003witha newVXNwhichusesadiffer- entconstructionmethod. VXNUSNASDAQ-100EuropeanYes(CBOE)22Septem- ber20032February2001tothe presentThesameastheVIX VXDUSDJIAEuropeanYes(CBOE)18March 20057October1997tothe presentThesameastheVIX VDAXGermanDAX-100EuropeanYes(DeutscheBorse)5December 19942January1992tothe presentPleaserefertoDeutsche Bourse(2003) VX1andVX6FrenchCAC-40EuropeanYes(MONEP)8October 19973January1994tothe presentPleaserefertoMoraux, NavatteandVilla(1999) GVIXGreekFTSE/ASE-20EuropeanNo—10February2000to 30December2002PleaserefertoSkiadopou- los(2004) AVIXAustralianS&P/ASX200EuropeanNo—11July2001to27 September2002PleaserefertoDowling andMuthuswamy(2003) VFTSEUKFTSEEuropean/AmericanNo—14June1993to17 March2000Describedinsection3
Table 2: Eight options’ implied volatilities used to compute theVXOindex
Nearby contract (N) Second nearby contract (SN)
Call Put Call Put
Xl(< S) σc,NXl σp,NXl σc,SNXl σp,SNXl Xu(≥ S) σc,NXu σp,NXu σc,SNXu σp,SNXu
Table 3: Eight options’ implied volatilities used to compute the FTSEAIS index
Nearby contract (N) Second nearby contract (SN)
Call Put Call Put
(St/Xt < 1) σc,NSt/Xt<1 σSp,Nt/Xt<1 σSc,SNt/Xt<1 σp,SNSt/Xt<1 (St/Xt > 1) σc,NSt/Xt>1 σSp,Nt/Xt>1 σSc,SNt/Xt>1 σp,SNSt/Xt>1
Table 4: Four options’ implied volatilities used to compute the FTSEOTM index
Nearby contract (N) Second nearby contract (SN)
Call Put Call Put
(St/Xt < 1) σc,NSt/Xt<1 σSc,SNt/Xt<1
(St/Xt > 1) σSp,Nt/Xt>1 σp,SNSt/Xt>1
Table 5: Number of option implied volatility bid/ask averages records greater than a realised volatility measure, and average implied volatility by strike price over the period from 14 June 1993 to 29 December 1998.
OS−10 5822 805 1785 220 4037 585
OS−9 8201 1038 2357 251 5844 787
OS−8 11209 1494 3119 388 8090 1106
OS−7 15270 2066 3991 464 11279 1602
OS−6 20476 3332 4713 646 15763 2686
OS−5 26604 5096 5503 815 21101 4281
OS−4 33489 8137 6171 1133 27318 7004
OS−3 40136 13440 6899 1792 33237 11648
OS−2 45860 22648 7197 3151 38663 19497
OS−1 47085 35410 7704 5146 39381 30264
OS0 137 148 20 17 117 131
OS1 38266 44425 5401 5988 32865 38437
OS2 24711 41818 3347 5974 21364 35844
OS3 15859 36318 2388 5104 13471 31214
OS4 9657 29371 1691 4997 7966 24374
OS5 6709 25090 1326 4507 5383 20583
OS6 4727 20722 1162 4105 3565 16617
OS7 3370 16628 882 3586 2488 13042
OS8 2224 13997 687 3384 1537 10613
OS9 1763 11659 600 2986 1163 8673
OS10 1309 9530 426 2729 883 6801
Average implied volatility by strike price
OS−10 0.180∗∗∗∗ 0.273∗∗∗∗ 0.193∗∗∗∗ 0.285∗∗∗∗ 0.174∗∗∗∗ 0.268∗∗∗∗
OS−2 0.144 0.152 0.151 0.171∗∗∗∗ 0.143 0.149
OS−1 0.146 0.151 0.149 0.158∗∗∗∗ 0.145 0.150
OS0 0.152 0.151 0.180 0.192 0.147 0.145
OS1 0.147 0.150 0.156∗∗∗ 0.162∗∗∗∗ 0.146 0.148
OS2 0.153 0.153 0.156∗ 0.154 0.152 0.153
OS3 0.157∗∗∗∗ 0.157∗∗∗∗ 0.168∗∗∗∗ 0.163∗∗∗∗ 0.155∗∗ 0.156∗∗∗∗
Percentage of records with implied volatility greater than realised volatility
OS−10 50.84 68.20 56.75 64.55 48.23 69.57
OS−9 49.25 63.78 50.32 60.56 48.82 64.80
OS−8 45.36 66.60 49.66 62.11 43.70 68.17
OS−7 48.11 62.05 48.58 62.07 47.95 62.05
OS−6 45.17 59.48 46.32 58.98 44.83 59.61
OS−5 44.01 54.65 45.90 56.44 43.52 54.31
OS−4 46.36 55.49 49.12 56.31 45.73 55.35
OS−3 48.65 52.09 52.60 54.69 47.83 51.69
OS−2 51.57 49.97 55.72 52.68 50.80 49.53
OS−1 54.87 51.81 58.18 54.06 54.22 51.43
OS0 59.12 55.41 65.00 52.94 58.12 55.73
OS1 56.85 57.16 59.91 58.53 56.35 56.95
OS2 63.76 61.77 67.02 65.57 63.25 61.13
OS3 65.89 64.91 69.22 67.36 65.30 64.51
OS4 71.42 69.03 72.97 72.74 71.09 68.27
OS5 76.55 72.73 77.90 75.93 76.22 72.03
OS6 79.37 75.62 78.57 80.10 79.64 74.51
OS7 83.38 79.04 81.07 80.20 84.20 78.72
OS8 86.02 81.17 84.57 85.22 86.66 79.87
OS9 84.00 83.80 84.17 86.00 83.92 83.04
OS10 86.63 84.74 85.92 87.69 86.98 83.56
FTSE-100 options implied volatility records for the period from 14 June 1993 to 29 December 1998. OSi stands for option, and the Si subscript indicates the option strike relative to the FTSE-100 spot value (Strike + ((i − 1) ∗ 25.0) > Spot ≤ (Strike + (i ∗ 25.0)). A positive i indicates an in-the-money call option or an out-of-the-money put option; and a negative i indicates an out-of-the-money call option or a in-the-money put option. The stars next to the option implied volatilities average indicates the level of significance of testing if the average of the implied volatility is higher than the realised volatility measure: (****) indicate a 1% level and (*) 20% level of significance.
Table 6: Number of days with incomplete data to compute VFTSE using different methodologies
Trades Bid-Ask averages VFTSEAIS
American 108 67
European 509 165
Complete 27 9
VFTSEOTM
American 114 67
European 489 173
Complete 32 12
VFTSEMF
American 339 284
European 980 566
The number of days of the trades sample are 1, 655, 1, 656 and 1, 658 for American, European and both options, respectively. The number of days of the bid-ask averages sample are 1, 660, 1, 662 and 1, 663 for American, European and both options, respectively.
Table7:VFTSEindexdescriptivestatisticsfortheVFTSEAIS (AlternativeInterpolationScheme)andtheVFTSEOTM (Out-The-Money options)withdifferentoptionssampledata. VFTSEAIS A,BAVFTSEAIS C,BAVFTSEAIS C,T ˆσln(ˆσ)∆ˆσˆσln(ˆσ)∆ˆσˆσln(ˆσ)∆ˆσ Average0.178218-1.7905970.0000870.180301-1.7795840.0000900.180838-1.7754050.000064 StandardDeviation0.0702030.3517110.0135800.0707120.3547580.0135420.0703970.3512170.014000 Skewness1.4304640.5994500.7332891.3284100.5328950.5815161.3470860.5509420.637822 Kurtosis5.304792.63636417.5679264.8708932.51167313.8592504.9457012.54121213.206244 Minimum0.090540-2.401966-0.1205290.087968-2.430780-0.1025930.095968-2.343740-0.102433 03/12/199603/12/199631/10/199705/07/199605/07/199631/10/199723/08/199623/08/199631/10/1997 Maximum0.506945-0.6793540.1269100.502662-0.6878380.1058630.501818-0.6895180.107573 30/10/199730/10/199728/10/199705/10/199805/10/199828/10/199705/10/199805/10/199828/10/1997 Numberofobservations159215921592165316531653163016301630 AugmentedDickey-Fullertest-3.0375075-2.6457982-14.0563905-3.2450016-2.7590863-13.8502952-3.1733335-2.7941086-13.8620456 Autocorrelations Lag10.9806610.984640-0.0745100.9809090.984560-0.1016290.9798590.983442-0.125130 Lag20.9640790.972574-0.0895080.9660470.973643-0.0229810.9646380.971900-0.061450 Lag30.9509700.962653-0.0475860.9516880.963234-0.1353900.9517220.962956-0.056752 Lag40.9398710.954821-0.0053260.9423500.9556120.0211460.9414040.955129-0.011595 Lag50.9288650.946735-0.0347420.9322280.947628-0.0192090.9311950.947188-0.002213 VFTSEOTM A,BAVFTSEOTM C,BAVFTSEOTM C,T ˆσln(ˆσ)∆ˆσˆσln(ˆσ)∆ˆσˆσln(ˆσ)∆ˆσ Average0.177338-1.7961210.0000710.179456-1.7845940.0000690.179401-1.7833010.000060 StandardDeviation0.0702680.3533230.0161970.0705170.3557710.0157550.0696840.3513650.015802 Skewness1.4845810.588367-0.0940021.3331260.5208570.8545091.3438800.5297860.567234 Kurtosis5.8270962.67717045.9830144.9251982.52386815.5863785.0013602.54552713.248832 Minimum0.079663-2.529956-0.2305700.079663-2.529956-0.0966950.090196-2.405775-0.115781 24/12/199624/12/199631/10/199724/12/199624/12/199606/10/199818/12/199518/12/199531/10/1997 Maximum0.611967-0.4910760.1813660.511407-0.6705900.1599160.507498-0.6782630.125854 30/10/199730/10/199730/10/199705/10/199805/10/199813/01/199905/10/199805/10/199813/01/1999 Numberofobservations159215921592165016501650162516251625 AugmentedDickey-Fullertest-3.1523398-2.6871403-14.5637135-3.1671818-2.7151269-14.8042450-3.1291445-2.7417603-14.4345751 Autocorrelations Lag10.9729780.978249-0.2163190.9746040.976537-0.1929040.9737170.976583-0.207403 Lag20.9574680.9667260.0118750.9589030.965686-0.0322630.9579920.964947-0.017176 Lag30.9414140.956145-0.0628110.9445490.954687-0.0943960.9431680.954335-0.068318 Lag40.9285970.946386-0.0287890.9349680.9463880.0131480.9319210.945523-0.028223 Lag50.9171330.936824-0.0193270.9246940.9375450.0007480.9220870.9374000.009716
Table 8: VFTSE index descriptive statistics for theVFTSEMF(Model Free) with bid-ask averages American sample data.
VFTSEMFA,BA ˆ
σ ln(ˆσ) ∆ˆσ
Average 0.171665 -1.809150 0.000088
Standard Deviation 0.057040 0.298967 0.036441
Skewness 1.700268 0.448458 1.030688
Kurtosis 7.723964 3.953622 18.037459
Minimum 0.045871 -3.081922 -0.219248
12/02/1998 12/02/1998 11/11/1997
Maximum 0.515286 -0.663034 0.318179
30/10/1997 30/10/1997 12/11/1997
Number of observations 1375 1375 1375
Augmented Dickey-Fuller test -3.9485659 -3.7783162 -15.5444323 Autocorrelations
Lag 1 0.795069 0.766525 -0.441214
Lag 2 0.773308 0.743958 0.012746
Lag 3 0.743779 0.712343 0.035649
Lag 4 0.700079 0.668775 -0.064255
Lag 5 0.683691 0.659180 -0.006676
Table9:Summaryofdescriptivestatisticsfordailyannualrealisedvolatilitymeasuresseries.Themeasureswereobtainedusinghigh frequencyFTSE-100futurestradessquaredreturnsfortheperiodfrom14June1993to17July1998. σSIVlnσSIVσJTlnσJTσNWlnσNWσOSFlnσOSF Average0.117248-2.1995970.114641-2.2359410.116218-2.2126380.116496-2.206565 Standarddeviation0.0444980.3258350.0495560.3652380.0464400.3373980.0431410.328892 Skewness3.64500.42504.47050.26564.19290.40952.39860.3810 Kurtosis43.23493.994959.99914.228254.31684.167618.77023.5949 Minimum0.034583-3.3644040.022102-3.8120700.029088-3.5374230.036721-3.304401 24/12/199624/12/199624/12/199624/12/199624/12/199624/12/199623/08/199323/08/1993 Maximum0.782584-0.2451550.923700-0.0793680.855281-0.1563250.609190-0.495625 28/10/199728/10/199728/10/199728/10/199728/10/199728/10/199728/10/199728/10/1997 NumberofObservations12901290129012901290129012901290 Autocorrelations Lag10.6034440.6622440.5156540.5767400.5697710.6411180.6481340.671497 Lag20.5104630.5801580.4228220.4830620.4752800.5515580.5525320.594181 Lag30.5220630.5808110.4523790.4869440.4966240.5530970.5435070.581527 Lag40.4424410.5397020.3486900.4360230.4020880.5049680.4939530.549210 Lag50.4463950.5647030.3348850.4503100.3987190.5269780.4825710.566252 SIVstandsforsimpleintegratedvolatility,JTforJiangandTian(2005b),NWforNeweyandWest(1987),andOSFforoptimalsamplingfrequency.With exceptionofthedailyoptimalsamplingfrequencyrealisedvolatilitymeasure(σOSF)allotherestimatesareobtainedusing5minutesquaredreturns.All measurestakesintoaccountthemarketopentocloseperiod.OnecorrectiontermisconsideredwhenestimatingtheJTandNWrealisedvolatilitymeasures. Dailyrealisedvolatilityestimateswereobtainedmultiplyingthedailyestimatesby√ 251.
Table10:Summaryofdescriptivestatisticsfordailyannualrealisedvolatilitymeasuresseries.Themeasureswereobtainedusing high-frequencyFTSE-100futuresbid-askaveragesquaredreturnsfortheperiodfrom14June1993to17July1998. σSIVlnσSIVσJTlnσJTσNWlnσNWσOSFlnσOSF Average0.110353-2.2624100.111400-2.2638240.111084-2.2591660.109992-2.266215 Standarddeviation0.0412220.3357450.0465830.3652430.0434150.3444650.0410660.336462 Skewness2.61860.25033.64380.20773.22910.26212.20310.3273 Kurtosis25.38813.705444.51003.928236.53233.878217.20383.3126 Minimum0.020470-3.8887830.019670-3.9286360.020079-3.9080940.038918-3.246293 24/12/199624/12/199624/12/199624/12/199624/12/199624/12/199624/12/199624/12/1996 Maximum0.641959-0.4432320.816362-0.2028970.733404-0.3100580.572212-0.558245 28/10/199728/10/199728/10/199728/10/199728/10/199728/10/199728/10/199728/10/1997 NumberofObservations12901290129012901290129012901290 Autocorrelations Lag10.6503010.6784030.5502730.5929460.6098450.6524500.6526730.671362 Lag20.5598670.6029110.4631150.5152060.5202000.5759290.5480110.594633 Lag30.5527720.6012860.4594300.5124870.5151430.5734390.5471870.594156 Lag40.4922430.5566140.3812250.4583500.4426700.5205110.4946140.553662 Lag50.4971930.5839190.3701750.4728250.4405190.5440680.4977710.580166 SIVstandsforsimpleintegratedvolatility,JTforJiangandTian(2005b),NWforNeweyandWest(1987),andOSFforoptimalsamplingfrequency.With exceptionofthedailyoptimalsamplingfrequencyrealisedvolatilitymeasure(σOSF)allotherestimatesareobtainedusing5minutesquaredreturns.All measurestakesintoaccountthemarketopentocloseperiod.OnecorrectiontermisconsideredwhenestimatingtheJTandNWrealisedvolatilitymeasures. Dailyrealisedvolatilityestimateswereobtainedmultiplyingthedailyestimatesby√ 251.
Table11:Univariateregressionresults,realisedvolatilityistheoptimalweights(OW)measure,andhistoricalvolatilityistheprevious 22tradingdayrealisedvolatilityobservation. β0β1AdjustedR2HRMSEF-test PanelA:σt LRV0.038(0.033)[0.257]{0.000}0.748(0.245)[0.004]{0.309}0.5550.166737873.99(0.000)JB,Wh,BP VFTSEAIS C,T0.060(0.017)[0.001]{0.000}0.567(0.122)[0.000]{0.001}0.4200.186183664.28(0.000)JB,Wh,BG,BP VFTSEAIS C,BA0.062(0.016)[0.000]{0.000}0.556(0.117)[0.000]{0.000}0.4050.188835647.58(0.000)JB,Wh,BG,BP VFTSEAIS A,BA0.061(0.017)[0.001]{0.000}0.562(0.121)[0.000]{0.001}0.4160.187391660.46(0.000)JB,Wh,BG,BP VFTSEOTM C,T0.062(0.015)[0.000]{0.000}0.558(0.107)[0.000]{0.000}0.4140.185913657.23(0.000)JB,Wh,BG,BP VFTSEOTM C,BA0.065(0.016)[0.000]{0.000}0.545(0.117)[0.000]{0.000}0.3870.188923627.55(0.000)JB,Wh,BG,BP VFTSEOTM A,BA0.063(0.017)[0.001]{0.000}0.559(0.127)[0.000]{0.001}0.3920.188327632.93(0.000)JB,Wh,BG,BP VFTSEMF A,BA0.073(0.020)[0.001]{0.000}0.454(0.118)[0.000]{0.000}0.3410.205891581.92(0.000)JB,Wh,BG,BP 0.5×LRV+0.5×VFTSEAIS C,T0.042(0.039)[0.282]{0.000}0.701(0.279)[0.015]{0.288}0.5210.168775809.85(0.000)JB,W,BP 0.5×LRV+0.5×VFTSEAIS C,BA0.043(0.039)[0.277]{0.000}0.697(0.282)[0.017]{0.288}0.5140.170182799.05(0.000)JB,W,BP PanelB:ln(σt) LRV-0.424(0.167)[0.014]{0.000}0.779(0.085)[0.000]{0.012}0.6030.0904334044.93(0.000) VFTSEAIS C,T-0.652(0.256)[0.014]{0.000}0.679(0.128)[0.000]{0.015}0.5120.1028303286.64(0.000)Wh,BP VFTSEAIS C,BA-0.689(0.221)[0.003]{0.000}0.657(0.107)[0.000]{0.002}0.4920.1047873157.80(0.000)BG VFTSEAIS A,BA-0.686(0.229)[0.004]{0.000}0.660(0.111)[0.000]{0.003}0.4970.1038193191.46(0.000)Wh,BG VFTSEOTM C,T-0.686(0.236)[0.005]{0.000}0.659(0.118)[0.000]{0.006}0.5030.1042113228.11(0.000)BP VFTSEOTM C,BA-0.686(0.229)[0.004]{0.000}0.656(0.111)[0.000]{0.003}0.4910.1063683149.34(0.000)BG,BP VFTSEOTM A,BA-0.667(0.256)[0.012]{0.000}0.665(0.124)[0.000]{0.009}0.4900.1062163146.26(0.000)BG,BP VFTSEMF A,BA-1.100(0.326)[0.001]{0.000}0.456(0.178)[0.013]{0.004}0.3050.1210782298.58(0.000)JB,Wh,BG,BP 0.5×LRV+0.5×VFTSEAIS C,T-0.456(0.279)[0.108]{0.000}0.773(0.139)[0.000]{0.109}0.5930.0929573945.75(0.000)W,BP 0.5×LRV+0.5×VFTSEAIS C,BA-0.472(0.168)[0.007]{0.000}0.764(0.086)[0.000]{0.008}0.5850.0937873866.47(0.000)W Thenumberinparenthesisbesidetheparameterestimateisthestandarderror,whicharecomputedfollowingarobustprocedurewheneverappropriatewhichtakesintoaccountheteroscedasticity usingthecorrectionofCribari-Neto(2004),orcorrectsforthepresenceofautocorrelationandheteroscedasticityusingtheproceduresuggestedbyNeweyandWest(1994).Thenumbersinsquare bracketsarethep-valuesassociatedwithat-testfortheregressioncoefficientbeingequaltozero,andvaluesinbracketsarethep-valuesassociatedwithat-testfortheregressioncoefficient beingequaltoone.JBstandsforJarqueandBera(1987)testfornormality,BPforBreuschandPagan(1979)testforheteroscedasticity,BGforBreusch(1978)andGodfrey(1978)testfor autocorrelation,andWhstandsforWhite(1980)testforheteroscedasticity.Ifanyoftheseinitialsappearnexttoaregressionresultsthismeansthatitispossibletorejectthosetestswith asignificancevalueof5%.ThereportedFteststatisticandthep-valueinparenthesis,testthejointhypothesisH0:β0=0andβ1=1.heteroscedasticityconsistentrootmeansquared error(HRMSE)isdefinedbyexpression18.
Table12:Univariateregressionresultsfrom22samples,realisedvolatilityistheoptimalweights(OW)measure,andhistoricalvolatility istheprevious22tradingdayrealisedvolatilityobservation. β0β1AdjustedR2HRMSE PanelB:ln(σt) LRV-0.391(0.174)[0.029]{0.000}0.797(0.088)[0.000]{0.025}0.631VFTSEAIS C,BA,VFTSEAIS A,BA,VFTSEOTM C,T, VFTSEOTM C,BA,VFTSEOTM A,BA,VFTSEMF A,BA
0.0884VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA, VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA, VFTSEMF A,BA VFTSEAIS C,T-0.644(0.228)[0.007]{0.000}0.684(0.113)[0.007]{0.007}0.523VFTSEOTM C,T,VFTSEMF A,BA0.1025VFTSEMF A,BA VFTSEAIS C,BA-0.653(0.223)[0.005]{0.000}0.677(0.110)[0.005]{0.005}0.522VFTSEMF A,BA0.1026VFTSEMF A,BA VFTSEAIS A,BA-0.658(0.226)[0.005]{0.000}0.675(0.111)[0.005]{0.005}0.523VFTSEMF A,BA0.1026VFTSEMF A,BA VFTSEOTM C,T-0.662(0.214)[0.003]{0.000}0.673(0.107)[0.003]{0.003}0.520VFTSEMF A,BA0.1028VFTSEMF A,BA VFTSEOTM C,BA-0.664(0.216)[0.003]{0.000}0.670(0.107)[0.003]{0.003}0.518VFTSEMF A,BA0.1031VFTSEMF A,BA VFTSEOTM A,BA-0.680(0.219)[0.003]{0.000}0.662(0.108)[0.003]{0.003}0.516VFTSEMF A,BA0.1034VFTSEMF A,BA VFTSEMF A,BA-0.971(0.299)[0.002]{0.000}0.529(0.162)[0.005]{0.005}0.3610.1169 0.5×LRV+0.5×VFTSEAIS C,T-0.435(0.272)[0.115]{0.000}0.785(0.136)[0.000]{0.119}0.614VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA, VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA, VFTSEMF A,BA 0.0918VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA, VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA, VFTSEMF A,BA 0.5×LRV+0.5×VFTSEAIS C,BA-0.441(0.239)[0.071]{0.000}0.781(0.120)[0.000]{0.074}0.613VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA, VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA, VFTSEMF A,BA
0.0918VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA, VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA, VFTSEMF A,BA Thenumbersreportedaretheaveragecoefficientsforthe22regressionsusing22non-overlappingseriesof22tradingdays,averagestandarderrors,averageadjustedR2andaverage heteroscedasticityconsistentrootmeansquarederror(HRMSE).Thenumberinparenthesisbesidetheparameterestimateistheaveragestandarderror,whicharecomputedfollowing arobustprocedurewheneverappropriatewhichtakesintoaccountheteroscedasticityusingthecorrectionofCribari-Neto(2004),orcorrectsforthepresenceofautocorrelationand heteroscedasticityusingtheproceduresuggestedbyNeweyandWest(1994).Thenumbersinsquarebracketsarethep-valuesassociatedwithat-testfortheaverageregressioncoefficient beingequaltozero,andvaluesinbracketsarethep-valuesassociatedwithat-testfortheaverageregressioncoefficientbeingequaltoone.Theheteroscedasticityconsistentroot meansquarederror(HRMSE)isdefinedbyexpression18.LRV,VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA,VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA,VFTSEMF A,BAindicatesthatthe specificadjustedR2(HRMSE)issignificantlylarger(smaller)thanthecorrespondingadjustedR2(HRMSE)forthelaggedrealisedvolatility,VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA, VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA,VFTSEMF A,BA,basedontheDieboldandMariano(1995)testforthestatisticaldifferencebetweentwolossfunctionsusinga5%levelof significance.
Table13:Encompassingregressionresults,realisedvolatilityistheoptimalweights(OW)measure,andhistoricalvolatilityisthe previous22tradingdayrealisedvolatilityobservation. InterceptLRVVFTSEAIS C,TVFTSEAIS C,BAVFTSEAIS A,BAVFTSEOTM C,TVFTSEOTM C,BAVFTSEOTM A,BAVFTSEMF A,BAAdj.R2RMSREF-test -0.403(0.168)0.620(0.168)0.174(0.158)——————0.6040.089914.883 [0.020]{0.000}[0.001]{0.028}[0.276]{0.000}(0.000) -0.408(0.168)0.653(0.164)—0.138(0.153)—————0.6010.090117.304 [0.019]{0.000}[0.000]{0.039}[0.370]{0.000}(0.000) -0.406(0.168)0.638(0.163)——0.154(0.151)————0.6030.089916.969W [0.019]{0.000}[0.000]{0.031}[0.315]{0.000}(0.000) -0.401(0.167)0.614(0.157)———0.181(0.145)———0.6070.089717.195W [0.020]{0.000}[0.000]{0.017}[0.217]{0.000}(0.000) -0.401(0.168)0.639(0.158)————0.155(0.147)——0.6040.090217.719 [0.021]{0.000}[0.000]{0.026}[0.297]{0.000}(0.000) -0.397(0.169)0.642(0.159)—————0.153(0.150)—0.6030.090116.930 [0.023]{0.000}[0.000]{0.028}[0.312]{0.000}(0.000) -0.421(0.169)0.766(0.121)——————0.016(0.098)0.5950.090450.596 [0.016]{0.000}[0.000]{0.059}[0.875]{0.000}(0.000) -0.423(0.176)0.631(0.179)1.002(1.067)-3.404(2.237)2.341(1.982)0.421(0.791)0.435(1.433)-0.616(1.052)-0.019(0.110)0.5850.0858— [0.021]{0.000}[0.001]{0.045}[0.353]{0.999}[0.135]{0.055}[0.244]{0.502}[0.597]{0.468}[0.763]{0.695}[0.561]{0.131}[0.866]{0.000} Thenumberinparenthesisbesidetheparameterestimateisthestandarderror,whicharecomputedfollowingarobustprocedurewheneverappropriatewhichtakesintoaccountheteroscedasticity usingthecorrectionofCribari-Neto(2004),orcorrectsforthepresenceofautocorrelationandheteroscedasticityusingtheproceduresuggestedbyNeweyandWest(1994).Thenumbersinsquare bracketsarethep-valuesassociatedwithat-testfortheregressioncoefficientbeingequaltozero,andvaluesinbracketsarethep-valuesassociatedwithat-testfortheregressioncoefficient beingequaltoone.JBstandsforJarqueandBera(1987)testfornormality,BPforBreuschandPagan(1979)testforheteroscedasticity,BGforBreusch(1978)andGodfrey(1978)testfor autocorrelation,andWhstandsforWhite(1980)testforheteroscedasticity.Ifanyoftheseinitialsappearnexttoaregressionresultsthismeansthatitispossibletorejectthosetestswitha significancevalueof5%.ThereportedFteststatisticandthep-valueinparenthesis,testthejointhypothesisH0:βLRV=0andβVFTSE=1.heteroscedasticityconsistentrootmeansquared error(HRMSE)isdefinedbyexpression18.
Table14:Encompassingregressionresultsfrom22samples,realisedvolatilityistheoptimalweights(OW)measure,andhistorical volatilityistheprevious22tradingdayrealisedvolatilityobservation. InterceptLRVVFTSEAIS C,TVFTSEAIS C,BAVFTSEAIS A,BAVFTSEOTM C,TVFTSEOTM C,BAVFTSEOTM A,BAVFTSEMF A,BAAdj.R2RMSRE (1)-0.372(0.174)0.649(0.161)0.161(0.155)——————0.635(8)0.0873 [0.037]{0.000}[0.000]{0.034}[0.303]{0.000} (2)-0.374(0.168)0.654(0.161)—0.155(0.152)—————0.634(7),(8)0.0874 [0.031]{0.000}[0.000]{0.036}[0.310]{0.000} (3)-0.375(0.168)0.651(0.161)——0.157(0.150)————0.634(2),(8)0.0874 [0.029]{0.000}[0.000]{0.034}[0.301]{0.000} (4)-0.374(0.174)0.650(0.158)———0.159(0.149)———0.636(1),(8)0.0872(1) [0.036]{0.000}[0.000]{0.031}[0.292]{0.000} (5)-0.373(0.167)0.654(0.159)————0.155(0.148)——0.635(7),(8)0.0873 [0.030]{0.000}[0.000]{0.034}[0.300]{0.000} (6)-0.376(0.170)0.658(0.158)—————0.150(0.147)—0.634(8)0.0873 [0.032]{0.000}[0.000]{0.035}[0.313]{0.000} (7)-0.380(0.169)0.758(0.124)——————0.047(0.108)0.6270.0879 [0.029]{0.000}[0.000]{0.056}[0.665]{0.000} (8)-0.373(0.178)0.642(0.177)0.096(0.976)-0.580(1.789)0.547(1.793)0.040(0.631)0.345(1.145)-0.242(1.097)-0.039(0.151)0.6290.0822 (1),(2),(3),(4),(5),(6),(7) [0.041]{0.000}[0.001]{0.049}[0.922]{0.359}[0.747]{0.381}[0.762]{0.802}[0.950]{0.135}[0.764]{0.570}[0.826]{0.263}[0.800]{0.000} Thenumbersreportedaretheaveragecoefficientsforthe22regressionsusing22non-overlappingseriesof22tradingdays,averagestandarderrors,averageadjustedR2andaverage heteroscedasticityconsistentrootmeansquarederror(HRMSE).Thenumberinparenthesisbesidetheparameterestimateistheaveragestandarderror,whicharecomputedfollowing arobustprocedurewheneverappropriatewhichtakesintoaccountheteroscedasticityusingthecorrectionofCribari-Neto(2004),orcorrectsforthepresenceofautocorrelationand heteroscedasticityusingtheproceduresuggestedbyNeweyandWest(1994).Thenumbersinsquarebracketsarethep-valuesassociatedwithat-testfortheaverageregressioncoefficient beingequaltozero,andvaluesinbracketsarethep-valuesassociatedwithat-testfortheaverageregressioncoefficientbeingequaltoone.Theheteroscedasticityconsistentrootmean squarederror(HRMSE)isdefinedbyexpression18.(1),(2),(3),(4),(5),(6),(7),(8)and(9)indicatesthatthisspecificregressionspecificationadjustedR2(HRMSE)issignificantly larger(smaller)thanthecorrespondingadjustedR2(HRMSE)regressionspecification,basedontheDieboldandMariano(1995)testforthestatisticaldifferencebetweentwoloss functionsusinga5%levelofsignificance.
Table15:Firststageoftheinstrumentalvariablesregressionresults,realisedvolatilityistheoptimalweights(OW)measure,and historicalvolatilityistheprevious22tradingdayrealisedvolatilityobservation. InterceptLRVLVFTSEAIS C,TLVFTSEAIS C,BALVFTSEAIS A,BALVFTSEOTM C,TLVFTSEOTM C,BALVFTSEOTM A,BALVFTSEMF A,BAAdj.R2HRMSE VFTSEAIS C,T0.057(0.136)0.643(0.095)0.369(0.093)——————0.7960.0712 [0.676]{0.000}[0.000]{0.000}[0.000]{0.000} VFTSEAIS C,BA0.072(0.139)0.632(0.095)—0.389(0.092)—————0.7910.0725 [0.609]{0.000}[0.000]{0.000}[0.000]{0.000} VFTSEAIS A,BA0.084(0.136)0.603(0.094)——0.424(0.090)————0.8010.0702 [0.541]{0.000}[0.000]{0.000}[0.000]{0.000} VFTSEOTM C,T0.032(0.151)0.646(0.105)———0.354(0.101)———0.7550.0787 [0.832]{0.000}[0.000]{0.001}[0.001]{0.000} VFTSEOTM C,BA0.035(0.149)0.634(0.102)————0.371(0.098)——0.7630.0777 [0.816]{0.000}[0.000]{0.001}[0.000]{0.000} VFTSEOTM A,BA0.040(0.142)0.599(0.097)—————0.409(0.095)—0.7800.0730 [0.780]{0.000}[0.000]{0.000}[0.000]{0.000} VFTSEMF A,BA-0.146(0.248)0.878(0.146)——————-0.001(0.121)0.4820.1079JB [0.560]{0.000}[0.000]{0.409}[0.993]{0.000} Thenumberinparenthesisbesidetheparameterestimateisthestandarderror,whicharecomputedfollowingarobustprocedurewheneverappropriatewhichtakesintoaccountheteroscedasticity usingthecorrectionofCribari-Neto(2004),orcorrectsforthepresenceofautocorrelationandheteroscedasticityusingtheproceduresuggestedbyNeweyandWest(1994).Thenumbersinsquare bracketsarethep-valuesassociatedwithat-testfortheregressioncoefficientbeingequaltozero,andvaluesinbracketsarethep-valuesassociatedwithat-testfortheregressioncoefficient beingequaltoone.JBstandsforJarqueandBera(1987)testfornormality,BPforBreuschandPagan(1979)testforheteroscedasticity,BGforBreusch(1978)andGodfrey(1978)testfor autocorrelation,andWhstandsforWhite(1980)testforheteroscedasticity.Ifanyoftheseinitialsappearnexttoaregressionresultsthismeansthatitispossibletorejectthosetestswitha significancevalueof5%.ThereportedFteststatisticandthep-valueinparenthesis,testthejointhypothesisH0:βLRV=0andβVFTSE=1.heteroscedasticityconsistentrootmeansquared error(HRMSE)isdefinedbyexpression18.
Table16:Secondstageoftheinstrumentalvariablesunivariateandencompassingregressionresults,realisedvolatilityistheoptimal weights(OW)measure,andhistoricalvolatilityistheprevious22tradingdayrealisedvolatilityobservation. InterceptLRVIVFTSEAIS C,TIVFTSEAIS C,BAIVFTSEAIS A,BAIVFTSEOTM C,TIVFTSEOTM C,BAIVFTSEOTM A,BAIVFTSEMF A,BAAdj.R2HRMSEF(a)F(b)H -0.365(0.171)0.831(0.089)——————0.6140.09274014.21411.900 [0.038]{0.000}[0.000]{0.063}(0.000)(0.001) -0.355(0.229)0.350(0.321)0.477(0.299)——————0.6160.09052.4411.309BP [0.128]{0.000}[0.280]{0.048}[0.117]{0.087}(0.097)(0.253) -0.378(0.171)—0.821(0.089)—————0.6110.09323988.02444.231 [0.031]{0.000}[0.000]{0.048}(0.000)(0.000) -0.358(0.230)0.377(0.304)—0.447(0.280)—————0.6170.09052.9391.502BP [0.126]{0.000}[0.221]{0.045}[0.117]{0.054}(0.062)(0.220) -0.397(0.170)——0.813(0.088)————0.6070.09403940.49134.198 [0.024]{0.000}[0.000]{0.039}(0.000)(0.000) -0.362(0.232)0.410(0.281)——0.412(0.257)————0.6170.09053.4841.260BP [0.125]{0.000}[0.151]{0.040}[0.115]{0.026}(0.038)(0.262) -0.344(0.171)———0.839(0.089)———0.6200.09224075.47594.164 [0.049]{0.000}[0.000]{0.076}(0.000)(0.000) -0.341(0.225)0.302(0.344)———0.532(0.317)———0.6190.09042.1061.907BP [0.136]{0.000}[0.383]{0.047}[0.099]{0.146}(0.132)(0.167) -0.339(0.173)————0.837(0.089)——0.6160.09294032.293-74.656 [0.055]{0.000}[0.000]{0.074}(0.000)(1.000) -0.337(0.224)0.343(0.332)————0.490(0.301)——0.6180.09052.3322.248BP [0.139]{0.000}[0.306]{0.053}[0.109]{0.096}(0.107)(0.134) -0.343(0.176)—————0.835(0.091)—0.6070.09403942.226-47.485 [0.057]{0.000}[0.000]{0.075}(0.000)(1.000) -0.337(0.226)0.408(0.308)—————0.423(0.281)—0.6160.09062.8021.557BP [0.142]{0.000}[0.190]{0.060}[0.138]{0.045}(0.070)(0.212) -0.279(0.183)——————0.898(0.098)0.6080.09093948.624-9.363 [0.132]{0.000}[0.000]{0.300}(0.000)(1.000) Thenumberinparenthesisbesidetheparameterestimateisthestandarderror,whicharecomputedfollowingarobustprocedurewheneverappropriatewhichtakesintoaccountheteroscedasticity usingthecorrectionofCribari-Neto(2004),orcorrectsforthepresenceofautocorrelationandheteroscedasticityusingtheproceduresuggestedbyNeweyandWest(1994).Thenumbersinsquare bracketsarethep-valuesassociatedwithat-testfortheregressioncoefficientbeingequaltozero,andvaluesinbracketsarethep-valuesassociatedwithat-testfortheregressioncoefficient beingequaltoone.JBstandsforJarqueandBera(1987)testfornormality,BPforBreuschandPagan(1979)testforheteroscedasticity,BGforBreusch(1978)andGodfrey(1978)testfor autocorrelation,andWhstandsforWhite(1980)testforheteroscedasticity.Ifanyoftheseinitialsappearnexttoaregressionresultsthismeansthatitispossibletorejectthosetestswitha significancevalueof5%.ThereportedF(a)teststatisticandthep-valueinparenthesis,testthejointhypothesisH0:βIntercet=0andβVFTSE=1.ThereportedF(b)teststatisticandthe p-valueinparenthesis,testthejointhypothesisH0:βLRV=0andβVFTSE=1.heteroscedasticityconsistentrootmeansquarederror(HRMSE)isdefinedbyexpression18.Thelastcolumn ofthetablereportstheHausman(1978)teststatisticandp-valueinparenthesis.
Table17:Univariateregressionresultsfrom22samples,realisedvolatilityistheoptimalweights(OW)measure,andhistoricalvolatility isthepreviousdayrealisedvolatilityobservation. β0β1AdjustedR2HRMSE PanelB:ln(σt) LRV-0.912(0.170)[0.000]{0.000}0.517(0.081)[0.000]{0.000}0.437VFTSEMF A,BA0.1078VFTSEMF A,BA VFTSEAIS C,T-0.645(0.230)[0.007]{0.000}0.683(0.114)[0.007]{0.007}0.526LRV,VFTSEOTM C,T,VFTSEOTM A,BA, VFTSEMF A,BA
0.1017VFTSEOTM A,BA,VFTSEMF A,BA VFTSEAIS C,BA-0.655(0.223)[0.005]{0.000}0.676(0.110)[0.005]{0.005}0.525LRV,VFTSEMF A,BA0.1018VFTSEMF A,BA VFTSEAIS A,BA-0.659(0.225)[0.005]{0.000}0.675(0.110)[0.005]{0.005}0.526LRV,VFTSEMF A,BA0.1018VFTSEMF A,BA VFTSEOTM C,T-0.663(0.216)[0.003]{0.000}0.673(0.107)[0.003]{0.003}0.522LRV,VFTSEMF A,BA0.1020VFTSEMF A,BA VFTSEOTM C,BA-0.666(0.216)[0.003]{0.000}0.669(0.106)[0.003]{0.003}0.520LRV,VFTSEMF A,BA0.1023VFTSEMF A,BA VFTSEOTM A,BA-0.682(0.217)[0.003]{0.000}0.661(0.107)[0.002]{0.002}0.518LRV,VFTSEMF A,BA0.1026VFTSEMF A,BA VFTSEMF A,BA-0.970(0.297)[0.002]{0.000}0.530(0.160)[0.005]{0.005}0.3650.1160 0.5×LRV+0.5×VFTSEAIS C,T-0.597(0.173)[0.001]{0.000}0.694(0.087)[0.000]{0.001}0.565LRV,VFTSEAIS C,T,VFTSEAIS C,BA, VFTSEAIS A,BA,VFTSEOTM C,T,VFTSEOTM C,BA, VFTSEOTM A,BA,VFTSEMF A,BA 0.0960LRV,VFTSEAIS C,T,VFTSEAIS C,BA, VFTSEAIS A,BA,VFTSEOTM C,T,VFTSEOTM C,BA, VFTSEOTM A,BA,VFTSEMF A,BA 0.5×LRV+0.5×VFTSEAIS C,BA-0.600(0.178)[0.001]{0.000}0.691(0.089)[0.000]{0.001}0.565LRV,VFTSEAIS C,T,VFTSEAIS C,BA, VFTSEAIS A,BA,VFTSEOTM C,T,VFTSEOTM C,BA, VFTSEOTM A,BA,VFTSEMF A,BA
0.0960LRV,VFTSEAIS C,T,VFTSEAIS C,BA, VFTSEAIS A,BA,VFTSEOTM C,T,VFTSEOTM C,BA, VFTSEOTM A,BA,VFTSEMF A,BA Thenumbersreportedaretheaveragecoefficientsforthe22regressionsusing22non-overlappingseriesof22tradingdays,averagestandarderrors,averageadjustedR2andaverage heteroscedasticityconsistentrootmeansquarederror(HRMSE).Thenumberinparenthesisbesidetheparameterestimateistheaveragestandarderror,whicharecomputedfollowing arobustprocedurewheneverappropriatewhichtakesintoaccountheteroscedasticityusingthecorrectionofCribari-Neto(2004),orcorrectsforthepresenceofautocorrelationand heteroscedasticityusingtheproceduresuggestedbyNeweyandWest(1994).Thenumbersinsquarebracketsarethep-valuesassociatedwithat-testfortheaverageregressioncoefficient beingequaltozero,andvaluesinbracketsarethep-valuesassociatedwithat-testfortheaverageregressioncoefficientbeingequaltoone.Theheteroscedasticityconsistentroot meansquarederror(HRMSE)isdefinedbyexpression18.LRV,VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA,VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA,VFTSEMF A,BAindicatesthatthe specificadjustedR2(HRMSE)issignificantlylarger(smaller)thanthecorrespondingadjustedR2(HRMSE)forthelaggedrealisedvolatility,VFTSEAIS C,T,VFTSEAIS C,BA,VFTSEAIS A,BA, VFTSEOTM C,T,VFTSEOTM C,BA,VFTSEOTM A,BA,VFTSEMF A,BA,basedontheDieboldandMariano(1995)testforthestatisticaldifferencebetweentwolossfunctionsusinga5%levelof significance.
Table18:Encompassingregressionresultsfrom22samples,realisedvolatilityistheoptimalweights(OW)measure,andhistorical volatilityisthepreviousdayrealisedvolatilityobservation. InterceptLRVVFTSEAIS C,TVFTSEAIS C,BAVFTSEAIS A,BAVFTSEOTM C,TVFTSEOTM C,BAVFTSEOTM A,BAVFTSEMF A,BAAdj.R2RMSRE (1)-0.532(0.182)0.255(0.094)0.474(0.119)——————0.579(7)0.0938(5),(7) [0.005]{0.000}[0.009]{0.000}[0.000]{0.000} (2)-0.538(0.175)0.256(0.091)—0.469(0.112)—————0.579(7)0.0938(7) [0.003]{0.000}[0.007]{0.000}[0.000]{0.000} (3)-0.540(0.176)0.256(0.090)——0.468(0.113)————0.580(7)0.0938(7) [0.003]{0.000}[0.007]{0.000}[0.000]{0.000} (4)-0.545(0.174)0.258(0.092)———0.464(0.114)———0.577(7)0.0940(7) [0.003]{0.000}[0.007]{0.000}[0.000]{0.000} (5)-0.546(0.171)0.259(0.092)————0.460(0.111)——0.576(7)0.0942(7) [0.002]{0.000}[0.007]{0.000}[0.000]{0.000} (6)-0.558(0.172)0.262(0.091)—————0.452(0.111)—0.575(7)0.0944(7) [0.002]{0.000}[0.006]{0.000}[0.000]{0.000} (7)-0.674(0.192)0.365(0.102)——————0.294(0.131)0.5080.1002 [0.001]{0.000}[0.001]{0.000}[0.029]{0.000} (8)-0.506(0.185)0.247(0.100)0.258(0.958)-0.330(1.747)0.486(1.671)0.045(0.637)0.404(1.154)-0.348(1.109)-0.020(0.155)0.581(7)0.0873 (1),(2),(3),(4),(5),(6),(7) [0.009]{0.000}[0.017]{0.000}[0.789]{0.442}[0.851]{0.450}[0.773]{0.760}[0.944]{0.141}[0.728]{0.608}[0.755]{0.230}[0.898]{0.000} Thenumbersreportedaretheaveragecoefficientsforthe22regressionsusing22non-overlappingseriesof22tradingdays,averagestandarderrors,averageadjustedR2andaverage heteroscedasticityconsistentrootmeansquarederror(HRMSE).Thenumberinparenthesisbesidetheparameterestimateistheaveragestandarderror,whicharecomputedfollowing arobustprocedurewheneverappropriatewhichtakesintoaccountheteroscedasticityusingthecorrectionofCribari-Neto(2004),orcorrectsforthepresenceofautocorrelationand heteroscedasticityusingtheproceduresuggestedbyNeweyandWest(1994).Thenumbersinsquarebracketsarethep-valuesassociatedwithat-testfortheaverageregressioncoefficient beingequaltozero,andvaluesinbracketsarethep-valuesassociatedwithat-testfortheaverageregressioncoefficientbeingequaltoone.Theheteroscedasticityconsistentrootmean squarederror(HRMSE)isdefinedbyexpression18.(1),(2),(3),(4),(5),(6),(7),(8)and(9)indicatesthatthisspecifregressionspecificationadjustedR2(HRMSE)issignificantlylarger (smaller)thanthecorrespondingadjustedR2(HRMSE)regressionspecification,basedontheDieboldandMariano(1995)testforthestatisticaldifferencebetweentwolossfunctions usinga5%levelofsignificance.
Table 19: VFTSE index descriptive statistics for theVFTSEfor the periods before and after July 1997.
VFTSE before July 1997 VFTSE after July 1997 ˆ
σ ln(ˆσ) ∆ˆσ σˆ ln(ˆσ) ∆ˆσ
Average 0.136790 -2.007598 0.000083 0.250774 -1.410280 0.000103
Standard Deviation 0.026934 0.188900 0.007076 0.062373 0.226369 0.019993
Skewness 0.733237 0.459522 0.540629 1.329665 0.772094 0.432906
Kurtosis 2.533629 2.172182 6.562079 4.651344 3.132518 7.473931
Minimum 0.087968 -2.430780 -0.031133 0.160586 -1.828928 -0.102593
05/07/1996 05/07/1996 01/12/1993 15/07/1997 15/07/1997 31/10/1997
Maximum 0.209998 -1.560658 0.040421 0.502662 -0.687838 0.105863
28/11/1994 28/11/1994 07/02/1994 05/10/1998 05/10/1998 28/10/1997
Number of observations 1022 1022 1022 631 631 631
Augmented Dickey-Fuller test -2.2335506 -2.4055172 -11.319301 -2.9955183 -3.0793577 -8.3974164 Autocorrelations
Lag 1 0.961806 0.959712 -0.173428 0.948146 0.947713 -0.086961
Lag 2 0.938388 0.935445 -0.063748 0.905436 0.905976 -0.013627
Lag 3 0.918936 0.915064 -0.030326 0.863448 0.863552 -0.154097
Lag 4 0.901692 0.896108 0.029962 0.837231 0.836698 0.019839
Lag 5 0.882410 0.877319 0.037927 0.809050 0.807096 -0.031624
Table 20: Descriptive statistics for the filtered series of the logarithm of the
VFTSEindex.
Filtered series of the logarithm of theVFTSE
Complete Sample Before July 1997 After July 1997
Average -0.000290 -0.004102 -0.001547
Standard Deviation 0.064750 0.054984 0.074887
Skewness 0.5974 0.7019 0.3403
Kurtosis 5.5913 6.4051 4.3916
Minimum -0.267734 -0.211021 -0.268784
15/02/2000 02/05/1997 15/02/2000
Maximum 0.327334 0.281273 0.256497
28/10/1997 06/12/1996 13/01/1999
Number of observations 1153 522 380
Autocorrelations
Lag 1 0.010104 0.009123 -0.017825
(0.3658) (0.4174) (0.3641)
Lag 2 0.033359 0.050454 0.029400
(0.1287) (0.1245) (0.2833)
Lag 3 -0.071279 0.035952 -0.184661
(0.0078) (0.2057) (0.0002)
Lag 4 0.013721 -0.002225 0.075670
(0.3206) (0.4797) (0.0701)
Lag 5 0.005416 0.061394 -0.006429
(0.4270) (0.0804) (0.4501)
Ljung-Box(25) 31.734236 20.381129 31.573523
(0.1658) (0.7266) (0.1707)
The volatility series was filtered using the fractional integration filter (1 − L)0.83for the complete sample and the sub-period after July 1993, and (1 − L)0.75for the sub-period before July 1997. The numbers in parentheses are the levels of significance of the autocorrelation statistics.
Figure 1: American and European, calls and puts daily implied volatilities during the period from 14 June 1993 to 17 March 2000
0 10 20 30 40 50 60 70
01/01/1994 01/01/1995 01/01/1996 01/01/1997 01/01/1998 01/01/1999 01/01/2000
Average daily volatility (%)
Date
European Calls European Puts
European Calls European Puts