Predicting financial volatility: High-frequency time -series forecasts vis-à-vis implied volatility

Transcription

1 Predicting financial volatility: High-frequency time -series forecasts vis-à-vis implied volatility Martin Martens* Erasmus University Rotterdam Jason Zein University of New South Wales First version: December 10, This version: October 9, 2003 Abstract Recent evidence suggests option implied volatilities provide better forecasts of financial volatility than time-series models based on historical daily returns. In this study both the measurement and the forecasting of financial volatility is improved using high-frequency data and long memory modeling, the latest proposed method to model volatility. This is the first study to extract results for three separate asset classes, equity, foreign exchange and commodities. The re sults for the S&P 500, YEN/USD and Light, Sweet Crude Oil provide a robust indication that volatility forecasts based on historical intraday returns do provide good volatility forecasts that can compete with, and even outperform implied volatility. JEL: C53, G14 Keywords: Volatility forecasting, high-frequency data, implied volatility. * Contact author: Martin Martens, Econometric Institute and Department of Finance, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands. Tel (+31) , Fax (+31) , mmartens@few.eur.nl We would like to thank an anonymous referee and seminar participants at the 15th Australasian Finance & Banking Conference in Sydney (2001), the NAKE day (2002) at the Dutch Central Bank, and the University of Maastricht for providing useful comments. All remaining errors are our own responsibility.

2 Predicting financial volatility: High-frequency time -series forecasts vis-à-vis implied volatility Abstract Recent evidence suggests option implied volatilities provide better forecasts of financial volatility than time-series models based on historical daily returns. In this study both the measurement and the forecasting of financial volatility is improved using high-frequency data and long memory modeling, the latest proposed method to model volatility. This is the first study to extract results for three separate asset classes, equity, foreign exchange and commodities. The results for the S&P 500, YEN/USD and Light, Sweet Crude Oil provide a robust indication that volatility forecasts based on historical intraday returns do provide good volatility forecasts that can compete with, and even outperform implied volatility. JEL: C53, G14 Keywords: Volatility forecasting, high-frequency data, implied volatility. 1

3 INTRODUCTION Predicting financial volatility is crucial for risk management, asset management and option pricing. An inspection of current literature points to a consensus that option implied volatilities produce superior volatility forecasts compared to forecasts produced from historical data. In many cases it is found that, for example, GARCH forecasts do not even provide any incremental information that is not already embedded in implied volatilities. Examples include Day and Lewis (1993) for oil futures, Jorion (1995) for foreign exchange rates, and Christensen and Prabhala (1998) for the S&P 100 index. In an efficient market the option price incorporates all available information and therefore the superiority of options volatility forecasts is in line with theoretical expectations. Practically, however, the question still remains; are option prices really efficient? Or is it simply that time-series forecasts based on daily data have failed to reveal their inefficiency? The use of high-frequency data has induced dramatic improvements in both measuring and forecasting volatility. Andersen and Bollerslev (1998) firstly introduced model-free realized, or integrated, volatility measures defined by the summation of highfrequency intraday squared returns. Most recently, Andersen et al. (2003) model these realized volatility measures for the Japanese Yen / US Dollar (YEN/USD) and Deutsche Mark / US Dollar (DEM/USD) exchange rates using a fractionally integrated autoregressive process, capturing the long memory in volatility. In these models volatility shocks manifest themselves for very long periods and decay at a hyperbolic rate. Forecasts from long memory models provide notable improvements over daily GARCH forecasts at the 1- and 10-day horizons. The modeling of the long memory property in volatility has the potential to improve forecasts particularly for much longer horizons, needed to compete 2

4 with option implied volatility forecasts. Clearly, the advent of high-frequency data and modeling of the long memory property of volatility series warrants a fresh investigation into the informational efficiency of option-implied volatilities. In addition to the present work, two independent and concurrent studies by Li (2002) and Pong et al. (2003) compare high-frequency time-series forecasts to implied volatilities. There are, however, some important differences between our study and its concurrent counterparts. First, whereas these studies focus solely on exchange rates, our study also considers the S&P 500 index and Light, Sweet Crude Oil. Second, we rely on publicly available information only, whereas these studies use Over -The-Counter (OTC) implied volatilities. Third, we are of the opinion that our methodology provides the cleanest contest between time-series forecasts and implied volatilities. For instance, Li (2002) uses overlapping data 1 in the forecast evaluation regressions, similar to Canina and Figlewski (1993). Fleming (1998) and Christensen and Prabhala (1998) show that the use of overlapping data favors time-series forecasts, and that Canina and Figlewski come to the wrong conclusion. The fact that Li (2002) concludes that the high-frequency time-series forecasts provide incremental information to that of implied volatilities is therefore not surprising, whereas the question remains whether they really do, or whether this is a statistical artifact 2. Moreover, Pong et al. (2003) find that forecasts from long memory models using high-frequency time-series do not provide additional information to that 1 For example, every day a volatility forecast is produced for the next 20 days. The ex-post 20-day volatility has 19 terms in common with the ex-post 20-day volatility of the previous day producing a very high autocorrelation in the ex-post 20-day volatilities. As a result regressing the ex-post volatilities on ex-ante volatility forecasts produces spurious results, including biased parameter estimates and inflated regression R 2 s. Computing robust standard errors does not change this. 2 The bias in Li s results is perhaps most clearly illustrated by looking at the regression R 2 s for the long memory model. For example, Andersen et al. (2003) report 10-day forecast R 2 s of and for the DEM/USD and YEN/USD exchange rates, respectively. Li, for an almost identical sample period, reports 1-3

5 already contained in implied volatilities. Pong et al. (2003) use non-overlapping data, but regress ex-post realized volatilities on implied volatilities, and use the regression coefficients to produce enhanced out-of-sample implied volatility forecasts. In contrast, we use implied volatilities directly as extracted from option prices, without further adjustments using historical volatilities. As such our results provide the clearest answer to the question of whether implied volatilities are in fact efficient or whether high-frequency time-series forecasts contain incremental information. This study is the first to report such results for equities and commodities. Our results for the S&P 500, YEN/USD and Light, Sweet Crude Oil show that long memory forecasts dramatically improve upon daily GARCH forecasts, confirming the results of Andersen et al. for exchange rates. More importantly, contrary to recent findings for daily data for equity, exchange rates and commodities, we find that high-frequency time-series forecasts do have incremental information over that contained in implied volatilities. Hence, the use of high-frequency data in combination with modeling the long memory characteristic of realized volatilities can be credited with altering the outcome of the competition between time-series volatility forecasts and implied volatilities. Our findings do not just apply to the 24-hour foreign exchange market, but also to assets with limited trading hours. Through using overnight futures trading in S&P 500 index-futures and Sweet Crude Oil futures we bridge the gap between the open and closing times of these assets respective exchanges. As a result the recent developments in time-series forecasting can be successfully applied to these markets, opening the way to extend the application of high frequency methodologies beyond exchange rates to many more assets. month R 2 s of and Similarly, Pong et al. (2003) report a 1-month horizon R 2 of for the GBP/USD, whereas Li s R 2 is

6 The remainder of this study is organized as follows. The next section will present an overview of the current literature that compares time-series with implied volatility forecasts, in the process motivating our choice of data and methodology. Subsequently the data are presented, including the computation of implied volatilities and integrated volatility measures. This is followed by the long memory model for realized volatilities. A fifth section presents the results and the final section concludes. LITERATURE REVIEW There is a sizeable amount of literature that compares volatility forecasts embedded in option prices with those from time-series models, see Poon and Granger (2003) for a comprehensive review. These studies, considering a variety of assets and asset classes, have sometimes reported conflicting results, although more recent literature indicates a consensus that implied volatility forecasts are superior to their time-series counterparts. In the discussion below we limit ourselves to the issues important for our work, in particular motivating our choice of data and implied volatility extraction. The first crucial decision in computing implied volatilities is the model choice. This potentially depends on the underlying asset and the need to adjust for dividend payments, account for the early exercise premium, and consider stochastic volatility. Popular model choices include Black and Scholes (1973), Black (1976), Barone-Adesi and Whaley (1987), and binomial trees. A second problem that affects implied volatilities are measurement errors. An important example is non-synchronous trading. This can occur in various forms. First, the 5

7 time of the closing prices of the option and underlying asset may differ 3. Second, Fleming et al. (1995) point out that an index lags its true value, since some of the quotes of the stocks underlying the index must be stale. Third, for infrequently traded options, the last trade may occur prior to the last trade of the underlying asset. Transaction costs also introduce measurement errors, either through the bid-ask bounce in option and asset prices, or simply because arbitrage is expensive in the case of an index. Finally, Jorion (1995) also points out that closing prices may be subject to clerical measurement errors. Over time several studies have proposed solutions to the aforementioned problems. The basic idea behind most of these solutions is that a good implied volatility estimate can be obtained through a weighted average of each of the available implied volatilities. Presently, the most common approach is to compute a weighted average of implied volatilities of a limited number of calls and puts that are closest to being at-the-money. Deep in- or out-of-the - money options tend to be traded less frequently and are therefore more prone to measurement errors. Ederington and Guan (2000b) is a recent study that compares various weighting schemes, finding a weighted average of the two closest-to-the-money calls and two closest-to-the-money puts provides the best results for S&P 500 futures options 4. In this study we will use options on futures, as they are arguably the least affected by any of the aforementioned problems. Jorion (1995) notes that since futures do not pay dividends 3 For example, the options on the S&P 100 index trade until 4:15 pm Eastern Standard Time (EST), whereas the NYSE closes at 4:00 pm EST. 4 The weights depend on the distance of the strike prices from the underlying futures price. This weighting scheme closely resembles that used by the CBOE for the Market Volatility Index (VIX) based on S&P 100 options, and described in detail in Fleming, Ostdiek and Whaley (1995). The next section will provide more detail on these weighting schemes, as something similar is used for the present study. 6

8 and are traded at the same location as the options, measurement errors associated with dividends and non-synchronous trading are significantly reduced. Most importantly, all futures and options closing quotes are carefully scrutinized by the exchange because they are used for daily settlement, and are therefore less likely to suffer from clerical measurement errors 5. Ederington and Guan (2000a) add that arbitrage and speculation on futures options are easier and less costly. In addition to the calculation of the implied volatilities, existing studies also differ in how these implied volatilities are compared with time-series volatility forecasts. The most popular approach is to regress the realized volatility from day t+1 to t+t, RLZ, t + 1 T, on the implied volatility forecast, IV t + 1, T, and the time-series forecast, TS t 1, T +, RLZ t+ 1, T = a + b1ivt+ 1, T + b2ts t+ 1, T + ε t + 1, T (1) Note that both forecasts are based on information up to and including day t, so that the implied volatility would be obtained from the closing option price(s) on day t. Studies differ in that some fix the forecast horizon, T, and then search for the option maturity closest to T or alternatively interpolate between options that have smaller and larger maturities. A possible fixed horizon is one day, for which the option is used which is 5 We computed implied volatilities both from closing prices and settlement prices. For each available maturity the implied volatilities of the two nearest-the-money calls and the two nearest-the-money puts are used. For closing prices the four resulting implied volatilities often differed considerably. In contrast, for the settlement prices the four implied volatilities are often very close, sometimes identical up to 4 decimals for calls and puts with the same exercise price. Hence, the availability of settlement prices is crucial. 7

9 closest to maturity 6. Jorion (1995) labels this the information test. To be fair to implied volatilities, the choice for T should be determined by the available option maturities. When the options natural forecasting horizon is used, Jorion labels the regression in equation (1) an efficiency test, as one should find that b 2 is not significantly different from zero if implied volatilities are efficient. Both types of tests serve distinct and important purposes. The efficiency test establishes whether the option price effectively impounds all available information. Practitioners, however, often require accurate volatility forecasts for a particular horizon each day, regardless of whether options with corresponding maturities are available. In this case the information test provides an indication from which practitioners can assess the forecasting method that is able to provide superior forecasts on a day-to-day basis. For these reasons, we conduct both types of tests. This also allows us to ascertain the extent to which the forecast horizon mismatch problem disadvantages implied volatility forecasts. An alternative way of comparing time-series and implied volatilities is the insample test introduced by Day and Lewis (1992), in which the GARCH model is extended with the implied volatility, r t ε h t t = µ + ε t t 1 ~ N(0, ht ) Φ (2) = ω + αε 2 t 1 + βh t 1 + γ IV t,1 6 Usually an additional requirement is that the maturity should be at least one week, as, for example, bid -ask problems get more serious for very short maturities. 8

10 where r t is the daily return, ε t is a residual which follows a normal distribution with mean zero and conditional variance h t, conditional upon the information set 1 Φ t on day t 1. The model in (2) is also sometimes used to produce out-of-sample forecasts, see for example Kroner et al. (1995). Perhaps the most puzzling tests arise when implied volatilities are not used directly in equation (1) as obtained from option prices, but are enhanced by historical time-series. For example Taylor and Xu (1997) and Blair et al. (2001), use the model in equation (2) with α set equal to zero. The resulting out-of-sample forecasts are treated as implied volatility forecasts. However, in reality they are based on a time-series of implied volatilities with exponentially declining weights, where the driving parameter, β, is based on the historical daily data. Hence, implied volatility is implicitly deemed to be biased, and using lagged implied volatilities and historical daily returns rectifies this. Strictly speaking, this is not a test on the quality and efficiency of implied volatility forecasts, as it is closer to being a composite forecast of (daily) time-series and implied volatility. Pong et al. (2003) use the regression in equation (1), with b 2 equal to zero, to correct the bias in implied volatilities. The forecasts are then compared with forecasts from the long memory model. The results show that long memory forecasts offer no incremental information to implied volatilities at the one and three month horizons for the GBP/USD exchange rate. As we mention above this is not strictly a test of the efficiency of implied volatilities, as the implied volatilities are enhanced by using historical daily data. Therefore Taylor and Xu, Blair et al. and Pong et al. do not conduct a test on the original implied volatilities. As such, it is again not possible to reach firm conclusions regarding the efficiency of implied volatilities. 9

11 The only other study that compares Long Memory and Implied Volatility Forecasts is Li (2002). He finds that for the DEM/USD, YEN/USD and GBP/USD exchange rates, implied volatility dominates over shorter horizons (1 month) and long memory forecasts are superior over the longer term. As noted previously, however, Li (2002) uses overlapping data in the forecast evaluation regressions given in (1). DATA AND IMPLIED VOLATILITIES The Futures Industry provided daily data for options on S&P 500 and YEN/USD futures and transaction data for S&P 500 and YEN/USD futures. Both futures and option contracts trade on the Chicago Mercantile Exchange (CME) floor from 8:30 a.m. to 3:15 p.m. (one hour behind Eastern Standard Time, EST 1). Since January 3, 1994, the S&P 500 futures also trade on GLOBEX, the overnight electronic trading system of the CME, from 3:30 p.m. to 8:00 a.m. (8:15 a.m. from February 26, 1996, onwards). The YEN/USD futures started trading on GLOBEX at the start of For both contracts our data set deliberately starts from the introduction of the contracts on GLOBEX, for reasons expressed below. The data spans to the end of This results in 1,763 and 1,258 daily observations for the S&P 500 and YEN/USD, respectively. Light, Sweet Crude Oil futures and options data come from a number of sources. The New York Mercantile Exchange (NYMEX) provided the daily data for the options. The Futures Industry provided the futures settlement prices, whereas CQG (Comprehensive Quotes and Graphics, an official NYMEX data vendor) provided the futures transaction data for both the floor trading session from 8:45 a.m. to 2:05 p.m. (EST 1) and trading on ACCESS from 3:00 p.m. to 6:55 a.m. (7:55 a.m. from September 8, 2000, onwards). 10

12 ACCESS is the overnight electronic trading system of NYMEX, and Light, Sweet Crude Oil futures started trading on ACCESS in June Again our data set starts from the point where the futures also trade overnight, and spans to the end of This results in 1,877 daily observations. The above contracts are selected since both the futures and options on futures are amongst the most actively traded contracts in their respective asset classes. For example the S&P 500 and Light, Sweet Crude Oil futures are the most actively traded equity and commodities contracts in the world. The YEN/USD is the second most actively traded currency contract behind the Euro/US Dollar contract, which, at this stage, has an insufficient history to make a useful comparison between time-series and implied volatility forecasts. Implied volatility of S&P 500 and YEN/USD futures options S&P 500 and YEN/USD futures options mature every 3 rd Friday of the month, each month of the year. These options are written on the nearest-to-maturity futures contract. The futures maturities are March, June, September, and December. The options that mature on the same day as the futures contract are called non-serial or ordinary options, whereas the other eight options are called serial options. Following Ederington and Guan (2000b) we do include the serial options in the analysis. They usually have a maximum maturity of about days, whereas the non-serial options are more popular and usually start trading about 1 year prior to maturity. We found that the inclusion of the serial options provides implied volatilities that are superior for forecasting volatility at shorter horizons as compared to using the non-serial options only. 11

13 For every day we observe both calls and puts at a variety of strike prices for options with the same expiration date. Implied volatilities are computed based on the Barone -Adesi and Whaley (1987) approximation to account for the early exercise premium of the American style S&P 500, YEN/USD and Light, Sweet Crude Oil futures options. For the risk-free rate we use the 1-month, 3-month and 6-month Eurodollar interest rates and a linear extrapolation to cover all maturities. Researchers have proposed a plethora of different weighted averages of implied volatilities calculated from numerous options with the same expiry date, to obtain a single best implied volatility. Ederington and Guan (2000b) study a whole set of weighting schemes for S&P 500 futures options. They find that averages using just a few at-the - money implied volatilities predict slightly better than the broader weighted averages using up to the first eight in- and out-of-money options for both calls and puts. In our case we use a weighted average of the two closest-to-the-money calls and two closest-to-the-money puts where the weights are chosen so that the average strike equals the underlying futures price. To formalize this, assume the relevant futures price is F, and the nearest available exercise prices for the call (put) options are Xc1 and Xc2 (Xp1 and Xp2) such that Xc1 Xc2 (X p1 X p2 ). Let IV c1 and IV c2 (IV p1 and IV p2 ) be the corresponding implied volatilities for the calls (puts). Then the implied volatility we will use is, IMPLIED t, T X = 0.25 X c2 c2 X X F X p2 p2 c1 F X IV p1 c1 IV X X p1 c2 c2 X X F X p2 p2 c1 F X IV p1 c2 IV p2, (3) 12

14 where T is the time-to-maturity translated into business days. Ederington and Guan (2000b) show that for S&P 500 futures options using this weighted average of the two closest-tothe-money calls and puts has the best out-of-sample forecasting performance. It performs slightly better than an equally weighted average of the same four options, and better than weighting schemes using up to 32 options as suggested in Latane and Rendleman (1976), Chiras and Manaster (1978), and Whaley (1982). On each day t a limited number of horizons are available 7, and the available horizons are different for each day. To be able to cater for any forecast horizon we interpolate between the available implied volatilities. Let T1 and T2 be two available maturities on day t such that T 1 T T 2. Then to match the forecast horizon we will use 8, IV _ VIX ( T T1) = IMPLIEDt, T + ( IMPLIED ) 1 t, T IMPLIED 2 t, (4) 1 ( T T ) t, T T 2 1 The implied volatility is similar to the Market Volatility Index (VIX) for S&P 100 index options described in detail in Fleming et al. (1995) and used by Blair et al. (2001). The main difference is that VIX uses weights such that the average time-to-maturity equals a 7 Options with maturities less than 6 business days are excluded. The time value of short-lived options relative to bid-ask spreads is small making the imp lied volatilities less stable. 8 If T is smaller than the shortest time-to-maturity available we simply use the implied volatility for the shortest time -to-maturity. If T is larger than the longest time -to-maturity available we use the implied volatility for the longest time-to-maturity. 13

15 constant 22 trading days. Obviously this is inappropriate for the whole range of forecasting horizons we will consider here 9. Implied volatility of Light, Sweet Crude Oil futures options Light, Sweet Crude Oil futures mature every month on the third business day before the 25 th of the month preceding the delivery month. Futures options stop trading three business days before the underlying futures contract. Prior to August 1997 the options expired on the Friday before the termination of futures trading. Implied volatilities are computed in the same way as described above by equations (3) and (4). Measuring realized volatility using intraday returns The most common method to measure realized volatility is to square the daily returns. Andersen and Bollerslev (1998) show that although the daily squared return is an unbiased estimator of the true volatility, it is also an extremely noisy estimator. Andersen and Bollerslev show both theoretically and empirically that the sum of the intraday squared returns provides the best measure of realized volatility. It is vitally important that our data sample covers the full 24-hour period, especially for the YEN/USD and Light, Sweet Crude Oil. For both assets, the volatility during non-us business hours is in excess of 50 percent of the total daily volatility. Hence, the use of, for example, the squared close-to-open return to measure overnight volatility would be subject to the same problems as using the daily squared return to measure daily volatility. 9 As a quality check we compared VIX for S&P 100 index options with our IV_VIX with a maturity of 22 days, finding a correlation of for the out-of-sample period from 1996 to The correlation should be 14

16 To avoid market microstructure problems, the last price in each 5-minute interval is selected for floor trading 10 and the last price in each 30-minute interval is selected for overnight trading 11. Prices are obtained from futures contracts that are closest to maturity. When futures contracts are very close to maturity a switch is made to the next maturing contract 12. The switch occurs when volume from the next maturing contract begins to exceed the volume of the closest to maturity contract. This was found to occur approximately five trading days from maturity. The measure for daily realized volatility is then REALIZED N D N 2 D 2 t+ 1,1 = rt + 1, n ) + ( rt + 1, d ) n= 1 d = 1 (, (5) where r, is the intra-night return on day t in intra-night period n (n = 1 N), and N t n r, is the D t d intraday return on day t for intraday period d (d = 1 D). Realized volatility for multiple days is then obtained by simply adding the daily realized measures, T REALIZED t REALIZED + 1, T = t+ i,1. (6) i= 1 high as even the 50 largest stocks already account for over 50% of the S&P 500 index. 10 Andersen and Bollerslev (1998) for exchange rate volatility and Blair et al. (2001) for S&P 100 index volatility also use the 5-minute frequency. 11 Overnight trading attracts smaller volume and the use of a higher frequency would result in many zero returns. Lower frequencies such as 60-minutes and 90-minutes provide very similar results. 12 For example the S&P 500 March 1997 contract matured on Friday March 21. Until March 14 the March futures prices are used up to 3:15 p.m. Then from 3:15 p.m. onwards the June futures prices are used. This 15

17 Sample characteristics of squared returns and realized volatilities Table I reports sample statistics for the daily squared return and realized volatility at the 1- day horizon and aggregated to a 10-day horizon. The results for the daily horizon confirm the findings of Andersen et al. (2001, 2003), with daily squared returns possessing higher volatilities and lower autocorrelations than the corresponding realized volatilities. As expected such differences are still very much present at the 10-day horizon, although the relative differences become smaller. -Insert Table I about here- The maximum daily volatility of 135.8% for Light, Sweet Crude Oil REALIZED is quite conspicuous compared with other oil volatility observations. It occurs on Monday March 23, 1998, and is driven by the Friday-close to Sunday evening-open return, following an OPEC meeting over the weekend. The meeting caused a jump in the oil futures price from $14.61 to $16.25 per barrel. While we do not wish to alter observed data for forecasting, we did set this value to 20% when using it as ex-post volatility to evaluate the various forecasts. After evaluating models with and without this outlier it was found that only the overall level of the performance of the models was affected, but not their relative performance. gives two prices at 3:15 p.m. to compute the correct last 5-minute return (3:10-3:15) of the March contract and the correct first return (3:15-3:30) for the June contract. 16

18 TIME-SERIES FORECASTS Following the work of Andersen et al. (2003) for exchange rates, the log daily realized standard deviation is modelled as a fractionally integrated process. See also Andersen, Bollerslev, Diebold and Ebens (2001), Andersen, Bollerslev, Diebold and Labys (2001) and Ebens (1999) for more detail on the properties of the realized volatilities for exchange rates, individual US stocks and the Dow Jones index, respectively. These studies also explain the motivation and rationale for using a long memory model for log-realized volatilities. This centres on the fact that autocorrelations in realized volatilities die at a hyperbolic rate, and a fractionally integrated process of order d, 0 < d < 1, captures this. In comparison, in a GARCH model a shock to the system dissipates at an exponential rate. Log transformed realized volatilities follow a more symmetric distribution than the original realized volatilities. Equation (7) displays the nature of this log transformation. y 1 log( REALIZED ) 1 (7) t = 2 t + 1, The following Autoregressive Fractionally Integrated Moving Average (ARFIMA), model is estimated for the log realized standard deviation, θ ( L)(1 L) d ( y t µ ) = ε t (8) where d is the degree of fractional integration, L is the lag operator, θ(l) is a polynomial in the lag operator, and ε t is white noise. The fractional differencing parameter, d, is estimated using the Robinson (1995) estimator. The differencing operator is applied to 100 lags. Next, an Autoregressive (AR) model is estimated for the differenced time -series, with the lag 17

19 length determined by the partial autocorrelation function. This provides the polynomial θ(l). Together with the estimated d the ARFIMA(p,d,0) model in equation (8) can be rewritten into an AR( ) representation, k= 1 y = µ + π ( y µ ) (9) t k t k where the coefficients π k are determined by multiplying the two polynomials θ(l) and d ( 1 L). Equations (7) and (9) are then used to produce a forecast for the volatility, which will be referred to as the long memory forecast. At any stage the summation in equation (9) can obviously only run for k smaller than t. The initial in-sample periods for the S&P 500 and Light, Sweet Crude Oil futures comprises of the first 500 observations. For the YEN/USD we use the first 506 observations from 2 January 1996 through 31 December Forecasts are then produced for various horizons. Subsequently, every day one further observation is added to the in-sample period, the model parameters are then re-estimated, and again forecasts are produced. Table II provides the parameter estimates for the initial in-sample period as well as the full sample period. The estimates for d are around 0.4 and are all significantly different from 0 and from 1, implying the log realized volatilities are fractionally integrated. -Insert Table II about here- 13 The Futures Industry could not provide the December 1997 option information. By using at least 506 observations for the in-sample period, and hence starting forecasting from the start of 1998, the December 1997 option data are not required. 18

20 The same procedure is also followed to produce forecasts for the daily GARCH(1,1) model (γ=0 in equation (2)), the daily GARCH model extended with implied volatility (α=0 in equation (2), as in Blair et al. (2001)), and the daily GARCH model extended with realized volatility (α=0 in equation (2) and the implied volatility replaced with realized volatility). Estimation of the various GARCH models will allow for a comparison with existing studies. In addition it will show the differences in performance (relative to implied volatility) of the GARCH model and the long memory model. Andersen et al. (2003) indicate that perhaps this difference is not so much due to the modeling of the long memory characteristic in the data, but more so to the effective incorporation of information contained in the high-frequency data. In the GARCH model based on daily data only a small weight is attached to the latest observation for the daily squared return, whereas in the long memory model the volatility forecast is to a much larger extent, based on the latest realized volatility. For example, a typical weight for the newest daily squared return in the GARCH forecast is five percent, whereas in the long memory model the weight for the newest realized volatility is equal to d if θ1 equals zero, i.e. around 40 percent (see Table II). A consequence is that the GARCH forecast is generally much closer to the unconditional mean and reverts much faster to it than the long memory forecast. The π k coefficients presented in Table II illustrate the slow hyperbolic decay resulting from volatility shocks in the long memory model. The various forecasts are compared using the encompassing regression in (1) to conduct both information and efficiency tests. Individual forecasts are also evaluated separately by regressing them as the single explanatory variable against realized volatility. Information test regressions are estimated using non-overlapping data points. For example, 19

21 for the 20-day horizon, non-overlapping data is used such that the regression is run 20 times, each time starting one day later to form 20 series of non-overlapping 20-day windows. We report the average coefficients, average standard errors and average R 2. Furthermore the Heteroskedasticity consistent Root Mean Squared Error (HRMSE) is computed as follows HRMSE T = FORECASTt, T 2 ( 1 (10) REALIZED N 1 ) N t= 1 t, T RESULTS Implied volatility vis-à-vis GARCH The research design of this study focuses on first replicating ke y results from past literature that have favoured implied volatility. The purpose is to verify the quality of our implied volatility estimates, in confirming that implied volatilities do appear to dominate when using daily data or when comparing them against GARCH forecasts. It will also act to ensure that results from this study are not just specific to our data and methodology. Table III presents the results of the encompassing regression in equation (1), where the implied volatility forecasts are compared with the daily GARCH(1,1) forecasts at the daily and 20- day horizons in an information test (fix horizon, find best corresponding forecasts) and an efficiency test for the 20-day horizon (use only days for which the option maturity is exactly 20 trading days) 14. -Insert Table III here- 14 All the results below are for standard deviations, taking the square root of realized volatility, implied volatility and ARFIMA and GARCH variance forecasts. This mitigates the impact of outliers. The results for variances are available from the authors upon request, and lead to similar conclusions. 20

22 The results indicate that in all cases the implied volatility subsumes most of the forecasting information and that in several cases the coefficient of the GARCH forecast is not significantly larger than zero, similar to findings in the existing literature. For the YEN/USD exchange rate, matching the forecast horizon to the option maturity actually favours GARCH which seems counterintuitive. However, in doing so, the sample size is reduced to just 36 observations, leaving the results more susceptible to chance. Moreover the option implied volatility is based on 2 calls and 2 puts instead of the 4 calls and 4 puts used when interpolating between two implied volatilities, potentially increasing measurement errors. On average over 20 regressions for the 20-day horizon GARCH has no incremental information similar to Jorion s (1995) findings. Blair et al. (2001) compare the GARCH model extended with REALIZED with the GARCH model extended with implied volatility, finding that for the S&P 100 the GARCH model extended with implied volatility is superior based on the regression R 2. We also find this result for the S&P 500 futures options (results not reported here), although not as striking as in Blair et al. Furthermore the implied volatilities provide marginally better forecasts than the GARCH model extended with implied volatility. Consequently the GARCH model extended with implied volatility will not be considered GARCH extended with implied volatility improves over implied volatility on its own when comparing the forecasts on HRMSE instead of the regression R 2. Using HRMSE, however, GARCH extended with REALIZED (and the forecasts from the long memory model) are better than the forecasts from GARCH extended with implied volatilities. 21

23 Implied volatility vis-à-vis long memory The results in Table III confirm earlier findings that the daily GARCH volatility forecasts have little or no incremental information over that already contained in implied volatilities. Table IV presents the results when GARCH forecasts are replaced with long memory forecasts in the encompassing regression with implied volatility. -Insert Table IV about here - The results in Table IV generally show much larger coefficients for the long memory forecasts compared to the corresponding forecasts for the GARCH model in Table III. For the YEN/USD exchange rate, for example, the coefficient for the 1-day time-series forecasts is now compared to the previous figure of Interestingly, however, for the 20-day horizon, the long memory coefficient is not significantly different from zero for both information and efficiency tests. This can be at least partly explained by a very high volatility period starting October 7, 1998 (Asian Currency Crisis). When forecasting for longer horizons, say H days, events that trigger sustained high levels of volatility over a temporary period of time, lead to the so-called ghost-effect. This effect entails a large initial increase in realized volatilities only later to be followed by a delayed adjustment in the time series forecasts, when the large volatility observation becomes part of the in-sample estimation period, H days later. This affects the regression R 2 s for longer forecast horizons. The resulting regression R 2 s can be contrasted to the results for the HRMSE below, which do not favour any model. The long memory Crude Oil coefficients are much higher than those in the corresponding regression for the GARCH model. For the daily horizon, the long memory coefficient, 0.678, is significantly larger than zero and larger in magnitude 22

24 than the coefficient for the implied volatilities, For the 20-day horizon, the long memory forecasts are significant at the five percent level and slightly smaller in magnitude than the coefficient for the implied volatilities. For the IV matched horizon results, the long memory coefficient is no longer significant. However regression coefficients and the R 2 are approximately of the same magnitude as the average coefficients found from the 20 comparable regressions in the information test. The S&P 500 long me mory coefficient for the IV matched 20-day horizon regression is significant at the five percent level. Panel B of Table IV reports, in a summarised qualitative form, the results for the IV matched regressions, for the 6- through to 40-day horizons. For these 35 regressions run for the fixed IV horizons, the long memory coefficient is significant 25 times for the S&P 500, 4 times for the YEN/USD and 28 times for Light Sweet Crude Oil. The results from our data, particularly for the S&P 500 and Crude Oil, suggest that option prices are not informationally efficient and that the time-series forecasts do contain information not already embedded in the option implied volatilities. In general these results show that the long memory model for realized volatilities constructed from high-frequency data improve considerably upon the daily GARCH model 16. This confirms the results of Andersen et al. (2003) for the YEN/USD and DEM/USD exchange rates. This study extends these findings further, to the equity and commodities markets. The results also provide evidence that both implied volatilities and the long memory forecasts have incremental information not captured by the other. This suggests a composite forecast, combining the implied volatilities and long memory 16 Also based on the regression R 2 and the HRMSE the long memory model volatility forecasts are superior to the daily GA RCH forecasts (results not reported here). 23

25 forecasts, has the potential to outperform the individual forecasts. Table V reports the information test regression R 2 s and HRMSEs for the implied volatilities, the long memory forecasts, and an equally weighted sum of the two forecasts. -Insert Table V about here- The results are mixed when comparing the implied volatilities with the long memory forecasts. The (average) regression R 2 s for the implied S&P 500 volatilities are higher than the long memory forecasts for all but the 30- and 40-day horizons. For the YEN/USD, implied volatility R 2 s are higher than the R 2 s for the long memory model, for all but the 1- day horizon and for Light, Sweet Crude Oil, implied volatility R 2 s are higher for all but the 1-, 5- and 40-day horizons. In total, across all three assets, implied volatility wins 12 times and the long memory model 6 times. As an illustration, the Light, Sweet Crude long memory model regression R 2 is at a 1-day horizon, compared to for the implied volatilities. The average R 2 s for the 20-day horizon are and in favour of the implied volatilities. The HRMSE is lower for the long memory forecasts on 11 occasions and for implied volatility on 6 occasions. -Insert Table VI about here - Table VI reports the corresponding results when only considering those days for which the option maturity matches the forecast horizon. The results for the 10-day to 40- day horizons are similar to those in Table V with implied volatility scoring 7 wins and 5 losses based on R 2 and long memory forecasts scoring 9 wins and 3 losses based on 24

26 HRMSE. Surprisingly, these statistics favour the long memory model to a larger extent than in Table V. As previously mentioned, matching the forecast horizon to the option maturity does not necessarily favour implied volatility. The conclusion of the current literature, that historical time-series forecasts cannot compete with implied volatilities, clearly does not hold when using a long memory model for realized volatilities based on high-frequency data. In formulating the composite forecast as the equally weighted average of the implied volatilities and the long memory model forecasts, no attempt has been made to optimise the weights. Doing this would obviously further reduce the size of the out-ofsample period, as another in-sample period would be needed to optimise the weights within the encompassing regression framework. The optimal ex-post weights are the estimated coefficients in Table IV, and there is no clear pattern across horizons and assets. The composite forecasts with equal weights appear very promising. Table V shows that, on aggregate over all three assets and six horizons, the (average) R 2 of the composite forecast is higher than that of both the individual forecasts in 14 out of 18 cases, including all horizons for the S&P500 and Crude Oil. In 11 of those 14 cases the R 2 is statistically significantly higher than both the R 2 s of the individual forecasts using Diebold and Mariano s (1995) asymptotic test for the difference between two loss functions. The HRMSE of the composite forecast is lower than the HRMSE of both the individual forecasts in 15 out of 18 cases. Most significantly there are no cases where the composite forecast under performs the individual forecasts on both the R 2 and the HRMSE. Table VI confirms the validity of these results with the IV matched results showing that the 25

27 composite forecast is superior to each of the individual forecasts in 8 out of 12 cases based on R 2 s, and superior in 11 out of 12 based on HRMSE. CONCLUSION For the S&P 500, YEN/USD and Light, Sweet Crude Oil futures and futures options we are able to confirm the results of existing studies based on daily data. Implied volatilities provide superior forecasts compared to the daily GARCH(1,1) model consistent with findings of Christensen and Prabhala (1998) and Fleming (1998) for the S&P 100 index, Jorion (1995) for exchange rates, and Day and Lewis (1993) for Crude Oil. Furthermore, in conducting the analysis with high-frequency returns it is found that GARCH extended with implied volatilities provides better forecasts than GARCH extended with realized volatility, consistent with Blair et al. (2001). Utilizing the most recent developments in time-series volatility forecasting, a fractionally integrated autoregressive model is estimated for realized volatilities computed from squared high-frequency returns. This dramatically alters the outcome of the contest between time-series models and implied volatilities. It is shown that the long memory volatility forecasts can compete with implied volatilities, and in some instances even outperform implied volatilities. It is also evident that both implied volatilities and long memory forecasts have information that the other does not contain. The consistency of these results is apparent, since the findings are similar across three different assets in three different asset classes. There are two possible explanations for our findings. First, it may be that option prices are inefficient, because while option traders may have been able to capture GARCH 26

28 effects, they seemingly do not effectively incorporate the long memory characteristic that is found in intraday volatility series. Second, option prices may in fact be efficient but we have been unable to precisely extract the market s expectation of volatility. Arguments against the latter reason would articulate that implied volatility measurement errors have been minimized in this research by using settlement prices of options on futures and that these same implied volatilities outperform GARCH forecasts. This points to the conclusion that our methodology has improved option price efficiency relative to previous research. The results in this study show that while options traders appear to have had the edge over time series analysts, the long memory model seems to have levelled the playing field. A composite forecast combining implied volatilities and the ARFIMA forecasts provides the optimal volatility forecast. 27

29 References Andersen, T.G., & Bollerslev, T. (1998). Answering the skeptics: Yes standard volatility models do provide accurate forecasts. International Economic Review, 39, Andersen, T.G., Bollerslev, T., Diebold, F.X., & Ebens, H. (2001). The distribution of realized stock return volatility. Journal of Financial Economics, 61, Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P. (2001). The distribution of realized exchange rate volatility. Journal of the American Statistical Association, 96, Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica, 71, Barone-Adesi, G., & Whaley, R.E. (1987). Efficient analytical approximation of American option values. Journal of Finance, 42, Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, Black, F., & Scholes, M.. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, Blair, B., Poon, S. -H., & Taylor, S.J. (2001). Forecasting S&P 100 volatility: The incremental information content of implied volatilities and high frequency returns. Journal of Econometrics 105, Canina. L., & Figlewski, S. (1993). The informational content of implied volatility. Review of Financial Studies, 6, Chiras, D.P., & Manaster, S. (1978). The information content of option prices and a test of market efficiency. Journal of Financial Economics, 6,

30 Christensen, B.J., & Prabhala, N.R. (1998). The relation between implied and realized volatility. Journal of Financial Economics, 50, Day, T., & Lewis, C. (1992). Stock market volatility and the information content of stock index options. Journal of Econometrics, 52, Day, T., & Lewis, C. (1993). Forecasting futures market volatility. Journal of Derivatives, 1, Diebold, F.X., & Mariano, R.S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13, Ebens, H. (1999). Realized stock volatility. Working paper Johns Hopskins University. Ederington, L.H., & Guan, W. (2000a). Is implied volatility an efficient and effective predictor of future volatility? Working paper University of Oklahoma. Ederington, L.H., & Guan, W. (2000b). Measuring implied volatility: Is an average better? Working Paper University of Oklahoma. Fleming, J. (1998). The quality of market volatility forecasts implied by S&P 100 index option prices. Journal of Empirical Finance, 5, Fleming, J., Ostdiek, B., & Whaley, R.E. (1995). Predicting stock market volatility: A new measure. Journal of Futures Markets, 15, Jorion, P. (1995). Predicting volatility in the foreign exchange market. Journal of Finance, 50, Kroner, K.F., Kneafsey, K.P., & Claessens, S. (1995). Forecasting Volatility in Commodity Markets. Journal of Forecasting, 14, Latane, H.A., & Rendleman, R.J. (1976). Standard deviations of stock price ratios implied in option prices. Journal of Finance, 31,