考慮信用風險及流動性風險之可轉債評價 - 政大學術集成

全文

(1)國立政治大學商學院金融學系碩士論文. 指導教授: 廖四郎博士林士貴博士. Pricing Convertible Bonds with Credit Risk and Liquidity Risk 考慮信用風險及流動性風險之可轉債評價. 研究生: 許典玉撰中華民國 103 年 6 月 1.

(2) Abstract There are some risks with convertible bonds, and we find that there are liquidity risks with convertible bonds in the Taiwan market. We consider the credit risk and liquidity risk in the model to price the convertible bonds. We construct the dynamic default intensity process by setting the function which is inverse to stock price to estimate the credit risk. We use two methods to estimate liquidity risk. One is to construct the liquidity factor table by separating the different volumes of the convertible bonds into different levels to estimate liquidity risk, the other method is using the average bid-ask spread over the average convertible bond price to estimate liquidity risk. In this thesis, we use three different methods including forward method, backward method and LSMC method to prices the convertible bonds. We find that under the same parameters, the prices of convertible bonds using the backward method are the highest, while prices of convertible bonds using the forward method are the lowest.. I.

(3) Contents Abstract ....................................................................................................... I Table Contents ..........................................................................................III Figure Contents ........................................................................................ IV 1. Introduction .............................................................................................1 1.1 Motivation ......................................................................................1 1.2 Research Structure .........................................................................2 2. Literature Review ...................................................................................4 3. Research Method ....................................................................................6 3.1 Risk Description.............................................................................6 3.1.1 Credit Risk ...........................................................................6 3.1.2 Liquidity Risk ....................................................................13 3.2 Pricing structure ...........................................................................18 3.2.1 Pricing structure under risk neutral ....................................18 3.2.2 Pricing convertible bonds with credit risk and liquidity risk .....................................................................................................20 3. 3 Simulation Method......................................................................24 3.3.1 Forward Method .................................................................25 3.3.2 Backward Method ..............................................................26 3.3.3 Least Square Monte Carlo Method (LSMC) .....................27 4. Empirical Analysis ................................................................................28 4.1 Data Description ..........................................................................28 4.2 Empirical Result...........................................................................31 5. Conclusion ............................................................................................38 6. Reference ..............................................................................................40. II.

(4) Table Contents. Table 3-1 Liquidity Factor Table……………………………………….16 Table 4-1 The Contract of YES Logistics Corporation Convertible Bond ……………………………………………………………….28 Table 4-2 The Contract of China Airlines Convertible Bond…………..29 Table 4-3 Liquidity Factor Table……………………………………….30 Table 4-4 The convertible bond price of YES Logistics Corporation….32 Table 4-5 The convertible bond price of China Airline………………...33 Table 4-6 The convertible bond price of YES Logistics Corporation….34 Table 4-7 The convertible bond price of China Airline………………...36. III.

(5) Figure Contents. Figure 1-1 Research Structure……………………………………….…3 Figure 3-1 Total volumes of convertible bonds in Taiwan market……13 Figure 3-2 Total volumes of government bonds and convertible bonds in 2013……………………………………………………...14 Figure 3-4 Term Structure…………………………………………….16. IV.

(6) 1. Introduction. 1.1 Motivation Convertible bonds combine the debt right and call option right. It gives the convertible bond holders an opportunity to be shareholders of the company by call option right. Since the convertible bonds have the characteristic of hybrid security, which makes the price of them are similar to the value of stock when the stock price is higher, and the price of them are similar to the value of pure bond when the stock price is lower. In addition, the contracts of different convertible bonds issued by different companies are not the same. Some of them contain puttable right for convertible bond holders, some of them contain callable right for issuers, some of them contain both rights, and some of them do not contain any additional rights. Moreover, the conditions of these rights are not the same, too. If no default occurs, the convertible bond holders can receive a certain cash flow in the future time. In addition, they are given a chance to be shareholders of the company if the company performs well. It seems a great financial product for the investor if there is no default risk. Obviously, the default risk is an important factor of the convertible bonds. Moreover, we find that the volume of the convertible bonds is small in Taiwan. Compare to the volume of the government bonds, its volume is almost 200 times greater than the volume of convertible bonds in Taiwan market. Clearly, the liquidity risk is also an important factor of convertible bonds. Besides, we can find the volumes of the convertible bonds are lower and lower during the financial crisis, which shows that both credit risk and liquidity risk are significant factors to the convertible bond. The fake financial statement of Asia Plastic occurs recently. Whether the fake financial statement has been released to the public or not, the real value of the company is difficult to evaluate. That is why we usually use the reduced-form model instead of the firm-value model to estimate the credit risk. There is almost no one to estimate the liquidity risk to price the convertible bond. Since it is important, I try to consider it into my model. On the other hand, there are some rules be constructed after the Enron event. Under these rules, both issuers and investments of some financial products such as convertible bonds, options, futures, and so on need to disclose the fare price in the market. Clearly, pricing convertible bonds become more and more important. 1.

(7) 1.2 Research Structure There are five parts in this thesis. First, we introduce the convertible bonds and discuss why we need to price the convertible bonds under the credit risk and liquidity risk. Then, construct the structure of this thesis in part 1. Second, we review some thesis about pricing convertible bond, estimating the credit risk and the liquidity risk written by previous scholars in part 2. Third, we describe the way we estimate the credit risk and the liquidity risk in further detail first. Then, derive the pricing model of the convertible bond. Finally, we introduce 3 simulate method including forward method, backward method, and least square Monte Carlo method to find the convertible bond price in part 3. In part 4, we describe the data first, then we price these two convertible bonds by these three simulation methods. Finally, we discuss the results in part 5.. 2.

(8) Motivation. Literature Review. Research Method. Risk Description. Pricing model. Empirical Analysis. Conclusion. Figure1-1. Research Structure. 3. Simulation Method.

(9) 2. Literature Review Most of the people use firm-value model to describe the credit risk to price the convertible bond in the early period. This method originated in Black and Scholes (1973) and Merton (1974). According to Ingersoll (1977), Brenaan and Schwartz (1977), and Brennan Schwartz (1980), they consider the firm value as the underlying asset, and see the contingent claim of convertible bond as a right of call option. Then they use the same way to price convertible bond as call option by Black and Scholes formula. However, it is difficult to know the real value of a company, and to estimate the volatility of the firm value correctly is also a hard work. Since the structures of the firm value are always complicated, we are required to simplify them into the model. Many years later, some scholars focus on the price of the company. Since the price of the company does not contain the default risk, they use the interest rate with risk instead of the risk-free value as discount factor. According to McConnell and Schwartz (1986), they use stochastic process of the price of the company instead of the value of the company, and adjust the interest rate to find an appropriate discount factor. According to Derman (1994), he priced the convertible bond by binomial tree method, and used the weighted average of interest rate with risk and risk-free interest rate for discount factor. The weight of interest rate with risk is higher when the convertible probability is lower because the characteristic of the convertible bond is closer to the pure bond; the weight of the interest rate with risk is lower when the convertible probability is higher because the characteristic of the convertible bond is closer to the stock. According to Tsiveriotis and Fernandes (1998), they improve Derman’s idea rigorously. They divided the convertible bond into two parts, one is debt right part, and the other is call option right part. They chose the interest rate with risk to be the discount factor for the debt right part, and choose the risk-free interest rate to be the discount factor for the call option right part. It is unreasonable that we just adjust the risk-free interest rate to describe the credit risk in the real market because this way implies the stock price of the company would not drop when the default occurred. So most of researchers use reduced-form model to describe the credit risk to price the convertible bond recently. We do not need to find the value and volatility of the company in this model, and it implies that the drop of the stock price would reflect the default probability of the company. According to Jarrow and Turnbull (1995), Jarrow, Lando, and Turnbull (1997), Davis and Lischka (1999), Hung and Wang (2002), Ayache, Forsyth and Vetzal (2003), and Chambers and Lu (2007), they used this model to describe the credit risk. 4.

(10) On the other hand, Amihud (2002) proposed an illiquidity measurement and be widely used. We can show it as follows: ILLIQiy =. 1. 𝐷𝑖𝑖. 𝐷. 𝑖𝑖 ∑𝑡=1. �𝑅𝑖𝑖𝑖 �. 𝑉𝑉𝑉𝑉𝑖𝑖𝑖𝑖. Where Diy is the number of days for which data are available for stock i in year y, Riyd is the return on stock i on day d of year y, and VOLDivyd is the respective daily volume in dollars. He said this illiquidity measurement is strongly related to the liquidity ratio known as the Amivest measure, the ratio of the sum of the daily volume to the sum of the absolute return (e.g., Cooper et al., 1985; Khan and Baker, 1993). In 2013, Photis M. Panayids, Neophytos Lambertides, and Kevin Cullinance used this measurement with Fama-French model to do the regression and estimate the liquidity to price assets in US water transportation. The contract of the convertible bond has become more and more complicated recently. In addition to the convertible right, the convertible bond usually combines the puttable right and callable right together. The more complicated the contract is, the more boundary conditions would be generated when we price it by Partial Differential Equation (PDE) method. The more boundary conditions would cause the more partial differential equations, which would make it more difficult to reach the closed form. So some people try to use the binomial tree method or Monte Carol simulation method to price the convertible bond. Both of M. Ammann, A. Kind and C. Wilde (2008) use Monte Carol simulation method to price the convertible bond.. 5.

(11) 3. Research Method. 3.1 Risk Description In this part, we will talk about the risk of convertible bond. Since convertible bond combines the right of the debts and the right of shareholders, the most important risk of bond are interest rate risk and credit risk. The company performance may be a major concern because the convertible bond holders have a right to transfer their bonds to stock. Though the convertible bond holders are given a chance to become one of the members of the company, it should be pointed out that the liquidity is small in Taiwan. In other words, we have another risk --- liquidity risk in Taiwan market. According to Brennan and Swhwartz (1980), stochastic interest rate in pricing will lower the efficiency because there is only a small difference between stochastic interest rate and constant rate. According to Zong-Min, Tu (2010), he uses BGM model to describe the stochastic process of interest rate, and the result under BGM model does not have significant outstanding performance than the result under constant interest rate. Sometimes, the results under constant interest rate are even better than the result under BGM model. The reason why there is almost no difference under these two situation is probably because convertible bond contains two kinds of right, one is debt right, and the other is call option right. The increase of the interest rate will reduce the bond price, so the interest rate has opposite relation to the bond price. On the other hand, the increase of interest rate will also increase the option price, so the interest rate has positive relation to the call option price. Then, the effect of the change of interest rate may be eliminated. To enhance the efficiency of our model, we will not consider the interest rate risk here. We will consider the credit risk and liquidity risk as follows.. 3.1.1 Credit Risk There are two kinds of model to describe credit risk. One is structural form model (also called as Firm –Value Model), the other is reduced-form model (also called as Intensity Model). The feature of structural form model is that we use the capital structure of a company to describe the credit risk of the company. If the company possesses more debts than its assets, then it will be defined as the default. According to Merton (1974), 6.

(12) he used the structure of a company’s capital to describe the credit risk of the company’s bond. If we regard the asset of the company as stock price, the debt of the company as strike price, and the maturity date of the bond as the maturity date of the option, we could consider the capital of a company as a call option. Then the way we value the credit risk of the company is similar to the way we price the call option that was used by Black and Scholes (1973). But in this model, we will face some problems: (1) It is quite difficult to measure the real value of the company. With limited information that we can find in the market, assessing the real asset and the real debt of the company becomes a difficult and even an impossible task. There is no assurance that each financial report that we found demonstrates the real situation of the company. (2) It does not consider the change of credit rating. Most of company’s credit rating would be lower before the default date. However, we could not include this factor into this model. (3) The default date is difficult to expect in real market. The way to describe the value of the company is a time process in this model, which means the default date can be expected. However, the default date is difficult to be expected because it is usually caused by big event. Due to the difficulty of solving these problems, we choose to use reduced-form model to describe the credit risk in this paper. In this model, we use the bond with credit risk or credit spread and with some parameters such as recovery rate, loss rate and so on to estimate the credit risk. In this model, we propose that the happen of default event is Poisson process, and the first drop of stock price can be seen as the occurring of default event. The parameters of default intensity is determined by the information in the real market. There are some features in this model: (1) There is no arbitrage conditions in the market. (2) The probability of default is a stochastic process. (3) The Price of the commodity with credit risk is determined by the evaluation of the credit risk. (4) The recovery rate is exogenous. (5) We can calculate the probability of default by the credit rating in the market. The advantages of this model can be summarized as follows: (1) We do not need to calculate the assets of the company. 7.

(13) (2) The recovery rate is exogenous. According to different ordinary of paying debt, different recovery rate would be given. (3) We can use the market information such as credit rating to estimate the probability of default. According to Jarrow & Turnbull (1995), they used this model to estimate the credit risk, and priced the derivatives. In Jarrow, Lando and Turnbull (1997), they used Time-Homogeneous Markov Chain to describe the change process of company’s credit rating. They used the change of the credit rating matrix made by credit rating company and adjusted the risk premium to construct the risk neutral matrix about the change of the credit rating.. a. Credit market We need to define some notation which we need in reduced-form model as follows: (1) τ: Default Time There are some features about the default time. If we set Q (.) as the measure of default probability, then we would have: Q(τ< ∞) = 1, which means there is no company could run forever.. ①. Q(τ= ∞) = 0, which means there is no company will default when it just opens.. ②. Q(τ> t) > 0, t ∈ R+, which means the default time is positive.. ③. (2) Ht: Default Stochastic Process For every t ∈ R+,. Ht = 1{τ< t}, the default occurs. Ht = 0{τ< t}, the default does not occurs.. (3) λ: Default intensity The default intensity needs to be satisfied the equation as follows: 𝑣. EQ [𝑒 ∫𝑜 𝜆𝜆 𝑑𝑑 ] < ∞ 8.

(14) (4) L : Loss Rate The loss rate refers to the percentage of the bond value the bondholders will lose when the default occurs. For every loss rate, 0 < loss rate < 1. If the loss rate = 1, the bondholders will not get any money back when the default occurs. If the loss rate = 0, the bondholders will get all money back when the default occurs. (5) δ : Recovery Rate. The recovery rate refers to the percentage of the bond value the bondholders will get back when the default occurs. For every recovery rate, 0 < recovery rate < 1. If the recovery rate = 1, the bondholders will get all money back when the default occurs. If the recovery rate = 0, the bondholders will not get any money back when the default occurs.. Clearly, recovery rate = 1 – loss rate. (δ = 1 – L ). b. Setting Default Intensity Default intensity (λt) is the default probability in time [t, t+1], and no default event occurred before time t. There are two common methods to estimate the default intensity (λt) : one is using the difference between the corporation bond interest rate and the government bond interest rate; the other is using the spread of credit default swap (CDS) to estimate it. (1) Using Corporation bond The most common commodity in credit market is corporation bond. The price of some public offering corporation bonds are negotiated by the buyer and seller in the market, so these prices imply the information of credit risk of these companies. In the same conditions, the most important reason why the price between the corporation bond and the government bond different is credit risk. We know the relation between the government bond interest rate and the corporation bond interest rate under the same maturity date in the bond market. The equation is as follows:. 9.

(15) 𝑇. 𝑒 − ∫𝑡. 𝑅𝑢 𝑑𝑑. 𝑇. = 𝑒 − ∫𝑡. 𝑟𝑢 𝑑𝑑. [1{τ> T} + (1 – Lt) {τ< T}]…….……….…(1). where Rt is the interest rate with risk (corporation bond interest rate), rt is the risk-free interest rate (government bond interest rate), τ is the default date, and T is the maturity date. 𝑄. Then, we set expected value (𝐸𝑡 (. )) for both sides of the equation (1), it will become: 𝑇. 𝑄. 𝐸𝑡 [𝑒 − ∫𝑡. 𝑅𝑢 𝑑𝑑. 𝑇. 𝑄. ] = 𝐸𝑡 [𝑒 − ∫𝑡. 𝑟𝑢 𝑑𝑑. [1{τ> T} + (1 – Lt){τ< T}]]…….….(2). Suppose the interest rate and the default time is independent, and the recovery rate is constant, then the equation (2) will become: 𝑄. 𝑇. 𝐸𝑡 [𝑒 − ∫𝑡. 𝑅𝑢 𝑑𝑑. 𝑇. 𝑄. ] = 𝐸𝑡 [𝑒 − ∫𝑡. 𝑟𝑢 𝑑𝑑. 𝑄. ]𝐸𝑡 [1{τ> T} + (1 – Lt){τ< T}]…….(3). This equation is presented in the form of bonds:. 𝑃(t,T) = P(t,T)[Pr(τ> T) + (1 – L)P(τ< T)]…………….…….(4) 𝑄. 𝑇. 𝑄. 𝑇. where 𝑃(t,T) = 𝐸𝑡 [𝑒 − ∫𝑡 P(t,T) = 𝐸𝑡 [𝑒 − ∫𝑡. 𝑅𝑢 𝑑𝑑. 𝑟𝑢 𝑑𝑑. ]. ]. The default time is exponential distribution, the parameter is default intensity (λt), then we can express the equation (4) as: 𝑇. 𝑃(t,T) = P(t,T) [𝑒 − ∫𝑡. 𝜆𝑢 𝑑𝑑. Then, we know that. λt =. 𝜕ln(. 𝑇. + (1 – L)(1 - 𝑒 ∫𝑡. 𝑃(t,T) −(1−𝐿) P(t,T) ) 𝐿. 𝜕𝜕. 𝜆𝑢 𝑑𝑑. )]….....(5). ………….………….(6). Clearly, we can estimate the default intensity form bond market directly. Making it simpler, if we assume L = 1, then λt = 10. 𝑃(t,T) ) P(t,T). 𝜕ln(. 𝜕𝜕. ,.

(16) So, λt =. 𝑃(t,T) ) P(t,T). 𝜕ln(. 𝜕𝜕. =. 𝜕[𝑙𝑙𝑃(𝑡,𝑇) − 𝑙𝑙 𝑃(𝑡,𝑇)] 𝜕𝜕. .. As can been seen, the default intensity is the difference of instantaneous interest rate from time t to maturity date T between corporation bond and government bond.. (2) Using Credit Default Swap (CDS) The credit default swap is a kind of derivatives that the buyer wants to transfer the risk to seller. For example, if a bank loans money to its borrowers, but is afraid that the borrowers will default, then the bank buys a CDS from the sellers. In this contract, the bank will give a portion of money to the sellers of CDS every time during a period. If the borrowers default during this period, the sellers of the CDS must compensate the loss of the bank. In fact, it seems like the bank buys an insurance from the sellers of the CDS. The amount of money that the bank would pay for the sellers of the CDS depends on the CDS spread, and this spread is negotiated by both sides. Clearly, the CDS spread implies the credit risk. In a fair contract, the expected value of the buyer of CDS loss need to be equal to the expected value of what it pays for the seller of CDS. So we know that λtLt = spread, and we can change this equation as : λt =. 𝑠𝑠𝑠𝑠𝑠𝑠 𝐿𝐿. …………………………… (7). where Lt is the loss rate of the convertible bond at time t. We can estimate the default intensity in both ways by equation (6) and (7). The reason why we need to estimate the default intensity is because the convertible bond is a kind of bonds with credit risk. When we price the value of it, we need to use the interest rate with risk to discount it. We will construct a dynamic default intensity process, but we need the initial value to construct it. Cleary, we just need to find the corporation bond interest rate and government bond interest rate in the market, or the CDS spread in the market, in order to be able to estimate the initial value easily by these two methods.. 11.

(17) c. Dynamic Default Intensity Process The most common way to construct the dynamic default intensity process is to regard the default intensity as a decreasing function of stock price. When the stock price decreases, it implies there are some bad information such as negative news and bad performance of the company, and this result of will increase the default probability of the company. In this model, we will use the stock price which we simulated to construct the dynamic default intensity process. According to A. Takahashi, T. Kobayashi, and N. Nakagawa (2001), they use Binomial Tree to value the convertible bond, and they use the stock function to estimate the default intensity at every time as follows: λ( St , t) = θ +. 𝑐. 𝑆𝑡𝑏. ………………………………(8). where θ, b, c are some constants and θ > 0. There are three parameters in equation (8) need to be correction. According to A. Takahashi, T. Kobayashi, and N, Nakagawa (2001), they set θ and c first, and then use the real convertible bond price in the market to correct the parameter b. After finding all the suitable parameters for this model, the dynamic default intensity process has been constructed. The problem of this model is that when we set different θ and c at first, we will find the different parameter b. Under different parametersθ, c, and b, we will have different dynamic default intensity process, then we will get the different results of the convertible bond value which we priced. It is a big problem because we do not know which value we found is correct. To solve this problem, we use another model to construct the dynamic default intensity process. According to E. Ayache, P.A. Forsyth and K.R Vetzal (2004), they use the model to estimate the default intensity as follows: 𝑆. λ( St , t) = λ( St ) = λ( S0 )(𝑆𝑡 )𝛼 …………………..(9) 0. where λ( S0 ) is the default intensity when the stock price is S0.. If fact, Muromachi (2000) has used equation (9) to price the bonds of which credit rating are lower than BB+ in Japan market. He found the appropriate parameter α is between -2 and -1.2. Compare to the model of equation (8), this model is better in constructing the dynamitic default intensity than because it does not contain so many parameters inside, so we use this model in this thesis.. 12.

(18) 3.1.2 Liquidity Risk First, we need to have more information about the situations of convertible bonds in Taiwan market. In figure 1, we can see the total volume of convertible bonds in Taiwan from 1997 to 2013.. Figure 3-1. Total trading volumes of convertible bonds in Taiwan market. Asia financial crisis occurred in 1997, and there were a lot of companies with financial crisis at that time. The credit risk of the financial products which relate to those companies were higher and higher, leading to the price of these financial products lower and lower. Since the investors were afraid that those companies would default, they would not buy these financial products. We can see that the total volume of convertible bond is just around 21.5 million dollars in 1997, and the total volumes are increasing during 1997 to 2004. The biggest total volume of convertible bonds is in 2007, and then the total volumes are decreasing during 2007 to 2010 because there was another financial crisis during 2008. Compared to the government bonds, the total volumes of government bonds are almost 200 times greater than the total volumes of convertible bonds in figure 2. Clearly, the total volumes in the convertible bonds market are really small in Taiwan, so the liquidity risk of convertible bonds in Taiwan market is important.. 13.

(19) Figure 3-2. Total trading volumes of government bonds and convertible bonds in Taiwan in 2013. There are some common indicators to see the liquidity of the market, we are going to introduce them as follows: (1) The Depth of the Market: It indicates the volume of the financial products. If the volume is large at the specific price, the liquidity of the product will be perceived as good. Otherwise, the liquidity will be perceived as not good in the market. (2) The Width of the Market: It indicates the bid-ask spread of the financial product. If the bid-ask spread is large, it means the willing to pay of the buyer is much lower than the willing to pay of the seller. So if they want to trade to each other, the willing to pay of the buyer should increase a lot or the willing to pay of the seller should decrease a lot. Then the trade could be done. Since they have to make a larger concession for the deal to be done, so the bid-ask spread is larger, the product in the market is less liquidity. (3) Immediacy of Trade: It indicates the time to make trade be done. If the trade could be done in a short time, which means the market is liquidity. The longer the time for trade is, the less liquidity the market is. 14.

(20) There is a widely used illiquidity measurement proposed by Amihud (2002), the ratio gives the absolute (percentage) price change per dollar of daily trading volume, or the daily price impact of the order ﬂow. It can be written as follows: ILLIQiy =. 1. 𝐷𝑖𝑖. 𝐷. 𝑖𝑖 ∑𝑡=1. �𝑅𝑖𝑖𝑖 �. 𝑉𝑉𝑉𝑉𝑖𝑖𝑖𝑖. ………….….……. (10). where Diy is number of days for which data are available for stock i in year y, Riyd is the return on stock i on day d of year y, and VOLDivyd: the respective daily volume in dollars. According to Panayids, P. M., Lambertides, N. and Cullinance, K. (2013), they use this measurement with Fama-French model to do the regression and estimate the liquidity to price assets in US water transportation. It is difficult to put this model into our model to estimate liquidity risk directly since we will face some problems as follows: (1) We do not have the data about the future volumes of the convertible bond on the pricing date. And simulate them is not a reasonable method when we estimate the liquidity risk because volumes should be fact rather than fake. (2) Compare with other bonds, the volume of the convertible bond is so small that will make the illiquidity measurement become too large. (3). It is impossible to know the number of the trading day of future years.. To solve these problems about volume, we use another two methods to estimate the liquidity risk as follows: a. Volume Method We construct a system to solve these problems. First, calculate the proportion of the trading volume of the convertible bond to all trading volumes of convertible bonds. Let Wi =. 𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑉𝑉𝑉𝑉𝑉𝑉𝑖. 𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑉𝑉𝑉𝑉𝑉𝑉. ……………..……… (11). where Trading Volumei is the trading volume of convertible bond of company I, and Total Trading Volume is the trading volumes of all convertible bonds in the market. Second, we put each proportion into nine groups which are [0, 0.02), [0.02, 0.04), [0.04, 0.06), [0.06, 0.08), [0.08, 0.010), [0.10, 0.12), [0.12, 0.14), [0.14, 0.16), and [0.16, 1). Then we need to find the liquidity factors to correspond to these groups. 15.

(21) According to liquidity preference theory, market participants need to be compensated for the interest rate risk associated with holding longer-term bonds. We can see this situation in figure 4.. Figure 3-3. Term Structure. So we choose to use the difference between the interest rate of the 10-year government bond and the interest rate of the 1-year government bond to estimate the liquidity risk. Let d = interest rate10 – interest rate01 where interest rate10 is the interest rate of 10-year government bond, and interest rate01 is the interest rate of 1-year government bond. d* =. 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟10 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟01. =. 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟10 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟01. 𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜 𝑔𝑔𝑔𝑔𝑔𝑔 9. ………………………(12).. , since we have 9 groups.. So we can get the liquidity factor table as follows: Table 3-1. Liquidity Factor Table Weight (Wi). Liquidity Factor. 0.02↓. 8d* + d*. 0.02~0.04. 7d* + d*. 0.04~0.06. 6d* + d*. 16.

(22) 0.06~0.08. 5d* + d*. 008~0.1. 4d* + d*. 0.1~0.12. 3d* + d*. 0.12~0.14. 2d* + d*. 0.14~0.16. d* + d*. 0.16↑. d*. Obviously, after we find the weight of the volume of the convertible bond, we will have the correspondent liquidity factor. We can write the liquidity factor function as follows: 9𝑑 ∗ , 𝑖𝑖 𝑊𝑖 ∈ [0, 0.02) ⎧ ∗ 8𝑑 , 𝑖𝑖 𝑊𝑖 ∈ [0.02, 0.04) ⎪ ∗ ⎪7𝑑 , 𝑖𝑖 𝑊𝑖 ∈ [0.04, 0.06) ⎪6𝑑 ∗ , 𝑖𝑖 𝑊𝑖 ∈ [0.06, 0.08) 5𝑑 ∗ , 𝑖𝑖 𝑊𝑖 ∈ [0.08,0.1) LIQi = ⎨ 4𝑑∗ , 𝑖𝑖 𝑊 ∈ [0.1, 0.12) 𝑖 ⎪ ∗ 3𝑑 , 𝑖𝑖 𝑊 𝑖 ∈ [0.12, 0.14) ⎪ ∗ ⎪ 2𝑑 , 𝑖𝑖 𝑊𝑖 ∈ [0.14,0.16) ⎩ 𝑑∗ , 𝑖𝑖 𝑊𝑖 ∈ [0.16,1] b. Bid-Ask Spread Method On the other hand, we also can use bid-ask spread to estimate the liquidity risk. We already know that the width of the market indicates the bid-ask spread. The bid-ask spread is larger, the convertible bond in the market is less liquidity. In this method, we use the average bid-ask spread over the average convertible bond price as liquidity factor directly. We can write it down as follows: LIQ =. 𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏−𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠. 𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑏𝑏𝑏𝑏 𝑝𝑝𝑝𝑝𝑝. 17. ……………(13).

(23) 3.2 Pricing structure In this part, we will discuss the structure which we use to price the convertible bonds. First, we will construct the pricing structure in general. Second, we will consider a situation of the default probability of the convertible bond, and add the default probability and the recovery rate into our model. Finally, we add the credit risk and liquidity risk into the model and improve the situation of the recovery rate of the model.. 3.2.1 Pricing structure under risk neutral The feature of convertible bond is that it is not only a bond, but has the right to convert it to the company stock. In general, we would choose the maximum of bond value and conversion value, and discount it to find the convertible bond value. This way can be written as: 𝐶𝐶 = 𝑒 −𝑟𝑡 ･EQ[Max(bond value, conversion value)]. But in the real market, the convertible bond usually also contains the puttable right for the convertible bond holders and callable right to the issuers. Clearly, we need to consider these situations in the pricing structure. Now, we will explain the puttable right and callable right in further detail.. a. Puttable Right Puttable right is a right for the convertible bond holders that they can sell the bond back to the issuer at specific prices during some specific period. The different specific prices and periods depend on different contracts of the convertible bonds. We let the puttable time is πt, where t < π < T, and define the puttable price function as Np (π) in this thesis. For example, if the convertible bond holders have the right to sell the bonds back to the issuer for 103% of the face value of the convertible bond at year two, and sell the bonds back to the issuer for 105% of the face value of the convertible bond at year three in the contract. Then the puttable price function is as follows:. 18.

(24) Np (π) = �. 0 , 𝑖𝑖 𝜋1 ∈ [0,2) ∪ (2,3) ∪ (3, 𝑇] 103, 𝑖𝑖 𝜋2 ∈ 2 105, 𝑖𝑖 𝜋3 ∈ 3. Note that the puttable prices in the contract usually are the nominal puttable price, adding accrued interest is real puttable price. The convertible bond holders will get real puttable price if they strike this right in the real market. b. Callable Right Callable right is a right for the issuers that they can use these right to buy the convertible bonds back form the convertible bond holders at a specific price during a specific periods. We let the callable time is φt , where t <φt < T, and the callable function is Nc(φ) in this thesis. Actually, the callable price does not usually depend on the time, it usually depends on the different situations in the real market. For example, the callable condition is when the company stock price is 1.3 times higher than the conversion price for 30 consecutive days. The issuer has the right to buy the convertible bonds back from the convertible bond holders at a specific price (i.e. the face value of convertible bond) if the condition is hold. We can write the callable price function as follows: Nc(φ) = face value, when the callable condition holds. Note that the callable prices in the contract usually are the nominal callable price, adding accrued interest is real callable price. The convertible bond holders will get real callable price if the issuer strikes this right in the real market. Clearly, if the callable conditions be satisfied, the issuers can choose whether they want to buy the convertible bonds form the convertible bond holders or not. In fact, when the callable condition be satisfied, the convertible bond holders still have the right to convert their bonds to stocks if the conversion value is higher than the callable value at that time. In addition, if the callable condition be satisfied during puttable time, the convertible bond holders even have can choose puttable right. As a result, we can write down the convertible bond pricing structure in general as follows: CB(t)=EQ 𝑒 −𝑟𝑡 max𝑡∈𝜋𝜋𝜋 [min𝑡∈𝜋𝜋𝜋 ( 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝑣, 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣𝑣), 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣𝑣, 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑣𝑎𝑙𝑙𝑙] ⎧ 𝑒 −𝑟𝑡 max𝑡∈𝜑 [min𝑡∈𝜑 ( 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝑣, 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣𝑣), 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣𝑣] [ ] 𝑒 −𝑟𝑡 max𝑡∈𝜋 ( 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝑣, 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣𝑣, 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑣𝑣𝑣𝑣𝑣) ⎨ 𝑒 −𝑟𝑡 max𝑡 ( 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝑣, 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣𝑣) ⎩ 19.

(25) 3.2.2 Pricing convertible bonds with credit risk and liquidity risk a. Considering Default Risk into the Model In general, the convertible bond holders just can get back the money which the issuer could give to them when the default occurs. This amount of money may be the recovery rate of the bond market value or the recovery rate of the bond face value. The recovery rate depends on the promise from the issuer in the contract or the convertible bond holders ordinary of the creditors. According to Dmitri Lvov, Ali Bora Yigitbasioglu, and Naoufel El Bachir (2004), the convertible bond holders have two choices when the default occurs: one is to convert the convertible bonds to stocks with recovery rate, and the other one is to get the recovery rate multiplier the face value of convertible bonds. We can write it down as follows:. Max ( a ∗ δ ∗ 𝑆̃(𝜏), 𝛿 ∗ 𝐹𝐹),. where a is the conversion ratio, δ is recovery rate, 𝑆̃ is stock price under risk neutral, τ is the default time, and FV is face value of the convertible bond.. In order to make the pricing steps simpler, we need to define some notations first. First, let ω = πΛφ, and ex(ω) = �. 𝑝, 𝑖𝑖 𝜔 = 𝜋 𝑐, 𝑖𝑖 𝜔 = 𝜑. Second, define the dividend process to express the total cash flow as follows: 𝑡 𝑡 Dt = ∫0 (1 − 𝐻𝑢 )1{𝑢<𝜔} 𝑑𝑑(𝑢) + ∫0 𝑀𝑀𝑀( a ∗ δ ∗ 𝑆̃(𝑢), 𝛿 ∗ 𝐹𝐹)1{𝑢≤ω} d𝐻𝑢. + Max (a ∗ δ ∗ 𝑆̃(𝜏), 𝑁 𝑒𝑒(𝜔) (𝑡))(1 - 𝐻𝑡 )1{𝑡=𝜔} …………………….(14). where A(t) is the amount of cash flow of bond form time 0 to time t, we can write it as 𝑡. A(t) = ∫0 𝑑𝑑(𝑢) . Hu is the default stochastic process, which can be show as. Hu = �. 1, 𝑖𝑖 𝑢 ∈ 𝜏 0, 𝑤𝑤ℎ𝑒𝑒𝑒𝑒𝑒𝑒. There are three parts in equation (14), we are going to explain it in detail as follows: 20.

(26) 𝑡. (1) ∫0 (1 − 𝐻𝑢 )1{𝑢<𝜔} 𝑑𝑑(𝑢):. This item means the amount interest which the convertible bond holders can receive during the time [0, t] if there is no default occurs (Hu = 0). And no conversion events, puttable events, and callable events occur. If the default occurs (Hu = 1), then this item is zero, which means there is no money for convertible bond holders about the amount interest.. 𝑡 (2) ∫0 𝑀𝑀𝑀( a ∗ δ ∗ 𝑆̃(𝑢), 𝛿 ∗ 𝐹𝐹)1{𝑢≤ω} d𝐻𝑢 :. This item means the cash flow of the convertible bond if no conversion events, puttable events, and callable events occur before default occurs.. (3) Max (a ∗ δ ∗ 𝑆̃(𝜏), 𝑁 𝑒𝑒(𝜔) (𝑡))(1 - 𝐻𝑡 )1{𝑡=𝜔} :. This item means the cash flow of the convertible bond if one of conversion events, puttable events, or callable events occurs before the default date.. Clearly, we have understood the cash flow of the convertible bond at every time. According to the rule of risk neutral, there is no arbitrage opportunity in the market. Since the cash flow of the convertible bond is certain, we can discount it with the risk-free interest rate to find the value of the convertible bond. The pricing formula under risk neutral is as follows: CB(t,T) = sup𝜋∈𝛹 inf𝜑∈Ф 𝐸. 𝑄. 𝑡. 𝑇 𝑒 ∫0 𝑟𝑥 𝑑𝑑 [∫𝑡 ∫𝑢 𝑟 𝑑𝑑 𝑒0 𝑥. dDu Iξt ] ………………………(15). where Ψ is the set which contains all convertible time and puttable time, and Ф is the set which contains all callable time.. b. Improving the Model of Recovery Rate In the previous model of equation (2), we assume that the convertible bond holders have the right to choose which right is better for them when the default occurs. But in fact, it is almost impossible that we have the opportunity to choose which one is better for us when the default occurs. So we need to improve our model to make it closer in the real market. According to previous thesis, someone use recovery rate of face value to describe the return when the default occurs, and someone use recovery rate of market value to describe the return when the default occurs. There is still no 21.

(27) conclusion which way to describe the return is more appropriate now. According to Duffie and Singleton (1999), they use recovery rate of market value to describe the return under risk neutral when the default occurs, we will follow this model in this thesis as follows: We consider a bond with credit risk. Now is at time t, and the maturity date is time T. There is no cash flow during [t, T), but the convertible bond holders will receive XT at time T. First, we need to define some notations as follows: (1) s: Time, where t < s < T (2) λs: The default probability during time [s, s+1] under risk neutral of Q measure. And there is no default occurs before time s. That is, λs = 𝑃𝑟𝑄 [𝜏 < 𝑠 + 1lτ> s ]. (3) Ωs: The recovery of time s when the default occurs at time s. (4) rs: The risk-free interest rate at time s. (5) Vt: Bond value at time t. Clearly, 𝑄 𝑄 Vt = λt 𝑒 −𝑟𝑡 𝐸𝑡 [𝛺𝑡+1 ] + (1 – λt) 𝑒 −𝑟𝑡 𝐸𝑡 [𝑉𝑡+1 ] ………………….. (16). Using the recursive method to solve equation (16), the bond value at time t can be expressed as follows: 𝑗. 𝑄. 𝑗. − ∑𝑘=0 𝑟𝑡+𝑘 Vt = 𝐸𝑡 [∑𝑇−1 𝛺𝑡+𝑗+1 ∏𝑙=0(1 − 𝜆𝑡+𝑙−1 )] 𝑗=0 𝜆𝑡+𝑗 𝑒 𝑄. 𝑇−1. + 𝐸𝑡 [𝑒 − ∑𝑘=0 𝑟𝑡+𝑘 𝑋𝑇 ∏𝑇𝑗=1(1 − 𝜆𝑡+𝑗−1 )] ……………………. (17). When the default occurs at time s, the payoff of the convertible bond 𝐸𝑠𝑄 [𝛺𝑠+1 ].. is Since we use the recovery rate of market value to describe the payoff of the convertible bond when the default occurs, 𝐸𝑠𝑄 [𝛺𝑠+1 ] = (1 − 𝐿𝑠 )𝐸𝑠𝑄 [𝑉𝑠+1 ] should hold, where 1-Ls is the recovery rate. Then the equation (17) can be expressed as follows: 𝑄 𝑄 Vt = (1 − 𝜆𝑡 )𝑒 −𝑟𝑡 𝐸𝑡 [𝑉𝑡+1 ] + 𝜆𝑡 𝑒 −𝑟𝑡 (1 − 𝐿𝑡 ) 𝐸𝑡 [𝑉𝑡+1 ] ………..… (18) 𝑄. 𝑇−1. = 𝐸𝑡 [𝑒 − ∑𝑗=0 𝑅𝑡+𝑗 𝑋𝑇 ]. 22.

(28) where 𝑒 −𝑅𝑡 = (1 − 𝜆𝑡 )𝑒 −𝑟𝑡 + 𝜆𝑡 𝑒 −𝑟𝑡 (1 − 𝐿𝑡 ) ………….…..…. (19). According to equation (19), we know that Rt ≅ rt + λtLt. Clearly, the interest rate with credit risk is similar to the risk-free interest rate plus credit spread. c. Adding liquidity risk into the model In the last part, we already know how to price the convertible bond with credit risk. We have mentioned that we still have liquidity risk in the Taiwan market, so we need to try to incorporate it into the model. Based on equation (18), we can add the liquidity risk factor into it. We can write it down as follows:. 𝑄 𝑄 Vt = (1 − 𝜆𝑡 ) ∙ 𝐿𝐿𝐿 ∙ 𝑒 −𝑟𝑡 𝐸𝑡 [𝑉𝑡+1 ] + 𝜆𝑡 ∙ 𝐿𝐿𝐿 ∙ 𝑒 −𝑟𝑡 (1 − 𝐿𝑡 ) 𝐸𝑡 [𝑉𝑡+1 ]… (20) 𝑄. where. 𝑇−1. = 𝐸𝑡 [𝑒 − ∑𝑗=0 𝑅𝑡+𝑗 𝑋𝑇 ] …………………..(21). 𝑒 −𝑅𝑡 = 𝐿𝐿𝐿 ∙ (1 − 𝜆𝑡 )𝑒 −𝑟𝑡 + 𝐿𝐿𝐿 ∙ 𝜆𝑡 𝑒 −𝑟𝑡 (1 − 𝐿𝑡 ) ………..… (22). According to equation (19), we know that Rt ≅ rt + λtLt + LIQ. Clearly, the interest rate with credit risk is similar to the risk-free interest rate plus credit spread and plus liquidity risk factor. We will use this interest rate as discount factor to price the value of convertible bonds in this thesis.. 23.

(29) 3. 3 Simulation Method We will introduce 3 kinds of methods to simulate and we will discuss some general parts first. With the discount factor including credit risk and liquidity risk, we can combine them with both the puttable and callable conditions together. First, we consider the stochastic process of stock prices under risk neutral of measure Q. We can write it down as follows: 𝑑𝑆̃𝑡 𝑆̃𝑡. 𝑄. 𝑄. = ( rt - qt + λt + LIQ )dt + σt d𝑊𝑡. …………………….. (23). where 𝑊𝑡 is a Brownian Motion under measure Q, qt is the dividends of stock at time t, and σt is the volatility of the stock. Second, in order to use simulation methods in this part, we need to use the discretization equation to express the value of convertible bonds. In addition, we need to consider some characteristic of time, so we will divide them into sets. Let Γ = [ t0, t1, t2, t3, t4, ……, tn]. Λ = [ ta1, ta2, ta3, ta4, ……, tak] , where Λ is a set of coupon date.. Clearly, Λ ⊂ Γ.. According to equation (15), we can change it to discretization form as follows: CB. (t,T). =. 𝐷𝑡𝑗−1 )Iξt ] ………………(24). sup𝜋∈Γ inf𝜑∈Γ 𝐸 𝑄 [∑𝑛𝑗=1. 𝑡. 𝑒 ∫0 𝑟𝑥 𝑑𝑑. 𝑡𝑗 ∫ 𝑟 𝑑𝑑 𝑒0 𝑥. (. 𝐷𝑡𝑗 −. According to equation (20), with the results about credit risk and liquidity risk we have constructed in the previous parts, we would like to add these factors into equation (24), which can be modified as follows: CB. (t,T). 𝐷𝑡𝑗−1 )Iξt ] ………………(25). =. sup𝜋∈Γ inf𝜑∈Γ 𝐸. 𝑄. 𝑡. 𝑒 ∫0 𝑅𝑥 𝑑𝑑 [∑𝑛𝑗=1 𝑡𝑗 ∫ 𝑅 𝑑𝑑 𝑒0 𝑥. (. 𝐷𝑡𝑗 −. Finally, we can show the value of the convertible bond at each time in detail: (1) At maturity date:. CB(T,T) = EQ[Max( a∗ 𝑆̃(T), FV+c)1{𝜏>𝑇} + Max(a∗ δ ∗ 𝑆̃ (T), δ ∗ FV)1{𝜏=𝑇} ] ...(26) 24.

(30) where a is the conversion ratio and c is the interest of coupon rate. (2) Before maturity date: a. At coupon date (tϵΛ): CB (𝑡𝑎ℎ , T) = �𝑀𝑀𝑀 �𝑀𝑀𝑀 �𝐸𝑄 �. 𝑡𝑎ℎ 𝑅𝑥 𝑑𝑑. 𝑒 ∫0. 𝑡𝑎ℎ+1 𝑅𝑥 𝑑𝑑 𝑒 ∫0. 𝐶𝐶(𝑡𝑎ℎ+1 , 𝑇)�ξ𝑡𝑎ℎ � , 𝑁 𝑐 (𝑡𝑎ℎ )� + 𝑐, 𝑎 ∗. 𝑆̃(𝑡𝑎ℎ ), 𝑁 𝑝 (𝑡𝑎ℎ ) + 𝑐�� 1{𝜏>𝑡𝑎ℎ } + Max�𝑎 ∗ 𝛿 ∗ 𝑆̃(𝑡𝑎ℎ ), 𝛿 ∗ 𝐹𝐹�1{𝜏=𝑡𝑎ℎ} ………. (27) b. Does not at coupon date (t∉ Λ): CB (th, T) =. �𝑀𝑀𝑀 �𝑀𝑀𝑀 �𝐸𝑄 �. 𝑡. ℎ 𝑒 ∫0 𝑅𝑥 𝑑𝑑. 𝑡ℎ+1 𝑅𝑥 𝑑𝑑 𝑒 ∫0. 𝐶𝐶(𝑡ℎ+1 , 𝑇)�ξ𝑡ℎ � , 𝑁 𝑐 (𝑡ℎ )� , 𝑎 ∗ 𝑆̃(𝑡ℎ ), 𝑁 𝑝 (𝑡ℎ )�� 1{𝜏>𝑡ℎ }. + Max�𝑎 ∗ 𝛿 ∗ 𝑆̃(𝑡ℎ ), 𝛿 ∗ 𝐹𝐹�1{𝜏=𝑡ℎ} ……… (28) where 𝐸𝑄 �. 𝑡. ℎ 𝑒 ∫0 𝑅𝑥 𝑑𝑑. 𝑡ℎ+1 𝑅𝑥 𝑑𝑑. 𝑒 ∫0. 𝐶𝐶(𝑡ℎ+1 , 𝑇)�ξ𝑡ℎ � is the holding value of the convertible bond,. we will use different ways to estimate it in the three methods. We are going to discuss it in detail in the following sections.. 3.3.1 Forward Method The idea of this method is simple, and it is close to the real situation. Although we have a lot of stock paths which we simulated before, we do not know how the price is going to change. In general situation, we compare with the conversion value, puttable value, and the bond value, and find the maximum one. If the maximum value is bond value, it means we will choose to hold this convertible bond. If the maximum value is either the conversion value or puttable value, we will execute our right. That is, we will regard either of values we indicate above as the cash flow value to find its present value. However, in a particular situation, if the callable conditions be satisfied, we 25.

(31) need to find the minimum of bond value and callable value first, and then find the maximum value of conversion value, puttable value, and the value which we have found first. Then, we will regard any of the three values as the cash flow value to find its present value. This is a method of Monte Carlo, so we simulate more times, the result is more accurate. If we simulate 10,000 times, we will have 10,000 present values. Then, we average these present values, the result is the theoretical price of the convertible bond. To make our method more efficient, we use antithetic method to make the variance become small with a higher speed, which means the value will converge to the same price faster. In short, in forward method, we need to simulate a lot of stock paths first, then use these stock paths and the market information to find the default factor paths. Third, with these default factor paths, we can see the result which we simulate to find when the bond will default, when will be the convertible date, puttable date, or callable date. And we see these values as the cash flow values and through the cash flow values, we can find the present values. Finally, we average these present values to get a mean value 1. In the same way, we use the same random variables but apply opposite signs to simulate the stock paths, too. As a result, we will get another mean value 2. Finally, we average mean value1 and mean value 2, then we find the theoretical price of the convertible bond.. 3.3.2 Backward Method The second method is backward method. The idea of this method is similar to the binomial tree method, we estimate the price from the last term, which means we discount it from the end to the front. In this method, we see the maximum value of bond value and conversion value as the cash flow, and use this value to discount. Then, we see this cash flow as holding value of the bond, and compare with conversion value again. We find the maximum value of these value, and discount it, we would get a new cash flow. We do these steps again and again, we will get the result. By the way, if the callable conditions or puttable conditions are satisfied, we need to find the minimum of bond value and callable value first, then compare this value with conversion value and puttable value, and find the maximum. After getting the maximum, we see it as cash flow of the bond. We do the same steps above again and again, and then we will get the result. In order to make this method more efficient, we use antithetic method which we mentioned in the previous part, too. 26.

(32) In short, we can write it down a follow: 𝐸𝑄 �. 𝑡. ℎ 𝑒 ∫0 𝑅𝑥 𝑑𝑑. 𝑡ℎ+1 𝑅𝑥 𝑑𝑑. 𝑒 ∫0. 𝐶𝐶(𝑡ℎ+1 , 𝑇)�ξ𝑡ℎ �. 𝑒 −𝑅𝑡 � max � conv (t + 1), bv(t + 1)��, if t ∉ (π ∪ φ) = � 𝑒 −𝑅𝑡 [max(𝑐𝑐𝑐𝑐(𝑡 + 1), 𝑏𝑏(𝑡 + 1), 𝑝𝑝(𝑡 + 1))], 𝑖𝑖 𝑡 𝜖 𝜋 𝑎𝑎𝑎 𝑡 ∉ 𝜑 𝑒 −𝑅𝑡 �max�min�𝑏𝑏(𝑡 + 1), 𝑐𝑐𝑐𝑐(𝑡 + 1)� , 𝑐𝑐𝑐𝑐(𝑡 + 1), 𝑝𝑝(𝑡 + 1)��, 𝑖𝑖 𝑡𝑡 (𝜋 ∩ 𝜑). where conv (t+1) is the conversion value at time t+1, bv (t+1) is the bond value at time t+1, pv (t+1) is the puttable value at time t+1, and calv (t+1) is the callable value at time t+1.. 3.3.3 Least Square Monte Carlo Method (LSMC) This method is proposed by Longstaff and Schwartz (2001). They use this method to value American option. American option can be executed before the maturity date, but we do not know whether we execute it earlier is better or not by traditional Monte Carlo Method. Least Square Monte Carlo Method can help us solve this problem. Since convertible bond is similar to American option, both of them can be executed before the maturity date. We can see the face value of bond as strike price, and see conversion value as stock price, then the convertible bond is almost the same as American option. However, convertible bonds usually have puttable and callable parts in the real situation, it is more complicated. In this case, we will compare the conversion value with puttable value, and find the bigger one to replace the role of stock price in American option. If the callable conditions are satisfied, we need to compare with conversion value, puttable value and callable value, and find the maximum value to replace the role of stock price in American option. First, we simulate a lot of stock paths, and find out a set which contains those in-the-money paths. Second, do the regression as follows: 𝐸𝑄 �. 𝑡. ℎ 𝑒 ∫0 𝑅𝑥 𝑑𝑑. 𝑡ℎ+1 𝑅𝑥 𝑑𝑑 𝑒 ∫0. 𝐶𝐶(𝑡ℎ+1 , 𝑇)�ξ𝑡ℎ � = αℎ + 𝛽ℎ 𝑆̃(𝑡ℎ ) + 𝛾ℎ 𝑆̃ (𝑡ℎ )2. Compare the convertision value and holding value, then we will know which strategy is better. We repeat these steps again and again, we will find all best strategies at every time. Finally, discount these values under the best strategies and average them, we will find the value of convertible bond. Note that if the puttable conditions or callable conditions are satisfied, both holding value and conversion value need to compare with them, too. 27.

(33) 4. Empirical Analysis. 4.1 Data Description (1) The contract of convertible bonds We choose two convertible bonds to do the empirical analysis. One is YES Logistics Corporation (2609), the other one is China Airlines (2610). We can see more information in the convertible bonds of these two companies as follows: Table 4-1. The Contract of YES Logistics Corporation Convertible Bond YES Logistics Corporation (26094) Items. Content. Issue Date. 2013/6/7. Face Value. NT$100,000. Initial Stock Price. 12.25. Conversion Price. 14.23. Volatility. 5.73%. Coupon Rate. 0%. Dividend Rate. 0%. Maturity Time. 5 years. Resetting Price. No. Credit rating. twBBB. Recover Rate. 0%. Puttable Value. Face value at third year. Callable Value. Face value Condition: the stock price is 1.3 times higher than the initial stock price 28.

(34) Table 4-2. The Contract of China Airlines Convertible Bond China Airlines (26105) Items. Content. Issue Date. 2013/12/26. Face Value. NT$100,000. Initial Stock Price. 10.7. Conversion Price. 12.24. Volatility. 4%. Coupon Rate. 0%. Dividend Rate. 0%. Maturity Time. 5 years. Resetting Price. No. Credit rating. twBBB. Recover Rate. 0. Puttable Value. 100.75% of face value at third year. Callable Value. Face value Condition: the stock price is 1.3 times higher than the initial stock price. (2) Credit risk According to equation (9), we need to find initial λ0 first. In 3.1.1, we know that we can use the spread of CDS or the different interest rate between corporation bond and government bond. Since sometimes we do not have the interest rate of the corporation bond, we can use the interest rate of the credit rating instead. For example, if the ranking of the company is twAAA, then we can use the interest rate of twAAA instead of the interest rate of the corporation bond. The credit rating of both YES Logistics Corporation and China Airlines are 29.

(35) twBBB, which has a 1-year interest rate of 1.469 % at pricing date (2014/4/30). At the same time, the interest rate of 1-year government bond is 0.11%. Clearly, the initial λ0 is λ0 =. =. 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟 𝑜𝑜 𝑡𝑡𝑡𝑡𝑡−𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟 𝑜𝑜 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑏𝑏𝑏𝑏 𝐿𝐿𝐿𝐿 𝑅𝑅𝑅𝑅. 1.469%−0.11% 1. = 1.458%. (3) Liquidity Risk a. Volume Method According to 3.1.2, we need to use the interest rate of 10-year government bond and 1-year government bond to construct the liquidity table. The interest rate of the 10-year government bond is 1.4845%, and the interest rate of the 1-year government bond is 0.11%. According to equation (12), we can find: d* =. =. 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟10 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟𝑟𝑟01 𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜 𝑔𝑔𝑔𝑔𝑔𝑔. 1.4845%−0.11% 9. = 0.001549 Then we can construct the liquidity factor table as follows: Table 4-3. Liquidity Factor Table Weight (Wi). Liquidity Factor. 0.02↓. 0.013945. 0.02~0.04. 0.012396. 0.04~0.06. 0.010846. 0.06~0.08. 0.009297. 008~0.1. 0.007747. 0.1~0.12. 0.006198 30.

(36) 0.12~0.14. 0.004648. 0.14~0.16. 0.003099. 0.16↑. 0.001549. Since WYES = 1.0023% → LIQYES = 0.013945 WCA = 4.0713% → LIQCA = 0.010846. b. Bid-ask Spread Method According to equation (13), we can find the different Liquidity for YES Logistics Corporation and China Airline. LIQYES =. LIQCA =. 𝑠𝑎𝑠𝑟𝑠𝑇𝑠 𝑛𝑇𝑑−𝑠𝜆𝑎 𝜆𝑠𝑟𝑠𝑠𝑑. ≈ 0.006. 𝑠𝑎𝑠𝑟𝑠𝑇𝑠 𝑛𝑇𝑑−𝑠𝜆𝑎 𝜆𝑠𝑟𝑠𝑠𝑑. ≈ 0.002. 𝑠𝑎𝑠𝑟𝑠𝑇𝑠 𝑐𝑉𝑙𝑎𝑠𝑟𝑡𝑇𝑛𝑙𝑠 𝑛𝑉𝑙𝑑 𝑠𝑟𝑇𝑐𝑠 𝑠𝑎𝑠𝑟𝑠𝑇𝑠 𝑐𝑉𝑙𝑎𝑠𝑟𝑡𝑇𝑛𝑙𝑠 𝑛𝑉𝑙𝑑 𝑠𝑟𝑇𝑐𝑠. 4.2 Empirical Result In this part, we simulate stock prices 10,000 times to price the convertible bonds of these two companies by using forward method, backward method and LSMC method. We use static analysis to find the appropriate α in these models. According to equation (9), finding the appropriate α means we can complete the dynamic default intensity process. The discount factor Rt ≅ r + λtLt + LIQ, where r is the risk-free interest rate, λtLt is default intensity multiplier loss rate, and LIQ is liquidity factor. Clearly, we find the discount factor for all time. According to equation (21), we complete our pricing model. Now we do the calibration with the price of convertible bond in the market to find the appropriate α. First, we price the convertible bond with the liquidity factor by volume method. We can see the result as follows:. 31.

(37) a. Volume Method Table 4-4 The convertible bond price of YES Logistics Corporation YES Logistics Corporation α. -2.5. -3. -3.5. -4. -5. -6. -7. -8. Real Price. 101700. 101700 101700 101700 101700 101700 101700 101700. Forward Method. 102070. 101570 101220 100340 98757. 96942. 96598. 95522. Standard Deviation. 356. 324. 328. 343. 354. 347. 352. 323. Percentage Error. 0.36. 0.13. 0.47. 1.34. 2.89. 4.68. 5.02. 6.07. Backward Method. 117610. 117430 116860 116740 116590 115790 117490 116230. Standard Deviation. 341. 331. 357. 346. 361. 353. 325. 330. Percentage Error. 15.64. 15.47. 14.91. 14.79. 14.64. 13.85. 15.53. 14.29. LSMC Method. 112700. 112120 111500 110660 107650 107620 107340 107270. Standard Deviation. 352. 342. 336. 337. 325. 321. 311. 291. Percentage Error. 10.82. 10.25. 9.64. 8.81. 5.85. 5.82. 5.55. 5.48. item. . . . α. -20. -21. -22. -23. -24. -25. Real Price. 101700. 101700. 101700. 101700. 101700. 101700. Forward Method. 92134. 91476. 90699. 89563. 89466. 88428. Standard Deviation. 317. 331. 307. 347. 323. 398. Percentage Error. 9.41. 10.05. 10.82. 11.93. 12.03. 13.05. Backward Method. 117900. 118580. 118210. 116550. 116010. 118780. item. 32.

(38) Standard Deviation. 363. 336. 315. 303. 372. 326. Percentage Error. 9.406. 10.053. 10.817. 11.934. 12.029. 13.050. LSMC Method. 103350. 110210. 102730. 101900. 108860. 101494. Standard Deviation. 311. 343. 342. 322. 384. 325. Percentage Error. 1.62. 8.37. 1.01. 0.20. 7.04. 0.20. In table 4-4, we find that the most appropriate α is -3 with forward method, the most appropriate α is -23 with LSMC method, and there is no appropriate α with backward method.. Table 4-5 The convertible bond price of China Airline China Airline α. -2.5. -3. -3.5. -4. -5. -6. -7. -8. item Real Price. 100200 100200 100200 100200 100200 100200 100200 100200. Forward. 104330 103590 103270 102810 101970 100850 100200 99106. Standard Deviation. 326. 334. 323. 329. 340. 360. 376. 349. Percentage Error. 4.12. 3.38. 3.06. 2.60. 1.77. 0.65. 0.00. 1.09. Backward. 119890 119940 119670 119500 118170 118520 118040 117430. Standard Deviation. 321. 335. 328. 337. 348. 339. 382. 338. Percentage Error. 19.65. 19.70. 19.43. 19.26. 17.93. 18.28. 17.80. 17.20. LSMC. 120230 119280 117420 109710 110050 108150 109020 108590. Standard Deviation. 342. 323. 320. 335. 336. 307. 317. 315. Percentage Error. 19.99. 19.04. 17.19. 9.49. 9.83. 7.93. 8.80. 8.37. . . . 33.

(39) α. -20. -21. -22. -23. -24. -25. Real Price. 100200. 100200. 100200. 100200. 100200. 100200. Forward. 90678. 90132. 89476. 89043. 88755. 88588. Standard Deviation. 417. 398. 406. 379. 378. 363. Percentage Error. 9.50. 10.05. 10.70. 11.13. 11.42. 11.59. Backward. 118310. 116010. 116340. 115430. 121150. 116420. Standard Deviation. 373. 273. 306. 280. 378. 370. Percentage Error. 18.07. 15.78. 16.11. 15.20. 20.91. 16.19. LSMC. 107070. 106900. 109240. 105020. 108910. 110690. Standard Deviation. 263. 332. 299. 242. 301. 340. Percentage Error. 6.86. 6.69. 9.02. 4.81. 8.69. 10.47. item. In table 4-5, we find that the most appropriate α is -7 with forward method, the most appropriate α is -23 with LSMC method, and there is no appropriate α with backward method.. b. Bid-ask Spread Method Second, we price the convertible bond with liquidity factor by bid-ask spread method. Table 4-6 The convertible bond price of YES Logistics Corporation YES Logistics Corporation α. -2. -3. -3.5. -3.75. -4. -5. -6. Real Price. 101700. 101700 101700 101700 101700 101700 101700. Forward. 103550. 103020 102370 101500 99902. item. 34. 99381. 97134.

(40) Standard Deviation. 391. 381. 462. 386. 427. 438. 445. Percentage Error. 1.82. 1.30. 0.66. 0.20. 1.77. 2.28. 4.49. Backward. 117760. 117170 117140 115570 116630 114960 115350. Standard Deviation. 377. 421. 354. 378. 325. 367. 394. Percentage Error. 15.79. 15.21. 15.18. 13.64. 14.68. 13.04. 13.42. LSMC. 107960. 109810 106060 106570 105380 104370 105760. Standard Deviation. 340. 354. 316. 293. 324. 299. 263. Percentage Error. 6.16. 7.97. 4.29. 4.79. 3.62. 2.63. 3.99. -6.5. -6.75. -7. -8. Real Price. 101700. 101700. 101700. 101700. Forward. 95432. 95310. 94505. 94429. Standard Deviation. 473. 422. 412. 382. Percentage Error. 6.16. 6.28. 7.07. 7.15. Backward. 114500. 116710. 115810. 115460. Standard Deviation. 377. 356. 401. 376. Percentage Error. 12.59. 14.76. 13.87. 13.53. LSMC. 106730. 114250. 105830. 107010. Standard Deviation. 294. 248. 309. 283. Percentage Error. 4.95. 12.34. 4.06. 5.22. α item. In table 4-6, we find that the most appropriate α is -3.75 with forward method, the most appropriate α is -5 with LSMC method, and there is no appropriate α with backward method.. 35.

(41) Table 4-7 The convertible bond price of China Airline China Airline α. -2. -3. -3.5. -3.75. -4. -5. -6. Real Price. 100200. 100200 100200 100200 100200 100200 100200. Forward. 106130. 106580 105010 104730 104440 102130 101630. Standard Deviation. 308. 321. 355. 325. 331. 365. 356. Percentage Error. 5.92. 6.37. 4.80. 4.52. 4.23. 1.93. 1.43. Backward. 119430. 119540 119030 119500 119520 118580 117720. Standard Deviation. 301. 318. 287. 300. 335. 325. 317. Percentage Error. 19.19. 19.30. 18.79. 19.26. 19.28. 18.34. 17.49. LSMC. 107830. 106370 105080 106030 104480 103990 107370. Standard Deviation. 319. 297. 298. 297. 279. 278. 330. Percentage Error. 7.61. 6.16. 4.87. 5.82. 4.27. 3.78. 7.16. -6.5. -6.75. -7. -8. Real Price. 100200. 100200. 100200. 100200. Forward. 101440. 100580. 99734. 99013. Standard Deviation. 390. 377. 388. 371. Percentage Error. 1.24. 0.38. 0.47. 1.18. Backward. 118130. 118300. 117210. 117640. Standard Deviation. 306. 344. 332. 381. Percentage Error. 17.89. 18.06. 16.98. 17.41. LSMC. 105250. 104390. 106080. 108220. item. α item. 36.

(42) Standard Deviation. 285. 250. 289. 332. Percentage Error. 5.04. 4.18. 5.87. 8.00. In table 4-7, we find that the most appropriate α is -6.75 with forward method, the most appropriate α is -5 with LSMC method, and there is no appropriate α with backward method.. 37.

(43) 5. Conclusion We considered the credit risk and liquidity risk in this thesis. First, we used the reduced-form model to describe the credit risk. We constructed the dynamic default intensity process by setting the function which is inverse to the stock price to estimate the credit risk. Second, we used two methods to estimate the liquidity risk. One is volume method, and the other one is bid-ask spread method. In volume method, we constructed the liquidity factor table by separating the different volumes of the convertible bond into different levels to estimate the liquidity risk. In bid-ask spread method, we used the average bid-ask spread over the average convertible bond price to estimate the liquidity risk. According to the results in Part 4, we found that under the same parameter, the prices of convertible bonds estimated by the backward method are the highest, and the prices of convertible bonds estimated by the forward method are the lowest. There were two methods used to estimate the liquidity risk in this thesis. For the volume method, with LSM method, we found that the most appropriate α = -23 for both underlying assets. The percentage error is 0.2 for YES Logistics Corporation and 4.81 for China Airline. In the forward method, we found that the most appropriate α = -3 for the YES Logistics Corporation convertible bond with a percentage error of 0.13, and the most appropriate α = -7 for the China Airline convertible bond with a percentage error of 0. Since the function of the price of a convertible bond looks like a decreasing function under the forward method when α is on the interval [-30, -2], we found an appropriate α for each convertible bond. There was no appropriate α found using the backward method. For the bid-ask spread method, with LSM method, we found that the most appropriate α = -5 for both underlying assets. The percentage error is 2.63 for YES Logistics Corporation and 3.78 for China Airline. Using forward method, we found that the most appropriate α = -3.75 for the YES Logistics Corporation convertible bond with a percentage error of 0.2, and the most appropriate α = -6.75 for the China Airline convertible bond with a percentage error of 0.38. Since the function of the price of convertible bonds looks like a decreasing function using forward method when α is on the interval [-8, -2], we found an appropriate α for each convertible bond. There was no appropriate α found using the backward method. The main problem with using the backward method is that the highest value must be chosen as the holding value, which causes the estimate price to always be higher than the real price. 38.

(44) In conclusion, for these two methods to estimate the liquidity risk, we can find the specific α by using LSMC method, and different α by using forward method. Clearly, we can price any convertible bond with the model in this thesis by using LSMC method directly. With forward method, we need to find the appropriate α for each convertible bond first.. 39.

(45) 6. Reference [1] 涂宗旻(2010), “考慮信用風險及利率風險下之可轉債評價”, 碩士論文, 國立政治大學金融研究所 [2] Takahashi, A., Kobayashi T. and Nakagawa, N. (2001), “Pricing convertible bond with default risk”, Journal of Fixed Income [3] Ammann, M., Kind, A. and Wilde C. (2003), “Are convertible bonds under priced? An analysis of the French market”, Journal of Banking and Finance 27, 635-653 [4] Black, F. and Scholes, M. (1973), “The Price of Option and Corporate Liabilities”, Journal of Political Economy 81, 637-659 [5] Brennan, M. J. and Schwartz, E. S. (1977), “Convertible bonds: valuation and optimal strategies for call and conversion”, Journal of Finance, 32, 5, 1699-1715 [6] Brennan, M. J., and Schwartz, E. S. (1980). “Analyzing convertible bonds”, Journal of Financial and Quantitative analysis, 15, 907-993 [7] Brennan, M. J. and Schwartz, E. S. (1988), “The case for convertibles”, Journal of Applied Corporate Finance [8] Derman, E. (1994), “ Valuing convertible bonds as derivatives”, Technical Report, Goldman Sachs [9] Davis, M. and Lischka, F. R. (1999), “ Convertible bonds with market risk and credit risk”, Technical Report, Tokyo-Mitsubishi International PLC [10] Chambers, D. R. and Lu, Q. (2007), “A Tree Model for Pricing Convertible Bonds with Equity, Interest Rate, and Default Risk.” Journal of Derivatives, 14, 4, 25-4 [11] Hang, M.W, and Wang J.Y. (2002), “Pricing convertible bonds subject to default risk”, Journal of Derivative, 10 (2) [12] Ingersoll, J. (1977), “A contingent-claims valuation of convertible securities”, Journal of Financial Economics, 4, 289-322 [13] McConnell, J. J. and Schwartz, E. S. (1986), “ LYON Taming”, The Journal of Finance, Volume XLI, No.3, July. 40.

(46) [14] Longstaff. F and Schwartz, E. (2001), “Valuing American Option by Simulation: A Simple Least Square Approach”, Review of Financial Studies, 14, Spring 2001, 113-147 [15] Merton, R.C. (1974), “On the pricing of corporate debt: The risk structure of interest rates”, Journal of Finance, 29 (2), 449-470 [16] Masaakikijima and Muromachi, Y. (2000), “Credit Events and the Valuation of Credit”, Review of Derivatives Research, 4, 55-79 [17] Ammann, M., Kind, A. and Wilde, C. (2008), “Simulation-Based Pricing of Convertible Bonds”, Journal of Empirical Finance, 15, 310-331 [18] Panayids, P. M., Lambertides, N. and Cullinance, K. (2013), “Liquidity risk premium and asset pricing in US water transportation”, Transportation Research, Part E, 52, 3-15 [19] Jarrow, R. A. and Turnbull, S. M. (1995), “Pricing Derivatives on Financial Securities Subject to Credit Risk”, The Journal of Finance, VOL L, No.1 [20] Jarrow, R. A., Lando, D., and Turnbull, S. M. (1997), “A Markov Model for the Term Structure of Credit Risk Spreads”, The Review of Financial Studies, V 10, NO.2 [21] Tsiveriotis, K. and Fernandes, C. (1998).”Valuing convertibles bonds with credit risk”, Journal of Fixed Income, 2, 95-102 [22] Amihud, Y. (2002), “Illiquidity and stock returns: cross-section and time- series eﬀects”, Journal of Financial Markets, 5, 31–56. 41.

(47)