Download Convertible Bonds Valuation based on Multiple

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 2t 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.
The empirical analysis shows that the theoretical values for the analyzed Chinese convertible
bonds are not always higher than the observed market prices, and this result is different from most
of the prior research. Another result is 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.
References
Ammann M., Kind A., Wilde C., 2003. Are convertible bonds under priced? Journal of Banking
and Finance 27(4), 549-792.
Ammann M., Kind A., Wilde C., 2008. Simulation-based pricing of convertible bonds, Journal of
Empirical Finance 15, 310-331.
Bardhan, I., Bergier, A., Derman, E., Dosemblet, C., Kani, I., Karasinski, P., 1993. Valuing
convertible bonds as derivatives. Goldman Sachs Quantitative Strategies Research Notes, July.
Brennan, M.J., Schwartz, E.S., 1977. Convertible bonds: Valuation and optimal strategies for call
and conversion. The Journal of Finance 32(5), 1699-1715.
Brennan, M.J., Schwartz, E.S., 1980. Analyzing convertible bonds. Journal of Financial and
Quantitative Analysis 15 (4), 907–929.
Buchan, M.J., 1998. The pricing of convertible bonds with stochastic term structures and
corporate default risk, working paper. Amos Tuck School of Business, Dartmouth College.
Carayannopoulos P., Kalimipalli Madhu, 2003. Convertible Bond Prices and Inherent Biases,
Journal of Fixed Income, 13(3):64-73.
Carayannopoulos, P., 1996. Valuing convertible bonds under the assumption of stochastic interest
rates: An empirical investigation. Quarterly Journal of Business and Economics 35 (3), 17–31.
Cheng-Few Lee , Kin-Wai Lee , 2009. Gillian Hian-Heng Yeo , Investor protection and
convertible debt design, Journal of Banking & Finance 33, 985–995.
Cox, J.C., Ross, S.A., Rubinstein, M., 1979. Option pricing: A simplified approach. Journal of
Financial Economics 7 (3), 229-263.
Craig M. Lewis, Patrick Verwijmeren , 2011. Convertible security design and contract innovation,
Journal of Corporate Finance 17, 809–831.
Davis M ， Lisehka F. Convertible Bonds with Market Risk and Credit Risk. American
Mathematical Society International Press，2002
García, D., 2003. Convergence and biases of Monte Carlo estimates of American option prices
using a parametric exercise rule. Journal of Economic Dynamics and Control 26, 1855–1879.
Ingersoll, J.E., 1977. A contingent claim valuation of convertible securities. Journal of Financial
Economics 4, 289–322.
King, R., 1986. Convertible bond valuation: An empirical test. Journal of Financial Research 9 (1),
53–69.
La Porta, R., Lopez-de-Silanes, F., Shleifer, A., Vishny, R., 2002. Investor protection and
corporate valuation. Journal of Finance 57, 1147–1170.
Lewis, M., Rogalski, R., Seward, J., 2003. Industry conditions, growth opportunities and market
reactions to convertible debt financing decisions. Journal of Banking and Finance 27, 153–181.
McConnell, J.J., Schwartz, E.S., LYON taming, 1986.The Journal of Finance 41 (3), 561–576.
Tsiveriotis, K., Fernandes, C., 1998. Valuing convertible bonds with credit risk. The Journal of
Fixed Income 8 (3), 95–102.
Zheng Zhenlong, Lin Hai, Valuation of Chinese convertible bonds, Journal of Xiamen
University(Arts & Social Sciences), 2004,2:93-99.
联系人: 梁朝晖
天津工业大学经济学院
电话：18602688195
邮件：[email protected]