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In Table 12, the ARs of stocks on the expiration dates are not as significantly negative as the results in Table 11. A possible explanation is that investors would certainly exercise their call warrants in the expiration dates, so the issuers need no more stocks to hedge their short positions of warrants. They would sell their present holding positions in the stock market.
Table 12
The Abnormal Returns of Stocks During the Expiration Dates of All Non-Deep In-the-money Warrants
t-3 t-2 t-1 t t+1 t+2 t+3
Mean 0.0810*** 0.1086*** -0.0100 0.0010 0.0440** -0.0169 -0.0363*
Std. 2.1529 2.2242 2.1400 2.1527 2.0526 2.1083 2.0585
t-value 3.87 5.01 0.48 0.05 2.20 0.82 1.81
N 10553
Day t is the expiration dates of warrants. The abnormal returns (ARs) of stocks are the daily returns of underlying stocks minus the daily market returns. The sample comprises 10553 non-deep in-the-money warrants over the period 2006-2010. ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
In the following sections, we discuss further how the warrant market affects the stock market in different levels of investor sentiment.
C. Liquidity Change and Cumulative Abnormal Returns
The results in Table 13 show that the mean DVOL is －0.0309 with t＝10.66, highly significant. The coefficient of VOLRWS is significantly positive. It means if the average daily trading volume of warrants relative to the average daily trading volume of the stock before the (days －30 to －1) warrant expired raise, the trading volume of stocks relative to the market volume would also raise. Even though the mean of DVOL is negative, the increase in average stock volume is significantly induced by the decrease of numbers of living warrants.
The mean DSPD is positive but insignificant. We could see the coefficient of
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VOLRWS is significantly negative. It is consistent with the results of DVOL which means
that the greater the trading volume of warrants is, the greater the trading volume of the underlying stock is. And it also improved the liquidity of the stock market since the bid-ask spread decreased.
Then we could see the mean DTRQ is －0.0493 with t＝16.22, highly significant.
However, the coefficient of VOLRWS is significantly positive which represents that the liquidity of the stock market is indeed improved.
Finally, the average Dstd is significantly positive. It means that the market depth of the stock market is poorer after the warrants expired, but the coefficient of VOLRWS is insignificant.
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Changes in The stock liquidity following the warrant Exercise Regression Estimation Results
Dependent Variable Mean Constant VOLRWS R2
DVOL -0.0309*** indicate stock j and the market, respectively. A indicates the period of 30 trading days after the warrant expiration, days ＋1 to ＋30, and B indicates the period of 30 days before expiration, days －30 to －1.
𝐷𝑆𝑃𝐷𝑗= 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐴− 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐵. Roll (1984) proposes that 𝑆𝑃𝑅𝐸𝐴𝐷𝑗= 2 ∙ �−𝐶𝑂𝑉𝑗, where COV is the ratio of the average daily trading volume of warrants to that of the stock in the period of 30 trading days (days －30 to －1) before the warrant expired. The sample comprises 3549 warrants exercise over the period 2006-2010.
The estimated models are
(7) 𝐷𝐿𝐼𝑄𝑈𝐼𝐷𝐼𝑇𝑌𝑗= 𝛾0+ 𝛾1𝑉𝑂𝐿𝑅𝑊𝑆𝑗+ 𝑢𝑗,
Where DLIQUIDITYj＝DVOLj, DSPDj, DTRQj, or Dstdj.
t-statistics are in parentheses. The t-statistics if the regression coefficients are calculated using robust estimation of the standard errors, following White (1980). ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
Table 14 shows how all the factors we use above affect the CAR. Our main test is whether stock prices are affected by the warrants expired.
Hypothesis H1 could be explained as following reasons. According to the results in Table 13, we have noticed that the trading volume would transfer from warrants to stocks. The liquidity of the stock market increased after the warrants expired. The ensuing improvement in liquidity should bring higher stock prices (Amihud and Mendelson (1986), Brennan and Subrahmanyam (1996), Amihud, Mendelson, and Lauterbach (1997)).
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However, according to the results in Table 11, the mean value of the CAR is significantly negative. So we further study the relation between the CAR and the liquidity of stocks by examining Hypothesis H3. In Table 14, the coefficients of DVOL are significantly positive in the model (1) and (3). It is consistent with Hypothesis H3 which suggests that the coefficient of DVOL should be positive. If investors anticipate that the consolidation of trading between the warrants and the stocks improves liquidity, the increase in stock price should be an increasing function of the increase in its trading volume.
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Determinants of the Cumulative Abnormal Return (CAR) on the Expiration of All Deep-in-the-Money Warrants
Model (1) (2) (3) (4) (5) (6) (7) (8) (9) 𝑙𝑜𝑔�𝑉𝑂𝐿𝑗𝐵∕ 𝑉𝑂𝐿𝐵𝑚�. VOL is the average daily volume, and j and m indicate stock j and the market, respectively. A indicates the period of 30 trading days after the warrant expiration, days ＋1 to ＋30, and B indicates the period of 30 days before expiration, days －30 to －1. 𝐷𝑆𝑃𝐷𝑗= 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐴− 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐵. Roll (1984) proposes that over the post-exercise increase in number of shares. The sample comprises 3549 warrants exercise over the period 2006-2010. t-statistics are in parentheses. The t-statistics if the regression coefficients are calculated using robust estimation of the standard errors, following White (1980). ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
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The second measure is bid-ask spread, DSPD. In model (2), (3), and (4), the coefficients of DSPD are all significantly positive. It is consistent with Hypothesis H2 but contrary to Hypothesis H3. Then we could see the coefficients of the third and forth measures in model (5), (6), (7), and (8). Hypothesis H3 suggests that the coefficients of DTRQ and Dstd should be positive and negative, respectively. The higher the turnover
rate is, the higher CAR is. It represents that the consolidation of trading improves liquidity, and further benefits CAR. Similarly, the deeper the market depth is, the higher CAR is.
The fifth measure is DVOLRAT, which is the difference between DVOL and RATWS. Since the number of shares of stocks naturally increases after the warrant
exercise, it may be expected that the trading volume of stocks would increase as well.
However, most of warrants use cash to implement the contracts instead of using stocks.
Besides, part of the trading volume in the stocks may be due to arbitrage or hedge transactions between stocks and warrants. We obtain that the mean of DVOLRAT is － 0.0335 with t＝11.57, significantly different from zero, and the median is －0.0363, implying that the trading volume of most stocks increases by less than the increase in the number of shares of stock after the warrants expired. The result inflects that the part of the trading volume before warrants exercise is due to arbitrage or hedge trading. The coefficients of DVOLRAT are significantly positive in model (4) and (9). The results are consistent with Hypothesis H3.
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D. Investor Sentiment
In this section, we want to see if there is any results change in different levels of investor sentiment. We use VIX as a proxy to measure investor sentiment. As we mentioned in the previous section, we use the mean value of VIX in the whole sample period as a criterion to define the period of high VIX and low VIX.
Table 15 shows changes in the stock liquidity following the warrant exercise in the high VIX period which we calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, ＋30).
According to the results in Table 15, the main difference to the results in Table 13 is the coefficient of VOLRWS in the regression using Dstd as the dependent variable which is significantly positive in Table 15. It suggests when investor sentiment is relatively high, the market depth would be worse apparently.
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Changes in The stock liquidity following the warrant Exercise in the Period of High VIX
Regression Estimation Results
Dependent Variable Mean Constant VOLRWS R2
DVOL -0.0138*** indicate stock j and the market, respectively. A indicates the period of 30 trading days after the warrant expiration, days ＋1 to ＋30, and B indicates the period of 30 days before expiration, days －30 to －1.
𝐷𝑆𝑃𝐷𝑗= 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐴− 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐵. Roll (1984) proposes that 𝑆𝑃𝑅𝐸𝐴𝐷𝑗= 2 ∙ �−𝐶𝑂𝑉𝑗, where COV is the ratio of the average daily trading volume of warrants to that of the stock in the period of 30 trading days (days －30 to －1) before the warrant expired. The sample comprises 1138 warrants in the period of high VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, ＋30)) exercise over 2006-2010.
The estimated models are
(7) 𝐷𝐿𝐼𝑄𝑈𝐼𝐷𝐼𝑇𝑌𝑗= 𝛾0+ 𝛾1𝑉𝑂𝐿𝑅𝑊𝑆𝑗+ 𝑢𝑗,
Where DLIQUIDITYj＝DVOLj, DSPDj, DTRQj, or Dstdj.
t-statistics are in parentheses. The t-statistics if the regression coefficients are calculated using robust estimation of the standard errors, following White (1980). ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
Besides, the coefficients of DVOL and DTRQ are insignificant different from zero. It means that the effect of improvements in the stock liquidity is getting insignificant in the period of high VIX.
Further, we could see the relation between CAR and variables during the period of high VIX. We obtain that the mean of CAR in Table 16 is －0.1405 with t＝1.46, insignificantly different from zero. The coefficients of the first measure of liquidity,
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DVOL, are still significantly positive in model (1) and (3) as well as the results in Table
14. However, the coefficients of DSPD are insignificant in model (2), (3), and (4). These results are different to those in Table 14. It means that Hypothesis H2 is weaker in the period of high VIX. Then we could check the coefficients of DTRQ, Dstd, and DVOLRAT, they are all consistent with the results in Table 14. Going a step further, we could see that the moduli are bigger than the results in the whole sample period. It suggests that investors are more sensitive in the period of high VIX.
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Determinants of the Cumulative Abnormal Return (CAR) on the Expiration of All Deep-in-the-Money Warrants in the Period of High VIX
Model (1) (2) (3) (4) (5) (6) (7) (8) (9) 𝑙𝑜𝑔�𝑉𝑂𝐿𝑗𝐵∕ 𝑉𝑂𝐿𝐵𝑚�. VOL is the average daily volume, and j and m indicate stock j and the market, respectively. A indicates the period of 30 trading days after the warrant expiration, days ＋1 to ＋30, and B indicates the period of 30 days before expiration, days －30 to －1. 𝐷𝑆𝑃𝐷𝑗= 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐴− 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐵. Roll (1984) proposes that over the post-exercise increase in number of shares. The sample comprises 1138 warrants in the period of high VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, ＋30)) exercise over 2006-2010. t-statistics are in parentheses. The t-statistics if the regression coefficients are calculated using robust estimation of the standard errors, following White (1980). ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
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In Table 17, we could find out that the mean value of DSPD is insignificantly negative during the period of low VIX which we calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, ＋30). Comparing with the results in the period of high VIX, we could see the coefficients of VOLRWS get much more significant in Table 17. It reveals that the stock liquidity is improved after the warrants expired. The results are more robust in the period of low VIX.
Table 17
Changes in The stock liquidity following the warrant Exercise in the Period of Low VIX
Regression Estimation Results
Dependent Variable Mean Constant VOLRWS R2
DVOL -0.0389*** indicate stock j and the market, respectively. A indicates the period of 30 trading days after the warrant expiration, days ＋1 to ＋30, and B indicates the period of 30 days before expiration, days －30 to －1.
𝐷𝑆𝑃𝐷𝑗= 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐴− 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐵. Roll (1984) proposes that 𝑆𝑃𝑅𝐸𝐴𝐷𝑗= 2 ∙ �−𝐶𝑂𝑉𝑗, where COV is the ratio of the average daily trading volume of warrants to that of the stock in the period of 30 trading days (days －30 to －1) before the warrant expired. The sample comprises 2411 warrants in the period of low VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, ＋30)) exercise over 2006-2010.
The estimated models are
(7) 𝐷𝐿𝐼𝑄𝑈𝐼𝐷𝐼𝑇𝑌𝑗= 𝛾0+ 𝛾1𝑉𝑂𝐿𝑅𝑊𝑆𝑗+ 𝑢𝑗,
Where DLIQUIDITYj＝DVOLj, DSPDj, DTRQj, or Dstdj.
t-statistics are in parentheses. The t-statistics if the regression coefficients are calculated using robust estimation of the standard errors, following White (1980). ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
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In Table 18, we obtain that the mean of CAR is － 0.0874 with t ＝ 1.56, insignificantly different from zero as well as the results in the period of high VIX.
Nevertheless, we could still see the coefficients of DVOL, DTRQ, Dstd, and DVOLRAT are significant in most of models, although the mean of CAR is insignificantly different form zero. Further, we could find out that the moduli of the coefficients are lower than those in the whole sample and high VIX period apparently.
The coefficients of DSPD get more insignificant in the period of low VIX comparing with the whole sample period. So we could infer that Hypothesis H2 is only established in the whole sample period. If we separate the sample into the periods of high and low VIX, the strength of Hypothesis H2 gets worse. In contrast, Hypothesis H3 is established in any period of the sample, especially more robust in the period of high VIX. Next section we will go further to separate the sample into different levels of investor sentiment around warrants expiration.
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Determinants of the Cumulative Abnormal Return (CAR) on the Expiration of All Deep-in-the-Money Warrants in the Period of Low VIX
Model (1) (2) (3) (4) (5) (6) (7) (8) (9) 𝑙𝑜𝑔�𝑉𝑂𝐿𝑗𝐵∕ 𝑉𝑂𝐿𝐵𝑚�. VOL is the average daily volume, and j and m indicate stock j and the market, respectively. A indicates the period of 30 trading days after the warrant expiration, days ＋1 to ＋30, and B indicates the period of 30 days before expiration, days －30 to －1. 𝐷𝑆𝑃𝐷𝑗= 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐴− 𝑆𝑃𝑅𝐸𝐴𝐷𝑗𝐵. Roll (1984) proposes that over the post-exercise increase in number of shares. The sample comprises 2411 warrants in the period of low VIX (we calculate the mean value of VIX during the period of 30 trading days before and after the expiration dates following the window of (－30, ＋30)) exercise over 2006-2010. t-statistics are in parentheses. The t-statistics if the regression coefficients are calculated using robust estimation of the standard errors, following White (1980). ***, **, and * indicate significant level 1%, 5%, and 10%, respectively.
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