FTSE-100 implied volatility index

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FTSE-100 implied volatility index

Nelson Areal

nareal@eeg.uminho.pt

NEGE,

School of Economics and Management University of Minho

4710-057 Braga Portugal

Phone: +351 253 604 100 Ext. 5523, Fax:+351 253 601 380

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.

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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.

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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)

1Pong, 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

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tested the information content of threeUSvolatility 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 theUKimplied 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 theVFTSE.

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 usingUKmarket data that compare the forecasting ability ofFTSE-100implied 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.

2For more on volatility forecasting please refer to Poon and Granger (2003) and references therein.

3For 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).

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historical methods is improved when the amount of past data used is matched to the forecast horizon.

Noh and Kim (2006) useFTSE-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 ofFTSE-100data, but when considering theS&P 500data set implied volatility performs better than historical volatility. TheirFTSE-100results 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 considersUKmarket 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 usingFTSE-100options 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 theFTSE-100 index.

In the next section we will describe the construction methodology of volatility indices on theUSmarket (VXO,VIX,VXNandVXD). 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-100option 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.

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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.

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The Chicago Board Options Exchange (CBOE) introduced in 1993 theCBOEmarket volatil-ity 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 toVXO. The newVIXis not only computed by a different method but is also based on S&P 500options data.4

Currently, there are several volatility indices being maintained and distributed: the Chicago Board Options Exchange (CBOE) Market Volatility Index (VIX) onS&P 500, theVXOon theS&P 100 stock index, theCBOE DJIAVolatility Index (VXD) on the Dow Jones Industrial Average (DJIA), the CBOE Nasdaq-100 Volatility Index (VXN); the German volatility index (VDAX); and the March ˜A c! des Options N ˜A c!gociables de Paris (MONEP) volatility indices (VX1 and

VX6) on theCAC-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 theUSmarket volatility indices and their differ-ent 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 oldVIXvolatility index, currently designated asVXO, was suggested by Whaley (1993) and introduced by theCBOEin 1993. 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 VIXand started to provide prices for two indices; the original-formula index was renamed

VXOand a newVIXwas introduced with a new construction methodology.5

On the same date, theCBOEchanged the computation methodology of theCBOENASDAQ

Volatility Index (VXN) which had been introduced in 1998. It was previously based on the

VXOindex construction methodology, and is now based on the VIX.6

On 26 March 2004, theCBOEFutures Exchange (CFE) started trading futures onVIX. VIX

options were launched in February 2006.

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): theCBOEDJIA Volatility

4From now on, unless stated otherwiseVIXdesignates the new volatility index andVXOthe index computed

under the original specification.

5TheCBOEprovide, on its website, figures dating back to 1990 computed with the new methodology for the

VIX.

6The newVXNsubstituted the old one and has a history dating back from 2 February 2001 to the present.

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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 TheCBOEFutures Exchange (CFE) introducedCBOE DJIAVolatility Index Futures on 25 April 2005.

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.

2.1.1 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 differentS&P 100options (OEX) and represents the market consensus of the expected volatility over the next 30 calendar days. At the time VXOwas 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.

TheVXOis based on eight near-the-money, nearby and second nearby tradedOEXoptions. It is constructed in a fashion such that at any given time it represents the implied volatility of an at-the moneyOEXoption 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 weight-ing schemes, includweight-ing the one used byVXO, 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 (Xl), and two more from a call and a put just above the current index level (Xu). 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: σXl

N = (σ Xl c,N + σ Xl p,N)/2, σ Xu N , σ Xl SN, and σ Xu SN.

After this, compute the at-the-money implied volatilities for the nearby (σN) and the sec-ond nearby (σSN) volatilities by interpolating the in and out-of-the-money implied volatilities:

7They also introduced on the same date theCBOE DJIABuyWrite 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).

8TheCBOEprovides daily data on this index from 7 October 1997 onwards.

9An eight-day-to-maturity limit is imposed since options with shorter maturities tend to have higher

volatil-ities. Such options may also induce liquidity related biases.

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σN = σXNl ! Xu−S Xu−Xl " + σXu N ! S−X l Xu−Xl " (1) σSN = σSNXl ! Xu−S Xu−Xl " + σXu SN ! S−X l Xu−Xl " (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 = σN ! N tN−22 NtSN−N tN " + σSN ! 22−N tN NtSN−N tN " (4) where NtN is the number of trading days to maturity of the nearby contract and NtSN is the number of trading days to maturity of the second nearby option series.

TheVXOis 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

σt= σc # √Nc √ Nt $ (5)

where σt is the trading day implied volatility rate, σc is the calendar day implied volatility rate, Nc is the number of calendar days to maturity, and Nt is the number of trading days to maturity given by:12

Nt= Nc− 2 × int(Nc/7) (6)

2.1.2 The S&P 500 volatility index (VIX) construction methodology

The new VIXaims, 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.

TheVXOwas based on S&P 100option prices whereas theVIXis based onS&P 500options data.13 The new VIX considers the entire range of strike prices, contrary to the VXO which

11This 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).

12This adjustment causes the VXOto be higher than actual volatility. Blair, Poon and Taylor (2001) and

Simon (2003), when studying theVXN, adjust the index values in order to remove this adjustment.

13TheS&P 100option ticker isOEXand theS&P 500option 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

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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

σ2= 2 T # rT −# S0 S∗ ert− 1 $ − lnS∗ S0 + erT % S 0 1 K2p(K)dK+ (7) + erT % ∞ S 1 K2c(K)dK $

where S0is 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 σ2= 2e rT T #% F0 0 p(K) K2 dK + % ∞ F0 c(K) K2 dK $ (8)

where F0 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 risk-neutral 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

14Please refer to Carr and Wu (2006) for a detailed comparison betweenVXOand the newVIX.

15For a detailed description of the valuation of this contract please refer to Demeterfi, Derman, Kamal and

Zou (1999).

16See also the related work of Carr and Madan (2002), Bakshi and Madan (2000), Bakshi, Kapadia and

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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 VFTSEas 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

VXOmethodology to compute intraday values for the index.

Therefore, we propose a method to construct the index that still considers implied volatili-ties on eight options but with a different interpolation scheme. The volatility index computed with this methodology will be designated as alternative interpolation scheme VFTSEAIS.

Since the out-of-the-money options are the ones on which trading is concentrated, infor-mation 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,

VFTSEOTM.

We will also compute yet another volatility index, based on model free option prices, using the same methodology as theVIX. Such an index will be designated as the model free volatility index VFTSEMF.17

For each index a daily figure will be computed using the most recent option prices available for each day.

As already mentioned, VFTSEAIS 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 (St/Xt) 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

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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 (σc,N, σp,N, σc,SN and σp,SN). For example, the at-the-money implied volatility of the call option of the nearby maturity σc,N will be given by:

σc,N = σSc,Nt/Xt<1+ (1 − St/XtSt/Xt<1) ×

(σSt/Xt>1c,N −σSt/Xt<1

c,N )

(St/XtSt/Xt>1−St/X

tSt/Xt<1) (9)

With these estimates we will compute the implied volatility value of the nearby contract (σN) and the second nearby contract (σSN) averaging the put and call implied volatilities of each contract:

σN = (σc,N + σp,N)/2 (10)

σSN = (σc,SN + σp,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 theVFTSEOTM 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 theUK market there are both European and American options on the FTSE-100stock index. Whenever possible, we will compute, for each volatility index methodology, a volatility index for American options (VFTSEA), another considering only the European options infor-mation (VFTSEE) and finally one with information from both datasets (VFTSEC), designated as the complete sample.18 For theVFTSEMF only two datasets are going to be used, one with American options (VFTSEMFA ) and another with European options (VFTSEMFE ) data.19

18A detailed description of these datasets will be given in the following section.

19The 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.

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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 (VFTSEA,VFTSEE,VFTSEC) we will have two versions: one for trades and another for bid-ask averages, e.g.: the VFTSEA will have a version based on trades (VFTSEA,T) and another based on bid-ask averages (VFTSEA,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 ex-ercise 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 recommen-dations when estimating the implied volatilities of FTSE-100stock index options.

Ideally we would like to use the trueFTSE-100index value, but this is not possible since in a broad index like the FTSE-100even 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 theFTSE-100spot 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 theS&P 100 market, theUK stock market has the same opening hours as the

options market. Despite this, there is a wildcard option embedded in the valuation ofFTSE-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 theUS market to the relatively quiet wildcard period in London.23

20Harvey and Whaley (1991) show that the infrequent trading of stocks of theS&P 100index at the closing

of the day is not a problem when estimating the option’s implied volatility using spot index values.

21A 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.

22This 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.

23For more details on the wildcard option please see: Fleming and Whaley (1994), Dawson (2000) and

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4

The dataset

For the current analysis we will need intraday data on the FTSE-100index spot value, records of all trades and quotes of theFTSE-100European and American options, a proxy for the risk-free interest rate, and theFTSE-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 theFTSE-100stock index.

4.1

FTSE-100 dividends

FTSE-100dividends were computed by FTSE and obtained from DataStream for the period from

14 June 1993 to 18 December 2001.24 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.25

4.3

Spot data

The spot data of the FTSE-100index 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 1996. 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 theFTSEdataset that have a zero value for the spot index,

and these were deleted. The total number of records of spot data given byFTSEis 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-100spot 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 theFTSEfor the period when this data was available.

24The start of the period is the first day of availability of this dataset.

25We performed all calculations using also an interest rate given by the interpolation of the two nearest

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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 byFTSEwe 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 byFTSEwhich 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 1999.

4.4

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 Inter-national 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 1990

26We also found a few days where there are records of trades and no records of quotes, and vice-versa, for

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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 2000.

5

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

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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, sepa-rately 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 1997. 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-the-money 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.

28Areal (2006) showed that these methodologies provided the best speed/accuracy relationship, when valuing

options on the FTSE-100 index, taking into account discrete dividends.

29After 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.

30This 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.

31Although it is not shown here, this effect is present even if we divide the period in two, before and after

1996.

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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 theVFTSE, we will only consider options with a moneyness level between 0.95 ≥ S/X ≤ 1.05.

Figure 3 about here

Finally figure 3 exhibit the average dailyFTSE-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 1998. 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

33The results for the trades sample are very similar.

34Again the results for the trades sample are very similar.

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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 intro-duction of derivatives on such an index more difficult, since this reduces the hedging effective-ness 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 pre-vent one of the possible uses of theVFTSE. 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

com-pute VFTSE using the three index construction methodologies (VFTSEAIS, VFTSEOTM, and

VFTSEMF), described in section 3. On these days the computation ofVFTSEis not possible

be-cause there is not enough data required by the construction methodology of the index. Given these results we will only present results for the VFTSEAIS methodology and bid-ask averages data for the American and complete samples, and the VFTSEAIS with trades data and the complete sample; for theVFTSEOTMmethodology and bid-ask averages data for the American and complete samples, and the VFTSEOTM with trades data and the complete sample; and finally for theVFTSEMF 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 theVFTSEindices resulting from the afore-mentioned 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 = ˆσt− ˆσt−1).

Table 7 about here

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All index methods, with the exception of theVFTSEMF, show some common characteristics such as a very high autocorrelation of the index level for the first lags, around 97%. A 97% figure is reported for the first order autocorrelation of the VXO (Fleming, Ostdiek and Whaley, 1995).36 When we analyse the autocorrelation of the changes in the index level it is clear that the VFTSEMFA,BA has the highest negative first lag autocorrelation level (-44%);

VFTSEOTMA,BA,VFTSEOTMC,BA andVFTSEOTMC,T have a similar negative first order autocorrelations of

around -22%, -19% and -21%, respectively. The VFTSEAIS 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 (VFTSEAIS) 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 theVFTSEAISA,BAwith theVFTSEAISC,BAseries we will resort to the comparison of the log of the index level descriptive statistics. Let ˆσ2t = σ2t(1 + ut) with ut the zero-mean measurement error. Then ln(ˆσ2

t) = ln(σt2)+ut−1/2u2t+. . . and hence a more accurate estimate has a higher value of E[ln(ˆσ2t)] and a lower value of var[ln(ˆσ2t)] (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.

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.

36Using the Augmented Dickey-Fuller test it is not possible to reject the hipothesis of unit root for most of

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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 100options 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 onS&P 500 futures, are able to measure realised volatility computed using high-frequency data and a long mem-ory 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 threeUSvolatility 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 theS&P 500index 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 theUK 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) useFTSE-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

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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 1995. 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-the-money 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 UKmarket 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 usingFTSE-100options data, including one that uses a model free implied volatility measure.

We will assess the information content of volatility forecasts usingVFTSE volatility mea-sures and realised volatility meamea-sures by univariate and encompassing regressions of the form:37

σ[t,t+22]=β0+ βLRVσLRV[t]+ βV F T SEAIS C,TV F T SE AIS C,T[t]+ (12) + βV F T SEAIS C,BAV F T SE AIS C,BA[t]+ βV F T SEAIS A,BAV F T SE AIS A,BA[t]+ + βV F T SEOT M C,T V F T SE OT M C,T[t]+ βV F T SEOT M C,BAV F T SE OT M C,BA[t]+ + βV F T SEOT M A,BAV F T SE OT M A,BA[t]+ βV F T SEM F A,BAV F T SE M F A,BA[t]

where σ2 is a realised volatility measure for the period of 22 trading days, from t to t + 22, and σLRV 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]=β0+ βLRV ln σLRV[t]+ βV F T SEAIS C,T ln V F T SE AIS[t] C,T + (13) + βV F T SEAIS C,BAln V F T SE AIS C,BA[t]+ βV F T SEAIS A,BAln V F T SE AIS A,BA[t]+ + βV F T SEOT M C,T ln V F T SE OT M C,T[t]+ βV F T SEOT M C,BAln V F T SE OT M C,BA[t]+ + βV F T SEOT M A,BAln V F T SE OT M A,BA[t]+ βV F T SEM F A,BAln V F T SE M F A,BA[t]

This latter specification is considered since the logarithm of volatility is usually closer to

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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 uni-variate 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 volatil-ity contained in the other volatilvolatil-ity 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]= θ0+ θ1(0.5σLRV[t]+ 0.5V F T SE) (14)

ln σ[t,t+22]= θ0+ θ1ln(0.5σLRV[t]+ 0.5V F T SE) (15)

where VFTSEwill 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 7.3.2.

7.1

Data and realised volatility measure

For the test of forecasting ability we will use the sample from 14 June 1993 to 17 July 1998. 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 obser-vations. 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 (σOW) 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

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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:

σ2JT [t,t+22]= 1 22 n & i=1 r2i + 2 22 l & h=1 # n n − h $n−h & i=1 riri+h (16)

where ri 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:

σN W [t,22]2 = 1 22 n & i=1 r2i + 2 22 l & h=1 # 1 − h l + 1 $n−h & i=1 riri+h (17)

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 A¨ıt-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 (σOSF) 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 (σOW), the realised variance correction procedure (σJT) used by Jiang and Tian (2005b), the Newey-West realised variance estimator (σN W) used by Hansen and Lunde (2005), a realised variance estimator using the daily optimal sampling frequency (σOSF), and finally a simple integrated volatility measure (σSIV) which is given by the summation of squared five minute

38For other approaches to bias correction of the realised volatility estimator see for instance Oomen (2005)

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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 alsoFTSE-100index futures trades data.

TheFTSE-100index futures data required to estimate the realised volatility measures

men-tioned 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 frequencyFTSE-100futures trades and bid-ask average square returns, respectively. The Jiang and Tian (2005b), henceforthJT, and the Newey-West, henceforthNWmeasures 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

7.2.1 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 het-eroscedasticity using the correction of Cribari-Neto (2004) which performs better in smaller samples than White (1980), or for the presence of autocorrelation and heteroscedasticity us-ing 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

σSIV is identical to σJT and σN W when zero correction terms are considered in expressions 16 and 17.

40Breusch 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.

Figure

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References

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Related subjects :
Outline : 9 Final remarks