Investor sentiment and value and growth stock index options

Transcription

1 Investor sentiment and value and growth stock index options Jerry Coakley*, George Dotsis, Xiaoquan Liu**, and Jia Zhai January 2013 * Essex Finance Centre and Essex Business School, University of Essex, Colchester CO4 3SQ, UK. jcoakley@essex.ac.uk. Department of Economics, University of Athens, 5 Stadiou Street, Athens 10562, Greece. gdotsis@econ.uoa.gr. ** Essex Finance Centre and Essex Business School, University of Essex, Colchester CO4 3SQ, UK. liux@essex.ac.uk. University of Ulster, Shore Road, Newtownabbey, Co. Antrim BT37 0QB. 0

2 Abstract The paper examines the relationship between both individual and institutional investor sentiment measures and the risk-neutral skewness of seven stock index options comprised of either growth or value stocks. It provides novel evidence that growth index option prices are affected by sentiment measures. The regression results indicate a significantly positive relationship between sentiment measures and the risk-neutral skewness estimated from four growth index options and a negative relationship with two value index options. The results are economically significant since an associated long-short trading strategy yields high abnormal returns with a Sharpe ratio of up to 1.1 and zero exposure to systematic risk. These high abnormal returns provide evidence of a value premium type anomaly in the index options markets. Keywords: Risk-neutral skewness; Growth options; Option market anomalies JEL classification: G12, G13 1

3 1. Introduction Increasing numbers of researchers investigate empirical anomalies in options markets that do not seem to conform to the predictions of classical asset pricing theories. These are exemplified by studies such as Lakonishok, Lee, Pearson, and Poteshman (2007). They investigate option market trading activity for several investor classes from 1990 to They report a distinctive trading pattern for the least sophisticated customer group (discount customers) who dramatically increase their call purchases and put sales primarily of growth stocks rather than value stocks during the dotcom bubble. By contrast, more sophisticated investors increase their trading only mildly at best. Constantinides, Jackwert and Perrakis (2009) examine the mispricing of S&P 500 options over the period from 1986 to They conclude that there is no evidence that the S&P 500 options market is becoming more rational over time. The objective of this paper is to address the question of whether investor sentiment affects stock index option styles. The central question the paper addresses is whether sentiment measures have a differential effect on the cross-section of index options and specifically on style (growth and value) index options. In this regard, it builds on the seminal paper of Han (2008) who establishes that sentiment affects the S&P 500 index options. We take a different tack by focusing on the impact of sentiment on style index options. We produce robust evidence which establishes that sentiment has a positive impact on the prices of all four growth index options and a negative effect on two out of three value index options. This is a novel finding which we also show to be economically significant. The sentiment measures employed include one broadly-based sentiment indicator the Baker-Wurgler (2006) measure. A second measure is the index of consumer sentiment (CS) based on the survey conducted by the University of Michigan. This is used as a measure of 2

4 individual sentiment. The final one is the bull-bear spread (BB) computed by Investors Intelligence which is employed as a proxy for institutional investor sentiment. The indices examined include the Russell 1000 and 2000 Growth and Value indices and the S&P SmallCap 600 Growth and Value indices. The component stocks for the Nasdaq 100 index tend to be large growth stocks (Asquith, Pathak and Ritter, 2005). The first contribution of this paper is the novel finding that investor sentiment measures exert a significantly positive impact on the prices of all four US growth stock index options. Using monthly options data from June 2001 to January 2010, the regression results show that either the CS or BW sentiment measures affects the risk-neutral skewness (hereafter RNS) impacts on the four growth stock index options irrespective of whether they are comprised of small or large cap stocks. They also show that investor sentiment measures exert a significantly negative impact on two out of three value stock index options and this is a small firm effect. We conjecture that the basis of this contrast between growth and value indices is related to a value premium type anomaly in options markets. One approach to explaining the latter is the behavioral biases of investors. Lakonishok, Shleifer, and Vishny (1994) argue that growth stocks are overpriced relative to value stocks due to investors overreaction. Barberis, Shleifer, and Vishny (1998) develop a model of investor sentiment consistent with a behavioral explanation of the value premium. Baker and Wurgler (2006) suggest that low capitalization stocks, younger, unprofitable, highly volatile, non-dividend paying, growth companies, or stocks of firms in financial distress, are more likely to be affected by investor sentiment. The second contribution is to show that our findings are economically significant. They can earn abnormal returns by trading simultaneously growth and value index options in longshort strategies. We employ a variance swaps trading strategy because Kozhan, Neuberger and 3

5 Schneider (2011) show that the skew premium depends on the same underlying factors as the variance risk premium and strategies designed to exploit one of the risk premiums and hedge out the other make zero excess returns. Our preferred long-short strategy has zero exposure to systematic variance risk and a significant bi-monthly alpha of up to 26% with a Sharpe ratio of 1.1. Our study builds on the seminal paper of Han (2008) who establishes that the optionimplied RNS of monthly S&P 500 index option returns becomes more negative when institutional market sentiment is more bearish and less negative when it is more bullish. However our results are contrary to his view that consumer (small investor) sentiment measures are not relevant in this context as we find that the CS index has a robust impact. Lemmon and Ni (2010) suggest that sentiment impacts on individual stock options rather than on stock index options. The rationale is that the former are actively traded by individual investors who are more susceptible to sentiment while the latter are mainly traded by sophisticated institutional investors. Our main results directly challenge their conclusion that sentiment has little or no impact on stock index options. Our findings are related to two recent studies of style index options. Blackburn, Goetzmann, and Ukhov (2009) find that investors in value index options tend to exhibit a higher level of risk aversion than those in growth index options. Their long-short trading strategies in value and growth index options generate modest positive returns over their sample period. He, Lee and Wei (2010) find that investor reaction to information in growth options proxied by the Nasdaq 100 and Russell 2000 growth indices are stronger than those to the Russell 2000 value. Their results imply that difference in investors behaviour and styles is one potential explanation 4

6 for the value premium. 1 Our high abnormal returns provide stronger evidence of a value premium type anomaly in index options markets than both of these studies. The reminder of this paper is organized as follows Section 2 outlines and discusses the concept of risk neutral skewness. Section 3 describes our data and analyzes the empirical results. Section 4 outlines our trading strategy and discusses its economic significance. A final section concludes. 2. Risk-neutral skewness and sentiment It is well-known that volatility smile inferred from index options is tantamount to the negative skewness in the risk-neutral distributions of the index returns (Bakshi, Kapadia and Madan, 2003). When limits to arbitrage exist in the options market, investor sentiment is found to bias option prices (Poteshman and Serbin, 2003 and Mahani and Poteshman, 2004) and this in turn affects the RNS of index options. Thus in a bearish (bullish) environment investors may bid up the prices of out-of-money put (call) options more than out-of-money call (put) option prices. This would lead to more negative (positive) RNS. In addition, the RNS of growth index options is more likely to be affected by sentiment measures, especially by individual sentiment measures, than is the RNS of options written on the broad market index or a value index. Individual investors, who are more susceptible to sentiment and who mainly trade in the growth options market, push up the prices for out-of-money (OTM) growth put options more than OTM growth call options, leading to more negative skewness in the growth options when compared with the skewness of options written on the broad market index or a value index. 1 Stein (1989) examines the term structure of implied volatility of S&P 100 index options and finds underreaction in the short run and overreaction in the long run. Poteshman (2001) reports similar results for S&P 500 index options. 5

7 Bakshi, Kapadia and Madan (2003) show that the index option volatility smile implies negative skewness of the risk-neutral density of index returns. Their method is used to derive the RNS of stock index options as functions of out-of-money call and put option prices. Empirically, their model-free approach allows one to accurately extract risk-neutral volatility, skewness and kurtosis from option prices. They suggest that any payoff can be spanned and priced using an explicit positioning across options of a wide range of different strike prices. Hence the method extracts information from out-of-money call and put options which enjoy better liquidity and information content than options at other moneyness levels. We use their method to derive the RNS of value and growth index options as functions of out-of-money call and put option prices. At date t for a given horizon τ, τhe RNS is expressed as a weighted sum of three contracts: the quadratic contract V ( t, τ), the cubic contract W ( t, τ), and the quartic contract X ( t, τ), respectively, as follows, q r 2 V( t, τ) E { e τ R t, τ} q r 3 W( t, τ) E { e τ R t, τ} X( t, τ) E where q denotes the expectation operator under the risk-neutral measure. These three contracts can be shown to be weighted sums of OTM call and put options, q { e rτ R 4 t,τ } 2(1 ln[ K / St]) V( t, τ) = C( t, τ; K) dk S 2 t K St 2(1 + ln[ S t / K ]) + P ( t, τ; K ) dk, 0 2 K 2 6ln[ K / St] 3(ln[ K / St]) W( t, τ) = C( t, τ; K) dk S 2 t K 2 St 6ln[ St / K] + 3(ln[ St / K]) P( t, τ; K) dk 0 2 K (1) (2) 6

8 2 3 12(ln[ K / St]) 4(ln[ K / St]) X( t, τ) = C( t, τ; K) dk S 2 t K 2 3 St12(ln[ St / K]) + 4(ln[ St / K]) + P( t, τ; K) dk, 0 2 K (3) where S t is the price of the underlying asset and C( t, τ, K) and P( t, τ, K) are OTM call and put option prices, respectively, with strike price K and τ is time to maturity. Using equations (1), (2) and (3), the risk-neutral variance and skewness can be expressed as follows, VAR( t, τ) E {( R E [ R ]) } = e V( t, τ ) µ ( t, τ) q q 2 rτ 2 t, τ t, τ (4) E {( R E [ R ]) } e W( t, ) 3 e ( t, ) V( t, ) + 2 ( t, ) SKEW( t, τ) = = E R E R e V t t q q 3 rτ rτ 3 t, τ t, τ τ µ τ τ µ τ q q 2 3/2 rτ 2 3/2 {( t, τ [ t, τ]) } [ (, τ ) µ (, τ) ] (5) rτ rτ rτ rτ e e e where µ ( t, τ) = e 1 V( t, τ) W( t, τ) X( t, τ) This method has gradually gained popularity in the literature and become the industry standard for estimating RNS from option prices (see, for example, Conrad, Dittmar, and Ghysels, 2013, Han, 2008, and Rehman and Vilkov, 2010). 2 This is largely due to the fact that the method is model-free and we make full use of information contained in traded option prices without the need to subscribe to a specific option pricing model. 3. Data and empirical results 3.1 Data and summary statistics Sentiment indices 2 Other methods that have been proposed to extract skewness information from option prices include the volatility skew of Xing, Zhang, and Zhao (2010) and the volatility spread of Bali and Hovakimian (2009) and Cremers and Weinbaum (2010). 7

9 Three sentiment measures are employed in this paper. The first is the Baker and Wurgler (2006) (BW) general sentiment measure. The BW measure is a general sentiment proxy because it is calculated as the first principal component of six alternative sentiment measures. Two alternative sentiment measures are used to distinguish between institutional and individual sentiment. Individual sentiment is proxied by the Michigan Consumer Confidence Index (CS), which is compiled from consumer confidence surveys. The Michigan Consumer Research Centre has been conducting the survey since 1947, assessing consumer confidence regarding personal finances, business conditions and purchasing power based on 500 surveys. Answers to each question are translated into a relative score to compile the index. The index is available on a monthly basis since 1978 and is considered one of the leading indicators of the perceived health of the US economy. The CS is viewed as a measure of individual investor sentiment since it is based on household surveys. It has been employed in empirical studies by researchers such as Lemmon and Portniaguina (2006), Lemmon and Ni (2010) and Han (2008). Han (2008) employs the bull-bear spread as indication of bullish (bearish) market sentiment. He finds that the bull-bear spread captures the sentiment of institutional investors and helps to explain the time variation in RNS of S&P 500 options. The bull-bear spread (BB), which is computed by Investors Intelligence, is based on regular surveys of over 120 market newsletters and measures the difference between the percentage of bullish advisors and the percentage of bearish advisors. This measure is based on the opinions of professional advisors and so we use it as a proxy for institutional sentiment. Summary statistics Option prices are collected from the Ivy Database of OptionMetrics which provides historical 8

10 prices of all US listed equity and index options based on the closing quotes at the Chicago Board Options Exchange (CBOE). We extract the security ID, trading date, expiration date, call or put flag, strike price, best bid, best offer and implied volatility from the option price file for the following indices and sample periods. The data are from June 2001 to January 2010 for the Russell SmallCap 2000 Growth (RSG) and Value (RSV) indices, the Nasdaq (NDAQ), and the S&P 500 (SPX) index 3, from December 2001 to January 2010 for the Russell LargeCap 1000 Growth (RLG) and Value (RLV) indices, and from October 2002 to August 2009 for the S&P SmallCap 600 Growth (SMLG) and Value (SMLV) indices. The underlying security prices and interest rates are taken separately from the security price file and the zero curve file. End-ofmonth observations are used to construct the risk-neutral moments. The following conventional selection criteria are applied to our data. First, we require positive trading volume to avoid stale quotes. We use the mid-point of the most competitive bid and ask quotes as the price for options to reduce the noise induced by the bid-ask bounce. Second, we use only OTM call and put options which are the most frequently traded and effectively cover the whole moneyness spectrum. Third, we focus only on the trading days that have at least two OTM put and call option prices for the purpose of interpolation. It is important to note that since we do not have a continuum of option prices, there are two possible sources of biases, namely discreteness and asymmetry of integration. Therefore, for each maturity, we follow Jiang and Tian (2005) and interpolate implied volatilities across moneyness levels (K/S) to obtain a continuum of implied volatilities. For moneyness below (above) the lowest (highest) available moneyness level in the market, we use the implied volatility at the lowest (highest) strike price. After this interpolation and extrapolation procedure, we are able to generate a fine grid of implied volatilities over a wide range of moneyness levels 3 While this is not a style index, it is included for information. 9

11 and they in turn provide us with OTM call and put option prices using the Black-Scholes formula as a mapping tool (Black and Scholes, 1973). Model-free skewness is calculated using options with two months to expiration (60 days), as this is one of the most actively traded maturities for the Russell 1000 and 2000 and the S&P style index options. We first compute model-free moments using options that have the nearest maturities (less than 60 days and greater than 60 days) to construct a constant two-month to maturity RNS. Then we linearly interpolate between the two maturities to synthesize the RNS with a constant 60 days to maturity. The final sample contains seven series of between 95 and 104 monthly observations of option-implied RNS for the style indices and 104 observations for the SPX. Table 1 reports the summary statistics of RNS inferred from option index prices. [Table 1 around here] The mean skewness of the risk-neutral density is negative for all indices, indicating a higher probability of downward movement than predicted by the lognormal distribution. Excluding the SPX index with a skewness of -1.77, the most negatively skewed style index is the RSV (-1.04) while the SMLG is the least skewed (-0.66). Interestingly, the NDAQ skewness lies in the midrange of the other three growth indices. The two Russell value indices are more negatively skewed than the two Russell growth indices while the S&P SmallCap indices both have similar levels of skewness. We formally test the hypothesis that the RSV and RSG and RLV and RLG have the same mean skewness using the time series of monthly observations. Standard errors are adjusted for heteroscedasticity and serial correlation following the Newey-West (1987) method. The results suggest that the RSV mean skewness is significantly lower (more negative) than the RSG skewness at the 1% significance level. Similarly, the hypothesis of equal mean skewness for 10

12 RLV and RLG is rejected at 10% level. This provides an initial indication of differences between the skewness of value and growth indices. The median skewness measures are very similar except for the NDAQ where the median is larger by a factor of 0.2. The standard deviations of the two Russell and S&P value indices are less than those for their corresponding growth indices and this is significant at 5% level for all three pairs. The NDAQ exhibits the highest standard deviation of all style indices at The skewness of all the indices is positively autocorrelated at the 5% significance with the exception of the SMLV and SMLG indices. In Figure 1 we show the time series of the risk-neutral skewness of these seven indices. [Figure 1 around here] The risk neutral skewness of the style indices varies over time but to a lesser extent than the NDAQ. The latter s RNS exhibits a downward trend over the whole sample period. The only interruption is a very sharp dip in The skewness of the RSG, RSV, RLG and RLV indices seems to be more stable over time and substantially so during the period of the banking and financial crisis. Indeed, the RLV index trends upwards since Table 2 presents descriptive statistics of the sentiment measures and control variables. [Table 2 around here] The average level of the Michigan Consumer Confidence Index (CS) is with a standard deviation and strong autocorrelation (0.95). Lemmon and Ni (2010) report similar results. The average value of the BB variable is with a standard deviation of While the mean and median values are quite similar for the BB and CS measures, they differ for the BW measure. Here the mean is but the median is marginally negative at The BW measure is 4 During the market turmoil that started in September 2008, the risk-neutral skewness of the SPX also increased substantially (became less negative) as noted in Birru and Figlewski (2011). 11

13 the most volatile in relative terms since its coefficient of variation is 10.8 compared with 0.82 for BB and just 0.15 for CS. Table 2 also reports that the implied volatility for the seven style indices falls in the range and their serial correlation coefficients are in the range. Figure 2 plots the sentiment proxies (BW, CS and BB) from June 2001 to January [Figure 2 around here] The BW index falls to a sample low in 2003, increases gently to 2007 before falling again and remaining relatively flat since The CS index shows little variation over the sample period. After September 2008, the CS reaches historical lows but starts to increase slowly since the middle of The BB spread exhibits the most variation. It declines from 2007 but recovers in late Table 3 reports the correlation coefficients of the variables used in the empirical analysis. [Table 3 around here] The unconditional pairwise correlation coefficients between BW and CS and BW and BB are and , respectively. Finally, that between CS and BB is The CS variable has an unconditional positive correlation with the RNS of all four growth indices and an unconditional negative correlation with the three value indices. The CS and BB indices yield similar results for the value indices also. The CS and BB indices exhibit a positive correlation with the RSG and NDAQ indices but a negative correlation with the RLG and SMLG indices. 3.2 Regression results In this section, we report time series regression results to examine how our three sentiment measures affect the skewness of the risk-neutral density of our seven style index options. The dependent variable is the RNS of the relevant style index returns. This is inferred from the contemporaneous index option prices evaluated on the last trading day of each month. The 12

14 sentiment proxies are measured on or prior to that date using the latest available data. The following baseline regression is employed, Skew = a + b Sent + b Vol + b Skew + ε (6) t 1 t 2 t 3 t 1 t where Skew t is the RNS computed following equation (5) at time t, Sent t is a sentiment variable at time t and measured either by a general sentiment index (BW), institutional sentiment (BB) or individual sentiment (CS). We include lagged skewness Skewt 1 as a control variable to take into account the positive autocorrelation in the dynamics of RNS. We also include the implied volatility Vol t as an independent variable because, in stochastic volatility models such as that of Heston (1993), volatility is an important determinant of skewness (Han, 2008). The implied volatility is constructed according to equation (4) and has a constant 60-day to maturity. Full sample In the regression analysis, we seek to highlight the contrast between the separate impact of the three sentiment measures on individual style indices. Table 4 reports the empirical results for the coefficients on the various style indices and their t-statistics for the full sample period. We employ the Newey-West method to account for heteroscedasticity and serial correlation of the standard errors (Newey and West, 1987). [Table 4 around here] The results are novel in the option anomalies literature. One sentiment measure has a significantly positive impact on the RNS derived from each of four growth index options at the 1% significance level and the economic impact is considerable in all four cases. The generic BW sentiment measure exerts a positive effect of similar economic magnitude on the RLG and SMLG indices or half of our growth style indices. This supports a cross sectional effect of general 13

15 sentiment on both large firm and small firm growth indices. The individual consumer sentiment measure CS has a positive effect on the NDAQ and RSG indices. The economic impact on the large firm NDAQ index is almost twice that on the small firm RSG index. In economic terms, when consumers become more (less) optimistic, the risk-neutral density of the Nasdaq 100 index is less (more) negatively skewed. This is a novel finding since Lemmon and Portniaguina (2006) show that the Michigan index is a good predictor mainly for the returns of just small capitalization stocks. 5 Finally, the coefficient on institutional sentiment measured by the bull/bear spread (BB) is statistically insignificant for all four indices. Overall our results suggest that individual and general (but not institutional) sentiment measures have a significant impact on growth style indices. These sentiment measures affect both large and small cap indices alike The results for the three value indices are also interesting but different. Now the RNS derived from two of the three value style indices are negatively impacted by a sentiment measure at the 1% significance level. Again the BB measure is statistically insignificant for all three value indices. There is evidence of a small firm effect in the case of value indices. The BW sentiment measure exerts a negative effect on the RSV index while the CS measure has a negative effect on the SMLV index. This finding is consistent with the results of Lemmon and Portniaguina (2006) and those of Lemmon and Ni (2010). The latter find that the impact of CS on the implied volatility smile of individual stock options is more pronounced in the case of small capitalization stocks. Finally we examine the impact of sentiment measures on the RNS of the S&P 500 index options to compare our results with those in the literature. Neither the coefficient on the BW measure nor that on the CS measure is statistically significant. Interestingly, the coefficient on 5 These small growth portfolios, whose returns are more difficult to explain by standard rational asset pricing models (see, for example, Hodrick and Zhang, 2001) see a bigger impact of CS on the implied volatility smile. 14

16 institutional sentiment measured by the bull/bear spread (BB) is also statistically insignificant. This finding contrasts with Han (2008) who finds that the BB is significantly positively related to the RNS of S&P 500 index options. 6,7 Our main empirical results can be summarized as follows. All of the four growth indices are impacted by either general (BW) or individual consumer sentiment (CS). Previous studies have shown that the returns of growth stocks are driven by sentiment and investor overreaction. Here our empirical evidence suggests that individual investor sentiment also has an impact on the option prices of growth portfolios. The latter exerts a significantly positive effect on the RNS of growth index options (Russell 2000 Growth index and Nasdaq 100 index) and especially on the latter. The finding of a significant CS impact on a large firm index is novel. Sub samples Since the sample period includes the banking and financial crisis that commenced in late 2007, we run the same regressions over two sub-samples ( and ) and report the results in Table 5. [Tables 5 around here] 6 We are able to replicate the results in Han (2008) using the BB variable for the earlier sample period in his paper ( ) and so the difference stems from the distinctive sample periods under investigation. To make sure that these results are not due to errors in our RNS measure, we used data on SPX risk-neutral skewness provided by the CBOE ( and the results are qualitatively the same with similar coefficients and t-statistics. 7 It would be interesting to formally test whether our empirical results are consistent with available option volume data by different types of investors. However, only data for the two-year period from October 2009 to October 2011 were obtained from the Options Clearing Corporation (OCC). These data show that 42% of trading of the NDAQ and RSG originated from brokerage or full-service customers compared with 31% for the SPX index options and 32% for the RSV index options. 15

17 The results for the pre-crisis sub-sample generally replicate those for the full sample. The four growth indices are significantly impacted by the same sentiment measures as was the case for the full sample period. The economic impact is stronger in all cases and especially so for the small firm indices (RSG and SMLG). The SMLG is now also negatively affected by the CS sentiment measure in line with the unconditional negative correlation between these two variables. Lastly, the results for the three value style indices are qualitatively similar to those for the full sample with the exception of a stronger economic impact of CS on SMLV. The empirical results for the crisis period are somewhat distinctive. The BW sentiment index positively impacts the NDAQ at the 5% significance level and the SMLG at the 10% significance level. The BB sentiment measure also positively impacts the SMLG at the 10% significance level albeit only marginally in economic terms. This is the only significant result for the BB measure in our study. Finally, the BW sentiment index negatively impacts the RLV at the 5% significance level and this is the only significant results for the value style indices in this period. The lack of significant results in this period is perhaps not surprising as positive investor sentiment wanes or disappears in periods of crisis. Summing up, the results for the pre-crisis sub-sample replicate and reinforce the novel results for the full sample. Growth style index options are positively affected by sentiment measures while value style indices are negatively affected. Not surprisingly, there is a paucity of significant results in the crisis period. Our results support the view that the impact of sentiment is time varying in line with Stambaugh, Yu and Yuan (2012). They argue that anomalies are stronger following high levels of sentiment. This is not surprising given that our sample period contains a bull market phase up to around 2007 when the financial and banking crisis began. 16

18 3.3 Robustness tests Additional control variable In this section, we perform robustness tests on the relation between the sentiment measures and the RNS of index options. The regression analysis now includes an additional control variable that is a proxy for the relative demand of index options (Relative Demand). Bollen and Whaley (2004) and Garleanu, Pedersen and Poteshman (2009) suggest that net buying pressure is an important determinant of the slope of the implied volatility smile. Following Han (2008) and Bollen and Whaley (2004), demand pressure is defined as the ratio of the open interest for outof-money put options to the open interest for near- and at-the-money call options. The OMT puts are options with Black-Scholes delta 3/ 8 p 1/ 8. The near and at-the-money options include calls with delta 1/2 c 5/8 and puts with delta 1/2 p 3/8. Table 6 reports the sentiment coefficient estimates and the corresponding t-statistics. [Table 6 around here] After the inclusion of relative demand in our baseline regression, we find that the coefficient on the individual sentiment measure CS is significantly positive for RSG and NDAQ. This is consistent with our results for the full sample reported in Table 4. However, neither the BW nor BB coefficient is significant for any growth indices. For value indices, the BW coefficient is significantly negative for RSV and that on CS is negative and significant for RLV. These results are qualitatively similar to those for the full sample in Table 4. Future return jumps Similar to Lemmon and Ni (2010), we test whether individual sentiment is correlated with investors assessment of future return jumps. If investors assess rationally the probability of 17

19 futures jumps, this information could be reflected in the RNS. They would buy more put options relative to call options if they expect a negative jump and more calls relative to puts if they expect a positive jump. To examine if jump expectations subsume the information contained in CS (and BW and BB), we use one-month future realized skewness (Future Realized Skewness) as a proxy for future return jumps. Realized skewness is calculated using daily returns within every month in our sample and used as an additional control variable with a one-month forward lag in regression (6). The results are reported in Table 7. [Table 7 around here] Consistent with the previous results, CS remains significantly positively related to the RNS of the RSG and NDAQ growth indices and BW is significant and positive for RLG. For value indices, BW is significant and negatively impacts the RSV. These robustness results are qualitatively similar to our main results in Table Economic significance In this section, we examine the economic significance of the previous empirical results. Since we have found that sentiment measures have a positive impact on growth index options but a weaker negative one on value index options, we test if it possible to generate abnormal profits by implementing implied skew trading strategies in the two option markets. Our implied skew strategy is based on variance swaps trading. We choose this approach because Kozhan, Neuberger and Schneider (2011) show that the skew premium depends on the same underlying factors as the variance risk premium and strategies designed to exploit one of the risk premiums and hedge out the other make zero excess returns. The return (RP) or payoff to the variance swap can be defined in two different ways. First, RP is the difference between the realized variance (RV) on the floating leg and the variance 18

20 swap rate (SW) on the fixed leg: RPt, T = RVt, T SWt, T. This is the profit per $1 investment in the fixed leg of the swap. Conversely, the payoff to the variance swap seller is given by SW RV. The payoff can also be defined in percentage terms as PRPt, T = RVt, T / SWt, T 1. t, T t, T On the last trading day of each month in our sample period we use the risk-neutral variance computed from equation (4) as the variance swap rate and we calculate the realized variance over the next two months using n 2 t, T = ln( t i t i 1), where S + + t is the price of the underlying i= 1 RV S S index at time t. The RV is then annualized by multiplying by 252/n, where n is the number of trading days. 8 We calculate the payoffs from the variance swap strategies for the Russell SmallCap 2000 Growth (RSG) and Value (RSV) indices, Nasdaq (NDAQ), and the S&P 500 (SPX) index from June 2001 to January 2010, for the Russell LargeCap 1000 Growth (RLG) and Value (RLV) indices from December 2001 to January 2010, and for the S&P SmallCap 600 Growth (SMLG) and Value (SMLV) indices from October 2002 to August The descriptive statistics are reported in Table 8. [Table 8 around here] The average payoff of the variance swap strategies is a direct estimate of the variance risk premium. Consistent with previous results in the literature, the variance risk premium is negative for all indices. The variance risk premium of the growth indices is higher (in absolute terms) than the variance risk premium of the value indices. For example, the variance risk premium of the growth index RSG is almost four times larger (in absolute terms) than the variance risk premium of the value index RSV and similar in magnitude to the variance risk premium of SPX. However, 8 Annualizing RV using 360/60 produces similar results. 19

21 the t-statistics suggest that only the variance risk premium of the RSG and NDAQ indices is statistically significant. The average realized two-month volatilities of the various pairs of growth/value indices are quite similar: RSG/RSV (23%, 22%), RLG/RLV (18%, 18%) and SMLG/SMLV (21%, 23%). The fact that variance risk is more heavily priced in the growth index options (and especially in the NDAQ) compared to value index options, despite the fact that the indices have similar return variances, suggests that either growth index return volatility is more exposed to some risk factor or that growth index options are mispriced. Note that the maximum percentage return on the variance swap strategies is quite high (e.g., 300%). These returns refer to the 2008 period after the collapse of Lehman Brothers when realized variance increased dramatically and writers of variance swaps incurred huge losses. SR is the annualized Sharpe ratio which is calculated by dividing the average PRP by its Newey-West standard deviation and then multiplying by 252/44 to annualize it. The highest Sharpe ratios (0.83 and 1.05) are generated by trading RSG and NDAQ variance swap strategies, respectively. We employ the Fama and French (1993) three factor model augmented by the Carhart (1997) momentum factor and the SPX variance risk premium where the latter is used as a proxy for systematic variance risk. The following equation is estimated to test whether the returns on the variance swap strategies in the value and growth index options can be explained by known risk factors: PRP = a + β R + β PRP + β SMB + β HML + β UMD + ε (7) M SPX t, T 1 t, T 2 t, T 3 t, T 4 t, T 5 t, T The excess returns on the market portfolio (R M ), the size factor (SMB), the value factor (HML) and the momentum factor (UMD) are all taken from Professor Ken French s website. 20

22 Table 9 reports the coefficient estimates of regression (7) with Newey-West t-statistics. [Table 9 around here] All variance swap strategies have a significant exposure to systematic variance risk and the estimated β 2 coefficient is highly significant. To test if it is possible to generate abnormal returns by trading simultaneously growth and value index options, we implement long/short strategies. In the long/short strategies we buy two-month variance swaps on value indices and we sell twomonth variance swaps on growth indices. We implement the long/short strategies for the following pairs of value/growth index options: RSV/RSG, RSV/NDAQ, RLV/ RLG and SMLV/ SMLG. We use regression (7) to test whether the returns of the long/short strategies can be explained by known risk factors. The coefficient estimates of the regression are reported in the lower panel of Table 9. The RSV/RSG strategy has insignificant exposure to systematic variance risk and a significant bimonthly alpha of 18% with an annualised Sharpe ratio of The RLV/ RLG strategy has also insignificant exposure to systematic variance, a Sharpe ratio of 0.95 and the bi-monthly alpha is 13%. The RSV/NDAQ strategy achieves the highest abnormal bi-monthly return of 26% with an impressive Sharpe ratio of 1.1 and zero exposure to systematic variance risk. These generally high abnormal returns provide stronger evidence of a value premium type anomaly in the index options markets than do the related studies of Blackburn et al. (2009) and He et al. (2010). 5. Concluding remarks This paper examines the relationship between three investor sentiment measures - the Baker- Wurgler index (BW), the Michigan Consumer Confidence Index (CS), the bull-bear spread (BB) - and style index options. The latter include the Russell 1000 and 2000 Growth and Value indices, 21

23 the S&P SmallCap 600 Growth and Value indices, and the Nasdaq 100 index. The primary issue it addresses is whether investor sentiment affects the risk-neutral skewness of value and growth index options in different ways. Our results indicate that sentiment measures exerts a significantly positive impact on the pricing of options written on the four growth indices, the Russell 1000 and 2000 Growth indices, the S&P SmallCap 600 Growth index and the Nasdaq 100 indices. They also show that sentiment measures negatively affect the skewness of the two of the three value indices. These are novel findings in the options literature. Our results are economically significant. They can earn high abnormal returns by trading simultaneously growth and value index options in long/short strategies. The RSV/NDAQ strategy achieves the highest abnormal bi-monthly return of 26% with an impressive Sharpe ratio of 1.1 and zero exposure to systematic variance risk. These results suggest a type of value premium anomaly in options markets. They support stronger evidence on this type of anomaly than do related studies such as those of He et al. (2010) and Blackburn et al. (2009). Finally, our findings are consistent with the behavioral theories of Lakonishok, Shleifer, and Vishny (1994) and Baker and Wurgler (2006) relating to the mispricing of growth stocks in the underlying equity markets. In future research, it would be interesting to explore the causes of the value premium type anomaly. Can it be explained by a clientele effect or by distinctive behaviour by different groups of investors? Or can it be rationalised by arbitrageurs like hedge funds taking advantage of mispricing in style index options? 22

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27 Table 1. Summary statistics for risk-neutral skewness This table reports summary statistics of monthly risk-neutral skewness of options written on growth/value/market-wide indices following Bakshi, Kapadia and Madan (2003). The sample period is from June 2001 to Jan 2010 for Russell SmallCap 2000 Growth/Value indices, Nasdaq, and S&P 500 index, from Dec 2001 to Jan 2010 for Russell LargeCap 1000 Growth/Value indices, and from Oct 2002 to Aug 2009 for S&P SmallCap 600 Growth/Value indices. Index Mean Median Stdev Serial Correlation Russell SmallCap 2000 Growth Russell LargeCap 1000 Growth S&P SmallCap 600 Growth Nasdaq Russell SmallCap 2000 Value Russell LargeCap 1000 Value S&P SmallCap 600 Value S&P

28 Table 2. Summary statistics for sentiment measures and control variables This table reports summary statistics of sentiment measures and control variables. The sentiment measures are the Baker and Wurgler (2006) sentiment (BW), the Consumer Confidence index of University of Michigan (CS), and the bull-bear spread computed by Investors Intelligence (BB). Implied volatility is derived following Bakshi, Kapadia and Madan (2003). The sample periods are from June 2001 to Jan 2010 for the sentiment measures and Russell SmallCap 2000 Growth/Value indices, Nasdaq, and S&P 500 index, from Dec 2001 to Jan 2010 for Russell LargeCap 1000 Growth/Value indices, and from Oct 2002 to Aug 2009 for S&P SmallCap 600 Growth/Value indices. Variables Mean Median Stdev Serial Correlation Baker and Wurgler index (BW) Consumer Confidence index (CS) Bull-bear spread (BB) Implied volatility Russell SmallCap 2000 Growth Russell LargeCap 1000 Growth S&P SmallCap 600 Growth Nasdaq Russell SmallCap 2000 Value Russell LargeCap 1000 Value S&P SmallCap 600 Value S&P

29 Table 3. Correlation coefficients of variables This table reports correlation coefficients of sentiment measures and the risk-neutral skewness of growth/value/market-wide indices. The sentiment measures are the Baker and Wurgler (2006) sentiment (BW), the Consumer Confidence index of University of Michigan (CS), and the bull-bear spread computed by Investors Intelligence (BB). The sample periods are from June 2001 to Jan 2010 for the sentiment measures and Russell SmallCap 2000 Growth/Value indices, Nasdaq, and S&P 500 index, from Dec 2001 to Jan 2010 for Russell LargeCap 1000 Growth/Value indices, and from Oct 2002 to Aug 2009 for S&P SmallCap 600 Growth/Value indices. BW CS BB BW CS BB Russell SmallCap 2000 Growth Russell LargeCap 1000 Growth S&P SmallCap 600 Growth Nasdaq Russell SmallCap 2000 Value Russell LargeCap 1000 Value S&P SmallCap 600 Value S&P

30 Table 4. Investor sentiment and risk-neutral skewness: full sample The dependent variable is the risk-neutral skewness of growth/value/market-side index options derived following Bakshi, Kapadia, and Madan (2003). The explanatory variables are the Baker and Wurgler (2006) sentiment (BW), the Consumer Confidence index of University of Michigan (CS), and the bullbear spread computed by Investors Intelligence (BB). The sample period is from June 2001 to Jan 2010 for Russell SmallCap 2000 Growth/Value indices, Nasdaq, and S&P 500 index, from Dec 2001 to Jan 2010 for Russell LargeCap 1000 Growth/Value indices, and from Oct 2002 to Aug 2009 for S&P SmallCap 600 Growth/Value indices. The HAC-adjusted Newey-West (1987) t-statistics are reported in the parentheses. The coefficients and t-statistics for two additional control variables (implied volatility and lagged skewness) are not reported but available upon request from the authors. Growth indices BW CS BB Russell SmallCap 2000 Growth (0.0795) (3.3461) ( ) Russell LargeCap 1000 Growth (2.5321) (0.1434) ( ) S&P SmallCap 600 Growth (2.5837) ( ) (1.1129) Nasdaq Value indices (0.5544) (2.7999) (1.4524) Russell SmallCap 2000 Value ( ) (0.0203) ( ) Russell LargeCap 1000 Value ( ) S&P SmallCap 600 Value Broad market index ( ) ( ) ( ) S&P (0.0435) ( ) ( ) 29