第23天文獻(xiàn)forecasting volatility roles of sampling frequency and horizon

1、FORECASTING VOLATILITY:ROLES OF SAMPLINGFREQUENCY AND FORECASTINGHORIZONWING HONG CHAN XIN CHENGJOSEPH K.W. FUNG*This study empirically tests how and to what extent the choice of the sampling frequency, the realized volatility (RV) measure, the forecasting horizon and the time-series mchronous execu

2、heWe would like to acknowledge with thain providing the data. The authors thank Futures Research Symposium, particularly Bo tions. This study is based, in part, on Chengs Pexpressed in this study are those of the authors, and Institute for Monetary Research, its Council of Advisors, o*Correspondence

3、 author, Department of Finance and Decisioand Clearing.at the 20th Asia Pacificelpful comments and sugges-Baptist University. The views sarily reflect those of theof Directors.Received June 2010; Accepted June 2010 Wing Hong Chan is an Associate Professor of Economics, Depa Laurier University, Ontar

4、io, Canada and also at Depanomics, Wilfrid University of Xin Cheng is a Ph.D. Candidat Baptist University, Joseph K.W. Fung is a Professor of Finance, Department of Finance and Decision Sciences,Baptist University, Council of Advisors,and also a Member of theInstitute for Monetary Research, Central,

5、.The Journal of Futures Markets, Vol. 30, No. 12, 1167 1191 (2010)© 2010 Wiley Periodicals, Inc.View this article online at DOI: 10.1002/fut.204761168Chan, Cheng, and Fungstudy avoids the influence of various market microstructure factors in measuring RV with high-frequency intraday data and in

6、 inferring implied volatility (IV) from option prices. The study shows that excluding non-trading-time volatility pro- duces significant downward bias of RV by as much as 36%. Quality of predictionis significantly affected by the forecasting horizon and RV m, but is largelyimmune from the choice of

7、sampling frequency. Consistent with prior research,IV outperforms time-series forecasts; however, the information content of histor- ical volatility critically depends on the choice of RV measure. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 30:11671191, 2010INTRODUCTIONForecasting volatility is

8、 of critical importance in option pricing and risk man- agement. Over the past two decades, much effort has been devoted to develo sophisticated forecasting ms, improving forecasting evaluation techniques, and testing the forecasting performance for various markets. This studyextends the literature

9、and examines how the choices of forecasting m realized volatility (RV) measure affect the quality of volatility forecast.andThere are two major types of forecasting mstime-series ms thatmake projections based on historical volatility (TS-HV), and implied volatility(IV) that is inferred from option p

10、rices. Poon and Grangers (2003) survey shows that IV, though it is biased under some circumstances, generally outper- forms TS-HV volatility forecasts.1On the other hand, actual volatility or RV is an abstract concept and can only be estimated subject to measurement error. Merton (1980) shows that t

11、he standard deviation of RV measure is a monotonic decreasing function of sam- pling frequency. However, Andersen, Bollerslev, Diebold and Labys (2000) find that measures of RV become unstable with extremely high sampling frequen- cies such as 5 and 10 seconds. Therefore, Andersen, Bollerslev, Diebo

12、ld and Labys (2001) argue that intraday returns provide better estimates of RV but5-minute sampling is perhaps optimal, taking intothe impact of marketmicrostructure factors (such as bidask bounce and non-trading) on measures based on high-frequency data. Similarly, Aït-Sahalia, Mykland and Zha

13、ng(2005) argue that if the microstructure noise is un sampling frequency is finite, say 5-minutes.ed for, the optimalCorsi, Zumbach, Muller, and Dacorogna (2001) and Zumbach, Corsi, and Trapletti (2002) show that the market microstructure impacts on RV estimation can be corrected by using pre-filter

14、ed return data that directly factor in the cor- relation structure in high-frequency data. This result is consistent with the1See also Day and Lewis (1988), Canina and Figlewski (1993), Figlewski (1997), Taylor and Xu (1997), Davidson, Kim, Ors, and Szakmary (2001), Szakmary, Ors, Kim, and Davidson

15、(2003), Koopman, Jungbacker, and Hol (2005).Journal of Futures MarketsDOI: 10.1002/futForecasting Volatility1169finding of Aït-Sahalia et al. (2005), who show that better volatility estimation can be obtained with high-frequency data as long as there is an adjustment forthe spurious correlation

16、 structure. On the other hand, usingtraniondata, Oomen (2005) also finds that the optimal sampling frequency can be reduced to 12 seconds, after incorporating an error correction scheme to allevi- ate the bias, from 2.5 minutes before the adjustment. Benchmarking on the “true” return variance, Bandi

17、 and Russell (2006, 2008) find the estimation error associated with the 5-minute sampling frequency is acceptable, but that associated with the 15-minute interval is highly volatile.In the presence of overnight market closure, the optimal sampling proce- dure is different. The overnight returns are

18、found to be noisy and are likely to erase the benefit of increasing the sampling frequency. To put it another way, the problem caused by sampling too frequently may be dwarfed by noisy overnight returns. Therefore, for the purpose of forecast performance evalua- tion, it is important to choose the a

19、ppropriate adjustment for non-trading-time returns as well as sampling frequency. The empirical studies about testingvolatility forecasting mshigh-frequency volatility are limited. Blair,Poon and Taylor (2001) and Martens and Zein (2004) measure RV with intradaydata and still find that IV dominates

20、TS-HV ms in forecasting volatility.Owing to the finite maturity of futures contracts, the characteristics of actual volatility may change when expiration day approaches. Samuelson (1965) hypothesizes that futures volatility rises as the contracts approach expiration. IfSamuelson is correct, then the

21、 quality of forecasts from different ms shoulddecline when the prediction horizon shortens. Fle(1998) finds a decayingpattern in IV forecasts as the horizon rises from 1 to 10 days. However, Jorion (1995) finds that IV prediction is more accurate for longer horizons.Although most recent studies find

22、 that IV outperforms TS-HV in forecast- ing volatility, many studies (Chan, Kalimipalli, & Jha 2009; Poon & Granger, 2003) show that TS-HV provides incremental information over IV. On theother hand, practitioners have been using both IV and TS-HV in fortheirforecasts. As the volatility struc

23、ture may change over different maturity hori- zons of the derivatives, the information share between IV and TS-HV may also vary as the options and futures expire.This study empirically tests how and to what extent the choice of samplingfrequency, RV measure, forecasting horizon, and time-series m qu

24、ality of volatility forecast. The data set from theaffect the market for anextended time period from July 2000 to December 2006 contains highly syn- chronous executable quotes retrieved from an electronic trading platform, which removes the influence of bidask price bounce and non-trading in measuri

25、ng RV with high-frequency intraday data and in inferring IV from option prices. The study shows that excluding non-trading-time volatility produces significantJournal of Futures Markets DOI: 10.1002/fut1170Chan, Cheng, and Fungdownward bias of RV by as much as 36%. Quality of prediction is significa

26、ntlyaffected by the forecasting horizon and RV m, but is largely immune fromthe choice of sampling frequency. Consistent with prior research, IV outper- forms time-series forecasts; however, the information content of historical volatility critically depends on the choice of RV measure.The rest of t

27、he study is organized as follows: Section 2 describes the data and research methodology; Section 3 summarizes and interprets the empirical results; and Section 4 concludes.DATA AND METHODOLOGYDataThe study uses complete bid and ask quotes of the Hang Seng Index (HSI) options and futures for the peri

28、od July 2000 to December 2006 obtained from the “Bid and Ask RecordAll Futures/Options” CDs published by the HongKong Stock Exchange. Themarket setting is very convenient fortesting the predictive power of IV. The HSI option is European style. It has the same trading time as HSI futures. It uses fut

29、ures-style margining so a modifiedBlacks (1976) mcan be used to further reduce the number of parameters.All of these can reduce the measurement error.The options and futures are both traded on the electronic trading platform and the quotes represent firm commitments of market participants and are po

30、tentially executable. As trading in both contracts is concentrated in the two nearest month maturity contracts, the study focuses on the spot and next month contracts. There are two trading sessions each day for both options and futures, namely, 9:45 a.m. to 12:30 p.m. and 2:30 p.m. to 4:15 p.m. The

31、 spot month contract ceases trading at 4:00 p.m. on the last trading day (or expira- tion day) of the contract. The contract expires on the day before the last busi- ness day of the month. There are no afternoon sessions on Christmas and New Year Eves, or when the area is under severe weather condit

32、ions.The data contain the best bid and ask prices and the corresponding quan- tities. The quotes are refreshed throughout the trading sessions whenever changes occur. The quotes are good until there are indications that a particular bid or offer is being lifted or withdrawn. Records associated with

33、price quotes of “0” such as “99999” or “999999” are deleted from the data.For option quotes which appear in the same trading session with the same maturity, the first bid (ask) is matched with the immediately following ask (bid) only if the bid is lower than the ask. If the updated bid (ask) is lowe

34、r (higher) than the ask (bid) in the current pair, the new pair is recorded. If the updated bid (ask) is higher (lower) than the ask (bid) in the current pair, the current pairJournal of Futures MarketsDOI: 10.1002/futForecasting Volatility1171is discarded and the updated quote should be matched wit

35、h the following ask (bid). In addition, we match each bid with the ask of the same contract that refreshes within 1 minute. Then each option pair is matched with the synchro- nous futures bidask pair that has the same time to maturity. The futures data are treated in a similar manner.Measures of RVF

36、our classes of RV are used in the empirical tests. The first measure is based only on intraday trading-hour returns; the second measure is estimated by a simple sum of trading-hour and non-trading-hour returns; the third measure is based on a weighted sum of trading-hour and non-trading-hour returns

37、; andthe fourth measure is the standard deviation of-to-returns.Intraday volatility measureThis measure only includes trading-hour returns and is estimated as the sum of squared intraday returns as follows:2annual_trading_timeTN2Intradayvolatility B remaining_futures_lifetime aa ri,j(1)i 1 j 1where

38、ri, j pi, jpi, j 1, pi , jis the natural logarithm of the middle futuresquotes at the end of the j-th interval on day i,3 annual_trading_time and remaining_ futures_lifetime are both in second, T is the number of trading days to futures maturity, and N is the number of sampling intervals within 1 da

39、y.Total volatility measureFollowing Blair et al., (2001), the total volatility is calculated as the sum of trading- hour squared returns and non-trading-hour squared returns as follows:242TTTNa ai 1 j 1Total volatilityar Br b222ai,Na ri,L (2)i,jTi 1i 1where rt,L and rt,N are the lunch-break and over

40、night returns respectively, 242 is the average number of trading days annually from 2000 to 2006.2A similar approach is used in Jiang and Tian (2005) to measure realized volatility with high-frequency data.3Although most of the studies use data sampled over regular time intervals, Oomen (2005, 2006)

41、 and some other studies use non-regular sampling intervals.Journal of Futures Markets DOI: 10.1002/fut1172Chan, Cheng, and FungScaled total volatility measureFollowing Hansen and Lunde (2005), the following scaled total volatility meas- ure is used to reduce the impact of the noisiness of the non-tr

42、ading-hour vari- ance on the measure:242TTN2Scaled total volatility Bac a ri,j(3)i 1j 1ddna awhere c(rk r)2 r2ais a scale that adjusts the weightings of thek,jk 1k 1 j 1trading-hour and non-trading-hour variances in thevolatility-to-measure, rtreturn dur-is the ing the s-to-return on day t, and r is

43、 the averageatility measureThis is the tradition the daily returns. The darithmic daily closing prices.a-Estimation of Option IVDaily IV for different option classes is extra within the last 30 min of a trading session. Th avoid bidask bounce. Option quotes that violatdiscarded. Option quotes that a

44、re below the minimu are arbitrarily large are discarded. Following Whaley (19Lastrapes (1993), the daily IV for each option class is obtain mean squared error between the market and mbserved order to ditions are ck and that moureux andminimizing the1Nmin N a (market_option_pricei Black_price iv(4)i

45、1where market_option_pricei represents the mid-quote of o is the modified Black (1976) commodity option pricing mof option quotes on a particular day, and iv is the daily IV estimate.iceAs HSI options and futures share the same trading schedule and expira- tion cycle (for short-dated options) and HS

46、I options are European style, the HSI options can be priced as if they are options on the futures (Duan & Zhang,4The intuitionis to scale the intraday volatility up to the expected value. Hansen and Lunde (2005)also demonstrate such estimator is approximately unbiased under reasonable assumption

47、s.Journal of Futures MarketsDOI: 10.1002/futForecasting Volatility11732001). The non-synchronous problem is minimized by using intraday HSI futures and option data. Besides, as the HSI options adopt future-style margin- ing, Blacks m can be further simplified as follows:5call FN(d1) XN(d2)put FN( d1

48、) XN( d2)d1 ln(F X) 1 2s2h (s2h)d2 d1 s2h(5)where F represents the mid-quote of futures, X is the exercise price, s is the volatility, and h is the time to maturity.Forecasting MsFour different time-series ms are used to forecast future volatility, namely,moving average m eroskedastic m(MA), general

49、ized autoregressive conditional het- (GARCH), autoregressive fractionally integrated movingaverage m(ARFIMA), and ARFIMA with Jump (ARFIMA_ J) m.Moving averageThe MA forecast is the simple average of the past 124 days daily RV.t 1 1 12422a(6)stsii t 124GARCHThe GARCH(1, 1) mhas often been found to o

50、utperform other time-seriesms for both in-sample and out-of-sample forecasts; see for exampleEderington and Guan (2005). Therefore, it is adopted in this study for m comparison. The error variance is specified as follows:s2 a a e2 b s2 .(7)t01 t 11 t 1The estimation window is 124 days and rolls forw

51、ard.ARFIMAThe ARFIMA(1, 1) specification is used to capture long range dependence (or long memory) in the volatility dynamics. The estimation window is 124 days and rolls forward.5See Lieu (1990).Journal of Futures Markets DOI: 10.1002/fut1174Chan, Cheng, and Fung(1 L)d ln s2 v g ln s2e(8)t 1ttwhere

52、 the fractional parameter d between zero and one represents a long mem- ory structure implying slow hyperbolic decay in autocorrelations.ARFIMA-JThe jump component J is defined according to Andersen, Bollerslev and Diebold (2007).Jt max(RVt BVt, 0)1 ¢BVt (22p) 20 rj¢, ¢t ( j 1)¢,

53、 ¢ 00 r22(9)at j 2(1 L)d ln s2 v g ln s2g J e1t 12 t 1ttwhere is the sampling frequency, RVt refers to the RV constructed by sum- high-frequency squared returns, and BVt is the bipower variation.Test Whether the Choice of RV Construction Method Affects the Forecasting PerformanceAs different RV

54、 measures also have different distribution characteristics, it is worth testing whether those measures, sampled at the same frequency, have similar forecasting performance. Therefore, we regress different RV measureswith the same sampling frequency on five forecasting m pare the regression results.s

55、 and then com-ln RVt 1, T a b ln forecastt et(10)where RVt 1,T refers to a certain RV measure sampled at 5 minutes, 3 minutes, 1 minute and 30 seconds, and forecastt is a certain m s forecast based on day t and can be IV, MA, GARCH, ARFIMA and ARFIMA-J.Test Whether Volatility S PerformanceGiven the

56、volatility sAffects the Forecastingpattern described in the previous section, we exam-ine to what extent different moneyness IV varies in their forecasting per- formance.ln RVt 1,T a b ln IVi,t etwhere i refers to certain option moneyness group.(11)Journal of Futures MarketsDOI: 10.1002/futForecasting Volatility1175Encompassing RegressionTo investigate whether time-series ms have additional information to IV, weemploy the classic encompassing regression framework.ln RVt 1,T a b1 ln IVt b2 ln time_series_forecastt et(12)where IVt is the volatilities implied by ATM calls and puts, and