The information content of implied volatility in the crude oil futures market

Transcription

1 The information content of implied volatility in the crude oil futures market Asyl Bakanova University of Lugano and Swiss Finance Institute February 28, 2011 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 realized volatility. We also find that implied volatility outperforms historical volatility as a predictor of future realized volatility and subsumes all information contained in historical data. Keywords: Volatility forecasts, implied volatility, extreme value, crude oil. asyl.bakanova@usi.ch 1 Introduction Financial market volatility plays a very important role in the theory and practice of asset pricing, risk management, portfolio selection and hedging. Because of its importance, both market participants and financial academics have long been interested in estimating and predicting future volatility. Volatility models that fall into one of two categories, the ARCH family and the stochastic volatility family, have been commonly used in modeling volatility for estimation and forecasting. These models are based on historical data. Recently there has been a growing interest in extracting volatility from prices of options. This is because 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. 1

2 The hypothesis that implied volatility (IV) is a rational forecast of subsequently realized 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. Early research on the predictive content of IV found that IV explains variation in future volatilities better than historical volatility (HV) (see, for example, Latane and Rendleman (1976), Chiras and Manaster (1978), Schmalensee and Trippi (1978) and Beckers (1981)). In subsequent research, Kumar and Shastri (1990), Randolph et al. (1991), Day and Lewis (1992), Lamoureuax and Lastrapes (1993), and Canina and Figlewski (1993) found that IV is a poor forecast of the subsequently RV over the remaining life of the option. Specifically, Day and Lewis (1992) 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 research designs, such as maturity mismatch and/or 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 forecasting 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. IV from options written on crude oil futures was examined by Day and Lewis (1993). They compare it to a simple HV and out-ofsample GARCH volatiilty forecasts and find extremely good forecasting perofmance for IV in this market. Although the results are somewhat mixed, the overall opinion seems to be that IV has predictive power and therefore is a useful measure of expected future volatility. However, given the equivocal results and conclusions across different options markets, it is clear that further research on the predictive power of IV is needed. Most of these 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. Crude oil is the biggest and most widely traded commodity market in the world and the light, sweet crude oil futures contract is the world s most liquid and largest-volume futures 1 Poon and Granger (2003) provide an extensive review of the literature on volatility forecasting. 2

3 contract trading on a physical commodity. Since one of the characteristics of prices in the oil markets is volatility, this market is a very promising area for testing volatility models. In particular, we test whether the IV is a better predictor of future RV and whether it reveals incremental information beyond that contained in historical returns. We calculate IV of light, sweet crude oil futures 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 various researchers (e.g., Carr and Madan (1998), Demeterfi et al. (1999), Britten-Jones and Neuberger (2000), Carr and Wu (2006) and Jiang and Tian (2005)). This measure is calculated directly from market observables, such as the market prices of options and interest rates, independent of any pricing model. Thus the measurement error resulting from model misspecification is reduced. As volatility proxy, we use 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), the log range is nearly Gaussian, more robust to microstructure noise and much less noisy than alternative volatility measures such as log absolute or squared returns. In addition, these estimators require the daily open, close, high and low price data that are readily available for most financial markets. To our knowledge, this is the first study to apply model-free methodology for implied volatility and the range-based estimators in the light, sweet crude oil futures market. 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. Finally, we find that implied volatility has better predictive power during more volatile subperiod following September 11 terrorist attacks. 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. In Section 5 we conduct robustness analysis. The conclusions drawn from this study are presented in Section 6. 3

4 2 Data description The dataset for this study contains daily time series of light, sweet crude oil futures and American-style options written on these futures which are both traded on the New York Mercantile Exchange (NYMEX) for the period from January 02, 1996 through December 14, On the NYMEX, futures and futures options are traded on the same floor, which facilitates hedging, arbitrage, and speculation. Moreover, both markets close at the same time and their prices are observed simultaneously which reduces the non-synchronicity biases and other measurement errors. 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. 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 $ 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 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 2 plots the closing prices of the futures data. Figure 2 plots the returns series actually used in this study, which is 100 ln(p t /P t 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, such as fat tails and excess kurtosis. 2 NYMEX does list European-style options. However, the trading history is much shorter and liquidity is much lower than for the American-style options. 3 The reason for requiring option prices to exceed the given thresholds is that crude oil options are quoted with a precision of 0.01 USD. 4

5 Figure 1: Closing prices of light, sweet crude oil futures Table 1: Descriptive statistics Statistic Returns Mean Std. Dev Skewness Kurtosis Jarque-Bera P-value Methodology 3.1 Range-based volatility estimators The idea of using information on daily high and low prices, as well as the opening and closing prices, goes back to Parkinson (1980) and Garman and Klass (1980) with further contributions by Beckers (1983), Ball and Torous (1984), Wiggins (1991), Rogers and Satchell (1991), Kunitomo (1992), Yang and Zhang (2000). Define the following variables: O t = the opening price of the trading day t, C t = the closing price of the trading day t, H t = the highest price of the trading day t, L t = the lowest price of the trading day t. Traditionally, the unconditional realized volatility of asset returns has been estimated using the series of closing prices as the 5

6 Figure 2: Returns of light, sweet crude oil futures daily squared return: ˆσ 2 C,t = (lnc t lnc t 1 ) 2 (1) However, using closing prices alone to estimate volatility ignores information contained in intraday high and low prices. Assuming an underlying geometric Brownian motion with no drift, Parkinson(1980) developed an estimator which uses the daily high and low prices of the asset for estimating its volatility: ˆσ 2 P,t = 1 4ln2 (lnh t lnl t ) 2 (2) Since the price path is not observable when the market is closed, Garman and Klass (1980) suggest a method to mitigate the effect of discontinuous observations by including opening and closing prices along with the highest and lowest prices as follows: ˆσ 2 GK,t = 0.5(lnH t lnl t ) 2 (2ln2 1)(lnC t lno t ) 2 (3) Rogers and Satchell (1991) relaxed the assumption of no drift and proposed an estimator which is given by: ˆσ 2 RS,t = (lnh t lno t )(lnh t lnc t ) + (lnl t lno t )(lnl t lnc t ) (4) Finally, Yang and Zhang (2000) proposed new improvements by presenting an estimator that is independent of any drift and consistent in the presence of opening 6

7 price jumps. This estimator can be interpreted as a weighted average of the Rogers and Satchell (1991) estimator, the close-open volatility and the open-close volatility with the weights chosen to minimize the variance of estimator: with: ˆσ 2 Y Z,t = 1 N [ N t=1 ln O t C t 1 ln O t C t 1 ] 2 + κ N κ = where ln Ot C t 1 = 1/N N t=1 ln Ot C t 1, ln Ct O t Satchell (1991) estimator. [ N t= N+1 N Model-free implied volatility ln C t O t ln C t O t ] 2 + (1 κ)σ 2 RS,t (5) = 1/N N t=1 ln Ct O t, and σ RS,t is the Rogers- Britten-Jones and Neuberger (2000) proposed an implied volatility measure derived entirely from no-arbitrage condition rather than from any specific model and calculated directly from market observables, independent of any pricing model. In addition, it incorporates information from the volatility skew by using a wider range of strike prices rather just in-the-money options. This methodology, based on the theory for variance swap contracts and with pricing obtained from the full cross-section of options prices, has been adopted by the Chicago Board of Exchange (CBOE) in constructing the monthly implied volatility index, known as VIX, from on S&P500 index option prices. Following the VIX methodology with some modifications, we calculate implied volatility using the following equation: σ 2 T = 2 T i K i e rt Q(K Ki 2 i, T ) 1 T ( F 1) 2 K 0 where: K i is the difference between strike prices defined as K i = K i+1 K i 1 ; K 2 i is the strike price of the i-th out-of-the-money option (a call if K i > F and a put otherwise); Q(K i, T ) is the settlement price of the option with strike price K i ; F is a forward index level derived from the nearest to the money option prices by using put-call parity, such that F = K + e rt (C(K, T ) P (K, T )); K 0 is the first strike below the forward index level F ; T is expiration date for all the options involved in this calculation; r is the risk free rate to expiration. σt 2 is calculated at two of the nearest maturities of the available options, T 1 and T 2. Then, we interpolate between σt 2 1 and σt 2 2 ) to obtain an estimate at 30-day maturity: σ MF t = 365 { 30 T 1 σ 2 T 1 [ NT2 30 N T2 N T1 7 ] [ ]} 30 + T 2 σt 2 NT1 2 N T2 N T1

8 where N T1 and N T2 denote the number of actual days to expiration for the two maturities. 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 volatility index has a significant amount of information over the historical volatility, 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: σ RV t σ RV t σ RV t = α 1 + β 1 σ IV t + ε 1t (6) = α 2 + β 2 σ HV t + ε 2t (7) = α 3 + β 1 σ IV t + β 2 σ HV t + ε 3t (8) If, as our hypothesis H1 earlier stated, IV is an unbiased predictor of the RV, we should expect α 1 = 0 and β 1 = 1 in regression (6). Moreover, if implied volatility is efficient, the residuals ε 2t from regression (6) 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 R 2 from regression (6) than regression (7). Finally, if hypothesis H3 is correct, then when IV and HV appear in the same regression, as in (8), we would expect β 2 = 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 nonoverlapping observations by computing realized volatility separately for each calendar month, following the example of Christensen and Prabhala (1998), since nonoverlapping data results in more robust econometric findings. 4 Empirical results Figures 4 shows the daily level of the implied volatilityfor the entire sample. In Figure 4 we present the various estimates of daily realized volatility. The peaks of 8

9 these estimates are approximately synchronous, but the general behavior of the series differs, both in the range of variances and persistence 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. Figure 3: Model-free implied volatility at daily frequency Table 3 reports the ordinary least-square estimates for regressions (6)-(8). From the first regression it is seen that implied volatility does contain information about realized volatility. However, we cannot conclude that the logarithm of implied volatility is an unbiased estimator of realized volatility. The coefficient is and is significantly different from zero, but also significantly less than unity although the intercept is statistically not different from zero at 5% significance level. An F -test rejects the joint hypothesis α 1 = 0 and β 1 = 1 at 1% significance level. This conclusion is found to 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. As Christensen and Prabhala (1998) suggests, the results may be affected by errors in variables (EIV) which induces a bias in both slope coefficients. 9

10 Figure 4: Daily level of realized volatilities 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, , but its predictive power is quite 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. 5 Robustness analysis We perform several exercises to verify the robustness of our results. We evaluate the impacts of error-in-variable problems on our regressions where the implied volatility is used as a regressor and we analyze whether the information efficiency of implied volatility varies significantly over different subsample periods. 10

11 5.1 Instrumental variable The fact that the slope estimate is significantly below the null hypothesis of one could be either due to implied volatility being a biased forecast or due to the bias induced by the error-in-variable problem. Christensen and Prabhala (1998) assume that the error-in-variables (EIV) problem causes implied volatility to appear both biased and inefficient 4 and propose to use instrumental variable framework as a way of correcting EIV problems in OEX implied volatility. Within this framework, to correct for EIV, the following equations is used: σ IV t = α + βσ IV t 1 + ε t (9) σ IV t = α + βσ IV t 1 + γσ HV t + ε t (10) Under this procedure, implied volatility σt IV is first regressed on an instrument σt 1, IV which 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. With σt 1 IV as the instrument, we estimate regression (9 using OLS. Then we reestimate specifications (6-8) by replacing implied volatility, σt IV, with fitted values from the regression (9). We use the same procedure for specification (8). First, we regress σt IV on both σt 1 IV and σt HV and use the fitted values of σt IV from this regression for specification (8). Table 4 reports estimates based on this IV Procedure. Panel A reports estimates of the first-step regressions (9) and (10), while Panel B reports the estimates of (6) and (8). The estimates in Panel B provide evidence that implied volatility is much less biased and efficient. The point estimates of β 1 in both specifications (6) and (8) are and 0.824, respectively. Also the IV estimate of β 2 is not significantly different from zero, indicating that implied volatility is efficient. 5.2 Subperiod analysis To control for the time period effect, we reestimate specifications (6) and (8) 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 5 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. These results suggest 4 The EIV problem has two effects. It generates a downward bias for the slope coefficient of implied volatility and the an upward bias for the slope coefficient of past volatility, which explains the underestimation of implied volatility and the over estimation of past volatility. As a result, the usual OLS will lead to false conclusions concerning implied volatility predictive power. 11

12 that there was a regime shift with implied volatility becoming less biased after the attacks. 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, although it is 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 following September 11 terrorist attacks. 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 giving biased parameter estimates or by inefficiency of the options market, i.e. implied volatility being an inefficient forecast of future volatility. 12

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16 Table 2: Descriptive statistics RV C RV P RV GK RV RS RV Y Z IV Panel A: Full period - 01/1996 to 12/1996 Mean St. Dev Kurtosis Skewness Panel B: Subperiod 01/1996 to 09/2001 Mean St. Dev Kurtosis Skewness Panel C: Subperiod 10/2001 to 12/2006 Mean St. Dev Kurtosis Skewness logrv C logrv P logrv GK logrv RS logrv Y Z logiv Panel A: Full period - 01/1996 to 12/1996 Mean St. Dev Kurtosis Skewness Panel B: Subperiod 01/1996 to 09/2001 Mean St. Dev Kurtosis Skewness Panel C: Subperiod 10/2001 to 12/2006 Mean St. Dev Kurtosis Skewness

17 Table 3: Information content of implied volatility: OLS estimates Intercept IV HV adj.r 2 Wald test Durbin-Watson (0.044) (0.1145) (0.0352) (0.0945) (0.0397) (0.2167) (0.1981) Table 4: Information content of implied volatility: Instrumental variables estimates Panel A: first stage regressions estimates Dependent variable: σt IV Intercept IV HV adj.r 2 Durbin-Watson (0.000) (0.000) (0.000) (0.000) (0.123) Panel B: second stage IV estimates Dependent variable: σt RV Intercept IV HV adj.r 2 Durbin-Watson (0.666) (0.000) (0.683) (0.000) (0.481) 17

18 Table 5: Information content of implied volatility: subperiod analysis Dependent variable: σt RV Panel A: 01/ /2001 Intercept IV HV adj.r 2 Durbin-Watson (0.046) (0.000) (0.000) (0.008) (0.072) (0.002) (0.500) Panel B: 10/ /2006 Intercept IV HV adj.r 2 Durbin-Watson (0.395) (0.000) (0.000) (0.000) (0.389) (0.000) (0.429) 18