The information content of implied volatility in the
crude oil market
Asyl Bakanova
University of Lugano and Swiss Finance Institute
April 5, 2010
Abstract
In this paper, we evaluate the information content of an option-implied volatility of the light, sweet crude oil futures traded at New York Mercantile Exchange (NYMEX). This measure of volatility is calculated using model-free methodology that is independent from any option pricing model. We do find that the option prices contain important information for predicting future re-alized volatility. We also find that implied volatility outperforms historical volatility as a predictor of future realized volatility and subsumes all informa-tion contained in historical data.
Keywords: Volatility forecasts, implied volatility, extreme value, crude oil.
1
Introduction
Financial market volatility plays a very important role in asset pricing, risk man-agement, portfolio selection and hedging. Because of its importance, both market participants and financial academics have long been interested in estimating and pre-dicting future volatility. Volatility models that fall into one of two categories, the ARCH family and the stochastic volatility family, have been commonly used in mod-eling volatility for estimation and forecasting. These models are based on historical data.
Recently, however, there has been a growing interest in extracting volatility from prices of options. This is because option prices are highly related to market expec-tations about the future volatility of the underlying asset over the remaining life of the option. Therefore, if markets are efficient and the option pricing model is correct, then the implied volatility calculated from option prices should be an unbiased and
efficient estimator of future realized volatility, that is, it should correctly subsume information contained in all other variables including the asset’s price history.
The hypothesis that implied volatility (IV) is a rational forecast of subsequently re-alized volatility (RV) has been frequently tested in the literature.1 Empirical research across countries and markets so far has failed to provide a definitive answer as the prior studies provide mixed evidence. Early research on the predictive content of IV found that IV explains variation in future volatilities better than historical volatility (HV). Latane and Rendleman (1976), Chiras and Manaster (1978), Schmalensee and Trippi (1978) and Beckers (1981) using stock options and the basic Black and Scholes (1973) option pricing model come to this conclusion. In subsequent research, Ku-mar and Shastri (1990), Randolph et al. (1991), Day and Lewis (1993), Lamoureuax and Lastrapes (1993), and Canina and Figlewski (1993), all of which examine either options on individual stocks or options on the cash S&P100 index, using more sophis-ticated methodologies found that IV is a poor forecast of the subsequently RV over the remaining life of the option. Some of these studies also find that IV has little power to predict short-run changes in the volatility of the underlying asset, e.g. over a one-week horizon, compared to predictions that could be derived from time-series models. Specifically, Day and Lewis (1993) and Lamoureux and Lastrapes (1993) find that IV has some predictive power, but that GARCH and/or HV improve this predictive power and Canina and Figlewski (1993) show the absence of correlation between IV and future RV over the remaining life of the option.
But the findings in the papers above are subject to a few problems in their re-search designs, such as maturity mismatch and overlapping samples, among others. Overcoming these problems, more recent papers (e.g., Jorion (1995), Fleming (1998), Moraux et al. (1999), Bates (2000), Blair et al. (2001), Simon (2003), Corrado and Miller (2005)) confirm that IV still outperforms other volatility measures in fore-casting future volatility, although there is some evidence that it is a biased forecast. Christensen and Prabhala (1998), using monthly non-overlapping data, find that IV in at-the-money one-month OEX call options is an unbiased and efficient forecast of ex-post RV after the 1987 stock market crash. Szakmary et al. (2003) find that for a large majority of the 35 futures options markets IV, though not a completely unbiased predictor of future volatility, outperforms the HV as a predictor of future volatility, and that HV is subsumed by IV for most of the 35 markets examined. Al-though the results are somewhat mixed, the overall opinion seems to be that IV has predictive power over future volatility and therefore is a useful measure of expected future volatility. Given the equivocal results and conclusions across different options markets, it is clear that further research on the predictive power of IV is needed.
In this paper, we test whether the implied volatility is a better predictor of future realized volatility and whether it reveals incremental information beyond that con-1Poon and Granger (2003) provide an extensive review of the literature on volatility forecasting.
tained in historical returns in the crude oil market. To do this, we first construct an implied volatility index of light, sweet crude oil futures traded at New York Mercan-tile Exchange (NYMEX) from options prices based on the concept of the fair value of future variance that appeared first in Dupire (1994) and Neuberger (1994) and was improved further by Carr and Madan (1998), Demeterfi et al. (1999), Carr and Wu (2006) and Jiang and Tian (2005), among others. This measure is calculated directly from market observables, such as the market prices of options and interest rates, independent of any pricing model. Then we estimate realized volatility using the range-based – or extreme value – estimators proposed separately by Garman and Klass (1980), Parkinson (1980), Rogers and Satchell (1991), and Yang and Zhang (2000). According to Alizadeh, Brandt and Diebold (2002) they are more robust to microstructure noise and much less noisy than alternative volatility measures such as log absolute or squared returns.
Our findings can be summarized as follows. We find strong indications that the implied volatility obtained from option prices, though slightly biased, indeed contains important information for predicting realized volatility at a monthly frequency. It is also significant in the multiple regression where historical volatility is included, which means that implied volatility subsumes the information content of historical volatility. The performance of option price based predictions of future volatility is substantially improved by applying the instrumental variable approach to correct for error in the predicted volatility variable suggested by Christensen and Prabhala (1998). Consistent with Christensen and Prabhala (1998) that document a regime shift after October 1987 crash, we also provide evidence that there was a similar shift after terrorist attacks of September 11, 2001, with implied volatility being less biased during more volatile subperiod.
Our study is motivated by three major shortcomings in the growing literature that deals with the forecast quality of implied volatility. Most of the earlier studies have focused on individual stocks and stock indices, bonds, and currencies. In this paper we analyze the implied volatility in the light, sweet crude oil market, which deserves an attention for a number of reasons. According to NYMEX, the light, sweet crude oil futures contract has become the world’s largest-volume futures contract trading on a physical commodity and it is used as a principal international pricing benchmark. Since one of the characteristics of prices in the oil markets is volatility, which is both relatively high and variable over time, this market is a very promising area for testing volatility models.
Second, studies based on stock options may not obtain consistent estimates of implied volatility because the option prices, underlying security price, and information about the underlying asset are not observed simultaneously. This is because options on individual stocks and cash market stock indices do not trade on the same exchange as their underlying asset and the closing prices are not synchronous. Thus, one major contribution of our study is that we use data from NYMEX, where the options on
futures and underlying futures contracts trade on the same floor and their prices are observed simultaneously which reduces the measurement errors.
Lastly, most of the studies used implied volatility obtained from Black-Scholes option pricing model, which is based on the assumptions that are in contrast with actual market observations. In our paper we construct an implied volatility index using the model-free methodology that does not depend upon any particular para-metric assumptions and thus helps to avoid measurement errors resulting from model misspecification. To our knowledge, this is the first study to thoroughly analyze the model-free implied volatility in the crude oil market.
The paper proceeds as follows. Section 2 describes the data. Section 3 presents the methodology used for construction of implied volatility, extreme value estimators and analysis of the information content of implied volatility. Section 4 describes statistical properties of implied and realized volatilities the results for the forecasting performance of the volatility index in terms of future realized volatility. Section 5 presents the results of an alternative specification. The conclusions drawn from this study are presented in Section 6.
2
Data and sampling procedure
The database contains daily time series of light, sweet crude oil futures and options written on these futures contracts traded on the NYMEX for the period from Novem-ber 1986 through DecemNovem-ber 2006.2 The light, sweet crude oil is the world’s most
actively traded physical commodity and, according to NYMEX, the crude oil future contract is the world’s most liquid and largest volume contract trading on a physical commodity. NYMEX lists futures contracts with monthly expirations several years into the future and American-style options on these futures.3 There are a number
of practical advantages to using the NYMEX options. First, the NYMEX options and futures close at the same time, therefore, there is no non-synchronicity biases. Second, it is easier to hedge options using highly liquid futures. On the NYMEX, futures and futures options are traded on the same floor, which facilitates hedging, arbitrage, and speculation. It also makes the market more efficient.
We consider only options at the two nearest maturities. When the time to the nearest maturity is less than seven calendar days, the next two nearest maturities are used instead. The reason for choosing options that expire the next calendar months 2The options expire three business days prior to the expiration of the underlying futures contract,
which in turn expires on the third business day prior to the 25th calendar day of the month preceding the delivery month. If the 25th calendar day is a non-business day, expiration is on the third business day prior to the business day preceding the 25th calendar day.
3NYMEX does list European-style options on both commodities. However, the trading history
is that they are the most liquid ones. We match all puts and calls by trading date, maturity, and strike. For each pair, we drop strikes for which put/call price is less than $0.01.4 Generally, a large number of options meet these selection criteria. Since the
options we use are American type, their prices could be slightly higher than prices of the corresponding European options. The difference, however, is very small for short maturities that we focus on (Whaley (1986)).
The futures high, low, open and closing prices are taken from the corresponding nearest futures contracts. We use only the futures contracts with the same contract months as options to ensure the best match between the implied volatility and the realized volatility calculated from subsequent futures prices. To eliminate any effect at the time of rollover, we compute daily futures returns using only price data from the identical contract. Therefore, on the day of rollover, we gather futures prices for both the nearby and first-deferred contracts, so that the daily return on the day after rollover is measured with the same contract month. For the proxy of the risk-free interest rate, we use the rates of the Treasury bill that expires closest to the option expiration date.
Figure 1 plots the closing prices of the futures data. Figure 2 plots the returns series actually used in this study, which is 100×ln(Pt/Pt−1) of the futures data. Table
1 reports summary statistics for daily returns. Crude oil futures returns conform to several stylized facts which have been extensively documented for financial variables. The distribution of the returns has fat tails and excess kurtosis, indicating that the distribution of daily returns is non-Gaussian. Figure 2 reflects another stylized fact, the clustering effect. Variances of returns change over time and large (small) changes tend to be followed by large (small) changes.
Table 1: Descriptive statistics Statistic Returns Mean 0.043 Std. Dev. 2.216 Skewness -0.296 Kurtosis 5.250 Jarque-Bera 612.04 P-value 0.000
4The reason for requiring option prices to exceed the given thresholds is that crude oil options
Figure 1: Closing prices of light, sweet crude oil futures
3
Methodology
3.1
Implied volatility
The method we follow to construct the implied volatility index is very similar to the construction of the new VIX.5 It is based on the concept of fair value of future
variance developed by Demeterfi et al. (1999), but, as Jiang and Tian (2005) show, it is theoretically equivalent to the model-free implied variance formulated in Britten-Jones and Neuberger (2000). This implied volatility measure is derived entirely from no-arbitrage conditions rather than from a specific model. Since it does not impose strong distributional assumptions, the forecast is common to all consistent processes; hence, this model is viewed as model-free implied volatility.
Following the model-free methodology with some modifications, we calculate the model-free implied variance using the following equation:
σM F2 = 2 T X i ∆Ki K2 i erTQ(Ki, T)− 1 T( F K0 −1)2
where: ∆Ki is the difference between strike prices defined as ∆Ki = Ki+1−2Ki−1; Ki
is the strike price of the i-th out-of-the-money option (a call if Ki > F and a put
otherwise); Q(Ki, T) is the settlement price of the option with strike price Ki; F is
5A detailed description of the new methodology of the implied volatility index can be found in
Figure 2: Returns of light, sweet crude oil futures
a forward index level derived from the nearest to the money option prices by using put-call parity; K0 is the first strike below the forward index level F; T is expiration
date for all the options involved in this calculation;ris the risk free rate to expiration. Then the implied volatility index is estimated on a particular trading day t by interpolating (and in a few instances extrapolating) the implied volatilities calculated at two nearest maturities of the available options, T1 and T2, to obtain an estimate
at 30-day maturity as follows:
σIV,t = 100× v u u t 365 30 × ( T1σT21 " NT2 −30 NT2 −NT1 # +T2σ2T2 " 30−NT1 NT2 −NT1 #)
where NT1 and NT2 denote the number of actual days to expiration for the two
ma-turities, and σ2
T1 and σ
2
T2 are implied volatilities for near- and next-term options,
respectively. The result forms a composite hypothetical option that is at-the-money and has 30 calendar (or about 22 trading) days to expiration.
3.2
Realized Volatility
In order to test the information content of the implied volatility index regarding realized volatility, we first need to estimate latter. Traditionally, the unconditional realized volatility of asset returns has been estimated using the series of closing prices as the daily squared return. However, this estimator is greatly noisy and does not
take into account the information given by the path of the price inside the period of reference.
When high-frequency data is unavailable, extreme values - open, high, low, and close prices - allow us to get close to the real underlying process, even if we do not know the whole path of asset prices. Indeed, Alizadeh, Brandt and Diebold (1999) demonstrate that the range, defined as the difference between the high and low prices (in logarithms), is a reasonable proxy for the true volatility. It is nearly Gaussian, robust to microstructure noise as a result of nonsynchronous trading, discrete price observations, intraday periodic volatility patterns and bid-ask bounce and much less noisy than alternative volatility measures such as log absolute or squared returns.
Parkinson (1980), was the first to propose an extreme value volatility estimator for a security following driftless geometric Brownian motion. His estimator is given by σP,t= s 1 4ln2[ln(Ht)−ln(Lt)] 2
where Ht is the highest and Lt is the lowest prices on a trading day t.
Garman and Klass (1980) improve on this procedure by constructing a volatility estimator measure that incorporates the opening and closing prices in addition to the high/low records. Their estimator assumes Brownian motion with zero drift and no opening jumps and is defined by:
σGK,t = q
0.5[ln(Ht)−ln(Lt)]2 −[2ln2−1][ln(Ct)−ln(Ot)]2
where Ct is the closing and Ot is the opening prices on a trading day t.
Yang and Zhang (2002) extend the Garman-Klass estimator by allowing for open-ing jump and assumopen-ing Brownian motion with zero drift. Thus, when the drift is nonzero, this estimator will tend to overestimate the volatility.
σGKY Z,t= s [ln(Ot)−ln(Ct−1)]2+ 1 2[ln(Ht)−ln(Lt)] 2−[2ln2−1][ln(C t)−ln(Ot)]2
Both the Parkinson and Garman-Klass estimators, despite being theoretically more efficient, are based on assumptions of driftless geometric Brownian motion pro-cess. Rogers and Satchell (1991) relaxed this assumption and proposed an estimator which is given by:
σRS,t = q
[ln(Ht)−ln(Ot)[ln(Ht)−ln(Ct)] + [ln(Lt)−ln(Ot)][ln(Lt)−ln(Ct)]
All of the above measures are one-period measures. Suppose we have data from
N trading days (t= 1, ..., N), then multiperiod measure would be:
σi = 1 N N X t=1
Finally, Yang-Zhang (2000) proposed new improvements by presenting an estima-tor that is independent of any drift and consistent in the presence of opening price jumps and calculated as a weighted average of the Rogers and Satchell estimator, the close-open volatility and the open-close volatility with the weights chosen to minimize the variance of estimator:
σY Z,t= v u u t 1 N N X t=1 " ln Ot Ct−1 −ln Ot Ct−1 #2 + κ N N X t=1 " lnCt Ot −lnCt Ot #2 + (1−κ)σ2 RS,t with: κ= 0.34 1.34 +NN+1−1 where ln Ot Ct−1 = 1/N PN t=1lnCOt−t1, ln Ct Ot = 1/N PN
t=1lnCOtt, and σRS,t is the Rogers-Satchell (1991) estimator.
3.3
The information content of implied volatility
IV has been regarded as an unbiased expectation of the RV under the assumption that the market is informationally efficient and the option pricing model is specified correctly. Consistent with the previous literature, to test whether the implied volatil-ity index has a significant amount of information over the historical volatilvolatil-ity, we examine the following three hypotheses:
H1. Implied volatility is an unbiased estimator of the future realized volatility. H2. Implied volatility has more explanatory power than the historical volatility in forecasting realized volatility.
H3. Implied volatility efficiently incorporates all information regarding future volatility; historical volatility contains no information beyond what is already included in implied volatility.
To test the above hypotheses, we use the following regression models commonly used in the literature:
σtRV =α1+β1σtIV +ε1t (1)
σtRV =α2+β2σHVt +ε2t (2)
σRVt =α3+β1σtIV +β2σtHV +ε3t (3)
If, as our hypothesis H1 earlier stated, IV is an unbiased predictor of the RV, we should expect α1 = 0 and β1 = 1 in regression (1). Moreover, if implied volatility is
efficient, the residuals ε2t from regression (1) should be white noise and uncorrelated
with any variable in the market’s information set. If, in accordance with hypothesis H2, IV includes more information (i.e., current market information) than HV, then IV should have greater explanatory power than HV, and we would expect a higher
R2 from regression (1) than regression (2). Finally, if hypothesis H3 is correct, then when IV and HV appear in the same regression, as in (3), we would expect β2 = 0
since HV should have no explanatory power beyond that already contained in IV. For the analysis on the information content of implied volatility, we use non-overlapping observations by computing realized volatility separately for each calen-dar month, following the example of Christensen and Prabhala (1998), since non-overlapping data results in more robust econometric findings. Both volatility mea-sures are expressed in annual terms to facilitate interpretation. We also run the regressions for log-volatilities since the logarithms of volatilities conform best to nor-mality, but also because it enables us to compare our results to those obtained in previous literature.
4
Results
Figures 3 and 4 shows the daily level and first differences of the implied volatility index for the entire sample. There is an evidence of heteroskedasticity in the implied volatility, as can be seen in Figure 4. In Figure 5 we present the various estimates of daily realized volatility. The peaks of these estimates are approximately synchronous, but the general behavior of the series differs, both in the range of variances and per-sistence phenomenon. Estimators using range data are less volatile than the classical estimator. The Augmented Dickey-Fuller test strongly rejects the presence of a unit root in all the series.
Descriptive statistics for the levels and logarithms of both realized and implied volatilities are provided in Table 2 for the entire sample in Panel A, for the first subperiod January, 1996, through September, 2001, in Panel B, and for the second subperiod October, 2001, through December, 2006, - in Panel C. Both average implied volatility and average log implied volatility exceed the means of the corresponding realized volatility. The implied volatility index is skewed to the right and displays excess kurtosis. The coefficients of skewness of all measures of realized volatilities are positive; the mass of probability in the right side of the distribution appears slightly larger than on the left side. All the realized volatility measures appear leptokurtic, while the distributions of the log-volatility series are less so. Skewness and kurtosis are much higher for realized volatilities in the second subperiod.
Table 3 reports the ordinary least-square estimates for regressions (1)-(3). From the first regression it is seen that implied volatility does contain information about re-alized volatility. However, we cannot conclude that the logarithm of implied volatility is an unbiased estimator of realized volatility. The coefficient is 0.6722 and is signifi-cantly different from zero, but also signifisignifi-cantly less than unity although the intercept is statistically not different from zero at 5% significance level. An F-test rejects the joint hypothesisα1 = 0 andβ1 = 1 at 1% significance level. This conclusion is found to
Figure 3: Daily level of implied volatility index
be robust across a variety of asset markets (see Neeley (2004)) and has thus provided the motivation for several attempted explanations of this common finding. Popular suggestions include computing RV with low-frequency data (Poteshman (2000)); that the standard estimation with overlapping observations produces inconsistent parame-ter estimates (Dunis and Keller (1995), Christensen et al. (2001)); and that volatility risk is not priced (Poteshman (2000) and Chernov (2002)). However, Neeley (2004) evaluates these possible solutions and finds that the bias in implied volatility is not removed. As Christensen and Prabhala (1998) suggests, the results may be affected by errors in variables (EIV) which induces a bias in both slope coefficients. Consistent estimation in presence of the possible errors in variables problem may be achieved using an instrumental variable method that we present in the next section.
Despite its biasedness, implied volatility remains a better predictor than past realized volatility. Indeed, taken alone, historical volatility is statistically significant, 0.5001, but its predictive power is quiet inferior to the implied volatility predictive power. If we put in the same regression implied volatility and past volatility, we obtain interesting results. We see that the slope coefficient for implied volatility remains statistically significant in the multiple regression. The coefficient on historical volatility decreases strongly and is insignificant, which indicates that it does not contain information beyond that in implied volatility. It is sufficiently precise that it subsumes the information content of historical volatility. In all specifications, the Durbin-Watson (DW) statistics are not significantly different from two, indicating
Figure 4: First differences of implied volatility index
that the regression residuals are not auto-correlated.
To control for the time period effect, we reestimate specifications (1)-(3) separately for two separate subperiods: pre-September 11 subperiod (January 1996 to August 2001) and a post-September 11 subperiod (October 2001 to December 2006). Panel A of Table 4 reports estimates for the first subperiod and Panel B for the second subperiod.
The estimate of the slope coefficient for implied volatility in the first subperiod are smaller than corresponding estimates in the second subperiod. The Chow (1960) test statistics for a structural change around the September 11, 2001, are significant at 10% significance level. Also the likelihood ratio statistics are both significant at 5%. These results suggest that there was a regime shift with implied volatility becoming less biased after the attacks.
5
The Instrumental Variable Approach
Following Christensen and Prabhala (1998) methodology, in this section we consider an alternative specification such as:
σtIV =α+βσtIV−1+εt (4)
Figure 5: Daily level of realized volatilities
The main goal is to use these equations in an instrumental variable framework in order to correct for error-in-variable problems that might occur in implied volatility. The EIV problem has a double effect. The first one is to generate a downward bias for the slope coefficient of implied volatility and the second one is an upward bias for the slope coefficient of past volatility. The usual OLS will lead to false conclusion concerning implied volatility predictive power. Efficient estimation is possible under an instrumental variable (IV) procedure. The implied volatility is first regressed on an instrument and fitted values from this regression replace implied volatility in equations (1) and (3). These specification are then estimated by OLS. We also want to check if past volatility is a good predictor of implied volatility. Since past volatility is related to future volatility and since implied volatility reflects future volatility information, implied volatility should endogenously depend on past volatility.
Under this procedure, implied volatility σtIV is first regressed on an instrument andσIV
t−1 seems to be a natural candidate for it since it is correlated with true implied
volatility at time t but is not correlated with the measurement error associated with implied volatility sampled one month later. WithσtIV−1as the instrument, we estimate regression (4 using OLS. Then we reestimate specifications (1-3) by replacing implied volatility,σIV
t , with fitted values from the regression (4). We use the same procedure
for specification (3). First, we regress σIVt on both σIVt−1 and σtHV and use the fitted values of σIV
t from this regression for specification (3).
of the first-step regressions (4) and (5), while Panel B reports the estimates of (1) and (3). The estimates in Panel B provide evidence that implied volatility is much less biased and efficient. The point estimates of β1 in both specifications (1) and (3) are
0.892 and 0.824, respectively. Also the IV estimate of β2 is not significantly different
from zero, indicating that implied volatility is efficient.
6
Conclusion
In this paper, we construct a model-free implied volatility from the options on light, sweet crude oil futures and different measures for realized volatility for these futures. The main question we address is whether volatility implied by the option prices predict future realized volatility. We find that implied volatility does predict future realized volatility alone as well as with the past volatility. We also find that historical volatility does not add any information beyond that in implied volatility. Hence, we cannot reject the hypothesis that the volatility implied by option prices is an efficient but slightly biased estimator of realized volatility. The implied volatility appears to be less biased and more efficient once we account for error-in-variables and apply instrumental variable approach. This result provides support for the use of option pricing theory even for light, sweet crude oil options. We also find that in the light, sweet crude oil market, the implied volatility performs better during the more volatile period.
The main issue that deserves attention in the future research is to understand whether in the crude oil market the bias is caused by OLS estimation method giv-ing biased parameter estimates or by inefficiency of the options market, i.e. implied volatility being an inefficient forecast of future volatility.
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Table 2: Descriptive statistics
RVC RVP RVGK RVGKY Z RVRS RVY Z IV Panel A: Full period - 01/1996 to 12/1996
Mean 0.335 0.315 0.328 0.365 0.330 0.307 0.372 St. Dev. 0.090 0.068 0.074 0.095 0.078 0.077 0.072 Kurtosis 6.040 5.988 7.066 8.748 8.149 5.574 4.367 Skewness 1.489 1.335 1.575 1.990 1.766 1.898 0.983 Panel B: Subperiod 01/1996 to 09/2001 Mean 0.344 0.318 0.328 0.370 0.331 0.309 0.354 St. Dev. 0.092 0.066 0.071 0.096 0.076 0.075 0.059 Kurtosis 5.509 4.165 4.385 7.180 5.065 5.374 3.599 Skewness 1.252 0.852 0.954 1.421 1.071 1.221 0.589 Panel C: Subperiod 10/2001 to 12/2006 Mean 0.326 0.312 0.326 0.360 0.329 0.306 0.392 St. Dev. 0.088 0.071 0.077 0.094 0.081 0.079 0.0794 Kurtosis 7.463 7.906 9.739 12.873 11.306 9.207 3.807 Skewness 1.849 1.802 2.168 2.726 2.440 2.604 0.951
logRVC logRVP logRVGK logRVGKY Z logRVRS logRVY Z logIV Panel A: Full period - 01/1996 to 12/1996
Mean -0.488 -0.511 -0.494 -0.449 -0.492 -0.524 -0.437 St. Dev. 0.106 0.088 0.089 0.099 0.093 0.097 0.080 Kurtosis 3.673 3.779 4.170 5.028 4.492 4.666 3.126 Skewness 0.627 0.536 0.683 0.867 0.746 0.838 0.412 Panel B: Subperiod 01/1996 to 09/2001 Mean -0.477 -0.507 -0.493 -0.444 -0.491 -0.522 -0.457 St. Dev. 0.109 0.087 0.091 0.104 0.095 0.100 0.072 Kurtosis 3.466 3.198 3.240 4.132 3.383 3.513 2.851 Skewness 0.368 0.218 0.291 0.410 0.305 0.408 0.115 Panel C: Subperiod 10/2001 to 12/2006 Mean 0.500 -0.515 -0.496 -0.455 -0.493 -0.526 -0.415 St. Dev. 0.103 0.089 0.089 0.093 0.091 0.094 0.084 Kurtosis 4.414 4.682 5.568 7.049 6.259 6.645 2.819 Skewness 0.963 0.888 1.162 1.556 1.328 1.424 0.477
Table 3: Information content of implied volatility: OLS estimates
Intercept IV HV adj.R2 Wald test Durbin-Watson
0.1151 0.6722 0.2561 0.005 2.0385 (0.044) (0.1145) 0.2076 0.5001 0.1246 0.000 2.1747 (0.0352) (0.0945) 0.1132 0.6547 0.0266 0.2504 0.004 2.068 (0.0397) (0.2167) (0.1981)
Table 4: Information content of implied volatility: subperiod analysis Dependent variable: σRVt
Panel A: 01/1996 - 08/2001
Intercept IV HV adj.R2 Durbin-Watson
0.117 0.701 0.22 2.04 (0.046) (0.000) 0.244 0.333 0.09 2.15 (0.000) (0.008) 0.108 0.631 0.092 0.21 2.17 (0.072) (0.002) (0.500) Panel B: 10/2001-12/2006
Intercept IV HV adj.R2 Durbin-Watson
0.037 0.822 0.48 2.04 (0.395) (0.000) 0.217 0.390 0.18 2.15 (0.000) (0.000) 0.038 0.907 -0.092 0.48 2.17 (0.389) (0.000) (0.429)
Table 5: Information content of implied volatility: Instrumental variables estimates Panel A: first stage regressions estimates
Dependent variable: σIV t
Intercept IV HV adj.R2 Durbin-Watson
0.128 0.656 0.428 2.04
(0.000) (0.000)
0.133 0.747 -0.127 0.434 2.1747
(0.000) (0.000) (0.123)
Panel B: second stage IV estimates Dependent variable: σRV
t
Intercept IV HV adj.R2 Durbin-Watson
-0.02 0.88 0.29 2.00
(0.666) (0.000)
-0.02 0.83 0.06 0.29 2.14
