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Convertible Bonds Valuation based on Multiple Options A Study from Chinese Market Zhaohui LIANG☆ Cheng LI Liang ZHANG College of Economics Tianjin Polytechnic University, China Abstract Convertible bonds (CBs) are corporate bonds that contain multiple exotic options. A component model and least-squares Monte Carlo approach were used to precisely price convertible bonds and to solve the pricing problem of exotic options consisting of American and path-dependent provisions. Furthermore, using the simulation model proposed, we present an empirical pricing study of the Chinese market. Our results do not confirm the evidence of previous studies that the market prices of convertible bonds are, on average, lower than the prices generated by a theoretical model. In addition, the results demonstrate that the provisions of the convert price revision and the put condition have a significant effect on a CB's theoretical value. These specified provisions reflect an issuer's preference for equity-like or debt-like bonds. JEL Classification: C22; G11; G14; G32 Keywords: multiple options; component model; convertible bonds; pricing 1. Introduction A convertible bond (CB) is a type of hybrid security combining the characteristics of bonds, stocks, and options. For demanders of capital, the advantages of issuing CBs include lower issuance costs and delayed equity dilution. From investors’ perspectives, a CB offers potential profits by conversion of the bond into a The work was done with financial support from the National Natural Foundation of China (70971098 71371136). Corresponding author at: Faculty of Economics, Tianjin Polytechnic University, No.399, Binshuixi Road Xiqing District, Tianjin, China. Tel.: +86 83956338. E-mail address: [email protected]. ☆ predetermined number of stocks when stock prices increase and could lock investors’ profit losses by put and redemption provisions. A CB can be seen as a derivative security, and it may contain multiple complex options, including path-dependent, exotic, and American options. Because these options interact with each other, the value of the CB is not just the sum of these individual options’ values. Thus, evaluating a CB’s price is an important topic for issuers and investors. Theoretical approaches to a CB’s valuation based on options are proposed by Ingersoll (1977) and Brennan and Schwartz (1977). In their models, “firm value” is the only underlying variable of options. The latter research develops a CB valuation model from two aspects. The first aspect considers more uncertainties in the real transaction process and expands the one-factor model to N-factor models by adding additional variables. For example, Brennan and Schwartz (1980) incorporate interest rates in their valuation model, but they find no significant effect on the CB’s value. Davis and Lischka (2002) further propose a three-factor model and involve a new variation of default risk, which was examined in combination with the drift rate of stock price. The other aspect focuses on analysis accuracy under a one-factor model. For example, noting that the variable “firm value” is not easily correctly measured, McConnell and Schwartz (1986) indicate that stock price is a good substitution for firm value. In their model, default risk is also considered a constant credit spread, and this approach is more operable and practical than the previous models with firm value. To precisely estimate default risk, Bardhan et al. (1993) and Tsiveriotis and Fernandes (1998) develop a component model by splitting the CB’s value into a stock component and straight bond component. Only the straight bond component contains default risk because it is related to cash payments. In contrast, the stock component has no relation with default risk because stocks can always be traded. Tsiveriotis and Fernandes (1998) simplify the conversion condition and ignore the call and put conditions, but this component model still contributes to calculating the CB’s value more reasonably while considering the default risk premium. Although the convertible bond is an important financial instrument, only a few studies focus on the pricing of convertible bonds, and most believe that a CB’s market price is lower than its intrinsic value. For example, both King (1986) and Carayannopoulos (1996) find that CBs’ values are undervalued to some extent in the U.S. using daily data from 1977 and monthly data from 1989, respectively. Ammann et al. (2003) also argue that the CBs’ market prices were underpriced in France during the period from 1999 to 2000. They find that out-of-money convertibles have larger price differences between market and theoretical prices than at- and in-the-money convertibles and that the difference is smaller with a shorter time to maturity. However, Buchan (1998) finds that for 35 Japanese convertible bonds, the observed market prices are slightly higher than the theoretical prices. Carayannopoulos and Kalimipalli (2003) show that market prices were only overpriced for out-of-money CBs during 2001-2002 in the United States. Using 32 convertible bonds and 69 months of daily market prices, Ammann et al.(2008) find that the US market prices of convertible bonds are not, on average, lower than the prices generated by a theoretical model. We choose to analyze the Chinese market. The first reason for this decision is that as hybrid instruments, convertible bonds are difficult to value because they depend on variables related to the underlying stock (price dynamics), the fixed income part (interest rates and credit risk), and the interaction between these components. Embedded options, such as conversion, call, and put provisions, often are restricted to certain periods, may vary over time, and are subject to additional path-dependent features of the state variables. Sometimes, individual convertible bonds contain innovative, pricing-relevant specifications that require flexible valuation models. However, most Chinese convertible bonds have unique conversion-price-reset clauses for the conversion price and a restricted put provision, and the trigger conditions of put and call are path dependent and American-style. These characteristics make CB pricing more complicated. The purpose of this study is to present a pricing model based on Monte Carlo simulation that can address these valuation challenges. We implement this model and use it to perform the first simulation-based pricing study of the Chinese convertible bond market that accounts for early-exercise and path-dependent features. Secondly, in China, only listed blue-chip companies have the right to issue convertible bonds, and most of these companies have a special controlling shareholder — the Chinese state. La Porta et al. (2002) and Lee (2009) indicate that in most countries, listed firms generally have controlling shareholders who have the ability and incentives to expropriate minority shareholders and creditors. Therefore, the Chinese market may be a good setting for verifying this viewpoint. Thirdly, in contrast to the CBs of other markets, the Chinese CB coupon is very low and the put provision is harsh, which means that the issuer's interest tends to be protected more than the investors. Therefore, we wonder if a Chinese CB's performance will be different other than that of other markets. Because most researchers believe that CBs’ market prices are lower than their fair values, one aim of this paper is to investigate how to estimate the theoretical values of CBs and whether CBs are underpriced in China. As for CB pricing, an approach with a closed-form solution is almost impossible because it fails to account for a number of real-world specifications. Dividends and coupon payments are often modeled continuously rather than discretely, early-exercise features are omitted, and path-dependent features are excluded. The second pricing approach that values convertible bonds uses a lattice-based method. The first theoretical model was introduced by Brennan and Schwartz (1977). Ammann et al. (2003) extend this approach by accounting for call features with various trigger conditions. In addition, Hung and Wang (2002) propose a tree-based model that accounts for both stochastic interest rates and default probabilities but loses the recombining property. Another tree-based model is presented by Carayannopoulos and Kalimipalli (2003), who use a trinomial tree and incorporates the reduced credit-risk model. The lattice-based methods provide an advantage when dealing with American options, but in the face of practical problems related to real convertible-bond specifications, these approaches present some general challenges: the computing time grows exponentially with the number of state variables, path dependencies cannot be incorporated easily, and the flexibility in modeling the underlying state variables is low. The third approach is Monte Carlo simulation and may overcome many of the drawbacks of the lattice-based approaches. Monte Carlo simulation is well suited for modeling discrete coupon and dividend payments, including more realistic dynamics of the underlying state variables, and taking into account path-dependent call features. Despite all the natural advantages of the Monte Carlo approach, pricing American-style options, such as those present in convertible bonds, within a Monte Carlo pricing framework is a demanding task. In recent years, a considerable number of important articles have addressed the problem of pricing American-style options using Monte Carlo simulation. For example, Barraquand and Martineau (1995) present methods based on backward induction that stratify the state space and find the optimal exercise decision for each subset of state variables. Garcia (2003) and Ammann et al. (2008) propose an enhanced Monte Carlo method (but they simplify the boundary conditions). A numerical comparison of different Monte Carlo approaches is provided by Fu et al. (2001). This paper contains theoretical and empirical contributions. One contribution is that we propose a least-squares Monte Carlo approach to solve the numerical pricing difficulty when facing a conflict between path-dependent and American-style options. Another contribution of this paper is that we undertake the first pricing study for the Chinese convertible bond market after the financial crisis. We find that the theoretical values for the analyzed convertible bonds are not always higher than the observed market prices, and this result is different from those of most previous studies. Another result is that the provisions of a convert price revision and a put condition might have an impact on a CB's theoretical value, and these specified provisions reflect the issuer’s preference for equity-like or debt-like bonds. The remainder of the paper is organized as follows. Section 2 describes the research models. Section 3 presents empirical results and further discussion. Section 4 concludes the paper. 2. Research Models 2.1. Pricing model Because the effect of the interest rate on a CB’s value is rather minimal (Brennan and Schwartz, 1980), this study gives weight to the value of a CB if the CB is regarded as a complex derivative consisting of multiple exotic and path-dependent options. We use a one-state variable model and incorporate state-dependent credit risk in the component model established by Tsiveriotis and Fernandes (1998). This model splits a CB’s value into two components, a stock and a straight bond, and both components are discounted separately with different discount rates. An important implication of the component model comes from the fact that the value of a CB has components with different default risks. For example, if the underlying equity is that of the issuer - as is usually the case - the equity upside has zero default risk because the issuer can always deliver its own stock. By contrast, coupon payments, principals, and any put provisions that allow the holder to put the CB back to the issuer for a pre-specified amount of cash depend on the issuer’s timely access to the required cash amounts and thus introduce credit risk. In fact, the future cash flows of the CB depends on the possibility of early termination of the CB due to conversion, call, or put, actions that, in turn, are contingent on the random behavior of the underlying stock. Generally, convertible bonds set different rights or obligations for the issuers and investors during various periods, as shown in Figure 1. Investors have the right to convert CBs to their underlying stocks. Investors have the right to execute put provisions if triggered. 0 tc tp In the early period after issuance, CBs' holders have no right to convert or put. tr Issuers have the rights to call or reset conversion prices if boundary conditions are triggered. T Issuers’ obligations to execute redemption provisions. Figure 1. Multiple options embedded in CBs Note: tc the starting time of convert, tp the starting time of call, tr the starting time of put, T the ending time of CB. Conditions such as the possibility of early conversion, callability and conversion-price-reset provisions by the issuer and putability by the holder introduce extra optionality that depends on both the equity and the fixed income parts. Typically, path dependencies arise from the fact that a call may only be allowed when the stock price exceeds a certain level for a pre-specified number of days in a pre-specified period, usually at least 20 out of the last 30 trading days. Monte Carlo simulation is just suitable for modeling the realistic dynamics of the underlying state variables and for taking into account such complex path-dependent features. Therefore, we view convertible bonds as complex derivatives consisting of multiple path-dependent and American exotic options and introduce an enhanced Monte Carlo method. 2.2. The Final and Boundary Conditions To account for the various complex characteristics of the bonds in our sample, such as embedded options and triggers, we extend the aforementioned approaches with several contract-specific boundary conditions. The terminal condition: Redemption is the provision that the issuer has the obligation to buy back all of the convertible bonds that have not been converted at the maturity date T . The provision is actually a European exotic option for the investors; they have the right to choose whether to convert or to wait for call back at time T . The terminal condition is given by the following: u(S ,T ) Max(nST , B) (1) and B, nS B; v( S , T ) 0, nS B, (2) where B and u are the par value and actual value of the CB, respectively, n denotes the conversion rate, v denotes the value of the “cash-only part of the CB”, and S denotes the price of the underlying stock. The conversion boundary condition: During the conversion period starting at tc and ending at time T , investors have the right to convert CBs to specified shares of the underlying stocks. It is a forward American call for investors. The conversion boundary condition is the following: ut nSt t [tc , T ] . (3) The value of the convertible bond cannot be less than the conversion value; otherwise, an arbitrage opportunity would occur. The call boundary condition: The call boundary condition states that during the callable period starting at t p , when the trigger condition is satisfied (say, the price of the underlying stock is lower than a certain level for a pre-specified number of days in a pre-specified period), the issuer has the right to call at a specified redemption price K . The provision means a path-dependent American call for the issuer. Due to the American-style and path-dependent characters of the instrument, it is necessary to check the following boundary condition: ut Max( K , nSt ) t [t p , T ] . (4) Otherwise, the issuer can arbitrage to make a profit by shorting CBs and calling them back at the same time. The put boundary condition: The put boundary condition states that during the putable period starting at tr , when the trigger condition is satisfied (say, the price of the underlying stock is lower than a certain level for a pre-specified number of days in a pre-specified period), the holders have the right to sell these CBs back to the issuer at a specified price P . Therefore, the provision is also a path-dependent American option. It is also necessary to check the boundary condition. The put boundary condition requires that ut P Here, if t [tr ,T ] . (5) ut P , then vt P If the convertible price was below the relevant put price, the investor could exercise the put option and realize a risk-free gain. Conversion-price-reset clause:The reset condition is usually satisfied when a put condition triggering, so the reset boundary condition is the same as that of the put and is also a path-dependent American option. When a put condition triggers, the issuer has the motive to reset the conversion price to a lower level to stop continued put. The issuer would not revise the conversion price downward for fear of damaging current shareholders' benefit unless the put triggered. Therefore, when triggered, the optimal strategy of the issuer is to reset the conversion price exactly equal to the CB's price (revising downward too much would damage shareholders' benefit). 2.3 Numerical solution A number of backward induction procedures have been proposed for valuing complex options, from Monte Carlo simulation to binomial and multinomial trees to finite difference procedures for solving partial differential equations. Among the three approaches, the first is well suited for path-dependent option, whereas the last two are well suited for American options. Despite the natural advantages of the Monte Carlo approach for taking into account path-dependent features and modeling the dynamics of the underlying state variable, pricing American-style options such as those present in convertible bonds within a Monte Carlo pricing framework is a demanding task. In this paper, we introduce a Least Square Monte Carlo procedure to solve the conflict between path-dependent and American-style features. Another advantage of this approach is that it easy to add or delete some specific provision because convertible bonds have different provision combinations and the individuation brings difficulties in valuation. A first step toward the Least-Square Monte-Carlo is to simulate the path of the underlying stock. Suppose it follows the stochastic process: dS Sdt Sdz (6 ) where dz is a Weiner process, is the expected return in a risk-neutral world, and is the volatility. We divide the life of the derivative into N short intervals of length t and approximate equation (6) as the following: S (t t ) S (t ) S (t )t S (t ) t (7) where is a random sample from a normal distribution with mean zero and standard deviation 1.0. This procedure enables the value of S at time t to be calculated from the initial value of S , the value at time 2t to be calculated from the value at time t , and so on. One simulation trial involves constructing a complete path for S using N random samples from a normal distribution. The second step begins from the above boundary conditions and solves simultaneously for the asset value and determining the optimal exercise policy. A drawback of Monte Carlo simulation is that it is computationally very time consuming and cannot easily handle situations in which there are early-exercise opportunities. Therefore, we combine the least square method with it to overcome the difficulty. When determining, we use the least square method to estimate the CB's value U ( St , t ) of every space, backwards, in turn, starting from the last node. Consider immediately executing the option when the boundary condition triggered. The investor can obtain a return of F ( St , t ) . If U (St , t ) F ( St , t ) . The investor executes an option: otherwise, he gives up. Then, at every space, we obtain the optimal exercise policy, the value of CB u , the value of the "cash-only part of the CB" v and the value of the non-cash part u v . The third step is to discount the expected payoffs of v and u v with different discounted rates considering credit risk, as discussed before. The average of the present value U from every simulation path is the fair theoretical value of the convertible bond. 3. Empirical Results 3.1 Data We have two aims in this empirical research. One aim is to investigate whether prices observed on Chinese secondary markets are below the theoretical fair value, as is believed in most other countries. The other aim is to explore how the provisions affect CBs' values. Because we view a CB as a complex derivative with multiple exotic options, it is easy to add or delete some specific provision/option to observe certain results. We consider 3 typical CBs with distinguishing features in the Chinese market, Shuangliang, Shihua, and Zhonghang. Shuangliang was issued in 2010 and terminated in 2012. With the underlying stock price continuously declining, the originally designed conversion price was too high. Although the revised provision was executed, it did not stop a large percentage of put by the investors. Shihuang and Zhonghang were issued in 2011 and 2010, respectively, and both are outstanding as of 2012. The circulation of the two CBs accounts for one third of the total volume in the Chinese CB market. To improve their capital adequacy ratio, the two firms' CBs set restriction on their put provision to hold cash flows. The put provision states that investors cannot execute put unless the capital uses change. Therefore, we select the three CBs as representative samples to research Chinese CBs’ valuations and provision designs. The main provisions of these CBs are listed below. Table 1. Main provisions of example CBs in China Issue time / Mature time Conversion provision Call provision Put provision Shuangliang May. 2010/ May. 2015 Shihua Mar. 2011/ Mar. 2017 Zhonghang Jun. 2010/ Jun. 2016 Conversion price: 21.11 RMB Starting time of conversion: Nov. 2010 (1) redemption at maturity The issuer has the obligation to call the remainder CBs at the sum of par value plus the last interest at maturity (2) conditional call After the starting time of conversion, if the underlying stock’s price is not lower than 130 percent of the conversion price at least 20 out of the last 30 trading days, the issuer has the right to call all or part of unconverted CBs at the price of 105% of the par value. Conversion price: 9.73 RMB Starting time of conversion: Aug. 2011 (1) redemption at maturity The issuer has the obligation to call the remainder CBs at the sum of 107 percent of par value at maturity (2) conditional call After the starting time of conversion, if the underlying stock’s price is not lower than 130 percent of the conversion price at least 15 out of the last 30 trading days, the issuer has the right to call all or part of unconverted CBs at the price of the sum of the par value plus interest. The holders have no right to put unless the firm changes the promised capital usage. Conversion price: 3.74 RMB Starting time of conversion: Dec. 2010 (1) redemption at maturity The issuer has the obligation to call the remainder CBs at the sum of 105 percent of par value at maturity (2) conditional call After the starting time of conversion, if the underlying stock’s price is not lower than 130 percent of the conversion price at least 15 out of the last 30 trading days, the issuer has the right to call all or part of unconverted CBs at the price of the sum of the par value plus interest. If the underlying stock's price is lower than 70 percent of the conversion price lasting for 30 The holders have no right to put unless the firm changes the promised capital usage. Conversion price-reset provision trading days after the starting time of conversion, the holders have the right to put the CBs back at the price of 103 percent of par value. During the conversion period, if the stock price is lower than 85% of the conversion price at least 20 out of the last 30 trading days, the issuer has the right to reset the conversion price. The revised price will be the higher value from the average price of the 20 trading days before the stockholders’ meeting and the price 1 day before the stockholders’ meeting. During the conversion period, if the stock price is lower than 80% of the conversion price at least 15 out of the last 30 trading days, the issuer has the right to reset the conversion price. The revised price will be the average value from the average price of the 20 trading days before the stockholders’ meeting and the price 1 day before the stockholders’ meeting. During the conversion period, if the stock price is lower than 80% of the conversion price at least 15 out of the last 30 trading days, the issuer has the right to reset the conversion price. The revised price will be the average value from the average price of the 20 trading days before the stockholders’ meeting and the price 1 day before the stockholders’ meeting. The parameter stock’s volatility and its return rate are estimated according to daily stock close prices from Jan. 2010 to Aug. 2012. The credit spread adopts the results from Zheng and Lin (2003), 0.90% for 3 years and 0.98% for 5 years. 3.2 Empirical results We find some interesting results from our empirical research: (1) The conversion-price-reset clause increases the CB's fair theoretical price. Shuangliang was issued in 2010. With the underlying stock's price continuously declining in a bear market, the originally designed conversion price was too high. Although the reset provision was executed, it did not stop a large percentage of put by the investors, and it was terminated in Jan. 2012. To investigate how the reset provision affects the CB's price, we make a comparative study between the CBs that contain a reset clause and those that do not (in Figure 2, the results do not consider a reset provision, and in Figure 3, the results add the provision). It is obvious that the theoretical price increases if the reset provision is added; otherwise, the market price would be overpriced. 135 the theory price the market price 130 125 120 price 115 110 105 100 95 90 0 50 100 150 200 250 300 days from the issue date Figure 2. Theoretical price of Shuangliang (not considering the reset provision) 160 the theory price the market price 150 price 140 130 120 110 100 0 50 100 150 200 250 300 days from the issue date Figure 3. Theoretical price of Shuangliang (considering the reset provision) (2) The theoretical values for the analyzed convertible bonds are not always higher than the observed market prices, and this result is different from most of the previous research. Although Shuangliang's market price is underpriced (see Figure 3) when the reset provision is considered, we find it is not a normal situation when valuing the others. Figure 4 and Figure 5 contrast the theoretical prices and observed market prices from Shihua and Zhonghang. It can be seen that their market prices are overvalued almost all the time. 115 the theory price the market price 110 105 price 100 95 90 85 80 0 50 100 150 200 250 300 days from the issue date Figure 4. Theoretical price and observed market price from Shihua 120 the theory price the market price 115 110 price 105 100 95 90 85 Figure 5. 0 50 100 150 200 days from the issue date 250 300 Theoretical price and observed market price from Zhonghang The results are not consistent with most prior research, which believes that CBs' market prices are, on average, lower than their fair values. Comparing prior research data and approaches, we use the new data from 2010 to 2012. Through the bear period, the Chinese stock market continuously falls, and the CBs are out-of-money for a long time. However, previous research did not use data from this period. This discrepancy could explain why we obtain different results. As for approach, some researchers simplified the complex path-dependent bound conditions, and some did not take credit risk into account. Our approach does not suffer from these limitations. By contrast, the result of overpriced values in the Chinese market could be related to its market characters. As an emerging market, the stock trading mechanisms in China are not completely liberalized. For example, the constraints of short sales lead to a result in which the biased stock prices cannot be corrected by arbitrage. Furthermore, the small percentage of institutional investors and insufficient investable objects could be possible reasons. Moreover, complex clauses make it difficult to judge the reasonable value for investors. (3) When imposing restriction on put clauses, with the decline of the underlying stocks’ prices, CBs' equity values disappear. For the purpose of improving the firms’ capital adequacy ratios, both Shihuang and Zhonghang set a restriction on their put provisions to hold cash flows. The provisions state that investors cannot put unless the capital uses change. However, when CBs are out-of-money for a long time, the equity values disappear and CBs merely present debt property, a straight line as shown in Figure 4 and Figure 5. The component model divides a CB's value into equity value and pure-debt value. We can conveniently observe how the CB's value varies with equity value and debt value by plotting a 3-D graph as in Figure 6. 120 theory price of CB 100 80 60 40 20 0 3 2.5 300 2 200 1.5 value of stock component Figure 6. 100 1 0 Time Equity Value and Debt Value for Zhonghang It can be seen from Figure 6 that with time passing and the underlying stock price falling, the CB becomes deeply out-of-money and its equity value is very small, but the put provision cannot be executed because of the limitation clause. Therefore, only the debt value remains. In real trading, Shuangliang was terminated in advance, being put by 90 percent investors on account of a too-high conversion price. By contrast, Zhonghang and Shihua impose restrictions on their put clauses except for their high conversion prices. Investors neither choose to invert nor execute the put provisions. The two CBs are equal to low-interest bonds. 3.3 Further discussion From the above discussion, convertible bonds contain both debt and equity characteristics. When firms design a convertible debt, they choose how debt-like or equity-like the offer will be by specifying security features such as the conversion ratio, maturity date and call period. Lewis et al. (2003) argue that the expected probability of converting the convertible debt to equity at maturity is a useful summary measure that simultaneously considers various security design features. Thus, the higher (lower) the expected probability that the convertible debt will be converted from debt to equity at maturity, the more equity-like (debt-like) the convertible's security design is. Issuing a debt-like CB can prevent dilution of a firm's profits and protect the current shareholders. This paper further discusses Chinese issuers’ intentions as measured by the expected probability of converting the convertibles at maturity. Following Lewis et al. (2003), we estimate the probability that a convertible debt issue is converted into equity at maturity based on observable characteristics at the time of issue. This probability is estimated using Black-Scholes option pricing as N (d2 ) , where N () is the cumulative probability under a standard normal distribution, and ln( S0 / X ) (r div 2 / 2)T , d2 T 0.5 (8) where S 0 is the underlying stock price on the CB's issue date, X is the conversion price, r is the interest rate yield on a 3-year government bond, div is the issuing firm’s dividend yield for the fiscal year before the convertible issue date, is the standard deviation of the common equity return estimated over 200 days prior to the issue date and T is the number of years until maturity of the convertible debt. One step further, we can estimate the expected time for conversion by employing the model of Lee et al. (2009). Suppose the underlying stock price follows a geometric Brownian motion with drift , the expected stock price E ( St ) at a future time t is expressed as the following: E ( St ) S0e t (9) A convertible becomes equity at the time of conversion and becomes in-the-money when the stock price exceeds the conversion price X . By setting Eq. (9) equal to the conversion price and estimating the future rate of stock price appreciation (i.e., the drift variable), we can solve for the expected time until profitable conversion. Thus, the shorter (longer) the expected conversion time, the more (less) equity-like the convertible is deemed at the time of issuance. Using the annualized growth estimates, we compute the expected time to conversion by setting the conversion price ( X ) equal to the expected future stock price in Eq. (9), X E ( St ) S0e t , (10) and then solve for the expected time for conversion t * : t* log( X ) log( S0 ) (11) We calculate the expected conversion ratio and time as Table 2. Table 2. expected conversion ratio and time for CBs Shuangliang Shihuang Zhonghang expected conversion ratio 0.3763 0.5954 0.7111 expected time of conversion (year) 3.36 2.60 2.58 Lee et al. find that firms in countries with stronger shareholder rights issue convertible debt with a higher expected probability of conversion. Our empirical results are consistent with this conclusion. Table 2 demonstrates that the CBs of Shuangliang, Shihua and Zhonghang, which have stronger state-owned shareholders, have higher expected conversion ratios and shorter expected times of conversion, which makes them equity-like bonds. However, this result is a contradiction with the "debt-like" characteristics shown in Figure 4 and Figure 5, which can be attributed to their too-high-conversion-price clause. 4. Conclusions A convertible bond (CB) is a hybrid security that contains both equity and debt characteristics. We view CBs as derivative products with multiple complex options. Despite the large sizes of international convertible bond markets, very little empirical research on the pricing of convertible bonds has been undertaken, and most believe that CBs' market prices are lower than their fair values. This paper investigates how to value the theoretical values of CBs and whether CBs are also underpriced in China. A least-squares Monte Carlo approach was introduced to solve the pricing problem of multiple exotic options consisting of American and path-dependent provisions. We applied the approach of a component model to value credit risk. Furthermore, we discuss an issuer's incentive through expected conversion probabilities and the conversion time. 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