Why are Derivative Warrants More Expensive than Options? An Empirical Study Gang Li and Chu Zhang This version: May, 2009 Li, garyli@hkbu.edu.hk, Hong Kong Baptist University (HKBU), Kowloon Tong, Kowloon, Hong Kong and Zhang, czhang@ust.hk, Hong Kong University of Science and Technology (HKUST), Clear Water Bay, Kowloon, Hong Kong. We would like to thank Du Du, George Jiang, Nengjiu Ju, Ming Liu, Sophie Ni, Qian Sun, Yexiao Xu, seminar participants at 2008 China International Conference in Finance and Universities of Fudan, HKBU, HKUST, Macau, and Xiamen, and especially the anonymous referee for helpful comments on earlier versions of the paper. Chu Zhang acknowledges financial support from HKUST research project competition grant, RPC06/07.BM28. All remaining errors are ours.
Why are Derivative Warrants More Expensive than Options? An Empirical Study Abstract Derivative warrants typically have higher prices than do otherwise identical options. Using data from the Hong Kong market during 2002-2007, we show that the price difference reflects the liquidity premium of derivative warrants over options. Newly issued derivative warrants are much more liquid than are options with similar terms. As a result, long-term derivative warrants are preferred by traders who trade frequently. In spite of their higher prices, short-term returns on long-term derivative warrants are, in fact, higher than the hypothetical short-term returns on options. The differences in price and liquidity measures decline as the contracts get closer to maturity.
I. Introduction The call and put derivative warrants traded in many markets are like usual call and put options traded in the US and elsewhere, except that they can be issued (i.e., sold short) only by certain financial institutions approved by regulators. It has been observed that derivative warrants tend to be priced higher than otherwise identical options. This phenomenon, which is in violation of the law of one price, a central theme in financial economics, is the focus of the current paper. Our objective is to understand what causes derivative warrants and options with identical payoffs to have different prices and how they can coexist in the market. The main thrust of our investigation is the difference in liquidity between derivative warrants and options. The effect of liquidity on asset pricing has received considerable attention in the literature in recent years. In their seminal work, Amihud and Mendelson (1986) provide a theoretical argument that illiquidity caused by higher bid-ask spreads leads to price discounts and higher expected returns. There are many empirical studies on primitive assets confirming this theory. 1 On the liquidity effects on derivative assets, recent works include Jarrow and Potter (2008), Cetin, Jarrow, Protter, and Warachka (2006), Garleanu, Pedersen, and Poteshman (2007) and Deuskar, Gupta, and Subrahmanyam (2008). What makes derivatives, and especially options, special is that derivatives are in zero net supply. As buyers require price discounts on illiquid derivatives, sellers require price premiums. Whether illiquid derivatives carry a price discount or a price premium, as Deuskar at el. (2008) argue, depends on whether the marginal trader is a buyer or a seller, on the risk appetite of the marginal trader, on the extent to which the 1 Amihud and Mendelson (1989) and Brennan and Subrahmanyam (1996) find that illiquid stocks are traded at lower prices and have higher expected returns when other factors are controlled. Amihud and Mendelson (1991) find that the less liquid Treasury notes have lower prices and higher yields-to-maturity. Silber (1991) find that restricted stocks which are prohibited to trade in the open market are sold at an average discount of 33.75%. Chan, Hong, and Subrahmanyam (2006) find that a higher premium on American Depositary Receipts (ADR) is associated with higher ADR liquidity and lower home share liquidity. 1
derivative cannot be replicated by primitive assets, and on the nature of the illiquidity. As liquidity is a broad concept with many dimensions, it is possible that some notions of illiquidity cause price discounts on derivative assets while other notions of illiquidity lead to price premiums. Brenner, Eldor, and Hauser (2001) find that non-tradable options are priced at a mean discount of 18% to 21% of the exchange-traded synthetic options with the same payoff and conclude that this form of illiquidity has an negative effect on derivative prices. On the other hand, some derivatives are demanded by investors for hedging and speculation, but there are few sellers around. Because of stochastic volatility, jumps, discrete rebalancing, transaction costs, and illiquidity of the underlying assets, derivatives cannot be replicated without costs. The buying pressure combined with illiquidity in this situation leads to price premiums. Deusker et al. (2008) document substantial price premiums on over-the-counter interest rate caps and floors as dealers need to cover costs in hedging their short positions. A similar phenomenon documented by Bollen and Whaley (2004) is the overpricing of out-of-the-money S&P 500 index put options which are bought by investors for hedging potential losses in large stocks, with few sellers. We investigate the difference in prices of derivative warrants and options written on the Hang Seng Index (HSI) in the Hong Kong market. The Hong Kong derivative warrants market is the largest in the world in terms of trading volume. 2 There is an advantage of our study of the liquidity effects on derivative prices over the existing studies. Unlike most other studies that involve inferring the price of the otherwise identical liquid asset for a given illiquid asset, both derivative warrants and options are traded in the market and their prices are directly observed. In addition, a variety of liquidity measures for both types of derivatives can be easily constructed. We use a matched sample of derivative warrants and options so that other factors affecting derivative prices, but unrelated to liquidity, are controlled. The quantitative relationship between the price difference and 2 Derivative warrants are traded in Germany, Switzerland, Italy, Britain, Australia, Hong Kong, Singapore, Korea and some other countries under different names. Derivative warrant is the term used in Hong Kong. 2
liquidity differences can be more easily examined. Our empirical results show that by the standard measures of liquidity, including bid-ask spread, trading volume, turnover ratio, contract size and the Amihud (2002) measure, long-term derivative warrants are much more liquid than are options. These results suggest that the overpricing of derivative warrants is explained by the liquidity differences between derivative warrants and options. The fact that derivative warrants are traded at higher prices than are options indicates that the illiquidity in these dimensions entails price discounts, consistent with Brenner et al. s (2001) findings. The results presented in this paper are also consistent with Deuskar et al. s (2008) argument regarding liquidity and price premiums/discounts. Since this is not the focus of the paper, we defer further discussions to the last section. The results on turnover ratios imply that about a half of the derivative warrants have a holding period of less than two weeks, and about 20% have a holding period of less than one day. On the contrary, about 80% of options contracts have a holding period of longer than one month. Derivative warrants are more frequently traded than are options. We further investigate the returns on derivative warrants and options for different holding periods, using ask prices for initial buying and bid prices for later selling. It turns out that short-term holding period returns on derivative warrants are mostly higher than those on the corresponding options, although derivative warrants are bought at higher prices. As the holding period gets longer, the return difference between derivative warrants and the corresponding options becomes narrower. Eventually, the returns on options are higher than those of derivative warrants. These results are a manifestation of the clientele effect, proposed by Amihud and Mendelson (1986), that investors with different holding periods maximize the after-transaction-cost expected returns, which leads to a phenomenon, in equilibrium, that the assets with larger bid-ask spreads are held by investors with longer holding periods. Our work is also related to Chan and Pinder (2000) who study the derivative warrants market in Australia and find overpricing of derivative warrants relative to options. They 3
show that the trading volume of derivative warrants relative to that of options can explain some of the overpricing among other variables such as days-to-maturity, the presence of market makers in the options market, and the identity of warrants issuers. We account for the overpricing of derivative warrants with a full range of liquidity measures. We also go one step further to analyze the holding period returns on derivative warrants and options. Our results indicate more clearly that two assets with identical cash flows but with different prices can coexist in the market if the transaction costs are different and that the more liquid derivative warrants market caters to the short-term trading needs of investors. There are other studies of derivative warrants, especially on the Hong Kong market, but they are unrelated to warrant overpricing. 3 The remainder of this paper is organized as follows. Section 2 describes the Hong Kong derivative warrants and options markets. Section 3 describes the data. Section 4 provides evidence of overpricing of derivative warrants relative to options and presents the main results on overpricing and liquidity. Section 5 compares holding period returns on derivative warrants and options. Section 6 concludes the paper. II. The Derivative Warrants and Options Markets in Hong Kong Trading of derivative warrants and options in Hong Kong is conducted in the Hong Kong Exchange and Clearing Limited (HKEx), which is divided into the Securities Market and the Derivatives Market. The Securities Market is also known as the Stock Exchange. For historic reasons, stocks and derivative warrants are traded on the Stock Exchange. The Derivatives Market is further divided into the Futures Exchange and the Stock Options 3 Duan and Yan (1999) use a semi-parametric approach to pricing derivative warrants that substantially improves upon the Black-Scholes model. Several papers in the literature focus on the effect of the introduction of derivative warrants on the price and trading volume of the underlying securities, for example, Chan and Wei (2001), Chen and Wu (2001) and Draper, Mak, and Tang (2001). A recent paper by Chow, Li, and Liu (2007) examines the trading records of market makers in the Hong Kong derivative warrants market to understand their inventory management. 4
Exchange. Index futures and index options, among others, are traded on the Futures Exchange, while options on individual stocks are traded on the Stock Options Exchange. There are two types of warrants listed on the Hong Kong Stock Exchange: equity warrants and derivative warrants. Equity warrants, issued by the underlying company itself, entitle holders to purchase equity securities from the underlying company at a predetermined price. Derivative call warrants are similar to equity warrants except that they are issued by a third party, usually a financial institution. Derivative put warrants, also issued by a third party, entitle holders to sell equity securities of an underlying company at a pre-determined price to the issuer. In Hong Kong, trading of equity warrants started in 1977, while trading of derivative warrants started in 1989. In recent years, the bulk of the warrants traded on the Hong Kong Stock Exchange are derivative warrants. The underlying assets of the derivative warrants are mostly blue-chip stocks. Other underlying assets include stock indexes, baskets of stocks, and some commodities. All the derivative warrants are of European style. The issuers of derivative warrants are typically largeand medium-sized financial institutions. Several major European and Australian banks, such as Societe Generale, KBC, Deutsche Bank, BNP Paribas, and Macquarie Bank, are among the most active issuers. Each underlying asset may have multiple issuers who compete with each other to offer popular contract specifications, lower prices and better liquidity. Options trading in Hong Kong started in March 1993. Initially, only options written on the HSI were introduced. Trading of options on individual stocks started in September 1995. The index options are of European style and settled by cash, while the stock options are of American style with physical delivery of the underlying assets upon exercise. The contract specifications of the options are set by the exchange. A system of market makers has been implemented whereby a handful of market makers is involved. At the end of 2007, there were 18 market makers in the Futures Exchange and about 30 market makers in the Stock Options Exchange. The market makers are required to provide liquidity to 5
the trading system. The requirements are not stringent, however. For options expiring in the nearest three months with near-the-money strikes, market makers are required to provide continuous quotes, while for all others, they only need to respond to requests for quotes. Panels A and B of Figure 1 plot the total trading volume of derivative warrants and options in terms of billions of Hong Kong dollars (HKD) over the period 2002-2007. The derivative warrants have gained much popularity over time with their trading volume increasing from about 7 billion HKD per month in 2002 to 397 billion HKD per month in 2007. By the end of 2006, the Hong Kong derivative warrants market had become the largest derivative warrants market in the world in terms of trading volume and it accounted for one-third of the total trading volume on the Hong Kong Stock Exchange. The trading volume of options in Hong Kong grew at a moderate pace from about 1.04 billion HKD per month in 2002 to 14.2 billion HKD per month in 2007, which pales in comparison to the trading volume of derivative warrants. Figure 1 here The rapid growth of the derivative warrants market in Hong Kong owes much to certain regulatory changes made in late 2001. Before these changes, issuers were required to place at least 85% of a derivative warrants issue with more than 100 investors on the issue date (or more than 50 investors if the size of the warrant issue was small). This exerted substantial pressure on the issuers. The 2001 rule repealed that requirement so that issuers could sell an entire issue gradually over time. The 2001 rule also required that each issuer appoint a liquidity provider to input bid and ask prices in the trading system, either continuously or on request. The new rules improved the liquidity of the derivative warrants market substantially. Another factor that contributes to the relative liquidity of derivative warrants over options is their minimum trading size. Set by the exchange, a round lot of options 6
on stocks is the same as that of underlying stocks. However, a round lot of derivative warrants, determined by the issuer, is typically only one-tenth of that of underlying stocks. This facilitates speculative trading by many small investors. There is much anecdotal evidence that options and derivative warrants have different clienteles. In a survey conducted by the HKEx, 50.7% of respondents revealed that they used the HSI options for pure trading, 37.4% for hedging, and 12% for arbitrage. Among the investors, only 24% were local and overseas retail investors and the remaining were either market makers, proprietary traders, or local and overseas institutional investors. In the derivative warrants market, most traders are individual investors and they tend to hold warrants for only short periods. A survey conducted in 2006 by the Hong Kong Securities and Futures Commission revealed that 86.8% of the respondents traded derivative warrants for short-term gains and only 0.4% of the respondents used derivative warrants for long-term investments. It should be noted that issuers of derivative warrants in Hong Kong are not required to hold the underlying assets, thus it is possible that derivative warrants may not be covered. On the other hand, options in Hong Kong are settled by the Hong Kong Futures Exchange Clearing Corporation and a margin is required for short positions. As a result, derivative warrants have higher credit risk than do options, therefore, other things being equal, derivative warrants should be traded at lower prices than options are traded. We show in Section 4.A, however, that derivative warrants tend to have higher prices than do options. III. Data Description A. Price Data We focus on derivative warrants and options written on the HSI in this paper. A comparison between derivative warrants and options on the index is clean as they are both 7
of European style and cash-settled. The HSI is the benchmark index in the Hong Kong stock market and the derivatives written on it are the most liquid ones. For the rest of the paper, we will refer to derivative warrants on the HSI as just warrants because the underlying is an index and there is no confusion. The data on warrants and options on the HSI are obtained from the HKEx. The warrants data include daily closing bid and ask prices, trading share volume, dollar volume, and other contract specifications such as maturity and strike price. The HKEx requires liquidity providers to disseminate the number of shares bought or sold, the average buying and selling prices, and the amount of warrants outstanding on a daily basis. Such information is available from the HKEx website. The options data include daily closing bid and ask prices, trading volume, open interest, maturity and strike price of the options. The HSI level is from Datastream. We use two samples of warrants and options matched with the same maturity and strike price. One sample, named TQ, consists of daily closing quotes with positive trading volume for warrants. In about one-third of this sample, the daily trading volume of options is zero. The other sample, named TT, consists of daily closing quotes with positive trading volumes for both warrants and options. Unless otherwise stated, all the results below are based on the TT sample. The sample period is from July 15, 2002 to December 31, 2007. Before this starting date, data on options are not available. There are many missing closing bid and ask quotes for the option data provided by the HKEx. We use the intraday option bid and ask quotes as supplements. 4 Panels C and D of Figure 1 plot the trading volume of warrants and options on the HSI during the sample period. The total trading volume of the warrants on the HSI increased from about 2.65 billion HKD per month in 2002 to about 81 billion HKD per month in 2007. However, the trading volume of the options on the HSI experienced only a moderate increase from about 0.75 billion HKD per month in 2002 to about 10 billion HKD per month in 2007. As we saw earlier for the whole market, the growth of the 4 We select the quote that is the closest to 4:00 pm, no earlier than 3:45 pm, and no later than 4:15 pm, because the warrants market is closed at 4:00 pm. 8
warrants on the HSI has been much faster than that of the options on the HSI. We denote the value of the HSI on business day t as H(t) and the strike price of a warrant or an option as K. Let p b w(t, κ, m) and p a w(t, κ, m) be the closing bid and ask prices, respectively, of a warrant (either a call or a put) on day t with moneyness, κ = 1 K/H(t) for a call and κ = K/H(t) 1 for a put, and maturity, m, measured by the number of days. Similarly, let p b o(t, κ, m) and p a o(t, κ, m) be the bid and ask closing prices, respectively, of the option (either a call or a put) on day t with moneyness, κ, and maturity, m. Warrants with the same (κ, m) issued by different issuers are treated as separate observations, but matched with the same option. For simplicity, the dependence of κ on t is suppressed unless it is necessary. We divide the entire matched sample into several groups according to moneyness and maturity. Moneyness is divided into the groups of κ 0.03 (out-of-the-money, OTM), 0.03 < κ 0.03 (at-the-money, ATM) and κ > 0.03 (in-the-money, ITM). Maturity is divided into the groups of short term (ST) with m 60 days, medium term (MT) with 60 < m 120 days and long term (LT) with m > 120 days. If there are less than eight pairs of matched warrants and options within a group in a week, we regard the week as a missing week for that group. Table 1 shows the weekly average number of matched pairs of warrants and options and the non-missing weeks in each of moneyness and maturity groups. There are about 235 matched pairs of warrants and options each week on average and the majority of the groups have more than 160 non-missing weeks in the sample. The average of bid-ask average price of warrants, P w = 100(p a w + p b w)/2h, and of options, P o = 100(p a o + p b o)/2h, of various moneyness-maturity groups are reported in Table 1. The reason for normalizing by H is to make the price data comparable across time, as the values of the HSI and the values of derivatives written on it contain an upward trend. Defined this way, the prices of warrants or options are expressed in terms of the percentage of the HSI level. In all the groups, the prices of warrants are higher than those of options, and the overpricing is the highest for the LT groups. 9
Table 1 here B. Liquidity Variables We employ a number of variables to measure the liquidity of warrants and options contracts. We focus on liquidity, rather than liquidity risk, in this paper. The former refers to the tradability of a security without adverse market impact, while the latter refers to the unpredictable changes in liquidity over time as well as their relationship with marketwide liquidity factors. The liquidity measures we examine include the bid-ask spread, the trading volume, the Amihud illiquidity measure, the turnover ratio, the contract size and the percentage of trading by liquidity providers. Among the liquidity measures, the bid-ask spread is based on quotes, while the other measures are based on transactions. Each variable measures one aspect of liquidity. The bid-ask spread is widely used in the literature. Table 1 reports the average bidask spread for warrants, S w = 100(p a w p b w)/h, and for options, S o = 100(p a o p b o)/h, in each moneyness-maturity group. Similar to the price measure, the bid-ask spread is expressed in the percentage of the HSI level. The bid-ask spreads of warrants are generally decreasing with maturity. However, they generally increase with maturity for options. The second measure we consider is trading volume. We report in Table 1 the group average of V w /1000H and V o /1000H, where V w and V o are the daily dollar trading volumes for warrant and option contracts, respectively. Similarly, we normalize the dollar volume by the HSI level to adjust for the upward trend. Warrants are much more actively traded than are options in most of the groups. The trading volume is distributed differently across maturity groups for the warrants and the options. Most of trading volume of the warrants is concentrated in the MT and LT groups. Options, however, have the largest volume in the ATM-ST group. A popular measure of illiquidity is the Amihud (2002) measure, which has been used 10
in numerous studies. The measure is based on Kyle s (1985) λ, the response of prices to order flow. The Amihud illiquidity measure, A(t, κ t, m), is defined as (1) A(t, κ t, m) = 1 11 5 i= 5 R(t + i, κ t+i, m i) V (t + i, κ t+i, m i)/1000, where R(t, κ t, m) is the daily absolute percentage return on a contract with (κ t, m) on day t. The Amihud measure is defined only for observations with positive trading volumes. We winsorize the measure at 2 for both the warrants and options, which roughly corresponds to the 99th percentile of the distribution. Table 1 shows that the Amihud measure is in general lower for the warrants than for the options for the MT and LT groups, but higher for the ST groups. Across moneyness, the OTM warrants and options have a higher Amihud measure than the ITM counterparts have. The trading in the warrants market is more active than that in the options market as evidenced by the difference in the daily dollar trading volumes between the two markets. Alternatively, we can measure the trading activity in each market by the turnover ratio, the frequency of a share changing hands, within a given period. The reciprocal of the turnover ratio is usually interpreted as the average holding period by investors. warrants and options, however, some modifications are needed because the outstanding amount changes over time, unlike the case of stocks and bonds. The turnover ratio of a warrant contract, T w (t, κ, m), is defined as (2) T w (t, κ, m) = [U w(t, κ t, m) N w (t, κ t, m)] 1[U L 2 w(t, κ t, m) N w (t, κ t, m)], O w (t 1, κ t 1, m + 1) where U w (t, κ t, m) is the trading volume of the warrant with (κ t, m) on day t expressed by the number of shares, U L w(t, κ t, m) is the number of the warrants with (κ t, m) on day t traded by the liquidity provider, O w (t, κ t, m) is the outstanding amount of the warrant with (κ t, m) on day t, and N w (t, κ t, m) = max[o w (t, κ t, m) O w (t 1, κ t 1, m + 1), 0] is the new issues of the warrants. The new issues are subtracted because the turnover ratio is defined for existing contracts. Half of the trading by liquidity providers is subtracted 11 For
to avoid double counting. 5 This definition assumes that liquidity traders always trade to provide liquidity to customers and never trade for themselves. Liquidity providers do trade for themselves sometimes, but we do not have the data to distinguish between the trades by liquidity providers for themselves and those for customers. The above definition may, therefore, underestimate the actual turnover ratio. The turnover ratio for an option contract, T o (t, κ, m), is defined as (3) T o (t, κ, m) = U o(t, κ t, m) N o (t, κ t, m), O o (t 1, κ t 1, m + 1) where U o (t, κ t, m) is the trading volume in contracts of the option with (κ t, m) on day t, O o (t, κ t, m) is the outstanding amount of the option with (κ t, m) on day t, and N o (t, κ t, m) = max[o o (t, κ t, m) O o (t 1, κ t 1, m + 1), 0] is the number of new contracts with the same terms. Like the definition of turnover ratio for warrants, N o (t, κ t, m) is subtracted to exclude the new option contracts in calculating the turnover ratio. However, we do not have the information on the amount of trading by option market makers. We assume that all the trades are between customers and, therefore, the turnover ratio of options defined this way may overestimate the actual turnover ratio. Table 1 reports the group average of T w and T o. The turnover ratio of the warrants tends to decrease as the maturity decreases, while the turnover ratio of the options tends to increase as the maturity decreases. In the MT and LT groups, the turnover ratio of the warrants is many times higher than that of the options. The turnover ratio in the ATM groups of warrants tends to be higher than that of the ITM and OTM groups. For the options, the ATM and OTM groups have a higher turnover ratio than the ITM groups. 6 These turnover ratios imply roughly that the average holding period of the warrants is between one day to one week and that the average holding period of the options is 5 A direct trade between two customers is counted just once. A trade between two customers through a liquidity provider is counted twice in U w and U L w. 6 Turnover ratios are highly positively skewed. For the warrants, the median of turnover ratios for the ATM are 0.23, 0.42 and 1.65 for the ST, MT and LT group, respectively, and they are higher than the ITM and OTM groups. For the options sample, the ATM-ST group has the highest median turnover ratio of 0.04. 12
about one month. Recall that the turnover ratio of the warrants may be underestimated, while the turnover ratio of the options may be overestimated. The actual situation may therefore be more extreme than what we describe here. The next measure we consider is the contract size, which is the number of underlying assets for one round lot of warrants, or one contract of options. The contract size matters because there is a large number of small, individual investors in terms of personal wealth who trade derivative warrants for increasing leverage. They can participate in the market only when they can afford to buy the minimum amount. A small contract size facilitates these trades. Let C w and C o denote the contract size for warrants and options, respectively. Table 1 shows that the average contract size for the warrants is about 2.4 HSI, while one contract of options always corresponds to 50 HSI. The minimum dollar amount for buying warrants is typically below five thousand HKD, which makes the warrants market accessible to small, individual investors. The options contracts, however, typically cost tens of thousands of HKD. In fact, 45.4 percent of warrants transactions have actual trading sizes below 50 HSI, and 24.8 percent are below 25 HSI. The trading volume of warrants with actual trading sizes below 25 HSI exceeds the entire trading volume of options on HSI in the TT sample. In this sense, the warrants market plays an indispensable role for small individual investors. The percentage of warrant trading by liquidity providers is calculated as the share volume traded by liquidity providers divided by the total share volume for a warrant contract on a day on which the total trading volume is positive, L w = Uw/U L w. The higher the value, the more actively liquidity providers supply liquidity to the market by quoting the best bid and ask prices. In general, liquidity providers supply liquidity quite actively. The average L w for the entire sample is 67.67%. Table 1 shows that, for the LT warrants, about 90% of trading involves liquidity providers. Across moneyness, liquidity providers trade the ITM warrants more actively than they trade the OTM warrants. There is no data on what percentage market makers of options trade. Therefore, L o is 13
unavailable. Panel A of Table 2 reports the correlations among price, P, maturity, m, moneyness, κ, and the liquidity measures. Since V and T are positively skewed, we define Ṽ = log(v/h+ 1) and T = log(t +1), which are also used in the regression analysis in the next section. We put a negative sign in front of S, C and A to define all the measures as liquidity measures. The upper triangle is for the TQ sample with a positive trading volume of warrants, but possibly no trading of options. A for the TQ sample is undefined. The lower triangle is for the TT sample with positive trading volumes of both warrants and options. The correlations reported are the time-series averages of the weekly cross-sectional correlations. The results show that the price is positively correlated with maturity and, especially, with moneyness. The liquidity measures are positively related to m and κ, except for S. All the liquidity measures are moderately, positively related. The highest pair is C and L. This is because all warrants have low values of C and high values of L, and options are the opposite. Comparatively, the correlations of A with other liquidity measures are lower. The magnitudes of the correlations across the two samples are comparable. Panel B of Table 2 reports the correlations among the price difference, D P, maturity, m, moneyness, κ, and the liquidity differences between the matched warrants and options pairs. The price difference is highly correlated with maturity, but not so with moneyness. The correlations between liquidity differences are mostly lower than the correlation between liquidity measures in Panel A. Table 2 here IV. Overpricing and Liquidity A. Overpricing In this subsection, we document the overpricing of warrants relative to options. Let Pw a = 100p a w/h, Pw b = 100p b w/h, Po a = 100p a o/h, and Po b = 100p b o/h. Panel A of Table 14
3 reports the proportion of the observations for which the warrant price is greater than the matched option price, where the price is set as the bid, ask, or bid-ask average. Most warrants are traded at higher prices than are their options counterparts. The proportions for P w > P o are above 80%, except for the ITM-ST group. The proportion is higher for the LT groups than for the ST groups and higher for the OTM groups than for the ITM groups. The patterns in the proportions of Pw a > Po a and of Pw b > Po b are similar. The case of Pw b > Po a is an arbitrage opportunity for those who can sell the warrant, buy the matched option, and keep the position to maturity. The proportions of Pw b > Po a for the LT groups are almost as high as those for P w > P o. While individual investors cannot sell warrants short, why don t they sell what they already own and buy the matched options? In addition, why don t warrant issuers sell more overpriced warrants and hedge their positions with the matched options? These are the questions that make the overpricing phenomenon nontrivial. We will return to these questions in the following sections. For completeness, the proportions of Pw a > Po b are also reported. These proportions are virtually 100%, meaning that profitable opportunities for buying warrants, selling the matched options, and holding them to maturity are rare, although these actions are feasible for individual investors. Such contracts are mainly concentrated in the ITM-ST group. Table 3 here Panel B of Table 3 reports the time-series averages of the cross-sectional average price differences for each moneyness-maturity group, with their t-ratios. The average price differences, P w P o, Pw a Po a, and Pw b Po b, are all significantly greater than zero, except for the ITM-ST group for Pw b Po b. Overall, call warrants are more overpriced than put warrants. When the transaction costs are taken into account, the averages of Pw b Po a are much smaller in magnitude and the significance is also reduced. In particular, overpricing tends to become insignificant, or even reversed, in the ST groups. For completeness and comparisons, the averages of Pw a Po b are also included. 15
B. Liquidity Differences In this subsection, we present the evidence that the liquidity measures are also different between the warrants and the matched options. In Panel A of Table 4, we report the proportion of which the liquidity (illiquidity) measure of a warrant is greater (less) than that of the matched option. For the LT and MT groups, the proportion is higher than 50% for all the liquidity measures. Only for the OTM-ST group is the proportion substantially less than 50% for S w < S o, V w > V o and T w > T o. The results also show that all warrants have smaller minimum contract sizes than do options. Table 4 here Panel B of Table 4 reports the time-series average of the cross-sectional average liquidity measure differences for each moneyness-maturity group, with their t-ratios. Negative signs are put in front of the illiquidity measures. For (V w V o )/1000H, T w T o and (C w C o ), the differences in the liquidity measures are positive for all the moneynessmaturity groups, and the majority of them are significant. For the other two liquidity measures, (S w S o ) and (A w A o ), the differences for the MT and LT groups are mostly positive and significant, but some of the differences for the ST groups are negative. C. Analysis-of-Variance In standard options pricing theory, warrants/options prices normalized by the price of the underlying asset depend on their moneyness, maturity, the stochastic volatility of the underlying asset, and the other state variables that govern the evolution of the price of the underlying asset. The central theme of this paper is that warrants/options prices also depend on their liquidity. To establish the dependence of the warrants/options prices on liquidity, we need to separate the effects of other variables. As we see in Table 2, both prices and liquidity measures are correlated with the maturity. It is important to remove 16
the maturity effect to avoid finding potential spurious relationships. To deal with this, we pool the matched warrants and options pairs together and run the following regression, (4) P = b1 + b 2 κ + b 3 m + b 4 σ + P, where P is P w or P o and σ is the stochastic volatility of the HSI estimated with daily returns on the HSI from 1998 to 2007 using the so-called GJR-GARCH model of Glosten, Jagannathan, and Runkle (1993). The regression is run for each moneyness-maturity group using pooled cross-section and time-series data. While the true relationship of warrants/options prices with moneyness, maturity, and stochastic volatility may be nonlinear, estimating a linear relationship within a moneyness-maturity group reduces the error of linear approximation. 7 The residual, P, mainly captures omitted liquidity effects, the approximation error from the linear term of the stochastic volatility, and other omitted state variables. We investigate the relationship between P and liquidity measures in the following subsections. In this subsection, we further characterize the overpricing of warrants relative to options by conducting an analysis-of-variance for the normalized price unexplained by moneyness, maturity and stochastic volatility. Panel A of Table 5 reports the mean of P for the warrants and the options separately. The means of price residuals for warrants are all positive, while the means of price residuals for options are all negative. The differences in means between the warrants and options reflect the between-type variation in the price residuals, P. Panel A also shows that the overpricing of warrants is the largest for the LT groups. Panel B of Table 5 reports the standard deviation of P for the warrants and the options separately and combined. The total variation of the combined sample is larger than that of separate warrants sample or separate options sample for most of the groups. The standard deviations of Pw and of P o indicate the within-type variation of P. For some LT groups, the variation of the combined sample is about two times as large as that of the separate samples. The last 7 As a robustness check, we also regress P on κ, m and σ nonparametrically across all the moneynessmaturity groups to obtain the price residuals P. The results are quantitatively similar. 17
part of Panel B in Table 5 reports the standard deviation of the price residual difference P w P o, which is the same as the price difference, P w P o, because warrants and options are matched pairs. The variation in the price residual difference is considerably smaller than that in the combined price residual sample, but it is non-trivial. In the next two subsections, we ask why warrants are more expensive than options and why some warrants are more overpriced than other warrants. Table 5 here D. Why are Warrants More Expensive than Options? In this subsection, we use regression analysis to quantify the overpricing of warrants relative to the options as contributed by the liquidity difference between the warrants and the options. We run regressions of the price residuals obtained in the last subsection on various measures of liquidity, (5) P = β1 + β 2 ( S) + β 3 ( C) + β 4 Ṽ + β 5 T + β6 L + β 7 ( A) + ε. We run the regressions with each liquidity measure separately, with all the liquidity measures together except for L, and with all the liquidity measures together where L for an option is arguably set to zero. We run the regressions for the TQ sample, in which A is undefined when there is no trading volume for options, and for the TT sample. The cross-sectional regressions are run for each moneyness-maturity group in each week. The slope coefficients are averaged across groups first and then averaged across weeks. The results are reported in Table 6. For each regression, the first row shows the coefficient estimates. The second row shows the coefficient estimates multiplied by the standard deviation of the corresponding independent variable, so that the number measures the change in the dependant variable for one standard deviation change in the independent variables. The third row reports the t-statistics of the coefficient adjusted for 10-week lags in the autocorrelations using the Newey and West (1987) procedure. In Panel A 18