FTSE-100 implied volatility index

Transcription

1 FTSE-100 implied volatility index Nelson Areal NEGE, School of Economics and Management University of Minho Braga Portugal Phone: Ext. 5523, Fax: February 2008 Abstract Three different methodologies to construct the UK implied volatility index (VFTSE) are suggested using high-frequency data on FTSE-100 index options. We consider construction methodologies similar to the VXO volatility measure based on the S&P 100 options and to the VIX model-free volatility measure based on the S&P 500 options. A detailed description of the database and some stylised facts about the FTSE-100 option implied volatilities are presented. An analysis of the statistical properties of the volatility indices that result from the use of different construction methodologies is performed as well as the analysis of their forecasting ability. We found that the realised volatility measure constructed using high-frequency data on FTSE-100 index futures is the best forecast of future 22 trading day volatility. All the volatility indices with the exception of one perform similarly well. Among the indices that show to have good information content the volatility index with the best statistical properties is chosen as the VFTSE index. An analysis of the VFTSE and its statistical properties is performed, where it is shown that the VFTSE series also exhibits long memory effects which can effectively be removed by using a filtering scheme. This work was supported by the Portuguese Foundation for Science and Technology. I am grateful for helpful comments from Stephen Taylor, Mark Shackleton and Martin Martens, on previous drafts of this work. All errors are the sole responsability of the author.

2 FTSE-100 implied volatility index Abstract Three different methodologies to construct the UK implied volatility index (VFTSE) are suggested using high-frequency data on FTSE-100 index options. We consider construction methodologies similar to the VXO volatility measure based on the S&P 100 options and to the VIX model-free volatility measure based on the S&P 500 options. A detailed description of the database and some stylised facts about the FTSE-100 option implied volatilities are presented. An analysis of the statistical properties of the volatility indices that result from the use of different construction methodologies is performed as well as the analysis of their forecasting ability. We found that the realised volatility measure constructed using high-frequency data on FTSE-100 index futures is the best forecast of future 22 trading day volatility. All the volatility indices with the exception of one perform similarly well. Among the indices that show to have good information content the volatility index with the best statistical properties is chosen as the VFTSE index. An analysis of the VFTSE and its statistical properties is performed, where it is shown that the VFTSE series also exhibits long memory effects which can effectively be removed by using a filtering scheme. 1 Introduction Estimates of future stock market volatility are important for practitioners and academics alike. Practitioners need this estimate to make decisions about asset allocation, option valuation and trading strategies. Academics are interested in studying the statistical properties of risk and return. Volatility indices use implied option volatilities information, thus representing the market consensus of the future stock market volatility. By their construction these indices are based on market data, are forward-looking and have a constant forecast horizon (Fleming, Ostdiek and Whaley, 1995). These volatility indices are sometimes referred as the investor fear gauge, the higher the index, the greater the fear (Whaley, 2000). Implied volatilities have been used to forecast future volatility, e.g.: Day and Lewis (1992), Lamoureux and Lastrapes (1993), Fleming (1998), Christensen and Prabhala (1998), and Christensen and Hansen (2002). There is also evidence that volatility indices provide efficient estimates of the short-term market volatility (Harvey and Whaley, 1992b; Fleming, Ostdiek and Whaley, 1995). More recently Blair, Poon and Taylor (2001) showed that volatility indices provide more accurate volatility forecasts than a measure of realised volatility obtained from high frequency returns. Martens and Zein (2004) compare the long memory forecasts with the implied option volatilities for the Standard and Poors 500 stock market index index, and find incremental information for the long memory forecasts. 1 Corrado and Miller Jr. (2005) 1 Pong, Shackleton, Taylor and Xu (2004) compare the long memory forecasts with short memory forecasts of pound, mark and yen exchange rates against the dollar using high frequency data for the historical volatility and conclude that the superior accuracy of historical volatility, relative to implied volatilities, comes from the 1

3 tested the information content of three US volatility indices and show that the forecasts based on those indices, although upward biased, are still more efficient in terms of mean squared forecast errors than historical realised volatility. 2 Recently several authors (Demeterfi, Derman, Kamal and Zou, 1999; Britten-Jones and Neuberger, 2000; Carr and Madan, 2002; and Jiang and Tian, 2005b) suggested the use of model free implied volatility, which are computed from option prices data without requiring the use of any option pricing model. The model free CBOE market volatility index (VIX) index has been used by Bollerslev, Gibson and Zhou (2005) to estimate the volatility risk premium. Giot (2003) provides evidence that volatility indices provide meaningful volatility information in Value-at-Risk (VaR) models. Jiang and Tian (2005b) compare the forecast ability of model free implied volatility for the Standard and Poors 500 stock market index (S&P 500) index options with a high frequency realised volatility measure and the Black and Scholes (1973) implied volatility and found that model free implied volatility subsumes all information contained in the Black and Scholes (1973) implied volatility and past realised volatility and is a superior forecast for future realised volatility. Volatility indices can give rise to the introduction of futures and options instruments on such indices, as recently occurred in the US market. These derivatives can be used in turn to create hedge strategies against changes in volatility, or to speculate on changes in the market volatility. 3 This study proposes for the first time, as far as we are aware, a volatility index for the United Kingdom (UK) market by using option s data on the FTSE-100 stock index. This new index is designated as the UK implied volatility index (VFTSE) and is going to rely on implied volatility estimates. Accordingly to Harvey and Whaley (1991, 1992a), in order to ensure a reliable implied volatility estimate, it is required that the valuation model takes into account the early exercise opportunity of American options, and the discrete cash dividends of the stock index; simultaneous (contemporaneous) stock index levels must be used; and finally, multiple option transactions must be used to estimate market volatility. We will follow all of these recommendations in order to reduce the measurement errors of implied volatilities when estimating the VFTSE. We will consider different construction methodologies for the volatility index, including a model free implied volatility index, investigate the statistical properties of these indices, and select the one which provides the best future 22 trading day volatility forecast. Our study also contributes to the literature since there are few studies using UK market data that compare the forecasting ability of FTSE-100 implied volatilities. Gwilym and Buckle (1999) find evidence that historical volatilities are more accurate predictors than implied volatility (based on the mean squared error and mean absolute error), and regression results suggest that implied volatilities are upward biased estimates of future volatility but superior in terms of information content. They consider different forecasting horizons and conclude that the accuracy is better for longer forecast horizons for all measures, and that the accuracy of use of high frequency returns and not from a long memory specification. 2 For more on volatility forecasting please refer to Poon and Granger (2003) and references therein. 3 For more on using futures and options on a volatility index to devise strategies to hedge volatility please refer to Brenner and Galai (1989), Whaley (1993), Carr and Madan (2002), and Psychoyios and Skiadopoulos (2004). 2

4 historical methods is improved when the amount of past data used is matched to the forecast horizon. Noh and Kim (2006) use FTSE-100 and S&P 500 data to conclude that historical volatility using high-frequency returns outperforms implied volatility in forecasting future volatility in the case of FTSE-100 data, but when considering the S&P 500 data set implied volatility performs better than historical volatility. Their FTSE-100 results also suggest that implied volatility can be an unbiased forecast for future volatility. Our study presents several differences when compared to previous studies. First it is one of the few studies that considers UK market data; second it compares several different realised volatility measures computed using high-frequency data; third it uses several implied volatility indices constructed for the first time using FTSE-100 options data for which great care was taken to avoid implied volatility measurement errors. In summary we find that high frequency historical volatility is a better forecast of future 22 day ahead volatility and that the volatility indices here proposed are a biased forecast of future volatility. We find that most of the volatility indices perform similarly well when forecasting future volatility, with the exception of the model free realised volatility index. These results are robust to the choice of the realised volatility measure. Based on the information content of the various volatility indices proposed and on the analysis of their statistical properties we recommend one of them to represent the implied volatility of the FTSE-100 index. In the next section we will describe the construction methodology of volatility indices on the US market (VXO, VIX, VXN and VXD). Section 3 describes the construction methodologies of various UK implied volatility indices. A detailed description of the database and some stylised facts about the FTSE-100 option implied volatilities are presented in sections 4 and 5. Section 6 presents the various implied volatility indices series and their statistical properties. In section 7 the information content of volatility indices and of realised volatility is put to test, a description of the data set, the methods used, the results and several robustness tests are also presented. Section 8 presents the VFTSE long memory properties. Final remarks about this study are presented in section 9. 2 Volatility indexes Volatility indices were first suggested by Gastineau (1977) and then followed by Cox and Rubinstein (1985), Brenner and Galai (1989) and Whaley (1993). Gastineau (1977) proposes the use of an average of at-the-money options on 14 stocks with three to six months to maturity combined with a measure of historical stock market volatility. Cox and Rubinstein (1985) suggest an improvement on this procedure by considering multiple call options on each stock, and introduce a weighting scheme where the volatilities are averaged in such a way that the index will be at-the-money and will have a constant time to expiration (Fleming, Ostdiek and Whaley, 1995). Brenner and Galai (1989) propose the construction of a volatility index for the equity, bond and foreign exchange markets based on the historical volatility, implied options volatilities, or some weighted combination of implied and historical volatility measures. 3

5 The Chicago Board Options Exchange (CBOE) introduced in 1993 the CBOE market volatility index (VIX) stock index with the same construction methodology as suggested by Whaley (1993) which uses Standard and Poors 100 stock market index (S&P 100) options implied volatilities. On 22 September 2003 the CBOE changed the construction method of the index, and renamed the index computed with the original methodology to VXO. The new VIX is not only computed by a different method but is also based on S&P 500 options data. 4 Currently, there are several volatility indices being maintained and distributed: the Chicago Board Options Exchange (CBOE) Market Volatility Index (VIX) on S&P 500, the VXO on the S&P 100 stock index, the CBOE DJIA Volatility Index (VXD) on the Dow Jones Industrial Average (DJIA), the CBOE Nasdaq-100 Volatility Index (VXN); the German volatility index (VDAX); and the Marchà c des Options Nà c gociables de Paris (MONEP) volatility indices (VX1 and VX6) on the CAC-40 index options. There is also a volatility index developed by Dowling and Muthuswamy (2003) for the Australian market designated as AVIX based on implied volatilities on the Australian stock exchange index options (S&P/ASX200) index options; and a Greek option market volatility index (GVIX) suggested by Skiadopoulos (2004). But such indices are not maintained or distributed in real time to the best of our knowledge. This section presents a brief description of the US market volatility indices and their different construction methodologies. Table 1 contains a summary of information related to those indices namely: their names, underlying option index and option exercise type, introduction date, data history, and any changes that affected them. Table 1 about here 2.1 The US market volatility indices The old VIX volatility index, currently designated as VXO, was suggested by Whaley (1993) and introduced by the CBOE in This index had a price history since 1986, and is based on the S&P 100 (OEX) options. On 22 September 2003, CBOE changed the computation methodology of VIX and started to provide prices for two indices; the original-formula index was renamed VXO and a new VIX was introduced with a new construction methodology. 5 On the same date, the CBOE changed the computation methodology of the CBOE NASDAQ Volatility Index (VXN) which had been introduced in It was previously based on the VXO index construction methodology, and is now based on the VIX. 6 On 26 March 2004, the CBOE Futures Exchange (CFE) started trading futures on VIX. VIX options were launched in February On 18 March 2005, the CBOE announced the beginning of the dissemination of a new volatility index based on the Dow Jones Industrial Average (DJIA): the CBOE DJIA Volatility 4 From now on, unless stated otherwise VIX designates the new volatility index and VXO the index computed under the original specification. 5 The CBOE provide, on its website, figures dating back to 1990 computed with the new methodology for the VIX. 6 The new VXN substituted the old one and has a history dating back from 2 February 2001 to the present. For more on the old VXN, its statistical properties and information content please refer to Simon (2003). 4

6 Index (VXD) based on option prices on the Dow Jones Industrial Average stock index. 7 The construction methodology of this index is also based on the same methodology as the VIX. 8 The CBOE Futures Exchange (CFE) introduced CBOE DJIA Volatility Index Futures on 25 April We will describe the computation process of the two indices and will adhere to the CBOE nomenclature, so that the old volatility index VIX will be designated as VXO, and the new index will be designated as VIX. As already mentioned, the VXN and VXD are based on the same construction methodology as the new VIX The S&P 100 volatility index (VXO) construction methodology The VXO was introduced by Whaley (1993) and is based on the implied volatilities of eight different S&P 100 options (OEX) and represents the market consensus of the expected volatility over the next 30 calendar days. At the time VXO was introduced, the OEX options were the most liquid index option instrument traded on the US market. The aim of this index was to create the foundations on which volatility derivatives could later be traded on the market. The VXO is based on eight near-the-money, nearby and second nearby traded OEX options. It is constructed in a fashion such that at any given time it represents the implied volatility of an at-the money OEX option with 30 calendar days to expiration. Such an index will move approximately linearly with changes in prices of options induced by changes in option volatility. Thus this maximises the hedging effectiveness of volatility index derivatives. Ederington and Guan (2002b) compared the forecasting ability of several different weighting schemes, including the one used by VXO, and conclude that this weighting scheme produces volatility estimates which are among the best forecasts of future volatility. The nearby option series (here designated as N) is defined as the series with shortest time to maturity, and with at least 8 calendar days to maturity. 9 The second nearby contract (SN) is the next contract to expire in relation to the nearby option series. 10 For each maturity, four options implied volatilities are used, two from a call and a put just below the current index level (X l ), and two more from a call and a put just above the current index level (X u ). The eight options are depicted in Table 2. Table 2 about here The first step is to average the call and put option implied volatilities for each of the four options categories, to define: σ X l N =(σx l c,n + σx l p,n )/2, σxu N, σx l SN, and σxu SN. After this, compute the at-the-money implied volatilities for the nearby (σ N ) and the second nearby (σ SN ) volatilities by interpolating the in and out-of-the-money implied volatilities: 7 They also introduced on the same date the CBOE DJIA BuyWrite Index (BXD). A buy-write, also called a covered call, is an investment strategy in which an investor buys a stock or a basket of stocks, and also sells call options that correspond to that stock or basket of stocks. For more on this index please refer to Whaley (2002). 8 The CBOE provides daily data on this index from 7 October 1997 onwards. 9 An eight-day-to-maturity limit is imposed since options with shorter maturities tend to have higher volatilities. Such options may also induce liquidity related biases. 10 For a thorough description of the VXO construction methodology please refer to Whaley (1993, 2000). 5

7 σ N σ SN = σ X l N = σ X l SN ( ) Xu S X u X l + σ Xu N ( ) Xu S X u X l + σ Xu SN ( ) S Xl X u X l ( ) S Xl X u X l (1) (2) (3) Finally, interpolate/extrapolate between the nearby and the second nearby volatilities to create a 30 calendar day (22 trading day) implied volatility: V XO ( ) ( ) NtN NtN = σ N N tsn N tn + σ SN N tsn N tn (4) where N tn is the number of trading days to maturity of the nearby contract and N tsn is the number of trading days to maturity of the second nearby option series. The VXO is based on trading days, but for the computation of options implied volatilities it uses calendar days. Whaley (1993) then suggests to transform the calendar day implied volatility into a trading day implied volatility by: 11 ( ) Nc σ t = σ c Nt (5) where σ t is the trading day implied volatility rate, σ c is the calendar day implied volatility rate, N c is the number of calendar days to maturity, and N t is the number of trading days to maturity given by: 12 N t = N c 2 int(n c /7) (6) The S&P 500 volatility index (VIX) construction methodology The new VIX aims, as the VXO, to represent the expected market volatility over the next 30 calendar days. It remains based on real time data on stock index options and is calculated and dessiminated every minute of each trading day. The three main differences in the new VIX formulation are: the change of options it is based upon; the consideration of a broader range of strike prices to compute the index; and finally the fact that the expected volatility is derived directly from option prices. data. 13 The VXO was based on S&P 100 option prices whereas the VIX is based on S&P 500 options The new VIX considers the entire range of strike prices, contrary to the VXO which 11 This assumes that total volatility over the options remaing life is the same whether time to maturity is measured using calendar days or trading days (Whaley, 1993). 12 This adjustment causes the VXO to be higher than actual volatility. Blair, Poon and Taylor (2001) and Simon (2003), when studying the VXN, adjust the index values in order to remove this adjustment. 13 The S&P 100 option ticker is OEX and the S&P 500 option ticker is SPX and is an European exercise style option. The SPX average daily transaction volume has grown in 2004 to about 196, 000 contracts, as opposed to the OEX average daily volume for the same year of about 65, 000 (CBOE, 2004). 6

8 considered only at-the-money strikes. Finally the estimation formula now in use does not require any method of option valuation, instead it uses only option prices. The CBOE claims that these changes improved the VIX, since it is based on options on the S&P 500 which is the primary U.S. stock market benchmark, closely followed by many stock funds; also it pools information from option prices over the whole volatility skew. 14 The VIX now uses out-of-the-money put and call options weighted by the inverse of the square of their strike prices. The new estimation procedure relies on the concept of fair value of future variance developed by Demeterfi, Derman, Kamal and Zou (1999). Volatility is going to be estimated by the square root of the price of variance, which is in turn estimated by valuing a variance swap which is a forward contract on realised volatility. The fair value of future variance is given by: 15 σ 2 = 2 T ( ( S0 rt e rt 1 S + e rt S 1 K 2 c(k)dk ) ) ln S S + e rt 1 p(k)dk+ (7) S 0 K2 where S 0 is the current underlying asset price, T is the option time to maturity, K is the option strike price, r the risk-free interest rate, c and p are the call and put prices, respectively, and S is an arbitrary stock price (usually chosen to be close to the forward price). Since the estimation of the risk-neutral expected volatility is derived using only option prices, the volatility estimate is a model free expected volatility. Besides the advantage of not assuming any option valuation model, there is also no need to estimate the dividends paid over the life of the option. Jiang and Tian (2005a) demonstrate that this measure is conceptually identical to the model free implied volatility developed by Britten-Jones and Neuberger (2000), which is defined as: 16 0 σ 2 = 2erT T ( F0 0 p(k) ) K 2 dk + c(k) F 0 K 2 dk (8) where F 0 is the forward price with the same maturity as that of the option. Jiang and Tian (2005b) show that this method yields the correct measure of total riskneutral expected integrated variance even in a jump-diffusion setting. They also show that the model free implied volatility subsumes all information contained in the Black and Scholes (1973) implied volatility and historical variance as is a more efficient forecast for future realised volatility. The new estimation procedure has nevertheless some limitations. First of all it should provide perfect expected risk neutral volatilities only when it is estimated using an infinite 14 Please refer to Carr and Wu (2006) for a detailed comparison between VXO and the new VIX. 15 For a detailed description of the valuation of this contract please refer to Demeterfi, Derman, Kamal and Zou (1999). 16 See also the related work of Carr and Madan (2002), Bakshi and Madan (2000), Bakshi, Kapadia and Madan (2003), Carr and Wu (2004) and Jiang and Tian (2005b). 7

9 number of strikes, when the strike price intervals approach zero. Since these assumptions are not verified in practice, this estimate may be a biased estimate of expected volatility. Carr and Wu (2004) and Jiang and Tian (2005b) demonstrate that necessary steps must be taken in order to minimize these implementation errors. Jiang and Tian (2005a) demonstrate that the CBOE construction methodology does not accomplish this which may lead to substantial bias in the calculated index values. 3 Construction methodologies for a FTSE-100 volatility index In the previous section we described several methodologies already used to construct volatility indices. In this section we will present different methods considered to construct a FTSE-100 volatility index (VFTSE), in section 6 the statistical properties of these indices are analysed, and in section 7 their information content are assessed. The UK option market, although a liquid market, is not as liquid as the US market. So we are unable to use the same methodology to construct the VFTSE as the one used for the VXO. If the same methodology was used there would be many days where it would not be possible to find eight options with the required characteristics to compute the index with simultaneous time stamps; and even for more recent data it would not be feasible to use the VXO methodology to compute intraday values for the index. Therefore, we propose a method to construct the index that still considers implied volatilities on eight options but with a different interpolation scheme. The volatility index computed with this methodology will be designated as alternative interpolation scheme VFTSE AIS. Since the out-of-the-money options are the ones on which trading is concentrated, information will be incorporated faster into the prices of such options. With this in mind we will consider another version of the index where only out-of-the-money options are considered, VFTSE OTM. We will also compute yet another volatility index, based on model free option prices, using the same methodology as the VIX. Such an index will be designated as the model free volatility index VFTSE MF. 17 For each index a daily figure will be computed using the most recent option prices available for each day. As already mentioned, VFTSE AIS will use eight options implied volatilities, four for the nearby contract and another four for the second nearby contract. We will include options with at least eight calendar days to expiration. The options considered are described in Table 3. The difference between these options and the ones used by VXO is that these eight options are not required to be traded/quoted simultaneously and, therefore, each of the at-the-money options and out-of-the money options are not required to have the same exercise prices. So the spot index level over the exercise price ratio (S t /X t ) is not necessarily the same for all options, since the moment in time (t) they were traded could be different for each of them. Table 3 about here 17 This construction methodology was already described in section

10 The use of puts and calls will reduce possible mis-measurements of the reported index level and the risk-free interest rate estimate. Having identified these eight options we will now interpolate to obtain the at-the-money (S/X = 1) call and put options implied volatility value, for each expiry contract (σ c,n, σ p,n, σ c,sn and σ p,sn ). For example, the at-the-money implied volatility of the call option of the nearby maturity σ c,n will be given by: σ c,n = σ St/Xt<1 c,n + (1 S t /X ) (σ St/Xt>1 c,n σ S t /X t <1 c,n ) tst /X t <1 (S t/x tst /X t >1 St/Xt S t /X t <1 ) (9) With these estimates we will compute the implied volatility value of the nearby contract (σ N ) and the second nearby contract (σ SN ) averaging the put and call implied volatilities of each contract: σ N =(σ c,n + σ p,n )/2 (10) σ SN =(σ c,sn + σ p,sn )/2 (11) If no data is available for either calls or puts on a given maturity, the implied volatility for that maturity is simply the implied volatility of the calls or the puts, depending on which data is available. With the value of the average between the implied volatility of the at-the-money option for each maturity we will interpolate to obtain the value of a 30 calendar days to maturity option implied volatility, using expression 4, but instead of 22 trading days we will use 30 calendar days. For computing the VFTSE OTM we only require four out-of-the-money options, a call and a put for the nearby contract and another call and put for the second nearby contract. With the call and put implied volatilities we will interpolate the value of an at-the-money option implied volatility. And with the estimates of the at-the-money implied volatilities for the nearby and second nearby, we will interpolate/extrapolate to obtain the implied volatility of an option with 30 calendar days to expiration. Table 4 summarises the four options implied volatilities required to construct this index. Table 4 about here On the UK market there are both European and American options on the FTSE-100 stock index. Whenever possible, we will compute, for each volatility index methodology, a volatility index for American options (VFTSE A ), another considering only the European options information (VFTSE E ) and finally one with information from both datasets (VFTSE C ), designated as the complete sample. 18 For the VFTSE MF only two datasets are going to be used, one with American options (VFTSE MF ) and another with European options (VFTSEMF ) data.19 A 18 A detailed description of these datasets will be given in the following section. 19 The model free methodology assumes the use of European options. The use of American options will introduce a bias in estimation of volatility, but this is expected to be small since the methodology uses only out-of-the-money options. 9 E

11 For each dataset we will use two series of option prices: prices on options traded on the market, and contemporaneous bid-ask quote averages. There are two potential advantages for the use of bid-ask averages. Firstly it should reduce the bouncing of prices resulting from the fact that a trade can be originated by a bid or quote, and secondly there is more data on quotes than on trades so information will be reflected faster in this dataset than in trades. For each index (VFTSE A, VFTSE E, VFTSE C ) we will have two versions: one for trades and another for bid-ask averages, e.g.: the VFTSE A will have a version based on trades (VFTSE A,T ) and another based on bid-ask averages (VFTSE A,BA ). Harvey and Whaley (1991, 1992a) concluded that, in order to ensure a reliable implied volatility estimate, it is required that the valuation model takes into account the early exercise opportunity of American options and the discrete cash dividends of the stock index; simultaneous (contemporaneous) stock index levels must be used; and finally, multiple option transactions must be used to estimate market volatility. We will follow all of these recommendations when estimating the implied volatilities of FTSE-100 stock index options. Ideally we would like to use the true FTSE-100 index value, but this is not possible since in a broad index like the FTSE-100 even the reported index level is a stale indicator of its true level because not all stocks in its portfolio are traded continuously. As the true value is unavailable, a proxy must be used. Whaley (1993) suggests the value of an actively traded futures contract on the index. Unfortunately, the FTSE-100 futures have a series of only four maturities over the year, therefore some sort of approximation would have to be used to imply future prices from the currently traded futures maturities. This would also introduce an estimation error problem. We opted to use the FTSE-100 spot index as an indicator of its actual level. 20 When calls and puts are used, any bias produced by the staleness of reported index levels will be reduced, as the bias will be approximately equal and opposite for call and put options implied volatilities. Contrary to the S&P 100 market, the UK stock market has the same opening hours as the options market. Despite this, there is a wildcard option embedded in the valuation of FTSE-100 index options. 21 For most of the period considered, the settlement time was 16:10 and the option holder had until 16:31 to communicate to the clearing house an early exercise decision. 22 Dawson (2000) studied the value of the wildcard option for the American options on the FTSE- 100 index over the period between 4 January 1993 and 31 January 1996, and concluded that the wildcard option has an insignificant value. The author attributes this contrasting result with the US market to the relatively quiet wildcard period in London Harvey and Whaley (1991) show that the infrequent trading of stocks of the S&P 100 index at the closing of the day is not a problem when estimating the option s implied volatility using spot index values. 21 A wildcard option arises when an investor has an interval following the determination of the day s settlement price during which he can decide whether or not to exercise (Dawson, 2000). Any information arriving during this period can influence the value of the option but not the settlement price. 22 This period later changed. For instance, after 25 May 2000 the buyer had up to 16:55 on any business day prior to the expiry date of the contract to give the clearing house an exercise notice. On the exercise day of a contract the final exercise time limit is 18:00 hours. 23 For more details on the wildcard option please see: Fleming and Whaley (1994), Dawson (2000) and references therein. 10

12 4 The dataset For the current analysis we will need intraday data on the FTSE-100 index spot value, records of all trades and quotes of the FTSE-100 European and American options, a proxy for the riskfree interest rate, and the FTSE-100 discrete dividends over the period. We will next describe in more detail each of these required parameter inputs to estimate the implied volatility of options on the FTSE-100 stock index. 4.1 FTSE-100 dividends FTSE-100 dividends were computed by FTSE and obtained from DataStream for the period from 14 June 1993 to 18 December The data was provided as an ex-dividend adjustment expressed in index points on a cumulative basis for the calendar year. The dividends that FTSE considered up to 7 July 1997 are the declared gross dividends and thereafter the net dividends. 4.2 Interest rates We will use the Euro Sterling currency rate as a proxy for the risk-free interest rate. The data was collected from DataStream, with a daily frequency for the analysed period for the overnight, one week, one month, three months, six months and one year maturities. Since we will consider options with one, two and three-months to maturity, we will use the rates with maturities which are closest to the option maturity Spot data The spot data of the FTSE-100 index has two sources. For the period from 01 May 1990 to 25 November 1994 and from 26 August 1996 to 17 March 2000 it was obtained from the FTSE company. The same source was unable to provide data for the period from 25 November 1994 to 26 August The dataset comprised high-frequency spot data. Up to 23 February 1998 we have records with 1 minute intervals and from then on we have intraday quotes at 15 seconds intervals. There are two records given by the FTSE dataset that have a zero value for the spot index, and these were deleted. The total number of records of spot data given by FTSE is 1,836,788. All of them are valid, corresponding to 2,113 days. To fill the void between 1994 and 1996, we used the information contained in the options dataset (described in detail in the next section). Up to the end of 1997, the Settlement field of the options database corresponds to the latest value of the FTSE-100 spot index. Using this information we were able to get the values for the spot index for all the minutes where there is an option traded for the missing period. To gather as much information as possible, we used both American and European options records. To test the accuracy of this information we compared it with the spot data given by the FTSE for the period when this data was available. 24 The start of the period is the first day of availability of this dataset. 25 We performed all calculations using also an interest rate given by the interpolation of the two nearest interest rates maturities and the results are almost identical. 11

13 There is only one record, on 13 May 1991, of an American option that has two different spot values for the index within the same minute. When we compare the values of the spot index given by FTSE we conclude that one of its values is clearly not correct. This record was deleted. There are 459 cases where the spot level given from the American options is different to that given by European options. In these cases we used the data from the European options. This decision results from the analysis of the period in which we have both spot data given by the options and spot data given by FTSE which enables us to conclude that in these cases the correct data is given by the European options. Please note that usually this difference is only 0.1 index points. There are 77,047 occasions where there are differences between the values of the spot index from the two sources. For 14,081 records the difference is due to a delay in one minute of the option s spot index in relation to the FTSE. Only in 4679 cases is the difference greater than 1 index point. After these procedures we ended up with a database with 1,963,177 spot records over 2,497 days. The trading hours over this period are from 8:00 to 16:30 hours. There is one day (27 July 1995) with only two records during the complete day, and there are four days missing from the dataset: 20 March 1992, 31 December 1998, 30 July 1999, and 18 November Options data FTSE-100 data was obtained from the floor option trades and quotes records for FTSE-100 options (European and American) contracts sold on compact disk (CD) by the London International Financial Futures Exchange (LIFFE). Each record in the LIFFE data files provides a time recorded to an accuracy of one second and also a trading volume when the price refers to a transaction. The options database comprises the following periods: from 02 January 1991 to 30 May 2000 for American options; and from 02 January 1991 to 06 June 2000 for the European exercise type contracts. The original database was subjected to screening to remove any record with a time stamp outside the opening hours (1,050,260 records), zero volume in the case of trades (1 record), a premium of zero (92, 362 records), and with dates beyond the expiry date of the contract (340 records). After deleting all such records we ended up with 3, 659, 330 records of American trades and quotes over the period of 2, 309 days; and with 1, 368, 257 records for European options over the period of 2, 313 days. The difference in the number of days is due not only to the different end dates for the two types of options but also to missing days in the American options database. There are some days which are not holidays missing from the dataset, 65 days in total, most of them are in February 1998 (10 days), July 1999 (20 days) and May 2000 (15 days). 26 The options trading hours changed during the analysed period: from 02 January We also found a few days where there are records of trades and no records of quotes, and vice-versa, for both European and American options. 12

14 to 17 July 1998 the market was open from 8:35:00 up to 16:10:00; from 20 July 1998 to 17 September 1999, there are records from 8:35:00 to 16:30:00; and finally from 20 September 1999 to the end of the period the opening was at 8:00:00 and the close at 16:30:00. During this period the timing of the expiry dates of the option contracts also changed. The last trading day of the January 1991 contract is 11 January 1991, which corresponds to the second Friday of the month; from the February 1991 contract to the May 1992 contract the last day of transactions is the last day of the month or the previous working day, if the last day of the month is a holiday; from the June 1992 contract onwards the last trading day is the third Friday of the expiry month, or the previous working day if it is a holiday. Only after 02 January 1998 are there valid volume data for the records of trades. Until then, all the trades have a volume of one. Up to 1998 the number of records on American options was significantly higher than on European options, both for trades and quotes. From 1998 onwards the number of records is similar for American and European options, but the average volume traded is substantially higher for European options. For the computation of the volatility index we will only require the records on the next to expiry contract, with at least 8 calendar days to maturity, and the second next to expiry contract. We opt to consider options with at least 8 calendar days to expiration because the time value of options with short lives are very small relative to their prices which makes the implied volatilities more volatile. As already mentioned we will construct a volatility index using trades and another one using the bid-ask average quotes records. In order to construct the bid-ask average records we matched the quote records by time stamp and averaged the bid and ask contemporaneous quotes for that option. All the options records were merged with the spot records in order to match each option record with the closest spot record available. After this we ended up with the following database: 468, 704 records of trades and 918, 346 bid-ask averages records, over the period from 14 June 1993 to 17 March Implied volatilities of options on the FTSE-100 stock index Within the Black-Scholes paradigm the price of an option (O(S, X, D, σ, r, t)) is dependent on the current asset value (S), the exercise price (X), discrete dividends over the life of the option (D), volatility of the underlying asset price (σ), the riskless interest rate (r) and time to maturity (T ) of the option. Implied volatilities can be estimated by inverting the valuation formula: σ = σ(o, S, X, D, r, t). To estimate the implied volatility of options on the FTSE-100 stock index, we will use a Brent algorithm to invert this valuation formula. 27 To value American options we will use: the recombining binomial tree of Schroder (1988) with quadratic interpolation (B-QI ) to value calls with a moneyness level of S/X < 1.1 and puts with a moneyness level of S/X > 0.9; for deep-in-the-money call options (S/X 1.1) the value will be given by the recombining tree suggested by Hull and White (1988) and Harvey 27 For more on this algorithm please refer to Press, Teukolsky, Vetterling and Flannery (1992). 13

15 and Whaley (1992a); and finally for deep-in-the-money put options (S/X < 0.9) we will use a recombining binomial tree with Wilmott, Dewynne and Howison (1998) parameters. All binomial trees were computed using 500 time steps. To value the European option s value we will use the Bos, Gairat and Shepeleva (2003) approximation. 28 Implied volatilities were estimated and after testing for boundary conditions and deleting the records which did not conform we ended up with: 345, 326 records of trades for American options (172, 295 calls and 173, 031 puts); and 116, 813 European options trades (59, 096 calls and 57, 717 puts). The bid-ask averages dataset totals 690, 829 records of American options (341, 908 calls and 348, 921 puts) and 209, 051 of European options (of which 107, 546 are calls and 101, 505 are puts). There is more information on quotes than on trades records, and on American than on European options. The records on European puts are fewer than on European calls, both for trades and quotes. 29 Figure 1 about here The implied volatility daily average over the analysed period is plotted on Figure 1, separately for the trades and the bid-ask averages samples. It is possible to observe the increase of the daily average options implied volatility during 1994, then after 1995 there is a decrease which is followed by another increase by the end of Put implied volatilities from both American and European options and from trades and quotes are higher than call implied volatilities. 30 This is particularly clear for the last part of the period considered. Figure 2 about here Figure 2 shows the average implied options volatility by moneyness level (S/X) for the same samples. There is a clear smile effect. 31 Implied option volatilities from bid-ask averages are smaller than the implied volatilities backed from trades data. This is more so for in-themoney put options implied volatilities. The smile effect is attributed by many authors to erroneous assumptions of the valuation model used to extract implied volatilities, or to the presence of measurement errors in the options parameters. 32 Ederington and Guan (2002a) show that for options on the S&P 500 futures index, the smile effect cannot be completely attributed to incorrect estimation of implied volatilities. They show that it is possible to devise a profitable trading strategy, without considering transaction costs, based on the Black and Scholes (1973) implied volatilities. Their results suggest that the true (or correctly calculated) smile is somewhat flatter than the smile but far from flat. After transaction costs the frequent rebalancing required by the strategy will eliminate the profits. Therefore, they conclude that the presence of the smile is not incompatible with market efficiency. 28 Areal (2006) showed that these methodologies provided the best speed/accuracy relationship, when valuing options on the FTSE-100 index, taking into account discrete dividends. 29 After careful review of the implied option volatilities we decided to delete 87 records of trades and 206 records of bid-ask averages, which had volatility estimates higher than 100%, and were most likely a result of recording errors or trading errors, since they are the only records on those days with high volatility estimates. 30 This finding implies that for European options the put-call parity fails. Nonetheless, an arbitrage trading strategy is not expected to be profitable due to the existence of trading costs and bid-ask spreads. 31 Although it is not shown here, this effect is present even if we divide the period in two, before and after See Hentschel (2003) for the treatment of measurement errors in option volatility estimates. 14

16 The existence of the smile effect on our implied volatility estimates limits the use of the linear interpolation to obtain the 30 calendar days at-the-money implied volatility estimate for the VFTSE, already discussed on section 3. Therefore, when computing the VFTSE, we will only consider options with a moneyness level between 0.95 S/X Figure 3 about here Finally figure 3 exhibit the average daily FTSE-100 implied volatility as a function of the moneyness level (S/X) and the option s time to maturity, by exercise type, considering the bid-ask average options samples. 33 It is possible to observe that options close to expiration have higher averaged implied volatilities than options with more time to expiration. Again it is possible to observe that in-the-money options, both calls and puts, have higher average implied volatilities. There are no significant differences in the averaged implied volatilities from American and European options. Recently Ederington and Guan (2002b) compared several different schemes for averaging implied option volatilities in order to choose the one with best volatility forecasting ability. They conclude that the question of the weighting scheme is not truly important, what is relevant is the fact that implied option volatilities are upward biased measures of expected volatility. They also found that this bias is stable over time, so it is possible to remove it from the implied volatility estimates. When this is done, the differences between the various weighting schemes are very small. Table 5 about here Following their study we compared the averaged implied option volatility by strike price for calls and puts with the measure of realised volatility obtained by Areal and Taylor (2002). Table 5 show the results for the bid-ask quote averages sample. 34 The results reported are for the period from 14 June 1993 to 29 December The total number of options records per strike price and the percentage of observations for which the implied volatility is higher than the average realised volatility are also reported. Again it is possible to observe that there are significant differences in the average implied volatilities by strike price. The null hypothesis that the average implied option volatilities is no greater than the average realised volatility over the period is rejected for several strikes. 35 An explanation for the upward bias in implied option volatility is that the volatility risk premium is negative. Bakshi and Kapadia (2003), using a delta-hedging strategy and S&P 500 options data for the period of 1 January 1988 to 30 December 1995, found evidence that volatility risk premium is on average negative. The intuition of this result is that a negative risk premium suggests an equilibrium where equity index options act as a hedge to the market portfolio, which is in accordance to the evidence that equity prices react negatively to positive volatility shocks (e.g.: Whaley, 2000 and Simon, 2003). Therefore investors would be willing to pay a premium to hold options and will make the option price higher than its price when 33 The results for the trades sample are very similar. 34 Again the results for the trades sample are very similar. 35 The results of Ederington and Guan (2002b) show a higher bias for the at-the-money options. 15

17 volatility is not priced. The results of Bakshi and Kapadia (2003) are in accordance to the evidence found previously by Chernov and Ghysels (2000), Pan (2002), Chernov (2003) and Jones (2003). Yet another explanation for the upward bias in implied volatility of index options is the existence of a large negative correlation risk premium as suggested by Driessen, Maenhout and Vilkov (2005). The authors demonstrate that a large fraction of changes in market volatility stem from correlation changes and that a model where the entire market risk premium is due to a correlation risk premium explains the data extremely well. This allows them to reconcile the findings that implied volatilities of options on stocks exhibit different stylized facts relative to implied volatilities of index options. Driessen, Maenhout and Vilkov (2005) conclude that index options are expensive and earn low returns relative to options on stocks because they hedge correlation risk and insure against the risk of a loss in diversification benefits. The use of a historical volatility measure within the volatility index would make the introduction of derivatives on such an index more difficult, since this reduces the hedging effectiveness of such derivatives (Whaley, 1993). Therefore, removing the implied volatility bias, as suggested by Ederington and Guan (2002b), is not an appealing procedure as this would prevent one of the possible uses of the VFTSE. Nevertheless, this analysis advises that we should focus on out-of-the-money calls and close-to-the money options implied volatility estimates. 6 FTSE-100 volatility indices Table 6 has the number of days per options data where it would not be possible to compute VFTSE using the three index construction methodologies (VFTSE AIS, VFTSE OTM, and VFTSE MF ), described in section 3. On these days the computation of VFTSE is not possible because there is not enough data required by the construction methodology of the index. Given these results we will only present results for the VFTSE AIS methodology and bid-ask averages data for the American and complete samples, and the VFTSE AIS with trades data and the complete sample; for the VFTSE OTM methodology and bid-ask averages data for the American and complete samples, and the VFTSE OTM with trades data and the complete sample; and finally for the VFTSE MF methodology and bid-ask averages data for the American sample. All other datasets would produce a volatility index with too many missing days to be of interest. Table 6 about here Tables 7 to 8 show descriptive statistics of all the VFTSE indices resulting from the aforementioned methods and options data. For each method and data set we report results for three series, the index level (ˆσ), the log of the index (ln( ˆσ)) and, finally, following Fleming, Ostdiek and Whaley (1995), the changes in the index level ( ˆσ =ˆσ t ˆσ t 1 ). Table 7 about here Table 8 about here 16

18 All index methods, with the exception of the VFTSE MF, show some common characteristics such as a very high autocorrelation of the index level for the first lags, around 97%. 97% figure is reported for the first order autocorrelation of the VXO (Fleming, Ostdiek and Whaley, 1995). 36 is clear that the VFTSE MF A,BA VFTSE OTM A,BA, VFTSEOTM C,BA When we analyse the autocorrelation of the changes in the index level it has the highest negative first lag autocorrelation level (-44%); and VFTSEOTM C,T around -22%, -19% and -21%, respectively. have a similar negative first order autocorrelations of The VFTSE AIS construction method shows the smallest level of autocorrelations on the changes of the index level with a value of -7% for the American bid-ask averages sample, -10% for the bid-ask averages sample and close to -13% for the complete sample of trades. Therefore, since more autocorrelation in the changes of the index level can be interpreted as the presence of more measurement errors in the implied volatility estimation (Harvey and Whaley, 1991), we will concentrate our analysis, from now on, on the alternative interpolation scheme (VFTSE AIS ) construction methodology. As for the sample data used to compute the index, there is not much difference between the trades data and the bid-ask averages sample when considering the complete (American and European) options sample. This is true for the index level, the logs of the index and even the changes of the index level series. Thus, since the use of bid-ask quotes averages results in more days where it is possible to compute the index we favour this series. When comparing the VFTSE AIS A,BA with the VFTSEAIS C,BA series we will resort to the comparison of the log of the index level descriptive statistics. Let ˆσ t 2 = σt 2 (1 + u t ) with u t the zero-mean measurement error. Then ln(ˆσ 2 t ) = ln(σ 2 t )+u t 1/2u 2 t +...and hence a more accurate estimate has a higher value of E[ln(ˆσ 2 t )] and a lower value of var[ln(ˆσ 2 t )] (Areal and Taylor, 2002). If the implied volatility indices were not a biased estimate of future realised volatility, we could say that the one with least variance would provide a better estimate of future volatility. But, as already mentioned, previous research has consistently found that implied volatility is an upward biased measure of future realised volatility. That can be due to measurement errors, or to the existence of negative volatility premium or even to the existence of a negative correlation premium. Therefore to decide which construction methodology creates the most informative measure of future realised volatility we need to run a horse race between these measures. The tests of forecasting ability for a constant 22 trading day ahead period of these volatility indices will be performed in the next section. A 7 The information content of FTSE-100 indices Many studies have analysed the information content of volatility indices or implied index option volatilities. For a detailed literature review about volatility forecasting see Poon and Granger (2003). Previous studies have found evidence that implied index options provide superior volatility forecasts compared to the ones obtained from historical data. 36 Using the Augmented Dickey-Fuller test it is not possible to reject the hipothesis of unit root for most of the the index levels series. 17

19 Canina and Figlewski (1993) using S&P 100 options implied volatility found that there is no correlation between implied volatility and future realised volatility. Christensen and Prabhala (1998) found for the S&P 100 options that the use of overlapping observations and mismatched sample periods can produce results that are not precise and can favor historical forecasts. When non-overlapping data is used they found that implied volatility is an upward biased estimator of future volatility but contains important information about future volatility, outperforming historical volatility forecasts. Fleming (1998), also using S&P 100 options data, reached similar conclusions for one month ahead volatility forecasts. More recently Blair, Poon and Taylor (2001) using S&P 100 data showed that volatility indices provide more accurate volatility forecasts than a measure of realised volatility obtained from high frequency returns. Martens and Zein (2004), using floor and electronic transaction data on S&P 500 futures, are able to measure realised volatility computed using high-frequency data and a long memory specification over periods of 24 hours. They find that historical volatility does provide good volatility forecasts, and contains incremental information over that contained in implied volatilities. They also find that a combination between the historical volatility measure and the implied volatility renders the best volatility forecast. Pong, Shackleton, Taylor and Xu (2004) using exchange rates data conclude that the superior accuracy of historical volatility, relative to implied volatilities, comes from the use of high frequency returns and not from a long memory specification. Corrado and Miller Jr. (2005) tested the information content of three US volatility indices (VIX, VXO and VXN) and show that the forecasts based on those indices, although upwardly biased, are still more efficient in terms of mean squared forecast errors than historical realised volatility. As previously mentioned Jiang and Tian (2005b) compare the forecast ability of model free implied volatility from the S&P 500 index options with a high frequency realised volatility measure and the Black and Scholes (1973) implied volatility and found that model free implied volatility subsumes all information contained in the Black and Scholes (1973) implied volatility and past realised volatility and is a superior forecast for future realised volatility. Fewer studies are available for the UK market. Gwilym and Buckle (1999) find evidence that historical volatilities are more accurate predictors than implied volatility (based on the mean squared error and mean absolute error), and regression results suggest that implied volatilities are upward biased estimates of future volatility but superior in terms of information content. They consider different forecasting horizons and conclude that the accuracy is better for longer forecast horizons for all measures, and that the accuracy of historical methods is improved when the amount of past data used is matched to the forecast horizon. Noh and Kim (2006) use FTSE-100 and S&P 500 data to conclude that historical volatility using high-frequency returns outperforms implied volatility in forecasting future volatility in the case of FTSE-100 data, but when considering the S&P 500 data set implied volatility does better than historical volatility. Their FTSE-100 results also suggest that implied volatility can be an unbiased forecast for future volatility. Noh and Kim (2006) use a non-overlapping monthly sample from January 1994 to June 1999, and measure the implied volatility as the 18

20 average of two European call options implied volatilities with expiry on the following month (with an average life of 17 trading days), one whose strike is closest to the index price from below and one whose strike is closest to the index from above. Our study differs from Gwilym and Buckle (1999) because we use a longer data set and high frequency data to measure realised volatility whereas they use daily data from June 1993 to May Furthermore they use American options data to obtain the implied option volatilities but they value them as European. Morever they only consider call and put at-themoney option implied volatilities and the average between them whilst we consider several diferent methods to estimate an implied option volatility index which are constructed in order to avoid measurement errors. Finally they use overlapping observations which can produce spurious regression results. The present work differs from Noh and Kim (2006) since we consider several different measures of realised volatility and implied volatilities over a similar period of time. Moreover not only do we use non-overlapping monthly data, but we also consider 22 different samples of non-overlapping monthly observations. In summary our study has several differences when compared to previous studies. First, it is one of the few studies that considers UK market data. Second, it compares several different realised volatility measures computed using high-frequency data. Third, it uses several implied volatility indices constructed for the first time using FTSE-100 options data, including one that uses a model free implied volatility measure. We will assess the information content of volatility forecasts using VFTSE volatility measures and realised volatility measures by univariate and encompassing regressions of the form: 37 σ [t,t+22] =β 0 + β LRV σ LRV [t] + β V F T SE AIS C,T V F T SEAIS C,T [t] + (12) + β V F T SE AIS C,BA V F T SEAIS C,BA[t] + β V F T SE AIS A,BA V F T SEAIS A,BA[t] + + β V F T SE OT M V F T SEOT M C,T C,T [t] + β V F T SEC,BA OT M V F T SEOT C,BA[t] M + + β V F T SE OT M A,BA V F T SEOT M A,BA[t] + β V F T SE MF A,BA V F T SEMF A,BA[t] where σ 2 is a realised volatility measure for the period of 22 trading days, from t to t + 22, and σ LRV is the lagged realised volatility, the proxy for historical volatility. Only encompassing regressions are specified since univariate regressions are a restricted case of the encompassing regressions. Another regression form is going to be considered using the natural logarithm of volatility: ln σ [t,t+22] =β 0 + β LRV ln σ LRV [t] + β V F T SE AIS C,T ln V F T SEAIS[t] C,T + (13) + β V F T SE AIS ln V F T SEAIS C,BA C,BA[t] + β V F T SEA,BA AIS ln V F T SEAIS A,BA[t] + + β V F T SE OT M C,T ln V F T SEC,T OT M [t] + β M V F T SEC,BA OT M ln V F T SEOT C,BA[t] + M + β V F T SE OT M ln V F T SEOT A,BA A,BA[t] + β V F T SEA,BA MF ln V F T SEMF A,BA[t] This latter specification is considered since the logarithm of volatility is usually closer to 37 These are tests of information content and not out-of-sample forecasting accuracy. 19

21 the Normal distribution than volatility itself. This regression approach closely follows the method used, among others, by Christensen and Prabhala (1998), Corrado and Miller Jr. (2005) and Jiang and Tian (2005b). The univariate regressions allows us to infer about the information content of one volatility forecast, whilst the encompassing regressions allows us to analyse the relative importance of two or more forecasts and also if one volatility forecast subsumes all information about future volatility contained in the other volatility forecasts. Since Martens and Zein (2004) found that a simple measure which combines a realised volatility with implied volatility measures can produce better forecasts, we are also going to consider the following specifications: σ [t,t+22] = θ 0 + θ 1 (0.5σ LRV [t] +0.5V F T SE) (14) ln σ [t,t+22] = θ 0 + θ 1 ln(0.5σ LRV [t] +0.5V F T SE) (15) where VFTSE will be substituted by each volatility indices here considered. Usually the proxy for historical volatility is taken to be the lagged realised volatility with a matching horizon taken from the previous period (Christensen and Prabhala, 1998). We will follow this when selecting our proxy for historical volatility, and will test the consequences of using a different proxy in section Data and realised volatility measure For the test of forecasting ability we will use the sample from 14 June 1993 to 17 July The start date of our sample is dictated by the availability of data to perform the computation of the volatility indices, and the terminal date is chosen in order to have a sample period were the futures exchange opening hours were constant. To avoid the use overlapping data which can result in spurious regression results we follow Christensen and Prabhala (1998) and create a series of 22 trading day non-overlapping observations. Since the starting date of such a series can influence the characteristics of this series and consequently the regression results, we also analyse the result of 22 regressions, with each regression running with data starting one day later than the previous regression forming a different non-overlapping series. For this forecasting exercise we will need a measure of realised volatility. We will use the optimal weight realised volatility measure (σ OW ) given by Areal and Taylor (2002). This realised volatility measure uses high frequency futures trades data, and has the advantage of taking into account the close to open period, the intradaily volatility pattern, as well as the weekday volatility pattern. Using the Bandi and Russell (2006) procedure we conclude that the choice of 5 minute returns is very close to the optimal sampling frequency. Since the optimal weight realised volatility measure uses the complete data sample to estimate the intradaily variance proportions this could imply a look ahead bias when estimating future realised volatility. Therefore we will also use different measures of realised volatility. We will consider the one used by Jiang and Tian (2005b) which will allow for a better comparison 20

22 of their results. The authors use two frequencies to compute realised volatility measure: 30 and 5 minute returns. They use 30 minute index returns to calculate the realised volatility over one month and 5 minute index returns to calculate the lagged realised volatility. To correct for the bias in realised volatility induced by the autocorrelation in index returns they use a correction method suggested by French, Schwert and Stambaugh (1987) for daily data and by Hansen and Lunde (2006) for intradaily data. Using this correction procedure the realised variance over the period [t, t + 22] is given by: σ 2 JT[t,t+22] = 1 22 n ri i=1 l h=1 ( ) n h n r i r i+h (16) n h where r i is the index return during the i-th interval, n is the total number of intervals in the period and l is the number of correction terms included. They use one correction term when computing the realised variance using 5 minute index returns and no correction terms (l = 0) when using 30 minute index returns. The use of this correction procedure has two drawbacks, first it increases the variance of the estimator, and second it can result in negative variance. For these reasons Hansen and Lunde (2005) use a different correction approach: i=1 σ 2 NW[t,22] = 1 22 n ri i=1 l h=1 ( 1 h ) n h r i r i+h (17) l +1 which is designated by the authors as the Newey-West modified realised variance, since it uses the Bartlett kernel proposed by Newey and West (1987). This estimator has the advantages of being non-negative and of being almost identical to the sub-sampled based estimator of Zhang, Mykland and Aït-Sahalia (2005) which is consistent with integrated variance when the noise is of the independent type as demonstrated by Barndorff-Nielsen, Hansen, Lunde and Shephard (2005) and pointed out by Hansen and Lunde (2005). 38 The choice of one frequency to measure returns may not be optimal, which in turn can result in either biased or excessive volatile variance forecasts. Therefore we will also use a realised volatility measure that is constructed using a daily optimal sampling frequency (σ OSF ) using the method proposed by Bandi and Russell (2006). Our study differs from Jiang and Tian (2005b) since we will use returns of the futures on the index, whereas they used returns of the spot index. Our choice has the advantage of avoiding the problem that an index is a lagged indicator of the actual index portfolio value, since not all stocks are traded continuously. Also we will use our optimal weight realised volatility measure (σ OW ), the realised variance correction procedure (σ JT ) used by Jiang and Tian (2005b), the Newey-West realised variance estimator (σ NW ) used by Hansen and Lunde (2005), a realised variance estimator using the daily optimal sampling frequency (σ OSF ), and finally a simple integrated volatility measure (σ SIV ) which is given by the summation of squared five minute 38 For other approaches to bias correction of the realised volatility estimator see for instance Oomen (2005) and Bandi and Russell (2005). i=1 21

23 returns. 39 With the exception of the optimal weight realised volatility measure, which is computed using trade data, all other realised volatility measures are going to be computed using bid-ask futures data, and also FTSE-100 index futures trades data. The FTSE-100 index futures data required to estimate the realised volatility measures mentioned previously is described at great length by Areal and Taylor (2002), which also includes a statistical analysis of the optimal weights realised volatility measure (OW). Table 9 about here Table 10 about here Tables 9 and 10 has a descriptive statistics summary for the above mentioned realised volatility measures estimated using high frequency FTSE-100 futures trades and bid-ask average square returns, respectively. The Jiang and Tian (2005b), henceforth JT, and the Newey-West, henceforth NW measures were obtained using one correction term (l = 1 in expressions 16 and 17). Daily realised volatility estimates were obtained multiplying the daily estimates by 251. The average and standard deviations of the daily annual realised volatility measures are usually higher using trades data than bid-ask averages data. All realised volatility measures are skewed to the right and highly leptokurtic, and this is more so when using futures trades data. The natural logarithm of these measures are closer to the normal distribution. All these measures of realised volatility are highly autocorrelated. 7.2 Results Univariate regressions Table 11 has the results for the univariate regressions, when realised volatility is measured using optimal weights (OW), and historical volatility is given by the previous 22 trading day realised volatility observation. This table has results for two panels: panel A uses regression specified by expression 12, whilst panel B uses the regression specified by expression 13. Table 11 about here We test the regression residuals using the Jarque and Bera (1987) test for normality, the White (1980) test for heteroscedasticity if the residuals are not normal, the Breusch and Pagan (1979) test for heteroscedasticity if the residuals are normal and the Breusch (1978) and Godfrey (1978), henceforth the Breusch-Godfrey, test for autocorrelation. 40 The reported standard errors are corrected whenever appropriate for the presence of heteroscedasticity using the correction of Cribari-Neto (2004) which performs better in smaller samples than White (1980), or for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). Numbers in square brackets are the p-values associated with a t-test for the regression coefficient being equal to zero, and values 39 σ SIV is identical to σ JT and σ NW when zero correction terms are considered in expressions 16 and Breusch and Pagan (1979) test is preferred to the more general test for heteroscedasticity of White (1980) but can only be used when residuals are normal (Greene, 2003). The Breusch-Godfrey test is used here since the Durbin and Watson (1951) is not valid when some of the regressors are lagged dependent variables. 22

24 in brackets are the p-values associated with a t-test for the regression coefficient being equal to one. The first result to note is that residuals of all regressions using volatility specification (expression 12) do not follow a Normal distribution. Thus great care must be taken when interpreting the probability of tests associated with these regressions. Moreover the adjusted R 2 are almost always smaller for panel A than for panel B. Therefore we will concentrate our analysis on the results of panel B regressions. If a volatility forecast does not contain any information about future realised volatility, its regression coefficient (β 1 ) must be zero. Table 11 in square brackets next to the β 1 coefficient estimate. The p-values associated with this test are in All regression coefficients (β 1 ) are positive, statistically different from zero for all univariate regressions, considering the conventional significance levels. Therefore all volatility measures contain information about future volatility. For a given volatility forecast to be unbiased the intercept (β 0 ) of the regression must be zero and the volatility coefficient estimate (β 1 ) must be one. The reported F test statistic and the p-value in parenthesis formally test the joint hypothesis H 0 : β 0 = 0 and β 1 = 1. It is possible to reject this hypothesis for all the volatility forecasts considered. Both of these results are in accordance with the previous literature. Also reported are adjusted R 2 and heteroscedasticity consistent root mean squared error (HRMSE) which is computed using the following expression: HRMSE = 1 n N i=1 ( 1 F ) 2 i (18) R i where n is the number of observations, F is the forecast and R is the realised volatility measure. When we analyse the regression adjusted R 2 and HRMSE it is possible to notice that the best results are given by the historical forecast, followed by VFTSE AIS C,T volatility index. Not surprisingly given the number of days where it is not possible to compute the VFTSE MF C,T index, the model free index provides the worst results of all the volatility indices considered. These results do not question the validity of previous findings (Corrado and Miller Jr., 2005; Jiang and Tian, 2005b), since they are most likely due to the scarce range of options data available in the UK market for the analysed period. So far it is possible to conclude that historical high-frequency realised volatility contains more information about future 22 trading days ahead volatility than all implied volatility indices; and that historical and implied volatilities are biased forecasts of future volatility. Since these results can be attributed to the start date of the non-overlapping 22 trading day sample, we also ran these regressions 22 times, with each regression running with data starting one day later than the previous regression forming a different non-overlapping series. The results are shown on table 12. We report the average coefficients, the average standard errors, the average adjusted R 2 and the average HRMSE. It is also shown which of the averaged adjusted R 2 /HRMSE regression results are statistically larger/smaller than the other forecasts using the Diebold and Mariano (1995) test. If any of the following: LRV, VFTSE AIS VFTSE AIS C,BA, VFTSEAIS A,BA, VFTSEOT C,T M, VFTSEOT M C,BA, VFTSEOT M A,BA 23 and VFTSEMF A,BA C,T,, appears next

25 to the adjusted R 2 (HRMSE) it indicates that this specific adjusted R 2 (HRMSE) is significantly larger (smaller) than the corresponding adjusted R 2 (HRMSE) for the lagged realised volatility, VFTSE AIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSEOT C,T M, VFTSEOT C,BA M, VFTSEOT A,BA M, VFTSEMF A,BA, respectively, based on the Diebold and Mariano (1995) test for the statistical difference between two loss functions using a 5% level of significance. Table 12 about here The results confirm our initial conclusions. They also allow us to conclude that the historical volatility forecast is statistically better than all other forecasts both in terms of adjusted R 2 and HRMSE. Also, when considering the HRMSE, the differences between all the other forecasts are not statistically different except for the model free volatility index which is worse than all the other measures. The results for the adjusted R 2 are similar, except that it is possible to conclude that VFTSE AIS C,T has an average adjusted R2 which is statistically higher than the one given by the VFTSE OT C,T M. The average adjusted R2 is 63, 1% when considering the historical forecast and around 52% for the volatility indices forecasts, with the exception of the model free volatility index which produces a much lower adjusted R 2 (36%). Both tables 11 and 12 also include results for a forecast resulting from the combination (average) of the historical volatility and two volatility indices VFTSE AIS C,T, VFTSEAIS C,BA. The univariate regressions are of the form specified by expression 15. The results show that we cannot reject the hypothesis of the intercept of these regressions being equal to zero and the hypothesis for the forecast coefficient being equal to one. Nevertheless a formal test for the joint hypothesis is rejected. Any of these combined forecasts yields an average adjusted R 2 of 61% Encompassing regressions The encompassing regressions will allow us to analyse the relative importance between two or more volatility forecasts and also if one volatility forecast subsumes all information about future volatility contained in the other volatility forecast(s). We will compare the information efficiency of the historical volatility forecast with the volatility indices forecast. If historical volatility subsumes all information about future volatility then β V F T SE = 0. If on the other hand the volatility indices contain all information about future volatility then β LRV = 0. It is also possible to express this as a joint hypothesis of β LRV = 0 and β V F T SE = 1. This will be tested using an F test. Tables 13 and 14 show the results of the encompassing regressions for both one series of 22 trading day non-overlapping observations, and the 22 series of 22 trading day nonoverlapping observations, respectively. Again the reported regression coefficients standard errors are corrected for heteroscedasticity and autocorrelation whenever appropriate. The results of all the regressions show that we can reject the hypothesis that H 0 : β LRV =0 (p-values are in square brackets) and that it is not possible to reject the hypothesis that H 0 : β V F T SE = 0 at a 5% significance level. We also tested the hypothesis that H 0 : β LRV =1 (p-values are in brackets) and the hypothesis that H 0 : β V F T SE = 1, and it is possible to reject 24

26 these hypotheses at a 5% significance level. The joint test of the hypothesis of β LRV = 0 and β V F T SE = 1 (F-test) can also be rejected for all regressions. Table 13 about here Table 14 about here When we compare the adjusted R 2 of the univariate regressions with the related encompassing regressions we can see that the adjusted R 2 are higher for the encompassing regressions. But they are very close to the adjusted R 2 given by the historical volatility forecast. There is also a decrease in the HRMSE, but again they are very similar to the HRMSE given by the univariate regression of the lagged realised volatility. This leads us to conclude that lagged realised volatility is a better forecast of future realised volatility, and it also seems that volatility indices do not contain information that the realised volatility does not contain. This supports the results of Noh and Kim (2006) which also found that historical high-frequency realised volatility outperform implied volatility forecasts. As for the information efficiency for the various indices, again the results show that there is not much difference between them with the exception of the VFTSE MF A,BA which again appears to be worse than the others. 7.3 Robustness tests Despite the results obtained in the previous section which strongly support the evidence that realised volatility forecast is a better forecast of future volatility than implied volatility indices, and that all indices perform similarly (with the exception of the VFTSE AIS which performs slightly better then the others, and of the VFTSE MF A,BA indices which is a worse forecast of future volatility) we will perform some robustness tests to ensure the validity of our results Instrumental variables regression Christensen and Prabhala (1998), Corrado and Miller Jr. (2005) and Jiang and Tian (2005b) among others recommend the use of an instrumental variables framework to account for possible errors in the errors-in-variable (EIV) problem associated with implied volatility. There can be many sources of measurement errors of implied volatilities, as previously described, and despite all efforts here taken to reduce them when constructing the volatility indices there can still remain errors in measurement such as the misspecification of the option valuation method, non accounting of the volatility risk premium or the correlation risk premium. Christensen and Prabhala (1998) found evidence supporting the existence of errors in the measurement of the estimated implied volatilities. Corrado and Miller Jr. (2005) found, for the S&P 100 index, that an instrumental variable procedure improved regressions results for the period of , and for the S&P 500 index it improved results for the period of , although after 1995 there was no improvement resulting from the use of instrumental variables. These results suggest that biases caused by EIV effects on implied volatilities used to construct the CBOE volatility indices largely disappeared after

27 The results of the OLS regressions, specifically the high adjusted R 2 reported previously and the procedures used to limit the effect of EIV when estimating the implied volatilities, lead us to think that except for the VFTSE MF A,BA measurement errors are not going be an issue in our sample data. At first we would expect the model free implied volatility index (VFTSE MF A,BA ) to be less prone to EIV, especially since Jiang and Tian (2005b) found no measurement errors for their model free volatility estimate. But the fact that we are using a different procedure to obtain the model free estimates, and our dataset is not as rich as it should to construct such an index, can result in substantial errors in measurement. 41 We will follow Christensen and Prabhala (1998) and Jiang and Tian (2005b) when defining the instruments. In the first stage of the procedure, implied volatilities indices and lagged realised volatilities are going to be regressed on an instrument. In the second stage these instrumental variables will replace implied volatilities indices and lagged realised volatility in regressions 12 and 13. going to be estimated using: Instrumental variable regression for the lagged realised volatility is ˆσ LRV [t] =α 0 + α 1 σ LRV [t 1] + e t (19) and for each volatility index an instrument must be estimated, for instance for the VFTSE AIS C,T using: V F Tˆ SE AIS C,T [t] =α 0 + α 1 σ LRV [t 1] + α 2 V F T SEC,T AIS [t 1] + e t (20) Similar specifications were used to obtain instruments for all other volatility indices. Results for the first stage are reported in table 15. Table 16 has results for the univariate and encompassing second stage instrumental variables regressions. Table 15 about here Table 16 about here The regression coefficients are higher for the univariate instrumental variable regressions when compared with the OLS univariate regressions, indicating that there might be EIV problems associated with the volatility indices. It is not possible to reject the hypothesis that the regression coefficients are equal to one, at a 5% significance level for two indices,vftse AIS C,BA. Nevertheless, the joint hypothesis of the intercept being equal to zero and and VFTSE AIS A,BA the volatility index coefficient being equal to one is rejected for all usual significance levels (F(a) test in the table). These results are consistent with the OLS regression results. For the encompassing regressions it is not possible to reject the hypothesis that the regression coefficient for the lagged realised volatility is equal to zero. It is also not possible to reject the hypothesis that the regression coefficient for the implied volatility index is equal zero. As for the joint hypothesis that H 0 : β LRV = 0 and β V F T SE = 1 (F(b) test in table), they can be rejected for all encompassing regressions but one, which considers the VFTSE AIS A,BA 41 For a more detailed comparison between the model free index construction methodology used by the CBOE and by Jiang and Tian (2005b), please refer to Jiang and Tian (2005a). 26

28 volatility index. This contradicts the OLS encompassing regressions results. The coefficients of the lagged realised volatility and the volatility indices are more similar in terms of magnitude than the OLS encompassing regression results. Table 16 also shows the Hausman (1978) test for the presence of EIV. The results of the test show that for univariate regressions the EIV problem does matter, but for encompassing regressions this is not a problem Alternative measurements of realised volatility As mentioned in section 7.1 we considered several different realised volatility measures computed using FTSE-100 index futures trades data and also bid-asks average data. We considered the following measures: the realised variance correction procedure (σ JT ) used by Jiang and Tian (2005b), the Newey-West realised variance estimator (σ NW ) used by Hansen and Lunde (2005), a realised variance estimator using the daily optimal sampling frequency (σ OSF ), and finally a simple integrated volatility measure (σ SIV ) which is given by the summation of squared five minute returns. As Jiang and Tian (2005b) did, we also considered the use of 35 minute returns to estimate realised volatility measure, although for our data that sampling frequency is not optimal. We considered the use of different number of correction terms for the estimation of σ JT (expression 16) and σ NW (expression 17). For the former we considered only one correction term l = 1, as the use of higher correction terms can result in negative realised volatility measures, and for the latter 4 different correction l = {1,..., 4} terms. The univariate regressions, multivariate regressions and the instrumental variable regressions were repeated for all these series and all the results confirm what we have previously concluded. That both realised volatility and implied volatilities are biased forecasts of future volatility, and that realised volatility has more information about future volatility than volatility indices. Another consistent result is the fact that our model free volatility index always has the worse performance among all the forecasts. All other indices have similar performance with a tendency for the VFTSE AIS to perform better than the VFTSE OT M indices. Andersen, Bollerslev, Diebold and Ebens (2001) recommend filtering the returns with an MA(1) filter before estimating the high frequency realised volatility measures. Bandi and Russell (2005), although using an MA(1) filter for their sample data that did not improved results, conclude that filtering can be beneficial. Jiang and Tian (2005b) also tested whether their results were the same using filtered and unfiltered returns. In light of this we also apply an MA(1) filter to all our returns and estimated new realised volatility measures. We reran all the previous regression analysis (univariate, encompassing and instrumental variables) and the results show that the parameter estimates and test statistics are very close to the ones obtained using the unfiltered returns. We opted, as is usual in these studies, to consider the previous 22 trading day realised volatility measure as the lagged realised volatility to be a proxy for historical volatility. Jiang and Tian (2005b) consider a different proxy, using the previous daily realised volatility. When the previous daily realised volatility observation is taken as the lagged realised volatility, the results do change quite substantially. Table 17 has the results for our optimal weights volatility 27

29 measure (all other measures yield similar results). Table 17 about here This table shows the results for the 22 univariate regressions using 22 different nonoverlapping series. When the proxy for historical volatility is the previous daily realised volatility observation, historical volatility is no longer the best forecast of future volatility with an average adjusted R 2 of 43,7%, and all volatility indices perform similarly well with an average adjusted R 2 of about 52% (except for the VFTSE MF A,BA which has an average adjusted R 2 of 36,5%). The simple combination of the historical volatility and the VFTSE AIS C,BA yields an average adjusted R 2 of 56% and an average HRMSE of , which are statistically better than any of the other univariate regression results. This is not surprising given the results of Gwilym and Buckle (1999) which lead the authors to conclude that the accuracy of historical methods is improved when the amount of past data used is matched to the forecast horizon. Table 18 show the average encompassing regression results for the 22 series of non-overlapping observations, when lagged realised volatility is the previous day observation. When comparing these results with the ones provided by table 14, it is possible to conclude than when historical volatility is the previous day realised volatility observation the regression coefficient for the encompassing regressions are higher, although still remaining statistically different from 1. Table 18 about here 8 VFTSE statistical properties Given the results obtained in the previous section, and the fact that none of the VFTSE OT M indices can predict future 22 trading day volatility better than VFTSE AIS indices, we will select VFTSE AIS C,BA which has the better statistical properties as the VFTSE and concentrate our analysis on it. Henceforth, VFTSE AIS C,BA will be designated only as VFTSE. Figure 4 about here Figure 4 shows the evolution of the VFTSE index level, first with a linear scale and then with a logarithmic scale, for the period from 14 June 1993 to 17 March It is possible to observe a clear increase in the VFTSE level after 1997, especially at the end of 1997 and at the end of 1998, which can be associated with the Asian financial crisis, the Russian default and the Long-Term Capital Management fund problems. It is also possible to observe that expected volatility has periodic jumps. These jumps are at times when unexpected news cause investors to expect an increase in future volatility. Table 19 shows descriptive statistics for the VFTSE index series before and after July The VFTSE for the period after July 1997 has a higher average volatility and a higher standard deviation. Table 19 about here 28

30 The VFTSE series does not follow a normal distribution. The skewness of 1.32 indicates a slightly longer right tail than the normal distribution, and the kurtosis of 4.87 indicates a slight leptokurtosis. These values compare to a skewness figure of 0.73 and 1.33, and a kurtosis figure of 2.53 and 4.7, for the periods before and after July 1997, respectively. Figure 5 shows the empirical distribution of the VFTSE and its logarithm. Figure 5 about here The first three lags of the changes in the VFTSE level autocorrelations, , and , compare with , and for the changes of the VXO reported by Fleming, Ostdiek and Whaley (1995). Figure 6 shows the autocorrelations of the index level, the logarithm of the index level and the changes in the index level, for the analysed period. The autocorrelations are very high, and positive for at least 250 lags. After 250 lags the autocorrelations are still positive and around 0.33 for the index level and 0.35 for the logarithm of the index level series. The changes in the VFTSE level series after the first few lags are not statistically different from zero. Figure 6 about here Figure 7 about here Figure 7 show the autocorrelations of the index level for the two VFTSE sub-periods. The VFTSE level of the two sub-periods have a very high positive autocorrelation for the first lags, but for the sub-period before July 1997 the decay of autocorrelations is slower than for the sub-period after July The autocorrelations of the VFTSE level before July 1997 are close to zero after 215 lags whilst for the sub-period after July 1997 they approach zero much faster after only 99 lags. Figure 8 shows the comparisons of the VFTSE index level and the realised volatility measure of FTSE-100 futures prices for the period from 14 June 1993 to 29 December These figures show that the differences between the two measures of volatility are higher after Figure 8 about here 8.1 Temporal dependence The Augmented Dickey-Fuller test cannot reject the hypothesis that VFTSE index level has a unit root. Schwert (1989) and DeGennaro, Kunkel and Lee (1994) argue that the Augmented Dickey-Fuller test lacks power, therefore making it difficult to reject the null hypothesis of unit root. Given this and the fact that the slow decay of the autocorrelations of the VFTSE series suggests a long memory process, we tested the series for fractional integration. 42 We will use the Geweke and Porter-Hudak (1983) (GPH) estimate of d to provide conclusive evidence that a long memory component exists in the volatility series. This method has been applied to the series of the logarithm of the VFTSE index level ln(ˆσ t ). 42 For more on long memory processes, please refer to Baillie (1996), and the references therein. 29

31 We use frequencies 2πj n, with j =1, 2,..., nθ, θ =0.8 and n =1, 653 observations, to obtain an estimated d equal to 0.83 with a standard error of Figure 9 shows, for the complete sample and also for the two sub-periods, the estimated degree of fractional integration d as a function of the number of periodogram ordinates n θ, and also displays 95% confidence intervals for d. Similar tests have been performed for the two sub-periods, with the resulting d estimate of 0.75 (and a standard error of 0.040) for the sub-period before July 1997, and 0.83 (and a standard error of 0.048) for the sub-period after July A long memory parameter higher than 0.5 indicates that the volatility index series is nonstationary. This is not the first time that a value higher than 0.5 is reported in the literature. For instance, Bollerslev and Mikkelsen (1996) estimate the fractional integration parameter for the S&P 500 index volatility as being around 0.6. Moreover Andrews and Guggenberger (2003) and Bandi and Perron (2005) find the d estimates to be consistently upwardly biased in finite samples. Bandi and Perron (2005), using simulation, demonstrate that the GPH estimators deliver point estimates for d higher for implied than for realised volatility. The estimates of d for the VFTSE data are quite different from the 0.42 estimated for the realised volatility measure of the FTSE-100 stock index. This could be explained in part by the results of Bandi and Perron (2005) and/or the fact that realised and implied volatility indices measure volatility in different horizons. Figure 9 about here A simple long memory filter (1 L) 0.83 explains almost all of the dependence in the VFTSE index. 43 The autocorrelations of (1 L) 0.83 [ln(ˆσ t )] are near zero, as shown on Figure 10. Table 20 contains some descriptive statistics for the filtered series for the complete sample period and also for the sub-periods. 44 Figure 10 about here Table 20 about here 9 Final remarks 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 43 The filter is truncated for 500 observations. 44 For 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. 30

32 construction methodology adapted for a market not as liquid as the US market; the out-ofthe-money (OTM) index, which uses only OTM options data; and finally the model free (MF) volatility index which uses the VIX construction 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 VFTSE indices using only options with a moneyness level (S/X) between 0.95 and 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, European, 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 VFTSE OTM and VFTSE MF indices have a much higher level of autocorrelation than the VFTSE AIS index. To recommend the construction of the VFTSE using 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-100 index options market available to construct such index. These results lead us to recommend the construction of the VFTSE using 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 VFTSE for 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. The VFTSE autocorrelations 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 practitioners 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 the VFTSE and the stock market index since, in the US 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. 31

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38 Table 1: Summary information about several volatility indices. Index Market Underlying option index Option type Maintained by a stock market? Introduction date History Methodology Observation VXO US S&P 100 American Yes (CBOE) January January 1986 to the present Described in section Originally designated when introduced as VIX and only on 22 September 2003 its name changed to VXO VIX US S&P 500 European Yes (CBOE) 22 September January 1990 to the present Described in section Old VXN US NASDAQ-100 European Yes (CBOE) January 1998 January 1995 to 19 September 2003 The same as the VXO This index has been replaced on 22 September 2003 with a new VXN which uses a different construction method. VXN US NASDAQ-100 European Yes (CBOE) 22 September February 2001 to the present The same as the VIX VXD US DJIA European Yes (CBOE) 18 March October 1997 to the present The same as the VIX VDAX German DAX-100 European Yes (Deutsche Borse) 5 December January 1992 to the present Please refer to Deutsche Bourse (2003) VX1 and VX6 French CAC-40 European Yes (MONEP) 8 October January 1994 to the present Please refer to Moraux, Navatte and Villa (1999) GVIX Greek FTSE/ASE-20 European No 10 February 2000 to 30 December 2002 Please refer to Skiadopoulos (2004) AVIX Australian S&P/ASX200 European No 11 July 2001 to 27 September 2002 Please refer to Dowling and Muthuswamy (2003) VFTSE UK FTSE European/American No 14 June 1993 to 17 March 2000 Described in section 3 37

39 Table 2: Eight options implied volatilities used to compute the VXO index Nearby contract (N) Second nearby contract (SN) Call Put Call Put X l (<S) σ X l c,n σ X l p,n σ X l c,sn σ X l p,sn X u ( S) σ Xu c,n σ Xu p,n σ Xu c,sn σ Xu p,sn Table 3: Eight options implied volatilities used to compute the FTSE AIS index Nearby contract (N) Second nearby contract (SN) Call Put Call Put (S t /X t < 1) σ St/Xt<1 c,n σ St/Xt<1 p,n σ St/Xt<1 c,sn σ St/Xt<1 p,sn (S t /X t > 1) σ St/Xt>1 c,n σ St/Xt>1 p,n σ St/Xt>1 c,sn σ St/Xt>1 p,sn Table 4: Four options implied volatilities used to compute the FTSE OTM index Nearby contract (N) Second nearby contract (SN) Call Put Call Put (S t /X t < 1) σ St/Xt<1 c,n σ St/Xt<1 c,sn (S t /X t > 1) σ St/Xt>1 p,n σ St/Xt>1 p,sn 38

40 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 Option strike Calls Puts European Calls European Puts American Calls American Puts O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S Average implied volatility by strike price O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S Percentage of records with implied volatility greater than realised volatility O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S FTSE-100 options implied volatility records for the period from 14 June 1993 to 29 December O S i 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. 39

41 Table 6: Number of days with incomplete data to compute VFTSE using different methodologies Trades Bid-Ask averages VFTSE AIS American European Complete 27 9 VFTSE OTM American European Complete VFTSE MF American European 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. 40

42 Table 7: VFTSE index descriptive statistics for the VFTSE AIS (Alternative Interpolation Scheme) and the VFTSE OTM (Out-The-Money options) with different options sample data. VFTSE AIS A,BA VFTSE AIS C,BA VFTSE AIS C,T ˆσ ln(ˆσ) ˆσ ˆσ ln(ˆσ) ˆσ ˆσ ln(ˆσ) ˆσ Average Standard Deviation Skewness Kurtosis Minimum /12/ /12/ /10/ /07/ /07/ /10/ /08/ /08/ /10/1997 Maximum /10/ /10/ /10/ /10/ /10/ /10/ /10/ /10/ /10/1997 Number of observations Augmented Dickey-Fuller test Autocorrelations Lag Lag Lag Lag Lag VFTSE OTM A,BA VFTSE OTM C,BA VFTSE OTM C,T ˆσ ln(ˆσ) ˆσ ˆσ ln(ˆσ) ˆσ ˆσ ln(ˆσ) ˆσ Average Standard Deviation Skewness Kurtosis Minimum /12/ /12/ /10/ /12/ /12/ /10/ /12/ /12/ /10/1997 Maximum /10/ /10/ /10/ /10/ /10/ /01/ /10/ /10/ /01/1999 Number of observations Augmented Dickey-Fuller test Autocorrelations Lag Lag Lag Lag Lag

43 Table 8: VFTSE index descriptive statistics for the VFTSE MF (Model Free) with bid-ask averages American sample data. VFTSE MF A,BA ˆσ ln(ˆσ) ˆσ Average Standard Deviation Skewness Kurtosis Minimum /02/ /02/ /11/1997 Maximum /10/ /10/ /11/1997 Number of observations Augmented Dickey-Fuller test Autocorrelations Lag Lag Lag Lag Lag

44 Table 9: Summary of descriptive statistics for daily annual realised volatility measures series. The measures were obtained using high frequency FTSE-100 futures trades squared returns for the period from 14 June 1993 to 17 July σ SIV ln σ SIV σ JT ln σ JT σ NW ln σ NW σ OSF ln σ OSF Average Standard deviation Skewness Kurtosis Minimum /12/ /12/ /12/ /12/ /12/ /12/ /08/ /08/1993 Maximum /10/ /10/ /10/ /10/ /10/ /10/ /10/ /10/1997 Number of Observations Autocorrelations Lag Lag Lag Lag Lag SIV stands for simple integrated volatility, JT for Jiang and Tian (2005b), NW for Newey and West (1987), and OSF for optimal sampling frequency. With exception of the daily optimal sampling frequency realised volatility measure (σ OSF ) all other estimates are obtained using 5 minute squared returns. All measures takes into account the market open to close period. One correction term is considered when estimating the JT and NW realised volatility measures. Daily realised volatility estimates were obtained multiplying the daily estimates by

45 Table 10: Summary of descriptive statistics for daily annual realised volatility measures series. The measures were obtained using high-frequency FTSE-100 futures bid-ask average squared returns for the period from 14 June 1993 to 17 July σ SIV ln σ SIV σ JT ln σ JT σ NW ln σ NW σ OSF ln σ OSF Average Standard deviation Skewness Kurtosis Minimum /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/1996 Maximum /10/ /10/ /10/ /10/ /10/ /10/ /10/ /10/1997 Number of Observations Autocorrelations Lag Lag Lag Lag Lag SIV stands for simple integrated volatility, JT for Jiang and Tian (2005b), NW for Newey and West (1987), and OSF for optimal sampling frequency. With exception of the daily optimal sampling frequency realised volatility measure (σ OSF ) all other estimates are obtained using 5 minute squared returns. All measures takes into account the market open to close period. One correction term is considered when estimating the JT and NW realised volatility measures. Daily realised volatility estimates were obtained multiplying the daily estimates by

46 Table 11: Univariate regression results, realised volatility is the optimal weights (OW) measure, and historical volatility is the previous 22 trading day realised volatility observation. β0 β1 Adjusted R 2 HRMSE F-test Panel A: σt LRV (0.033) [0.257] {0.000} (0.245) [0.004] {0.309} (0.000) JB, Wh, BP VFTSE AIS C,T (0.017) [0.001] {0.000} (0.122) [0.000] {0.001} (0.000) JB, Wh, BG, BP VFTSE AIS C,BA (0.016) [0.000] {0.000} (0.117) [0.000] {0.000} (0.000) JB, Wh, BG, BP VFTSE AIS A,BA (0.017) [0.001] {0.000} (0.121) [0.000] {0.001} (0.000) JB, Wh, BG, BP VFTSE OT C,T M (0.015) [0.000] {0.000} (0.107) [0.000] {0.000} (0.000) JB, Wh, BG, BP VFTSE OT C,BA M (0.016) [0.000] {0.000} (0.117) [0.000] {0.000} (0.000) JB, Wh, BG, BP VFTSE OT A,BA M (0.017) [0.001] {0.000} (0.127) [0.000] {0.001} (0.000) JB, Wh, BG, BP VFTSE MF A,BA (0.020) [0.001] {0.000} (0.118) [0.000] {0.000} (0.000) JB, Wh, BG, BP 0.5 LRV VFTSE AIS C,T (0.039) [0.282] {0.000} (0.279) [0.015] {0.288} (0.000) JB, W, BP 0.5 LRV VFTSE AIS C,BA (0.039) [0.277] {0.000} (0.282) [0.017] {0.288} (0.000) JB, W, BP Panel B: ln(σt) LRV (0.167) [0.014] {0.000} (0.085) [0.000] {0.012} (0.000) VFTSE AIS C,T (0.256) [0.014] {0.000} (0.128) [0.000] {0.015} (0.000) Wh, BP VFTSE AIS C,BA (0.221) [0.003] {0.000} (0.107) [0.000] {0.002} (0.000) BG VFTSE AIS A,BA (0.229) [0.004] {0.000} (0.111) [0.000] {0.003} (0.000) Wh, BG VFTSE OT C,T M (0.236) [0.005] {0.000} (0.118) [0.000] {0.006} (0.000) BP VFTSE OT C,BA M (0.229) [0.004] {0.000} (0.111) [0.000] {0.003} (0.000) BG, BP VFTSE OT A,BA M (0.256) [0.012] {0.000} (0.124) [0.000] {0.009} (0.000) BG, BP VFTSE MF A,BA (0.326) [0.001] {0.000} (0.178) [0.013] {0.004} (0.000) JB, Wh, BG, BP 0.5 LRV VFTSE AIS C,T (0.279) [0.108] {0.000} (0.139) [0.000] {0.109} (0.000) W, BP 0.5 LRV VFTSE AIS C,BA (0.168) [0.007] {0.000} (0.086) [0.000] {0.008} (0.000) W The number in parenthesis beside the parameter estimate is the standard error, which are computed following a robust procedure whenever appropriate which takes into account heteroscedasticity using the correction of Cribari-Neto (2004), or corrects for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). The numbers in square brackets are the p-values associated with a t-test for the regression coefficient being equal to zero, and values in brackets are the p-values associated with a t-test for the regression coefficient being equal to one. JB stands for Jarque and Bera (1987) test for normality, BP for Breusch and Pagan (1979) test for heteroscedasticity, BG for Breusch (1978) and Godfrey (1978) test for autocorrelation, and Wh stands for White (1980) test for heteroscedasticity. If any of these initials appear next to a regression results this means that it is possible to reject those tests with a significance value of 5%. The reported F test statistic and the p-value in parenthesis, test the joint hypothesis H0 : β0 = 0 and β1 = 1. heteroscedasticity consistent root mean squared error (HRMSE) is defined by expression

47 Table 12: Univariate regression results from 22 samples, realised volatility is the optimal weights (OW) measure, and historical volatility is the previous 22 trading day realised volatility observation. β0 β1 Adjusted R 2 HRMSE Panel B: ln(σt) LRV (0.174) [0.029] {0.000} (0.088) [0.000] {0.025} VFTSE AIS C,BA, VFTSEAIS A,BA, VFTSEOT C,T M, VFTSE OT C,BA M, VFTSEOT A,BA M, VFTSEMF A,BA VFTSE AIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSE OT M, VFTSEOT M M C,T M C,BA M VFTSEOT A,BA M, VFTSEOT VFTSE MF A,BA VFTSE AIS C,T (0.228) [0.007] {0.000} (0.113) [0.007] {0.007} VFTSE OT C,T M, VFTSEMF A,BA VFTSE MF A,BA VFTSE AIS C,BA (0.223) [0.005] {0.000} (0.110) [0.005] {0.005} VFTSE MF A,BA VFTSE MF A,BA VFTSE AIS A,BA (0.226) [0.005] {0.000} (0.111) [0.005] {0.005} VFTSE MF A,BA VFTSE MF A,BA VFTSE OT C,T M (0.214) [0.003] {0.000} (0.107) [0.003] {0.003} VFTSE MF A,BA VFTSE MF A,BA VFTSE OT C,BA M (0.216) [0.003] {0.000} (0.107) [0.003] {0.003} VFTSE MF A,BA VFTSE MF A,BA VFTSE OT A,BA M (0.219) [0.003] {0.000} (0.108) [0.003] {0.003} VFTSE MF A,BA VFTSE MF A,BA VFTSE MF A,BA (0.299) [0.002] {0.000} (0.162) [0.005] {0.005} LRV VFTSE AIS C,T (0.272) [0.115] {0.000} (0.136) [0.000] {0.119} VFTSE AIS C,T,VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSE OT C,T M, VFTSEOT C,BA M, VFTSEOT A,BA M, VFTSE MF A,BA VFTSE AIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSE OT M, VFTSEOT M M C,T M C,BA M VFTSEOT A,BA M, VFTSEOT VFTSE MF A,BA 0.5 LRV VFTSE AIS C,BA (0.239) [0.071] {0.000} (0.120) [0.000] {0.074} VFTSEAIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSE OT M, VFTSEOT M M C,T M C,BA M VFTSEOT A,BA M, VFTSEOT VFTSE MF A,BA VFTSE AIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSE OT M, VFTSEOT M M C,T M C,BA M VFTSEOT A,BA M, VFTSEOT VFTSE MF A,BA The numbers reported are the average coefficients for the 22 regressions using 22 non-overlapping series of 22 trading days, average standard errors, average adjusted R 2 and average heteroscedasticity consistent root mean squared error (HRMSE). The number in parenthesis beside the parameter estimate is the average standard error, which are computed following a robust procedure whenever appropriate which takes into account heteroscedasticity using the correction of Cribari-Neto (2004), or corrects for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). The numbers in square brackets are the p-values associated with a t-test for the average regression coefficient being equal to zero, and values in brackets are the p-values associated with a t-test for the average regression coefficient being equal to one. The heteroscedasticity consistent root mean squared error (HRMSE) is defined by expression 18. LRV, VFTSE AIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSEOT C,T M, VFTSEOT C,BA M, VFTSEOT A,BA M, VFTSEMF A,BA indicates that the specific adjusted R 2 (HRMSE) is significantly larger (smaller) than the corresponding adjusted R 2 (HRMSE) for the lagged realised volatility, VFTSE AIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSE OT C,T M, VFTSEOT C,BA M A,BA M VFTSEMF A,BA, VFTSEMF significance., based on the Diebold and Mariano (1995) test for the statistical difference between two loss functions using a 5% level of 46

48 Table 13: Encompassing regression results, realised volatility is the optimal weights (OW) measure, and historical volatility is the previous 22 trading day realised volatility observation. Intercept LRV VFTSE AIS C,T VFTSE AIS C,BA VFTSE AIS A,BA VFTSE OT C,T M VFTSE OT C,BA M VFTSE OT A,BA M VFTSE MF A,BA Adj. R 2 RMSRE F-test (0.168) (0.168) (0.158) [0.020] {0.000} [0.001] {0.028} [0.276] {0.000} (0.000) (0.168) (0.164) (0.153) [0.019] {0.000} [0.000] {0.039} [0.370] {0.000} (0.000) (0.168) (0.163) (0.151) W [0.019] {0.000} [0.000] {0.031} [0.315] {0.000} (0.000) (0.167) (0.157) (0.145) W [0.020] {0.000} [0.000] {0.017} [0.217] {0.000} (0.000) (0.168) (0.158) (0.147) [0.021] {0.000} [0.000] {0.026} [0.297] {0.000} (0.000) (0.169) (0.159) (0.150) [0.023] {0.000} [0.000] {0.028} [0.312] {0.000} (0.000) (0.169) (0.121) (0.098) [0.016] {0.000} [0.000] {0.059} [0.875] {0.000} (0.000) (0.176) (0.179) (1.067) (2.237) (1.982) (0.791) (1.433) (1.052) (0.110) [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} The number in parenthesis beside the parameter estimate is the standard error, which are computed following a robust procedure whenever appropriate which takes into account heteroscedasticity using the correction of Cribari-Neto (2004), or corrects for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). The numbers in square brackets are the p-values associated with a t-test for the regression coefficient being equal to zero, and values in brackets are the p-values associated with a t-test for the regression coefficient being equal to one. JB stands for Jarque and Bera (1987) test for normality, BP for Breusch and Pagan (1979) test for heteroscedasticity, BG for Breusch (1978) and Godfrey (1978) test for autocorrelation, and Wh stands for White (1980) test for heteroscedasticity. If any of these initials appear next to a regression results this means that it is possible to reject those tests with a significance value of 5%. The reported F test statistic and the p-value in parenthesis, test the joint hypothesis H0 : β LRV = 0 and β V F T SE = 1. heteroscedasticity consistent root mean squared error (HRMSE) is defined by expression

49 Table 14: Encompassing regression results from 22 samples, realised volatility is the optimal weights (OW) measure, and historical volatility is the previous 22 trading day realised volatility observation. Intercept LRV VFTSE AIS C,T VFTSE AIS C,BA VFTSE AIS A,BA VFTSE OT C,T M VFTSE OT C,BA M VFTSE OT A,BA M VFTSE MF A,BA Adj. R 2 RMSRE (1) (0.174) (0.161) (0.155) (8) [0.037] {0.000} [0.000] {0.034} [0.303] {0.000} (2) (0.168) (0.161) (0.152) (7),(8) [0.031] {0.000} [0.000] {0.036} [0.310] {0.000} (3) (0.168) (0.161) (0.150) (2),(8) [0.029] {0.000} [0.000] {0.034} [0.301] {0.000} (4) (0.174) (0.158) (0.149) (1),(8) (1) [0.036] {0.000} [0.000] {0.031} [0.292] {0.000} (5) (0.167) (0.159) (0.148) (7),(8) [0.030] {0.000} [0.000] {0.034} [0.300] {0.000} (6) (0.170) (0.158) (0.147) (8) [0.032] {0.000} [0.000] {0.035} [0.313] {0.000} (7) (0.169) (0.124) (0.108) [0.029] {0.000} [0.000] {0.056} [0.665] {0.000} (8) (0.178) (0.177) (0.976) (1.789) (1.793) (0.631) (1.145) (1.097) (0.151) [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} (1),(2),(3),(4),(5),(6),(7) The numbers reported are the average coefficients for the 22 regressions using 22 non-overlapping series of 22 trading days, average standard errors, average adjusted R 2 and average heteroscedasticity consistent root mean squared error (HRMSE). The number in parenthesis beside the parameter estimate is the average standard error, which are computed following a robust procedure whenever appropriate which takes into account heteroscedasticity using the correction of Cribari-Neto (2004), or corrects for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). The numbers in square brackets are the p-values associated with a t-test for the average regression coefficient being equal to zero, and values in brackets are the p-values associated with a t-test for the average regression coefficient being equal to one. The heteroscedasticity consistent root mean squared error (HRMSE) is defined by expression 18. (1), (2), (3), (4), (5), (6), (7), (8) and (9) indicates that this specific regression specification adjusted R 2 (HRMSE) is significantly larger (smaller) than the corresponding adjusted R 2 (HRMSE) regression specification, based on the Diebold and Mariano (1995) test for the statistical difference between two loss functions using a 5% level of significance. 48

50 Table 15: First stage of the instrumental variables regression results, realised volatility is the optimal weights (OW) measure, and historical volatility is the previous 22 trading day realised volatility observation. Intercept LRV LVFTSE AIS C,T LVFTSE AIS C,BA LVFTSE AIS A,BA LVFTSE OT C,T M LVFTSE OT C,BA M LVFTSE OT A,BA M LVFTSE MF A,BA Adj. R 2 HRMSE VFTSE AIS C,T (0.136) (0.095) (0.093) [0.676] {0.000} [0.000] {0.000} [0.000] {0.000} VFTSE AIS C,BA (0.139) (0.095) (0.092) [0.609] {0.000} [0.000] {0.000} [0.000] {0.000} VFTSE AIS A,BA (0.136) (0.094) (0.090) [0.541] {0.000} [0.000] {0.000} [0.000] {0.000} VFTSE OT C,T M (0.151) (0.105) (0.101) [0.832] {0.000} [0.000] {0.001} [0.001] {0.000} VFTSE OT C,BA M (0.149) (0.102) (0.098) [0.816] {0.000} [0.000] {0.001} [0.000] {0.000} VFTSE OT A,BA M (0.142) (0.097) (0.095) [0.780] {0.000} [0.000] {0.000} [0.000] {0.000} VFTSE MF A,BA (0.248) (0.146) (0.121) JB [0.560] {0.000} [0.000] {0.409} [0.993] {0.000} The number in parenthesis beside the parameter estimate is the standard error, which are computed following a robust procedure whenever appropriate which takes into account heteroscedasticity using the correction of Cribari-Neto (2004), or corrects for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). The numbers in square brackets are the p-values associated with a t-test for the regression coefficient being equal to zero, and values in brackets are the p-values associated with a t-test for the regression coefficient being equal to one. JB stands for Jarque and Bera (1987) test for normality, BP for Breusch and Pagan (1979) test for heteroscedasticity, BG for Breusch (1978) and Godfrey (1978) test for autocorrelation, and Wh stands for White (1980) test for heteroscedasticity. If any of these initials appear next to a regression results this means that it is possible to reject those tests with a significance value of 5%. The reported F test statistic and the p-value in parenthesis, test the joint hypothesis H0 : β LRV = 0 and β V F T SE = 1. heteroscedasticity consistent root mean squared error (HRMSE) is defined by expression

51 Table 16: Second stage of the instrumental variables univariate and encompassing regression results, realised volatility is the optimal weights (OW) measure, and historical volatility is the previous 22 trading day realised volatility observation. Intercept LRV IVFTSE AIS C,T IVFTSE AIS C,BA IVFTSEAIS A,BA IVFTSEOT C,T M IVFTSE OT C,BA M IVFTSEOT A,BA M IVFTSEMF A,BA Adj. R2 HRMSE F(a) F(b) H (0.171) (0.089) [0.038] {0.000} [0.000] {0.063} (0.000) (0.001) (0.229) (0.321) (0.299) BP [0.128] {0.000} [0.280] {0.048} [0.117] {0.087} (0.097) (0.253) (0.171) (0.089) [0.031] {0.000} [0.000] {0.048} (0.000) (0.000) (0.230) (0.304) (0.280) BP [0.126] {0.000} [0.221] {0.045} [0.117] {0.054} (0.062) (0.220) (0.170) (0.088) [0.024] {0.000} [0.000] {0.039} (0.000) (0.000) (0.232) (0.281) (0.257) BP [0.125] {0.000} [0.151] {0.040} [0.115] {0.026} (0.038) (0.262) (0.171) (0.089) [0.049] {0.000} [0.000] {0.076} (0.000) (0.000) (0.225) (0.344) (0.317) BP [0.136] {0.000} [0.383] {0.047} [0.099] {0.146} (0.132) (0.167) (0.173) (0.089) [0.055] {0.000} [0.000] {0.074} (0.000) (1.000) (0.224) (0.332) (0.301) BP [0.139] {0.000} [0.306] {0.053} [0.109] {0.096} (0.107) (0.134) (0.176) (0.091) [0.057] {0.000} [0.000] {0.075} (0.000) (1.000) (0.226) (0.308) (0.281) BP [0.142] {0.000} [0.190] {0.060} [0.138] {0.045} (0.070) (0.212) (0.183) (0.098) [0.132] {0.000} [0.000] {0.300} (0.000) (1.000) The number in parenthesis beside the parameter estimate is the standard error, which are computed following a robust procedure whenever appropriate which takes into account heteroscedasticity using the correction of Cribari-Neto (2004), or corrects for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). The numbers in square brackets are the p-values associated with a t-test for the regression coefficient being equal to zero, and values in brackets are the p-values associated with a t-test for the regression coefficient being equal to one. JB stands for Jarque and Bera (1987) test for normality, BP for Breusch and Pagan (1979) test for heteroscedasticity, BG for Breusch (1978) and Godfrey (1978) test for autocorrelation, and Wh stands for White (1980) test for heteroscedasticity. If any of these initials appear next to a regression results this means that it is possible to reject those tests with a significance value of 5%. The reported F(a) test statistic and the p-value in parenthesis, test the joint hypothesis H0 : β Intercet = 0 and β V F T SE = 1. The reported F(b) test statistic and the p-value in parenthesis, test the joint hypothesis H0 : β LRV = 0 and β V F T SE = 1. heteroscedasticity consistent root mean squared error (HRMSE) is defined by expression 18. The last column of the table reports the Hausman (1978) test statistic and p-value in parenthesis. 50

52 Table 17: Univariate regression results from 22 samples, realised volatility is the optimal weights (OW) measure, and historical volatility is the previous day realised volatility observation. β0 β1 Adjusted R 2 HRMSE Panel B: ln(σt) LRV (0.170) [0.000] {0.000} (0.081) [0.000] {0.000} VFTSE MF A,BA VFTSE MF A,BA VFTSE AIS C,T (0.230) [0.007] {0.000} (0.114) [0.007] {0.007} LRV, VFTSE OT C,T M, VFTSEOT A,BA M, VFTSE MF A,BA VFTSE OT A,BA M, VFTSEMF A,BA VFTSE AIS C,BA (0.223) [0.005] {0.000} (0.110) [0.005] {0.005} LRV, VFTSE MF A,BA VFTSE MF A,BA VFTSE AIS A,BA (0.225) [0.005] {0.000} (0.110) [0.005] {0.005} LRV,VFTSE MF A,BA VFTSE MF A,BA VFTSE OT C,T M (0.216) [0.003] {0.000} (0.107) [0.003] {0.003} LRV,VFTSE MF A,BA VFTSE MF A,BA VFTSE OT C,BA M (0.216) [0.003] {0.000} (0.106) [0.003] {0.003} LRV,VFTSE MF A,BA VFTSE MF A,BA VFTSE OT A,BA M (0.217) [0.003] {0.000} (0.107) [0.002] {0.002} LRV,VFTSE MF A,BA VFTSE MF A,BA VFTSE MF A,BA (0.297) [0.002] {0.000} (0.160) [0.005] {0.005} LRV VFTSE AIS C,T (0.173) [0.001] {0.000} (0.087) [0.000] {0.001} LRV,VFTSE AIS C,T, VFTSEAIS C,BA, VFTSE AIS, VFTSEOT M A,BA, VFTSEOT C,T M C,BA M VFTSE OT M A,BA M A,BA LRV, VFTSE AIS C,T, VFTSEAIS C,BA, VFTSE AIS, VFTSEOT M A,BA, VFTSEOT C,T M C,BA M VFTSE OT M A,BA M A,BA 0.5 LRV VFTSE AIS C,BA (0.178) [0.001] {0.000} (0.089) [0.000] {0.001} LRV, VFTSEAIS C,T, VFTSEAIS C,BA, VFTSE AIS, VFTSEOT M A,BA, VFTSEOT C,T M C,BA M VFTSE OT M A,BA M A,BA LRV, VFTSE AIS C,T, VFTSEAIS C,BA, VFTSE AIS, VFTSEOT M A,BA, VFTSEOT C,T M C,BA M VFTSE OT M A,BA M A,BA The numbers reported are the average coefficients for the 22 regressions using 22 non-overlapping series of 22 trading days, average standard errors, average adjusted R 2 and average heteroscedasticity consistent root mean squared error (HRMSE). The number in parenthesis beside the parameter estimate is the average standard error, which are computed following a robust procedure whenever appropriate which takes into account heteroscedasticity using the correction of Cribari-Neto (2004), or corrects for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). The numbers in square brackets are the p-values associated with a t-test for the average regression coefficient being equal to zero, and values in brackets are the p-values associated with a t-test for the average regression coefficient being equal to one. The heteroscedasticity consistent root mean squared error (HRMSE) is defined by expression 18. LRV, VFTSE AIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSEOT C,T M, VFTSEOT C,BA M, VFTSEOT A,BA M, VFTSEMF A,BA indicates that the specific adjusted R 2 (HRMSE) is significantly larger (smaller) than the corresponding adjusted R 2 (HRMSE) for the lagged realised volatility, VFTSE AIS C,T, VFTSEAIS C,BA, VFTSEAIS A,BA, VFTSE OT C,T M, VFTSEOT C,BA M A,BA M VFTSEMF A,BA, VFTSEMF significance., based on the Diebold and Mariano (1995) test for the statistical difference between two loss functions using a 5% level of 51

53 Table 18: Encompassing regression results from 22 samples, realised volatility is the optimal weights (OW) measure, and historical volatility is the previous day realised volatility observation. Intercept LRV VFTSE AIS C,T VFTSE AIS C,BA VFTSE AIS A,BA VFTSE OT C,T M VFTSE OT C,BA M VFTSE OT A,BA M VFTSE MF A,BA Adj. R 2 RMSRE (1) (0.182) (0.094) (0.119) (7) (5),(7) [0.005] {0.000} [0.009] {0.000} [0.000] {0.000} (2) (0.175) (0.091) (0.112) (7) (7) [0.003] {0.000} [0.007] {0.000} [0.000] {0.000} (3) (0.176) (0.090) (0.113) (7) (7) [0.003] {0.000} [0.007] {0.000} [0.000] {0.000} (4) (0.174) (0.092) (0.114) (7) (7) [0.003] {0.000} [0.007] {0.000} [0.000] {0.000} (5) (0.171) (0.092) (0.111) (7) (7) [0.002] {0.000} [0.007] {0.000} [0.000] {0.000} (6) (0.172) (0.091) (0.111) (7) (7) [0.002] {0.000} [0.006] {0.000} [0.000] {0.000} (7) (0.192) (0.102) (0.131) [0.001] {0.000} [0.001] {0.000} [0.029] {0.000} (8) (0.185) (0.100) (0.958) (1.747) (1.671) (0.637) (1.154) (1.109) (0.155) (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} (1),(2),(3),(4),(5),(6),(7) The numbers reported are the average coefficients for the 22 regressions using 22 non-overlapping series of 22 trading days, average standard errors, average adjusted R 2 and average heteroscedasticity consistent root mean squared error (HRMSE). The number in parenthesis beside the parameter estimate is the average standard error, which are computed following a robust procedure whenever appropriate which takes into account heteroscedasticity using the correction of Cribari-Neto (2004), or corrects for the presence of autocorrelation and heteroscedasticity using the procedure suggested by Newey and West (1994). The numbers in square brackets are the p-values associated with a t-test for the average regression coefficient being equal to zero, and values in brackets are the p-values associated with a t-test for the average regression coefficient being equal to one. The heteroscedasticity consistent root mean squared error (HRMSE) is defined by expression 18. (1), (2),(3), (4), (5), (6), (7), (8) and (9) indicates that this specif regression specification adjusted R 2 (HRMSE) is significantly larger (smaller) than the corresponding adjusted R 2 (HRMSE) regression specification, based on the Diebold and Mariano (1995) test for the statistical difference between two loss functions using a 5% level of significance. 52

54 Table 19: VFTSE index descriptive statistics for the VFTSE for the periods before and after July VFTSE before July 1997 VFTSE after July 1997 ˆσ ln(ˆσ) ˆσ ˆσ ln(ˆσ) ˆσ Average Standard Deviation Skewness Kurtosis Minimum /07/ /07/ /12/ /07/ /07/ /10/1997 Maximum /11/ /11/ /02/ /10/ /10/ /10/1997 Number of observations Augmented Dickey-Fuller test Autocorrelations Lag Lag Lag Lag Lag

55 Table 20: Descriptive statistics for the filtered series of the logarithm of the VFTSE index. Filtered series of the logarithm of the VFTSE Complete Sample Before July 1997 After July 1997 Average Standard Deviation Skewness Kurtosis Minimum /02/ /05/ /02/2000 Maximum /10/ /12/ /01/1999 Number of observations Autocorrelations Lag (0.3658) (0.4174) (0.3641) Lag (0.1287) (0.1245) (0.2833) Lag (0.0078) (0.2057) (0.0002) Lag (0.3206) (0.4797) (0.0701) Lag (0.4270) (0.0804) (0.4501) Ljung-Box(25) (0.1658) (0.7266) (0.1707) The volatility series was filtered using the fractional integration filter (1 L) 0.83 for the complete sample and the sub-period after July 1993, and (1 L) 0.75 for the sub-period before July The numbers in parentheses are the levels of significance of the autocorrelation statistics. 54

56 Figure 1: American and European, calls and puts daily implied volatilities during the period from 14 June 1993 to 17 March Average daily volatility (%) /01/ /01/ /01/ /01/ /01/ /01/ /01/2000 Date European Calls European Puts (a) Trades American Calls American Puts Average daily volatility (%) /01/ /01/ /01/ /01/ /01/ /01/ /01/2000 Date European Calls European Puts American Calls American Puts (b) Bid-ask averages 55

57 Figure 2: Average implied volatilities by moneyness level during the period from 14 June 1993 to 17 March Volatility Moneyness level S/X European Calls European Puts (a) Trades American Calls American Puts Volatility Moneyness level S/X European Calls European Puts American Calls American Puts (b) Bid-ask averages 56

58 Figure 3: Average implied volatilities by moneyness level and time to maturity, during the period from 14 June 1993 to 17 March American and European bid-asks average sample. Implied volatility 0.65 Implied volatility Time to maturity Time to maturity Moneyness (a) American Calls Moneyness (b) American Puts Implied volatility 0.55 Implied volatility Time to maturity Time to maturity Moneyness (c) European Calls Moneyness (d) European Puts 57

59 Figure 4: The VFTSE index level during the period from 14 June 1993 to 17 March FTSE-100 spot index /01/ /01/ /01/ /01/ /01/ /01/ /01/2000 Date (a) Index level FTSE-100 spot index /01/ /01/ /01/ /01/ /01/ /01/ /01/2000 Date (b) Index level in logarithmic scale 58

60 Figure 5: Distribution of the VFTSE index level and the logarithm of the index level from 14 June 1993 to 17 March Normal Relative frequency / Density (a) Index level Normal Relative frequency / Density (b) Index level in logarithmic scale The continuous curves overlaying the histogram are firstly a normal density that matches the mean and standard deviation (dotted curve) and secondly the density estimate based upon Gaussian kernels and bandwidth equal to 0.1 (solid curve) for the levels and changes in the VFTSE index level series, and 0.25 for the log of the VFTSE level series; the kernel estimate is smoother than that provided by the standard bandwidth of 0.06 for this data (Silverman, 1986, page 48). 59

61 0 Figure 6: Autocorrelations of the VFTSE index level, the logarithm of the index level, and changes of the index level, from 14 June 1993 to 17 March Autocorrelations Lags (a) Index level Autocorrelations Lags (b) Logarithm of the index level Autocorrelations Lags (c) Changes in the index level 60

62 0 Figure 7: Autocorrelations of the VFTSE index level from 14 June 1993 to 30 June 1997 and from 01 July 1997 to 17 March Autocorrelations Lags (a) 14 June 1993 to 30 June Autocorrelations Lags (b) 1 July 1997 to 17 March

63 Figure 8: Comparison of VFTSE index level with the realised volatility of FTSE-100 futures prices from 14 June 1993 to 29 December VFTSE Realized Volatility /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/1998 Date VFTSE Realised volatility 0.30 VFTSE-RV 0.20 VFTSE - Realised Volatility /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/ /01/ /07/1998 Date 62

64 Figure 9: GPH estimates of the degree of fractional integration, d of the logarithm of the VFTSE as a function of the number of periodogram ordinates, n θ, used in their calculations Lower 95% bound Upper 95% bound d m (a) Complete Sample Lower 95% bound Upper 95% bound d m (b) Before July Lower 95% bound Upper 95% bound d m (c) After July

65 Figure 10: Autocorrelations of (1 L) 0.83 [ln(ˆσ t )] Lower 95% bound Upper 95% bound Autocorrelations Lags 64