How to do the implied volatility forecast

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PREDICTING STOCK

MARKET VoLATinT A NEW MEASURE

JEFF FLEMING BARBARA OSTDIEK ROBERT E. WHALEY

INTRODUCTION

The CBOE Market Volatility Index (VIX) is an average of S&P 100 option (OEX) implied volatilities. As such, it represents a market- consensus estimate of future stock market volatility.' The computation and dissemination of VIX on a real-time basis offers practitioners and academics an important new source of information. Practitioners, for

This research was supported by the Futures and Options Research Clcnlt-r al the Fuqua Scbool of Business. Duke University. We gratefully acknowledge the beJpful comments and suggestions of Fischer Blaek. Mark Rubinstein, and two anonymous referees. We also tbank participants at the University (if Pennsylvania, the University of Texas al Dallas, and tbe UniversJly of Waterloo/KPMG Peal Marwick Tborne seminars, as well as attendees of the 1993 Conference on Financial Innovation: 20 Years of HIack/Scboles and Merton (Duke University) and the 1994 Berkeley Program in Finance, Ojai Valley, California.

' Since OEX options are the most actively traded index options, VIX promises to be the U.S. stock market volatilily "standard" upon whicb derivative conlracts may be written. A varlely of option or embedded-option security positions are sensitive to ehanges in expected stock market volalility. Volatility derivatives, therefore, could be used to offset tbis exposure. Whaley (1995) describes the economic benefits of volatilitv derivatives as well as some risk management applications.

• J^fJ lleniing is an Assistant Professor of Administrative Science at Rice University.

m Barbara Ostdiek is an Assistant Professor of Administrative Science at Rice University.

• Rohert E. Whaley is the T. Austin Einch Eoundation Professor of Business Administration at the Euqua School of Business, Duke University.

266 Fleming, Ostdiek, and Whaiey

example, face a variety of day-to-day decisions such as asset allocation, covered-call writing, and portfolio insurance, all of which require an up-to-the-minute estimate of future stock market volatility. On the other hand, academics are interested in studying temporal patterns in expected return and risk. Changes in the level of VIX represent changes in conditional stock market volatility. By virtue of the inde.x construction, this conditional volatility measure is market-determined, forward-looking, and has a constant (one-month) forecast horizon.

The usefulness of the volatility index as a measure of expected volatility depends on its behavior. This study investigates the statistical properties ot VIX and evaluates its predictive power. The assessment proceeds along three levels. First, the time-series history of VIX is examined to provide a description of its univariate properties and seasonalities. Second, the study considers the consistency of VIX with the empirical literature regarding conditional volatility, such as the tem- poral relationship between volatility and stock market returns. Finally, the article evaluates the performance of VIX as a forecast of stock market volatility.

The evidence reported in this study indicates that VIX is well behaved with little evidence of seasonality. Daily changes in VIX display a slight first-order autocorrelation. For weekly changes, significant mean reversion is detected. VIX also has a strong negative and asymmetric association with contemporaneous stock market returns. And, regarding forecast performance, the index exhibits a strong relationship to future realized stock market volatility. On the basis of this evidence, the volatility index indeed appears to be a useful proxy for expected stock market volatility.

The methodology and findings of this study are presented as follows. The second section describes the construction of VIX. The third section examines the CBOF's Market Volatility Index over a seven-year period (1986-1992).^ Properties of daily and weekly changes in VIX are documented and the relationship between VIX changes and S&P 100 index returns is investigated. The fourth section focuses on intraday and intraweek seasonalities in VIX changes. The performance of VIX as a stock market volatility forecast is considered in the fifth section. The final section provides a summary of the major findings of the article.

is bistorical VIX series Is conslructed ex post (ratber tban in rcal-timf) from tbe CBOE's MDR system and was made available to researcbers by tbe CBOE wben tbe volatility index was Introduced in February 1993.

Market Volatility Index 267

COMPOSITION OF THE CBOE MARKET VOLATILITY INDEX

The idea of creating a volatility index irom option prices emerged soon after the introduction of exchange-traded options in April 1973. Gastineau (1977), for example, created a "volatility-index" by averaging the implied volatilities of the at-the-money call options of 14 stocks. Cox and Ruhinstein (1985, Appendix 8A) refined the Gastineau procedure by including multiple call options on each stock and by weighting the volatilities in such a manner that the index is at the money and has a constant time to expiration.

The GBOE Market Volatility Index captures the spirit of these earlier efforts and extends the concept in two important ways. First, VIX is based on index options rather than stock options. While the "average" level of individual stock volatilities may be of passing interest, market participants are more concerned with portfolio risk—the level of risk after the idiosyncratic risks of the individual stocks have been diversified away. Second, VIX is based on the implied volatilities of both call and put options. This not only increases the amount of information incorporated into the index but also mitigates concerns regarding call/put option clienteles and possible mismeasurements of the reported index level and the short-term interest rate. Since the focus of this study is to evaluate VIX as a measure of future stock market volatility, the study begins with a description of the index's construction.

Implied Volatility Computation

The CBOE Market Volatility Index is constructed from the implied volatilities of eight OEX options.^ Computing these component implied volatilities requires three types of information: (i) an option valuation model, (ii) the values of the model's determinants (except for volatility), and (iii) an observed option price. With respect to the model, VIX construction is based on the Black-Scholes (1973)/Merton (1973b) option valuation framework. Since the OEX options are American-style, the cash-dividend adjusted binomial method is used to compute the component implied volatilities.

Within the Black—Scholes/Merton model, option value is deter- mined by the current index value, the option's exercise price and time to expiration, the riskless interest rate, and the amount and timing of the anticipated cash dividends paid during the option's life. To generate the

*A more detailed description of VI.X cons t ru t t ion is provided in tbe appendix in Wbaley (1993).

268 Fleming, Ostdiek, and Whaiey

historical VIX series, 1986-1992, the CBOE used actual cash dividends to proxy for anticipated dividends."* The source of the dividend data prior to June 1988 is from Harvey and Whaley (1992a). The source of the dividend data from June 1988 through December 1992 is from the S^P 100 Information Bulletin. The interest rate is the effective yield (based on the average bid/ask discount) of the T-bill whose maturity most closely matches the option expiration and has at least 30 days to maturity. The source of the T-bill discounts is The Wall Street Journal. Of the three remaining option determinants, the exercise price and time to expiration are known. The reported S&P 100 index level is used to proxy for the current S&P 100 index value."^ The CBOE automatically records into its Market Data Retrieval (MDH) system the reported index level whenever an OEX option trades or has price quotes issued.

The third type of information required to compute the component implied volatilities is observed option prices. Here, the midpoint of the most recent bid/ask price quote is used. Using actual transaction prices would induce negative first-order autocorrelation in implied volatility changes as option prices bounce randomly between bid and ask levels. The implied volatility for each option series, then, is determined by equating the theoretical option value, given the procedures outlined above, with the most recent bid/ask price midpoint.

Trading-Day Adjustment

The implied volatilities for the OEX option series used in computing VIX are stated in trading days rather than calendar days. If the time to option expiration were measured in calendar days, the implied volatility would be a rate per calendar day. The return variance over a weekend (from Eriday close to Monday close), then, should be three times greater than it is over any other pair of trading days (say, Monday close to Tuesday close). Empirically, however, weekend volatility is approximately the same as volatility during trading days.^ Eor this reason, each (calendar- day) implied volatility is adjusted to a trading-day basis. Eirst, based on

Tbe real-time VfX computation uses a forecast of future dividends to compute tbe component implied volatililies. Tbe bislnrical series for tbese forecasts were unavailable wben tbe CBOE introduced tbe index. •"To avoid tbe problems assnt iated witb extreme staleness of tbe reported index level, computation and dissemination of VIX generally begins at 9:00 A.M (CST), 30 minutes after tbe stock market bas opened. Also, since tbe stock market closes at .^:00 P.\1 (CST), IS minutes prior to tbe OEX option market. VIX computation ends at .̂ :C)0 PM eacb day. *• Frencb and Roll (1986) estimate lb;it tbe weekend return variance for all NYSE and AMEX stocks is only 10.7% greater tban tbt- trading day variance. For tbe largest market capitalisation quintiie (whJcb includes all S&P 100 stocks), tbe weekend return variance is only 8.2% bigber.

Market Volatility Index 269

tbe number of calendar days to expiration, N^, the number of trading days, N|, is computed as

N, = Nc - 2 X int(N,/7) (1)

since expiring OEX options always cease trading on a Eriday (except for a rare holiday). Next, the implied volatility rate is multiplied hy the ratio of the square root of the number of calendar days to the square root of the number of trading days, that is.

where criicTc) is the annualized trading-day (calendar-day) implied volatility.

This trading-day adjustment of implied volatilities is different than simply using the number of trading days in valuing the option. The option s time to expiration parameter affects valuation not only through total volatility, but also through the expected rate of appreciation in the index level and through the length of time over which the option s expected payoff is discounted to the present. Both of these latter considerations are more appropriately measured using calendar days.

Weighting the Component Implied Volatilities

The implied volatilities of the eight near-the-money, nearby, and second nearby OEX call and put options provide the inputs for constructing the volatility index. Some definitions and notation will facilitate the discussion of the weighting procedure used to determine VIX. The nearby OEX series is defined as the series with the fewest, but with at least eight, calendar days to expiration. The second nearby OEX series is the series of the following contract month. To represent the component implied volatilities, subscript the volatility rate with two terms, separated by a comma, e.g., (TC,\- The first subscript is either c or p, depending on whether the implied volatility is computed from a call or a put. The second subscript is either " 1 " or "2," depending on whether the implied volatility is computed from the first (nearby) or second (second nearby) option series. Einally, denote the exercise price just below (above) the current index level, S, as XyiXa). This notation is used as a superscript, e.g., a^'.

The weighting of the component implied volatilities has three steps. Eirst, the implied volatilities of the call and the put in each of the four categories of options are averaged, (T] , for example, denotes the

270 Fleming, Ostdiek, and Whaley

average of the call and put implied volatilities for the nearby options V V \ '

series with the lower exercise price, that is, (rj '' = [CTCJ + (T«\)l2. The V Y Y

other three averages are denoted o-'i', (T\ ", and (TI\ Averaging the call and put implied volatilities mitigates possible bias due to staleness in the observed index level. When the market rises quickly during the trading day, for example, the most recently observed stock prices for some component stocks may lag their true values. This, in turn, causes the reported S&P 100 index level to lag its true value. OEX options, on the other hand, trade more frequently than the S&P 100 basket of stocks. Consequently, an OEX call (put) implied volatility, computed from the reported S&P 100 index level during a rising market is upward- (downward-) biased. Since the upward (downward) bias of the call implied volatility is approximately equal to the downward (upward) bias of the put implied volatility, the effect of infrequent trading of index stocks on the level of VIX is mitigated.'^

Second, "at-the-money" implied volatilities are created by interpo- lating between the nearby and second nearby average implied volatilities.

and

In this way, VIX is always "at-the-money."^ Einally, a constant time to expiration is maintained. The nearby and

second nearby at-the-money volatilities are weighted to create a constant 30-ealendar day (22-trading day) time to expiration. Letting N,, and N,, represent the number of trading days to expiration of nearby and second nearby contracts, the CBOE Market Volatility Index is

Tbis averaging also mitigates tbe possible effects of mismeasuremenl in tbf riskless rate. Tbanks are due Mark Rubinstein for pointing tbis out.

rbc linear interpolation of in-tbe-money and oul-of-tbe-moncy implied volatilities to create an at-tbe-money implied volatility implicitly assumes tbat tbe "volatility smile" is well approximated by a line. For tbe small nmge over wbicb tbe interpolation is made, tbis approximation is reasonable. [See, for example. Dumas. Fleming, and Wbaley (1994).] Similarly, tbe nearby and second nearby implied volatilities arc interpolated to create a constant .^0-calendar day (22-trading day) time to expiration. Hence, tbe dejjree of bias in VIX as an estimate of tbe iO-day volatility rate depends upon tbe linearity of tbe term structure of volatility between tbese two nearby maturities.

Market Volatility index 271

The CBOE uses this procedure to compute VIX on a real-time basis every minute of the trading day. With this construction, VIX provides investors with an up-to-the-minute assessment of the market's perception of future volatility. This study now turns to investigating the empirical properties of changes in the level of VIX.

DAILY AND WEEKLY VIX MOVEMENTS

This study examines daily and weekly movements of the CBOE Market Volatility Index during the seven-year sample period, January 1986 through December 1992. This section begins by describing VIX and S&P 100 index movements during the period and their behavior sur- rounding some important economic events. The univariate properties of VIX changes are then discussed, together with VIX cross-correlations with the S&P 100 index.

Historical Account

Historical data are available on the CBOE Market Volatility Index since the beginning of 1986. Figure 1 plots the weekly closing index levels against the absolute magnitude of weekly S&P 100 returns. Several periods of relatively high volatility and large stock market moves stand out. The October 1987 stock market crash is accompanied by an end- of-week volatility level in excess of two and a half times the next largest historical level. (Figure I is truncated. VIX closed the week at 98.81 on October 23, 1987, after reaching its historical high of nearly 173 in the last hour of trading on October 20.) As the figure shows, it took several months for VIX to return to pre-crash levels. 1 he index declined somewhat erratically throughout 1988, attaining a plateau around 20 by early 1989 before spiking up during the October 1989 mini-crasb. (The index opened on Monday, October 16, 1989, at 51.71 but closed the day at 24.29.) The index then remained somewhat erratic for several months before settling around 20 just before the Iraq invasion of Kuwait in August 1990. News of the invasion brought a large stock market move (the S&P 100 lost nearly 15% that week) as well as a sharp increase in VIX (from 20 to 35). After falling sharply to the low 2O's at the end of 1990, the index again spiked when Allied forces attacked Iraq in January 1991. During the remainder of 1991 and in 1992, the index fell to historically low levels with a moderate spike in late 1991 and another

272 Fieming, Ostdielc, and Whaiey

CBOE Market Volatility Index S&P 100 Index Return (Absolute Value, %)

1986 1987 1988 1989 1990 1991 1992 1993

FIGURE 1. Weekly closing CBOE niiirkel volatility index levels and S&P 100 index return absolute values during the period from 1986 through 1992. The data points for each index reflect the closing level for the lasl day of each week. Thf S&P 100

returns are reported on the basis of the log index relative multiplied by 100.

in November 1992 around the U.S. presidential election before closing the year under 1 5.

During the period covered by Figure 1, spikes in the volatility index are usually accompanied hy large moves, up or down, in the stock index level. There are exceptions, however. In mid-1987 and again in November 1992, for example, VIX spiked sharply during periods when S&P 100 index movement remained fairly moderate.

Statistical Properties of the Volatility Index

The empirical analysis of the volatility index is based on volatility "changes" (i.e., A V , = YJ - "M-i, where V, denotes the level of the volatility index at time t) rather than levels. Three considerations dictate this. First, the variable of interest for academics and practitioners is changes or innovations to expected volatility. They want to know how changes in expected volatility influence changes in security valuation. Second, if stock prices follow a random walk, estimation of the relation- ship between the stock and volatility indexes in levels may be spurious. Third, VIX levels also appear to be near-random walk. Daily VIX levels have a first-order autocorrelation of nearly 97%. Although a unit root

Market Volatility Index 273

can he rejected for this series, such high autocorrelation may affect inference in finite samples.

The univariate properties of the volatility index are considered first. Summary statistics are calculated for daily and weekly VIX changes over the entire sample period and for calendar-year suhperiods. In each case, results are presented with and without the 1987 and 1989 market crashes. The 1987 crash period is defined, in most analyses, as October 19-30, 1987, and the 1989 mini-crash period is defined as Octoher 13-16, 1989.

Table I summarizes the properties of daily closing volatility index changes. The mean daily changes range from —0.0827% in 1988 to 0.0243% in 1990. The mean volatility change over the entire sample period is —0.0026. Apparently, over the seven-year period, market volatility did not drift in one direction or another. The standard deviation of the daily volatility changes (volatility of the volatility) also is fairly stable, ranging from 0.7421 in 1992 to 1.8014 in 1988. Not surprisingly, including the 1987 crash observations increases the 1987 standard deviation of volatility changes by more than a factor of five, from 1.6013 to 8.8886 on a daily basis.

Table I also provides the autocorrelation structure of the daily volatility index changes for one through three lags. Unlike the mean and standard deviation, the autocorrelation structure of VIX changes varies quite substantially from year to year. The first-order autocorrelations, for example, range from -0.301 in 1988 to 0.094 in 1987 without the crash. Over the noncrash sample, the first- and second-order coefficients, —0.073 and —0.104, reveal a significant negative autocorrelation. This degree of correlation, however, is much smaller than the autocorrelation reported in Harvey and Whaley (1991) for individual S&P 100 options. Based on 3:00 PM prices, they report an autocorrelation structure of -0 .33 and -0 .13 (-0.33 and -0.09) for daily volatility changes implied by the nearby at-the-money call (put).

The smaller autocorrelations for VIX relative to those for individual OEX implied volatilities arc attributable to index construction. Kirst, VIX is a time-to-expiration weighted average of tbe volatilities implied by the nearby and second nearby options. This weighting places less weight on the near-expiration implied volatilities, which tend to be more volatile as expiration approaches. Second, the component implied volatilities are computed using bid/ask price midpoints rather than transaction prices. This eliminates the spurious negative correlation induced by bid/ask bounce. Finally, the index is an average of eight option prices which reduces the rebounding effect on the implied volatility following a day

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Market Volatility Index 275

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Table I also summarizes the properties of daily S&P 100 index returns. Most notably, the autocorrelation for one through three lags is negligible except when the Octoher 1987 crash data is included. The effect of infrequent trading on close-to-close S&P 100 index returns appears to he insignificant.

The final five columns of Tahle I summarize the cross-correlation structure hetween VIX changes and S&P 100 index returns. The rela- tionship hetween changes in volatility and changes in the stock market is important for a number of reasons. The Sharpe (1964)/Lintner (1965) and Merton (1973a) capital asset pricing models predict that stock prices will fall as expected volatility rises.'' In addition, Black (1976) and Christie (1982) suggest that leverage can induce an inverse relationship between stock returns and changes in future volatility. When stock prices fall (relative to hond prices), leverage increases and causes expected volatility to increase."*" If the volatility index proxies for future stock market volatility, VIX movement should be inversely related to the contemporaneous stock market return.

The estimated contemporaneous correlation hetween changes in expected volatility and S&P 100 returns reported in Table I is large and negative across the sample and within each subperiod. The contempora- neous cross correlation in the noncrash sample is —0.61 5. As the theory predicts, stock prices fall (rise) when expected volatility rises (tails). The correlation reverses sign and dampens out quickly to more moderate levels at the noncontemporaneous lags. Slight positive correlation shows up hetween the current change in expected volatility and hoth past and future stock index returns, but the stronger of the two effects appears between VIX changes and past stock market returns.

Tahle II summarizes the statistical properties of weekly volatility changes. Tbe weekly data exhibit patterns in mean and standard devia- tion similar to those apparent in the daily data. The mean weekly index

^Merlon's thcorv' motivates many recent .AiitoRef^ressivt- Conditional Hfleniskcdaslifity In Means (ARCH-M) models of stock market volatility. See, for example, t'rench, Schwert. iind Stiimbaugh (1987). Bollcrslev. Engle, and Wooldridge (1988). and Nelson (1989, 1991). '"Black (1992) argues that ihe inverse relationship between stock prices and volatility also may be attributable to changes in operating leverage. Por example, a downward revision in the firm's prospects without a commen.surate reduction in costs (other [ban interest) eaiises the lirm's operating leverage In increase and the stock price to fall. "More recent studies, such as Krench, Schwert, and Stamhaugh (19871 and Schwert 11989, 1990). find an even stronger relationship between returns and volatility than suggested by tbe leverage arguments of Black and Cbristie.

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Market Volatility Index 277

changes range from -0.4027 in 1988 to 0.0909 in 1990, approximately five times their corresponding levels in Table I. The volatility of the weekly volatility index changes ranges from a low of 1.2426 in 1992 to 3.1973 in 1990. Again, including the crash periods sharply affects the volatility of the volatility index. The standard deviation of 1987 volatility changes, including the Octoher market crash, is over 20—more than six times greater than the largest standard deviation reported for the noncrash years.

Table II also provides the autocorrelation structure of the weekly volatility index changes. The weekly changes have significantly stronger negative first-order autocorrelation, -0.259 over the noncrash period, than evidenced in daily changes, -0 .073 . Unlike the daily autocorrela- tion, the weekly autocorrelation is negative in all years. The magnitude of these lag one autocorrelation coefficients, relative to those in Table I, re- flect the persistence of longer-term stock market volatility. Negative first- order autocorrelation also appears in the weekly S&P 100 index returns. This autocorrelation, however, largely dissipates by the second lag.

The strong temporal association between volatility changes and stock market returns, reported in Table I for daily observations, is also present in Table II for weekly observations. Over the noncrash sample, the contemporaneous cross-correlation is —0.464. This relationship is fairly strong in all years. Positive relationships between current expected volatility and past-week and future-week stock returns also are evident. The correlation between the current volatility change and the lagged and lead stock market returns, respectively, are 0.205 and O.I 14. The cross-correlations estimated for the other noncontemporaneous lags, however, exhibit greater variability from year to year and are generally insignificant.

Co-Movement of the Volatility Index with Stock Market Returns

A number of researchers have detected intertemporal relationships between stock market prices and expected volatility. This section focuses more explicitly on these relationships. In particular, VIX is used as a proxy for expected volatility and the S&P 100 proxies for the stock market in validating several stylized facts reported in previous research with alternative volatility proxies. First, Black (1976) and Christie (1982) documented a strong negative association between stock market returns and changes in ex post future volatility. French, Schwert, and Stambaugh (1987) provided similar evidence using changes in expected

278 Fleming, OstdEek, and Whaley

stock market volatility. Second, Schwert (1989, 1990) found asymmetry in the relationship between returns and expected volatility, that is, the increase in expected volatility corresponding to a given negative stoek market return is larger than the decrease in expected volatility corresponding to a similar size positive return. Third, French, Schwert, and Stambaugh reported a positive relationship between the predictahle component of stock market volatility and expected stock returns. If VIX represents the market volatility prediction, changes in the volatility index may be positively related to future S&P 100 returns. Finally, to the extent VIX is mean reverting, the strong negative contemporaneous relationship between stock returns and expected volatility may induce a slight positive association between changes in expected volatility and past stock market returns.

The degree to which these past findings are reflected in the relationship between VIX and the S&P 100 index can be assessed with the following multivariate regression of the volatility change on two lag, two lead, and the contemporaneous stoek market returns, as well as the magnitude (absolute value) of the contemporaneous return.

i + /3\s\\Rs.,\ + e, (5) i = - 2

This specification focuses on the intertemporal relationship between stock market returns and changes in expected volatility. A signifi- cant negative (positive) /3s.,,' ^ - 2 , . . . , 2, coefficient indicates that increases in expected volatility at time t are accompanied by stock market declines (increases) at time t + i.\ significant positive (negative) /3|s| coefficient indicates that stock market moves, independent of their direction, are accompanied hy increases (decreases) in expected volatility.'^ The sum oi /3\s\, and ^s.o, therefore, measures the asymme- try of the relationship between changes in expected volatility and stock market returns.

Several factors may influence the estimation of regression (5). First, the reported S&P 100 index level is based on the last trade prices of the index stocks. Since stocks do not trade continuously, reported index levels will always be stale relative to true index levels. Consequently, stock index returns, measured over short intervals, will be positively autocorrelated and the estimated contemporaneous correlation

Coefficients on lead and lag absolute stock index returns are prcdominamly insignificant, so they are nnl included in regression (5).

Market Volatility Index 279

between changes in VIX and S&P 100 returns will be reduced. Table I, however, reports that the autocorrelation in daily S&P 100 returns is insignificant, indicating that infrequent trading effects are negligihie. Second, the estimation of (5) may be affected by the correlation between R,s and \Rs\- Empirically, however, the correlation between Rs and |R.sl is inconsequential (i.e., —5.7%). Finally, Table I suggests tbat autocorrelation and heteroskedasticity in volatility changes may influ- ence estimation. To allow for these residual properties, the t-statistics provided for regression (5), and for all subsequent regressions in this study, are based on heteroskedasticity and autocorrelation consistent standard errors using Hansens (1982) method of moments estimation and Par/en weights [Gallant (1987), p. 533]. Since the length of residual autocorrelation in regression (5) is unknown, Andrews's (1991) method of automatic bandwidth selection is employed.

Table III reports the regression (5) results for daily VIX changes and S&P 100 returns. In the noncrash sample, the coefficient of the contemporaneous, signed change is large and negative, —75.55, with a (-statistic of —14.21. This coefficient is consistent with the cross- correlation analysis reported in Fahle I and confirms the strong negative contemporaneous relationship between changes in expected volatility and stock market returns. The lag one and two and lead two coefficients are positive and significant but much smaller in magnitude than the contemporaneous coefficient. So, while a positive relationship exists between changes in expected volatility and past and future stock returns, the relationship is dominated by the negative contemporaneous associ- ation. Nonetheless, the sign and magnitude of these coefficients offer some support for the hypothesized noncontemporaneous relationships described above. Finally, the (3\s[ coefficient is large and positive, 30.11, with a (-statistic of 3.87. Apparently, a positive relationship exists between the size of a stock market move and the contemporaneous change in expected stock market volatility.

The estimates of f3\s\ reported in Table III indicate a significant asymmetry in the relationship between volatility changes and contem- poraneous stock market returns. If the stock market return is positive, the coefficient impacting the change in volatility is /3j; ^ ^,s,() + ^|sh or —45.43, across the noncrash sample. A stock market increase is ex- pected to accompany a decrease in the volatility index. On the other hand, if the stock return is negative, the coefficient is ^^ = ^,s,o ^ fi\s\> or —105.66. A stock market decline is expected to accompany an in- crease in volatility. The difference in the magnitudes of the coefficients {/3s is more than two times the size of f3s ). however, clearly indicates

280 Fleming, Ostdiek, and Whaley

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Market Volatility Index 281

asymmetry. Negative stock market moves are associiited with changes in the volatility index that are much larger than those associated with positive stock market moves of similar size.

The regression (5) results for weekly VIX changes and S&P 100 returns are presented in Tahle IV. These results are similar in sign to the daily results hut, for signed returns, only the contemporaneous and lag one coefficients are significant. The contemporaneous coefficient for the noncrash sample, -51.78 (t-statistic = -6.41), reveals a strong relationship hetween volatility and returns at the weekly level. The ^ |s | coefficient, 39.53 ((-statistic = 3.52), indicates a positive relationship also exists hetween the size of a weekly stock market move and the contemporaneous weekly change in expected stock market volatility. /3s and ^s ' ys defined ahove, show an even more exaggerated asymmetric relationship hetween the direction of the stock market move and the change in expected volatility than apparent in the daily data. For the weekly regression,/3s = -12.25 and ^s ^ -91 .31 , nearly seven times the size of ^s •

The documented relationship hetween VIX and stock index returns calls into question the reasonahleness of using the Black-Scholes (constant volatility) option valuation framework. Its use, however, can he defended on two grounds. First, there is no commonly accepted alternative model. While the Cox and Ross (1976) constant elasticity of variance (CEV) model captures the inverse relationship hetween returns and volatility, it does not account for asymmetry in the relationship nor does it account for changing hehavior in the relationship through time. Second, even if an alternative specification were availahle, estimation is practically impossihle. Relatively error-free option pricing data are availahle only for at-the-money options. Estimating more than one option valuation parameter, however, requires option prices for a wide range of exercise prices.

INTRADAY AND INTRAWEEK VOLATILITV INDEX EFFECTS

Finding seasonalities in stock returns is commonplace in the empiri- cal finance literature. Among them are intraday effects such as those documented hy Wood, Mclnish, and Ord (1985), where return volatil- ity appears higher at the open and the close of trading than it does mid-day, and intraweek effects such as the day-of-the-week effect first documented by Cross (1973), where average stock returns from Friday close to Monday close are negative. Since implied volatility is based on, among other things, a stock index level, it may he the case that patterns

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Market Volatility Index 283

also appear in the volatility index. This section investigates intraday and intraweek patterns in the voiatiiity index.

Intraday Volatility Index Behavior

To identify intraday patterns in the volatility index, minute-by-minute volatility index levels are examined. First, the mean volatility index level is eomputed for each day and is used to construct minute-hy-minute deviations from the daily mean. The minute-by-minute deviations are then averaged longitudinally across all days during the sample period. Figure 2 displays the average deviation hy time of day.

In Figure 2, the volatility index begins the day about 10 basis points above the daily mean. The index then slowly decays until about 2:00 PM. when it quickly drops about 25 basis points below the mean. The slow decay until 2:00 may be an artifact of index construction. The number of days to expiration used to compute the VIX component implied volatilities remains constant throughout the trading day, even though the time to expiration does, in fact, decrease. To illustrate the effect this may have on the volatility index, consider the value oi a

Average Deviation {Basis Points)

-20 -

9:00 10:00 11:00 12:00 100 2:00 3:00

FIGURE 2. Average intradav deviations in CBOE market volatility index levels. The average

deviations from llie daily mean voiatiiity index level are plotted for each minute of" the trading day. The averages are computed for the noncrash sample period from

1986 through 1992 which excludes 10/19/87-10/30/87 and 10/1.3/89-10/16/89.

284 Fleming, Ostdiek, and Whaley

30-day, at-the-money OEX call at 9:00 AM. If the S&P 100 index is 400, the interest rate is 6%, the dividend yield rate is 3%, and volatility is 20%—the theoretical value of an American-style call is 9.615. Now, suppose the only thing that changes between 9:00 AM and 2:00 PM is that the time to option expiration shortens hy five hours. The value the call using .̂ 0 - 5/24 = 29.79 days to expiration is 9.579. If 9.579 is the call's price, but the implied volatility is still computed using 30 days to expiration, the implied volatility becomes 19.91%. In other words, because the time to expiration parameter is not continuously adjusted during tbe trading day, the implied volatility must decrease to pick up the reduction in the option's time premium. Under this explanation, the reduction in the volatility index should be smootb throughout the day, as is generally shown in Figure 2 (until 2:00 PM}, and should be approximately 10 basis points since the parameter values used in the illustration above closely match the average parameters realized during the sample period.

Figure 2 also shows that the volatility index drops by about 25 basis points after 2:00 PM. This, too, may be attributable to integer measurement of days to expiration. Since market makers bave no control over tbe erosion in option time premium overnight, they may choose to mark down option prices prior to the market close on day t to reflect the number of days to expiration for the following day, ( + 1. Using the parameters from the illustration above, this means the call option price should fall throughout the trading day from 9.615 (at 30 days) to 9.446 (at 29 days). If the call price is 9.446 at the end of the day and the number of days to expiration used is 30, the implied volatility is 19.62%, approximately 38 basis points below its true level. Not coincidentally, perhaps, the observed drop from the beginning to the end of the day in Figure 2 is about 35 basis points.

Although this illustration provides an explanation for the intraday patterns observed in the volatility index, it also appears to signal the possibility of arbitrage trading profits, particularly between 2:00 PM and 3:00 PM. But, consider the magnitude of the profits signaled in the illustration. At 3:00 PM, the option would be 30 - 6/24 = 29.75 days from expiration and its value would be 9.573. Arbitrage profits seem to exist because, by 3:00 PM, the option appears to be valued at 9.446 with a time to expiration of 29 days. The economic magnitude of the disparity, however, is just 9.573 - 9.446 = 0.127, almost exactly an eighth. Transaction costs, including those encountered in futures trades to complete the delta-neutral arbitrage, would seem to render these profits economically insignificant.

Market Volatility Index 285

Despite the economic insignificance of the end-of-day decline in option prices, the decline implies a statistically significant effect on the volatility index. This significance can be shown in a regression of hourly deviations from the mean (DEV) on hourly dummy variables.

The coefficients ^i, i = i . . . . , 7 , represent the average deviation from the mean daily volatility for each hour during the trading day. The regression results are reported in Table V.

Tahle V confirms the results depicted in Figure 2 by showing that the intraday deviations from the mean are quite persistent in a statistical sense. At 9:00 AM, for example, the coefficient estimate is 0.0972 witb a (-statistic of 6.46. The volatility index starts the day about 10 basis points above the daily mean. By the end of the trading day, the volatility index is significantly below the daily mean. At 3:00 PM, the coefficient estimate is -0.2418, with a 1-statistic of -14 .43 . Ihe time-of-day pattern evidenced by the volatility index, then, is a systematic decline throughout the day as the time premiums of the component options erode.

It would appear that a more precise construction which accounts for the deterministic time decay could remove the intraday trend in VIX. The decay, however, is not constant throughout the day. Although the time to expiration appears to decline by a full day from 9:00 to 3:00, most of tbe decay occurs in the final hour. The steeper decline in the index during the final hour indicates the OEX option traders do not simply account for the overnight time decay proportionally throughout the trading day. Instead, the time erosion is not impounded in option prices until traders believe there is a high probability that they will hold a given position overnight. Because the intraday trend in VIX is not a constant function of time, correcting it with a time to expiration adjustment would prove difficult.

Intraweek Volatility Index Behavior

Intraweek behavior is displayed in Figure 3. The average closing volatility index level is computed across the days of the sample for both the trading-day and calendar-day volatility indexes. Recall from the description of VIX construction that the component implied volatilities are adjusted from calendar days to trading days to reflect the fact that total weekend volatility is approximately the same as trading day volatilily. Figure 3 demonstrates what would happen if

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Market Volatility index 287

Average Trading-Day Index Average Calendar-Day Index

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computed For the sample period from 1986 through 1992 (excliidinj^ 10/19/87-10/30/87 and 10/13/89-10/16/89) are plotted for each day of the week. Curves arc

provided for the trading-day volatility index and the calendar-day volatility index.

no adjustment is made." Note, among other things, that the scale on the right shows that calendar-day volatilities are considerably lower than trading-day volatilities.

Figure 3 shows the importance of the trading-day adjustment. When calendar days are used, the average index level starts high on Monday and systematically falls during the week. The total drop in the calendar-day index is about 70 basis points, with about half of that amount occurring on Friday. Such behavior is perplexing, since there is little economic reason to believe expected market volatility should decline throughout the week. The average trading day index, on the other hand, displays a more sensible pattern, remaining fairly stable at a level of about 21%. The average level increases slightly on Wednesday and again on Thursday; however, the total size of the increase is less than 20 basis points.

Tables VI and VII provide a closer scrutiny of the day-of-the- week effects in intraweek VIX changes shown in Figure 3. The tables report results for the regressions of VIX changes on five day-of-the-week

' 'The calendar-day volatility index is not publicly available. It was computed when VIX was designed to demonstrale the virtues of using the trading-day adjustment.

288 Fleming, Ostdiek, and Whaley

TABLE VI

Day-of-the-Week Effects in CBOE Market Volatility Index Level Changes^

Period

Noncrash Sample'̂ 1986-1992

1986

1987

1988

1989

1990

1991

1992

Entire Sample 1986-1992

1987

1989

1755

252

242

253

249

253

252

254

1769

253

252

Monday

0.1836 (-2.44)

0.2857 (-2.92) -0.0172

(-0.07) -0.4319

( 1.60) -0.0493

(-0.41) -0.2080

(-0,76) -0.1286

(-0.68) -0.1561

(-1.56)

0.1663 (0.48) 2.4702

(1.06) -0.1323

(-0.92)

Coefficients (Day of Week)

Tuesday

0.0598 (1.12) 0.1188

(1.40) 0.1912

(0.94) -0.2010

(-1.46) -0.0316

(-0.25) -0.0141

(-0.08) 0.2188

(1.94) 0.1367

(1.38)

-0.0072 (-0.10) -0.3163

(-0.77) 0.0071

(0.05)

Weiincsday

0.0363 (0.66) 0.0165

(0.16) - 0.2322

(-1.33) 0.3904

(2.19) 0.0656

(0.71) 0.0596

(-0.36) -0.0410

(-0.26) 0.1008

(0.99)

-0.1914 (-0.98) -1.8071

(-1.39) 0.0656

(0.71)

Th iirsday

0.1180 (1.67) 0.2445

(2.18) 0.0233

(0.09) 0.0682

(0.28) 0.2220

(1.66) 0.3814

(1.66) -0.0736

(-0.46) -0.0390

(-0.40)

0.1504 (1.30) 0.2524

(0.37) 0.2220

(1.66)

Friday

0.1357 (-1.60) -0.0876

(-0.77) 0.0832

(0.32) -0.2735

(-0.75) -0.4294

(-2.76) 0.0129

(0.06) -0.0538

(-0.25) -0.1896

(-1.67)

-0.1208 (-1-32) -0.0561

(-0.21) -0.1963

(-0.71)

0.0059

0.0419

-0.0089

0.0101

0.0394

-0.0002

-0.0059

0.0158

-0.0007

0.0081

-0.0008

^Parameter estimaies are presented for the regression of daily volalility index changes on day-of-the-week dummy variables [regression (7)]. f-statistics are provided in parentheses and are based on Hansen's (1982) heteroskedasticity consistent method of moments estim_ation. Andrews's (t99I) method of automatic bandwidth selection identifies the length of residual autocorrelation as zero. R is Ihe coefficient of determination, adjusted for 5 degrees of freedom. ''The noncrash sample excludes index observations during the periods 10/19/87-10/30/87 and 10/13/89-10/16/89.

dummy variables,

-I- (7)

where A V , is the close-to-close change in volatility and Dit is a dummy variable for each day of the week, Monday through Friday. Table VI contains the regression results for VIX computed on the basis of trading days and Table VII contains the results for VIX computed using calendar days.

The estimated day-of-the-week coefficients confirm the patterns illustrated in Figure 3. The Table VI results, for example, show little

Market Volatility Index 289

TABLE VII Day-of-the-Week Effects in Calendar-Day Volatility Index Level Changes^

Period Ok.

Noncrash Sample'' 1986-1992

1986

1987

1988

1989

1990

1991

1992

Entire Sample 1986-1992

1987

1989

1755

252

242

253

249

253

252

254

1769

253

252

Monday

0.4056 (6.27) 0.2773 (3.36) 0.6704

(3.29) 0.3034 (1.31) 0.4583 (4.46) 0.4712

(2.00) 0.3867 (2.31) 0.2873

(3.30)

0.7134 (2.38) 2.8551

(1.43) 0.3823 (3.05)

Coefficients (Day of Week]

Tuesday

-0.0406 (-0.85)

0.0067 (0.09) 0.0382 (0.22)

-0.2696 (2.17) -0.0892

(-0.75) -0.1122

( 0.69) 0.0996

(1.00) 0.0417 (0.46)

-0.0980 (-1.49) -0.3948

(-1.11) 0.0573

(-0.47)

Wednesday

-0.1094 (-2.31) -0.1112

{-1.26) -0.3714

(-2.44) 0.1698

(1,11) -0.0704

(-0.89) -0.2149

(-1.51) -0.1635

( -1.20) -0.0173

(-0.20)

-0.3079 (-1.81) -1.7435

(-1.55) -0.0704

(-0.89)

Thursday

0.0442 (-0.73)

0.0755 (0.79)

-0.1569 (-0.71) -0.1178

(-0.57) 0.0641 (0.56) 0.1610

(0.82) -0.1918

(-1-38) 0.1438

(-1.69)

-0.0181 (-0.18)

0.0292 (0.05) 0.0641

(0.56)

Friday

-0.2751 (-3.78)

0.2236 ( 2.29) -0.1268

( 0.57) 0.4167

(-1.33) -0.5102

( 3.82) -0.1678

(-0.87) 0.1892

(-1.02) -0.2816

(-2.91)

-0.2631 (-3.34) -0.2545

(-1.11) -0.3067

(-1.28)

0.0364

0.0525

0.0458

0.0136

0.1170

0.0197

0.0289

0.0652

0.0117

0.0223

0.0274

^The calendar-day volatility index ievel is computed using the same procedure used to compute VIX, but witfiout the adjustment to trading days. Parameter estimates are presented for the regression of daily calendar-day volalility index changes on day-of-the-week dummy variables [regression (7)]. See notes to Table VI regarding estimation of the standard errors. R^ is the coefficient of determination, adjusted for 5 degrees of freedom. "The noncrash sample excludes index observations during the periods 10/19/87-10/30/87 and 10/13/89-10/16/89.

change in the volatility index from day to day. Only the Monday dummy variahle is significant in the overall noncrash sample, with the volatility index falling hy less than 20 hasis points from Friday close to Monday close. On the other hand, the results for the volatility index using calendar days (Tahle VII) show pronounced seasonality. Volatility increases significantly from Friday to Monday and decreases significantly from Tuesday to Wednesday and Thursday to Friday. On the hasis of these results, the trading-day adjustment to implied volatility is clearly warranted and it removes almost all of the intraweek seasonality that is seen without the adjustment.

Finally, the influence of stock index futures and option expiration days is worth noting. A great deal of puhlic controversy has heen

290 Fleming, Ostdiek, and Whaiey

focused on the volatility of stock index returns on "triple-witching" days when stock index futures, options, and futures options expire. StoU and Whaley (1987. 1991), for example, document increased volatility on the days surrounding expirations. To investigate the magnitude of expiration-day effects on VIX. regression (7) is estimated with additional dummy variahles to identify the expiration day, the day hefore expiration, and two days hefore expiration.'** Table VIII reports the regression results. While the estimations vary considerahly from year to year, the overall noncrash sample indicates that VIX falls by 45 hasis points the day hefore expiration and hy 49 hasis points on expiration day. One interpretation of these results is that index futures and option expiration days are truly high voiatiiity days. Since VIX represents the average volatility rate over the next 30 days, it should not he surprising that the average rate falls as the high volatility rate of an expiration day is dropped from the average and is replaced hy the volatility rate of a typical day. This evidence is consistent with Day and Lewis (1988).

THE VOLATILITV INDEX AS A FORECAST OF STOCK MARKET VOLATILITV

The previous two sections focus on characterizing the movements of VIX. This section evaluates how well VIX forecasts subsequent, realized volatility. This interpretation of the volatility index may appear, at first, inconsistent with the option valuation model which determines VIX. In particular, the Black-Scholes/Merton framework assumes a known and constant volatility rate. The true volatility rate, on the other hand, is unknown and time-varies. This contradiction requires that the implied volalility he related to some notion of an expected average volatility. For example, for the volatility processes descrihed in Hull and White (1987) (the volatility risk premium is either zero or constant), Feinstein (1989) demonstrates that the implied volatility approximates the market expectation of the average volatility over the life of the option.''' The level of VIX. then, hecause it is hased on the implied volatilities of OFX options, represents an estimate of expected S&P 100 index (or "stock market") volatility.

The analysis of VIX as a forecast of stock market volatility proceeds in two steps. The first step is the consideration of the hias in VIX

•"Post-expiration dummy variabk-s were also considered in the expiration-day regression but were insignificant and are not presented. ""Feinstein (1489) showed that ihc approximation is most accurate for at-thc-money and near-

expiration options.

Market Volatility Index 291

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as a forecast, that is, the degree to which VIX is below or ahove the subsequent realizations. Unbiasedncss, while desirable, is not a crucial property since forecast hias can be corrected if the degree of bias is known or can he estimated. The second step is the examination of the orthogonality of the VIX forecast error with other available information. If VIX incorporates all available information, then additional parameters will not improve the forecasting of future volatility. In this section, the analysis of the relationship hetween the volatility index and future stock market volatility uses the time-series methodology developed in Fleming (1994).

The volatility index is constructed for a constant .SO-calendar- day horizon (22 trading days) and, therefore, should represent the expectation of stock market volatility during the following montb. The unbiasedness and orthogonality tests, then, require an estimate of the volatility realized during the following month. In this section, the estimation of realized volatility is hased on a constant 28-calendar- day horizon which closely corresponds to the 30-calendar-day interval on which VIX is hased. At the same time, a 28-day realized volatility avoids the measurement problems attrihutable to intraweek and weekend effects in stock market returns hecause a 28-day horizon nearly always includes an equal number of stock returns from each day of the week. Specifically, for both unbiasedness and orthogonality tests, tbe realized volatility rate is given hy

y; = 19 T

rff (8)

wbere

is tbe sample variance and R, is the mean S&P 100 index return over tbe interval from ( to i + 20."" The term in the squared hrackets of (8) corrects for the bias attrihutahle to using the square root of tbe sample variance as an estimator for the standard deviation.'^ With a constant

r(a) — Jn .v"" e •" dx is the gamma function.

'This correction factor is identical to the correction provided in Cox and Rubinstein (1985, p. 256), and is bast-d on tbt- assumption that the stock index level is lognormally distributed. For the results provided below, the effect of the correction is fairly small. For example, for a volatility hased on 20 returns, the correction factor is 1.013.

Maritet Voiatiiity Index 293

28-calendar-day inter\al, 20 daily returns are involved in estimating V, ; this explains the constants 20 (N) and 19 (N - 1).'**

In conducting the unhiasedness and orthogonality tests, an AR(1) model of volatility will serve as the standard for evaluating tbe forecast performance of VIX. Poterba and Summers (1986) and Stein and Stein (1991), for example, argue that volatility follows an AR(1). In addition, Canina and Figlewski (1993) suggest that an AR(1) specification using the historical volatility rate dominates tbe OEX implied volatility as a forecast of realized volatility. Tbe historical volatility rate, defined as Vl, t = Vt*20i is the realized volatility over the preceding 28 calendar days (20 trading days). In terms of notation, Vi., will he used to denote either VIX (i.e., i ^ VIX) or historical volatility (i.e., i = h).

Unbiasedness Tests

The unhiasedness of a prox7 such as V , , for, say Vj , is frequently assessed with the regression.

V * ^ a + fiV, + et (9)

Unhiasedness requires that the estimates of a and ^ be indistin- guishable from zero and one, respectively. The volatility index and the computed realized and historical volatility levels, bowever. are near random walk processes. As a result, eq. (9), specified in volatility levels, potentially represents a spurious regression in finite samples. Subtracting the lagged independent variahle from hoth sides oi the regression provides a testable relationship,

V,* - X . 1 - « + /3(X, - X<-i) + ^i

tbat is free of the spurious regression problem. To account for pos- sible residual heteroskedasticity and autocorrelation, Hansen's (1982) generalized method of moments (GMM) is used to estimate the vector,

Since vector (11) involves two equations and two unknowns, the es-

timates of a and j0 are identically equal to those in eq. (10). GMM

estimation of eq. ( I I ) , however, provides consistent covariance matrix

'*'N = 20 is used for oxpositional convenience. During intervals in which holidays e.xist, i\ would be less than 20. The empirical results are hased on the actual number of trading days in the 28-calentiar-day interval.

294 Fleming, Ostdiek, and Whaiey

estimation and facilitates a framework for conducting the orthogonality tests. Parzen weights, with a lag length of 19 (the length of the overlap between consecutive dally volatilities), are used to estimate the covariance matrix.

The first two lines for each sample in Tahle IX summarize the unhiasedness regression results for the volatility index and the historical volatility. The CS-^-statistics provide a Wald test of the joint restrictions on a and /3,

(12)

TABLE IX Relationship between the CBOE Market Volatility

Index and Future S&P 100 Index Volatility

Unbiasedness Tests'̂ Voiatiiity index Historical volatility

Orthogonality Tests'̂ Volalility index Historical volatility

Unbiasedness Tests'̂ Volatility index Historical volatility

Orthogonality Tests'' Volalility index Historical volatility

0.0584 - 0.0028

0.0148 0.0710

--0.0513

- 0.0009

0.0536 0.0796

-15.19 -0.65

0.98 3.93

-5.58 -0.09

3.37

6.91

Noncrash Sample (1679 Days)''

0.6283 0.6314

0.6483

0.4802

( , IS

4.30 2.25

4.51

4.10

0.0315

0.0186

0.4530

0.2738

Entire Sample (1770 Days)

0.6842

0.8926

0.5047

0.4931

5.51

0.46

6.68 6.79

0.0449 0.0215

0.1519

0.0815

CS'

291.00 6.36

323.99 17.11

CS-

188.53 0.27

93.11 56.96

—

0.24

17.87

—

0.05 6.42

^The table considers the unbiasedness and orthogonality of VIX and the 28-day historical volatility as a forecast of the S&P 100 voiatiiity realized over the following 28 calendar days. For each set of results, f-stalistics report the significance of departures from zero and one, respectively, for « and ji. The (-statistics are based on Hansen's (1982) method of moments estimation of the covariance matrix, using Parzen weights and a residual autoregressive lag length of 19. Ti^ is the coefficient of determination, adjusted for 2 degrees of freedom.

"The noncrash sample excludes index observations during the periods 9/21/87-11/30/87 and 9/15/89-11/10/89 •̂ The unbiasedness tests involve estimation of regression (11). The unbiasedness test statistic is CS^, which is a Wald test that « and p. respectively, are nondifferent from zero and one. Under the null, CS^ is asymtotically distribuled xh The orthogonality tests are based on regression (13), using the histohcal volatility as an instrument in the VIX regression,

and using VIX as an instrument in the historical voiatiiity regression. The orthogonality test statistic, 01^ is the Hanseri (1982) test of overidentitying restrictions and is asymptotically distributed ^ f under the null of orthogonality.

Market Volatility Index 295

where O is the GMM estimate of the covariance matrix of the pa- rameter estimates, a and ^. On the basis of the CS'-statistics, the unbiasedness of the volatility index as a forecast of realized S&P 100 volatility is strongly rejected. In fact, the average difference between the volatility index and the subsequent, realized stock market volatility is 584 hasis points. The historical volatility incurs a weaker, but signilicant, unbiasedness rejection. Its unconditional bias is just 28 hasis points. Of course, the difference in means between the historical and realized measures is low by construction. Since the two samples are simply offset by 20 trading days, the difference in means is confined to the effect of differences between the first 20 historical volatility obser\'ations and the last 20 realized volatility obser\'ations.

Several factors may contribute to the forecast bias of the volatility index. First, the option pricing model may he misspecified. This may produce systematic bias in implied volatility levels while still producing an estimate that is highly correlated with the true implied volatility. Second, the estimation of the component implied volatilities ignores the wildcard option imbedded in the OEX option. Ignoring the wildcard option biases the level of VIX upward. Fleming and Whaley (1994) estimate that the wildcard option imbedded in an at-the-money OEX option may bias implied volatility, on average, by about 60 basis points. Third, the calendar-day to trading-day adjustment employed to account for weekend volatility may overcompensatc for lower weekend volatil- ity. The absence of day-of-the-week effects in the trading day index, however, indicates that the magnitude of this adjustment is appropriate. Fourth, the infrequent trading of S&P 100 stocks may bias downward the realized volatility estimate. The low autocorrelation in daily S&P 100 returns reported in Table I, however, indicates that this would not account for a large share of the bias in VIX. Finally, a portion of the upward bias in VIX may result from the over-pricing of OEX options. This finding is consistent with the evidence reported in Eongstaff (1993), for OEX call options, and Fleming (1994), for both calls and puts. Fleming, however, also found that the degree of forecast bias in OEX implied volatilities is not large enough to generate arbitrage trading profits after transaction costs.

While the forecast hias of the volatility index is strong, it is not problematic if the bias is constant and/or its magnitude is known. Given the constant 30-calendar-day horizon of the index, the bias may be reasonably constant. Below, a bias-adjusted VIX which assumes constancy is evaluated as a forecast. The more interesting result of the unhiasedness tests, though, involves the explanatory power of the

296 Fleming, Ostdiek, and Whaley

volatility index relative to historical volatility. The R^ statistics reported in Table IX indicate that VIX provides a much stronger relationship with realized volatility than does historical volatility. The orthogonality tests below confirm this evidence.

Orthogonality Tests

ln this section, the orthogonality of the volatility index and historical volatility are measured relative to one another. For example, the or- thogonality of the volatility index would be rejected if its forecast error covaries with historical volatility. If this were the case, historical volatility could be used to supplement VIX to provide a hetter composite forecast of realized volatility. Orthogonality is assessed with GMM estimation of the vector,

1 J-'

where V-,., represents the alternative forecast to "Vi^. Recall that eq. (U), for the unbiasedncss test, is specified in differences as a precaution against the spurious regression problem. Inference regarding orthogonality using eq. (13), however, is unaffected by the spurious regression problem, as is documented in Fleming (1994). Since eq. (13) remains specified in levels. Table IX provides the results of the ba- sic regression stated in both levels (orthogonality) and differences (unbiasedness).

Unlike eq. (11), the system in (13) is overidentified with three equations and two unknowns and cannot be set equal to zero. The closeness of the solution to zero, however, directly reflects the orthogo- nality of forecast "Vj., to the alternative forecast, V-;.,. The Hansen (1982) test of overidentifying restrictions measures the closeness to zero and provides a formal orthogonality test. In Table IX, this test statistic is labeled Ol ' and is asymptotically distributed xf under the null hypothesis of orthogonality.

The final two lines for each sample in Table IX summarize the orthogonality tests. As with the direct unbiasedness tests, unbiasedness is again rejected (on the basis of the CS^-statistics) for both the volatility index and the historical volatility. Again, the R^ statistics indicate that the explanator\' power of the volatility index dominates that of the historical volatility. The orthogonality test results provide further confirmation of this evidence. Orthogonality of the volatility index as a

Market Volatility Index 297

forecast of realized stock market volatility is not rejected (Ol" = 0.24) when the bistorical volatility is available in the specification. However, when the test is reversed and the voIatiUty index is included as an instrument in the estimation of the historical volatility system, strong orthogonality rejections result (OI^ = 17.87). On the basis of this comparison, the volatility index dominates the historical volatility as a forecast of stock market volatility.

The results in lable IX are consistent with the recent literature regarding the forecast quality of OEX implied volatilities. Fleming (1994) provided a more complete set of orthogonality tests using the implied volatility of at-the-money, nearby expiration OEX options. Those tests included instruments such as stock market returns, interest rate variables, and the ARCH-based parameters. While several instruments induced orthogonality rejections for the historical volatility, none pro- duced rejection for the implied volatility. Day and Eewis (1992) provided a more direct comparison between OEX implied volatilities and weekly GARCH and EGARCH forecasts, finding that the implied volatihty contributes information incremental to that contained in GARGH- based forecasts. They also found, however, that in-sample GARGH forecasts contribute information incremental to the implied volatility. Out-of-sample tests, though, provided inconclusive orthogonality evi- dence. Given these results and the fact that VIX is a function of the component OEX implied volatilities, the results reported in Table IX are not surprising.

Bias-Adjusted Volatility Index Forecasts

The evidence of the volatility index's explanatory power and orthogonality demonstrate that the index represents a quality forecast of future stock market volatility, provided its upward bias is constant or estimable. Two factors suggest that the VIX forecast bias may be reasonably well behaved. First, VIX represents the interpolation of eight OEX implied volatilities to maintain a constant moneyness level and a constant 30-calendar-day horizon. Because these parameters are fixed, VIX does not incur the usual time-variation in implied volatilities resulting from moneyness and time-to-expiration effects. Second, a portion of the VIX forecast bias is attributable to the OEX wildcard option which is ignored in computing the VIX component implied volatilities. Fleming and Whaley (1994) show that the wildcard's value is approximately linear in moneyness and time to expiration. Since VIX is always constructed from options straddling the money and straddhng 30 days to expiration.

298 Fleming, Ostdiek, and Whaley

and with an equal weight on calls and puts, the wildcard's effect on the VIX forecast bias should remain fairly constant.

To the extent the VIX forecast bias is relatively constant, a naive adjustment based on a rolling average of past forecast errors may sufficiently correct the bias. The forecast performance of two naive adjustments is considered below. A 253-day bias adjustment is computed by averaging the most recently realized year (or 253 trading days) of daily forecast hiases; a 20-day bias adjustment uses the most recent month of daily biases. Recause these adjustments are made using only information available at the time of the VIX obser\ation, the hias-adjusted VIX levels represent out-of-sample forecasts of stock market volatility.'''

Table X summarizes the forecast performance of the bias-adjusted VIX, as well as the unadjusted VIX and historical volatility over the same sample period. The unadjusted VIX and historical volatility results provide a control and are not quahtatively different from those reported in Table IX. Ihe bias-adjusted VIX results, on the other hand, reflect evidence regarding the constancy of the VIX forecast bias. For each adjustment, the unconditional forecast bias (a) is just 4 basis points. This measure of hias is tautological, however, since the mean error of a rolling average adjusted VIX goes to zero, by construction, as the sample size increases.

The more mearnngful assessment of the bias adjustments concerns the reaction of the R and the conditional unbiasedness statistic, CS^. For the 20-day adjustment, for example, tbe GS^-statistic falls from 252.25 to 8.97, but the explanatory power of VIX Is reduced from 0.0356 to 0.0250. Although the 20-day adjustment removes bias, it entails a dramatic loss of information regarding future volatility. The lower (but, still significant) GS^-statistic of 15.56 indicates that the 253-day adjustment also removes a large degree of the bias. Unlike the 20-day adjustment, however, the 253-day adjustment retains the explanatory power of VIX (R^ = 0.0341). The performance of the 253- day hias adjustment, which is a fairly constant adjustment since it is based on a long history of past biases, suggests the forecast bias of VIX is also fairly constant. The 253-day-bias-adjusted volatility index behaves similar to the unadjusted VIX, but with a lower magnitude of forecast bias.

VIX forecast at time ( is of the volatility rcali/[(?cl over the next 20 trading days (i.e., ' + ' ' + 20). Thf mast recent past forecast bias available at time 1, then, is from the VIX forecast at ( - 20. Ihe 2T^-day bias adjustment, for example, averages the biases for the 2^.^ trading days ending at I - 20.

Market Volatility Index 299

TABLE X Relationship between the Rias-Adjusted CBOE Market Volatility Index and Future S&P 100 Index Volatility^

Unbiasedness Tests'̂ Unadjusted volatility index Historical volatility 253-day-bias-adjusted volatility index 20-day-bias-adiusted volatility index

Unbiasedness Tests Unadjusted volatility index Historical volatility 253-day-bias-adjusted volatility index 20-day-bias-adjusted volatility index

(I'

-0.0601 -0.0032

0.0004 0.0004

-0.0515 -0.0009 -0.0002

0.0001

Noncrash

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-14.26 0.65 0.09 0.08

Entire

t.,

-4.80 -0.07

0.02 0.00

Sample

0.6327 0.6435 0.6389 0.6184

Sample (

0.6856 0.9068 0.6887 0.6919

(1407

ii-li

4.12 1.90 3.89 3.00

149H I

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5.37 0.41 5.27 1.76

Days)^

R-

0.0356 0.0187 0.0341 0.0250

0.0456 0.0216 0.0431 0.0282

cs-

252.25 4.73

15.56 8.97

CS-

157.01 0.19

51.47 3.32

^The table considers the out-of-sample performance of two bias-adjusted volatility indexes as a (orecast of the realized 28-day S&P 100 volatility. The bias-adjusted index is constructed by subtracting the average of the most recently realized 253 (or 20) daily forecast biases from the current VIX level. See notes to Table IX regarding estimation of standard errors. "•The noncrash sample (and the bias adjustment during this sample) excludes index observations during ttie periods 9/21/87-11/30/87 and 9/15/89-11/10/89. î See Table IX notes for an explanation of the unbiasedness tests and the definition of CS^.

SUMMARY

The CBOE Market Voiatiiity Inde.x (VIX) offers the market place and academic researchers a new measure of stock market volatility. This study describes how the index is constructed from the implied volatilities of eight S&P 100 index options, and then assesses its behavior over a seven-year period. Univariate time-series properties and seasonalities of the index are documented as well as the temporal relationship hetween the volatility index and stock market returns. Finally, the performance of VIX as a forecast of future slock market volatility is evaluated.

The time-series description shows that daily and weekly VIX changes are reasonably well behaved. The first-order autocorrelation of daily changes is significantly negative, although its magnitude is small (—0.07) and much lower than is observed for individual OEX implied volatilities. For weekly changes, the negative first-order autocorrelation is more pronounced (-0.26), perhaps reflecting mean reversion in the index over this longer interval of time.

300 Fleming, Ostdiek, and Whaley

The investigation of seasonalities reveals a slight intraday decline in the volatility index, with the index then falling sharply at the end of the day. This downward trend appears to be induced in the volatility index by assuming a constant time to expiration throughout the trading day while OEX option traders appear to price intraday time decay. The magnitude of the trend is about J55 basis points. However, it does not occur at a constant rate throughout the day. The analysis of intraweek patterns reveals no seasonalities in VIX, with the index trading within a range of about 20 basis points throughout the week. These descriptions of the time-series properties of the volatility index should be useful to researchers and practitioners in developing methods for valuing derivatives on the volatility index.

The volatility index also exhibits a strong temporal relationship with stock market returns. A large negative contemporaneous correlation exists between VIX changes and S&P 100 index returns, suggesting an inverse relationship between expected volatility and stock market prices. Furthermore, the relationship shows significant asymmetry. Negative stock market moves are accompanied by larger absolute changes in expected volatility than are positive market moves. VIX, to a much lesser extent, also exhibits a positive association with noncontemporaneous (both lag and lead) stock market returns.

Finally, VIX performs well as a volatility forecast, demonstrating a strong relationship with future realized stock market volatility. While the index is not unbiased, it is a significantly more accurate proxy for expected stock market volatility than is the commonly used first-order autoregressive volatility model. Because it imbeds market expectations, VIX appears to be a useful instrument for forecasting volatility.

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