Is Systematic Risk Priced in Options?, Slides of French

Download Is Systematic Risk Priced in Options? and more Slides French in PDF only on Docsity! Is Systematic Risk Priced in Options? Jin-Chuan Duan and Jason Wei∗ (January 17, 2006) Abstract In this empirical study, we challenge the prevalent notion that systematic risk of the underlying asset has no effect on option prices as long as the total risk remains fixed, a long cherished prediction of the Black-Scholes option pricing theory. We do so by examining two testable hypotheses relating both the level and slope of implied volatility curves to the systematic risk of the underlying asset. Using daily option quotes on the S&P 100 index and its 30 largest component stocks, we show that after controlling for the underlying asset’s total risk, a higher amount of systematic risk leads to a higher level of implied volatility and a steeper slope of the implied volatility curve. The findings are robust to various alternative specifications and estimations. Our empirical conclusions turn out to be consistent with the newly emerged GARCH option pricing theory. JEL classification code: G10, G13 ∗Both authors acknowledge the financial support from the Social Sciences and Humanities Research Council of Canada. We are grateful to N. Kapadia, G. Bakshi and D. Madan for supplying the data set and thank Baha Circi and Jun Zhou for their research assistance. We also thank seminar/conference participants at McMaster University, Queen’s University, CFTC, Institute of Economics of Academia Sinica, and the 2005 annual meeting of the Northern Finance Association for their comments. Both authors are with the Joseph L. Rotman School of Management, University of Toronto. Duan’s email: jcduan@rotman.utoronto.ca; Wei’s email: wei@rotman.utoronto.ca. Is Systematic Risk Priced in Options? Abstract In this empirical study, we challenge the prevalent notion that systematic risk of the underlying asset has no effect on option prices as long as the total risk remains fixed, a long cherished prediction of the Black-Scholes option pricing theory. We do so by examining two testable hypotheses relating both the level and slope of implied volatility curves to the systematic risk of the underlying asset. Using daily option quotes on the S&P 100 index and its 30 largest component stocks, we show that after controlling for the underlying asset’s total risk, a higher amount of systematic risk leads to a higher level of implied volatility and a steeper slope of the implied volatility curve. The findings are robust to various alternative specifications and estimations. Our empirical conclusions turn out to be consistent with the newly emerged GARCH option pricing theory. JEL classification code: G10, G13 risk. In fact, we argue that the implied volatility, risk-neutral skewness, and kurtosis are all tied to the systematic risk. Thus, finding that the risk-neutral skewness and kurtosis are capable of explaining variations in the implied volatility can be expected. We use option quotes for the S&P 100 index and its 30 largest component stocks from January 1, 1991 to December 31, 1995, a data set identical to the study by BKM (2003). The key variable employed in our study is the systematic risk proportion, which is defined as the ratio of the systematic variance over the total variance. We test two specific null hypotheses: (1) the level of implied volatility is not related to the systematic risk proportion, and (2) the slope of the implied volatility curve is not related to the systematic risk proportion. Both hypotheses are strongly rejected, indicating that the systematic risk plays an important role in determining option prices. Our empirical findings are robust in sub-samples and to differ- ent specifications and estimations. Interestingly, these empirical results are consistent with the GARCH option pricing model, which predicts that a higher systematic risk proportion leads to (1) a higher level of implied volatility and (2) a steeper negative slope in the implied volatility smile/smirk curve. The remainder of this paper is organized as follows. Section 2 lays out the hypotheses and testing procedures, and reports the main results. The data and test results are given in three subsections. Various robustness checks are reported in Section 3. The GARCH option pricing theory and its specific predictions concerning systematic risk are discussed in Section 4. Section 5 concludes the paper. 2 Empirical relation between systematic risk of the un- derlying asset and option prices According to the Black-Scholes (1973) option pricing theory, option prices do not depend on how much systematic risk is contained in the underlying asset as long as its total risk is fixed. To illustrate, imagine two stocks that are identical in every aspect except for the level of systematic risk or risk premium. The prices of options on these two stocks must be equal if the terms of the options are identical. When these option prices are converted into implied volatilities, they should not be related to systematic risk at all.2 It is difficult to find two stocks that are identical in every respect except for the systematic risk. In the 2Here we distinguish the general Black-Scholes option pricing theory from the specific Black-Scholes formula which is valid only under the geometric Brownian motion assumption. In other words, one can actually have the volatility smile/smirk phenomenon under the general Black-Scholes option pricing theory by discarding the geometric Brownian motion assumption. 3 empirical analysis, we must therefore control for the difference in total risk in studying the option pricing behavior across different underlying stocks. The key variable used in differentiating stocks in terms of systematic risk is the systematic risk proportion. For the j-th stock, we define its systematic risk proportion bj as the ratio of the systematic variance over the total variance. The two testable hypotheses based on the general Black-Scholes option pricing theory are formalized as follows: • Hypothesis 1: The implied volatility level of the options on the j-th stock is unrelated to the systematic risk proportion bj. • Hypothesis 2: The slope of the implied volatility smile/smirk curve of the options on the j-th stock is unrelated to the systematic risk proportion bj. Several empirical issues need to be sorted out before we proceed to the tests. To begin with, how do we estimate the average volatility, or the overall level of total risk? Since we use the Black-Scholes implied volatility to characterize the option pricing structure, it is natural to use some versions of historical volatility to proxy the future average volatility. The key issue is how far back we should go in estimating the historical volatility. Balancing between estimation efficiency from a larger sample and the relatively shorter options maturities in the data sample, we opt for a one-year (250 days) rolling window in calculating the volatility on a daily basis. Later in the robustness checks, we repeat the tests using a five-year rolling window and a weekly frequency. Another issue is the empirical characterization of the implied volatility curve. BKM (2003) assumed a constant slope on the logarithmic scale for the curve. While this strategy greatly simplifies the testing procedures and enhances the testing power (by lumping more observations together), it tends to mix the intricate features of the curve in different regions of the moneyness spectrum. To reveal potentially different features for different moneyness re- gions, we piecewise linearize the implied volatility curve into four distinct moneyness buckets, i.e., K/S = [0.9, 0.95), [0.95, 1.0), [1.0, 1.05) and [1.05, 1.10], and conduct tests within each bucket. As discussed earlier, we use time series of daily returns to estimate the systematic risk proportion. Specifically, we run daily, one-year rolling window, OLS regressions for stock j: Rjt = αj + βjRmt + ξjt, (1) from which the systematic risk and total risk can be calculated as β2jσ 2 m and σ 2 j . The system- atic risk proportion is simply bj ≡ β2jσ 2 m/σ 2 j for a particular day, which can in this case be 4 viewed as the regression R2. If we need a measure of systematic risk proportion for a period of, say, 4 weeks, we need to somehow average the daily estimates. In our study, we first average the daily variances over the period, and then calculate a bj. For robustness checks, we later repeat the tests by first computing the daily proportions and then averaging them over the period in question. To test our hypotheses, we follow BKM (2003) and perform the Fama-MacBeth (1973) type two-pass regressions. We need to obtain time series of estimates for the level and slope of the implied volatility curve, which are used to run the cross-sectional regressions to determine whether they are related to the systematic risk proportion. The cross-sectional regression is repeated over time and the time-averaged regression coefficients are used to determine whether a hypothesis is rejected or not. In order to estimate the level and slope of the implied volatility curve in the first-pass regressions, we need to decide on the length of non-overlapping regression windows. While a weekly window provides sufficient number of options in the study by BKM (2003), we must increase the window length because the option data have been further divided into four moneyness buckets. This is particularly necessary in ensuring reasonable estimates for the risk-neutral skewness and kurtosis. We adopt a window of one month (4 weeks). Thus, the second-pass regression (for testing the effect of the systematic risk proportion on the level and slope of the implied volatility curve) is performed on a monthly basis. The risk-neutral skewness and kurtosis are estimated in the same way as in BKM (2003). With the above in mind, we proceed with hypothesis testing as follows. In the first-pass regression, for each stock and moneyness bucket, we lump all the observations in a four-week period and repeat the following regression for the j-th stock: σimp jk − σhisj = a0j + a1j(yjk − ȳj) + εjk, k = 1, 2, ..., Ij, (2) for 65 times (260 weeks divided by 4). In the above, Ij is the number of options in a particular moneyness bucket for the j-th stock, yjk = Kjk/Sjk, and ȳj is the sample average of yjk.The intercept α0j and regression coefficient a1j are measures of the level and the slope of the implied volatility for a particular moneyness bucket, after adjusting for the j-th stock’s total risk, σhisj . 3 In the second pass, we perform three versions of cross-section regressions for each of the 65 non-overlapping periods using the intercept from the first-pass regressions as the dependent 3Historical volatility for the j-th stock is actually day-specific. The time subscript is omitted to simplify notation. The moneyness variable yjk is adjusted by its mean to ensure that the intercept α0j is the average difference between the implied volatility and the historical volatility for each month/bucket. 5 present for all stocks. The curve is downward sloping for most stocks when the option maturity is medium-term (71 − 120 days) or long-term (121 − 180). However, for short- term options (20− 70 days), the implied volatility tends to curve up in the last moneyness bucket, K/S = 1.05− 1.10. Second, it is apparent that, within the same moneyness bucket, the implied volatility is generally lower for longer term options. Third, the average implied volatility and the average historical volatility are generally close, and the former is higher than the latter for more than half of the stocks (19 out of 30), reinforcing the third bias mentioned at the beginning of the paper, i.e., implied volatilities are usually higher than the historical volatilities. The S&P 100 index has the highest volatility differential which is 0.0327. Finally, excluding the S&P 100 index, the systematic risk proportions range from 0.089 for MCI Communications to 0.380 for General Electric (GE). The average proportion across all stocks excluding the S&P 100 index is 0.235. To see the general association between the stocks’ key characteristics and the systematic risk proportion, we sort the stocks into quintiles by their systematic risk proportions, and calculate the average value of the characteristic variables for each quintile. The variables we examine are the ones used for later tests, namely, a) the average implied volatility minus the average historical volatility, b) the average slope of the implied volatility curve, c) the average risk-neutral skewness, and d) the average risk-neutral kurtosis. Since the last two variables do not change across moneyness, we only divide the sample into maturity buckets. Given the magnitude of the S&P 100 index’s systematic risk proportion, we put it in a separate group, quintile 5. The first quintile contains 6 stocks and the other three contain 8 stocks each. Since the estimations are done monthly as described before, the sorting is also done monthly, and the average variables are calculated for each quintile. We then average the monthly quantities for each quintile over 65 months. Table 1C contains the results. The most striking is the association between the systematic risk proportion and the implied volatility differential. A higher systematic risk proportion is associated with a higher implied volatility differential. For the other three variables, although not entirely monotonic, we see a clear positive association between the systematic risk proportion and the magnitude of the slope of the implied volatility curve, the risk-neutral skewness and kurtosis.6 Therefore, the sorting results already indicate a strong rejection of the two null hypotheses. Finally, before proceeding to the formal tests, we carry out two preliminary investiga- tions. First, we perform a crude parametric test of Hypothesis 1. Second, we demonstrate why the systematic risk proportion is a better measure than beta for our tests. To this end, 6One should not be alarmed by the seemingly smaller skewness and kurtosis of the index for the long-term maturity. This is mainly due to the lack of enough observations, as apparent in Table 1A. 8 we first regress the difference between the average implied volatility and the average histori- cal volatility on the average systematic risk proportion; we then do the same regression using average beta as the explanatory variable. The average volatilities and systematic risk propor- tions are from Table 1B. Average betas are calculated separately. OLS regressions are done for the entire sample and for various moneyness and maturity buckets. For each bucket, we run two versions of the regression: one with the S&P 100 index and the other without. The results are reported in Table 2. The R2 and t-values overwhelmingly show that the adjusted implied volatilities are positively related to the systematic risk proportions, while having no statistical relation to betas. This observation applies to all moneyness/maturity buckets, with or without the index. Thus, Hypothesis 1 is rejected with a high level of confidence. The fact that beta is not a good measure of systematic risk for our purpose is not surprising. A higher beta doesn’t always mean that the systematic risk accounts for most of the total risk. By the same token, equal betas doesn’t mean equal systematic risk proportions. This point can be illustrated by a simple example. Suppose the market volatility is σm = 0.2 and there are two stocks, A and B, with σA = 0.4 and σB = 0.5. If the stocks’ correlations with the market are ρA = 0.75 and ρB = 0.60, then the two stocks will have the same beta, 1.50, yet very different systematic risk proportions, 0.563 versus 0.360. 2.2 Level effect tests We now proceed to the formal tests. Table 3 reports the test results for the level effect, i.e., tests pertaining to Hypothesis 1. To conserve space, we omit the intercepts from the second-pass regressions. Panel A reveals a strong rejection of Hypothesis 1. The coefficient γ1 is positive across all moneyness and maturity buckets, and all the corresponding t-values save one are significant. In fact, almost all of them are significant at the 1% level. Not only positive on average, the vast majority of the 65 γ1 estimates are positive, as indicated by the percentages under γ1 > 0. Moving to Panel B where we control for the effects of the risk- neutral skewness and kurtosis, the γ1 estimates are still significant for most of the moneyness and maturity buckets. Comparing with the unconditional tests in Panel A, the significance level for the lower moneyness range (K/S = 0.9− 1.0) goes down slightly. Nonetheless, just as the unconditional tests in Panel A, only one t-value is insignificant, and almost all of them are significant at the 1% level. Overall, the unconditional and conditional tests both show a strong level effect. The implied volatility levels, controlling for the stock specific total volatilities, are significantly and positively related to the systematic risk proportion of the underlying stock. 9 In terms of economic significance, the R2 shows that the systematic risk proportion does a better job for the lower moneyness range in explaining the cross-sectional differences in the level of implied volatilities. For the univariate regressions covering all maturities, the systematic risk proportion alone explains 14.5%, 7.8%, 7.3% and 5.4% of the cross-sectional variations in the implied volatility for the four moneyness buckets respectively. When the risk-neutral skewness and kurtosis are added to the regressions, the corresponding numbers are 24.8%, 18.8%, 17.9% and 15.2%. Obviously, the implied volatilities are also affected by many other firm-specific variables not examined in this study, such as the ones examined by Dennis and Mayhew (2002). The focus of this paper is to establish the linkage between the option prices and the systematic risk. We therefore do not go further to exhaustively investigate all the potential factors affecting the implied volatility. The regression results also offer some other interesting insights. First of all, judging by the magnitude and t-value of the regression coefficient γ1 as well as the percentage of positive entries, we see that the effect of systematic risk proportion itself also takes a smirk pattern across moneyness. The effect is much stronger for the lower moneyness buckets. As the exercise price becomes higher, the level effect becomes weaker. This is consistent with the pattern of the implied volatilities. Second, in terms of maturities, it is clear that the effect is stronger for short-term (20−70 days) options, and it becomes weaker as the maturity gets longer. This is true for both the unconditional and conditional tests. The fact that the long-term options see the weakest effect is remarkably consistent with the predictions of the GARCH option pricing theory (viz, the implied volatility curve flattens out for very long-term options), a point to be addressed later in Section 4. Finally, in both the unconditional and conditional tests, the coefficients for the risk- neutral skewness and kurtosis are mostly insignificant and the signs are mixed. Nevertheless, as shown in Panel B, the effect of the systematic risk proportion on the implied volatility level remains significant, even after controlling for the risk-neutral skewness and kurtosis. 2.3 Slope effect tests Table 4 reports the results for the slope effect tests, i.e., tests pertaining to Hypothesis 2. The results are very similar to those in Table 2 in terms of rejecting the hypothesis. For most parts, the slope of the implied volatility curve is related to the systematic risk proportion in a statistically significant fashion. The bigger the systematic risk proportion, the steeper the slope. The significance remains after controlling for the risk-neutral skewness and kurtosis. 10 both the level and slope effects are weaker with long-term options and for the upper region of the moneyness range (K/S = 1.0 − 1.1). Taken together, the sub-sample tests clearly demonstrate that the impact of systematic risk on option prices are quite robust across sub-sample periods. 3.3 Data frequency and sample size for the systematic risk esti- mation In estimating the historical volatility and its composition, we run the OLS regression in (2) using a one-year rolling window with daily frequency. As mentioned before, our choice of daily frequency and one-year rolling window is a balanced consideration of estimation efficiency and the relatively short maturity of options. However, the shorter window and higher data frequency raise the concern that the resulting risk estimates may be highly time-varying and do not necessarily reflect changes in the systematic risk proportion. This concern may be alleviated by realizing that the risk measure we use in the second pass regression is the ratio of the systematic risk over the total risk and that this ratio may be stable despite the variation in the two absolute risk measures. Nonetheless, in order to assess the potential impact, we repeat the tests using a five-year rolling window at a weekly frequency, a frequency used by such institutions as Datastream and Standard and Poor’s when estimating betas. The weekly frequency is implemented by using data points on Wednesdays. Once we obtain the weekly risk estimates, we match them back to the original data and run the two-pass regressions as before. In other words, we still utilize all available option data. To conserve space, we report the level and slope test results in one table, Table 7. For brevity, we only report the regression coefficient and its t-value together with the R2 for the univariate regression (with the systematic risk proportion being the only explanatory variable) and the multivariate regression (with the risk-neutral skewness and kurtosis as well as the systematic risk proportion as the explanatory variables). It is clear from the table that the previous conclusions hold up for both the level effect and the slope effect. Overall, the statistical significance weakens slightly, but this is by no means uniform. In some cases, the t-values actually go up slightly. Since the longer window period and the lower data frequency do not alter the qualitative conclusions, for consistency and ease of comparison, we will continue to use the one-year rolling window at the daily frequency for subsequent robustness checks.7 7We have also re-run the regressions using a five-year rolling window at daily frequency. The results are virtually the same as those using a one-year rolling window. We omit the results for brevity. 13 3.4 Exclusion of the S&P 100 index Table 1B clearly shows that the S&P 100 index is an underlying asset with a substantially higher systematic risk than ordinary stocks. Although Table 2 crudely demonstrates that the general association between implied volatilities and the systematic risk proportion holds no matter whether the index is included or not, it is useful to ascertain the precise influence of the index on the overall conclusions. To this end, we repeat the tests by excluding all options written on the index. Again, to conserve space, we report the results in Table 8 in the same fashion as in Table 7. Comparing Panel A with Table 3, it is seen that the t-values for γ1 decrease slightly for most cases, but the significance remains in all cases. Therefore, the level effect also holds strongly with stock options. Comparing Panel B of Table 8 with Table 4, we see that the regression coefficient γ1 retains the right sign, but its significance has reduced substantially. Some t-values are still significant and many have their magnitudes larger than one. Thus, the slope effect is weaker among stock options. This is consistent with the well- established empirical regularity: the slope of the implied volatility curve for stock options is much flatter than that for index options. The flatter slope impedes the testing power in our case. Nevertheless, the results in Table 8 demonstrate that our general conclusions hold with or without index options. In fact, one may even argue that the stronger influence of the index options actually reinforces our conclusions, since the index has the highest systematic risk proportion and the largest difference between the implied and the historical volatilities, as apparent in Table 1B. At any rate, for consistency and ease of comparison, we keep index options in the analysis for subsequent robustness checks. 3.5 Panel regressions In our two-pass regressions, we use an estimated parameter from the first pass as a dependent variable for the second pass. This may give rise to several econometric issues such as the asymptotic properties of the second-pass estimators, which in turn could cast doubt about the statistical inferences we have drawn. To address this concern, we run a single-pass, panel regression and test the two hypotheses therein. Specifically, we run the following panel regression for each moneyness/maturity bucket: σimp ij − σhisij = [α0 + α1(bij − b̄i)] + [β0 + β1(bij − b̄i)](yij − ȳj) + εij = α0 + α1(bij − b̄i) + β0(yij − ȳj) + β1(bij − b̄i)(yij − ȳj) + εij, (6) where b̄i is the observation—weighted, cross-sectional average of the systematic risk proportion for each day, ȳi is the sample average of moneyness for stock j or the index within the bucket. 14 Broadly speaking, α0 can be understood as the average differential between the implied volatility and the historical volatility over all stocks and the index within the entire sample period. Similarly, β0 can be understood as the average slope of the implied volatility curve. They are not exactly the said quantities due to the interaction term bij ∗ yij. The coefficient α1 picks up the level effect. If the systematic risk proportion doesn’t affect the price level or the adjusted implied volatility, then α1 should not be different from zero, statistically speaking. A positive α1 would confirm the level effect. By the same token, the coefficient β1 picks up the slope effect. If the systematic risk proportion does not affect the slope of the implied volatility curve, then β1 should be zero. A negative β1 would imply that a stock with a higher than average systematic risk proportion will have a slope steeper than the average slope of all implied volatility curves, confirming the slope effect. Table 9 contains the results. Judging by the t-values of the coefficient α1, Hypothesis 1 is rejected at an extraordinary level of significance, reaffirming the level effect. As for the coefficient β1, except for three cases, the t-values are also significant and large for many cases. Therefore, Hypothesis 2 is also rejected, confirming the slope effect. If anything, the panel regression results indicate that our two-pass regression tests err on the conservative side. We have also repeated the tests by calculating b̄i as the simple average of the stocks’ systematic risk proportions (i.e., not weighted by the number of observations). The results are almost identical to those in Table 9.8 3.6 Systematic risk estimation using Fama-French factors So far, all the tests use systematic risk estimates from a single factor model, the market model. Insofar as different stocks may have different exposures to certain systematic risk factors, it is imperative to ascertain if our results are robust to the multi-factor model. To this end, we re-estimate the systematic risk by adding the two Fama-French factors, i.e., SMB and HML, to the market factor.9 By definition, the systematic risk proportion estimated with the two additional factors will be higher than the previous one. The question is, will it increase proportionally across stocks so that our level and slope effects would hold up? To this end, we repeat the two-pass regressions using the newly estimated systematic risk 8Incidentally, it is seen that the coefficient α0 is negative for the moneyness measure K/S beyond 1.0. This should be intuitive given the downward sloping feature of a typical implied volatility curve: implied volatilities in the moneyness range beyond 1.0 are lower than the average volatility at the mid-point or the at-the-money point. 9The daily factors are downloaded from the web-page of Kenneth French. 15 three implied volatility curves by varying the level of the asset risk premium. The GARCH parameters used to generate these graphs are α0 = 8 × 10−6, α1 = 0.85, α2 = 0.08 and θ = 0.5. These parameter values imply a physical stationary return volatility of 20% per annum. We assume that the initial conditional volatility is at this stationary level and then let it evolve according to the GARCH system. We compute the option prices using 50,000 sample paths in a Monte Carlo simulation coupled with the empirical martingale adjustment of Duan and Simonato (1998). We fix the maturity at 60 business days while varying the strike price. Once the option prices are computed, they are converted to the Black-Scholes implied volatilities. It is evident from these graphs that the GARCH option pricing model yields the smile/smirk pattern typically observed for the exchange-traded option contracts. These graphs show that a higher λ (due to a higher systematic risk without changing the total physical risk) leads to a higher implied volatility curve across all strike prices, which implies a positive level effect. A higher λ also makes the curve sloped more steeply, which is clearly a negative slope effect. The above predictions in effect reflect the fact that a higher risk premium simply leads to higher volatility and kurtosis and more negative skewness for the risk-neutral cumulative return distribution. Assuming a constant λ > 0 and leverage effect (θ > 0), Theorem 3.1 of Duan (1995) can be used to conclude that the risk-neutral stationary return variance is α0/[1− α1 − α2(1 + (θ + λ)2)], an increase from α0/[1− α1 − α2(1 + θ2)] under measure P . A higher risk-neutral volatility is actually due to the fact that the risk-neutral volatility is governed by a larger persistence parameter α1 + α2(1 + (θ + λ)2) in comparison to the one under measure P (i.e., α1+α2(1+θ2)). The risk-neutral volatility dynamic thus has a slower mean-reversion, implying a slower flattening of the volatility smile/smirk. Furthermore, the risk-neutral cumulative return becomes more skewed because the correlation between the one-period return and volatility becomes −2α2(θ + λ) as opposed to −2α2θ under measure P . The risk-neutral cumulative return also has fatter tails than the cumulative return under the physical measure. We illustrate these features in Figures 3-5 using the same parameter values as in Figure 2. In these plots, we compute the risk-neutral cumulative return’s volatility, skewness and kurtosis for different maturities using the same simulation procedure as for Figure 2. The volatilities in Figure 3 are annualized in the usual manner, i.e., dividing by the square root of the maturity (in years). When λ = 0, the risk-neutral cumulative return volatility equals the physical volatility. Since the initial conditional volatility is set to the physical stationary volatility of 20%, it is not surprising to see it staying at 20% for different maturities. When 18 λ > 0, the stationary risk-neutral volatility will be higher than the physical stationary volatility. Naturally, the risk-neutral cumulative return volatility will be increasing with maturity. If we had used an initial conditional volatility much higher than the stationary level, these curves would be downward sloping but the curve corresponding to a higher λ would continue to stay above the one under a lower λ. In short, the risk-neutral volatility is monotonically increasing in λ for a given maturity. Figure 4 indicates that the risk-neutral cumulative return’s skewness is a decreasing function of λ, meaning the risk-neutral distribution becomes more negatively skewed when λ is larger.11 Corresponding to a given λ, we see an interesting pattern in relation to maturity. Since the one-period conditional return distribution is normal, the skewness has to start from zero. The negative correlation between return and volatility leads to a negative skewness for the cumulative return distribution. The skewness will, however, diminish with maturity, a result due to the central limit theorem. Similarly, we observe the risk-neutral cumulative return’s kurtosis increases with λ for a given maturity. If we fix λ and examine how the kurtosis behaves in relation to maturity, it is clear that the risk-neutral kurtosis begins at 3, i.e., conditional normality, and then increases with maturity to a point (i.e., a maturity of roughly 50 days). After that, it begins to decline toward 3, again due to the central limit theorem. Figures 4-5 suggest that under the GARCH option pricing theory, the risk-neutral skew- ness and kurtosis are functions of λ, which is in turn a function of the systematic risk proportion b. We now empirically verify this claim. We regress cross-sectionally the risk- neutral skewness and kurtosis of 30 companies and the S&P100 index on their systematic risk proportion. That is, for j = 1, · · · , 31, Skew (rn) j = γ0 + γ1bj + ej (10) Kurt (rn) j = γ0 + γ2bj + ej (11) Again we run the cross-sectional regressions on a monthly basis as in Section 2 and report the average regression coefficients over all months in our sample. The t-statistics are computed from these monthly regression coefficients after taking into account their potential autocor- relation. For the risk-neutral skewness, we find γ̂0 = −1.549 with a t-value of −26.834 and γ̂1 = −1.336 with a t-value of −5.113. This result indicates that the risk-neutral return distributions are on average negatively skewed and the degree of the negative skewness is proportional to the systematic risk proportion. Our regression results for the risk-neutral 11Monte Carlo errors are more evident in Figures 4-5 in comparison with Figures 2-3. The difference in the magnitude is of course due to the fact that skewness and kurtosis are of power 3 and 4. 19 kurtosis are γ̂0 = 3.307 with a t-value of 10.323 and γ̂1 = 6.884 with a t-value of 3.892. This finding suggests that the stocks in our sample have on average leptokurtic risk-neutral return distributions and the kurtosis is increasing with the systematic risk proportion.12 In BKM (2003), the level and slope of the implied volatility curve have been found to be related to the risk-neutral skewness and kurtosis. Our results suggest that the level and slope of the implied volatility curve and the risk-neutral skewness and kurtosis are all largely influenced by the systematic risk proportion. 5 Summary and Conclusions In this study, we empirically examine the relationship between option prices and the sys- tematic risk of the underlying asset. The study is motivated by the realization that the risk premium or systematic risk of the underlying asset may play a role in determining the em- pirically observed difference between the risk-neutral and physical return distributions. The original Black-Scholes option pricing theory assumes a constant volatility which captures the riskiness of the underlying. Although there have been many subsequent studies generalizing the constant volatility assumption and the return distribution in general, almost all of the studies stipulate that volatility is the only measure of the underlying asset’s risk profile. The decomposition of the total risk into systematic and non-systematic risks plays no role in the valuation of options. We empirically invalidates this point in the current study. We show conclusively that option prices are related to the amount of systematic risk. After controlling for the overall level of total risk, a higher amount of systematic risk leads to a higher level of implied volatility and a steeper implied volatility curve. The effect remains strong after controlling for the risk-neutral skewness and kurtosis. The results are also robust to various alternative estimations of the variables and specifications of the tests. In summary, we have shown that the implied volatility smile/smirk phenomenon is predictably related to how the total risk is decomposed into systematic and non-systematic risks, a result that is fundamentally contradictory to the essence of the general Black-Scholes option pricing theory. We offer a potential explanation to the findings using the recently emerged GARCH option pricing theory. When volatility is stochastic and depends on the return innovation, the underlying asset’s risk premium becomes an integral part of option valuation by entering 12The regressions are also repeated by excluding the index. For the skewness regression, γ̂0 = −1.592, t = −13.068, γ̂1 = −0.943, t = −2.842; for the kurtosis regression, γ̂0 = 3.614, t = 6.855, γ̂1 = 4.460, t = 3.332. The t-values do go down, but are still significant. 20 [28] Lehar, A., M. Scheicher and C. Schittenkopf, 2002, “GARCH vs. Stochastic Volatility Option Pricing and Risk Management”, Journal of Banking and Finance, 26, 323-345. [29] Lehnert, T., 2003, “Explaining Smiles: GARCH Option Pricing with Conditional Lep- tokurtosis and Skewness”, Journal of Derivatives, 27-39. [30] Peña, I., G. Rubio and G. Serna, 1999, Why Do We Smile? On the Determinants of the Implied Volatility Function, Journal of Banking and Finance, 23, 1151-1179. [31] Peña, I., G. Rubio and G. Serna, 2001, Smiles, Bid-ask Spreads and Option Pricing, European Financial Management, 7(3), 351-374. [32] Ritchken, P. and R. Trevor, 1999, “Pricing Options Under Generalized GARCH and Stochastic Volatility Processes”, Journal of Finance, 54, 377-402. [33] Stentoft, L., 2005, “Pricing American Options when the Underlying Asset Follows GARCH Processes”, to appear in Journal of Empirical Finance. 23 24 Table 1A: Summary statistics – number of observation All Ticker Stock [0.90-0.95) [0.95-1.00) [1.00-1.05) [1.05-1.10] [0.90-1.10] [0.90-0.95) [0.95-1.00) [1.00-1.05) [1.05-1.10] [0.90-1.10] [0.90-0.95) [0.95-1.00) [1.00-1.05) [1.05-1.10] [0.90-1.10] Options 1. AIG American Int'l 1351 1635 1700 1096 5782 635 640 662 478 2415 706 753 781 547 2787 10984 2. AIT Ameritech 775 1039 1150 558 3522 409 413 455 371 1648 490 478 553 418 1939 7109 3. AN Amoco 761 902 1037 663 3363 442 397 448 399 1686 513 451 558 423 1945 6994 4. AXP American Express 677 745 686 610 2718 236 347 250 224 1057 382 321 336 317 1356 5131 5. BA Boeing Company 871 687 976 703 3237 401 298 454 321 1474 449 364 515 378 1706 6417 6. BAC BankAmerica Corp. 917 688 874 719 3198 328 264 319 302 1213 436 307 405 352 1500 5911 7. BEL Bell Atlantic 855 842 977 728 3402 343 363 346 362 1414 508 347 491 372 1718 6534 8. BMY Bristol-Myers 1103 1127 1252 932 4414 483 512 540 443 1978 610 557 663 559 2389 8781 9. CCI Citicorp 709 677 650 704 2740 258 278 230 274 1040 335 321 290 326 1272 5052 10. DD Du Pont 967 881 942 892 3682 381 387 349 380 1497 512 409 474 445 1840 7019 11. DIS Walt Disney 1205 1276 1271 1199 4951 511 520 524 513 2068 585 621 616 551 2373 9392 12. F Ford Moter 823 788 829 725 3165 393 394 360 380 1527 450 448 444 433 1775 6467 13. GE General Electric 1268 1331 1401 1125 5125 608 576 640 533 2357 706 688 758 614 2766 10248 14. GM General Motors 894 780 794 911 3379 419 334 362 424 1539 467 433 426 465 1791 6709 15. HWP Hewlett-Packard 1256 1254 1294 1121 4925 601 550 588 490 2229 667 642 663 564 2536 9690 16. IBM Int. Bus. Machines 1294 1300 1467 1287 5348 517 508 579 504 2108 663 625 684 606 2578 10034 17. JNJ Johnson & Johnson 1105 1017 1142 879 4143 459 406 435 360 1660 611 505 546 432 2094 7897 18. KO Coca Cola Co. 1052 952 952 892 3848 458 476 400 433 1767 600 464 502 536 2102 7717 19. MCD McDonald's Corp. 896 563 914 685 3058 379 284 392 334 1389 506 292 511 352 1661 6108 20. MCQ MCI Comm. 741 555 628 697 2621 290 258 218 299 1065 382 251 319 312 1264 4950 21. MMM Minn Mining 1268 1519 1548 1281 5616 568 598 603 549 2318 663 719 703 660 2745 10679 22. MOB Mobil Corp. 1037 1277 1376 901 4591 630 645 626 551 2452 741 690 758 676 2865 9908 23. MRK Merck & Co. 1189 1177 1178 889 4433 515 497 473 369 1854 610 542 529 425 2106 8393 24. NT Northern Telecom 659 576 675 565 2475 272 214 254 235 975 341 249 328 267 1185 4635 25. PEP PepsiCo Inc. 692 740 611 718 2761 252 285 240 288 1065 336 359 269 349 1313 5139 26. SLB Schlumberger Ltd. 893 1069 1119 907 3988 382 430 451 377 1640 444 507 492 463 1906 7534 27. T AT&T 969 739 992 807 3507 392 261 415 280 1348 459 362 416 435 1672 6527 28. WMT Wal-Mart 973 714 786 699 3172 508 285 398 300 1491 527 429 438 370 1764 6427 29. XON Exxon Corp. 1044 1000 1151 875 4070 462 427 414 435 1738 587 442 526 539 2094 7902 30. XRX Xerox Corp. 1333 1520 1569 1202 5624 563 584 599 521 2267 687 687 730 596 2700 10591 31. OEX S&P 100 Index 8206 8707 8766 3814 29493 5149 5997 6035 2597 19778 117 146 152 77 492 49763 Total 37783 38077 40707 29784 146351 18244 18428 19059 14326 70057 16090 14409 15876 13859 60234 276642 Long-term Options: 121 - 180 days in Maturity Moneyness, K/SMoneyness, K/S Short-term Options: 20 - 70 days in Maturity Medium-term Options: 71 - 120 days in Maturity Moneyness, K/S Notes: This table reports the number of observations within each moneyness bucket under a particular maturity range for options on the 30 largest component stocks in the S&P100 index and on the S&P100 index itself. Each observation is the last quote prior to 3:00pm (CST). The far right column under each maturity range is simply the sum of the preceding four columns. The last column of the table contains the total number of observations for each firm. The last row contains the total of each column. The sample period is from January 1, 1991 to December 31, 1995. All options are American style. 25 Table 1B: Summary statistics – implied volatility, historical volatility and systematic risk proportion Average Average Systematic Implied Historical Risk [0.90-0.95) [0.95-1.00) [1.00-1.05) [1.05-1.10] [0.90-1.10] [0.90-0.95) [0.95-1.00) [1.00-1.05) [1.05-1.10] [0.90-1.10] [0.90-0.95) [0.95-1.00) [1.00-1.05) [1.05-1.10] [0.90-1.10] Volatility Volatility Proportion 1. AIG American Int'l 0.2371 0.2281 0.2125 0.2146 0.2231 0.2282 0.2277 0.2126 0.2125 0.2207 0.2253 0.2268 0.2109 0.2099 0.2187 0.2214 0.2093 0.275 2. AIT Ameritech 0.2226 0.2056 0.1710 0.1806 0.1941 0.2189 0.2176 0.1684 0.1664 0.1928 0.2233 0.2273 0.1602 0.1583 0.1923 0.1933 0.1824 0.229 3. AN Amoco 0.2197 0.1927 0.1717 0.1910 0.1920 0.2003 0.1978 0.1676 0.1715 0.1842 0.2020 0.2028 0.1660 0.1662 0.1841 0.1879 0.1922 0.127 4. AXP American Express 0.3140 0.2935 0.2868 0.3009 0.2986 0.3060 0.2962 0.2979 0.3064 0.3010 0.3047 0.2948 0.2898 0.2959 0.2966 0.2986 0.2995 0.207 5. BA Boeing Company 0.2734 0.2539 0.2372 0.2481 0.2528 0.2563 0.2537 0.2316 0.2343 0.2434 0.2528 0.2498 0.2302 0.2292 0.2401 0.2473 0.2408 0.165 6. BAC BankAmerica Corp. 0.3078 0.2924 0.2664 0.2662 0.2838 0.2977 0.2989 0.2632 0.2588 0.2792 0.2929 0.2877 0.2564 0.2515 0.2723 0.2800 0.2700 0.257 7. BEL Bell Atlantic 0.2324 0.2084 0.1794 0.1978 0.2038 0.2219 0.2160 0.1816 0.1788 0.1995 0.2227 0.2227 0.1796 0.1723 0.1995 0.2017 0.2076 0.214 8. BMY Bristol-Myers 0.2304 0.2143 0.1884 0.2039 0.2088 0.2157 0.2110 0.1801 0.1849 0.1979 0.2147 0.2170 0.1783 0.1800 0.1970 0.2031 0.2003 0.290 9. CCI Citicorp 0.3403 0.3156 0.3058 0.3045 0.3168 0.3326 0.3241 0.3105 0.3033 0.3177 0.3279 0.3123 0.3006 0.2982 0.3101 0.3153 0.3357 0.208 10. DD Du Pont 0.2512 0.2430 0.2188 0.2254 0.2347 0.2438 0.2451 0.2134 0.2151 0.2298 0.2433 0.2429 0.2117 0.2112 0.2273 0.2317 0.2211 0.261 11. DIS Walt Disney 0.2975 0.2820 0.2608 0.2603 0.2751 0.2921 0.2835 0.2629 0.2588 0.2743 0.2827 0.2807 0.2568 0.2547 0.2689 0.2733 0.2540 0.268 12. F Ford Moter 0.3200 0.3014 0.2807 0.2867 0.2974 0.3118 0.2974 0.2763 0.2752 0.2906 0.3089 0.3040 0.2723 0.2718 0.2895 0.2936 0.2928 0.237 13. GE General Electric 0.2402 0.2141 0.1849 0.1899 0.2073 0.2257 0.2187 0.1809 0.1788 0.2012 0.2253 0.2216 0.1789 0.1736 0.2002 0.2040 0.1862 0.380 14. GM General Motors 0.3125 0.2918 0.2880 0.2904 0.2960 0.3031 0.2846 0.2875 0.2852 0.2905 0.3008 0.2940 0.2869 0.2864 0.2921 0.2937 0.3010 0.234 15. HWP Hewlett-Packard 0.3323 0.3251 0.3095 0.3121 0.3199 0.3260 0.3232 0.3094 0.3094 0.3173 0.3127 0.3154 0.2935 0.2980 0.3051 0.3154 0.3230 0.212 16. IBM Int. Bus. Machines 0.2874 0.2675 0.2589 0.2616 0.2685 0.2787 0.2703 0.2527 0.2513 0.2630 0.2696 0.2647 0.2453 0.2452 0.2562 0.2642 0.2544 0.218 17. JNJ Johnson & Johnson 0.2531 0.2406 0.2243 0.2259 0.2363 0.2437 0.2425 0.2205 0.2153 0.2312 0.2416 0.2390 0.2135 0.2112 0.2274 0.2329 0.2336 0.303 18. KO Coca Cola Co. 0.2605 0.2382 0.2157 0.2142 0.2331 0.2403 0.2344 0.2096 0.1987 0.2216 0.2381 0.2334 0.2096 0.1951 0.2193 0.2267 0.2148 0.326 19. MCD McDonald's Corp. 0.2687 0.2416 0.2229 0.2287 0.2411 0.2504 0.2413 0.2236 0.2163 0.2328 0.2513 0.2448 0.2259 0.2219 0.2361 0.2378 0.2255 0.230 20. MCQ MCI Comm. 0.3574 0.3285 0.2983 0.3137 0.3255 0.3368 0.3253 0.2995 0.3051 0.3175 0.3311 0.3208 0.2980 0.3015 0.3134 0.3207 0.4037 0.089 21. MMM Minn Mining 0.2252 0.2044 0.1819 0.1883 0.1992 0.2147 0.2057 0.1761 0.1744 0.1928 0.2106 0.2057 0.1743 0.1721 0.1908 0.1956 0.1783 0.270 22. MOB Mobil Corp. 0.2079 0.1920 0.1675 0.1788 0.1856 0.1928 0.1933 0.1633 0.1625 0.1786 0.1969 0.1966 0.1572 0.1587 0.1773 0.1815 0.1777 0.122 23. MRK Merck & Co. 0.2710 0.2545 0.2392 0.2547 0.2549 0.2579 0.2552 0.2345 0.2413 0.2479 0.2530 0.2491 0.2309 0.2403 0.2439 0.2506 0.2332 0.356 24. NT Northern Telecom 0.3172 0.3013 0.2825 0.2902 0.2979 0.2955 0.2900 0.2766 0.2744 0.2843 0.3057 0.2937 0.2784 0.2809 0.2900 0.2930 0.2764 0.216 25. PEP PepsiCo Inc. 0.2732 0.2302 0.2258 0.2316 0.2404 0.2684 0.2375 0.2320 0.2194 0.2387 0.2533 0.2359 0.2118 0.2143 0.2297 0.2373 0.2438 0.272 26. SLB Schlumberger Ltd. 0.2567 0.2507 0.2344 0.2403 0.2451 0.2474 0.2484 0.2270 0.2241 0.2367 0.2459 0.2495 0.2227 0.2240 0.2356 0.2409 0.2506 0.118 27. T AT&T 0.2319 0.2035 0.1865 0.2019 0.2062 0.2185 0.2034 0.1839 0.1897 0.1990 0.2173 0.2020 0.1855 0.1836 0.1973 0.2024 0.1961 0.260 28. WMT Wal-Mart 0.3022 0.2819 0.2556 0.2698 0.2790 0.2818 0.2843 0.2524 0.2676 0.2716 0.2825 0.2778 0.2614 0.2556 0.2705 0.2749 0.2581 0.349 29. XON Exxon Corp. 0.1991 0.1710 0.1525 0.1649 0.1717 0.1831 0.1726 0.1425 0.1449 0.1613 0.1807 0.1770 0.1424 0.1377 0.1592 0.1661 0.1688 0.166 30. XRX Xerox Corp. 0.2715 0.2623 0.2361 0.2345 0.2512 0.2626 0.2630 0.2303 0.2263 0.2458 0.2612 0.2618 0.2203 0.2198 0.2412 0.2475 0.2333 0.180 31. OEX S&P 100 Index 0.1846 0.1470 0.1162 0.1171 0.1444 0.1716 0.1503 0.1209 0.1136 0.1421 0.1667 0.1523 0.1256 0.1188 0.1422 0.1435 0.1108 0.952 Long-term Options: 121 - 180 days in Maturity Moneyness, K/SMoneyness, K/S Short-term Options: 20 - 70 days in Maturity Medium-term Options: 71 - 120 days in Maturity Moneyness, K/S Notes: This table reports the average implied volatilities within each moneyness bucket under a particular maturity range for options on the 30 largest component stocks in the S&P100 index and on the S&P100 index itself. The third last column of the table contains the average implied volatility for the entire sample, while the second last column contains the average historical volatility over the sample period. The last column contains the average proportion of systematic variance over the total variance. 28 Table 3: Regression tests for the level effect Panel A: Separate Regressions on Systematic Risk Proportion, and Skewness and Kurtosis avg. t γ1 > 0 R2 avg. t avg. t R2 All maturities 0.077 16.061 100.0% 0.145 -0.013 -1.644 -0.002 -1.067 0.111 Moneyness Short-term 0.074 16.233 100.0% 0.159 -0.014 -2.182 -0.001 -0.446 0.155 K/S Medium-term 0.064 14.245 100.0% 0.231 -0.030 -3.285 -0.006 -2.978 0.200 0.90 - 0.95 Long-term 0.104 3.978 79.6% 0.131 -0.009 -0.972 -0.003 -1.377 0.179 Moneyness All maturities 0.051 11.979 100.0% 0.078 -0.011 -1.353 -0.002 -1.207 0.095 K/S Short-term 0.044 11.920 98.5% 0.073 -0.004 -0.680 0.000 0.239 0.113 0.95 - 1.00 Medium-term 0.036 6.364 93.6% 0.083 -0.010 -0.953 -0.003 -1.162 0.213 Long-term 0.032 1.552 63.9% 0.077 -0.004 -0.343 -0.002 -0.767 0.268 Moneyness All maturities 0.047 5.424 98.5% 0.073 0.014 2.158 0.003 2.770 0.094 K/S Short-term 0.037 5.386 96.9% 0.056 0.018 3.235 0.004 3.737 0.120 1.00 - 1.05 Medium-term 0.027 4.786 90.9% 0.090 0.011 1.689 0.001 1.149 0.238 Long-term 0.093 4.164 78.6% 0.149 0.010 1.087 0.001 0.363 0.219 Moneyness All maturities 0.037 4.636 96.9% 0.054 0.014 2.070 0.003 2.507 0.082 K/S Short-term 0.024 4.061 78.5% 0.042 0.008 1.412 0.002 2.024 0.102 1.05 - 1.10 Medium-term 0.023 3.725 84.0% 0.055 0.003 0.265 0.000 0.136 0.197 Long-term 0.051 2.744 71.4% 0.110 0.017 2.742 0.003 1.585 0.236 γ1 γ2 γ3 Panel Β: Combined Regressions on Systematic Risk Proportion, Skewness and Kurtosis avg. t γ1 > 0 avg. t avg. t R2 All maturities 0.088 10.044 100.0% -0.017 -1.521 -0.004 -1.652 0.248 Moneyness Short-term 0.085 4.905 89.2% -0.014 -1.780 -0.003 -1.312 0.287 K/S Medium-term 0.066 11.120 100.0% -0.010 -0.855 -0.001 -0.503 0.408 0.90 - 0.95 Long-term 0.102 3.802 81.8% -0.014 -1.464 -0.005 -1.986 0.301 Moneyness All maturities 0.067 5.536 96.9% -0.014 -1.244 -0.004 -1.640 0.188 K/S Short-term 0.057 3.902 87.7% -0.003 -0.478 -0.001 -0.684 0.191 0.95 - 1.00 Medium-term 0.037 6.363 93.6% -0.004 -0.365 -0.002 -0.639 0.301 Long-term 0.034 1.531 61.1% -0.003 -0.268 -0.002 -0.809 0.345 Moneyness All maturities 0.056 5.769 93.9% 0.010 1.167 0.002 0.904 0.179 K/S Short-term 0.045 3.455 81.5% 0.016 2.512 0.003 1.905 0.200 1.00 - 1.05 Medium-term 0.033 4.718 87.9% 0.021 3.181 0.004 2.666 0.343 Long-term 0.091 3.183 78.6% 0.012 1.515 0.002 0.825 0.364 Moneyness All maturities 0.049 5.257 87.7% 0.012 1.255 0.002 0.999 0.152 K/S Short-term 0.033 2.429 72.3% 0.010 1.373 0.002 1.267 0.177 1.05 - 1.10 Medium-term 0.034 6.165 88.0% 0.011 0.969 0.001 0.594 0.287 Long-term 0.058 2.237 60.0% 0.011 2.114 0.001 0.716 0.347 γ1 γ2 γ3 Notes: This table contains two-pass regression results for the level effect tests. In the first pass, for each firm, we regress the difference between the implied volatility and the historical volatility on moneyness for non-overlapping periods of one month (i.e., 4 weeks): .)(10 ii his i imp i yyaa εσσ +−+=− We thus obtain a monthly time-series of the intercept 0a and the slope coefficient 1a for all firms including the S&P100 index. The moneyness variable is adjusted by the sample mean within the month so that the intercept 0a is the average of the difference between the implied volatility and the historical volatility. In the second pass, we cross-sectionally regress the intercept 0a on the systematic risk proportion b , the risk-neutral skewness and kurtosis. This regression is done every month in three different forms: (1) ,100 jjj eba ++= γγ (2) j rn j rn jj eKurtSkewa +++= )( 3 )( 200 γγγ and (3) .)( 3 )( 2100 j rn j rn jjj eKurtSkewba ++++= γγγγ The monthly regression coefficients are then averaged, and the corresponding t-values calculated with a first-order serial correlation correction. The results for regressions (1) and (2) are reported in Panel A, while those for regression (3) are in Panel B. To conserve space, we omit the regression intercept and its t-value. The t-values in bold type are significant at least at the 10% level, for two-tailed tests. The entries under 01 >γ are percentages of the monthly coefficient 1γ that are positive. The reported 2R is the average 2R from monthly cross-sectional regressions. The risk-neutral skewness and kurtosis are estimated using the same procedure as in Bakshi, Kapadia and Madan (2003). The maturity ranges for short-term, medium-term and long-term are, respectively, 20-70 days, 71-120 days, and 121-180 days. The regressions are performed separately for four moneyness buckets. 29 Table 4: Regression tests for the slope effect Panel A: Separate Regressions on Systematic Risk Proportion, and Skewness and Kurtosis avg. t γ1 < 0 R2 avg. t avg. t R2 All maturities -0.431 -5.394 86.2% 0.047 0.455 7.995 0.074 4.622 0.101 Moneyness Short-term -0.363 -5.123 78.5% 0.032 0.322 3.836 0.040 2.487 0.107 K/S Medium-term -0.411 -7.813 93.6% 0.100 0.257 3.838 0.045 2.679 0.153 0.90 - 0.95 Long-term -0.183 -1.099 54.6% 0.092 0.057 1.124 0.006 0.435 0.195 Moneyness All maturities -0.441 -6.163 92.3% 0.048 -0.016 -0.180 -0.013 -0.625 0.092 K/S Short-term -0.583 -10.989 95.4% 0.061 -0.081 -1.509 -0.037 -2.856 0.093 0.95 - 1.00 Medium-term -0.534 -14.137 100.0% 0.158 0.134 1.140 0.026 0.967 0.135 Long-term -0.212 -2.002 63.9% 0.056 0.060 1.139 0.012 1.109 0.139 Moneyness All maturities -0.557 -6.343 98.5% 0.055 0.015 0.240 -0.009 -0.583 0.073 K/S Short-term -0.612 -6.825 93.9% 0.060 -0.096 -1.082 -0.032 -1.655 0.098 1.00 - 1.05 Medium-term -0.500 -9.634 97.0% 0.167 0.108 1.277 0.034 1.910 0.223 Long-term -0.563 -2.629 73.8% 0.087 0.034 0.524 0.007 0.369 0.189 Moneyness All maturities 0.003 0.054 49.2% 0.016 -0.124 -1.583 -0.026 -1.388 0.090 K/S Short-term -0.053 -0.971 56.9% 0.021 -0.271 -2.747 -0.066 -2.816 0.114 1.05 - 1.10 Medium-term -0.158 -2.038 68.0% 0.060 -0.107 -1.258 -0.032 -2.337 0.149 Long-term -0.311 -1.633 54.3% 0.090 -0.154 -2.115 -0.045 -1.875 0.181 γ1 γ2 γ3 Panel B: Combined Regressions on Systematic Risk Proportion, Skewness and Kurtosis avg. t γ1 < 0 avg. t avg. t R2 All maturities -0.349 -4.086 76.9% 0.453 7.511 0.079 5.022 0.139 Moneyness Short-term -0.250 -1.656 56.9% 0.347 3.861 0.049 2.838 0.144 K/S Medium-term -0.528 -6.436 93.5% 0.033 0.228 -0.008 -0.217 0.264 0.90 - 0.95 Long-term -0.276 -1.322 59.1% 0.027 0.438 -0.004 -0.249 0.302 Moneyness All maturities -0.433 -4.742 81.5% -0.040 -0.424 -0.007 -0.364 0.134 K/S Short-term -0.511 -5.847 78.5% -0.131 -2.340 -0.035 -2.626 0.140 0.95 - 1.00 Medium-term -0.556 -8.017 93.5% 0.017 0.175 0.003 0.119 0.281 Long-term -0.259 -2.244 63.9% 0.067 1.471 0.019 1.792 0.186 Moneyness All maturities -0.479 -5.326 86.2% -0.017 -0.320 -0.007 -0.530 0.120 K/S Short-term -0.518 -3.926 76.9% -0.127 -1.624 -0.029 -1.656 0.150 1.00 - 1.05 Medium-term -0.434 -6.625 90.9% -0.011 -0.144 0.011 0.695 0.361 Long-term -0.619 -2.248 69.0% 0.045 0.568 0.014 0.674 0.276 Moneyness All maturities -0.007 -0.099 52.3% -0.139 -1.734 -0.030 -1.537 0.113 K/S Short-term 0.052 0.515 50.8% -0.293 -2.823 -0.071 -2.907 0.148 1.05 - 1.10 Medium-term -0.160 -1.788 64.0% -0.154 -1.269 -0.045 -1.816 0.224 Long-term -0.376 -1.556 57.1% -0.128 -1.574 -0.040 -1.430 0.280 γ1 γ2 γ3 Notes: This table contains two-pass regression results for the slope effect tests. In the first pass, for each firm, we regress the difference between the implied volatility and the historical volatility on moneyness for non-overlapping periods of one month (i.e., 4 weeks): .)(10 ii his i imp i yyaa εσσ +−+=− We thus obtain a monthly time-series of the intercept 0a and the slope coefficient 1a for all firms including the S&P100 index. The moneyness variable is adjusted by the sample mean within the month so that the intercept 0a is the average of the difference between the implied volatility and the historical volatility. In the second pass, we cross-sectionally regress the slope 1a on the systematic risk proportion b , the risk-neutral skewness and kurtosis. This regression is done every month in three different forms: (1) ,101 jjj eba ++= γγ (2) j rn j rn jj eKurtSkewa +++= )( 3 )( 201 γγγ and (3) .)( 3 )( 2101 j rn j rn jjj eKurtSkewba ++++= γγγγ The monthly regression coefficients are then averaged, and the corresponding t-values calculated with a first-order serial correlation correction. The results for regressions (1) and (2) are reported in Panel A, while those for regression (3) are in Panel B. To conserve space, we omit the regression intercept and its t-value. The t-values in bold type are significant at least at the 10% level, for two-tailed tests. The entries under <1γ are percentages of the monthly coefficient 1γ that are negative. The reported 2R is the average 2R from monthly cross-sectional regressions. The risk-neutral skewness and kurtosis are estimated using the same procedure as in Bakshi, Kapadia and Madan (2003). The maturity ranges for short-term, medium-term and long-term are, respectively, 20-70 days, 71-120 days, and 121-180 days. The regressions are performed separately for four moneyness buckets. 30 Table 5: Sub-sample regression tests for the level effect Panel A: Regressions on Systematic Risk Proportion avg. t γ1 > 0 avg. t γ1 > 0 avg. t γ1 > 0 All maturities 0.067 12.683 100.0% 0.103 0.070 13.825 100.0% 0.107 0.065 6.468 100.0% 0.099 Moneyness Short-term 0.066 11.814 100.0% 0.098 0.067 12.080 100.0% 0.100 0.064 6.490 100.0% 0.095 K/S Medium-term 0.060 12.727 100.0% 0.169 0.063 9.175 100.0% 0.165 0.058 8.939 100.0% 0.172 0.90 - 1.00 Long-term 0.077 6.001 83.1% 0.119 0.079 6.784 87.5% 0.113 0.075 3.245 78.8% 0.125 Moneyness All maturities 0.045 5.114 96.9% 0.064 0.059 6.337 96.9% 0.099 0.031 3.249 97.0% 0.029 K/S Short-term 0.038 4.910 96.9% 0.048 0.053 6.584 96.9% 0.078 0.024 2.421 97.0% 0.019 1.00 - 1.10 Medium-term 0.039 8.910 95.1% 0.102 0.051 8.138 93.1% 0.125 0.029 5.910 96.9% 0.081 Long-term 0.082 5.826 84.6% 0.127 0.098 9.478 90.6% 0.176 0.067 2.528 78.8% 0.080 γ1 γ1 γ1 Whole Sample, 01/01/91 - 31/12/95 Sub-sample, 01/01/91 - 30/06/93 Sub-sample, 01/07/93 - 31/12/95 R2 R2R2 avg. t avg. t avg. t avg. t avg. t avg. t All maturities -0.012 -1.541 -0.002 -1.143 0.091 -0.005 -0.535 -0.002 -0.971 0.067 -0.018 -1.516 -0.002 -0.600 0.115 Moneyness Short-term -0.011 -2.213 -0.001 -0.733 0.110 -0.005 -0.755 0.000 -0.211 0.061 -0.018 -2.298 -0.001 -0.644 0.156 K/S Medium-term -0.021 -3.208 -0.005 -3.417 0.171 -0.016 -1.544 -0.004 -2.101 0.192 -0.025 -3.208 -0.005 -2.680 0.152 0.90 - 1.00 Long-term -0.018 -2.369 -0.007 -3.135 0.136 -0.019 -1.469 -0.008 -2.201 0.165 -0.018 -2.105 -0.005 -2.505 0.108 Moneyness All maturities 0.014 2.058 0.003 2.480 0.082 0.023 3.873 0.004 2.622 0.075 0.005 0.418 0.003 1.126 0.089 K/S Short-term 0.014 2.729 0.003 3.358 0.081 0.024 5.049 0.005 5.142 0.070 0.004 0.566 0.002 1.190 0.091 1.00 - 1.10 Medium-term 0.002 0.322 0.000 -0.173 0.169 0.003 0.302 -0.001 -0.356 0.204 0.001 0.129 0.000 0.159 0.137 Long-term 0.007 1.370 0.001 0.449 0.154 0.009 1.028 0.001 0.248 0.173 0.005 0.890 0.001 0.476 0.136 γ2 R2 R2 R2 Panel B: Regressions on Skewness and Kurtosis γ3 Whole Sample, 01/01/91 - 31/12/95 Sub-sample, 01/01/91 - 30/06/93 Sub-sample, 01/07/93 - 31/12/95 γ2γ2 γ3 γ3 γ2 γ3 γ2 γ3 γ2 γ3 t γ1 > 0 t t t γ1 > 0 t t t γ1 > 0 t t All maturities 7.945 100.0% -1.500 -1.742 0.200 15.112 100.0% -0.685 -1.490 0.194 3.934 100.0% -1.199 -1.050 0.206 Moneyness Short-term 5.436 90.8% -1.967 -1.714 0.201 10.507 100.0% -0.401 -1.266 0.165 2.514 81.8% -2.069 -1.298 0.235 K/S Medium-term 17.608 100.0% -1.999 -2.242 0.300 10.732 100.0% -1.282 -1.717 0.322 15.554 100.0% -1.505 -1.364 0.280 0.90 - 1.00 Long-term 6.000 86.2% -2.053 -2.681 0.253 7.276 93.8% -1.355 -2.163 0.275 3.156 78.8% -1.662 -1.857 0.231 Moneyness All maturities 5.759 93.9% 1.055 0.791 0.158 8.435 100.0% 2.470 1.109 0.191 2.929 87.9% 0.059 0.218 0.126 K/S Short-term 3.553 81.5% 1.962 1.526 0.146 9.420 100.0% 4.952 3.056 0.165 1.116 63.6% 0.154 0.258 0.128 1.00 - 1.10 Medium-term 11.217 96.7% 1.199 0.773 0.275 8.502 96.6% 0.507 0.030 0.317 8.412 96.9% 1.284 1.148 0.238 Long-term 5.701 81.5% 1.138 0.570 0.278 11.537 90.6% 0.665 -0.022 0.335 2.394 72.7% 0.961 1.247 0.223 R2 R2 Panel C: Combined Regressions on Systematic Risk Proportion, Skewness and Kurtosis γ1 γ1 Whole Sample, 01/01/91 - 31/12/95 Sub-sample, 01/01/91 - 30/06/93 Sub-sample, 01/07/93 - 31/12/95 γ1 R2 Notes: This table contains two-pass regression results for the level effect tests. The regressions are run for the whole sample (01/01/91-31/12/95) as well as two sub-samples: (01/01/91-30/06/93) and (01/07/93-31/12/95). In the first pass, for each firm, we regress the difference between the implied volatility and the historical volatility on moneyness for non-overlapping periods of one month (i.e., 4 weeks): .)(10 ii his i imp i yyaa εσσ +−+=− We thus obtain a monthly time-series of the intercept 0a and the slope coefficient 1a for all firms including the S&P100 index. The moneyness variable is adjusted by the sample mean within the month so that the intercept 0a is the average of the difference between the implied volatility and the historical volatility. In the second pass, we cross-sectionally regress the intercept 0a on the systematic risk proportion b , the risk-neutral skewness and kurtosis. This regression is done every month in three different forms: (1) ,100 jjj eba ++= γγ (2) j rn j rn jj eKurtSkewa +++= )( 3 )( 200 γγγ and (3) .)( 3 )( 2100 j rn j rn jjj eKurtSkewba ++++= γγγγ The monthly regression coefficients are then averaged, and the corresponding t-values calculated with a first-order serial correlation correction. The results for regressions (1), (2) and (3) are reported in Panels A, B and C, respectively. To conserve space, we omit the regression intercept and its t- value. The t-values in bold type are significant at least at the 10% level, for two-tailed tests. In Panel C, the coefficients are omitted for brevity. The entries under 01 >γ (in Panels A and C) are percentages of the monthly coefficient 1γ that are positive. The reported 2R is the average 2R from monthly cross-sectional regressions. The risk-neutral skewness and kurtosis are estimated using the same procedure as in Bakshi, Kapadia and Madan (2003). The maturity ranges for short-term, medium-term and long-term are, respectively, 20-70 days, 71-120 days, and 121-180 days. The regressions are performed separately for two moneyness buckets. 33 Table 8: Level and slope effect tests without the index avg. t R2 avg. t avg. t avg. t R2 All maturities 0.102 3.513 0.099 0.119 4.078 -0.022 -1.979 -0.005 -2.261 0.215 Moneyness Short-term 0.102 4.033 0.120 0.109 4.054 -0.018 -2.165 -0.003 -1.780 0.257 K/S Medium-term 0.087 2.912 0.138 0.113 3.700 -0.007 -0.448 -0.001 -0.356 0.348 0.90 - 0.95 Long-term 0.104 3.978 0.131 0.102 3.802 -0.014 -1.464 -0.005 -1.986 0.301 Moneyness All maturities 0.090 4.196 0.084 0.102 4.656 -0.017 -1.602 -0.005 -2.142 0.196 K/S Short-term 0.078 5.375 0.076 0.079 4.787 -0.005 -0.607 -0.001 -0.897 0.193 0.95 - 1.00 Medium-term 0.087 2.376 0.084 0.097 2.213 0.002 -0.142 -0.001 -0.289 0.335 Long-term 0.031 1.512 0.078 0.033 1.511 -0.003 -0.252 -0.002 -0.782 0.346 Moneyness All maturities 0.087 3.200 0.098 0.096 3.799 0.006 -0.595 0.000 0.249 0.204 K/S Short-term 0.072 3.620 0.091 0.079 3.978 0.013 -2.033 0.002 -1.374 0.233 1.00 - 1.05 Medium-term 0.045 1.913 0.104 0.068 2.457 0.014 -1.191 0.002 -0.721 0.382 Long-term 0.093 4.164 0.149 0.091 3.183 0.012 -1.515 0.002 -0.825 0.364 Moneyness All maturities 0.076 2.909 0.097 0.087 3.099 0.008 -0.792 0.001 -0.502 0.193 K/S Short-term 0.057 2.756 0.101 0.057 2.533 0.006 -0.711 0.001 -0.375 0.244 1.05 - 1.10 Medium-term 0.117 2.659 0.139 0.147 2.647 0.006 -0.357 0.001 -0.230 0.363 Long-term 0.051 2.744 0.110 0.058 2.237 0.011 -2.114 0.001 -0.716 0.347 Multivariate RegressionsUnivariate Regressions Panel A: Level Effects γ1 γ2 γ3γ1 avg. t R2 avg. t avg. t avg. t R2 All maturities -0.145 -0.643 0.049 -0.148 -0.625 0.407 6.458 0.067 3.948 0.147 Moneyness Short-term -0.261 -1.050 0.054 -0.129 -0.410 0.319 2.964 0.036 1.525 0.175 K/S Medium-term -0.274 -0.817 0.105 -0.271 -0.730 0.046 0.436 -0.010 -0.359 0.276 0.90 - 0.95 Long-term -0.183 -1.099 0.092 -0.276 -1.322 0.027 0.438 -0.004 -0.249 0.302 Moneyness All maturities 0.071 0.478 0.031 0.056 0.358 -0.032 -0.321 -0.005 -0.208 0.123 K/S Short-term -0.082 -0.648 0.032 -0.066 -0.464 -0.105 -1.471 -0.026 -1.389 0.120 0.95 - 1.00 Medium-term -0.164 -0.625 0.087 -0.023 -0.084 0.098 0.428 0.033 0.644 0.257 Long-term -0.205 -1.932 0.057 -0.248 -2.120 0.065 1.428 0.018 1.675 0.187 Moneyness All maturities -0.551 -3.395 0.045 -0.506 -2.609 -0.051 -0.868 -0.014 -0.941 0.114 K/S Short-term -0.758 -3.448 0.068 -0.784 -3.156 -0.163 -1.798 -0.036 -1.772 0.167 1.00 - 1.05 Medium-term 0.263 1.036 0.114 0.309 0.991 0.003 0.031 0.007 0.415 0.331 Long-term -0.563 -2.629 0.087 -0.619 -2.248 0.045 0.568 0.014 0.674 0.276 Moneyness All maturities -0.229 -1.261 0.036 -0.302 -1.579 -0.153 -1.816 -0.035 -1.695 0.138 K/S Short-term -0.323 -1.229 0.051 -0.259 -1.060 -0.369 -3.973 -0.093 -4.668 0.177 1.05 - 1.10 Medium-term -0.432 -1.542 0.105 -0.723 -1.703 -0.141 -0.943 -0.048 -1.498 0.264 Long-term -0.311 -1.633 0.090 -0.376 -1.556 -0.128 -1.574 -0.040 -1.430 0.280 Univariate Regressions Panel B: Slope Effects Multivariate Regressions γ1 γ1 γ2 γ3 Notes: This table contains two-pass regression results for the level and slope effect tests by excluding the S&P 100 index from the sample. The testing procedures are the same as those in Tables 3 and 4. Panel A corresponds to Table 3 and Panel B corresponds to Table 4. Please refer to those tables for further explanations. In Tables 3 and 4, the second-pass cross-sectional regressions are run over the 30 stocks and the S&P 100 index; in this table, the cross-sectional regressions are run over the 30 stocks only. To conserve space, we only report the regression coefficients together with their t-values and the average 2R . For brevity, we also omit the results for regressions whose explanatory variables are only the skewness and kurtosis. The t-values in bold type are significant at least at the 10% level, for two-tailed tests. 34 Table 9: Level and slope effect tests based on panel regressions α0 t α1 t β0 t β1 t R2 All maturities 0.032 225.24 0.068 143.46 -0.361 -36.74 -0.730 -21.88 0.237 Moneyness Short-term 0.038 188.15 0.066 100.35 -0.566 -39.50 -0.615 -13.25 0.238 K/S Medium-term 0.032 129.59 0.064 85.19 -0.275 -16.16 -0.636 -12.06 0.295 0.90 - 0.95 Long-term 0.016 55.14 0.088 33.45 -0.021 -1.07 -0.660 -3.60 0.066 Moneyness All maturities 0.018 141.22 0.037 85.62 -0.229 -24.92 -0.616 -20.58 0.106 K/S Short-term 0.018 95.65 0.036 60.20 -0.318 -24.25 -0.654 -15.68 0.105 0.95 - 1.00 Medium-term 0.022 97.10 0.038 55.64 -0.202 -12.33 -0.524 -10.87 0.155 Long-term 0.015 51.96 0.053 21.06 -0.026 -1.29 -0.511 -2.79 0.030 Moneyness All maturities -0.007 -53.81 0.028 63.67 -0.057 -6.22 -0.409 -13.47 0.053 K/S Short-term -0.006 -30.28 0.022 36.59 -0.050 -3.84 -0.428 -10.26 0.035 1.00 - 1.05 Medium-term -0.004 -18.34 0.029 43.31 -0.090 -5.56 -0.427 -8.92 0.095 Long-term -0.014 -48.34 0.075 28.78 -0.026 -1.26 -0.098 -0.53 0.050 Moneyness All maturities -0.008 -53.31 0.021 31.73 0.070 6.40 0.050 0.95 0.018 K/S Short-term -0.003 -13.91 0.014 15.16 0.201 12.50 -0.117 -1.59 0.013 1.05 - 1.10 Medium-term -0.011 -37.30 0.021 20.60 -0.010 -0.50 0.240 2.87 0.029 Long-term -0.017 -56.35 0.066 22.52 -0.055 -2.66 -0.415 -1.96 0.036 Notes: This table contains panel regression results for the level and slope effect tests. For each moneyness / maturity bucket, instead of running the Fama-MacBeth two pass-regressions, we lump the entire sample and run the following panel regression: ijjijiijiij his ij imp ij yybbbb εββαασσ +−−++−+=− )]))(([()]([( 1010 , where ib is the cross-sectional average of the systematic risk proportion for each day, and jy is the sample average of moneyness for stock j or the index within the bucket. This panel regression tests the level and slope effects simultaneously. Specifically, if the systematic risk proportion doesn’t affect the price level or the level of the implied volatility (after adjusting for the historical volatility), then the coefficient 1α should not be significantly different from zero; likewise, if the systematic risk proportion doesn’t affect the slope of the implied volatility curve, then the coefficient 1β should not be significantly different from zero. The t-values in bold type are significant at least at the 10% level, for two-tailed test. 35 Table 10: Level and slope effect tests using systematic risk estimates derived from Fama-French factors avg. t R2 avg. t avg. t avg. t R2 All maturities 0.069 13.009 0.120 0.074 8.601 -0.017 -1.552 -0.004 -1.618 0.221 Moneyness Short-term 0.069 15.816 0.142 0.072 5.887 -0.013 -1.787 -0.002 -1.230 0.262 K/S Medium-term 0.064 14.167 0.219 0.065 10.874 -0.013 -1.158 -0.002 -0.836 0.397 0.90 - 0.95 Long-term 0.074 5.916 0.089 0.070 5.758 -0.011 -1.092 -0.004 -1.622 0.262 Moneyness All maturities 0.045 10.776 0.063 0.057 5.218 -0.014 -1.251 -0.004 -1.604 0.169 K/S Short-term 0.041 10.568 0.066 0.047 3.877 -0.003 -0.401 -0.001 -0.432 0.176 0.95 - 1.00 Medium-term 0.032 5.375 0.076 0.032 5.484 -0.008 -0.682 -0.002 -0.937 0.296 Long-term 0.010 0.661 0.065 0.021 1.873 -0.003 -0.248 -0.002 -0.716 0.328 Moneyness All maturities 0.040 4.076 0.061 0.046 4.103 0.011 1.307 0.002 1.199 0.165 K/S Short-term 0.032 3.980 0.052 0.034 2.849 0.016 2.771 0.003 2.432 0.192 1.00 - 1.05 Medium-term 0.025 3.295 0.089 0.032 4.067 0.020 3.176 0.003 2.600 0.342 Long-term 0.071 3.621 0.134 0.074 3.120 0.014 1.796 0.002 0.986 0.362 Moneyness All maturities 0.030 3.735 0.045 0.037 3.998 0.013 1.501 0.002 1.420 0.138 K/S Short-term 0.020 3.361 0.039 0.025 1.878 0.011 1.490 0.002 1.585 0.171 1.05 - 1.10 Medium-term 0.020 2.720 0.056 0.030 4.683 0.009 0.808 0.001 0.443 0.279 Long-term 0.024 1.544 0.106 0.036 1.832 0.012 2.288 0.001 0.766 0.334 Multivariate RegressionsUnivariate Regressions Panel A: Level Effects γ1 γ2 γ3γ1 avg. t R2 avg. t avg. t avg. t R2 All maturities -0.464 -6.628 0.051 -0.400 -5.260 0.454 7.500 0.080 5.041 0.142 Moneyness Short-term -0.372 -5.766 0.033 -0.230 -1.819 0.346 3.791 0.048 2.762 0.142 K/S Medium-term -0.430 -8.799 0.101 -0.528 -8.403 0.067 0.536 0.001 0.046 0.262 0.90 - 0.95 Long-term -0.231 -1.459 0.104 -0.287 -1.417 0.049 0.801 0.002 0.105 0.310 Moneyness All maturities -0.426 -7.016 0.048 -0.396 -5.342 -0.045 -0.497 -0.011 -0.528 0.134 K/S Short-term -0.558 -11.463 0.055 -0.446 -5.854 -0.130 -2.272 -0.038 -2.745 0.135 0.95 - 1.00 Medium-term -0.522 -16.269 0.149 -0.526 -7.930 0.046 0.437 0.008 0.328 0.271 Long-term -0.133 -1.303 0.052 -0.144 -0.906 0.075 1.431 0.030 1.478 0.187 Moneyness All maturities -0.507 -5.781 0.048 -0.421 -5.020 -0.003 -0.059 -0.005 -0.367 0.113 K/S Short-term -0.564 -6.132 0.055 -0.446 -3.190 -0.113 -1.370 -0.027 -1.450 0.146 1.00 - 1.05 Medium-term -0.492 -9.356 0.156 -0.424 -6.733 0.000 0.004 0.013 0.873 0.351 Long-term -0.481 -2.459 0.089 -0.488 -2.032 0.039 0.555 0.013 0.660 0.287 Moneyness All maturities -0.007 -0.159 0.017 -0.011 -0.173 -0.144 -1.809 -0.030 -1.570 0.112 K/S Short-term -0.047 -0.800 0.023 0.029 0.265 -0.298 -2.838 -0.070 -2.777 0.151 1.05 - 1.10 Medium-term -0.158 -2.124 0.059 -0.162 -1.835 -0.161 -1.326 -0.047 -1.904 0.222 Long-term -0.255 -1.553 0.089 -0.386 -2.050 -0.116 -1.566 -0.038 -1.458 0.278 Univariate Regressions Panel B: Slope Effects Multivariate Regressions γ1 γ1 γ2 γ3 Notes: This table contains two-pass regression results for the level and slope effect tests using systematic risk estimates derived from the Fama-French factors. The testing procedures are otherwise the same as those in Tables 3 and 4. Panel A corresponds to Table 3 and Panel B corresponds to Table 4. Please refer to those tables for further explanations. In Tables 3 and 4, the systematic risk is estimated by regressing the stock’s returns on the market returns (S&P 500). Here, the systematic risk is estimated by regressing the stock’s returns on the two Fama-French factors as well as on the market returns. The daily Fama-French factors are downloaded from Kenneth French’s website. To conserve space, we only report the regression coefficients together with their t-values and the average 2R . For brevity, we also omit the results for regressions whose explanatory variables are only the skewness and kurtosis. The t-values in bold type are significant at least at the 10% level, for two-tailed tests