Hypothesis - Data Description and Research Methodology

Chapter 3. Data Description and Research Methodology

3.3. Hypothesis

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%∆𝑉𝐼𝑋_𝑡= 𝛼₀+ 𝛼₁𝑅_𝑡+ 𝛼₁₁𝑅_𝑡²+ 𝛼₁₂𝐷_𝑡−1𝑅_𝑡+ 𝛼₁₃𝐷_𝑡−1+ 𝜀_𝑡

And we also separate returns into positive and negative returns to see the investor sentiment effect in different parts.

The above-mentioned is also implemented into VXN and NASDAQ-100 returns.

3.3. Hypothesis

Our hypothesis is investor sentiment has impact on the return-implied volatility relation. Previous literatures have proven the negative relation between return and implied volatility, and that lagged returns and lagged changes in implied volatility are also factors to determine changes in implied volatility. Here we care about whether investor sentiment would affect the negative relation.

Hypothesis 1: Investor sentiment intensifies the negative relation between return and change in implied volatility.

Hypothesis 2: Investor sentiment mitigates the negative relation between return and change in implied volatility.

Hypothesis 3: When returns are positive, investor sentiment intensifies the negative relation between return and change in implied volatility.

When positive returns occur, investors would decrease the implied volatility.

When in high sentiment periods, investors are more optimistic about the future, and the occurrence of positive returns fit their expectation. This raise their confident about the future, making them more certain that market will go up. Thus, they would decrease the implied volatility more than without sentiment.

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Hypothesis 4: When returns are negative, investor sentiment mitigates the negative relation between return and change in implied volatility.

When negative returns occur, investors would be panic and are afraid of carry on decline of the market, and this would raise the implied volatility. When in high sentiment periods, investors are more optimistic and have much more confident about the future, so they are not so scared by negative returns. Thus, the addition of implied volatility would be mitigated by investor sentiment.

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4. Empirical Results

4.1. Summary statistics of return, volatility and investor sentiment

4.1.1. Daily summary statistics

Table 1 shows the summary statistics of daily index returns, implied volatility and sentiment dummy of S&P 500 index and VIX. There are 5315 observations. The mean and standard deviation of daily S&P 500 return is 0.031% and 1.169, and for daily VIX is 20.381 and 8.2230. The mean of ∆𝑉𝐼𝑋_𝑡 and %𝑉𝐼𝑋_𝑡 are 0 and 0.182%, and the standard deviation is 1.509 and 6.103. For daily S&P 500 and VIX data, there are 45.8% days of 5315 days are classify as high sentiment ones.

Table 2 shows the summary statistics of VXN and NASDAQ-100 index return daily data. The mean and standard deviation of daily NASDAQ-100 return is 0.000%

and 1.894, and the VXN is 29.673 and 13.623. The mean of ∆𝑉𝑋𝑁_𝑡 and %𝑉𝑋𝑁_𝑡 are -0.01 and 0.089%, and the standard deviation is 1.698 and 5.079 respectively. For NASDAQ-100 and VXN’s daily data, 46.5% of 2512 days are classified as high-sentiment days.

4.1.2. Weekly summary statistics

In weekly data, Table 3 shows the summary statistics of weekly S&P 500 index return, VIX, and sentiment dummy. The mean and standard deviation of weekly S&P 500 return is 0.155% and 2.552, and for weekly VIX is 20.353 and 8.231. The weekly mean of ∆𝑉𝐼𝑋_𝑡 and %𝑉𝐼𝑋_𝑡 are 0.003 and 0.688%, and the standard deviation are 2.317 and 12.08. For daily S&P 500 and VIX data, there are 45.4% weeks of 1063 weeks are classify as high sentiment ones.

Table 4 shows the summary statistics of VXN and NASDAQ-100 index return weekly data. The mean and standard deviation of weekly NASDAQ-100 return is 0.051% and 3.991, and the VXN is 29.710 and 2.711. The mean of ∆𝑉𝑋𝑁_𝑡 and

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%𝑉𝑋𝑁_𝑡 are -0.066 and 0.395%, and the standard deviation are 3.737 and 11.152 respectively. For NASDAQ-100 and VXN’s weekly data, 46.9% of 503 weeks are classified as high-sentiment weeks.

4.2. Empirical result for daily data

4.2.1. Daily results of changes in VIX

Table 5 shows the regression result for daily changes in the VIX. Our main findings suggest that sentiment is a significant factor that mitigated the negative return-implied volatility relation. The negative return-implied volatility relation is mitigated significantly during high sentiment period, and support the hypothesis 2.

The coefficients of interaction of investor sentiment and return, 𝛼₁₂ in our models, are all positive. The coefficients are 18.141, 21.805, 1.440 and 1.429 in model 1, 2, 3, and 4 respectively, and the t-values are 5.89, 7.12, 10.95 and 10.88.

In order to investigate the whether the interaction of investor sentiment and return has different impact on market decline or flourish, we segregate S&P 500 daily returns into positive and negative returns. Panel A and B of Table 6 report the results of the four regression models for daily change in VIX and support hypothesis 4. The results show that investor reduce their risk averse when negative returns occur. The mitigate effect of interaction of investor sentiment and return is stronger when the market decline, for the coefficients in panel B are all higher than in panel A. We can say that when the market falls, investor sentiment reduced their panic. The VIX still raise, but would increase less when in high sentiment period. Investors are more optimistic and have much more confident about the future, so they are not so scared by negative returns and the addition of implied volatility would be smaller.

Table 7 and Table 8 report the sentiment effect of investor sentiment on changes in daily VIX. In overall daily data, investor sentiment has no sentiment effect on

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return-implied volatility relation, as in Table 7 the coefficients of sentiment dummy aren’t significant. While in Table 8 we segregate the returns and regress the sentiment effect again, and find that the investor sentiment has sentiment effect, though not much, when in positive returns. Why investor sentiment make the VIX raise slightly when in positive returns? Yu and Yuan (2011) conjecture that individual traders are the primary candidates for sentiment traders, and they are more active during high-sentiment periods. These individual traders tend to be inexperienced and naive investors, and are more likely to misestimate variance. And De Long et al. (1990) find noise trading is associated with increased price volatility. Brown (1999) also finds that unusual levels of individual investor sentiment are associated with greater volatility. So here in our study, we suppose that in high sentiment period, the sentiment effect of investor sentiment during positive returns is caused by individual traders who are inexperienced and noise trading, making the implied volatility slightly high.

4.2.2. Daily results of changes in VXN

Table 9 shows the regression results of daily changes in VXN. Contract to changes in VIX, the changes in VXN don’t be affected by the interaction of investor sentiment and return, for the coefficients of interaction of investor sentiment and return are not significant. While in Table 10, after returns are segmented, the interaction of investor sentiment and return weakly intensified the return-implied volatility changes relation in negative returns, as in panel B the coefficient in model 1 is -10.66, and t-value -1.72. This result is inconsistent with our hypothesis, since we support sentiment would mitigate the relation between return and changes in implied volatility.

Table 11 and Table 12 report the sentiment effect of investor sentiment on

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changes in daily VXN. In overall daily data, investor sentiment has no sentiment effect on return-implied volatility relation, as in table 11 the coefficients of sentiment dummy aren’t significant. While in Table 12, we find a weakly positive sentiment effect in positive returns and a stronger negative sentiment effect in negative returns.

The weakly positive sentiment effect in positive returns only significant in model 4 for the coefficient 0.007, and t-value 2.02. However, in negative return part, panel B of Table 12 shows investor sentiment has a consistent negative sentiment effect on the changes in VXN, and this means that in NASDAQ market, the negative return-implied volatility relation would be mitigated by the sentiment effect of investor sentiment.

4.2.3. Conclusions of daily results

For changes in VIX, high investor sentiment mitigates the negative return-volatility relation by interaction of investor sentiment and return, especially when in negative returns period. This means that sentiment changes the risk attitude of investors, and they are not so risk averse during high sentiment periods. The sentiment effect only exist in positive return, and we suspect that it’s due to the noise traders who are more active in high sentiment periods, especially when the positive returns occur.

For changes in VXN, the interaction of investor sentiment and return don’t has impact on the overall return-implied volatility relation, and only a weakly intensified effect when in negative returns. There is positive sentiment effect in positive returns, the same as VIX daily data. The investor sentiment mitigates the negative return-implied volatility relation through sentiment effect, since the sentiment effect is negative when negative returns occur.

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4.3. Empirical result for weekly data

4.3.1. Weekly results of changes in VIX

Table 13 shows the regression result for weekly changes in the VIX. We find that the negative return-implied volatility relation is mitigated significantly during high sentiment period, and support the hypothesis 2. The coefficients of interaction of investor sentiment and return, 𝛼₁₂, are positive in model 3 and model 4, the coefficients are 0.873 and 0.942 respectively, and the t-values are 3.51 and 3.77.

Again, we separate returns to investigate whether interaction of investor sentiment and return has different impact on positive returns and negative returns.

Table 14 reports the result. Contrary to daily data, we find the mitigation effect only exist in negative returns, as in panel B the coefficients of interaction of investor sentiment and return are significant in model 3 and 4, for t-value 3.28 and 2.68 respectively, while in panel A they are not significant at all. This consistent with our hypothesis 4 that investor sentiment mitigates the negative relation between return and change in implied volatility when returns are negative.

Then we test the sentiment effect of investor sentiment. Table 15 and Table 16 show the results of the sentiment effect regressions. For all weekly data, there is no sentiment effect of investor sentiment, as the sentiment dummies aren’t significant at all. But after segmenting the returns, as the result of daily data, we discover there is positive sentiment effect during positive returns. We again suspect this is due to noise trader or sentiment trader who increase the implied volatility slightly when market bloom.

In panel A of Table 15, we also discover that in positive returns period, the interaction of investor sentiment and return intensifies return-implied relation, consistent with our hypothesis 3 that when in high sentiment periods, investors are

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more optimistic about the future and the occurrence of positive returns fit their expectation. This raise their confident about the future, making them more certain that market will go up. Thus, they would decrease the implied volatility more than without sentiment.

Moreover, in panel B of Table 15, we discover in negative returns period, interaction of investor sentiment and return mitigates return-implied relation, consistent with our hypothesis 4 that when in high sentiment periods, investors are more optimistic and have much more confident about the future, so they are not so panic when undergo negative returns. And this make investors decrease the addition amount of implied volatility during high sentiment period.

4.3.2. Weekly results of changes in VXN

Table 17 reports the regression results of weekly changes in VXN. Sentiment weakly reduce the size of change in VXN in model 3 and model 4, the interaction of investor sentiment and return coefficients are 0.429 and 0.547, and t-value are 2.02 and 2.7 respectively. In table 18, we find the interaction of investor sentiment and return only has a significant impact on the changes in implied volatility when market decline. In panel B when returns are negative, the investor sentiment reduces the negative relation between return and implied volatility in model3 and model 4. Table 19 and Table 20 test the sentiment effect of investor sentiment. We find investor sentiment has a weakly positive sentiment effect in both positive and negative returns, and we suspect it’s due to noise traders who increase the volatility slightly during high sentiment periods.

4.3.3. Conclusions of weekly results

For changes in weekly VIX, investor sentiment mitigates the negative return-implied volatility relation, especially in the negative returns part. Investors are

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less risk aversion during high sentiment periods. The VIX decrease more when positive return, and increase less when negative returns. Since VIX has been regarded as the fear index, the reduction of VIX indicates that investors are less panic during high sentiment periods.

For changes in weekly VXN, we obtain similar result to weekly VIX data. The investor sentiment mitigates the negative return-implied volatility relation, especially when market decline.

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5. Conclusion

The VIX tends to increase during stock market declines and decrease when the market advances. We examine how investor sentiment affects the changes in implied volatility, testing the short-term relation between the S&P 500 index return and the changes of VIX from January 1990 to January 2011, and between the NASDAQ-100 index return and the changes of VXN from February 2001 to January 2011 with proxy for beginning-of-period investor sentiment at both daily and weekly level.

Our main findings suggest that investor sentiment is a significant factor in explaining the negative return-implied volatility relation. We find that during high sentiment periods, the negative and asymmetric relation of return to changes in implied volatility can be mitigated significantly by interaction of investor sentiment and return. That is investor sentiment is associated with fewer implied volatility in index returns, except the daily changes in VXN.

When returns are segregated into positive and negative returns, investor sentiment has different impact on the size of changes in implied volatility. In weekly positive return data, investor sentiment leads to a downward revision in both changes in VIX and VXN. That is when market rise, implied volatility decrease more when investors are in high sentiment.

Furthermore, we find that the investor sentiment has a significant impact on the changes in implied volatility when returns are negative. Our results consistently show that when market decline, changes in implied volatilities are positively correlated with interaction of investor sentiment and return, except daily changes in VXN. Implied volatility tends to increase during stock market declines, while the investor sentiment changes investors’ reaction, making them less risk averse, and reduce the size of the addition in VIX and VXN. In other words, in negative returns, investors are more

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panic than in positive returns, but the panic can be mitigated significantly when investors are in high sentiment. Thus, sentiment can alter the risk attitude of investors and reduce their panic in the future, especially when market has negative performance.

The sentiment effect of investor sentiment is positive when returns are positive.

That is, when market rise, the sentiment effect of investor sentiment causes the implied volatility to slightly increase more in high sentiment periods. We suspect that it’s due to sentiment traders and noise traders who are inexperienced individual traders and noise trading, making the implied volatility slightly high.

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Table 1

Summary statistics of VIX and S&P 500 index return daily data

The S&P 500 index return and VIX sample period is January 1990 to January 2011. The Chicago Board Option Exchange VIX index is a market implied volatility of S&P 500 index options, reflecting market expectations of next 30 days volatility conveyed by S&P 500 stock index option prices. S&P 500 return is the return of the index includes 500 leading companies in leading industries of the U.S. economy. ∆𝑉𝐼𝑋_𝑡 is the change in the VIX from the close on day t minus the close on day t-1. %∆𝑉𝐼𝑋_𝑡 is the percentage change in VIX at time t. 𝑅_𝑡 is the return in the S&P 500 index from day t-1 to day t. |𝑅_𝑡| is the absolute value and 𝑅_𝑡² is the square of 𝑅_𝑡. D is the dummy variable for the high-sentiment periods, which from investor sentiment index in Baker and Wurgler (2007) based on first principal component of six (standardized) sentiments. 𝐷_𝑡 equals one if the beginning-of-period sentiment index value of the current month t is positive.

Mean Median Std. Deviation Min. Max. Observations

VIX 20.381 18.97 8.230 9.31 80.86 5315

∆𝑉𝐼𝑋_𝑡 0.000 -0.05 1.509 -17.36 16.54 5314

%∆𝑉𝐼𝑋_𝑡(%) 0.182 -0.301 6.103 -29.573 64.215 5314

𝑅_𝑡(%) 0.031 0.053 1.169 -9.035 11.580 5315

|𝑅_𝑡|(%) 0.788 0.540 0.864 0 11.580 5315

𝑅_𝑡²(%) 0.014 0.003 0.045 0 1.341 5315

D 0.458 0 0.498 0 1 5315

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Table 2

Summary statistics of VXN and NASDAQ-100 index return daily data

The sample period is February 2001 to January 2011. The Chicago Board Option Exchange VXN index is a market implied volatility of NASDAQ-100 index options, reflecting market expectations of next 30 days volatility conveyed by NASDAQ-100 index option prices.

%∆𝑉𝑋𝑁_𝑡 is the percentage change in VXN at time t. 𝑅_𝑡 is the return in the NASDAQ-100 index from day t-1 to day t. |𝑅_𝑡| is the absolute value and 𝑅_𝑡² is the square of 𝑅_𝑡. D is the dummy variable for the high-sentiment periods, which from investor sentiment index in Baker and Wurgler (2007) based on first principal component of six (standardized) sentiments. 𝐷_𝑡 equals one if the beginning-of-period sentiment index value of the current month t is positive.

Mean Median Std. Deviation Min. Max. Observations

VXN 29.673 25.19 13.623 12.61 80.64 2512

∆𝑉𝑋𝑁_𝑡 -0.01 -0.08 1.698 -12.96 12.71 2511

%∆𝑉𝑋𝑁_𝑡(%) 0.089 0.0358 5.079 -26.879 43.742 2511

𝑅_𝑡(%) 0.000 0.094 1.894 -10.519 12.580 2512

|𝑅_𝑡|(%) 1.309 0.878 1.369 0 12.580 2512

𝑅_𝑡²(%) 0.036 0.008 0.090 0 1.583 2512

D 0.465 0 0.499 0 1 2512

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Table 3

Summary statistics of VIX and S&P 500 index return weekly data

The S&P 500 index return and VIX sample period is January 1990 to January 2011. The Chicago Board Option Exchange VIX index is a market implied volatility of S&P 500 index options, reflecting market expectations of next 30 days volatility conveyed by S&P 500 stock index option prices. S&P 500 return is the return of the index includes 500 leading companies in leading industries of the U.S. economy. ∆𝑉𝐼𝑋_𝑡 is the change in the VIX from the close on week t minus the close on week t-1. %∆𝑉𝐼𝑋_𝑡 is the percentage change in VIX at time t. 𝑅_𝑡 is the return in the S&P 500 index from week t-1 to week t. |𝑅_𝑡| is the absolute value and 𝑅_𝑡² is the square of 𝑅_𝑡. D is the dummy variable for the high-sentiment periods, which from investor sentiment index in Baker and Wurgler (2007) based on first principal component of six (standardized) sentiments. 𝐷_𝑡 equals one if the beginning-of-period sentiment index value of the current month t is positive.

Mean Median Std. Deviation Min. Max. Observations

VIX 20.353 19 8.231 9.31 80.86 1063

∆𝑉𝐼𝑋_𝑡 0.003 -0.065 2.317 -25.58 21.03 1062

%∆𝑉𝐼𝑋_𝑡(%) 0.688 -0.430 12.080 -35.182 77.874 1062

𝑅_𝑡(%) 0.155 0.270 2.552 18.340 19.111 1063

|𝑅_𝑡|(%) 1.787 1.293 1.828 0.003 19.111 1063

𝑅_𝑡²(%) 0.065 0.017 0.209 0.000 3.652 1063

D 0.454 0 0.498 0 1 1063

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Table 4

Summary statistics of VXN and NASDAQ-100 index return weekly data

The sample period is February 2001 to December 2011. The Chicago Board Option Exchange VXN index is a market implied volatility of NASDAQ-100 index options, reflecting market expectations of next 30 days volatility conveyed by NASDAQ-100 index option prices.

NASDAQ-100 return is the return of the index includes 100 of the largest domestic and international non-financial securities listed on The NASDAQ Stock Market based on market capitalization. ∆𝑉𝑋𝑁_𝑡 is the change in the VXN from the close on week t minus the close on week t-1. %∆𝑉𝑋𝑁_𝑡 is the percentage change in VXN at time t. 𝑅_𝑡 is the return in the NASDAQ-100 index from week t-1 to week t. |𝑅_𝑡| is the absolute value and 𝑅_𝑡² is the square of 𝑅_𝑡. D is the dummy variable for the high-sentiment periods, which from investor sentiment index in

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