The Predictive Power of the Implied
Volatility Skew for Realized Volatility
Bachelor Thesis Econometrics
Victor Monas
Student number: 10214992
Advisor
: Prof. dr. H. P. Boswijk
Date: 20-12-2013
Abstract
Previous research has described in detail the relationship between at-the-money option implied volatility and future realized volatility. Recent studies imply that there might be a predictive value of the volatility skew for future realized volatility. Using data from the S&P 500-index from April 2006 till May 2013 I find that there are only weak indications that the implied volatility skew indeed does have any predictive power for the future realized volatility.
1
Introduction
An option is a contract that gives the buyer the right to sell or buy an asset at a certain price before a certain time. A call option gives the buyer the right to buy an asset while a put option gives the buyer the right to sell it.
In the market for options the implied volatility is regarded as the forecast of future realized volatility and therefore plays a crucial role in financial economics (Christensen and Prabhala, 1998). The predictive value of the implied volatility gives traders and investors an insight in the amount of the future realized volatility, which is a measure of the magnitude of price movements of the underlying equity over a certain period. This is important because volatility comprises an integral part of the pricing of options: when the volatility is high, the premium on an option should also be relatively large and when the volatility is low, which means the stock is less risky, the option premium should also be small. An accurate prediction of the volatility will thus lead to more certainty about whether an option is priced correctly in the market.
The search for models that describe the option pricing process correctly, thereby leading to a correct estimation of the implied volatility, has been an important field of research in the fi-nancial economics in the past decades. An extensive catalogue of literature concerning the topic has already been written, in which many different models for calculating the implied volatility are considered. Most well-known is arguably the Black-Scholes formula, which is a closed form formula that can be used to calculate the implied volatilities from European options. The formula was first postulated by Fischer Black and Myron Scholes in their paper “The Pricing of Options and Corporate Liabilities” (1973).
One of the main advantages of the Black-Scholes formula is that all of its parameters, ex-cept for the volatility, are directly observable (Canina and Figlewski, 1993). This makes it easy to compute the ‘true’ price of an option. However the simplicity of the Black-Scholes formula also is its main downside. Using such a comprehensible formula for the calculation of option prices ineluctably leads to oversimplification of reality. To achieve this level of simplicity some assumptions are required for the formula. It is for instance assumed that the underlying stock
underlying equity is assumed to follow a log-normal diffusion process with constant parameters (Brockman and Chowdhury, 1997). The assumption of deterministic volatilities however is very restrictive because it assumes that implied volatilities depend only on the current stock price and the time, thereby neglecting the possibility for volatility shocks (Britten-Jones and Neuberger, 2000). Furthermore, Buraschi and Jackwerth (2001) show that the deterministic volatility is not able to capture all the dynamics that determine option prices and that models based on the assumption of stochastic volatility are more accurate in this regard.
New models have been postulated to correct for these deficiencies, such as the Heston model, which assumes stochastic volatility and Merton’s Jump Diffusion model which corrects for the assumption of continuity of the underlying equity’s price.
One of the consequences of the incorrect assumptions necessary for the Black-Scholes formula is that they possibly falsely suggest the existence of a so-called ‘volatility smile’. This smile is constructed by plotting the implied volatility on the vertical axis against the strike price of the option on the horizontal axis. It can then be found that the implied volatility is minimal around the at-the-money strike price of the option and increases the deeper the option is out-the-money or in-out-the-money. A smile is displayed in Figure 1. From empirical studies, such as that of Jackwerth and Rubinstein (1996), it is known that when equity options are considered the smile will have a largely skewed form, decreasing in volatility for higher strike prices. This particular form of the volatility smile is called the volatility skew.
The most popular explanation of the smile has been that it arises because of the afore-mentioned misspecifications required for the Black-Scholes formula and that when calculated with more precise models the volatility smile will actually be nothing more than a horizontal line (Rubinstein, 1994).
However, other research disputes these findings. Ederington and Guan (2000) find that the smile is only partially due to the erroneous specification in the Black-Scholes formula. They hypothesize that at least one of the other factors explaining the smile are hedging strategies. Hedgers will buy out-the-money puts (located left of the at-the-money options on the smile) to insure their portfolios against a future possible stock market crash. This will lead to excessive demand for these options that in turn will increase their prices. The higher prices will lead to larger implied volatilities for options with strike prices below the at-the-money strike price. Further evidence that supports this theory is that prior to the stock market crash of 1987 no volatility smile was observed (Hull, 2009, page 386). It was only after this massive financial cataclysm that investors became aware of the need to hedge their portfolios against such unex-pected events, thus leading to the existence of a volatility smile.
Implied volatility, as previously mentioned, gives investors an idea of future realized volatility. If option markets are efficient, then the implied volatility is an effective predictor of future realized volatility (Christensen and Prabhala, 1998). Intensive research about this has been conducted, often leading to contradicting results and conclusions. Canina and Figlewski (1993) find that implied volatility is not a significant predictor of the future realized volatil-ity and that a prediction based on historical realized volatilvolatil-ity leads to better predicted values. These results would shake the foundations of option pricing theory if they were found to be true. Their findings however seem to be highly unlikely since option prices, and thus option implied volatility, are based on the value of the underlying equity and therefore a relationship between option implied volatility and the volatility of the underlying asset should exist in effi-cient markets (Shu and Zhang, 2003).
Christensen en Prabhala on the other hand conclude that the implied volatility is the best predictor for future realized volatility and that the results of Canina and Figlewski’s research are due to their sampling method. Because Canina and Figlewski sample on a daily basis instead of a monthly basis there is great overlap between two consecutive observations, leading to a strong predictive power of historic volatility. However, when Christensen and Prabhala use monthly sampling intervals they find that the implied volatility is a significant predictor of future real-ized volatility while historic volatility is not. There have in fact been numerous studies that
and conclude that realized volatility is indeed best forecastable by the implied volatility. They further find that in 76% of the studies implied volatility is found to be a more accurate predictor of future realized volatility than historical realized volatility, thereby giving more evidence for the overall consensus that the results found by Canina and Figlewski are based on incorrect assumptions.
Shu and Zhang (2003) attempt to quantify the relation between implied volatility and realized volatility when different calculation methods for the two volatilities are used. To do so they use different calculation methods to find the values of the implied and realized volatility and then substitute the found values in the regression model
V olrealized(t) = α + β ∗ V olimplied(t) + ε(t) (1)
to calculate the α and the β When α = 0 the predictor is unbiased and when β = 1 the predictor is efficient. However for all calculation methods used they never find that the model is unbiased and efficient. Furthermore the R2 is at most 0.4364, implying that there are many factors
un-considered in the calculation of realized volatility.
Buraschi and Jackwerth (2001) use data from the S&P 500-index to test whether the away-from-the-money options are redundant securities when it comes to the spanning of the pricing kernel, meaning that all risk factors required for the correct calculation of the prices are already subsumed in the at-the-money option and the underlying equity itself. Their results indicate that the away-from-the-money options price additional risk factors that aren’t spanned by at-the-money option and underlying equity. This result suggests that the returns on the away-from-the-money options are driven by different economic factors than those relevant for at-the-money options.
In this paper it will be tested if the findings of Baruschi and Jackwerth can be extended to the relationship between implied and realized volatility. Since it is found that the at-the-money options do not subsume all risk factors necessary for option pricing and that the away-from-the-money options are required for spanning options of all risk factors, it might be true that the at-the-money implied volatilities also do not subsume all information required for realized volatility and that here too away-from-the-money options are needed to get a complete overview of factors responsible for the realized volatility. Therefore it could be true that when the options on the volatility skew are taken into consideration the future realized volatility might be better
predicted.
Further indication that the volatility skew might have predictive value is provided by the definition of the model-free implied volatility. The model-free implied volatility is different from other implied volatilities since the calculation does not require a specific parametric model (Rouah and Vainberg, 2007, page 323). The model-free implied volatility can be calculated, as specified by Britten-Jones and Neuberger (2000), using:
M odelf reeIV = E0 s Z t2 t1 dSt St 2 = s 2 Z ∞ 0 C(t2, K) − max(S0− K, 0) K2 dK
The model-free implied volatility is thus theoretically calculated using the whole range of strike prices from 0 to infinity, thereby suggesting that all strike prices are needed for the calculation of the implied volatility and that thus predictive value can be found in the volatility skew. Nat-urally, only a certain subset of options with these strike prices is traded in the market instead of the whole infinite range. This difference in range does however not lead to problems since the infinite integral can simply be truncated giving it integral boundaries ranging from the lowest strike price available in the market to the highest (Rouah and Vainberg, 2007, Page 324).
This paper will expand the work of Shu and Zhang, attempting to answer the question if the option implied volatility skew has any predictive value for future realized volatility. To do so the regression model in (1) will be extended so as to also include the implied volatility obtained from other options at the volatility skew as regressors. As far as I am aware there has been no prior research concerning this topic, since most of the current research focusses merely on the at-the-money implied volatilities.
The rest of the paper is organized as follows. Section 2 will describe the data, give a clearer view of our sampling methods and explain the methods used in this study. In section 3 the results of the conducted research will be presented and the robustness of these results will be tested. The final section 4 will conclude this paper.
2
Data and Methods
In this section the data and the sampling methods used for obtaining the volatility series will be described. Futhermore, the design of the research conducted in this paper is outlined in the second subsection.
2.1 Data and sampling methods
The data that will be used in this study are options on the S&P 500-index. A major advantage of these options compared to some other index options, such as the S&P 100-index options, is that the S&P 500-index options are European style. Therefore there will be no need to correct for early exercise. The sample period will be from April 26, 2006 until May 22, 2013. The data will be sampled on a monthly basis, which sums to a total of 85 data points. Each month seven different options on the volatility skew will be sampled to give a good representation of the volatility skew, being the options that are: at-the-money, 5% money, 7.5% in-the-money, 10% in-the-in-the-money, 2% out-the-in-the-money, 4% out-the-money and 6% out-the-money. We will obtain a total of 595 observations.
The S&P 500-index options by convention expire on the morning of the third Friday of every calendar month. The time-to-maturity is therefore calculated as the number of trading days from the day that the index option was traded till the Thursday that precedes the Friday the option contract ends. On the Wednesday following the Friday that the contract expires the price of various new call options Ct+1,K with around one month maturity and different strike prices
Kt+1 are recorded. The choice for Wednesday is based on the fact that the smallest number
of public holidays are held on Wednesdays. If a certain Wednesday is not a trading day, the following Thursday will be used.
It should be noted that the estimate of the implied volatility will be prone to measurement errors. These errors can arise for multiple reasons. First, the option prices and closing level of the S&P 500 may be nonsynchronous. This can be due to the fact that the prices are recorded at disparate times or because some of the prices of the 500 equities of the S&P 500 do not reflect the most recent information (Christensen & Prabhala, 1998). When such errors in index prices are supposed to be 0.25%, the error in the calculated implied volatility is estimated to be around
1.2% (Jorion, 1995). Further measurement errors for instance arise because option prices have a bid-ask spread.
The implied volatility for each of the options will be recorded from DataStream. These implied volatilities are calculated using the Black-Scholes formula. The underlying options will expire the next month at time t + 1 after which the implied volatilities for each of the options at time t + 1 are reported. In this way a non-overlapping series of implied volatility will be obtained. This sampling method is chosen because it omits any overlap between the data, which will thus exclude the correlation between consecutive observations. Therefore sampling problems like the ones encountered by Canina and Figlewski will be minimized.
Another series of implied volatility that will be obtained is the Volatility Index (VIX), which is an index that aggregates the expectation of future volatility over the next 30 days. It is often referred to as the ‘fear index’ since it gives a measurement of sentiment in the market. A high VIX will indicate a bearish market, while a low VIX implies bullish market conditions.
The Volatility Index is based on the model-free implied volatility extracted from the S&P 500-index. The value of the VIX is recorded on the same date as the values for the index options, however since the VIX is calculated over 365 calendar days while the realized volatility is calculated over 252 trading days a correction needs to be performed. Therefore the values of the VIX are multiplied with factor
q
252
365 to solve for this mismatch in time period.
Lastly the monthly realized volatility, which is the volatility during the past month, needs to be calculated. To perform this calculation firstly the daily 5-minutes realized volatilities of the S&P 500-index are obtained and summed to daily volatilities. Then the daily realized volatilities are aggregated to the realized volatility with to-maturities that match the respective time-to-maturity of the options used to calculate the implied volatility. Since our implied volatility is over a period of around a month we need to sum the daily realized volatilities to a monthly duration. To do so the following formula is used:
V olrel,monthly = v u u t 252 n ∗ n X i=1 V ol2 DailyRealized,i
2.2 Research Design
The goal of this paper is to examine if multiple options from the option implied volatility skew are indeed a better predictor for future realized volatility than solely the at-the-money option implied volatility. To determine this, the model postulated by Shu and Zhang in (1) will be extended to
V olrealized(t) = α + β1∗ V olimplied,atm(t) + β2∗ V olimplied,itm(10%)(t)
+ β3 ∗ V olimplied,otm(6%)(t) + ε(t)
(2)
In section 2.1 it is clarified how the required data is obtained. After obtaining the data and calculating the different volatilities the regression in (2) will be estimated. A problem that is particularly likely to be found is multicollinearity. Since it is likely that a high correlation will be found between the different option implied volatilities, the multicollinearity will also be strongly present. A negative consequence of multicollinearity is that the regressor’s stan-dard errors will sharply increase, thereby leading to lower t-statistics and corresponding higher p-values. The risk then becomes that regressor will incorrectly be dismissed as being not sig-nificant. To decrease the multicollinearity between the at-the-money option implied volatility and the away-from-the-money option implied volatilities (2) will be adjusted into
V olrealized(t) = α + β1∗ V olimplied,atm(t)
+ β2∗ (V olimplied,itm(10%)(t) − V olimplied,atm(t))
+ β3∗ (V olimplied,otm(6%)(t) − V olimplied,atm(t)) + ε(t)
(3)
Initially only 2 options, other than the at-the-money, are chosen instead of all 6 in order to find out if there is at least some minimal extra predictive value on the most extreme sides of the volatility skew.
After executing these regressions, a test will be performed to show whether the in-the-money and out-the-in-the-money implied volatilities have a jointly significant effect on the realized volatility additional to the at-the-money options.
It will also be tested if the implied volatility skew has a significant effect when the VIX is used instead of the at-the-money implied volatility. Since the VIX already includes implied volatilities extracted from theoretically all options along the implied volatility curve it can be expected that the extra predictive power of the implied volatility skew will not be significant when tested with the VIX instead of the at-the-money options implied volatility. Further checks will finally be performed to determine the robustness of our results.
3
Research
The calculation of the implied volatilities central to this paper are done by using the Black-Scholes formula. For dividend paying stocks the formula is the following:
C(S, K, T, t, δ, σ, r) = e−δ(T −t)S ∗ N (d1) − Ke−r(T −t)N (d2) where d1 = 1 σ√T − t ln S K + r − δ + 1 2σ 2 (T − t) and d2 = d1− σ √ T − t
K is the strike price of the option, S the price of the underlying equity, δ the dividend rate, σ the (implied) volatility and T-t the time to maturity. The risk free rate r is the theoretical rate of return for an investment with zero risk. However in reality there are no investments available that do not carry any risk. Most suitable for approximating the risk free rate are the return rates from short-term government bonds from governments that are unlikely to fail their payment obligations. For that reason the rate of return of the 3-month time-to-maturity bond issued by the United States government is often considered as the risk free interest rate.
N(.) is the cumulative standard normal distribution, N (t) = √1 2π
Rt
−∞e
−s2 2 ds.
We notice that all the necessary parameters are known and can be obtained from the stock indexes and option contracts except for σ. Therefore σ can be easily calculated using the Black-Scholes formula. However computational methods such as Gauss-Newton need to be used to calculate the value of the implied volatility since the Black-Scholes formula cannot be inverted analytically.
option. Furthermore it is observed that the average value of the realized volatility is almost 0.03 (3 percent points) lower than the at-the-money implied volatility. A double-sided t-test to check whether the means are statistically different from each other has a p-value of 0.0105, thus showing that the difference is significant at a 5% level. The difference between the two volatilities is not surprising and is a widely documented phenomenon called the volatility risk premium.
Table 1: Descriptive Statistics
The standard deviation of the volatilities is respectively 0.107, 0.098, 0.112, 0.100, 0.1331, 0.103, 0.098, 0.089 for the realized, at-the-money, 5% money, 7.5% in-the-money, 10% in-the-in-the-money, 2% out-the-in-the-money, 4% out-the-money and 6% out-the-money op-tions. Furthermore it is observed that the deeper the option is in-the-money, the higher the implied volatility becomes. This thus shows the existence of a volatility skew where volatility decreases when strike price increases, similar to that predicted by the literature. The volatility skew is also represented visually. This is done by taking a subset of the data, selecting only the months for which the at-the-money option implied volatility is below 11%, and plotting all the different options of that particular month. Figure 2 displays the skews.
Several graphs almost similar to what would be expected for the volatility skew are displayed in the figure. However, the lowest volatility in most months is found at the 4% out-the-money options instead of the expected 6% out-out-the-money options. But these results cannot be generalized for the complete dataset since in 48 out of 85 months the 6% out-the-money op-tion implied volatility is the lowest compared to 37 out of 85 months for the 4% out-the-money options.
Figure 2: Several Volatility Skews
The correlations between the different regressors in (2) can be found in the correlation matrix displayed in Table 2. It is observed that the regressors are highly correlated. The rea-son for this is that when for instance the at-the-money call options increases in value because of an increase in the value of the underlying asset the 10% the-money option will also in-crease, since it is based on the same underlying asset. Thus all option prices tend to move in the same direction. The correlation between the prices of the two options leads to a correla-tion between the different implied volatilities. The correlacorrela-tion between the at-the-money opcorrela-tion implied volatility and the 10% in-the-money and 6% out-the-money implied volatility is respec-tively 0.789 and 0.976. These results indicate that multicollinearity is highly likely between the different options and thus that it is necessary to use (3) instead of (2) for the regressions.
Table 2: Correlation Matrix
The regression of (1) is performed to see what the predictive value of the at-the-money implied volatilities is. If the coefficient of β differs significantly from zero, then the at-the-money option implied volatility has predictive power for the future realized volatility. Further-more, if the joint hypothesis of α=0 and β=1 cannot be rejected, then the at-the-money option
the regression are shown in the first column of Table 3, with the p-values reported in parenthe-ses.
Table 3: Benchmark Regression Results
The results show that the at-the-money implied volatility is a highly significant estimator of the realized volatility, having a p-value of 0.0000. However, just like in the research of Shu and Zhang, the joint hypothesis of unbiasedness and efficiency of the regressor is rejected with p-value 0.0001.
The results of the regression specified in (3) are displayed in the second column of Table 3. The results still show a high degree of significance for the at-the-money implied volatilities. Furthermore the 10% in-the-money option implied volatility is found to be a significant predic-tor for future realized volatility on a 10% significance level. However the 6% out-the-money and 10% in-the-money option implied volatilities individually are not significant on a 5% con-fidence level. Also when the two options are jointly tested using a Wald-test with H0 that the
coefficient of 10% in-the-money as well as the 6% out-the-money option implied volatility is equal to 0, it is found that the null hypothesis cannot be rejected since the p-value of the test is 0.1329. The options on the volatility skew are thus found to not be a significant predictor for future realized volatility. Furthermore the hypothesis that the at-the-money implied volatility is an unbiased and efficient predictor of the future realized volatility, thus the joint hypothesis that α=0 and that β1=1, cannot be rejected. This implies that all information required for predicting
Robustness Checks
To check the robustness of the results we extend the model in a couple of ways. First we check if the at-the-money implied volatility is still a significant predictor of realized volatility and the options on the volatility skew are still not significant when (3) is extended to include all sampled implied volatilities. The new regression model will then be the following:
V olrealized(t) = α + β1∗ V olimplied,atm(t) + β2∗ (V olimplied,itm(5%)(t) − V olimplied,atm(t)) + β3∗ (V olimplied,itm(7.5%)(t) − V olimplied,atm(t)) + β4∗ (V olimplied,itm(10%)(t) − V olimplied,atm(t)) + β5∗ (V olimplied,otm(2%)(t) − V olimplied,atm(t)) + β6∗ (V olimplied,otm(4%)(t) − V olimplied,atm(t)) + β7∗ (V olimplied,otm(6%)(t) − V olimplied,atm(t)) + ε(t) (4)
The results of the regression are displayed in the Table 4. The p-values are again reported in parentheses. The results do not differ much from the earlier results. The at-the-money option implied volatility again is a highly significant estimator of the implied volatility. And it is again found that the hypothesis that α=0 and β1=1 cannot be rejected, thus the hypothesis that the
at-the-money option implied volatility is an unbiased and efficient estimator for realized volatility cannot be rejected.
Table 4: Regression Results of Extended Regression Model
The implied volatilities from the volatility skew all fail to be significant individually at a 5% significance level. However the 10% in-the-money option implied volatility again is significant at a 10% level. Moreover, the Wald-test used to determine if the volatilities of the volatility skew have predictive worth for realized volatility has a p-value 0.3320, thus the hypothesis that the influence of all volatilities is zero cannot be rejected. These findings give a very weak
indication that the volatility skew does have some predictive value, since the hypothesis that the 10% in-the-money option implied volatility does not have any predictive value again can not be rejected at a 10% significance level.
To check if there is any difference between the predictive value of the implied volatility skew before and after the subprime mortgage crisis the dataset is divided in two subsets. The first subset includes data from May 2006 till August 2008, the month were the subprime crisis started having major consequences leading to the bankruptcy of several major financial firms, such as Lehman Brothers. The subprime mortgage crisis can be seen in the data by a sudden spike in the implied volatilities from August 2008 till June 2009, almost quadrupling compared to the mean value of the realized volatility. The second subset includes data from July 2009 till May 2013. The months were the subprime crisis was most intense are excluded from the dataset, since in a major crisis the normal rules of the economy do not apply and including these months might distort the results.
On the subsets the regression from (1), (3) and (4) are executed and the results are displayed in Table 5. It should be noted that the amount of observation in every subset is relatively small, containing only 27 observations in the first subset and 47 in the second.
The results show that the predictive value of the at-the-money option implied volatility is significant for both subsets when tested following (1), however the predictive value is larger before the crisis. When the regression on (4) is performed, only the first subset still maintains a significant predictive value for the at-the-money implied volatility. In the subset of the data from after the crisis the at-the-money implied volatility loses its predictive significance. An-other result shared in both time periods is that the implied volatility skew has no predictive power. Not only are all options that lie on the volatility skew individually not significant on a 5% level, but also the joint hypothesis that together they have no predictive value is not rejected for all different regressions in both the time frames. Furthermore the 10% in-the-money option implied volatility is only significant on a 10% level in the regression of (3) in the first time period. Moreover the assumption of unbiasedness and efficiency cannot be rejected when (3) and (4) are performed in the first time period. In all other regressions this hypothesis is however rejected.
Table 5: Regression Results of Subperiod Analysis
VIX and by regressing the realized volatility on the VIX as well as the 10% in-the-money option implied volatility and the 6% out-the-money implied volatility. The results of these regressions are displayed in Table 6 with the p-values shown in parentheses.
The VIX has a highly significant predictive power when it is the only regressor, as shown in the first column of Table 6. However the hypothesis that it is an unbiased and efficient predictor is rejected.
In column two of the table the realized volatility is regressed on the VIX as well as the 6% out-the-money option implied volatility and the 10% in-the-money implied volatility. The results again show a highly significant predictive value of the VIX. It is noteworthy that the hypothesis that the 10% in-the-money option implied volatility is not a significant predictor for the realized volatility is rejected. The hypothesis that the whole volatility skew is not a significant predictor of future realized volatility is also rejected. These results are unexpected since the VIX already includes information of options over all strike prices.
The last test for the robustness of the results will check if the results still hold when the implied volatility is corrected for the measurement errors. To correct for these errors an instrument variable (IV) procedure needs to be followed. Therefore a regressor needs to be
Table 6: Regression Results of the Volatility Index Regressions
specified that satisfies the requirements for an instrument. Past month’s implied volatility is possibly a good instrument since it is likely correlated with current implied volatility as well as having no correlation with the measurement error of εt the sampled implied volatilities one
month later. The Hausman χ2(1)-statistic to test the presence of measurement errors is 6.54, thus the H0 hypothesis that measurement errors are not present at a 5% significance level is
rejected. IV estimation is therefore justified.
Table 7: Instrumental Variables Regressions
rejected. The results of the regression of (3) are displayed in the second column of Table 7 and again show a highly significant predictive value for the at-the-money option implied volatility. Furthermore the hypothesis that the at-the-money implied volatility is an efficient and unbiased predictor of future realized volatility cannot be rejected. The null hypothesis that the volatility skew does not have any predictive power is again not rejected.
4
Conclusions
The question that was researched in this paper was if the options on the implied volatility skew have additional predictive value for the future realized return on top of that of the at-the-money option implied volatility.
Recent literature by Baruschi and Jackwerth implied that not all information required for the derivation of the future realized volatility was subsumed within the at-the-money implied volatil-ity and that thus away-from-the-money options might be required for better predictions. In this article I have therefore tested this predictive value of the option implied volatility from S&P 500-index options with strike prices from across the volatility skew using the Black-Scholes formula.
The results of the regressions performed show that the hypothesis that the implied volatility skew does not contain any predictive value for future realized volatility, cannot be rejected at a 5% significance level for almost all of the regressions. These results were found when the volatility skew was represented by two options as well as when represented by six options. It was also found that there is no difference in predictive value between time periods before and after the subprime mortgage crisis. However when the VIX was used instead of the at-the-money option implied volatility the null hypothesis of no predictive power for the volatil-ity skew was rejected. This is unexpected since the VIX uses the model-free implied volatilvolatil-ity and should thus already include the possible extra predictive value of options with different strike prices. Further research is required to determine if these results are merely coincidental or due to any misspecifications within the VIX. Lastly when corrected for errors-in-variables by using IV estimation the results showed a strong predictive value for the ATM implied volatility and none for the implied volatility of the volatility skew.
The conclusion of all my results taken together is that there are weak indications of some predictive power of the volatility skew for future realized volatility. Although the hypothesis that the away-from-the-money option implied volatilities do not have any predictive power in-dividually could not be rejected for all different strike prices in all different regressions at a 5%
these indications for predictive power of the volatility skew are only weak is remarkable since more advanced calculation methods for the implied volatility, such as the model-free implied volatility, do use the whole range of strike prices.
The internal validity of the research might be questioned because of its relatively small amount of data points. Also the calculation method using the Black-Scholes formula might be a detrimental factor for the attainment of correct results. However, even though the results might be distorted for these reasons, the results found in this paper should not be seen as meaning-less and should neverthemeaning-less be interpreted as bringing serious doubt to the significance of the implied volatility of the volatility skew for future realized volatility.
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