How Do Options Add Value? Evidence from the Convertible Bond Market*

Abstract

This paper studies the value relevance of the options market by focusing on convertible bond pricing. Pricing convertible bonds requires essentially the same set of information necessary to price options. Using a regression discontinuity design based on minimum stock price requirements for option listings, we find that the availability of stock options helps issuers attract more convertible bond buyers and reduces convertible issuers’ cost of financing. Our results highlight that the availability of individual stock options can add value to security issuers.

1. Introduction

The options market has grown tremendously in the past four decades. An important and highly debated question about stock options is how such derivative assets add value to the financial market.¹ The empirical evidence thus far remains mixed. On the one hand, studies examining the lead–lag relationship between stocks and options find that options contain information about future stock prices, suggesting that the options market contributes to price discovery and is informationally more efficient (e.g., Easley, O’Hara, and Srinivas, 1998; Chakravarty, Gulen, and Mayhew, 2004; Pan and Poteshman, 2006; Ge, Lin, and Pearson, 2016). On the other hand, studies have shown that: (i) option trading increases stock volatility and bid–ask spreads without conveying new information (e.g., Fedenia and Grammatikos, 1992); (ii) informed traders initiate trade in the stock market (e.g., Chan, Chung, and Fong, 2002); (iii) price discovery in the options market is economically insignificant (Muravyev, Pearson, and Broussard, 2013); and (iv) the option price-based stock return predictability is mainly driven by stock price pressure rather than information (Goncalves-Pinto et al., 2020). Moreover, even if informed traders do prefer the options market, the overall information environment does not necessarily improve if investors just migrate across markets without producing more information.

In this paper, we focus on the question regarding whether options add value to issuers of underlying stocks. We examine this in the context of convertible bond issues. A convertible bond is a security whose holder has the right to convert the bond into a pre-specified number of equity shares of the underlying company. In other words, a convertible bond is essentially a straight bond combined with stock options.² This “bond + options” payoff structure makes convertibles an interesting empirical setting to study the role of options. If the options market does not provide any additional information to what is available in stock and bond markets (or any other opportunities), the availability of listed options should not affect the valuations of convertible bonds. If the options market provides additional information that is useful in pricing convertibles (e.g., implied volatilities) and convertible bond issuers and investors actively utilize it, the availability of listed options will reduce adverse selection problems and affect convertible bond valuation.³

Our investigation of the convertible market further provides us with an opportunity to investigate the channel through which options may matter. The first possible channel is additional information provided by options and the second possible channel is the expanded hedging opportunity set available to investors. A large fraction of convertible bonds is held by convertible arbitrageurs, who combine long positions in convertibles with short positions in convertible issuers’ stocks (Calamos, 2003; Mitchell, Pedersen and Pulvino, 2007; Brown et al., 2012). Because options reduce short-sale constraints (Danielsen and Sorescu, 2001), convertible arbitrageurs could prefer convertibles issued by issuers with listed options. We study both channels.

We collect a sample of 1,357 convertible bond offerings issued by 815 unique US public firms and document that the issuer does not have listed options available at the time of issuance in 47% of the cases. Consistent with prior studies, such as Chan and Chen (2007) and Henderson and Tookes (2012), convertible bonds in our sample are offered at significant discounts. We find that the offering discount is significantly smaller for issues with listed options (14%) than for issues without (19%), which is an economically sizable reduction in relative underpricing of about 26%. The difference in discounts persists if we control for issuer characteristics and security design features.

Of course, option exchanges do not randomly select underlying stocks for individual stock option listing. In particular, firms that have lower information asymmetries or receive more investor attention are more likely to have listed options (Mayhew and Mihov, 2004). As such, the smaller discount for issues with listed options could be due to unobserved factors that simultaneously influence the availability of options and the pricing of convertible bonds. Our main conclusions are based on analyses that exploit the Securities and Exchange Commission (SEC)’s minimum stock price requirement for option listing. The closing price of the company’s shares must have been at least $3.00 (or $7.50 until 2004) for a majority of trading days during the three calendar months preceding the date of selection. This ad hoc price cutoff creates a discontinuity in the likelihood of option listing. Firms with average stock prices just above the cutoff have a higher unconditional likelihood of being selected for option listing than firms with average prices just below the cutoff.

Using the distance between an issuer’s average stock price and the cutoff price as the forcing variable, we employ a fuzzy regression discontinuity design (RDD) and use the subsample of convertible issues with average stock prices just around the cutoff price to investigate the role of options. We find that the likelihood of option listing increases from below 30% to above 50% as soon as the forcing variable passes the threshold. Next, we use this listing eligibility as an instrumental variable for the availability of options and estimate two-stage least squares (2SLS) regressions controlling for the forcing variable and other firm characteristics. The 2SLS results indicate that the availability of listed options significantly reduces the offering discount of convertible bond issues. In additional tests, we show that our results are reliable and robust. For example, we discuss potential estimation errors and check the robustness of our findings to a different valuation model and to varying valuation assumptions. Moreover, our conclusions are strengthened by placebo tests in which we employ fake minimum stock price requirements.

After establishing the mitigating effect of options on convertible underpricing, we investigate through which channels options affect the convertible market. First, to obtain insights into the information-provision channel, we examine whether the observed effects are stronger when the information environment of issuers is poor. We find that the relation between the availability of listed options and offering discounts is indeed stronger in case of a relatively poor information environment. In line with a continuing provision of implied volatility estimates through option prices, we further find that transaction prices of convertible bonds with listed options converge more quickly to their theoretical prices in the years after issuance.

If options improve the overall information environment for issuers and thereby reduce information asymmetry between issuers and investors, the availability of options could attract more capital suppliers who are concerned about adverse selection problems arising from information asymmetry. In addition, the availability of options could also attract more convertible arbitrageurs, who might value the reduction in short-sale constraints through options. We exploit data on convertible buyers and find that the availability of listed options significantly increases the number of buyers and allows convertible bond issuers to raise capital at a lower cost. The increase in demand comes from both hedge funds who can use options to implement convertible arbitrage strategies and long-only institutional investors who benefit from the information provided by options. Overall, these findings accord with the notion that individual stock options enable issuers of underlying stocks to attract more capital suppliers when they issue convertible bonds through both an improved information environment and a facilitation of arbitrage strategies.

This paper contributes to several strands of literature. First, our paper provides novel evidence on options having real effects on issuers of underlying stocks, contributing to the emerging literature on the real effects of options markets. Earlier work by Roll, Schwartz, and Subrahmanyam (2009) finds evidence that firms’ investments become more sensitive to stock prices as option trading increases. Cao et al. (2019) find that bank loan issuance is negatively related to option listings, even though Do, Truong, and Vu (2021) document that firms with listed options obtain lower bank loan spreads. Hong, Park, and King (2020) show that option listings lead to an increase in equity issuance, which relates to the finding in Naiker, Navissi, and Truong (2013) that firms with listed options have a lower implied cost of equity. Bernile et al. (2021) present evidence that options increase investment and innovation, while decreasing straight debt and the need for payout. Our paper focuses on convertible bonds, whose link to options is relatively direct through the option component embedded in a convertible security. Exploiting RDD, we document a real effect of option listings on the convertible security market. Through our ability to observe the identity of convertible investors, we also present evidence that the effect is through increased demands from both long-only institutional investors who pay attention to the information environment of issuers and hedge fund managers who care more about short-sale constraints and expanded hedging opportunities.

Second, we provide evidence that options facilitate information acquisition, in line with the informational role predicted by a large theoretical literature going back to Black (1975), Grossman (1988), Back (1993), Huang and Wang (1997), Cao (1999), and Massa (2002). On the empirical side, we use a unique approach based on convertible bonds, which allows us to circumvent potential concerns regarding the use of lead–lag relationships between options and stocks for testing the informational efficiency of the options market, as discussed in Muravyev, Pearson, and Broussard (2013) and Goncalves-Pinto et al. (2020). Some recent papers examine whether options increase stock price informativeness (e.g., Hu, 2018; Cao et al., 2019) in alternative ways.⁴ Our analysis provides an additional piece of evidence that the options market improves the overall information environment of the underlying company.

Finally, our findings contribute to literature on the initial underpricing of convertible bonds. Prior work has shown that convertible bond underpricing is influenced by bond ratings (Chan and Chen, 2007), capital constraints (Mitchell, Pedersen, and Pulvino, 2007), use of combined offerings with stock repurchases (De Jong, Dutordoir, and Verwijmeren, 2011), investment banks’ relationships with investors (Henderson and Tookes, 2012), and anticipated hedging costs of arbitrageurs (Grundy, Verwijmeren, and Yang, 2021). Our findings show that offering discounts are associated with information asymmetry and option availability. Moreover, we add to the literature on the interplay between convertible issuers and their investors. The existing literature has studied the impact of investor demand on issuance decisions and security design choices (e.g., Loncarski, ter Horst, and Veld, 2009; Choi et al., 2010; Brown et al., 2012; Grundy and Verwijmeren, 2018). We find that options help convertible bond issuers to attract more buyers.

The remainder of the paper is organized as follows. Section 2 describes the data and explains the variables used in the analyses. Section 3 presents our main results regarding whether options add value in the context of convertible bonds issuance using fuzzy RDD analyses. Section 4 investigates the impact of listed options on investor demand for convertibles. Section 5 concludes.

2. Data and Summary Statistics

2.1 Data

We collect convertible bond offerings issued by US public firms from 2000 onward from the SDC database and PlacementTracker. To allow for a post-issuance examination in the secondary market of up to 4 years, the issuance data stop at the end of 2014 and our secondary market analysis runs until the end of 2018. Issues by financial firms and utilities are excluded. We obtain bond-specific information from the Mergent Fixed Income Securities Database, financial statements information from Compustat, stock return and price data from CRSP, and analyst coverage data from IBES. We collect buyer information from PlacementTracker and registration statements available from the SEC’s EDGAR database, as in Brown et al. (2012). We also search through Bloomberg, Lipper TASS, and Hedge Fund Research databases to check whether the identified buyers can be classified as hedge funds. Our final sample includes 1,357 convertible offerings issued by 815 unique firms for which we are able to compute the offering discount or which provide information about the buyers of the convertible issue.

To examine whether convertible bond issuers have listed stock options, we rely on the OptionMetrics database and classify a firm as having options if the firm has option trading data available in the OptionMetrics database during the month of the convertible issue date. Out of our sample, 713 issues (52.5%) are made by issuers with listed options and 644 issues are made by issuers without available options. Figure 1 shows the number of convertible bond issues and the percentage of issues by issuers with listed options for each year over our sample period. As is shown, the annual number of convertible issues peaked between 2003 and 2007 and remained relatively constant after the financial crisis. The percentage of offerings made by issuers with options varies over time, which highlights the importance of including year-fixed effects in our regression specification.

Figure 1.

Number of convertible bond issues and number and percentage of issuers with listed options. The sample covers US convertible bond issues with available bond and firm characteristics from various sources, which are issued during 2000–14. The bars (left scale) show the total number of convertible bond issues (lighter bars) and the number of issues made by issuers with listed options written on the offering firms’ stocks (darker bars) in each year, and the line shows the percentage of issuers with listed options (right scale) in each year.

Open in new tab Download slide

2.2 Main Variables

Following the literature on convertible underpricing, the offering discount is calculated as the difference between the theoretical price and the offering price divided by the theoretical price:

Offering discount = \frac{Theoretical price - Offering price}{Theoretical price} .

To calculate the theoretical value of convertibles, we use the Tsiveriotis and Fernandes (1998) (TF) model, which has also been used by Ammann, Kind, and Wilde (2003); Chan and Chen (2007); Loncarski, ter Horst, and Veld (2009); and De Jong, Dutordoir, and Verwijmeren (2011), and which is known to be the most popular convertible bond valuation method among practitioners (Zabolotnyuk, Jones, and Veld, 2010). Our theoretical price calculations need the following inputs: (i) risk-free interest rate; (ii) stock price; (iii) stock return volatility; (iv) conversion ratio; (v) issue, settlement, and maturity dates; (vi) dividend yield; (vii) coupon rate and coupon payment frequency; (viii) call and put schedule; and (ix) credit spread. As such, the theoretical price is based on actual conversion features and is not the result of ad hoc assumptions about the effective maturity of convertible bonds. Note that the first six variables are also required to price options.

The risk-free interest rate is collected from Datastream and is the yield on US Treasury securities with a maturity date closest to the maturity of each convertible bond. To avoid a possible impact of (arbitrage-related) short-selling activity on stock prices, we use stock prices five trading days prior to the issue date. The stock return volatility is defined as the annualized volatility estimated from daily stock returns over the period, –240 to –40 trading days, as in Lewis, Rogalski, and Seward (1999). Credit spreads are based on yields on corporate industrial bonds with the same credit rating and the maturity closest to that of the convertible bond, which are collected from Datastream. Since convertible bonds are popular among small and unrated issuers, a substantial part of our sample offerings does not have a rating at issuance. Following previous studies (e.g., Loncarski, ter Horst, and Veld, 2009), we assign a BBB rating to unrated bonds for the purpose of calculating credit spreads.⁵ In total, there are 872 convertibles for which we have sufficient information to calculate the offering discount: We mostly lose observations when we are unable to match to required convertible design information in Mergent and are additionally unable to find this information in the issue prospectus.⁶ To avoid outlier effects, we winsorize offering discounts at the 1% and 99% levels.⁷

In our multivariate regression analyses, we control for a rich set of bond and firm characteristics. Bond characteristics are the stated maturity of the convertible bond and Proceeds/MV, which is the amount of proceeds raised at the offering divided by the market value of equity at the fiscal year-end preceding the issue date. We also include dummy variables indicating whether the issuer combines the convertible offering with stock repurchases (De Jong, Dutordoir, and Verwijmeren, 2011), whether the convertible is privately placed under Rule 144A, and whether the issuer has a credit rating. For each issue, we numerically measure the sensitivity of the convertible bond price to stock price movements, which results in the convertible’s delta. We define delta as the average of log theoretical convertible bond price changes over differences in log issuers’ stock prices when stock prices change:

Delta = \frac{1}{2} (\frac{\log (CV (S + d S)) - \log (CV (S))}{\log (S + d S) - \log (S)} + \frac{\log (CV (S)) - \log (CV (S - d S))}{\log (S) - \log (S - d S)}),

where CV is the convertible bond price estimated using the Tsiveriotis and Fernandes model, S is the issuer’s stock price, and dS indicates the change in S. We use a 5% change (i.e., dS = 0.05×S). The log-transformation normalizes the value of delta between 0 and 1.⁸

Moreover, we control for various firm characteristics: (i) firm size measured by the natural logarithm of total assets⁹; (ii) growth opportunities measured by the market-to-book ratio; (iii) stock liquidity proxied by the Amihud (2002) illiquidity measure (Amihud × 10⁶); and (iv) firm risk measured by the stock return volatility. Total assets and market-to-book ratios are estimated at the end of the fiscal year prior to the issue date. Amihud × 10⁶ is calculated as the average daily ratio of absolute value of stock return to dollar trading volume in millions during the window [–120, –20]. Volatility is defined as the daily stock return volatility calculated from stock returns over the window [–240, –40] relative to the convertible issue date. Table AI lists all variables in our analyses.

2.3 Summary Statistics

Table I reports the descriptive statistics of the variables used in the analyses. The average number of total institutional investors in the convertible offering is 41 and on average there are twenty-two hedge fund buyers. We observe that the average difference between the theoretical price and the offering price (the “offering discount”) is substantial (15.43%). The magnitude of our discount estimates is comparable to the offering discount estimates reported in prior studies (e.g., Loncarski, ter Horst, and Veld, 2009; Zabolotnyuk et al., 2010). Besides representing an offering discount, our estimates can also partly reflect estimation error in obtaining theoretical convertible values, due to incorrectly estimating input values or due to an incorrect theoretical model. To obtain some insights into the reliability of our theoretical value estimates, we examine whether the theoretical estimates converge to market prices post-issuance. Chan and Chen (2007) find that seasoned convertible bond prices gradually converge to their theoretical prices within 2 years after issuing. For each convertible bond with available trading data in TRACE, we collect its average market price in the 24th month after issuance and use the TF model to update the corresponding theoretical prices at that point in time. We find that after 2 years of trading, our sample issuers’ transaction prices do not significantly deviate from their theoretical prices anymore (with a 95% confidence interval of [–0.3%, 6%]). We study post-issuance underpricing more extensively in Section 3.1.b. Overall, we believe that our estimates largely represent offering discounts, as in earlier studies on convertible bond underpricing.¹⁰ Still, estimation error might be present and could reduce the probability of finding evidence for the hypothesized relations. We study the robustness of our findings to a different pricing model and different parameter estimations in Section 3.4.

Table I.

Open in new tab

Summary statistics

This table presents summary statistics of variables used in the paper. We describe how we estimate each variable in Appendix A. The sample covers US convertible bond issues with available bond and firm characteristic from various sources during the sample period, 2000–14. Eligible is defined for the RDD sample only including those with prices within the bandwidth of [–3, 3] around the price cutoff points.

	N	Mean	Std. Dev.	Min.	p25	Median	p75	Max.
Dependent variables
Offering discount (%)	872	15.43	19.71	–49.85	4.57	15.06	28.61	63.24
Total buyers	1,078	40.68	45.17	1	3	29	64	320
Hedge fund buyers	1,016	22.11	19.94	0	4	20	34	138
Other buyers	1,016	20.99	29.14	0	1	9	31	250
Main explanatory variables
Option	1,357	0.53	0.50	0	0	1	1	1
Eligible (RDD)	338	0.43	0.50	0	0	0	1	1
Bond-level variables
Delta	872	0.67	0.18	0	0.58	0.69	0.79	1
Proceeds/MV	1,344	0.29	3.15	0.00	0.08	0.15	0.25	115.32
Maturity (years)	1,357	8.15	7.04	0	5	5	7	40
Combined offering	1,357	0.10	0.31	0	0	0	0	1
Rule 144A	1,357	0.61	0.49	0	0	1	1	1
Rated	1,357	0.38	0.48	0	0	0	1	1
Rated $\times$ Investment grade	1,357	0.10	0.30	0	0	0	0	1
Firm-level variables
Log(Assets)	1,357	6.23	2.04	–1.78	5.03	6.39	7.62	12.50
Market to book	1,348	3.56	16.78	0.52	1.31	1.82	2.95	502.11
Volatility	1,314	0.04	0.03	0.01	0.02	0.04	0.05	0.53
Amihud $\times 10^{6}$	1,276	0.31	2.64	0.00	0.02	0.04	0.13	88.65
Nasdaq listing	1,357	0.46	0.50	0	0	0	1	1
Analyst coverage	1,357	6.36	6.86	0	0	5	9	38

	N	Mean	Std. Dev.	Min.	p25	Median	p75	Max.
Dependent variables
Offering discount (%)	872	15.43	19.71	–49.85	4.57	15.06	28.61	63.24
Total buyers	1,078	40.68	45.17	1	3	29	64	320
Hedge fund buyers	1,016	22.11	19.94	0	4	20	34	138
Other buyers	1,016	20.99	29.14	0	1	9	31	250
Main explanatory variables
Option	1,357	0.53	0.50	0	0	1	1	1
Eligible (RDD)	338	0.43	0.50	0	0	0	1	1
Bond-level variables
Delta	872	0.67	0.18	0	0.58	0.69	0.79	1
Proceeds/MV	1,344	0.29	3.15	0.00	0.08	0.15	0.25	115.32
Maturity (years)	1,357	8.15	7.04	0	5	5	7	40
Combined offering	1,357	0.10	0.31	0	0	0	0	1
Rule 144A	1,357	0.61	0.49	0	0	1	1	1
Rated	1,357	0.38	0.48	0	0	0	1	1
Rated $\times$ Investment grade	1,357	0.10	0.30	0	0	0	0	1
Firm-level variables
Log(Assets)	1,357	6.23	2.04	–1.78	5.03	6.39	7.62	12.50
Market to book	1,348	3.56	16.78	0.52	1.31	1.82	2.95	502.11
Volatility	1,314	0.04	0.03	0.01	0.02	0.04	0.05	0.53
Amihud $\times 10^{6}$	1,276	0.31	2.64	0.00	0.02	0.04	0.13	88.65
Nasdaq listing	1,357	0.46	0.50	0	0	0	1	1
Analyst coverage	1,357	6.36	6.86	0	0	5	9	38

Table I.

Open in new tab

Summary statistics

	N	Mean	Std. Dev.	Min.	p25	Median	p75	Max.
Dependent variables
Offering discount (%)	872	15.43	19.71	–49.85	4.57	15.06	28.61	63.24
Total buyers	1,078	40.68	45.17	1	3	29	64	320
Hedge fund buyers	1,016	22.11	19.94	0	4	20	34	138
Other buyers	1,016	20.99	29.14	0	1	9	31	250
Main explanatory variables
Option	1,357	0.53	0.50	0	0	1	1	1
Eligible (RDD)	338	0.43	0.50	0	0	0	1	1
Bond-level variables
Delta	872	0.67	0.18	0	0.58	0.69	0.79	1
Proceeds/MV	1,344	0.29	3.15	0.00	0.08	0.15	0.25	115.32
Maturity (years)	1,357	8.15	7.04	0	5	5	7	40
Combined offering	1,357	0.10	0.31	0	0	0	0	1
Rule 144A	1,357	0.61	0.49	0	0	1	1	1
Rated	1,357	0.38	0.48	0	0	0	1	1
Rated $\times$ Investment grade	1,357	0.10	0.30	0	0	0	0	1
Firm-level variables
Log(Assets)	1,357	6.23	2.04	–1.78	5.03	6.39	7.62	12.50
Market to book	1,348	3.56	16.78	0.52	1.31	1.82	2.95	502.11
Volatility	1,314	0.04	0.03	0.01	0.02	0.04	0.05	0.53
Amihud $\times 10^{6}$	1,276	0.31	2.64	0.00	0.02	0.04	0.13	88.65
Nasdaq listing	1,357	0.46	0.50	0	0	0	1	1
Analyst coverage	1,357	6.36	6.86	0	0	5	9	38

	N	Mean	Std. Dev.	Min.	p25	Median	p75	Max.
Dependent variables
Offering discount (%)	872	15.43	19.71	–49.85	4.57	15.06	28.61	63.24
Total buyers	1,078	40.68	45.17	1	3	29	64	320
Hedge fund buyers	1,016	22.11	19.94	0	4	20	34	138
Other buyers	1,016	20.99	29.14	0	1	9	31	250
Main explanatory variables
Option	1,357	0.53	0.50	0	0	1	1	1
Eligible (RDD)	338	0.43	0.50	0	0	0	1	1
Bond-level variables
Delta	872	0.67	0.18	0	0.58	0.69	0.79	1
Proceeds/MV	1,344	0.29	3.15	0.00	0.08	0.15	0.25	115.32
Maturity (years)	1,357	8.15	7.04	0	5	5	7	40
Combined offering	1,357	0.10	0.31	0	0	0	0	1
Rule 144A	1,357	0.61	0.49	0	0	1	1	1
Rated	1,357	0.38	0.48	0	0	0	1	1
Rated $\times$ Investment grade	1,357	0.10	0.30	0	0	0	0	1
Firm-level variables
Log(Assets)	1,357	6.23	2.04	–1.78	5.03	6.39	7.62	12.50
Market to book	1,348	3.56	16.78	0.52	1.31	1.82	2.95	502.11
Volatility	1,314	0.04	0.03	0.01	0.02	0.04	0.05	0.53
Amihud $\times 10^{6}$	1,276	0.31	2.64	0.00	0.02	0.04	0.13	88.65
Nasdaq listing	1,357	0.46	0.50	0	0	0	1	1
Analyst coverage	1,357	6.36	6.86	0	0	5	9	38

Regarding bond characteristics, the average delta of the convertibles in our sample is 0.67, indicating that on average, convertible bond prices increase by $0.67 as stock prices increase by $1.00 at the time of issuance. On average, convertible issuers raise 29% of the market value of equity by the convertible issuance. The average stated bond maturity is about 8 years. About 10% of our sample offerings are combined offerings that involve simultaneous announcements of share repurchases and 61% of our sample offerings are Rule 144A private placement offerings.

Regarding firm characteristics, the average log-transformed total assets value in $million is 6.23 (i.e., $507.76 million), the average market-to-book asset value is 3.56, the average volatility of daily stock returns is 4%, and the average Amihud illiquidity measure (daily absolute return over dollar trading volume in millions) is 0.31. Almost half (46%) of our sample offerings are made by Nasdaq-listed issuers. Finally, the average number of financial analysts covering our issuing firms is 6.

2.4 Univariate Tests

In Table II, we report summary statistics for issuers with and without listed options. In the last column, t-statistics for differences of means are presented. Comparing issues without listed options to issues with options, the univariate test shows that offering discounts are significantly larger (19% versus 14%) for issuers without listed options. Figure A1 shows that the differences in offering discount are significant in all three subperiods, 2000–04, 2005–09, and 2010–14.

Table II.

Open in new tab

Differences between issues with and without listed options

This table presents summary statistics for issues with and without listed individual stock options written on offering firms’ stocks. We describe how we estimate each variable in Appendix A. The last column reports t-statistics for the test of the difference between issues with and without listed options. The sample covers US convertible bond issues with available bond and firm characteristic from various sources during the sample period, 2000–14. *, **, and *** indicate significance at the 1%, 5%, and 10% levels, respectively.

	No option			With options			Difference
	N	Mean	SD	N	Mean	SD	t-stat
Dependent variables
Offering discount	312	0.19	0.19	560	0.14	0.20	–3.55***
Total buyers	598	28.14	34.05	480	56.31	51.98	10.70***
Hedge fund buyers	586	15.93	16.89	430	30.54	20.68	12.38***
Other buyers	586	12.77	20.22	430	32.20	35.12	11.12***
Deal characteristics
Delta	312	0.66	0.19	560	0.68	0.17	1.29**
Proceeds/MV	634	0.41	4.58	710	0.18	0.21	–1.33***
Maturity (years)	644	7.27	6.88	713	8.94	7.08	4.38***
Combined offering	644	0.06	0.23	713	0.15	0.35	5.38***
Rule 144A	644	0.49	0.50	713	0.72	0.45	8.81***
Rated	644	0.22	0.41	713	0.52	0.50	12.17***
Rated $\times$ Investment grade	644	0.04	0.20	713	0.15	0.35	6.76***
Firm characteristics
Log(Assets)	644	5.11	2.00	713	7.24	1.46	22.48***
Market to book	635	4.63	23.95	713	2.62	4.44	–2.20**
Volatility	605	0.05	0.03	709	0.04	0.02	–7.85***
Amihud $\times 10^{6}$	565	0.62	3.94	711	0.06	0.34	–3.78***
Nasdaq listing	644	0.49	0.50	713	0.44	0.50	–1.96*
Analyst coverage	644	3.43	4.65	713	9.01	7.44	16.37***

	No option			With options			Difference
	N	Mean	SD	N	Mean	SD	t-stat
Dependent variables
Offering discount	312	0.19	0.19	560	0.14	0.20	–3.55***
Total buyers	598	28.14	34.05	480	56.31	51.98	10.70***
Hedge fund buyers	586	15.93	16.89	430	30.54	20.68	12.38***
Other buyers	586	12.77	20.22	430	32.20	35.12	11.12***
Deal characteristics
Delta	312	0.66	0.19	560	0.68	0.17	1.29**
Proceeds/MV	634	0.41	4.58	710	0.18	0.21	–1.33***
Maturity (years)	644	7.27	6.88	713	8.94	7.08	4.38***
Combined offering	644	0.06	0.23	713	0.15	0.35	5.38***
Rule 144A	644	0.49	0.50	713	0.72	0.45	8.81***
Rated	644	0.22	0.41	713	0.52	0.50	12.17***
Rated $\times$ Investment grade	644	0.04	0.20	713	0.15	0.35	6.76***
Firm characteristics
Log(Assets)	644	5.11	2.00	713	7.24	1.46	22.48***
Market to book	635	4.63	23.95	713	2.62	4.44	–2.20**
Volatility	605	0.05	0.03	709	0.04	0.02	–7.85***
Amihud $\times 10^{6}$	565	0.62	3.94	711	0.06	0.34	–3.78***
Nasdaq listing	644	0.49	0.50	713	0.44	0.50	–1.96*
Analyst coverage	644	3.43	4.65	713	9.01	7.44	16.37***

Table II.

Open in new tab

Differences between issues with and without listed options

	No option			With options			Difference
	N	Mean	SD	N	Mean	SD	t-stat
Dependent variables
Offering discount	312	0.19	0.19	560	0.14	0.20	–3.55***
Total buyers	598	28.14	34.05	480	56.31	51.98	10.70***
Hedge fund buyers	586	15.93	16.89	430	30.54	20.68	12.38***
Other buyers	586	12.77	20.22	430	32.20	35.12	11.12***
Deal characteristics
Delta	312	0.66	0.19	560	0.68	0.17	1.29**
Proceeds/MV	634	0.41	4.58	710	0.18	0.21	–1.33***
Maturity (years)	644	7.27	6.88	713	8.94	7.08	4.38***
Combined offering	644	0.06	0.23	713	0.15	0.35	5.38***
Rule 144A	644	0.49	0.50	713	0.72	0.45	8.81***
Rated	644	0.22	0.41	713	0.52	0.50	12.17***
Rated $\times$ Investment grade	644	0.04	0.20	713	0.15	0.35	6.76***
Firm characteristics
Log(Assets)	644	5.11	2.00	713	7.24	1.46	22.48***
Market to book	635	4.63	23.95	713	2.62	4.44	–2.20**
Volatility	605	0.05	0.03	709	0.04	0.02	–7.85***
Amihud $\times 10^{6}$	565	0.62	3.94	711	0.06	0.34	–3.78***
Nasdaq listing	644	0.49	0.50	713	0.44	0.50	–1.96*
Analyst coverage	644	3.43	4.65	713	9.01	7.44	16.37***

	No option			With options			Difference
	N	Mean	SD	N	Mean	SD	t-stat
Dependent variables
Offering discount	312	0.19	0.19	560	0.14	0.20	–3.55***
Total buyers	598	28.14	34.05	480	56.31	51.98	10.70***
Hedge fund buyers	586	15.93	16.89	430	30.54	20.68	12.38***
Other buyers	586	12.77	20.22	430	32.20	35.12	11.12***
Deal characteristics
Delta	312	0.66	0.19	560	0.68	0.17	1.29**
Proceeds/MV	634	0.41	4.58	710	0.18	0.21	–1.33***
Maturity (years)	644	7.27	6.88	713	8.94	7.08	4.38***
Combined offering	644	0.06	0.23	713	0.15	0.35	5.38***
Rule 144A	644	0.49	0.50	713	0.72	0.45	8.81***
Rated	644	0.22	0.41	713	0.52	0.50	12.17***
Rated $\times$ Investment grade	644	0.04	0.20	713	0.15	0.35	6.76***
Firm characteristics
Log(Assets)	644	5.11	2.00	713	7.24	1.46	22.48***
Market to book	635	4.63	23.95	713	2.62	4.44	–2.20**
Volatility	605	0.05	0.03	709	0.04	0.02	–7.85***
Amihud $\times 10^{6}$	565	0.62	3.94	711	0.06	0.34	–3.78***
Nasdaq listing	644	0.49	0.50	713	0.44	0.50	–1.96*
Analyst coverage	644	3.43	4.65	713	9.01	7.44	16.37***

Issuers with listed options are more likely to have credit ratings (0.52 versus 0.22) and to be rated as investment grade (0.15 versus 0.04). Their convertible issues have slightly higher delta (0.68 versus 0.66) and longer maturities (8.94 versus 7.27) than their counterparts without listed options. Firms with listed options are also more likely to privately place the convertibles under Rule 144A (0.72 versus 0.49) and combine the bond issuance with simultaneous stock repurchases (0.15 versus 0.06).

Regarding firm characteristics, the average total assets (natural log transformed values of $7.24 versus $5.11) and the average number of financial analysts covering issuing firms (9.01 versus 3.43) are significantly greater for issuers with listed options, whereas the average market-to-book asset ratio (2.62 versus 4.63), the stock return volatility (0.04 versus 0.05), and the Amihud illiquidity measure (0.06 versus 0.62) are significantly lower for issuers with listed options. These substantial differences between the two groups highlight the importance of multivariate analyses and of our RDD analyses. The fraction of offerings made by Nasdaq-listed firms is marginally lower for issuers with listed options (0.44 versus 0.49).

3. Options and Convertible Bond Underpricing

This section examines whether the availability of options affects convertible underpricing upon issuance as well as during the post-issuance period in the secondary market.

3.1 Baseline Results

3.1.a. Offering discount

Before exploiting our RDD analyses, we provide insights into the relation between having options and the pricing of convertible bonds by estimating the following regression specification with year-fixed effects:

y_{i} = α + β \times Optio n_{i} + γ_{1}^{'} X_{i}^{deal} + γ_{2}^{'} X_{i}^{firm} + {Year F E_{i} + ε}_{i},

(1)

where

Option

is a dummy variable indicating issuers with listed options at the time of issuance, and

X^{deal}

and

X^{firm}

are vectors of bond- and firm-level characteristics, respectively. The year-fixed effects capture any common factors that affect all convertible issues in a given year, such as the rise of convertible arbitrage hedge funds (Grundy and Verwijmeren, 2018). The dependent variable is the offering discount of convertible bonds. We use clustered standard errors at the firm level in calculating t-statistics. Under the null hypothesis that options do not affect the pricing decisions of convertible bond issuers,

β

will not be different from zero.

Table III reports the estimation results. In Column (1), where we only control for deal characteristics, the $β$ estimate is –0.049 and statistically significant at the 1% level, implying that having listed options is related to a significant reduction in offering discounts of 4.9 percentage points, which is an economically sizable reduction of 26% relative to the average discount offered by firms without options (19%). This relation is statistically significant at the 5% level when we additionally control for firm-level characteristics in Column (2). The coefficient estimates of all control variables have the expected signs. Consistent with previous studies on convertible bonds (e.g., Loncarski, ter Horst, and Veld, 2009; De Jong, Dutordoir, and Verwijmeren, 2011), we find that the offering discount is smaller for issues with lower delta (i.e., more debt-like convertible bonds), issues combined with stock repurchases, issues with credit ratings, and issues by firms that are larger, have higher growth opportunities, and have less volatile but more liquid stocks.

Table III.

Open in new tab

Listed options and convertible offering discounts

This table presents the relation between having listed options and convertible underpricing. The dependent variable is the offering discount, calculated as (Theoretical Price–Offering Price)/Theoretical Price, where we use the model of TF to compute the theoretical price. Option is a dummy variable indicating issuers with listed options at the time of issuance. In Columns (3) and (4), we include analyst coverage to measure firms’ information environment and interact it with the Option dummy variable. All regressions include year-fixed effects to control for unobserved common factors in each year. All control variables are defined in Appendix A. $T$ -statistics based on clustered standard errors at the firm level are reported in parentheses. ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively. Bold values indicates the main estimates of interest.

	Offering discount
	(1)	(2)	(3)	(4)
Option	–0.049***	–0.035**	–0.084***	–0.067***
	(–3.393)	(–2.376)	(–4.382)	(–3.356)
Option $\times$ Analyst coverage			0.008***	0.006**
			(3.028)	(2.253)
Delta	0.525***	0.511***	0.533***	0.518***
	(12.190)	(10.759)	(12.106)	(10.800)
Proceeds/MV	0.001***	0.001***	0.000	0.000**
	(3.472)	(3.596)	(1.564)	(2.066)
Log(Maturity)	–0.027*	–0.019	–0.027*	–0.021
	(–1.888)	(–1.311)	(–1.961)	(–1.467)
Combined offering	–0.049***	–0.038**	–0.042**	–0.036**
	(–2.806)	(–2.153)	(–2.511)	(–2.137)
Rule 144A	–0.050***	–0.029**	–0.038***	–0.026*
	(–3.270)	(–2.064)	(–2.640)	(–1.885)
Rated	–0.070***	–0.035**	–0.057***	–0.036**
	(–5.473)	(–2.244)	(–4.450)	(–2.309)
Rated $\times$ Investment grade	–0.011	0.017	0.005	0.020
	(–0.491)	(0.769)	(0.237)	(0.941)
Log(Assets)		–0.024***		–0.018***
		(–4.108)		(–3.078)
Market to book		–0.007*		–0.006
		(–1.882)		(–1.616)
Volatility		1.098**		0.949*
		(2.326)		(1.962)
Amihud $\times 10^{6}$		0.021**		0.019**
		(2.220)		(2.551)
Nasdaq listing		–0.031***		–0.020
		(–2.646)		(–1.647)
Analyst coverage			–0.010***	–0.007***
			(–4.438)	(–2.993)
Constant	–0.033	0.069	0.003	0.062
	(–0.878)	(1.341)	(0.085)	(1.200)
Observations	869	847	869	847
R-squared	0.390	0.426	0.420	0.439
Year FE	Yes	Yes	Yes	Yes

	Offering discount
	(1)	(2)	(3)	(4)
Option	–0.049***	–0.035**	–0.084***	–0.067***
	(–3.393)	(–2.376)	(–4.382)	(–3.356)
Option $\times$ Analyst coverage			0.008***	0.006**
			(3.028)	(2.253)
Delta	0.525***	0.511***	0.533***	0.518***
	(12.190)	(10.759)	(12.106)	(10.800)
Proceeds/MV	0.001***	0.001***	0.000	0.000**
	(3.472)	(3.596)	(1.564)	(2.066)
Log(Maturity)	–0.027*	–0.019	–0.027*	–0.021
	(–1.888)	(–1.311)	(–1.961)	(–1.467)
Combined offering	–0.049***	–0.038**	–0.042**	–0.036**
	(–2.806)	(–2.153)	(–2.511)	(–2.137)
Rule 144A	–0.050***	–0.029**	–0.038***	–0.026*
	(–3.270)	(–2.064)	(–2.640)	(–1.885)
Rated	–0.070***	–0.035**	–0.057***	–0.036**
	(–5.473)	(–2.244)	(–4.450)	(–2.309)
Rated $\times$ Investment grade	–0.011	0.017	0.005	0.020
	(–0.491)	(0.769)	(0.237)	(0.941)
Log(Assets)		–0.024***		–0.018***
		(–4.108)		(–3.078)
Market to book		–0.007*		–0.006
		(–1.882)		(–1.616)
Volatility		1.098**		0.949*
		(2.326)		(1.962)
Amihud $\times 10^{6}$		0.021**		0.019**
		(2.220)		(2.551)
Nasdaq listing		–0.031***		–0.020
		(–2.646)		(–1.647)
Analyst coverage			–0.010***	–0.007***
			(–4.438)	(–2.993)
Constant	–0.033	0.069	0.003	0.062
	(–0.878)	(1.341)	(0.085)	(1.200)
Observations	869	847	869	847
R-squared	0.390	0.426	0.420	0.439
Year FE	Yes	Yes	Yes	Yes

Table III.

Open in new tab

Listed options and convertible offering discounts

	Offering discount
	(1)	(2)	(3)	(4)
Option	–0.049***	–0.035**	–0.084***	–0.067***
	(–3.393)	(–2.376)	(–4.382)	(–3.356)
Option $\times$ Analyst coverage			0.008***	0.006**
			(3.028)	(2.253)
Delta	0.525***	0.511***	0.533***	0.518***
	(12.190)	(10.759)	(12.106)	(10.800)
Proceeds/MV	0.001***	0.001***	0.000	0.000**
	(3.472)	(3.596)	(1.564)	(2.066)
Log(Maturity)	–0.027*	–0.019	–0.027*	–0.021
	(–1.888)	(–1.311)	(–1.961)	(–1.467)
Combined offering	–0.049***	–0.038**	–0.042**	–0.036**
	(–2.806)	(–2.153)	(–2.511)	(–2.137)
Rule 144A	–0.050***	–0.029**	–0.038***	–0.026*
	(–3.270)	(–2.064)	(–2.640)	(–1.885)
Rated	–0.070***	–0.035**	–0.057***	–0.036**
	(–5.473)	(–2.244)	(–4.450)	(–2.309)
Rated $\times$ Investment grade	–0.011	0.017	0.005	0.020
	(–0.491)	(0.769)	(0.237)	(0.941)
Log(Assets)		–0.024***		–0.018***
		(–4.108)		(–3.078)
Market to book		–0.007*		–0.006
		(–1.882)		(–1.616)
Volatility		1.098**		0.949*
		(2.326)		(1.962)
Amihud $\times 10^{6}$		0.021**		0.019**
		(2.220)		(2.551)
Nasdaq listing		–0.031***		–0.020
		(–2.646)		(–1.647)
Analyst coverage			–0.010***	–0.007***
			(–4.438)	(–2.993)
Constant	–0.033	0.069	0.003	0.062
	(–0.878)	(1.341)	(0.085)	(1.200)
Observations	869	847	869	847
R-squared	0.390	0.426	0.420	0.439
Year FE	Yes	Yes	Yes	Yes

	Offering discount
	(1)	(2)	(3)	(4)
Option	–0.049***	–0.035**	–0.084***	–0.067***
	(–3.393)	(–2.376)	(–4.382)	(–3.356)
Option $\times$ Analyst coverage			0.008***	0.006**
			(3.028)	(2.253)
Delta	0.525***	0.511***	0.533***	0.518***
	(12.190)	(10.759)	(12.106)	(10.800)
Proceeds/MV	0.001***	0.001***	0.000	0.000**
	(3.472)	(3.596)	(1.564)	(2.066)
Log(Maturity)	–0.027*	–0.019	–0.027*	–0.021
	(–1.888)	(–1.311)	(–1.961)	(–1.467)
Combined offering	–0.049***	–0.038**	–0.042**	–0.036**
	(–2.806)	(–2.153)	(–2.511)	(–2.137)
Rule 144A	–0.050***	–0.029**	–0.038***	–0.026*
	(–3.270)	(–2.064)	(–2.640)	(–1.885)
Rated	–0.070***	–0.035**	–0.057***	–0.036**
	(–5.473)	(–2.244)	(–4.450)	(–2.309)
Rated $\times$ Investment grade	–0.011	0.017	0.005	0.020
	(–0.491)	(0.769)	(0.237)	(0.941)
Log(Assets)		–0.024***		–0.018***
		(–4.108)		(–3.078)
Market to book		–0.007*		–0.006
		(–1.882)		(–1.616)
Volatility		1.098**		0.949*
		(2.326)		(1.962)
Amihud $\times 10^{6}$		0.021**		0.019**
		(2.220)		(2.551)
Nasdaq listing		–0.031***		–0.020
		(–2.646)		(–1.647)
Analyst coverage			–0.010***	–0.007***
			(–4.438)	(–2.993)
Constant	–0.033	0.069	0.003	0.062
	(–0.878)	(1.341)	(0.085)	(1.200)
Observations	869	847	869	847
R-squared	0.390	0.426	0.420	0.439
Year FE	Yes	Yes	Yes	Yes

To better understand the channel through which options affect offering discounts, in Columns (3) and (4) of Table III, we investigate how the relation between options and underpricing varies with the level of information asymmetry surrounding the issuer. Having listed options can be beneficial possibly because it provides implied volatility estimates and potential insight into future growth opportunities derived from differences in demand between call and put options. If options indeed reduce underpricing through the information channel, then they will add more value in a poorer information environment. Therefore, we would expect a stronger effect for issuers with higher information asymmetry.

Empirically, we measure information asymmetry as the number of analysts covering the issuer in the year prior to the issue date, which reflects firms’ information environment (e.g., Kelly and Ljungqvist, 2012; Derrien and Kecskes, 2013). We interact the $Option$ dummy variable with the analyst coverage variable while controlling for other bond and firm characteristics. We find that the coefficient estimates of $Option$ remain significantly negative and become larger in magnitude when compared with the estimates in Columns (1) and (2). The interaction term between $Option$ and analyst coverage is significantly positive, which implies that the negative relation between options and underpricing is more pronounced for firms with a worse information environment (i.e., lower analyst coverage). For firms without any analyst coverage, having options relates to a reduction in underpricing by 6.7 to 8.4 percentage points. The relation between options and offering discounts is close to zero for issuers with more than ten analysts, which is in line with the notion that listed options reduce convertible underpricing by facilitating information acquisition. To further investigate the role of an information channel, we examine how fast transaction prices converge to theoretical prices after issuance in the next sub-section.

3.1.b. Post-issuance underpricing

Existing literature shows that, after the initial underpricing, seasoned convertible bond prices gradually converge to their theoretical prices in the years after issuing (e.g., Chan and Chen, 2007; Grundy, Verwijmeren, and Yang, 2021). If listed options improve firms’ overall information environment, the convergence to theoretical prices could be better facilitated by the continuing provision of implied volatility estimates. Hence, we expect the prices of convertible bonds offered by issuers with options to converge more quickly to their theoretical prices, and therefore, expect those bonds to have a smaller underpricing in the secondary market.

To test these predictions, we compute the post-issuance underpricing of convertible bonds issued by firms with and without options, separately. For each convertible bond with available trading data in TRACE, we collect its monthly average market prices, and use the TF model to update the corresponding theoretical prices at each month-end. We examine trading data until the end of 2018 and calculate the potential underpricing as (Theoretical Price—Market Price)/Theoretical Price. We obtain a panel of 34,565 bond-month observations from 617 unique issues.¹¹

In Figure 2, we plot the average underpricing after 6, 12, 18, 24, and from 30 to 48 months relative to the issue month and the corresponding 90% confidence intervals. We consider convertible bond prices as having converged to their theoretical prices when the underpricing is not significantly different from zero. As is shown, the underpricing of convertible bonds in our sample disappears over time. If issuers have listed options at the time of issuance, their convertible bond prices converge to the theoretical prices within the first 18 months, on average. In contrast, if issuers do not have listed options, the convergence tends to take more than 48 months.

Figure 2.

Convertible underpricing after issuance with 90% confidence intervals. This graph plots the average post-issuance convertible bond underpricing of issues with and without listed options, illustrating their difference in the speed of convergence of the market price to the theoretical price. For each convertible bond issue with trading data available in TRACE, we obtain its average monthly market price after six (12, 18, 24, and 30–48) months relative to the issue month, use the TF model to update the corresponding theoretical price over time, and calculate the potential underpricing as (Theoretical Price—Market Price)/Theoretical Price. We also plot the corresponding 90% confidence intervals.

Open in new tab Download slide

In Table IV, we use the full panel to estimate the relation between listed options and post-issuance underpricing more formally, controlling for unobserved time-trends and other factors. More specifically, we control for issuer characteristics and evolving bond characteristics, such as delta, absolute moneyness, log of the time to first call, and log of the time to maturity.¹² As shown in Columns (1)–(3), the coefficient estimates of the $Option$ dummy imply that the underpricing of convertible bonds from issuers with options is significantly lower (3.0–4.5 percentage points), and this difference is not driven by other bond- or firm-characteristics, such as time-to-maturity and bond trading volumes.

Table IV.

Open in new tab

Listed options and post-issuance convertible underpricing

This table presents the relation between listed options and monthly convertible bond underpricing in the secondary market. The dependent variable is monthly convertible underpricing calculated as (Theoretical Price—Market Price)/Theoretical Price at the end of each month. Option is a dummy variable indicating issuers with listed options in the given month. While Columns (1)–(3) use the full sample, Column (4) focuses on a subsample of convertible bonds that experienced option listing during their lifetime, where we restrict the sample to [ $τ$ –12., $τ$ +12] (⁠ $τ$ represents option listing month) and include bond-fixed effects instead of bond characteristics to exploit this time-series variation. All regressions include year–month fixed effects to control for unobserved common factors in each month. All control variables are defined in Appendix A. $T$ -statistics based on clustered standard errors at the firm level are reported in parentheses. ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively. Bold values indicates the main estimates of interest.

Dependent variable	Post-issuance underpricing
	(1)	(2)	(3)	(4)
Option	–0.030**	–0.045***	–0.034**	–0.030**
	(–2.037)	(–2.854)	(–2.338)	(–2.197)
Delta		0.057***	0.054***	–0.075
		(3.425)	(3.363)	(–1.147)
Absolute moneyness		0.070***	0.083***	–0.007
		(6.842)	(8.352)	(–0.150)
Log(TTC)		–0.009	0.007	0.052
		(–0.846)	(0.699)	(1.025)
Log(TTM)		–0.014**	0.080***	0.173***
		(–2.157)	(7.216)	(2.958)
Volatility (Monthly)		0.101***	0.089***	0.040
		(8.185)	(7.828)	(0.888)
Amihud $\times 10^{6}$ (Monthly)		–0.068*	–0.029	–0.070*
		(–1.805)	(–0.841)	(–1.811)
Bond trading volume		0.011***	0.004	0.002
		(4.022)	(1.387)	(0.668)
Log(Assets)		–0.043***	–0.024***	0.067***
		(–7.584)	(–4.346)	(2.936)
Market to book		–0.012***	–0.013***	0.049
		(–3.780)	(–4.102)	(1.528)
Log(Maturity)			–0.151***
			(–9.494)
Combined offering			–0.009
			(–0.330)
Rule 144A			0.006
			(0.499)
Rated			–0.054***
			(–5.869)
Rated $\times$ Investment grade			0.022
			(1.256)
Nasdaq listing			–0.001
			(–0.102)
Constant	0.069***	0.167***	0.329***	–0.774***
	(5.159)	(3.678)	(6.443)	(–4.034)
Observations	34,565	20,383	20,128	402
R-squared	0.076	0.345	0.417	0.948
Year–month FE	Yes	Yes	Yes	Yes
Bond FE	No	No	No	Yes

Dependent variable	Post-issuance underpricing
	(1)	(2)	(3)	(4)
Option	–0.030**	–0.045***	–0.034**	–0.030**
	(–2.037)	(–2.854)	(–2.338)	(–2.197)
Delta		0.057***	0.054***	–0.075
		(3.425)	(3.363)	(–1.147)
Absolute moneyness		0.070***	0.083***	–0.007
		(6.842)	(8.352)	(–0.150)
Log(TTC)		–0.009	0.007	0.052
		(–0.846)	(0.699)	(1.025)
Log(TTM)		–0.014**	0.080***	0.173***
		(–2.157)	(7.216)	(2.958)
Volatility (Monthly)		0.101***	0.089***	0.040
		(8.185)	(7.828)	(0.888)
Amihud $\times 10^{6}$ (Monthly)		–0.068*	–0.029	–0.070*
		(–1.805)	(–0.841)	(–1.811)
Bond trading volume		0.011***	0.004	0.002
		(4.022)	(1.387)	(0.668)
Log(Assets)		–0.043***	–0.024***	0.067***
		(–7.584)	(–4.346)	(2.936)
Market to book		–0.012***	–0.013***	0.049
		(–3.780)	(–4.102)	(1.528)
Log(Maturity)			–0.151***
			(–9.494)
Combined offering			–0.009
			(–0.330)
Rule 144A			0.006
			(0.499)
Rated			–0.054***
			(–5.869)
Rated $\times$ Investment grade			0.022
			(1.256)
Nasdaq listing			–0.001
			(–0.102)
Constant	0.069***	0.167***	0.329***	–0.774***
	(5.159)	(3.678)	(6.443)	(–4.034)
Observations	34,565	20,383	20,128	402
R-squared	0.076	0.345	0.417	0.948
Year–month FE	Yes	Yes	Yes	Yes
Bond FE	No	No	No	Yes

Table IV.

Open in new tab

Listed options and post-issuance convertible underpricing

Dependent variable	Post-issuance underpricing
	(1)	(2)	(3)	(4)
Option	–0.030**	–0.045***	–0.034**	–0.030**
	(–2.037)	(–2.854)	(–2.338)	(–2.197)
Delta		0.057***	0.054***	–0.075
		(3.425)	(3.363)	(–1.147)
Absolute moneyness		0.070***	0.083***	–0.007
		(6.842)	(8.352)	(–0.150)
Log(TTC)		–0.009	0.007	0.052
		(–0.846)	(0.699)	(1.025)
Log(TTM)		–0.014**	0.080***	0.173***
		(–2.157)	(7.216)	(2.958)
Volatility (Monthly)		0.101***	0.089***	0.040
		(8.185)	(7.828)	(0.888)
Amihud $\times 10^{6}$ (Monthly)		–0.068*	–0.029	–0.070*
		(–1.805)	(–0.841)	(–1.811)
Bond trading volume		0.011***	0.004	0.002
		(4.022)	(1.387)	(0.668)
Log(Assets)		–0.043***	–0.024***	0.067***
		(–7.584)	(–4.346)	(2.936)
Market to book		–0.012***	–0.013***	0.049
		(–3.780)	(–4.102)	(1.528)
Log(Maturity)			–0.151***
			(–9.494)
Combined offering			–0.009
			(–0.330)
Rule 144A			0.006
			(0.499)
Rated			–0.054***
			(–5.869)
Rated $\times$ Investment grade			0.022
			(1.256)
Nasdaq listing			–0.001
			(–0.102)
Constant	0.069***	0.167***	0.329***	–0.774***
	(5.159)	(3.678)	(6.443)	(–4.034)
Observations	34,565	20,383	20,128	402
R-squared	0.076	0.345	0.417	0.948
Year–month FE	Yes	Yes	Yes	Yes
Bond FE	No	No	No	Yes

Dependent variable	Post-issuance underpricing
	(1)	(2)	(3)	(4)
Option	–0.030**	–0.045***	–0.034**	–0.030**
	(–2.037)	(–2.854)	(–2.338)	(–2.197)
Delta		0.057***	0.054***	–0.075
		(3.425)	(3.363)	(–1.147)
Absolute moneyness		0.070***	0.083***	–0.007
		(6.842)	(8.352)	(–0.150)
Log(TTC)		–0.009	0.007	0.052
		(–0.846)	(0.699)	(1.025)
Log(TTM)		–0.014**	0.080***	0.173***
		(–2.157)	(7.216)	(2.958)
Volatility (Monthly)		0.101***	0.089***	0.040
		(8.185)	(7.828)	(0.888)
Amihud $\times 10^{6}$ (Monthly)		–0.068*	–0.029	–0.070*
		(–1.805)	(–0.841)	(–1.811)
Bond trading volume		0.011***	0.004	0.002
		(4.022)	(1.387)	(0.668)
Log(Assets)		–0.043***	–0.024***	0.067***
		(–7.584)	(–4.346)	(2.936)
Market to book		–0.012***	–0.013***	0.049
		(–3.780)	(–4.102)	(1.528)
Log(Maturity)			–0.151***
			(–9.494)
Combined offering			–0.009
			(–0.330)
Rule 144A			0.006
			(0.499)
Rated			–0.054***
			(–5.869)
Rated $\times$ Investment grade			0.022
			(1.256)
Nasdaq listing			–0.001
			(–0.102)
Constant	0.069***	0.167***	0.329***	–0.774***
	(5.159)	(3.678)	(6.443)	(–4.034)
Observations	34,565	20,383	20,128	402
R-squared	0.076	0.345	0.417	0.948
Year–month FE	Yes	Yes	Yes	Yes
Bond FE	No	No	No	Yes

In Column (4), we restrict the sample to convertible bonds where the issuer experienced the initiation of option listing after issuance of the convertible bond. We estimate post-issuance underpricing during the period between 12 months before and 12 months after the option listing month. For this subsample of 402 monthly post-issuance underpricing estimates using forty-two unique bonds, we include bond-fixed effects, instead of bond issue characteristics variables, to better exploit the time-series variation in option availability. The results again indicate a significantly negative relation between option listing and underpricing. The underpricing of convertible bonds decreases by 3.0 percentage points during the 12-month period after option listing, on average, compared with the underpricing during the 12-month period before option listing. The result clearly suggests that the availability of options significantly affects underpricing in the secondary market. Therefore, these results support the idea that options add value to convertible bond issuers by improving the overall information environment of issuing firms.

The above finding rules out selection stories in which some unobserved time-invariant factors lead issuers with options to have smaller convertible underpricing. Of course, exchanges do not randomly select firms for option listing. In particular, firms that have less information asymmetry or receive more investor attention are more likely to have listed options (Mayhew and Mihov, 2004). As such, our main conclusions are drawn from analyses that deal with potential endogeneity concerns by exploiting one of the SEC’s individual stock option listing requirements, which we present in the next sub-section.

3.2 Fuzzy RDD Analyses

Among other requirements, the SEC mandates a minimum stock price requirement for option listing, which states that the closing price of the company’s shares must have been at least $3.00 (or $7.50 until 2004) for a majority of trading days during the three calendar months preceding the date of selection (Hu, 2018). We utilize the discontinuity in the likelihood of option listing created by this ad hoc price cutoff to reexamine our baseline result.

Using the distance between an issuer’s average stock price and the cutoff price as the forcing variable, we employ a fuzzy RDD and use the subsample of convertible issues with average stock prices just around the cutoff to estimate the role of options in a more robust way. Specifically, we define the forcing variable, Price distance, for each issuing firm

i

in issuing year

t

⁠, as follows:

{Price distance}_{i, t} = (\begin{matrix} p r c_{i, t} - 7.5, i f t \leq 2004 \\ p r c_{i, t} - 3, i f t > 2004 \end{matrix}),

(2)

where

pr c_{i, t}

is the average stock price of firm

i

over the first 3 months in year

t

⁠. Crossing the price threshold makes a firm eligible to have listed options, so our treatment assignment variable, Eligible, is defined as

Eligible = 1 (Price distance > 0) .

(3)

where 1() is one when the condition in parentheses is satisfied and zero otherwise.

Exchanges could have some other considerations when selecting stocks for option listing, which we do not observe. Given the uncertainty of treatment, we choose to implement a fuzzy RDD instead of a sharp RDD. Specifically, we estimate a 2SLS regression model with the following first-stage specification where Eligible is used as the instrument variable:

Optio n_{i} = γ_{0} + γ_{1} \times Eligibl e_{i} + γ_{2} \times Price distanc e_{i} + γ^{'} X_{i} + Year F E_{i} + u_{i} .

(4)

We then use the predicted value

\hat{Option}

to test for the effect of options on convertible pricing in the following second-stage specification:

y_{i} = β_{0} + β_{1} \times \hat{Optio n_{i}} + β_{2} \times Price distanc e_{i} + β^{'} X_{i} + Year F E_{i} + ε_{i},

(5)

where y is the offering discount and

β_{1}

provides the 2SLS estimate of the treatment effect of listed options. The vector of control variables

X_{i}

is the vector of issuer characteristics.¹³ We follow Imbens and Kalyanaraman (2012) to derive the asymptotically optimal bandwidth using a squared-error loss criterion and estimate all regressions using the Weighted Least Squares method, where the weights are inversely proportional to the price distance from the price cutoff point.

Figure 3 shows the annual number of convertible bond issues, the number of issues with stock prices eligible for option listing, and the percentage of issues made by issuers with listed options. We illustrate the presence of a discontinuity in the probability of having options in Figure 4. More specifically, Figure 4 shows the percentage of convertibles with listed options among those in each price distance bin together with fitted lines and corresponding 90% confidence intervals. Around the threshold, we can clearly observe a jump in the likelihood of having options. The percentage of issuers with option listing increases from below 30% to over 50% as soon as the forcing variable passes the threshold. The differences between average percentages in each side of the threshold are statistically significant and the difference also remains significant when we narrow the bandwidth around the threshold, as shown by the Option variable in Table V, where we present univariate analyses results.¹⁴

Figure 3.

Distribution of RDD sample. For the fuzzy RDD analysis, we focus on the sample that covers the convertible bond issues with price distances smaller than 3 but larger than –3, that is, bandwidth [–3, +3]. The price difference is between the average stock price during the first 3 months of the calendar year of issuance and the minimum stock price ($7.5 until 2004 and $3.5 since 2005) for option listings required by the SEC. Issuers with positive price distance are defined as being eligible for option listings. The bars (left scale) show the total number of convertible bond issues in this sample (lighter bars) and the number of issues made by issuers being eligible for option listing (darker bars) in each year, and the line shows the percentage of issuers with listed options (right scale) in each year.

Open in new tab Download slide

Figure 4.

Percentage of convertible bond issues with listed options among issuers with average prices around the price cutoff point. The x-axis plots the price distance, the difference between average stock price during the first 3 months of the calendar year of issuance and the minimum stock price ($7.5 until 2004 and $3.5 since 2005) for option listings required by the SEC. Issuers with price distances to the right of zero are eligible for option listings and issuers with price distances to the left of zero are ineligible for option listings. The y-axis shows the percentage of convertible bond issues with listed options among issuers with average prices around the price cutoff points within each price distance bin. The straight lines fit local linear regressions within $3 bandwidth on both sides of the price cutoff point, and the dashed curved lines represent the corresponding 90% confidence interval.

Open in new tab Download slide

Table V.

Open in new tab

Fuzzy RDD: univariate results

This table presents summary statistics of the main variables for issues made by issuers with average stock prices around the RDD price cutoff point. Panels A and B, respectively, consider issues made by issuers with average stock prices within the bandwidths of [–2, +2] and [–1, +1] around the price cutoff point. Issuers with average prices above the price threshold are eligible for option listing. We report t-statistics for the test of the differences between issues with and without option listing eligibility. We also report the same statistics for the complier group, that is, the group of issues with (without) options among those satisfying (not satisfying) the price eligibility condition. The last column reports t-statistics for the test of the differences between issues with and without options in the complier group. *, **, and *** indicate statistical significance of the difference (one-sided t-tests) at the 1%, 5%, and 10% levels, respectively.

	Eligible					Compliers
	for option listing					Ineligible and		Eligible and
	No		Yes		Diff.	No Option		Option		Diff.
	N	Mean	N	Mean	t-stat	N	Mean	N	Mean	t-stat
Panel A: Bandwidth [–2, +2]
Option	129	0.22	101	0.49	4.28^***
Offering discount	90	0.23	80	0.15	–2.62^***	63	0.25	44	0.11	–3.89^***
Total buyers	118	18.27	85	26.51	1.97^**	98	13.43	34	39.41	5.05^***
Hedge fund buyers	117	11.43	85	16.33	2.04^**	98	8.65	34	23.41	4.96^***
Other buyers	117	6.99	85	10.18	1.57^*	98	4.78	34	16.00	4.48^***
Panel B: Bandwidth [–1, +1]
Option	52	0.29	49	0.49	2.10^**
Offering discount	33	0.27	35	0.17	–2.05^**	19	0.28	19	0.12	–2.79^***
Total buyers	43	12.19	44	25.61	2.16^**	35	12.83	20	34.60	2.50^**
Hedge fund buyers	42	8.07	44	14.95	2.04^**	35	8.29	20	20.30	2.54^***
Other buyers	42	4.38	44	10.66	1.97^**	35	4.54	20	14.30	2.29^**

	Eligible					Compliers
	for option listing					Ineligible and		Eligible and
	No		Yes		Diff.	No Option		Option		Diff.
	N	Mean	N	Mean	t-stat	N	Mean	N	Mean	t-stat
Panel A: Bandwidth [–2, +2]
Option	129	0.22	101	0.49	4.28^***
Offering discount	90	0.23	80	0.15	–2.62^***	63	0.25	44	0.11	–3.89^***
Total buyers	118	18.27	85	26.51	1.97^**	98	13.43	34	39.41	5.05^***
Hedge fund buyers	117	11.43	85	16.33	2.04^**	98	8.65	34	23.41	4.96^***
Other buyers	117	6.99	85	10.18	1.57^*	98	4.78	34	16.00	4.48^***
Panel B: Bandwidth [–1, +1]
Option	52	0.29	49	0.49	2.10^**
Offering discount	33	0.27	35	0.17	–2.05^**	19	0.28	19	0.12	–2.79^***
Total buyers	43	12.19	44	25.61	2.16^**	35	12.83	20	34.60	2.50^**
Hedge fund buyers	42	8.07	44	14.95	2.04^**	35	8.29	20	20.30	2.54^***
Other buyers	42	4.38	44	10.66	1.97^**	35	4.54	20	14.30	2.29^**

Table V.

Open in new tab

Fuzzy RDD: univariate results

	Eligible					Compliers
	for option listing					Ineligible and		Eligible and
	No		Yes		Diff.	No Option		Option		Diff.
	N	Mean	N	Mean	t-stat	N	Mean	N	Mean	t-stat
Panel A: Bandwidth [–2, +2]
Option	129	0.22	101	0.49	4.28^***
Offering discount	90	0.23	80	0.15	–2.62^***	63	0.25	44	0.11	–3.89^***
Total buyers	118	18.27	85	26.51	1.97^**	98	13.43	34	39.41	5.05^***
Hedge fund buyers	117	11.43	85	16.33	2.04^**	98	8.65	34	23.41	4.96^***
Other buyers	117	6.99	85	10.18	1.57^*	98	4.78	34	16.00	4.48^***
Panel B: Bandwidth [–1, +1]
Option	52	0.29	49	0.49	2.10^**
Offering discount	33	0.27	35	0.17	–2.05^**	19	0.28	19	0.12	–2.79^***
Total buyers	43	12.19	44	25.61	2.16^**	35	12.83	20	34.60	2.50^**
Hedge fund buyers	42	8.07	44	14.95	2.04^**	35	8.29	20	20.30	2.54^***
Other buyers	42	4.38	44	10.66	1.97^**	35	4.54	20	14.30	2.29^**

	Eligible					Compliers
	for option listing					Ineligible and		Eligible and
	No		Yes		Diff.	No Option		Option		Diff.
	N	Mean	N	Mean	t-stat	N	Mean	N	Mean	t-stat
Panel A: Bandwidth [–2, +2]
Option	129	0.22	101	0.49	4.28^***
Offering discount	90	0.23	80	0.15	–2.62^***	63	0.25	44	0.11	–3.89^***
Total buyers	118	18.27	85	26.51	1.97^**	98	13.43	34	39.41	5.05^***
Hedge fund buyers	117	11.43	85	16.33	2.04^**	98	8.65	34	23.41	4.96^***
Other buyers	117	6.99	85	10.18	1.57^*	98	4.78	34	16.00	4.48^***
Panel B: Bandwidth [–1, +1]
Option	52	0.29	49	0.49	2.10^**
Offering discount	33	0.27	35	0.17	–2.05^**	19	0.28	19	0.12	–2.79^***
Total buyers	43	12.19	44	25.61	2.16^**	35	12.83	20	34.60	2.50^**
Hedge fund buyers	42	8.07	44	14.95	2.04^**	35	8.29	20	20.30	2.54^***
Other buyers	42	4.38	44	10.66	1.97^**	35	4.54	20	14.30	2.29^**

Next, we formally estimate the first-stage relationship between the eligibility for option listing and the availability of options at the time of convertible issuance. The results are reported in Column (1) and (2) of Table VI. Consistent with the graphic pattern and univariate test, we find that the listing eligibility significantly increases the likelihood of having options. The result in Column (1) shows that convertible bond issuers who we classify as eligible for option listings are about 30% more likely to have listed options available than those that we classify as not eligible. This positive relation is robust to controlling for other firm characteristics in Column (2). The $F$ -statistics from the first-stage regression without and with controls are 16.5 and 19.9, respectively, suggesting that the listing eligibility is not a weak instrument.

Table VI.

Open in new tab

Fuzzy RDD: regression results

This table presents the regression results of the fuzzy RDD analysis. Columns (1) and (2) report the first-stage results, where the dependent variable is Option, a dummy variable indicating whether the issuer had listed options at the time of its convertible bond issuance. The forcing variable, Price distance, is defined as the difference between the average stock price over the first 3 months of the calendar year of convertible bond issuance and the minimum stock price for option listings required by the SEC ($7.5 until 2004 and $3 from 2005). Eligible is a dummy variable indicating the issuer with eligible stock prices for option listing, that is, Price distance > $0$ ⁠. Columns (3)–(5) report the second-stage results, where the dependent variable is convertible offering discount. We use the listing eligibility indicator as the instrument variable of Option. The forcing variable, Price distance, is included in both the first and second stages. We use the optimal bandwidth following Imbens and Kalyanaraman (2012). In Column (5), we restrict our sample to offerings with offering discount values ranging from –30% to 30%. All regressions include year-fixed effects to control for unobserved common factors in each year, and are estimated using the Weighted Least Squares method, where the weights are inversely proportional to the price distance from the cutoff point. All control variables are defined in Appendix A. $T$ -statistics based on clustered standard errors at the firm level are reported in parentheses. ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively. Bold values indicates the main estimates of interest.

Dependent variable	Option		Offering discount
Model	First-stage		Second-stage
Sample restriction					\|OD\| ≤ 30%
	(1)	(2)	(3)	(4)	(5)
Eligible	0.295**	0.337***
	(2.414)	(2.612)
Option			–0.351***	–0.365***	–0.175**
			(–2.648)	(–3.140)	(–2.008)
Price distance	–0.029	–0.084	–0.004	0.004	0.012
	(–0.608)	(–1.596)	(–0.214)	(0.261)	(0.804)
Log(Assets)		0.113***		–0.005	–0.018
		(4.558)		(–0.306)	(–1.310)
Market to book		0.023***		0.009*	0.004
		(2.910)		(1.936)	(0.223)
Volatility		–0.211		3.030**	0.260
		(–0.319)		(2.419)	(0.180)
Amihud*10^6		–0.018		0.002	0.017*
		(–1.121)		(0.144)	(1.829)
Nasdaq listing		0.029		–0.018	–0.026
		(0.364)		(–0.455)	(–0.766)
Constant	–0.139	–0.854***	0.369***	0.275*	0.438***
	(–1.293)	(–5.019)	(6.241)	(1.928)	(3.291)
Observations	258	224	183	170	105
F–stat	16.487	19.905	5.825	6.477	16.984
Year FE	Yes	Yes	Yes	Yes	Yes

Dependent variable	Option		Offering discount
Model	First-stage		Second-stage
Sample restriction					\|OD\| ≤ 30%
	(1)	(2)	(3)	(4)	(5)
Eligible	0.295**	0.337***
	(2.414)	(2.612)
Option			–0.351***	–0.365***	–0.175**
			(–2.648)	(–3.140)	(–2.008)
Price distance	–0.029	–0.084	–0.004	0.004	0.012
	(–0.608)	(–1.596)	(–0.214)	(0.261)	(0.804)
Log(Assets)		0.113***		–0.005	–0.018
		(4.558)		(–0.306)	(–1.310)
Market to book		0.023***		0.009*	0.004
		(2.910)		(1.936)	(0.223)
Volatility		–0.211		3.030**	0.260
		(–0.319)		(2.419)	(0.180)
Amihud*10^6		–0.018		0.002	0.017*
		(–1.121)		(0.144)	(1.829)
Nasdaq listing		0.029		–0.018	–0.026
		(0.364)		(–0.455)	(–0.766)
Constant	–0.139	–0.854***	0.369***	0.275*	0.438***
	(–1.293)	(–5.019)	(6.241)	(1.928)	(3.291)
Observations	258	224	183	170	105
F–stat	16.487	19.905	5.825	6.477	16.984
Year FE	Yes	Yes	Yes	Yes	Yes

Table VI.

Open in new tab

Fuzzy RDD: regression results

Dependent variable	Option		Offering discount
Model	First-stage		Second-stage
Sample restriction					\|OD\| ≤ 30%
	(1)	(2)	(3)	(4)	(5)
Eligible	0.295**	0.337***
	(2.414)	(2.612)
Option			–0.351***	–0.365***	–0.175**
			(–2.648)	(–3.140)	(–2.008)
Price distance	–0.029	–0.084	–0.004	0.004	0.012
	(–0.608)	(–1.596)	(–0.214)	(0.261)	(0.804)
Log(Assets)		0.113***		–0.005	–0.018
		(4.558)		(–0.306)	(–1.310)
Market to book		0.023***		0.009*	0.004
		(2.910)		(1.936)	(0.223)
Volatility		–0.211		3.030**	0.260
		(–0.319)		(2.419)	(0.180)
Amihud*10^6		–0.018		0.002	0.017*
		(–1.121)		(0.144)	(1.829)
Nasdaq listing		0.029		–0.018	–0.026
		(0.364)		(–0.455)	(–0.766)
Constant	–0.139	–0.854***	0.369***	0.275*	0.438***
	(–1.293)	(–5.019)	(6.241)	(1.928)	(3.291)
Observations	258	224	183	170	105
F–stat	16.487	19.905	5.825	6.477	16.984
Year FE	Yes	Yes	Yes	Yes	Yes

Dependent variable	Option		Offering discount
Model	First-stage		Second-stage
Sample restriction					\|OD\| ≤ 30%
	(1)	(2)	(3)	(4)	(5)
Eligible	0.295**	0.337***
	(2.414)	(2.612)
Option			–0.351***	–0.365***	–0.175**
			(–2.648)	(–3.140)	(–2.008)
Price distance	–0.029	–0.084	–0.004	0.004	0.012
	(–0.608)	(–1.596)	(–0.214)	(0.261)	(0.804)
Log(Assets)		0.113***		–0.005	–0.018
		(4.558)		(–0.306)	(–1.310)
Market to book		0.023***		0.009*	0.004
		(2.910)		(1.936)	(0.223)
Volatility		–0.211		3.030**	0.260
		(–0.319)		(2.419)	(0.180)
Amihud*10^6		–0.018		0.002	0.017*
		(–1.121)		(0.144)	(1.829)
Nasdaq listing		0.029		–0.018	–0.026
		(0.364)		(–0.455)	(–0.766)
Constant	–0.139	–0.854***	0.369***	0.275*	0.438***
	(–1.293)	(–5.019)	(6.241)	(1.928)	(3.291)
Observations	258	224	183	170	105
F–stat	16.487	19.905	5.825	6.477	16.984
Year FE	Yes	Yes	Yes	Yes	Yes

After finding strong results from the first-stage analyses, we proceed to the second-stage analyses in which the availability of options is instrumented by listing eligibility. Before presenting the multivariate analysis, we illustrate the results by a simple comparison of the offering discount between issues with and without option listing eligibility. As shown in Panel A of Table V, the offering discount of issues with listing eligibility is about 15%, which is about 42% smaller relative to the offering discount of those without listing eligibility (23%). This difference is even larger (11% versus 25%) for the complier group, that is, issues made by issuers with (without) options and average stock prices that are eligible (ineligible) for option listing. This difference also remains significant when we narrow the bandwidth to focus on issues closer to the regression discontinuity threshold in Panel B of Table V.

Columns (3)–(5) of Table VI report the 2SLS regression results where we formally use the listing eligibility indicator as the instrumental variable of $Option$ ⁠. The 2SLS estimate of $Option$ is significantly negative in Column (3) and both its statistical significance and economic magnitude increase when we additionally control for issuer characteristics in Column (4). The seemingly large coefficient estimates, –0.351 in Column (3) and –0.365 in Column (4), are to some extent driven by extreme discount values in this small sample. When we restrict the sample to less extreme discount value estimates ranging from –30% to 30%, we find a coefficient estimate of 0.175 in Column (5), which implies a reduction in offering discounts of 17.5% by having listed options.

The IV estimates are overall larger than our baseline OLS estimates, which is common in papers using the IV method (Jiang, 2017). Note that our IV compliers are convertible bond issuers with listed options and with stock prices that are just above the minimum stock price requirement. This subgroup is likely to exhibit larger local average treatment effects. This is because issuers with stock prices around the price listing requirement tend to be smaller and riskier with a poorer information environment than others, suggesting that investors are likely to ask for a larger offering discount for the issues made by those issuers. The average (median) discount of this subsample is 19% (24%), indeed larger than the average (median) of the full sample, 15% (15%). Those issuers would also receive more benefit from having listed options through a reduction in information asymmetry and a facilitation of hedging.¹⁵ Given the firm characteristics and large offering discounts of this subsample, the estimated local average treatment effects based on the 2SLS result is likely representing an upper bound for the average treatment effect of having listed options. Interestingly, earlier studies have also documented that the economic impact of option listings can be sizable. For example, through option listings, Naiker, Navissi, and Truong (2013) find a relative reduction in the implied cost of equity of around 12%, Do, Truong, and Vu (2021) find a 9% relative reduction in loan spreads, Cao et al. (2019) find a 20% relative reduction in the bank debt ratio, and Hong, Park, and King (2020) document a 14.5% increase in equity issuance.

Overall, the 2SLS estimation results of the fuzzy RDD indicate that the availability of listed options significantly reduces convertible bond underpricing even after addressing concerns regarding the endogeneity of option listing decisions.

3.3 Placebo and Robustness Tests

The validity of our fuzzy RDD analysis roots critically in the smoothness of all factors besides the treatment (option listing) around the price cutoff point. In Table AII, we test for the comparability of all observed issue- and issuer-characteristics around the price cutoff. We run the RDD regression from Equation (4) using each observed characteristic as the dependent variable. As is shown, none of them is significantly different around the price threshold, which confirms the smoothness of other issue- and issuer-characteristics at the cutoff point.

Nevertheless, balance check tests only address concerns regarding observable factors and unobservable confounding factors may still affect our results. To address this concern, we perform a placebo test by repeating the fuzzy RDD analyses using fake minimum stock price requirements, $2, $5, and $9. If the real eligibility is not the main driver behind our main findings but other unobservable factors drive our results, we may find significant results even when we use fake price cutoff points.

Table VII presents these placebo results. The coefficient estimates of $Eligible$ are not statistically significant and sometimes even negative in Columns (1), (3), and (5), indicating no visible disparity in the options availability across the falsified price cutoffs. In Columns (2), (4), and (6), we use the placebo listing eligibility as an IV to estimate the effect of option listing on offering discount and find insignificant 2SLS coefficient estimates of $Option$ in all three cases. Overall, the placebo results show no effect of option listings on offering discounts when falsified price thresholds are used in the RDD analyses and thereby strengthen the validity and reliability of our fuzzy RDD results.

Table VII.

Open in new tab

Fuzzy RDD: Placebo tests

This table presents the regression results of the fuzzy RDD analysis using fake eligibility conditions instead of the actual eligibility condition, that is, fake minimum stock price requirements of $2, $5, and $9. Columns (1), (3), and (5) report the first-stage results, where the dependent variable is Option, a dummy variable indicating whether the issuer had listed options at the time of its convertible bond issuance. Columns (2), (4), and (6) report the 2SLS results, where the dependent variable is convertible offering discount. We employ a bandwidth of [–2, +2] around each falsified price cutoff. All regressions include year-fixed effects to control for unobserved common factors in each year, are estimated using weighted least squares, where the weights are inversely proportional to the distance from the placebo cutoff points. All control variables are defined in Appendix A. $T$ -statistics based on clustered standard errors at the firm level are reported in parentheses. ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.

Placebo RD	Cutoff at $2		Cutoff at $5		Cutoff at $9
Dependent variable	Option	Offering discount	Option	Offering discount	Option	Offering discount
Model	First-stage	Second-stage	First-stage	Second-stage	First-stage	Second-stage
	(1)	(2)	(3)	(4)	(5)	(6)
Eligible (placebo)	–0.050		0.160		0.130
	(–0.683)		(1.010)		(0.636)
Option		4.418		–0.145		0.767
		(0.561)		(–0.338)		(0.215)
Price distance (placebo)	0.126***	–0.317	–0.049	–0.025	–0.025	–0.043
	(3.880)	(–0.528)	(–0.672)	(–1.221)	(–0.251)	(–0.285)
Constant	0.185***	–3.452	0.334***	0.239	0.409***	0.381***
	(3.833)	(–0.498)	(3.774)	(0.562)	(3.854)	(19.777)
Observations	260	86	196	117	154	112
F-stat	11.551	15.287	0.599	20.709	0.404	47.139
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

Placebo RD	Cutoff at $2		Cutoff at $5		Cutoff at $9
Dependent variable	Option	Offering discount	Option	Offering discount	Option	Offering discount
Model	First-stage	Second-stage	First-stage	Second-stage	First-stage	Second-stage
	(1)	(2)	(3)	(4)	(5)	(6)
Eligible (placebo)	–0.050		0.160		0.130
	(–0.683)		(1.010)		(0.636)
Option		4.418		–0.145		0.767
		(0.561)		(–0.338)		(0.215)
Price distance (placebo)	0.126***	–0.317	–0.049	–0.025	–0.025	–0.043
	(3.880)	(–0.528)	(–0.672)	(–1.221)	(–0.251)	(–0.285)
Constant	0.185***	–3.452	0.334***	0.239	0.409***	0.381***
	(3.833)	(–0.498)	(3.774)	(0.562)	(3.854)	(19.777)
Observations	260	86	196	117	154	112
F-stat	11.551	15.287	0.599	20.709	0.404	47.139
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

Table VII.

Open in new tab

Fuzzy RDD: Placebo tests

Placebo RD	Cutoff at $2		Cutoff at $5		Cutoff at $9
Dependent variable	Option	Offering discount	Option	Offering discount	Option	Offering discount
Model	First-stage	Second-stage	First-stage	Second-stage	First-stage	Second-stage
	(1)	(2)	(3)	(4)	(5)	(6)
Eligible (placebo)	–0.050		0.160		0.130
	(–0.683)		(1.010)		(0.636)
Option		4.418		–0.145		0.767
		(0.561)		(–0.338)		(0.215)
Price distance (placebo)	0.126***	–0.317	–0.049	–0.025	–0.025	–0.043
	(3.880)	(–0.528)	(–0.672)	(–1.221)	(–0.251)	(–0.285)
Constant	0.185***	–3.452	0.334***	0.239	0.409***	0.381***
	(3.833)	(–0.498)	(3.774)	(0.562)	(3.854)	(19.777)
Observations	260	86	196	117	154	112
F-stat	11.551	15.287	0.599	20.709	0.404	47.139
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

Placebo RD	Cutoff at $2		Cutoff at $5		Cutoff at $9
Dependent variable	Option	Offering discount	Option	Offering discount	Option	Offering discount
Model	First-stage	Second-stage	First-stage	Second-stage	First-stage	Second-stage
	(1)	(2)	(3)	(4)	(5)	(6)
Eligible (placebo)	–0.050		0.160		0.130
	(–0.683)		(1.010)		(0.636)
Option		4.418		–0.145		0.767
		(0.561)		(–0.338)		(0.215)
Price distance (placebo)	0.126***	–0.317	–0.049	–0.025	–0.025	–0.043
	(3.880)	(–0.528)	(–0.672)	(–1.221)	(–0.251)	(–0.285)
Constant	0.185***	–3.452	0.334***	0.239	0.409***	0.381***
	(3.833)	(–0.498)	(3.774)	(0.562)	(3.854)	(19.777)
Observations	260	86	196	117	154	112
F-stat	11.551	15.287	0.599	20.709	0.404	47.139
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

As an additional robustness check, we repeat our RDD analyses using narrower bandwidths and higher-order polynomials since treatment effect estimates in the RDD analysis are often sensitive to the choice of bandwidth and polynomial order. In Panel A of Table VIII, we re-estimate 2SLS regressions with two narrower bandwidths of [–2, +2] and [–1, +1]. The 2SLS estimates remain to be significant even when we narrow the bandwidth, which re-enforces the validity of our fuzzy RDD setting. We also include higher order (up to second or third) polynomials of the forcing variable, Price distance, as additional controls in our RDD regressions. The estimates remain to be similar to our baseline estimate.

Table VIII.

Open in new tab

Robustness tests

This table presents robustness test results for the effect of listed options on convertible underpricing. The baseline result refers to the 2SLS estimate in Column (4) of Table VI. For brevity, we only report the main coefficient estimate. In Panel A, we re-estimate 2SLS regressions of the fuzzy RDD analysis using two narrower bandwidths of [–2, +2] and [–1, +1] or including higher order (up to second or third) polynomials of the forcing variable, $Price distance$ ⁠, as additional controls. In Panel B, we use credit spreads based on LIBOR interest rates as an alternative credit spread estimate in computing offering discounts. In Panel C, we use four alternative volatility measures in computing offering discounts: (i) historical volatility estimates using daily stock returns over trading days −120 to −1; (ii) historical volatility estimates using daily stock returns over trading days −520 to −1, as in Ammann, Kind, and Wilde (2003); (iii) historical volatility estimates using monthly stock returns over the 5 years preceding the issue date, as in King (1986); and (iv) GARCH(1,1) volatility estimates using daily stock returns over trading days −1,300 to −40, as in Figlewski (1997). In Panel D, we use the valuation model of Finnerty (2015), instead of the TF model, to compute offering discounts. In Panel E, we use two alternative dependent variables based on all offering discount estimates in Panels A–D: Small (large) discount is a dummy variable equal to one if all offering discount estimates of an issuer are smaller than 5% (larger than 15%). Bold values indicates the main estimates of interest.

	2SLS estimates of Option
	Coeff.	(t-stat)	Obs.
Baseline	–0.365***	(–3.140)	170
Panel A: Alternative RD estimation
Narrower bandwidth [–2, +2]	–0.355***	(–3.042)	157
Narrower bandwidth [–1, +1]	–0.543**	(–2.522)	63
Running variable polynomial order: quadratic	–0.371***	(–3.201)	170
Running variable polynomial order: cubic	–0.395**	(–2.469)	170
Panel B: Alternative credit spread estimates
LIBOR interest rate and spread	–0.353***	(–2.993)	170
Panel C: Alternative volatility measures
Estimation window [–120, –1]	–0.355***	(–3.004)	170
Estimation window [–520, –1]	–0.333***	(–2.966)	170
5Y monthly	–0.308***	(–2.837)	164
GARCH(1, 1)	–0.321***	(–2.820)	170
Panel D: Alternative valuation model
Finnerty (2015) model	–0.325***	(–3.354)	170
Panel E: Alternative dependent variable
Small discount (OD < 5%)	0.440**	(2.463)	164
Large discount (OD > 15%)	–0.964***	(–3.386)	164

	2SLS estimates of Option
	Coeff.	(t-stat)	Obs.
Baseline	–0.365***	(–3.140)	170
Panel A: Alternative RD estimation
Narrower bandwidth [–2, +2]	–0.355***	(–3.042)	157
Narrower bandwidth [–1, +1]	–0.543**	(–2.522)	63
Running variable polynomial order: quadratic	–0.371***	(–3.201)	170
Running variable polynomial order: cubic	–0.395**	(–2.469)	170
Panel B: Alternative credit spread estimates
LIBOR interest rate and spread	–0.353***	(–2.993)	170
Panel C: Alternative volatility measures
Estimation window [–120, –1]	–0.355***	(–3.004)	170
Estimation window [–520, –1]	–0.333***	(–2.966)	170
5Y monthly	–0.308***	(–2.837)	164
GARCH(1, 1)	–0.321***	(–2.820)	170
Panel D: Alternative valuation model
Finnerty (2015) model	–0.325***	(–3.354)	170
Panel E: Alternative dependent variable
Small discount (OD < 5%)	0.440**	(2.463)	164
Large discount (OD > 15%)	–0.964***	(–3.386)	164

Table VIII.

Open in new tab

Robustness tests

	2SLS estimates of Option
	Coeff.	(t-stat)	Obs.
Baseline	–0.365***	(–3.140)	170
Panel A: Alternative RD estimation
Narrower bandwidth [–2, +2]	–0.355***	(–3.042)	157
Narrower bandwidth [–1, +1]	–0.543**	(–2.522)	63
Running variable polynomial order: quadratic	–0.371***	(–3.201)	170
Running variable polynomial order: cubic	–0.395**	(–2.469)	170
Panel B: Alternative credit spread estimates
LIBOR interest rate and spread	–0.353***	(–2.993)	170
Panel C: Alternative volatility measures
Estimation window [–120, –1]	–0.355***	(–3.004)	170
Estimation window [–520, –1]	–0.333***	(–2.966)	170
5Y monthly	–0.308***	(–2.837)	164
GARCH(1, 1)	–0.321***	(–2.820)	170
Panel D: Alternative valuation model
Finnerty (2015) model	–0.325***	(–3.354)	170
Panel E: Alternative dependent variable
Small discount (OD < 5%)	0.440**	(2.463)	164
Large discount (OD > 15%)	–0.964***	(–3.386)	164

	2SLS estimates of Option
	Coeff.	(t-stat)	Obs.
Baseline	–0.365***	(–3.140)	170
Panel A: Alternative RD estimation
Narrower bandwidth [–2, +2]	–0.355***	(–3.042)	157
Narrower bandwidth [–1, +1]	–0.543**	(–2.522)	63
Running variable polynomial order: quadratic	–0.371***	(–3.201)	170
Running variable polynomial order: cubic	–0.395**	(–2.469)	170
Panel B: Alternative credit spread estimates
LIBOR interest rate and spread	–0.353***	(–2.993)	170
Panel C: Alternative volatility measures
Estimation window [–120, –1]	–0.355***	(–3.004)	170
Estimation window [–520, –1]	–0.333***	(–2.966)	170
5Y monthly	–0.308***	(–2.837)	164
GARCH(1, 1)	–0.321***	(–2.820)	170
Panel D: Alternative valuation model
Finnerty (2015) model	–0.325***	(–3.354)	170
Panel E: Alternative dependent variable
Small discount (OD < 5%)	0.440**	(2.463)	164
Large discount (OD > 15%)	–0.964***	(–3.386)	164

3.4 Reliability of Our Offering Discount Estimates

Our baseline measure of offering discounts is based on the model of TF, which is the most popular valuation method among both academics and practitioners (e.g., Zabolotnyuk, Jones, and Veld, 2010). Still, market participants use various pricing models and even when one uses the same pricing model, estimated prices can be different depending on the parameter choices. Therefore, we examine the robustness of our findings to a different pricing model and different parameter estimations.

We start by examining the robustness of our results with respect to different parameter estimates used in the TF valuation model. In Panel B of Table VIII, we consider an alternative credit spread estimate in computing offering discounts based on LIBOR rates. We consider the LIBOR term structure because the derivatives business has traditionally used LIBOR rates to obtain discount rates. We obtain LIBOR term structure data estimated using Eurodollar futures contracts from OptionMetrics. The resulting offering discounts have a mean of 14.57% and the corresponding 2SLS estimate is similar to the estimate from our baseline analysis.

Next, we focus on volatility estimates. Column 4 of Table VI shows a positive relation between offering discounts and our volatility estimate, which is defined as the daily stock return volatility calculated using stock returns over the window [–240, –40] relative to a convertible bond offering announcement date. A likely explanation for this relation is that firms with more volatile returns decide to offer a higher discount to make sure that they are able to raise the required funds. A similar relation between volatility and offering discounts is typically found in seasoned equity offerings. For example, Altinkilic and Hansen (2003) and Corwin (2003) report that the costs of seasoned equity offerings are significantly higher for issuers with more volatile stock returns. However, stock return volatility is also an important input variable to our convertible bond valuation model, making it useful to study the dependence of our main results on the choice of a volatility estimate. For this purpose, in Panel C, we use four alternative volatility measures in computing offering discounts: (i) historical volatility estimated using daily stock returns over trading days −120 to −1; (ii) historical volatility estimated using daily stock returns over trading days −520 to −1, as in Ammann, Kind, and Wilde (2003); (iii) historical volatility estimated using monthly stock returns over the 5 years preceding the issue date, as in King (1986); and (v) GARCH(1,1) volatility estimates using daily stock returns over trading days −1,300 to −40, as in Figlewski (1997). It can be seen that the results based on alternative volatility measures are again qualitatively similar to our baseline result.

In Panel D, we use the theoretical convertible bond price model developed by Finnerty (2015) as an alternative valuation method to estimate convertible bond underpricing. In Finnerty (2015), the value of a convertible bond is the sum of the value of a regular non-convertible bond and the value of an option to exchange the bond into stocks. The model incorporates stochastic interest rates and credit spreads, whereas the TF model assumes these to be constant. Finnerty (2015) derives a closed-form solution for the value of the exchange option. The model uses iterative procedures to adjust the value of the convertible bond for call and put provisions. We find that the offering discounts estimated from the Finnerty model have a mean of 15.35% and are highly correlated with our baseline measures from the TF model (correlation of 95%). The 2SLS coefficient estimate using the Finnerty model also remains similar to our baseline estimate.¹⁶

In Panel E, we use two new binary dependent variables, which are based on the offering discount estimates used in Panels A–D of Table VIII. Small (Large) discount of an issue is equal to one if all offering discount estimates in Panels A–D of the issue are smaller than 5% (larger than 15%) and zero otherwise.¹⁷ As such, we classify an issue as having a small (large) offering discount only if discount estimates based on all different valuation models and parameter choices suggest so. Thus, this approach utilizes information from all discount estimates and minimizes the impact of potential pricing errors from a single valuation model or parameter choice. The corresponding 2SLS estimates are statistically significant and imply that listed options significantly increase (reduce) the likelihood of offering discounts being small (large).

Taken together, our results show a clear effect of option availability on offering discounts independent of the choices that we make regarding valuation models and parameter choices for the calculation of theoretical convertible values.

4. Options and Capital Supply for Convertible Bonds

The results so far show that offering discounts are significantly lower for convertible bond issues made by issuers with options. Earlier findings, including the quicker convergence of transaction prices to theoretical prices for issues made by issuers with options, hint at the possibility of options improving the information environment of issuers. If options improve the overall information environment for issuers and thereby reduce information asymmetry between issuers and investors, the availability of options may attract more capital suppliers who want to avoid adverse selection problems arising from information asymmetry. That is, despite lower discounts at offerings, investors might be interested in investing in convertible bonds issued by firms with listed options since they have access to useful information such as implied volatility estimates and potential insight into future growth opportunities derived from differences in demand between call and put options (e.g., Pan and Poteshman, 2006). In addition, options can attract investors who are interested in hedging their long positions in convertibles since options provide additional hedging opportunities and relax short-sale constraints. In this section, we exploit data on convertible buyers to examine whether the availability of options affects the capital supply for convertible bonds, which will help us better understand the channels through which options affect convertible bond issuance decisions.

The univariate tests in Tables II and V suggest that issuers with options (or with option listing eligibility) attract twice as many buyers as those without options, 56 versus 28 (or 27 versus 18 around the option listing eligibility threshold). In Table IX, we formally test this effect. Our main conclusion results from employing our RDD analysis, but we also report OLS regression results. In Columns (1) and (2), the dependent variable is the natural log-transformed total number of institutional investors who purchased convertible bonds. In Column (1), we estimate an OLS regression of this dependent variable on our Option variable. Our sample contains 984 observations with all relevant information and we find a significantly positive coefficient estimate of $Option$ ⁠, implying that issues with options attract more capital suppliers than those without options do, after controlling for year-fixed effects and firm characteristics.¹⁸ In Column (2), we employ our fuzzy RDD setup using the instrumented option availability variable, comparable to our offering discount analysis in Table VI. The second-stage 2SLS estimate of $Option$ is again positive and statistically significant, consistent with the notion that options enable convertible bond issuers to attract significantly more investors. The sample size is considerably smaller for the 2SLS estimate (192 observations), but the 2SLS results indicate that the OLS result is not driven by the endogeneity of option listing decisions.

Table IX.

Open in new tab

Regression analysis of investor participation

This table presents the effects of listed options on investor participation in convertible bond offerings. The dependent variable is the log-transformed number of total institutional investors in Columns (1) and (2), the log-transformed number of hedge fund buyers in Columns (3) and (4), and the log-transformed number of non-hedge fund buyers in Columns (5) and (6). We report both the OLS regression results using the full sample and the second-stage 2SLS results using the fuzzy RDD subsample. Option is a dummy variable indicating issuers with listed options at the time of issuance. In Columns (2), (4), and (6), we use the instrumented Option variable using the eligibility indicator, Eligible, and the forcing variable, Price distance, is included in both the first and second stages. We use the optimal bandwidth following Imbens and Kalyanaraman (2012). All regressions include year-fixed effects to control unobserved common factors in each year. All control variables are defined in Appendix A. $T$ -statistics based on clustered standard errors at the firm level are reported in parentheses. ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively. Bold values indicates the main estimates of interest.

Dependent variable	Log(Total buyers)		Log(HF buyers)		Log(Non-HF buyers)
Model	OLS	2SLS	OLS	2SLS	OLS	2SLS
	(1)	(2)	(3)	(4)	(5)	(6)
Option	0.226**	1.670**	0.208**	1.638**	0.293***	1.384*
	(2.536)	(1.992)	(2.407)	(2.033)	(2.958)	(1.779)
Price distance		–0.030		–0.011		–0.039
		(–0.442)		(–0.153)		(–0.537)
Log(Assets)	0.321***	0.046	0.310***	–0.011	0.407***	0.171
	(10.399)	(0.374)	(10.020)	(–0.083)	(12.708)	(1.628)
Market to book	0.031*	–0.037*	0.039**	–0.047**	0.049***	0.001
	(1.767)	(–1.857)	(2.048)	(–1.995)	(3.126)	(0.066)
Volatility	–4.278**	0.785	–4.006**	–0.144	–5.911***	1.870
	(–2.408)	(0.321)	(–2.117)	(–0.058)	(–2.658)	(0.857)
Amihud $\times 10^{6}$	0.004	–0.005	–0.004	–0.007	0.012	0.008
	(0.434)	(–0.134)	(–0.529)	(–0.179)	(1.281)	(0.245)
Nasdaq listing	0.087	0.118	0.150**	0.056	0.061	–0.141
	(1.158)	(0.632)	(2.148)	(0.287)	(0.746)	(–0.758)
Constant	0.982***	–0.117	0.679***	–0.087	–0.268	–0.807
	(4.056)	(–0.182)	(2.905)	(–0.193)	(–1.044)	(–0.956)
Observations	984	192	922	179	922	179
R-squared	0.527		0.501		0.507
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

Dependent variable	Log(Total buyers)		Log(HF buyers)		Log(Non-HF buyers)
Model	OLS	2SLS	OLS	2SLS	OLS	2SLS
	(1)	(2)	(3)	(4)	(5)	(6)
Option	0.226**	1.670**	0.208**	1.638**	0.293***	1.384*
	(2.536)	(1.992)	(2.407)	(2.033)	(2.958)	(1.779)
Price distance		–0.030		–0.011		–0.039
		(–0.442)		(–0.153)		(–0.537)
Log(Assets)	0.321***	0.046	0.310***	–0.011	0.407***	0.171
	(10.399)	(0.374)	(10.020)	(–0.083)	(12.708)	(1.628)
Market to book	0.031*	–0.037*	0.039**	–0.047**	0.049***	0.001
	(1.767)	(–1.857)	(2.048)	(–1.995)	(3.126)	(0.066)
Volatility	–4.278**	0.785	–4.006**	–0.144	–5.911***	1.870
	(–2.408)	(0.321)	(–2.117)	(–0.058)	(–2.658)	(0.857)
Amihud $\times 10^{6}$	0.004	–0.005	–0.004	–0.007	0.012	0.008
	(0.434)	(–0.134)	(–0.529)	(–0.179)	(1.281)	(0.245)
Nasdaq listing	0.087	0.118	0.150**	0.056	0.061	–0.141
	(1.158)	(0.632)	(2.148)	(0.287)	(0.746)	(–0.758)
Constant	0.982***	–0.117	0.679***	–0.087	–0.268	–0.807
	(4.056)	(–0.182)	(2.905)	(–0.193)	(–1.044)	(–0.956)
Observations	984	192	922	179	922	179
R-squared	0.527		0.501		0.507
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

Table IX.

Open in new tab

Regression analysis of investor participation

Dependent variable	Log(Total buyers)		Log(HF buyers)		Log(Non-HF buyers)
Model	OLS	2SLS	OLS	2SLS	OLS	2SLS
	(1)	(2)	(3)	(4)	(5)	(6)
Option	0.226**	1.670**	0.208**	1.638**	0.293***	1.384*
	(2.536)	(1.992)	(2.407)	(2.033)	(2.958)	(1.779)
Price distance		–0.030		–0.011		–0.039
		(–0.442)		(–0.153)		(–0.537)
Log(Assets)	0.321***	0.046	0.310***	–0.011	0.407***	0.171
	(10.399)	(0.374)	(10.020)	(–0.083)	(12.708)	(1.628)
Market to book	0.031*	–0.037*	0.039**	–0.047**	0.049***	0.001
	(1.767)	(–1.857)	(2.048)	(–1.995)	(3.126)	(0.066)
Volatility	–4.278**	0.785	–4.006**	–0.144	–5.911***	1.870
	(–2.408)	(0.321)	(–2.117)	(–0.058)	(–2.658)	(0.857)
Amihud $\times 10^{6}$	0.004	–0.005	–0.004	–0.007	0.012	0.008
	(0.434)	(–0.134)	(–0.529)	(–0.179)	(1.281)	(0.245)
Nasdaq listing	0.087	0.118	0.150**	0.056	0.061	–0.141
	(1.158)	(0.632)	(2.148)	(0.287)	(0.746)	(–0.758)
Constant	0.982***	–0.117	0.679***	–0.087	–0.268	–0.807
	(4.056)	(–0.182)	(2.905)	(–0.193)	(–1.044)	(–0.956)
Observations	984	192	922	179	922	179
R-squared	0.527		0.501		0.507
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

Dependent variable	Log(Total buyers)		Log(HF buyers)		Log(Non-HF buyers)
Model	OLS	2SLS	OLS	2SLS	OLS	2SLS
	(1)	(2)	(3)	(4)	(5)	(6)
Option	0.226**	1.670**	0.208**	1.638**	0.293***	1.384*
	(2.536)	(1.992)	(2.407)	(2.033)	(2.958)	(1.779)
Price distance		–0.030		–0.011		–0.039
		(–0.442)		(–0.153)		(–0.537)
Log(Assets)	0.321***	0.046	0.310***	–0.011	0.407***	0.171
	(10.399)	(0.374)	(10.020)	(–0.083)	(12.708)	(1.628)
Market to book	0.031*	–0.037*	0.039**	–0.047**	0.049***	0.001
	(1.767)	(–1.857)	(2.048)	(–1.995)	(3.126)	(0.066)
Volatility	–4.278**	0.785	–4.006**	–0.144	–5.911***	1.870
	(–2.408)	(0.321)	(–2.117)	(–0.058)	(–2.658)	(0.857)
Amihud $\times 10^{6}$	0.004	–0.005	–0.004	–0.007	0.012	0.008
	(0.434)	(–0.134)	(–0.529)	(–0.179)	(1.281)	(0.245)
Nasdaq listing	0.087	0.118	0.150**	0.056	0.061	–0.141
	(1.158)	(0.632)	(2.148)	(0.287)	(0.746)	(–0.758)
Constant	0.982***	–0.117	0.679***	–0.087	–0.268	–0.807
	(4.056)	(–0.182)	(2.905)	(–0.193)	(–1.044)	(–0.956)
Observations	984	192	922	179	922	179
R-squared	0.527		0.501		0.507
Year FE	Yes	Yes	Yes	Yes	Yes	Yes

Hedge funds started to become the principal buyers of convertibles in the early 2000s (e.g., Mitchell, Pedersen and Pulvino, 2007; Grundy and Verwijmeren, 2018). In contrast to other types of investors, hedge funds care less about the firms’ fundamentals or information environment because they mostly engage in convertible arbitrage strategies that hedge the equity risk. Since listed options could relax short-sale constraints (Danielsen and Sorescu, 2001, Grundy, Lim and Verwijmeren, 2012), listed options will make it easier for hedge fund managers to engage in convertible arbitrage strategies. In addition, as pointed out by Calamos (2003), a practitioner text, convertible arbitrageurs can utilize listed options to create hedge profiles that go beyond a simple strategy of being long in convertibles and short in underlying stock. Thus, the hedging opportunities provided by options are likely to be attractive to hedge fund managers. Even though offering discounts are lower for issuers with options, they are still substantial, with the average discount being 14% for the convertibles issued by firms with options.

In addition, convertible arbitrageurs might also care about information asymmetry, although to a lesser extent than long-only investors. Hedge fund managers are likely to pay attention to information asymmetry when they cannot perfectly hedge their positions, for example, when credit default swaps are not available to hedge their credit risk exposures. In addition, hedge fund managers may pay attention to information asymmetry if short-selling costs increase as information asymmetry increases.¹⁹

Following Brown et al. (2012), we identify hedge fund buyers among the institutional investors participating in convertible bond issues. As shown in Table II, on average, there are about thirty-one hedge funds involved when issuers have options outstanding, while there are about sixteen hedge funds when issuers do not have listed options. In Column (3) of Table IX, we estimate a multivariate regression of the log-transformed number of hedge fund buyers on the Option variable. The coefficient estimate of $Option$ is 0.158 and statistically significant at the 5% level, implying that more hedge funds participate in convertible offerings when issuers have options outstanding. The positive and significant 2SLS estimate of $Option$ in Column (4), employing our RDD analysis, is consistent with the OLS result, suggesting that the effect is not driven by the endogeneity of option listing. Because convertible arbitrageurs are less concerned about information asymmetry through their delta-hedged positions than long-only buyers, the strength of the relation between available options and hedge fund participation highlights the usefulness of options for hedging opportunities.

To provide some more insights into hedging opportunities, we examine whether option open interests and option trading volume increase around convertible issue dates. Figure 5 plots abnormal open interests and trading volume of options between 30 days prior to issuance and 30 days following issuance relative to the averages calculated over trading days –60 and –31 prior to issuance dates. Consistent with an increased use of options by convertible arbitrage hedge fund managers around convertible issuance dates, both open interest and trading volume in options increase significantly around issue dates.

Figure 5.

Option trading around convertible issue days. This graph plots the average log change of open interests (left) and of trading volume of options written on issuers’ stocks around convertible issue dates. For each issue, we use the averages of open interest and trading volume over the window [–60, –31] prior to the issue day as benchmarks, —OI and —TV, respectively. We then calculate the daily log change as: log(OI_t) — log(—OI) and log(TV_t) — log(—TV), for each trading day t in the window [–30, +30] around the issue day.

Open in new tab Download slide

Most non-hedge funds, who are long-only convertible investors, would only benefit from the information role of options, as they are unlikely to use options to hedge their positions. Table II shows that about thirty-two non-hedge fund investors are involved when issuers have options outstanding, relative to about thirteen when issuers do not have listed options. In Column (5) of Table IX, we estimate an OLS regression to study the relation between option availability on non-hedge fund participation when controlling for firm characteristics. We find an increase in non-hedge fund investor participation when convertible issuers have options outstanding, and this relation is statistically significant at the 1% level. The 2SLS estimate in Column (6) confirms that option availability attracts significantly more non-hedge fund buyers. Still, the t-statistic for the effect in Column (6) is 1.779, indicating that we can only reject the null hypothesis of no relation with a 92% probability when focusing on the RDD setup. Note that the sample size in Column 6 is substantially smaller than in Column 5, with 179 versus 922 observations.

Overall, we find that the availability of listed options significantly increases the number of convertible bond buyers. The finding that options help convertible bond issuers attract more hedge funds and also seems to attract other long-only investors suggests that the options market contributes to the capital markets not only by providing improved hedging opportunities but also by reducing the information asymmetry between convertible issuers and investors, which corroborates our earlier results on the importance of the information environment. Since hedge fund managers engaging in convertible arbitrages do not bet on the direction of underlying stock while long-only investors are likely to do so, the results are consistent with the findings in Hu (2018) that option listing increases both informed and uninformed trading of underlying stocks.

5. Conclusion

In this paper, we use convertible bond issues as a setting to investigate whether stock options affect the value of other securities and if so, how they do so. We believe the convertible market is well suited to provide new information on this issue due to the resemblance between convertible securities and stock options. In addition, the popular use of convertible arbitrage strategies and our ability to observe the identity of convertible investors allow us to investigate whether the channel through which options have an effect is limited to an information channel.

We show that the availability of listed options significantly reduces convertible bond underpricing, especially for issuers with a relatively poor information environment. Consequently, firms with options are able to raise capital at lower costs through convertible bond offerings. We document that listed options attract more capital suppliers to convertible bond offerings. Our findings suggest that the positive role of options is both information- and hedging-related. Overall, we provide evidence that individual stock options significantly affect corporate financing decisions since they help issuers of underlying stocks attract more capital providers by reducing adverse selection problems arising from information asymmetry and providing expanded hedging opportunities.

Footnotes

In the frictionless world of Black and Scholes (1973), options are viewed as redundant because they can be perfectly replicated by the underlying assets and risk-free bonds. In the real world, with all types of frictions, options potentially make markets more complete and reduce information asymmetries. As the theoretical literature suggests, options incentivize and facilitate informed trading by expanding hedging opportunities (Ross, 1976; Hakansson, 1982), providing lower transaction costs and higher leverage (Black, 1975; Back, 1993; Cao, 1999), and relaxing short-sale constraints (Diamond and Verrecchia, 1987; Figlewski and Webb, 1993).

To be more precise, a convertible bond is a straight bond plus warrants, as the number of shares outstanding increases when the conversion option is exercised.

Zeitsch et al. (2020) show how to link the convertible bond implied volatility to the implied volatility surface of listed options.

Hu (2018) shows that option listing significantly increases both informed and unformed trading, and the stock market reactions to earnings surprises decrease after option listing. Cao et al. (2019) show that option trading increases the information content of prices using several proxies for the stock price informativeness such as stock return synchronicity to market return and idiosyncratic volatility.

Our conclusions in this paper are robust to excluding unrated bonds from our analysis. To further check the robustness of our results, we use an alternative way to measure the credit spread known as the “option-adjusted spread” (OAS). The OAS adjusts the credit spread for embedded options, such as call provisions. We obtain OAS data from the St. Louis FRED database. The average offering discount drops from 15.4% to 12.8% when OAS is used as the credit spread measure, but our overall conclusions remain similar.

To obtain some insights into the importance of sample selection bias, we use a two-step Heckman model. In the first step, we model the probability of observing the offering discount and estimate the inverse Mills ratio. The availability of Mergent data is used as an instrumental variable, which is likely to be related to whether there is data available to compute offering discounts, satisfying the relevance condition, while (post-issue) Mergent data are not likely to directly affect offering discounts, satisfying the exclusion condition. We then include the estimate of the inverse Mills ratio as an independent variable in the second stage regression of the offering discount on the availability of options. We find that our conclusions are robust to this control for sample selection bias.

Our results are robust to using different outlier treatments, such as removing observations in the 10% tails.

The delta estimates remain similar if we use changes of 1% or 10%. To further check the robustness of our results, we alternatively estimate deltas using the Black–Scholes formula, as in De Jong, Dutordoir, and Verwijmeren (2011). We find that the correlation between the numerical delta estimates and the Black–Scholes delta estimates is 74.8% and our results are qualitatively similar when we use these alternative delta estimates.

Our results are materially unaffected when we use alternative proxies for firm size such the natural logarithm of total sales or the natural logarithm of market capitalization. All robustness tests in this paper are available upon request. We report a range of additional robustness tests in Section 3.4.

Note that offering discounts are not limited to convertible bond issues. For example, Brophy, Ouimet and Sialm (2009) report an average offering discount of privately placed common stock of 11%.

Underpricing in the secondary market is estimated at the end of each month after issuance until the end of 2018. The minimum post-issuance period examined in this analysis among issues with available transaction prices is 48 months, that is, for those offerings made in December 2014, which corresponds to the 4-year convergence period in Grundy, Verwijmeren, and Yang (2021).

The absolute moneyness variable captures whether conversion options are well in- or out-of-the-money. We define all variables in this study in Table AI. The average time to first call is 4.97 years from issuers with listed options and 4.72 for those without listed options. This difference is not statistically significant (t-statistics of –0.741).

Bond characteristics are not relevant for the first-stage regression regarding option-listing decisions and are thus not included. In the second stage, we include all exogenous variables used in the first-stage regression, in addition to the instrumented variable. We have confirmed that our results are robust to including bond characteristics in the vector of control variables.

We find no statistical evidence of systematic manipulation of the forcing variable (test-statistic is 0.624 with a p- $p$ value of 0.533) when following McCrary (2008) to test whether there is a discontinuity in the density of the assignment variable.

Another reason is that, in our IV setting, both the treatment variable and instrument are binary, so the IV estimate is essentially the ratio between the reduced-form effect on the IV and the first-stage effect on the IV. If we had a perfectly complying group with the first-stage eligibility effect being close to 1 instead of 0.295 as in Column (1) of Table VI (i.e., all those firms with stock prices above the minimum required prices for option listing indeed have listed options), we get the reduced-form effect of a 14.7–18.3% reduction in discount.

We have confirmed that our OLS results are also robust to all robustness tests in this sub-section. In addition, we have examined special convertible bond features that theoretical models do not fully capture, related to soft callability features, floating coupons, and restrictive covenants (more specifically, negative pledge, cross default, cross acceleration, and indebtedness restrictions). We find that our results are robust to excluding convertible bonds with these special features.

The 5% cutoff represents the first quartile value of the average of the various discount estimates in Table VIII, whereas the 15% cutoff represents the median value.

We have confirmed that our conclusions are robust to also including (endogenous) bond issue characteristics.

D’Avolio (2002) shows that the cost of borrowing stock is increasing when opinions are dispersed.

* We would like to thank Marcin Kacperczyk (the editor), an anonymous referee, Darwin Choi, Tim Loughran, Neil Pearson, Xiao Xiao, Antti Yang, and seminar participants at KAIST, Seoul National University, Yonsei University, and University of Sydney for useful comments.

References

Altinkilic

Hansen

R. S.

(

2003

Discounting and underpricing in seasoned equity offers

Journal of Financial Economics

285

–

323

Variables	Definitions
Dependent variables
Offering discounts (in%)	Offering discounts in percentage are measured as the difference between theoretical and offering prices over theoretical prices, and are winsorized at the 1st and 99th percentiles. Theoretical prices are calculated using the method of TF, a binomial approach. Inputs used in the model are described in detail in Section 3.2.
HF buyers	Number of hedge funds buyers who are identified as hedge funds from the full list of hedge fund managers in Bloomberg, Lipper TASS, and Hedge Fund Research (HFR) databases. Log(HF buyers) is the natural log of the number of hedge funds buyers.
Total buyers	Number of buyers identified from registration statements available at the SEC’s Edgar database as in Brown et al. (2012). Log(Total buyers) is the natural log of the number of all buyers.
Main explanatory variables
Option	A dummy variable to indicate that stock options are available for trading at the time of convertible issuance. It is set to be one if the option trading data are available for the issuer in OptionMetrics in the month of the convertible issue date.
Bond characteristics
Delta	A measure of the sensitivity of the convertible bond price to a stock price movement. The exact definition is described in Section 2.2.
Proceeds/MV	Amount of capital raised at a convertible bond offering over the market capitalization (csho × prcc_f) at the end of fiscal year prior to the announcement of a convertible bond offering.
Maturity	Maturity of a bond in number of years.
Combined offering	A dummy variable to indicate combined offerings with a concurrent buyback announcement (or an intention to use the proceeds to buy back shares) within the window of 11 days around the convertible bond announcement date
Rule 144A	A dummy variable to indicate a Rule 144A private placement offering.
Rated	A dummy variable to indicate offerings with an available S&P credit rating
Investment grade	A dummy variable to indicate offerings with an investment grade credit rating (an S&P credit rating of BBB or higher)
Firm characteristics
Log(Assets)	Natural log of total assets of a firm (at) at the fiscal year end prior to the announcement of convertible bond issuance.
Market-to-book	Market-to-book asset ratio that is defined as total assets (at) plus market value of equity (csho × prcc_f) minus book value of equity (ceq) divided by total assets (at) as of the fiscal year end prior to a convertible announcement date.
Volatility	Volatility is defined as the daily stock return volatility calculated using stock returns over the window [–240, –40] relative to a convertible bond offering announcement date.
Amihud × 10⁶	An illiquidity measure based on Amihud (2002) times 1,000,000. It is calculated as the average absolute value of daily stock returns over trading volume in millions during the window [–120, –20] relative to a convertible bond offering announcement date.
Nasdaq listing	A dummy variable to indicate issuers listed in the Nasdaq.
Analyst coverage	Number of financial analysts covering the offering firm, which is collected from the IBES database.
Post-issuance bond characteristics
Delta	Convertible delta numerically computed with a 5% change of stock price, as explained in Section 2.2, at the end of a month.
Absolute moneyness	Absolute value of moneyness measured by the natural logarithm of the ratio of the month-end share price of the underlying stock to the conversion price (as in Van Marle and Verwijmeren, 2017).
Log(TTC)	Natural logarithm of the time to first call date (in years) from a given month.
Log(TTM)	Natural logarithm of the time until maturity (in years) from a given month.
Stock volatility (monthly)	Standard deviation of historical daily stock returns measured over a moving window of [–240; –40] prior to a month-end.
Amihud × 10⁶ (monthly)	Amihud (2002) illiquidity measure calculated using a moving window [–120, –20] prior to a month-end.
Bond trading volume	Natural logarithm of the total dollar trading volume of the convertible during a given month.
Fuzzy RDD-related variables
Price distance	Difference between the average stock price during 3 months from the beginning of the calendar year of an offering announcement and the minimum stock price of $3.0 ($7.5 until 2004) required by the SEC for option listing as specified in Equation (1).
Eligible	A dummy variable to indicate that “Price distance” of the offering firm is positive and therefore, the offering firm is likely to be eligible for listing of options written on its stock.

Article Contents

How Do Options Add Value? Evidence from the Convertible Bond Market*

Abstract

1. Introduction

2. Data and Summary Statistics

2.1 Data

2.2 Main Variables

2.3 Summary Statistics

2.4 Univariate Tests

3. Options and Convertible Bond Underpricing

3.1 Baseline Results

3.1.a. Offering discount

3.1.b. Post-issuance underpricing

3.2 Fuzzy RDD Analyses

3.3 Placebo and Robustness Tests

3.4 Reliability of Our Offering Discount Estimates

4. Options and Capital Supply for Convertible Bonds

5. Conclusion

Footnotes

References

Appendix

Citations

Views

Altmetric

Email alerts

Citing articles via

Latest

Most Read

Most Cited

This Feature Is Available To Subscribers Only

How Do Options Add Value? Evidence from the Convertible Bond Market^*