Volatility Forecasting in Crude Oil Futures Option Markets
An Analysis and Comparison of various Volatility Forecasting Methods for
WTI and Brent Futures Prices between 2006 and 2011
Thesis
Master of Science in Business Administration
Specialization: Finance
University of Groningen
Faculty of Economics and Business
Author: Torben Seikowsky
Student number: 1938886
Supervisor: Peter Smid
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Abstract
This thesis tests the information content of implied volatilities about future realized volatility in the WTI and Brent futures market using regression analysis. For the period from March 2006 to February 2011 it is found that implied volatility is not an unbiased volatility forecast in the given markets. Further, several forecasting methods based on historical data including a GARCH (1, 1) and EGARCH model hold additional information to implied volatility when included in the regression model. The forecasting error of all methods is measured for the entire sample period and three sub samples including a period before, during and after the recent financial crisis. A simple historical method is superior to GARCH (1, 1) and EGARCH where EGARCH is an improvement to GARCH (1, 1) only in the WTI market in the period after the crisis.
JEL classification
G17Keywords
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Table of Contents
1. Introduction
4
2. Theoretical Foundation
7
2.1 Futures and Options
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2.2 Option Pricing
8
2.3 Behavior of Asset Returns
10
2.4 Measuring Volatility
12
2.5 Standard Historical Approach
14
2.6 GARCH and EGARCH
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2.7 Implied Volatility
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3. Review of Empirical Results
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1. Introduction
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Modeling and forecasting volatility has been under vast investigation by academics and practitioners alike and arguably is one of the most important concepts in the whole world of finance (Brooks 2008). Chu and Freund (1996) indicate, that the forecast of volatility is the most difficult and controversial aspect in the whole process of option pricing. There are two general ways to generate a volatility forecast as an input for the Black-Scholes option valuation model as developed by Black and Scholes (1973) and its modifications. One approach relies on historical data of the underlying asset which can be used in different forecasting models to obtain the future volatility. The second approach is to calculate the implied volatility from current option prices observed in the market by solving the given option valuation model for the volatility that sets the model equal to the market price. As pointed out by Day and Lewis (1992), the ability of the implied volatilities to predict the future volatility of an underlying asset is considered to be a measure of the information content of the option prices. If it holds that the financial markets are efficient and are using all available information under a properly working option valuation model, the implied volatilities should, on average, contain all the information about the future volatilities during the remaining life of the options. With respect to the options on crude oil futures it is very relevant to know whether the implied volatilities do indeed hold all the information about future volatilities, which raises the first research question this study intends to answer: Do implied volatilities
reveal all information about future volatilities in the WTI and Brent crude oil futures option
markets? Even when a definitive answer to this question does not exist, it should reveal how well implied
volatility predicts future volatility in the given markets and whether options are valued under Black-Scholes specifications. It is fair to say and supported by Figlewski (1997) that the implied volatility should impound all relevant available information which can be obtained through historical data as well as any additional information about future expectations available at the time of valuation. In this theory all information that can be gained through forecasts based on historical methods should be incorporated in the implied volatility, which can only be improved through adjustments for relevant additional information. This brings rise to the second research question this study intends to answer: Does historical volatility
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and Brent crude oil futures option markets? Further one seeks to know which method of all the
forecasting techniques actually yields the most accurate estimate of future realized volatility. Therefore this paper intends to answer the more general question regarding forecasting performance: Which
forecasting method is the most accurate in the WTI and Brent crude oil futures option markets?
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2. Theoretical Foundation
2.1 Futures and Options
A forward contract represents the obligation to buy or sell a financial security or commodity at a pre-specified price at some future date. Forward contracts can be found in many business contracts and have been in existence for hundreds of years (Hillier et al. 2008). Futures contracts are similar to forward contracts, only that they trade on organized exchanges known as futures markets. There are several reasons for the popularity of futures contracts: The standardized contracts tend to be very liquid and easy to trade on open exchanges. Many times the futures contract is much simpler to trade than the underlying asset where, especially in the commodity markets, the delivery of the futures contract is often much easier than the delivery of the physical assets themselves. Furthermore as indicated in Hull (2008), the futures price is immediately known from trading while the spot price may not be so readily available. Schwartz (1997) goes so far to say that in many cases the spot price of a commodity is so uncertain that the corresponding futures contract closest to maturity is used as a proxy for the spot price.
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contracts, where one is obligated to buy or sell the underlying asset. The cost one pays for this right is known as the option premium. Since futures are the underlying assets of interest in this thesis, the options contracts that are relevant to this study are known as futures options which give the holder a position in a futures contract upon exercise. A call futures option is the right to enter into a long futures contract whereas a put futures option is the right to enter into a short futures contract. Futures options are generally American allowing them to be exercised any time during the life of the contract (Hull 2008).
2.2 Option Pricing
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Compared to the original Black-Scholes model the spot price is replaced by the futures price at time
zero and for the risk-free rate drops out of parts of the equation, resulting in the model as specified above. The variables and are the European call and put option prices, is the strike price, is the
time to maturity of the option and is the volatility of the futures price over the life of the option. The function ( ) is the cumulative probability distribution function for a standardized normal distribution for
the variables , , and respectively as defined in equation (3) and (4). A nice feature of the valuation of futures options is the fact that the convenience yield and cost of carry are already incorporated in the futures price. Under the assumption that both are functions only of time the volatility of the futures price is the same as the volatility of the underlying asset (Hull 2008). The Black model can be used to value any standard European call or put futures option. A precise valuation formula however does not exist to value an American option. In this case the technique called binomial approximation can be employed to approximate the value of an American option because it is possible to incorporate the value of the possibility of early exercise. This technique makes use of a binomial tree which models the possible future paths of the underlying asset. A binomial tree has an abstract number of time steps which makes it possible to allow for the option of early exercise at any given step. In case of specific choices of the parameters, in the limit, as the number of time steps approaches infinity the technique leads to the same process that is modeled by the Black model (Hull 2008). Binomial approximation relies on the same underlying assumptions as the Black model and for the purpose of this study can be regarded as a modification of the model in order to value the American type futures options.
In order to derive the Black-Scholes model and its modifications, the necessary assumption has to be made that the underlying asset price follows a geometric Brownian motion requiring that all future prices of the underlying asset will be lognormally distributed. This process has constant drift and volatility parameters and is shown in equation (5):
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is the limiting case of the so called random walk of the change of the underlying asset . implies
that has an expected drift rate of per unit of time , being the expected return. adds variability to the path followed by where is the volatility of the futures price and is a Wiener process. Going back to the Black model in equations (1) through (4) above we can see that the expected return does not enter into the model and does not have to be estimated which is another great feature of the model. Although the Black-Scholes model assumes the volatility to be a constant parameter, which can be argued to be an adequate assumption for a very short period of time, for longer periods the volatility can be expected to be time-varying (Figlewski 1997). Since volatility is not constant, it has to be forecasted and is not directly observable. Given the futures price as well as the strike price and time to maturity in
the Black model above, a futures option’s theoretical value today depends on the volatility that will be experienced over the lifetime of the option contract, which has to be forecasted. In summary, the Black-Scholes model that was developed for European options assumes a constant future volatility of the underlying asset prices. Although constant volatility is rarely observed in the real world, the model and its modifications are still theoretically correct if the average annualized future volatility for the lifetime of the option can be estimated at the time of valuation.
2.3 Behavior of Asset Returns
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terms, and that observed price changes are not lognormally distributed. Engle (2004) reports the same feature of financial returns which he names fat tails. A distribution with fat tails encounters more large changes and more small changes as would be expected under a normal distribution. Further he reports that these periods of extreme values tend to cluster over time being followed by periods of lower volatility, which gives rise to another characteristic of financial returns known as volatility clustering. The feature that a period of more extreme values tends to be followed by less pronounced price movements is also known as mean reversion since volatility seems to revert towards a more moderate long term level. The characteristics of the commodity markets increase the likeliness of mean reversion. When prices are relatively high, supply will increase, putting a downward pressure on prices. Relatively low prices will decrease supply and put an upward pressure on prices, which results in an impact of the relative prices on the supply of commodities that induces mean reversion (Schwartz 1997). Considering the characteristics of financial data of regular stocks, the volatility is generally higher when prices are falling where large price declines forecast greater volatility than similar price increases (Engle 2004). Doran and Ronn (2006) note that while in equity markets crises are characterized by lower prices, in the energy markets crises are reflected in higher product prices and higher volatilities. Lastly, although the mean return or expected return of an asset does not have to be estimated for the given option pricing models, the volatility itself is calculated by the deviations from a given mean. Any model that uses an estimate for the mean return of a financial time series can distort the measure of the volatility, if the estimate differs significantly from the true mean.
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2.4 Measuring Volatility
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volatility measure where 365 days would assume that volatility occurs on every day of the year even if markets are closed. As long as all volatility measures are annualized with the same amount of days the choice of days is not a problem for the measurement of forecasting accuracy which is the interest of this study. The following equations are specified with 252 days to annualize the volatility measures. Equation (6) displays the measure of the realized annualized volatility from time to time days as it is used in all other studies in this area (Ederington and Guan 2006):
( ) √
∑
( )
where equals the number of days that are used to calculate ( ) at time and . is measured as the mean of over the entire data period with days and is the continuously compounded
daily return defined as in equation (7) below for the given futures price at day and the previous day :
(
⁄ ) ( )
Dividing by rather than is a statistical procedure to adjust for the degrees of freedom resulting in
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2.5 Standard Historical Approach
The two main sources of volatility forecasts are time-series models and implied volatilities calculated from observed option prices (Ederington and Guan 2004). This section looks at the first time-series model using the historical standard deviation. The simple historical method for forecasting future volatility uses the annualized standard deviation for a certain historical period as the forecast for the standard deviation of the desired future period. Many names are in existence for this procedure and this study follows Ederington and Guan (2006) calling the method ( ) at time as specified in equation (8) below:
( ) √
∑
( )
where , the return deviation, as in equation (6). Equation (6) and (8) are essentially the same showing that the historical realized volatility is used as the forecast for future realized volatility. It is common to set equal to the number of days that are to be forecasted. If one wishes to obtain a volatility forecast for the next 20 trading days one sets obtaining a measure for STD(20) which is calculated from the previous 20 observations ending one day before the desired forecasting period where RLZ(20) begins. Figlewski (1997) finds that forecasting errors are generally lower if STD(n) is measured over a longer period, making it reasonable to use a STD(40), STD(60) and STD(120) or other desired periods even when forecasting the next 20 days. The most common known benefits of the STD(n) method are its simplicity and ease of calculation, often making it the benchmark for all other forecasting methods.
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instance, the trading days from 21 days ago or further back are not considered. When choosing a measure for one encounters two of the known characteristics of financial returns data. As volatility is changing over time, relatively older observations might not be very relevant for the recent future forecast. On the other hand, extreme observations as described by the distribution with fat tails might distort the estimate of a very short period of and shorter periods tend to be very noisy (Engle 2004), motivating the use of a
longer period. Further, the characteristics of mean reversion and volatility clustering are not addressed by the STD(n) method at all and could have a significant effect on forecasting accuracy. Lastly, using the historical standard deviation as a forecast has the additional shortcoming that only the information in past returns is considered, ignoring other possible information sets such as events in the news that are likely to move the markets (Ederington and Guan 2006).
2.6 GARCH and EGARCH
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reveals mean reversion, a forecast using the GARCH model allows the future volatility to revert back to its long term average as the forecast horizon increases.
For the purpose of monitoring daily volatility and the application of the GARCH framework, the formulas are generally simplified by making the following adjustments. The return of the daily futures prices is no longer measured as the logarithm but simply as the percentage change now denoting:
( )
Where is the given futures price on day and the previous day The mean of all the used observations is assumed to be zero which has very little effect on estimates of the standard deviation because the expected change of a variable in one day is very small when compared with its standard deviation (Hull 2008). The square of the standard deviation denoted as the variance rate can now, for any period of days, be expressed as:
∑
( )
Since the mean is zero and , in equation (10) and in the context of the GARCH framework is
equal to as defined in equation (9). Equation (10) is now adjusted for the distinctive features of the GARCH model. The variance rate is calculated from a long run average variance rate as well as from
and . Using the most recent observations of and results in a model that is specified as a
GARCH (1, 1) model as exhibited in equation (11) below:
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The GARCH forecast variance is basically a weighted average of three different variance forecasts, the long-run average, the forecast of the previous period and the new information that was not available when the previous forecast was made (Engle 2004). The more general GARCH (p, q) model calculates from
the most recent p observations on and the most recent q estimates of the variance rate. Many adjustments have been made to the model but by far the most popular of all the GARCH-type models is the GARCH (1, 1) model as specified above (Ederington and Guan 2006). The weights , and must
sum to one and are estimated through the procedure of maximum likelihood on a set of historical data. Once estimated it is common to say that the model has been fit to the data. This in-sample model can now be used to forecast the variance for any given period of days by using equation (12) below:
( ) ( ) ( ) ( )
The GARCH forecast, which is commonly obtained through econometric software, is a one step-ahead forecast where a daily variance is estimated for every individual day in the forecast period of n days. The average of the n variances has thus to be annualized and the square root is taken in order to obtain the forecast of the annualized volatility. It is important to understand that the parameters are estimated on the historical data and give the best statistical fit of the model to the data in-sample. These parameters are now used for the out-of-sample forecast but as volatility is known to be changing over time it is reasonable to expect that the parameters change as well resulting in inaccurate forecasts. This is supported by Figlewski (1997) who states that the behavior of a financial market is much different from in-sample estimation because the underlying structure is changing. Further he argues that in-sample goodness of-fit statistics as the parameters of the GARCH (1, 1) model are not a guarantee of how successful the model will be in forecasting.
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process with long run instability of the system (Hull 2008) and (Figlewski 1997). If statistical procedures do not allow a proper estimation of the parameters the model should not be used for volatility forecasts with the given data or if they are, inaccurate forecasts are likely. Further as mentioned the GARCH models focus on one step-ahead forecasts which makes them less suitable for long term forecasts which would simply converge to the long run variance rate Figlewski (1997). As is the case with the STD(n) method, only the information in past returns is considered, ignoring other possible information sets which is a general problem of all forecasting methods based on historical data.
Nelson (1991) developed an extension of the GARCH model called the EGARCH model, standing for exponential general autoregressive conditional heteroskedastic, which recognizes that volatility could respond asymmetrically to past forecast errors. The EGARCH model is the second to the GARCH (1, 1) model in popularity and allows equal positive and negative shocks to have different impacts on the conditional variance (Ederington and Guan 2004). EGARCH is based on the same intuition of the GARCH model and is solely estimated with the help of econometric software. The EGARCH model as estimated by Eviews 7 is provided in the Appendix B. Once again the forecast of the EGARCH forecast as obtained by Eviews 7 has to be converted into an average annualized volatility for the given forecast period.
2.7 Implied Volatility
A completely different approach to obtain a volatility forecast makes use of exchange traded options to calculate the so-called implied volatility which is called the (IV) method in my study. This approach does not rely on historical data to calculate a volatility forecast but uses option prices available at the time that the forecast is needed. With the assumption that the financial markets are pricing options using a given option pricing model like the Black-Scholes model and its modifications in this paper, it is possible to use any given option price, along with all other input variables, to solve the model backwards in order to determine the volatility input of the market.1 This volatility that sets the price given by the valuation model equal to the market price is the implied volatility (Figlewski 1997). In theory and under the same
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assumption as above, the future actual realized volatility during the lifetime of the option should, on average, equal the implied volatility at the time the option is valued (Szakmary et al. 2003). The complication of incorporating an adequate dividend yield or a convenience yield and cost of carry are not of concern when inverting the Black model for the valuation of futures options to obtain the implied volatility. Unfortunately this procedure only works in this simple way for European options because the possibility of early exercise in American options brings a lot of complexity to the valuation process. Since the options on WTI and Brent futures are all American the method of inverting a model of binomial approximations has to be used accounting for the early-exercise feature of the American options, which is based on the same underlying assumptions as the Black-Scholes model.
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3. Review of Empirical Results
While section 2 discusses the different forecasting methods and the underlying theories which are employed in this thesis, the review of empirical findings in this section reports the results and conclusions of the most relevant existing studies in the related fields of volatility forecasting. Contradicting evidence has been found in previous studies by various researchers regarding the information content of implied volatility for future realized volatility and whether historical volatility can add additional relevant information to implied volatility about future realized volatility.
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the option. Canina and Figlewski (1993) state in their results from an extensive study of the S&P 100 stock index options that implied volatility appears not to contain any information at all about future volatility whereas historical volatility does. The same result is supported by Figlewski (1997), who finds that historical volatility does contain some information about future realized volatility. A similar and very relevant finding for this study is documented by Agnolucci (2009), who reports that historical volatility contains more information about future volatility than implied volatility for WTI crude oil futures up to 2005. He concludes that GARCH-type models perform better than implied volatility in terms of predictive accuracy.
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4. Research Methodology
4.1 Information Content
The review of past empirical results shows that a great contradiction exists in the literature which is related to the research questions of this study. This section formulates a set of hypotheses and methodologies that are used to answer the proposed research questions. The first research question, whether implied volatilities reveal all information about future volatilities in the WTI and Brent crude oil futures option markets, results in the following testable hypothesis:
H1: Implied volatility is an unbiased estimator of future realized volatility of WTI and Brent futures
prices.
The hypothesis H1 is tested by the rationality test regression which is used in Figlewski (1997) measuring
the information content of implied volatilities as shown in equation (13) below:
( ) ( ) ( )
Where ( ) is defined as in equation (6), is the regression constant coefficient, the slope coefficient and an error term. ( ) is the implied volatility at time which is used as a forecast for
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K/F of the options. Implied volatilities for a K/F of 90%, 95%, 100%, 105% and 110% can be obtained and are collected for maturities of 1 month, 2 months, 3 months and 6 months. Only the implied volatilities with a K/F of 100% are used for ( ) because at the money (ATM) contracts tend to be the most liquid and are less impacted by volatility structures such as smiles or skews (Figlewski 1997). Further he states that in the money (ITM) or out of the money (OTM) implied volatilities can be much more sensitive to changes in option prices which further motivates the use of ATM and thus the 100% Bloomberg implied volatilities. Since option prices are not available for this study I do rely on the accuracy of the implied volatility that is provided by Bloomberg. Christensen and Prabhala (1998) show that their non-overlapping sample yields more reliable regression estimates than an overlapping sample does, motivating the use of a non-overlapping sample in my study. Starting on March 1st 2006, RLZ(20), (40), (60) and (120) are calculated moving forward through the sample period to the beginning of April, May and so on in steps of one month ending in February 2011. RLZ(n) is calculated as in equation (6) choosing a mean of zero as suggested by Figlewski (1997). Thus the sum of the squared returns is divided by n since no adjustment for the degrees of freedom is necessary. All volatility measures in this paper are annualized using 252 days including the implied volatilities where Bloomberg allows the choice of days.
Besides the analysis of the slope coefficient and corresponding significance and the interpretation of the Adjusted R-Square, the statistical test for un-biasedness which is of interest is the joint test that coefficients and in equation (13). The Adjusted R-Squared gives an indication
about how much of RLZ(n) is explained by the corresponding IV(m) measure. The test for un-biasedness is carried out by the Wald Coefficient test in Eviews 7. If the Null-Hypothesis of the Wald test of and as measured by the F-Statistic can be rejected, H1 can be rejected as well indicating that implied
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serial correlation the regression is estimated using the Newey-West HAC covariances. In the presence of both heteroskedasticity and serial correlation again the Newey-West HAC covariances are used because the covariance estimator is consistent in the presence of both heteroskedasticity and autocorrelation of unknown form.
4.2 Encompassing Regression
The second research question, whether historical volatility provides any additional information content to implied volatility about future realized volatility in the WTI and Brent crude oil futures option markets, is examined by the following testable hypothesis:
H2: Historical volatility does not hold any additional information content to implied volatilities about
future realized volatility of WTI and Brent crude oil futures prices.
Again following Figlewski (1997), the hypothesis H2 can be tested by the encompassing regression as
specified by equation (14) below:
( ) ( ) ( )
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with instability thus including negative parameters has been estimated, the estimation period is extended backwards until stability is reached. This is adequate since a GARCH forecast that is obtained today would include as many historical observations as necessary in order to obtain stable parameters.
Regression diagnostics are once again performed as described above and the White or Newey-West covariances used if adequate. Along with an analysis of the coefficients and and their corresponding significance, the comparison of the Adjusted R-Squared of equation (13) and (14) reveals whether
has improved the fit of the model and thus explanatory power of RLZ(n). If both implied and historical volatilities are rational forecasts, but historical volatility is based on a subset of information, the regression should yield coefficient estimates of , , and . Again the Wald coefficient test is conducted with Eviews 7 where a rejection of the Null Hypothesis through the according F-Statistic leads to the rejection of H2 indicating that historical volatility does hold additional information content to
implied volatilities about future realized volatility of WTI and Brent crude oil futures prices. In the other possible case when the historical volatility measure is the superior forecast of actual volatility the encompassing regression could yield coefficient estimates of , , and . From an economic point of view this would be a surprising result implying that implied volatility does not seem to be a good forecast of realized volatility at all. Another time the Wald coefficient test is conducted with Eviews 7, where an F-Statistic that does not lead to a rejection of the Null Hypothesis indicates that one cannot claim that , which means that one cannot reject that the historical volatility measure holds all information about future realized volatility.
4.3 Forecasting Accuracy
In addition to testing H1 and H2 the question arises which forecasting method is the most accurate in the
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ex post during the historical period that is necessary to calculate the desired measure. As I mention the sample is non-overlapping and all forecasts are repeated at the beginning of each month in the sample period starting in March 2006 and ending February 2011. Once again the GARCH (1, 1) and EGARCH measures start in March 2007. 1, 2, 3 and 6 months (IV) is used as a forecast for the following according days of realized volatility. All forecast measures, that are IV(m), STD(n), GARCH (1,1) and EGARCH, are compared on forecasting accuracy through the root mean squared error as defined in equation (15):
√[( ) ∑ ( ( ) ( ) )
] ( )
Where ( ) is the forecast measure that is compared to ( ) including IV(m), STD(n), GARCH (1,1)
and EGARCH and X is the number of forecasts during the sample period used for each forecast measure. The smaller the RMSE the more accurate is the volatility forecast. RMSE is chosen because it is used in most previous studies that compare forecast accuracy which ensures that the results of this study are comparable.
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Table 1
Overview of Forecasting Methods
Matching the 4 different forecast horizons of 1 month, 2 months, 3 months and 6 months with the corresponding RLZ(n) measure for the realized volatility and the different forecast methods including the corresponding STD(n), GARCH (1, 1), EGARCH and the corresponding IV(m) forecast.
Forecast Horizon
1 Month 2 Months 3 Months 6 Months
Realized Volatility RLZ(20) RLZ(40) RLZ(60) RLZ(120) Forecast Method STD(20) STD(20) STD(20) STD(20) STD(40) STD(40) STD(40) STD(40) STD(60) STD(60) STD(60) STD(60) STD(120) STD(120) STD(120) STD(120)
GARCH (1, 1) GARCH (1, 1) GARCH (1, 1) GARCH (1, 1)
EGARCH EGARCH EGARCH EGARCH
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5. Data Analysis
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Bera statistics normality can be rejected at the 99% significance level for both returns series. Lastly the number of observations is slightly higher for the Brent returns series because the ICE has more yearly trading days than the NYMEX. Figures 4 and 5 exhibit the annualized daily volatility for the period from March 1st 2006 until February 11th 2011 of the WTI and Brent futures prices respectively. Once again, both asset returns have very similar volatility patterns. From Figure 4 and 5 it becomes clear that volatility is changing over time for WTI and Brent futures with periods of higher and periods of lower volatility. A period of extreme volatility is observed during the recent financial crisis peaking at 137.5% for the WTI futures and at 131.1% for the Brent futures.
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Figure 1
WTI and Brent futures prices of the nearest expiration date observed at the end of each month measured in US dollars for the period of March 1st 2006 until February 11th 2011 as collected from Bloomberg.
0 20 40 60 80 100 120 140 160 P rice in $ US 3.1.2006 - 2.11.2011
WTI and Brent Monthly Futures Price
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Table 2
Descriptive Statistics
Descriptive statistics for the daily log returns of WTI and Brent futures for the period from March 1st 2006 until February 11th 2011 obtained with Eviews 7.
Log returns are computed as in equation (7): (
⁄ ).
WTI Returns Brent Returns
Mean 0.0003 0.0004 Median 0.0006 0.0009 Maximum 0.0869 0.0829 Minimum -0.0732 -0.0673 Std. Dev. 0.0172 0.0166 Skewness -0.0730 -0.0251 Kurtosis 5.10 4.94 Jarque-Bera 230.3 199.8 Probability 0.00** 0.00** Observations 1250 1279
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Figure 2
WTI futures daily log returns for the period from March 1st 2006 until February 11th 2011. Log returns are computed as in equation (7): ( ⁄ ). -10.0% -8.0% -6.0% -4.0% -2.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Da ily L o g Ret urn 3.1.2006 - 2.11.2011
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Figure 3
Brent futures daily log returns for the period from March 1st 2006 until February 11th 2011. Log returns are computed as in equation (7): ( ⁄ ). -10.0% -8.0% -6.0% -4.0% -2.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Da ily L o g Ret urn 3.1.2006 - 2.11.2011
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Figure 4
WTI futures annualized daily volatility for the period from March 1st 2006 until February 11th 2011. Daily volatility is calculated by a simple form of equation (6) for where the adjustment for the degrees of freedom is not necessary. The volatility is annualized using 252 days where , being the mean of over the entire data
period: ( ) √ 0% 20% 40% 60% 80% 100% 120% 140% 160% Da ily Vo la tility 3.1.2006 - 2.11.2011
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Figure 5
WTI futures annualized daily volatility for the period from March 1st 2006 until February 11th 2011. Daily volatility is calculated by a simple form of equation (6) for where the adjustment for the degrees of freedom is not necessary. The volatility is annualized using 252 days where , being the mean of over the entire data
period: ( ) √ 0% 20% 40% 60% 80% 100% 120% 140% Da ily Vo la tility 3.1.2006 - 2.11.2011
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Table 3
Results of Test for ARCH Effects
The F-statistics with according p-values for Engle (1982)’s test for ARCH are shown for the daily log returns of WTI and Brent futures. The test is performed for an OLS (ordinary least squares) regression, GARCH (1, 1) estimation as well as an EGARCH model on 10 previous observations in Eviews 7. Log returns are computed as in equation (7): (
⁄ )
WTI Returns Brent Returns
OLS 21.85 19.92 (0.000)** (0.000)** GARCH (1, 1) 2.03 1.50 (0.028)* (0.135) EGARCH 2.31 1.63 (0.011)* (0.092)
(**statistically significant at 99% level; *statistically significant at 95% level)
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to maturity. Thus for both underlying assets, the WTI and Brent futures, for the given time period the highest implied volatility is, on average, found for the 90% K/F and 1 month maturity options, the lowest percentage of K/F and shortest maturity of the sample. Further a so called term structure between shorter and longer maturities can be observed. For both assets, the implied volatilities are on average consistently decreasing as time to maturity is increasing. Thus on average for the given time period financial markets use a higher volatility input for options of closer maturity. A possible explanation for this observation could be that unexpected periods of high volatility have less of an impact on an annualized volatility of six months whereas a few days of relatively high volatility can have a substantial upwards effect on an annualized volatility of only one month. Figure 6 and 7 go one step further and take the average of the four different maturities in tables 4 and 5 for the given K/F. Thus figure 6 displays the average of all implied volatilities of WTI futures options with a K/F of 90%, 95%, 100% 105% and 110% and figure 7 shows the average of all implied volatilities of Brent futures options with a K/F of 90%, 95%, 100% 105% and 110% respectively. The same pattern, which can be called a volatility skew, is observed in both figures where implied volatility is a decreasing function of the K/F of the options. With the observed volatility skew, OTM call options and ITM put options are valued higher than other options for both underlying assets. A possible explanation for the skew could be that oil prices and futures prices accordingly have been shown to increase sharply in time of crisis or a stock market crash. Since the possibility of a crash is always given where crude oil futures prices are likely to sharply increase, an OTM call option could be valued higher by increasing the volatility parameter in order to incorporate the chance of being ITM after a crash. The observed skew for Brent and WTI futures options is the exact opposite of the commonly known volatility skew of equity options.
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Table 4
Volatility Surface WTI
The average annualized daily implied volatility of WTI futures options from Bloomberg for the period from January 10th 2006 until February 11th 2011 for options expiring close to 1 month, 2 months, 3 months and 6 months as well as for a K/F of 90%, 95%, 100%, 105% and 110% which is specified as the strike price K of the options divided by the current price F of the WTI futures with 1282 observations for each match of K/F and time to maturity. The numbers in the table are the averages of all observations for the given K/F and time to maturity.
K/F 90% 95% 100% 105% 110% 1 month 40.71 40.16 39.01 38.65 38.66 2 months 38.90 38.56 37.73 37.33 37.26 3 months 37.79 37.53 36.84 36.47 36.39 6 months 35.58 35.32 34.72 34.38 34.33
Figure 6
The average annualized daily volatility of WTI futures options for the period from January 10th 2006 until February 11th 2011. The average of all observations of a given K/F in table IV is taken, combining all maturities for the given strike over spot which equals the strike price K divided by the current price F of WTI futures. 35.5 36.0 36.5 37.0 37.5 38.0 38.5 90 95 100 105 110 Av er a g e Vo la tility in % K/F in %
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Table 5
Volatility Surface Brent
The average annualized daily implied volatility of Brent futures options from Bloomberg for the period from January 10th 2006 until February 11th 2011 for options expiring close to 1 month, 2 months, 3 months and 6 months as well as for a K/F of 90%, 95%, 100%, 105% and 110% which is specified as the strike price K of the options divided by the current price F of the Brent futures with 1312 observations for each match of K/F and time to maturity. The numbers in the table are the averages of all observations for the given K/F and time to maturity.
K/F 90% 95% 100% 105% 110% 1 month 41.01 40.55 39.35 38.64 38.53 2 months 39.34 39.07 38.23 37.61 37.47 3 months 38.47 38.25 37.53 36.95 36.83 6 months 36.47 36.32 35.75 35.28 35.19
Figure 7
The average annualized daily volatility of Brent futures options for the period from January 10th 2006 until February 11th 2011. The average of all observations of a given K/F in table V is taken, combining all maturities for the given strike over spot which equals the strike price K divided by the current price F of Brent futures. 36.0 36.5 37.0 37.5 38.0 38.5 39.0 90 95 100 105 110 Av er a g e Vo la tility in % K/F %
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Table 6
Volatility Surface WTI: Sub-Samples
The average annualized daily implied volatility of WTI futures options from Bloomberg are split in three sub periods representing a sample before the crisis from January 10th 2006 to October 31th 2007 with 455 observations for each K/F and time to maturity, during the crisis from November 1st 2007 to June 30th 2009 with 418 observations and after the crisis from July 1st 2009 until February 11th 2011 with 409 observations. The implied volatilities are for options expiring close to 1 month, 2 months, 3 months and 6 months as well as for K/F of 90%, 95%, 100%, 105% and 110% which is specified as the strike price K of the options divided by the current price F of the Brent futures. The numbers in the table are the averages of all observations in the sub period for the given K/F and time to maturity.
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Table 7
Volatility Surface Brent: Sub-Samples
The average annualized daily implied volatility of Brent futures options from Bloomberg are split in three sub periods representing a sample before the crisis from January 10th 2006 to October 31th 2007 with 464 observations for each K/F and time to maturity, during the crisis from November 1st 2007 to June 30th 2009 with 428 observations and after the crisis from July 1st 2009 until February 11th 2011 with 420 observations. The implied volatilities are for options expiring close to 1 month, 2 months, 3 months and 6 months as well as for K/F of 90%, 95%, 100%, 105% and 110% which is specified as the strike price K of the options divided by the current price F of the Brent futures. The numbers in the table are the averages of all observations in the sub period for the given K/F and time to maturity.
45
Figures 8-10:
The average annualized daily volatility of WTI futures options for the three sub samples from January 10th 2006 until October 2007 for the pre-crisis period, from November 2007 until June 2009 for the crisis period and from July 2009 until February 2011 for the post-crisis period. The average of all observations of a given K/F in table VI is taken, combining all maturities for the given strike over spot which equals the strike price K divided by the current price F of WTI futures.
Figure 8
Figure 9
27.5 28.0 28.5 29.0 29.5 30.0 90 95 100 105 110 Av er a g e Vo la tility in % K/F in %Volatility Smile WTI: Pre Crisis
46.5 47.0 47.5 48.0 48.5 49.0 90 95 100 105 110 Av er a g e Vo la tility in % K/F in %
46
Figure 10
Figures 11-13:
The average annualized daily volatility of Brent futures options for the three sub samples from January 10th 2006 until October 2007 for the pre-crisis period, from November 2007 until June 2009 for the crisis period and from July 2009 until February 2011 for the post-crisis period. The average of all observations of a given K/F in table VII is taken, combining all maturities for the given strike over spot which equals the strike price K divided by the current price F of WTI futures.
Figure 11
33.0 34.0 35.0 36.0 37.0 38.0 90 95 100 105 110 Av er a g e Vo la tility in % K/F in %Volatility Smile WTI: Post Crisis
29.46 29.48 29.50 29.52 29.54 29.56 29.58 90 95 100 105 110 Av er a g e Vo la tility in % K/F in %
47
Figure 12
Figure 13
46.0 47.0 48.0 49.0 50.0 51.0 52.0 90 95 100 105 110 Av er a g e Vo la tility in % K/F in %Volatility Skew Brent: Crisis
34.0 34.5 35.0 35.5 36.0 36.5 37.0 37.5 90 95 100 105 110 Av er a g e Vo la tility in % K/F in %
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6. Results
6.1 Rationality Test
Table 8 shows the results of the rationality test regression as specified in equation (13) for the WTI futures market for each measure of RLZ(n) and the according IV(m). The value of the coefficients can be interpreted as the percentage of realized volatility that is explained by the according implied volatility. A first look at the coefficients indicates that all are significantly different from zero at the 99% level as indicated by their low p-values underneath in parentheses but are so small that one can expect that their true value is not equal to one. This is confirmed by the Wald Test of the coefficients that and . By observing its F-Statistics and their according p-values underneath in parentheses in the very right
column one can infer that the Null hypothesis can be rejected at the 99% level for all four forecasting horizons. With the evidence in table 8 one can reject H1 of this study and say that implied volatility is not
an unbiased estimator of future realized volatility of WTI futures prices. Table 9 shows the results of the rationality test regression for the Brent futures market. Here the values for the are on average even smaller and all are significant at least at the 95% level providing first evidence against H1. Again the
49
Table 8
Results Rationality Test Regression WTI
( ) ( )
Results of the rationality test regression as specified in equation (13) for the WTI market where
RLZ(n) is defined as in equation (6) and IV(m) is the according implied volatility from Bloomberg.
The p-value of the coefficients α and β is provided in parentheses underneath their values. The F-Statistic along with the according p-value for the Wald Test of the coefficients that α = 0 and β = 1 is shown in the very right column:
RLZ(n) IV(m) α β Wald Test: α = 0, β = 1
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Table 9
Results Rationality Test Regression Brent
( ) ( )
Results of the rationality test regression as specified in equation (13) for the Brent market where
RLZ(n) is defined as in equation (6) and IV(m) is the according implied volatility from Bloomberg.
The p-value of the coefficients α and β is provided in parentheses underneath their values. The F-Statistic along with the according p-value for the Wald Test of the coefficients that α = 0 and β = 1 is shown in the very right column.
RLZ(n) IV(m) α β Wald Test: α = 0, β = 1
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6.2 Encompassing Regression
Table 10 shows the results of the encompassing regression as defined in equation (14) for the WTI futures market. The regression is estimated for all four forecast horizons with RLZ(20), (40), (60) and (120) accordingly, the corresponding IV(m) forecast as well as all six different historical forecasts, that is STD(20), (40), (60), (120), GARCH (1,1) and EGARCH. The values of the coefficients , and are shown for each regression along with the according p-value underneath in parentheses. The column second to the right entails the results of the Wald Test of the coefficients that , , and ,
showing the value of the F-Statistic and the p-value of the test underneath in parentheses. For all four forecasting horizons and all six different measures of HIST the Null Hypothesis of the Wald Test can be rejected at the 99% level indicating that the inclusion of a historical volatility measure does not change the inference that implied volatility is a biased estimate of future realized volatility. Further the results of the Wald Test let us infer that H2 can be rejected since IV(m) does not hold all information content about
future realized volatility in the WTI futures market. A closer analysis of the values and significance levels of the individual coefficients provides further evidence against H2. Again the value of the coefficients can
52
significant at the 99% level besides GARCH (1, 1) at the 95% level. When looking at the fourth forecast horizon of 6 months or 120 trading days neither of the of IV(6) are significant in combination with the six different historical methods. Now STD(40) has the highest coefficient of .6333, STD(60) .6273 and
STD(20) .4918, again all being significant at the 99% level. The very right column in table 10 shows the results of the second Wald Test of the coefficients that , , and , showing the value of
the F-Statistic and the p-value of the test underneath in parentheses. This additional test reveals some very interesting results. When STD(60) is added to IV(1) in the encompassing regression in order to explain RLZ(20) as the dependent variable the Null hypothesis of the Wald Test of interest cannot be rejected at the 95% level, meaning that one cannot say that the coefficient of STD(60) is not equal to one at the given level of significance. The same result is found when STD(40) is added to IV(2) with RLZ(40) as the dependent variable and when STD(60) is added to IV(3) with RLZ(60) being the dependent variable. In these certain cases one cannot say that the given historical volatility does not hold all information about future realized volatility when added to implied volatility in a regression model making it a rational and informational efficient forecast. In summary of the WTI futures markets, STD(60) has the most significant explanatory power of realized volatility for 20, 40 and 60 trading days and STD(40) for 120 trading days when combined with the according implied volatility in the regression. When IV(m) is significant it explains less than 1% of RLZ(n). GARCH (1,1) only is significant when used to forecast RLZ(60) but has less explanatory power than STD(n). EGARCH is not significant for any forecast horizon and one cannot say that EGARCH is an improvement to GARCH. In three exceptional cases STD(n) not only holds additional information to implied volatility about future volatility, but cannot be ruled out to hold all information about future volatility where implied volatility is not adding any valuable information content.
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Table 10
Results Encompassing Regression WTI
( ) ( )
Results of the encompassing regression as defined in equation (14) for the WTI market where RLZ(n) is defined as in equation (6), IV(m) is the according implied volatility from Bloomberg and HIST is the historical volatility method that is used. The p-value of the coefficients α, β1 and β2 is provided underneath their values in parentheses. The F-Statistic along with the according p-value for the Wald Test of the coefficients that α = 0 and β1 = 1 and β2 = 0 is shown in the column second to the right. The F-Statistic along with the according p-value for the Wald Test of the coefficients that α = 0 and β1 = 0 and β2 = 1 is shown in the very right column. All regressions are estimated for the entire sample period from March 2006 until February 2011 besides the regressions including GARCH (1, 1) and EGARCH which start in March 2007.
54 EGARCH 0.0676 0.0032 0.2582 2469631.2 16.4 (0.107) (0.103) (0.116) (0.000)** (0.000)** RLZ 60 IV 3 Month STD 20 0.0744 0.0017 0.4561 4293241.3 4.9 (0.009)** (0.106) (0.009)** (0.000)** (0.004)** STD 40 0.0777 0.0001 0.6739 4479005.5 3.2 (0.007)** 0.963 (0.001)** (0.000)** (0.031)* STD 60 0.0743 0.0000 0.6995 3934344.1 2.8 (0.024)* 0.995 (0.000)** (0.000)** (0.051) STD 120 0.0809 0.0035 0.1809 1594683.7 3.2 (0.051) (0.077) (0.533) (0.000)** (0.031)* GARCH (1,1) 0.0858 0.0000 0.6378 2102190.8 3.8 (0.018)* 0.982 (0.025)* (0.000)** (0.017)* EGARCH 0.1008 0.0025 0.2447 2370055.5 29.2 (0.018)* (0.207) (0.117) (0.000)** (0.000)** RLZ 120 IV 6 Month STD 20 0.1231 0.0003 0.4918 2670502.0 11.4 (0.003)** (0.815) (0.001)** (0.000)** (0.000)** STD 40 0.1325 -0.0010 0.6333 2755408.5 9.7 (0.006)** (0.639) (0.002)** (0.000)** (0.000)** STD 60 0.1326 -0.0009 0.6273 2278077.2 8.7 (0.010)* (0.699) (0.004)** (0.000)** (0.000)** STD 120 0.1215 0.0039 0.0346 979642.1 7.0 (0.026)* (0.225) (0.912) (0.000)** (0.000)** GARCH (1,1) 0.1562 -0.0003 0.4486 911330.2 5.5 (0.003)** (0.931) (0.159) (0.000)** (0.003)** EGARCH 0.1518 0.0025 0.1091 2591417.6 170.9 (0.022)* (0.308) (0.297) (0.000)** (0.000)** (**statistically significant at 99% level; *statistically significant at 95% level)
that H2 can be rejected since IV(m) does not hold all information content about future realized volatility in
the Brent futures market. Again a closer analysis of the values and significance levels of the individual coefficients provides further evidence against H2. For the encompassing regressions with a forecast
horizon of 1 month or 20 trading days the most explanatory power of RLZ(20) is found in of STD(60)
55
power at the 99% level when combined with STD(20) at .0028 and when combined with STD(120) at .0028 and at the 95% level when combined with GARCH (1, 1) at .0016 which are all less than 1%. For the regressions with a forecast horizon of 2 months or 40 trading days the most explanatory power of RLZ(40) is found again in of STD(60) with a value of .8030 significant at the 99% level. STD(40) and GARCH (1, 1) are significant at the 99% level as well with coefficients of .7820 and .7454 respectively. Lastly the of EGARCH is .4532 significant at the 95% level. IV(2) is not significant in any of the six
estimated regressions. In the models for a forecasting horizon of 3 months or 60 trading days with RLZ(60) as the dependent variable all are significant besides for STD(120) and EGARCH. STD(40) has the highest with .7344 followed by STD(60) with .6965 and GARCH (1, 1) with .6523 all significant at
the 99% level. STD(20) has a value .4380 which is significant at the 95% level. IV(3) is only significant when combined with STD(20) and STD(120) with a value of of .0017 with a 95% significance level and .0036 with a 99% significance level respectively. Lastly for the forecast horizon of 6 months or 120 trading days IV(6) is not significant when combined with the six different historical methods to forecast RLZ(120). Again STD(40) has the highest value of .7389 being followed by STD(60) with .6752. STD(20) has a value of .5019 and GARCH (1,1) .4873 where all four are significant at the 99% level. The very right column in table 11 shows the results of the second Wald Test of the coefficients that , ,
56
encompassing regression. When IV(m) is significant it again only explains less than 1% of RLZ(n). GARCH (1,1) is significant when combined with IV(m) for all forecasting horizons but has less explanatory power than STD(n). EGARCH is only significant once for a 40 trading day forecast horizon but one cannot say that EGARCH is an improvement to GARCH in the Brent futures market. As for the WTI markets, in three exceptional cases STD(n) not only holds additional information to implied volatility about future volatility, but cannot be ruled out to hold all information about future volatility where implied volatility is not adding any valuable information content.
Table 11
Results Encompassing Regression Brent
( ) ( )
Results of the encompassing regression as defined in equation (14) for the Brent market where RLZ(n) is defined as in equation (6), IV(m) is the according implied volatility from Bloomberg and HIST is the historical volatility method that is used. The p-value of the coefficients α, β1 and β2 is provided underneath their values in parentheses. The F-Statistic along with the according p-value for the Wald Test of the coefficients that α = 0 and β1 = 1 and β2 = 0 is shown in the column second to the right. The F-Statistic along with the according p-value for the Wald Test of the coefficients that α = 0 and β1 = 0 and β2 = 1 is shown in the very right column. All regressions are estimated for the entire sample period from March 2006 until February 2011 besides the regressions including GARCH (1, 1) and EGARCH which start in March 2007.
58
Tables 12 and 13 show the Adjusted R-Squared for both the rationality test regressions and the encompassing regressions of the WTI and Brent futures markets respectively. The Adjusted R-Squared reveals the overall explanatory of a regression model of the dependent variable. The intention of tables 12 and 13 is to reveal how much of RLZ(n) is explained by the according IV(m) in a regression and whether the fit of the model is improved when each historical method is added to the regression. First looking at table 12 for the WTI market one can see that the Adjusted R-Squared for the rationality test regression with RLZ(20) as the dependent and IV(1) as the independent variable is .4952 meaning that about 49.5% of RLZ(20) is explained by the model. The values of the Adjusted R-Squared for the historical methods that improve the overall fit of the model are marked with an asterisk in the very column of the table. All historical methods improve the fit of the model when added in the encompassing regression besides STD(120) where the Adjusted R-Squared is reduced to .4759. For the second rationality test regression with RLZ(40) as the dependent and IV(2) as the independent variable the Adjusted R-Squared is .5063. Four of the historical methods improve the Adjusted R-Squared when added to the regression including STD(20), (40) and (60) as well as GARCH (1,1), whereas STD(120) and EGARCH result in a decrease of the fit of the model to .4734 and .4888 respectively. The Adjusted R-Squared of the regression with RLZ(60) as the dependent variable and IV(3) as the independent variable is .4371. Again the value is only lowered when adding STD(120) or EGARCH to the model being .4045 and .4096 respectively. Lastly for the rationality test regression with RLZ(120) and IV(6) the Adjusted R-Squared is found to be .2829. This time STD(120), GARCH (1, 1) and EGARCH give a lower value when added to the regression at .2286, .2328 and .2081 in that order. Overall one can say that in most cases the historical methods add explanatory power of the realized volatility and clearly hold additional information content to implied volatility. This is additional evidence against H2 for the WTI futures market.The evidence is even more
59
IV(6) as the independent variables. These findings complement the evidence against H2 for the Brent
futures market.
Table 12
Adjusted R-Squared WTI
The Adjusted R-Squared for each rationality test regression and encompassing regression for the WTI futures market. The Adjusted R-Squared of the encompassing regressions that improve the rationality test regression are marked with an asterisk in the very right column.
RLZ(n) IV(m)
Rationality Test
Regression HIST
Encompassing Regression
Adjusted R-Squared Adjusted R-Squared
RLZ(20) IV(1) 0.4952 STD 20 0.5107* STD 40 0.5240* STD 60 0.5280* STD 120 0.4759 GARCH (1,1) 0.5004* EGARCH 0.5137* RLZ(40) IV(2) 0.5063 STD 20 0.5458* STD 40 0.5546* STD 60 0.5489* STD 120 0.4734 GARCH (1,1) 0.5144* EGARCH 0.4888 RLZ(60) IV(3) 0.4371 STD 20 0.5276* STD 40 0.5401* STD 60 0.5247* STD 120 0.4045 GARCH (1,1) 0.4682* EGARCH 0.4096 RLZ(120) IV(6) 0.2829 STD 20 0.3980* STD 40 0.3812* STD 60 0.3477* STD 120 0.2286 GARCH (1,1) 0.2328 EGARCH 0.2081
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Table 13
Adjusted R-Squared Brent
The Adjusted R-Squared for each rationality test regression and encompassing regression for the Brent futures market. The Adjusted R-Squared of the encompassing regressions that improve the rationality test regression are marked with an asterisk in the very right column.
RLZ(n) IV(m)
Rationality Test
Regression HIST
Encompassing Regression
Adjusted R-Squared Adjusted R-Squared
RLZ(20) IV(1) 0.4867 STD 20 0.5030* STD 40 0.5253* STD 60 0.5294* STD 120 0.5051* GARCH (1,1) 0.5463* EGARCH 0.4960* RLZ(40) IV(2) 0.4249 STD 20 0.5016* STD 40 0.5623* STD 60 0.5408* STD 120 0.4543* GARCH (1,1) 0.6038* EGARCH 0.4752* RLZ(60) IV(3) 0.4488 STD 20 0.5283* STD 40 0.5666* STD 60 0.5415* STD 120 0.4598* GARCH (1,1) 0.6072* EGARCH 0.4657* RLZ(120) IV(6) 0.3001 STD 20 0.4218* STD 40 0.4488* STD 60 0.4001* STD 120 0.2866 GARCH (1,1) 0.3766* EGARCH 0.2649
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6.3 Results RMSE
Tables 14 and 15 provide the results and answers to the question which forecasting method has the lowest forecasting error in the WTI and Brent futures markets as measured by the RMSE. The tables exhibit the results for the entire sample period and in addition provide the RMSE for three sub samples including a pre-crisis, a crisis and a post-crisis period. For all four forecasting horizons the lowest value for the RMSE and thus the least forecasting error is marked in bold. In table 14 one can see that for the entire sample in the WTI futures market the lowest RMSE when forecasting RLZ(20) belongs to STD(60) at .076995. Clearly IV(1) performs the worst and has the highest forecast error at .173267. STD(40) and (120) perform better than GARCH (1,1) and the EGARCH model does not achieve an improvement. For RLZ(40) again STD(60) has the lowest RMSE value at .070668 and IV(2) has the highest value at .152114 being the least accurate forecast. STD(40) performs better than GARCH (1, 1) which in turn is superior to STD(20) and STD(120). EGARCH does not improve the forecast of the GARCH (1, 1) model. For RLZ(60) one more time STD(60) has the lowest RMSE value with .071507. All STD(n) perform better that the GARCH (1, 1) forecast which is not improved by EGARCH. Once again IV(3) has the highest RMSE with a value of .153672. For RLZ(120), STD(40) yields the most accurate forecast with a value of .086141. This time EGARCH has the highest RMSE with a value of .173575 being followed by IV(6) and GARCH (1,1) where again all STD(n) have the four lowest values. To summarize the RMSE for the entire sample period for the WTI futures market one can say that STD(60) seems to be the most accurate volatility forecasting method in the WTI futures market only being outperformed once by STD(40) for the longest forecast of 120 trading days. GARCH (1,1) only improves STD(20) when forecasting RLZ(20) as well as STD(20) and STD(120) when forecasting RLZ(40). EGARCH is consistently the least accurate historical forecasting method and IV(m) performs worst for all forecasting horizons besides RLZ(120) where it is superior to EGARCH.
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63
STD(60) and STD(120) for the four forecast horizons. Also during the post-crisis period for the only time EGARCH outperforms GARCH (1, 1) and both models fare relatively better than during the crisis.
Table 14
Results RMSE WTI
The RMSE of each forecasting horizon with the according RLZ(n) matched with all seven forecasting methods employed in this study in the WTI futures market for the entire sample period from March 2006 to February 2011, a pre-crisis period from March 2006 to October 2007, a crisis period from November 2007 to June 2009 and a post-crisis period from July 2009 until February 2011. For the GARCH (1, 1) and EGARCH measure the entire sample goes from March 2007 until February 2011 and pre-crisis values cannot be provided. The lowest value of the RMSE is marked in bold which is defined as in equation (15): √[( ) ∑ ( ( ) ( ) ) ]
Forecast Horizon
1 Month 2 Month 3 Month 6 Month
Realized Volatility
RLZ(20) RLZ(40) RLZ(60) RLZ(120)
Entire Sample
Forecast Method RMSE
