3. 1
CONTENT
ABSTRACT...................................................................................................................2
中文摘要........................................................................................................................4
1 Introduction.................................................................................................................5
1.1 Background and Significance of Research .......................................................5
1.2 Research Framework and Content....................................................................6
1.3 Literature Summary ..........................................................................................7
2 Convertible Bond and Value Composition..................................................................8
2.1 Introduction on Convertible Bond ....................................................................8
2.2 Valuation of Convertible Bond .........................................................................9
3 Black-Scholes Model and Modification ...................................................................15
3.1 Black-Scholes Model and Application............................................................15
3.2 Modification on Black-Scholes Model ...........................................................18
4 Value Analysis Based on Black-Scholes Model .......................................................19
4.1 Data Selection .................................................................................................19
4.2 Parameter Estimation......................................................................................23
4.3 Empirical Study ..............................................................................................26
5 Conclusion ................................................................................................................37
Reference .....................................................................................................................38
4. 2
ABSTRACT
As one of financial derivatives, convertible bonds are quite complicated, which are
equipped with the features belonging to both bonds and options. Treated as bond,
convertible bonds is a vehicle for financing in the term of issuers and it is also an
investment tool to avoid risk in financial market for investors, because of the fact that
at the maturity, the bond will pay back principle and interest. However, it is quite
different from the ordinary bond, for, under certain circumstances, it can be transferred,
called, sold, etc. That is the reason to explain why the convertible bond is far more
complex than ordinary bond, and therefore, its pricing procedure is not the same with
ordinary corporate bond. Since year 2006, after convertible bonds are widely accepted
in United States since 1980s, China started to establish its own regulated financial
market for convertible bonds. In spite of that, history of convertible bonds are still so
short that convertible bonds cannot be recognized as a mature investment tools, mainly
because the mechanism is not understood deeply by investors. In order to get profound
understanding of this complicated financial derivatives, pricing is a good direction to
discover the core of convertible bonds.
Under such background, the paper is designed to address issue of pricing of
convertible bonds in Chinese market and conduct the empirical analyses based on
Black- Scholes model, which is very classic in the academic circle.
In the conclusion part of the paper, the results of empirical study is pointed out, and
based on that, I put forward several reasons that might explain the difference between
the market value and theory value.
7. 5
Empirical Study of Pricing of Convertible Bonds Based on
Black- Scholes Model
1 Introduction
1.1 Background and Significance of Research
The first convertible bond is issued by New York ERIE RAILWAY, an American
company in 1843. Afterwards, convertible bond is recognized by the investors
gradually due to its special features, that is, combination of bond and stock. The scale
of the market becomes larger and larger: in 2004, the global market for convertible bond
is approximate to 610 billion dollars with the scale of issuance close to 100 billion
dollar per year. Exposed to the financial crisis and euro debt crisis, the scale has shrink
a notch, but it still stay around at the level of 10 billion dollar.
The establishment of Chinese financial market is just twenty years or so, so that the
history of convertible bonds is very short. In the late 1992, China Baoan Corp. issued
convertible bond valued 0.5 billion Yuan to the investor in the society. It is the first
convertible bond whose issuer is listed in Shenzhen Stock Exchange. However, owing
to the incomplete regulation in the financial market, there are barely issuance in the 8
or 9 years afterwards; the whole market for convertible bond is in the statue of
stagnation. Until 2001, the convertible market witnessed a rapid development, as is
shown in the apparent increase in issuance scale and the number of issuance. During
two bull markets in 2006 and 2007 separately, one kind of convertible bond called
packaged convertible bond, a bond that bears two financing opportunities, appeared in
8. 6
the market, and it occupied the large percentage of the market suddenly. Unfortunately,
due to the adjustment in the financial market in 2008, the large percentage of packaged
convertible bond was not executed successfully, which leaded to the total amount of
asset of investor declined to zero, making the packaged convertible bond started to fade
away from 2009, but the situation also brought the new opportunity for the traditional
convertible bond. From 2010, with the large amount of issuance of convertible bond to
extent of several 10 billion dollar by China Bank, Sinopec, and ICBC, Chinese financial
market is largely extended. Until April, 2013, the whole scale of the convertible bond
market reach the level of more than 140 billion Yuan. The issuer is willing to finance
in the way of convertible bond, which provides a method to get capital in low cost.
Accordingly, the investors are enthusiastic about the convertible bond, for they can both
enjoy the certainty of the bond and harvest high rate of return of financial derivatives.
Therefore, these years have witnessed the great success in the financial market for
convertible bond.
Under the circumstances where the convertible bond market is expanding rapidly,
the problem has been raised that which the best way to price the convertible bond is.
Undoubtedly, this is what the investor mainly focuses on, because it relates to the
decision of investment and identification of risk level. What’s more, the pricing
problem is beneficial to the effectiveness of pricing model of our financial market in
China. The paper can be treated as a try in pricing of convertible bond in Chinese market.
1.2 Research Framework and Content
This paper is arranged in the following way. Frist of all, the paper gives brief
introduction of convertible bond and also the do analysis on its value components in
three parts. Next comes the other important component of this paper, Black-Scholes
model, including the background of model, basic assumption and of course the equation.
Also in order to get deeply understanding of Black-Scholes model, the paper explains
it in a qualitative way. Plus, in this chapter, it introduces the modification on the Black-
9. 7
Scholes model to make it more suitable and accurate to convertible bond valuation. Last
but not the least, in last section, the paper conducts the empirical research based on all
the theories and analyses above and analyze the results of theory value and market price.
1.3 Literature Summary
The earliest research on pricing convertible bond can be dated back to 1977, Ingersoll
(1977) disintegrates the zero- coupon convertible bond into three parts: bond, callable
part and convertible part, using Black-Scholes model to achieve the best trading
strategy for investor. However, their study is based on the value of the company, which
is hardly available in practice. Brennan and Schwartz (1977) also attained the solution
of partial differential equation based on valuation of the company using finite difference
method. Later on, McConnell and Schwartz (1986) attained the price of convertible
bond based on the price of stocks. However, their research is only suitable to convertible
bond with call provision, not to those with put provision and redressal provision.
Teiveriotis and Fernandes (1998) separate the value of convertible bond into two parts:
risk free equity and risky bond, producing a set of two partial difference equation, using
risk free rate and risk rate. The theory is therefore named as TF single factor model.
The method considers the interest rate as a fixed value, which is not realistic. Hence,
David and Lischka (1999) made some modification on the previous method---using
Vasicek model as the model for changeable future interest rate.
From 2000, various methods of pricing start to spring out. Takahashi et al (2001) and
Ammann et al (2003) applied binominal tree model and triple tree model, while
Bermudz and Webber (2003) used finite element method. Application of Monte Carlo
Simulation maturing in the field of option pricing, it is also used in the pricing of
convertible bond. Among all the researchers, Ammann (2005) et al did some
improvising on tradition Monte Carlo Simulation method, getting more precise result
of the price of convertible bond, which is quite innovative.
The domestic research started in relatively late years, among which stands out
10. 8
achievement from Zhenlong Zheng, Hai Lin (2004). They put up with several important
conclusion according to the current situation of Chinese convertible bond market and
used binominal tree model, finite difference method and Monte Carlo Simulation and
several other methods to price the 11 convertible bond in the market, drawing the
conclusion of severe undervaluation of convertible bond in Chinese market.
Yang Zhao, Lichen Zhao (2009) priced convertible bond via least square Monte
Carlo Simulation put forward by Longstaff (2001). Meanwhile, some classic models on
estimation of parameters are put into practice by them. And they successfully arrived
at the conclusion that the value of convertible bond is underestimated by 2% to 3%.
2 Convertible Bond and Value Composition
2.1 Introduction on Convertible Bond
In finance, a convertible bond or convertible note or convertible debt (or a
convertible debenture if it has a maturity of greater than 10 years) is a type of bond that
the holder can convert into a specified number of shares of common stock in the issuing
company or cash of equal value. It is a hybrid security with debt- and equity-like
features. It originated in the mid-19th century, and was used by early speculators such
as Jacob Little and Daniel Drew to counter market cornering1
.Convertible bonds are
most often issued by companies with a low credit rating and high growth potential.
To compensate for having additional value through the option to convert the bond to
stock, a convertible bond typically has a coupon rate lower than that of similar, non-
convertible debt. The investor receives the potential upside of conversion into equity
while protecting downside with cash flow from the coupon payments and the return of
1 In finance, to corner the market is to get sufficient control of a particular stock, commodity, or other asset to
allow the price to be manipulated. Another definition: "To have the greatest market share in a particular industry
without having a monopoly.
11. 9
principal upon maturity. These properties lead naturally to the idea of convertible
arbitrage, where a long position in the convertible bond is balanced by a short position
in the underlying equity.
From the issuer's perspective, the key benefit of raising money by selling convertible
bonds is a reduced cash interest payment. The advantage for companies of issuing
convertible bonds is that, if the bonds are converted to stocks, companies' debt vanishes.
However, in exchange for the benefit of reduced interest payments, the value of
shareholder's equity is reduced due to the stock dilution2
expected when bondholders
convert their bonds into new shares.
2.2 Valuation of Convertible Bond
When studying the valuation composition, we can simplify this complicated work
into a simple and perspicuous equation:
Value of Convertible Bond = Coupon Bond + Call Option + Put Provision- Call
Provision + Downredressal Provision3
Therefore, the analysis of convertible bond can be divided into three separate parts:
pure bond value, value of option and the value of special provision.
2.2.1 Value of Pure Debt
As mentioned before, when the convertible bond cannot be exercised, it is the equal
to the ordinary vanilla bond. And so is the value of convertible bond, which is also
called pure value. The pure bond value is equivalent to the present value of all the
expected fixed future cash flow before the maturity. The classic equation below is also
2 Stock dilution is an economic phenomenon resulting from the issue of additional common shares by a company.
This increase in the number of shares outstanding can result from a primary market offering (including an initial
public offering), employees exercising stock options, or by conversion of convertible bonds, preferred shares or
warrants into stock. This dilution can shift fundamental positions of the stock such as ownership percentage, voting
control, earnings per share, and the value of individual shares. A broader definition specifies dilution as any event
that reduces an investor's stock price below the initial purchase price.
3 All the components in the equation are introduced in the following.
12. 10
the valuation model used in the valuation of convertible bond.
B = ∑
𝐼𝑡
(1+𝑖) 𝑡
𝑛
𝑡=1 +
𝑃
(1+𝑖) 𝑛
（Equation1）
In the equation,
B stands for the value of ordinary corporate bond；
𝐼𝑡 stands for the interest rendered by bond annually4
；
𝑃 stands for the principle of bond / face value；
𝑖 stands for specific discount rate；
𝑛 stands for the time to maturity of the bond.
2.2.2 Value of Option
Once comes conversion term, convertible bonds are equipped with the value derived
from conversion function. Just as option, investors of convertible bond can shift from
the holding of bond to the purchasing of stocks at the price of their stock price. The
value of option are consisted of two parts: intrinsic value and time value.
Intrinsic value is defined as the difference between the market value of the underlying,
and the strike price of the given option. In detail, options can be separated into three
categories based on the difference of intrinsic value. First, in-the-money. For call option,
it means the spot price of underlying asset is larger than strike price; for put option, the
spot price is lower than strike price. Fortunately, profit can be made when it is in-the-
money. Second part is called out-of-money, which means for a call option the spot price
is lower than strike price. Under this circumstances, there is no need for investor to
exercise the option. By the same token, for a put option, spot price is higher compared
4 Convertible Bonds usually pay the interest annually.
13. 11
with strike price. Last one is named at-the-money. For both call and put, the spot price
is equal to strike price; there is no difference bet I en them.
Time value depends on a set of other factors which, through a multi-variable, non-
linear interrelationship, reflect the discounted expected value of that difference at
expiration. The reason why options have time value is that price can be violated during
the period to maturity, so that the possibility of making profit becomes bigger with the
longer lasting time before maturity. All in all time value of option represents
expectation of investor, which is derived from fluctuation of price of underlying asset.
The value of an option can be estimated using a variety of quantitative techniques
based on the concept of risk neutral5
pricing and using stochastic calculus. More
sophisticated models are used to model the volatility smile. These models are
implemented using a variety of numerical techniques. In general, standard option
valuation models depend on the following factors: the current market price of the
underlying security; the strike price of the option, particularly in relation to the current
market price of the underlying (in the money vs. out of the money); the cost of
holding a position in the underlying security, including interest and dividends; the
time to expiration together with any restrictions on when exercise may occur; an
estimate of the future volatility of the underlying security's price over the life of the
option. More advanced models can require additional factors, such as an estimate of
how volatility changes over time and for various underlying price levels, or the
dynamics of stochastic interest rates.
As the title indicated, the model we relied on to price convertible bond is Black-
Scholes model, which is will be introduced in detail in the following section. Here,
the paper gives a brief introduction of some other principal valuation techniques used
in practice to evaluate option contracts.
5 In economics and finance, risk neutral preferences are neither risk averse nor risk seeking. A risk neutral party's
decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indifferent
between choices with equal expected payoffs even if one choice is riskier.
14. 12
Stochastic volatility models: Since the market crash of 1987, it has been observed
that market implied volatility for options of lo I r strike prices are typically higher
than for higher strike prices, suggesting that volatility is stochastic, varying both for
time and for the price level of the underlying security. Stochastic volatility models
have been developed including one developed by S.L. Heston. One principal
advantage of the Heston model is that it can be solved in closed-form, while other
stochastic volatility models require complex numerical methods.
Binomial tree pricing model: Closely following the derivation of Black and
Scholes, John Cox, Stephen Ross and Mark Rubinstein developed the original version
of the binomial options pricing model. It models the dynamics of the option's
theoretical value for discrete time intervals over the option's life. The model starts
with a binomial tree of discrete future possible underlying stock prices. By
constructing a riskless portfolio of an option and stock (as in the Black–Scholes
model) a simple formula can be used to find the option price at each node in the tree.
This value can approximate the theoretical value produced by Black Scholes, to the
desired degree of precision. However, the binomial model is considered more
accurate than Black–Scholes because it is more flexible; e.g., discrete future dividend
payments can be modeled correctly at the proper forward time steps, and American
options can be modeled as well as European ones. Binomial models are widely used
by professional option traders. The Trinomial tree is a similar model, allowing for an
up, down or stable path; although considered more accurate, particularly when fewer
time-steps are modelled, it is less commonly used as its implementation is more
complex.
Monte Carlo models: For many classes of options, traditional valuation techniques
are intractable because of the complexity of the instrument. In these cases, a Monte
Carlo approach may often be useful. Rather than attempt to solve the differential
equations of motion that describe the option's value in relation to the underlying
security's price, a Monte Carlo model uses simulation to generate random price paths
15. 13
of the underlying asset, each of which results in a payoff for the option. The average of
these payoffs can be discounted to yield an expectation value for the option.[25] Note
though, that despite its flexibility, using simulation for American styled options is
somewhat more complex than for lattice based models.
Finite difference models: The equations used to model the option are often expressed
as partial differential equations (see for example Black–Scholes equation). Once
expressed in this form, a finite difference model can be derived, and the valuation
obtained. A number of implementations of finite difference methods exist for option
valuation, including: explicit finite difference, implicit finite difference and the Crank-
Nicholson method. A trinomial tree option pricing model can be shown to be a
simplified application of the explicit finite difference method. Although the finite
difference approach is mathematically sophisticated, it is particularly useful where
changes are assumed over time in model inputs – for example dividend yield, risk free
rate, or volatility, or some combination of these – that are not tractable in closed form.
2.2.3 Value of Special Provision
In the contract of convertible bond, lots of treaties and items are regulated. However,
there are three agreements are too essential to be analyzed: call provision, put provision
and downredressal provision.
(1) Call Provision
Call provision is a right belonging to issuers that when the price of convertible bond
exceed the call price, aka redemption price, the company who issue the convertible
bond can purchase back those convertible bond. That way, the value of convertible bond
is decreased, but the integrated value of company is not influenced. Subsequently, the
rights and benefit of stock holder of the company is improved, yet the rights of investors
is harmed to a certain degree. Therefore, once the price of convertible bond is higher
than redemption price, issuers have the motivation to redeem the bond; nevertheless,
when the bond price is lower than call price, issuer would never execute the action. To
16. 14
investors, under the condition that issuers are calling the bond, they can decide whether
or not to be called or convert the bond based on the comparison between call price and
conversion price.
(2) Put Provision
Put provision grant the investors a right to choose whether to hold the convertible
bond continually or to sell back the convertible bond to the company as the regulated
price, put price. The intention of bond holder is to maximum the value of their
investment: if the bond price is higher that put price, investor would give up the put
price or just suspending the use of that right temporarily; on the contrary, if the bond
price is lower than put price, the holder of convertible bond can arbitrage through
buying convertible bond and execute the put action.
(3) Downredressal Provision
Downredressal provision, aka conversion adjustment provision, defines that during
the conversion term, if the performance of the underlying stock is not satisfied enough
for the investor to exercise the option, the company who issue the convertible bond
have the right to adjust the conversion price to a lower level, which is always defined
as a certain ratio of previous conversion price. The use of downredressal provision is
triggered when certain condition is met, which always regulate that the time period and
the percentage by which the spot price of stock is lower than conversion price. Also,
the range of adjustment on price is strictly regulated. Downredressal provision not only
protect the rights and interests of investors but also have a shield on issuers, because
without downredressal provision, investors will indirectly deteriorate the financial
condition of company via selling back the convertible, that is executing put provision.
In the end of this section, there are some points I need to clarify here in order to
continue our discussion. First of all, when the situation happens such as distribution of
dividend, increase of stock capital, issue of additional new stocks and distribution of
cash dividend, the conversion price will also be changed accordingly because of the
17. 15
change in spot price of common stock. However, the change in the conversion price is
negligible in value of convertible bond, so that we do not take into consideration those
situations above. Secondly, in the previous study, it is proved that for the sake of the
protection of company’s own rights and interests, the issuer would not adjust the
conversion price unless they face the pressure of investor’s selling back the convertible
bond. Otherwise, although the condition of downredressal provision is satisfied, issuer
will not make adjustment on the conversion price. Thirdly, obviously, it is not that
appropriate to use Black-Scholes model to price the convertible bond because of the
neglect of those special provision, yet it still can be meaningful reference due to the fact
that the value underlying call provision, put provision and downredressal provision is
small, making no big difference in valuation of convertible bond despite neglect.
3 Black-Scholes Model and Modification
The valuation of pure bond have been introduced previously. Compared with that,
the method of option valuation is more complicated, so that the paper introduces it in
independent chapter.
3.1 Black-Scholes Model and Application
The Black-Scholes or Black-Scholes-Merton model is a mathematical model of a
financial market containing derivative investment instruments. From the model, one
can deduce the Black-Scholes formula, which gives a theoretical estimate of the price
of European-style options. The formula led to a boom in options trading and
legitimized scientifically the activities of the Chicago Board Options Exchange and
other options markets around the world. The model is widely used, although often
with adjustments and corrections, by options market participants. Many empirical
18. 16
tests have shown that the Black-Scholes price is "fairly close" to the observed prices,
although there are well-known discrepancies such as the "option smile".
The Black-Scholes model was first published by Fischer Black and Myron Scholes
in their paper in 1973, "The Pricing of Options and Corporate Liabilities", published
in the Journal of Political Economy. They derived a partial differential equation, now
called the Black-Scholes equation, which estimates the price of the option over time.
The key idea behind the model is to hedge the option by buying and selling the
underlying asset in just the right way and, as a consequence, to eliminate risk. This
type of hedging is called delta hedging and is the basis of more complicated hedging
strategies such as those engaged in by investment banks and hedge funds.
Robert C. Merton was the first to publish a paper expanding the mathematical
understanding of the options pricing model, and coined the term "Black-Scholes
options pricing model". Merton and Scholes received the 1997 Nobel Memorial Prize
in Economic Sciences for their work. Though ineligible for the prize because of his
death in 1995, Black was mentioned as a contributor by the Swedish Academy.
The model's assumptions have been relaxed and generalized in many directions,
leading to a plethora of models that are currently used in derivative pricing and risk
management. It is the insights of the model, as exemplified in the Black-Scholes
formula, that are frequently used by market participants, as distinguished from the
actual prices. These insights include no-arbitrage bounds and risk-neutral pricing. The
Black-Scholes equation, a partial differential equation that governs the price of the
option, is also important as it enables pricing when an explicit formula is not possible.
The Black-Scholes model assumes that the market consists of at least one risky
asset, usually called the stock, and one riskless asset, usually called the money
market, cash, or bond. The model makes assumptions on the assets:
(1) The rate of return on the riskless asset is constant and thus called the risk-free
interest rate.
19. 17
(2) The instantaneous log return of stock price is an infinitesimal random walk with
drift; more precisely, it is a geometric Brownian motion, and it is assumed that
its drift and volatility is constant (if they are time-varying, it is can be deduced a
suitably modified Black-Scholes formula quite simply, as long as the volatility
is not random).
(3) The stock does not pay a dividend.
There are also some assumptions on the market:
(1) There is no arbitrage opportunity (i.e., there is no way to make a riskless profit.)
(2) It is possible to borrow and lend any amount, even fractional, of cash at the
riskless rate.
(3) It is possible to buy and sell any amount, even fractional, of the stock (this
includes short selling).
(4) The above transactions do not incur any fees or costs (i.e., frictionless market).
Here comes the equation itself:
BSCall = S ∗ N(𝑑1) − 𝑋 ∗ N(𝑑2) ∗ 𝑒−𝑟𝑡
（Equation2）
and，
𝑑1 =
1
σ√ 𝑡
∗ [ln (
𝑆
𝑋
) + (𝑟 +
σ2
2
) ∗ 𝑡]
𝑑2 = 𝑑1 − σ ∗ √ 𝑡
Notation:
BSCall stands for the price of call option；
20. 18
S stands for the spot price of the stock underlying the convertiblebond；
𝑋 stands for executive price, that is, conversion price；
𝑟 stands for risk free rate；
𝑡 stands for time to maturity ( year)；
σ stands for the volatility of annualized return rate of stock ；
N( ) stands for the standard normal cumulative probability distribution function
of the variable.
At the first sight, equation seems very complicated. That’s why the paper would give
some qualitative understanding of the model. Suppose at the maturity day, the spot price
is S, so that the price of option is S − X. If we try to know the price of the option before
maturity, we need to extrapolate that what the possibility is for spot price to be S, which
is N(𝑑1). Also, I need to discount the exercise price to the value in the time 𝑡 at the
discount rate of N(𝑑2) ∗ 𝑒−𝑟𝑡
.
3.2 Modification on Black-Scholes Model
If I give the Black- Scholes model a second thought when applying it to the valuation
of convertible bond, I will find that the model can be more precise if taking equity
dilution into consideration. After the conversion, because of the conversion price is not
equal to spot price, the stock price actually changed, which is more obvious when
studying packaged convertible bond, which means that the bond and option can be
separately traded in the market by investors. Therefore, I need to make some adjustment
to the stock price.
Suppose after conversion, the stick price is 𝑆`
, so
𝑆`
=
𝑁∗S+𝜆∗𝑀∗𝑋
𝑁+𝜆∗𝑀
（Equation 3）
21. 19
And,
𝑁 stands for the total amount of stock；
S stands for the stock price before the conversion；
𝜆 stands for debt to equity transformation ratio；
𝑀 stands for the amount of amount of convertible bond.
Therefore, the price of the call option after the modification is,
BSCall =
𝑁
𝑁+𝜆∗𝑀
∗ [
𝑁∗S+𝜆∗𝑀∗𝑋
𝑁+𝜆∗𝑀
∗ N(𝑑1) − 𝑋 ∗ N(𝑑2) ∗ 𝑒−𝑟𝑡
]（Equation 4）
Pay attention to the factor beyond square brackets. Although the stock price is
adjusted due to conversion, the option price should be restored to the level before the
conversion. That’s why I use a factor to make adjustment, adjusting the option price to
the statues prior to stock dilution.
4 Value Analysis Based on Black-Scholes Model
4.1 Data Selection
Before the demonstration, the trading day, February 26, 2016, should be mentioned.
As you can see in the whole passage, all the information is updated to that trading day.
Secondly, because of the quite limited number of the packaged convertible bonds, the
passage mainly focuses on unpackaged convertible bond, but for convenience, I still
call it convertible bond.
Until February 26, 2016, the number of convertible bonds traded on the market is 11,
22. 20
including Gree CB (110030), Dianqi CB (1130080), GoerTek CB (128009), Sanyi CB
(110032), Guomao CB (110033), Jiuzhou CB (110034), Baiyun CB (110035),
GuanqiCB (113009), Lanbiao CB(123001), Shunchang CB (128010), and Hangxin CB
(110031). Hangxin CB is suspended from trading because of general meeting of stock
holder; therefore, it is not included in the range of study. Among all the other
10convertible bonds, Gree CB, Dianqi CB, and GoerTek CB have already enter the
conversion period, but the other 7 have not yet. The following charts (Table 1) gives
summary of information in the contracts of those 10 convertible bonds to illustrate the
condition, date and statues of special provision including put provision, call provision
and redressal provision, by that I can specify the premise that the neglect of those
special provision in the study of the value of those convertible bonds is appropriate in
the following research. And it is clear that all the special treaties is not triggered.
Apart from that, I also use the table (Table 2) to make clear all the important basic
elements of the bonds, including code, name, term of the bond, conversion price,
conversion start date, conversion end data and also the coupon rate for every year with
the term of bond.
Table 1 Summary of Special Provision (Cont.)
Code Name
Call Provision
Condition Date Statues
110030 Gree CB 30/30,70% 25-Dec-16 Untriggered
113008 Dianqi CB 30/30,70% 4-Jan-20 Untriggered
128009 GoerTek CB 30/30,70% 5-Jan-20 Untriggered
110032 Sanyi CB 30/30,70% 15-Jan-20 Untriggered
110033 Guomao CB 30/30,70% 26-Feb-19 Untriggered
110034 Jiuzhou CB 30/30,70% 2-Feb-19 Untriggered
110035 Baiyun CB 30/30,70% 22-Jan-20 Untriggered
23. 21
113009 Guangqi CB 30/30,70% 18-Dec-19 Untriggered
123001 Lanbiao CB 30/30,70% 12-Dec-18 Untriggered
128010 Shunchang CB 30/30,70% 22-Jan-20 Untriggered
Data Source: Wind Information
Table 1 Summary of Special Provision (Cont.)
Code Name
Put Provision
Condition Date Statues
110030 Gree CB 15/30,130% 30-Jun-15 Untriggered
113008 Dianqi CB 15/30,130% 4-Jul-16 Untriggered
128009 GoerTek CB 15/30,130% 5-Jul-16 Untriggered
110032 Sanyi CB 20/30,130% 21-Jul-16 Untriggered
110033 Guomao CB 15/30,130% 5-Sep-16 Untriggered
110034 Jiuzhou CB 15/30,130% 3-Aug-15 Untriggered
110035 Baiyun CB 15/30,130% 22-Jul-16 Untriggered
113009 Guangqi CB 15/30,130% 27-Jun-16 Untriggered
123001 Lanbiao CB 15/30,130% 19-Jun-15 Untriggered
128010 Shunchang CB 15/30,130% 29-Jul-16 Untriggered
Data Source: Wind Information
Table 1 Summary of Special Provision
Code Name
Redressal Provision
Condition Date Statues
110030 Gree CB 10/20,90% 25-Dec-14 Untriggered
25. 23
Table 2 Basic Information of 10 Convertible Bond
Code Name
Coupon Rate (%)
1yr 2yr 3yr 4yr 5yr 6yr
110030 Gree CB 0.6 0.8 1 1.5 2
113008 Dianqi CB 0.2 0.5 1 1.5 1.5 1.6
128009 GoerTek CB 0.5 0.7 1 1.6 1.6 1.6
110032 Sanyi CB 0.2 0.5 1 1.5 1.6 2
110033 GuomaoCB 0.3 0.5 0.9 1.4 1.7 2
110034 Jiuzhou CB 0.2 0.4 0.6 0.8 1.6 2
110035 Baiyun CB 0.2 0.4 1 1.2 1.5
113009 Guangqi CB 0.2 0.5 1 1.5 1.5 1.6
123001 Lanbiao CB 0.5 0.7 1 1.5 1.8 2
128010 Shunchang CB 0.5 0.7 1 1.6 1.6 1.6
Data Source: Wind Information
4.2 Parameter Estimation
4.2.1 Time to Maturity
Generally, the calculation of the time to maturity have two ways. First, count all the
trading days prior to the maturity; then divide all the working days by 252, which is
commonly recognized as the whole working days in one year. Second way is always
applied in the financial industry, which is to calculate the number of calendar days
before maturity, and then to divide it by 365. To be more accurate the paper use the first
method get the time to maturity in the form of year.
26. 24
4.2.2 Risk Free Rate and Expected Return on Pure Bond
Risk free rate and expected return on bond are two essential parameter in the whole
empirical research. Risk free rate is used as discount rate in the calculation of value of
option. Taking into consideration that the risk free rate should ensure liquidity and
certainty, I choose the yield to maturity of government bond which have the same
remaining time to maturity as the convertible bonds involved in the study (Table 3).
Expected return on pure bond is the discount rate for the pure bond value. For sure,
the uncertainty, that is, risk level, is higher than the government bond, so that higher
required rate of return is necessary. Hence, I choose the yield to maturity of corporate
bonds, which ranked in AAA, as the substitute for the discount rate of convertible bond
that have the same remaining time as the corporate bonds (Table 4). Also, as I
mentioned before, all the data presented here is updated to February 26, 2016, because
there is a little fluctuation in the price every day.
Table 3 Government Bond Yield
Time to Maturity YTM (%)
1 2.2811
2 2.434
3 2.4171
4 2.5506
5 2.7767
6 2.801
7 2.917
10 2.8545
Data source: Wind Information
27. 25
Table 4 Corporate Bond Yield
Time to Maturity YTM (%)
0 2.3447
0.25 2.7315
0.5 2.7765
0.75 2.7771
1 2.817
3 3.0888
5 3.3173
7 3.595
10 3.7891
15 3.9448
20 4.1261
30 4.3046
Data source: Wind Information
4.2.3 Volatility of Stock Return
What I need to lay emphasis on is that the volatility mentioned here is not that of
stock price, but the rate of return of the stock. And in the equation, the volatility is
annually based. As usual, the volatility is measured by the standard deviation of data.
Because volatility changed all the time, there is no need to use the data that is too old
to reflect the reasonable change of the stock; therefore, in the study, I extract the stock
price for the last six months until February 26, 2016. First step is to calculate the daily
rate of return. Take 𝜇 𝑡 as continuously compounded daily returns.
28. 26
𝜇 𝑡 = ln(
𝑆𝑡
𝑆𝑡−1
)
Second, calculate the standard deviation of 𝜇 𝑡. Define𝜎𝜇 as daily return standard
deviation.
𝜎𝜇 = √
∑ （𝜇 𝑡 − 𝜇̅）
2
（n − 1）
Generally, I assume that 252 days a year, so
σ = √252 ∗ 𝜎𝜇
4.3 Empirical Study
4.3.1 Process of Calculation
To illustrate how the study is conducted in practice, especially in EXCEL, I provide
one of them, Gree CB (110030) as example. Cree CB is issued for 5 years initially with
the coupon rates 0.6%, 0.8%, 1%, 1.5%, 2%, respectively for every year. Its conversion
can be executed at price 20.9 Yuan starting from June 30, 2015 and ending on December
24, 2019. In EXCEL, I use the function NETWORKDAYS () to easily calculation the
29. 27
working days bet I en the trading day and maturity day. Finally, the time to maturity is
3.96 years, approximate value of 4 years. Based on that, I can look up the right risk free
rate and expected rate of return on pure bond in the tables of YTM of government bond
and YTM of corporate bond, and it comes to be 2.55% and 3.232% individually.
Here comes the calculation of volatility of the rate of return in stock. The underlying
stock of Gree CB is called Gree Real Estate (600185). Based on the excerpt of the stock
price from September 1, 2015 to February 26, 2016 (Table 5), I can calculate the
standard deviation of daily rate of return of the stock is 4.1374%, so that volatility
annually based in 65.6792%. Of course, I can easily get the closing price of the stock
on the trading day, which is 15.97 Yuan.
Table 5 Closing Price of Gree Estate
Date
Closing
Price
Daily
Rate of
Return
1-Sep-15 17.74 N/A
2-Sep-15 16.13 -9.51%
7-Sep-15 16.17 0.25%
8-Sep-15 16.87 4.24%
9-Sep-15 17.78 5.25%
10-Sep-15 17.02 -4.37%
11-Sep-15 17.22 1.17%
14-Sep-15 16.27 -5.67%
15-Sep-15 14.74 -9.88%
16-Sep-15 16.07 8.64%
17-Sep-15 15.17 -5.76%
18-Sep-15 15.3 0.85%
21-Sep-15 15.75 2.90%
22-Sep-15 15.95 1.26%
23-Sep-15 15.33 -3.96%
24-Sep-15 15.65 2.07%
25-Sep-15 15.67 0.13%
28-Sep-15 15.87 1.27%
29-Sep-15 15.18 -4.45%
30-Sep-15 15.35 1.11%
8-Oct-15 16.8 9.03%
9-Oct-15 17.63 4.82%
12-Oct-15 18.49 4.76%
13-Oct-15 19.65 6.08%
14-Oct-15 19.02 -3.26%
15-Oct-15 19.57 2.85%
16-Oct-15 20.04 2.37%
31. 29
8-Jan-16 17.33 0.12%
11-Jan-16 15.84 -8.99%
12-Jan-16 16.46 3.84%
13-Jan-16 15.76 -4.35%
14-Jan-16 16.35 3.68%
15-Jan-16 15.47 -5.53%
18-Jan-16 15.56 0.58%
19-Jan-16 16.28 4.52%
20-Jan-16 15.92 -2.24%
21-Jan-16 15.96 0.25%
22-Jan-16 16.91 5.78%
25-Jan-16 16.98 0.41%
26-Jan-16 15.99 -6.01%
27-Jan-16 15.81 -1.13%
28-Jan-16 15.86 0.32%
29-Jan-16 16.39 3.29%
1-Feb-16 16.26 -0.80%
2-Feb-16 16.97 4.27%
3-Feb-16 17.39 2.44%
4-Feb-16 17.37 -0.12%
5-Feb-16 17 -2.15%
15-Feb-16 16.59 -2.44%
16-Feb-16 17.37 4.59%
17-Feb-16 17.4 0.17%
18-Feb-16 17.24 -0.92%
19-Feb-16 17.2 -0.23%
22-Feb-16 17.58 2.19%
23-Feb-16 17.21 -2.13%
24-Feb-16 17.26 0.29%
25-Feb-16 15.68 -9.60%
26-Feb-16 15.97 1.83%
Data source: Wind Information
With all the parameters and variables decided, the calculation of the value of option
can be done just via putting all the number into equation. However, in practice,
especially in EXCEL, it is obvious that the whole calculation is a tremendous amount
of work and the mistakes easily happen. To simplify the essential and complicated final
deal, I can hire VBA in the study. Visual Basic for Applications (VBA) is an
implementation of Microsoft's discontinue event-driven programming language, Visual
Basic 6, and its associated integrated development environment (IDE).Visual Basic for
Applications enables building user-defined functions (UDFs), automating processes
32. 30
and accessing Windows API and other low-level functionality through dynamic-link
libraries (DLLs). It supersedes and expands on the abilities of earlier application-
specific macro programming languages such as Word's WordBasic. It can be used to
control many aspects of the host application, including manipulating user interface
features, such as menus and toolbars, and working with custom user forms or dialog
boxes.
Now I define functions dOne, dTwo, BSCall:
With all the effort above, the calculation of the option price using Black-Scholes
model is as easy as pie, because you can just apply it as the normal formula inserted in
the EXCEL. Therefore, the value of option part of Gree CB is 7.121378 Yuan. Value of
pure debt is easily got applying the equation of valuation of ordinary bond, which is
Function dOne(Stock, Exercise, Time, Interest, sigma)
dOne = (Log(Stock / Exercise) + Interest * Time) / (sigma * Sqr(Time)) + 0.5 * sigma
* Sqr(Time)
End Function
Function dTwo(Stock, Exercise, Time, Interest, sigma)
dTwo = dOne(Stock, Exercise, Time, Interest, sigma) - sigma * Sqr(Time)
End Function
Function BSCall(Stock, Exercise, Time, Interest, sigma)
BSCall = Stock * Application.NormSDist(dOne(Stock, Exercise, Time, Interest,
sigma)) - Exercise * Exp(-Time * Interest) * Application.NormSDist(dTwo(Stock,
Exercise, Time, Interest, sigma))
End Function
33. 31
92.8902. Hence, the theoretical value of the convertible bonds was 100.0133 Yuan.
Compared with the theory value, the closing price 120.68 is 20.7 percentage higher.
Next part of the study is to recalculate the value of option part based on Modified
Black-Scholes model. Here, I assume that all the convertible bond issued is converted.
The point in this part is to adjust the stock price, which actually changed because of
stock dilution. The ingredients I need here is total number of stocks, the number of
bonds issued and conversion ratio. The first two variables are available from Wind
Information. Total number of stock is 577680899, and that of convertible bond is
9800000. The conversion ratio needs a little bit calculation. Based on the equation that
conversion ratio is equal to par value of the bond divided by the conversion price, I can
get the conversion ratio of Gree CB is 4.78. Also, I need to get a coefficient in the
equation to adjust the option value to the statues before the conversion, which is
0.924925 here. It can be calculated according to the formula after the conversion price
of 16.34 Yuan. Final theoretical value of convertible bonds 99.72 Yuan, and the actual
closing price is 21.0% higher than that.
4.3.2 Result of Empirical Analysis
In order to present the whole results of the study clearly, I presents the results both
in traditional Black-Scholes model and modified Black-Scholes model, in the form of
table.
Table 6 Theoretical Value of Convertible Bond
Code Name
Pure
Bond
Value
Option
Value
Theory
Value
Closing
Price
Differ
ence
Deviation
Degree
110030 Gree CB 92.89 7.12 100.01 120.68 20.67 20.66%
113008 Dianqi CB 90.40 4.15 94.55 117.55 22.99 24.31%
35. 33
110035
Baiyun
CB
0.81
3.76 92.38 100.00 7.62 8.25%
113009
Guangqi
CB
0.87
9.08 96.62 118.96 22.34 23.12%
123001
Lanbiao
CB
0.86
4.89 93.49 111.30 17.82 19.06%
128010
Shuncha
ng CB
0.95
2.99 91.18 125.05 33.87 37.15%
4.3.3 Analysis on the results
Graph 1 Comparison bet I en Actual Price and Theory Value
The graph above show the comparison among traditional theory value, modified
theory value and actual closing price for each convertible bonds. Basically, the trend of
the value I calculated from the model is consistent with market price, which proves that
36. 34
the Black-Scholes model can be used to value the convertible bond. Second, the results
of traditional model and model with modification is close, as I can see from the graph.
As I have mentioned above, the modification is more apparent and useful in the
packaged convertible bond, because it can be traded separately, and after the conversion
to the common stock, the pure bond can also be held. Unpacked convertible bond is
different form that, once the conversion is executed, the debt is no longer existed. If I
explain this from the perspective of balance sheet, it seems like the value of liability is
conveyed to equity part, so the price of stock do not change that much after the increase
in the number of common stock. However, for packaged convertible bond, with the
continuous holding of debt, the conversion increase the number of stock with no change
in the value of equity. Therefore, the value per stock is decreased to a large extent. Third,
within 10 convertible bonds, the valuation of first three have trends that are more
consistent with market price that the other seven. The reason is that the last seven
convertible bonds have not entered into conversion period, so that the deviation is more
obvious.
Forth, there are differences bet I en market price and theory value. The deviation
might be caused by several reasons:
(1) Investors’ insufficient awareness of market of convertible bonds
Convertible bonds is still a newly- developing financial products in domestic capital
market, so that investors have not gain an intimate knowledge of features that combine
both bond and option, some of who do not learn all kinds of treaties well, which
contributes to the consequence that they cannot hold the opportunity to conversion
timely. However, early or late execution will lead to deviation for the true value. Due
to complexity of convertible bonds, individual investors are not willing to trade
convertible bond, which directly lead to low trading volume in the market and small
total amount. All in all, complex substance of convertible bond make some difference
in the value deviation from true value.
37. 35
(2) Deficient regulation on issuance, transaction mechanism
Compared with the overseas market for convertible bonds, issuance in domestic
market is comparatively short. According to Measure for Implementation in Issuance
of Convertible Bonds by the Listed Company, shortest period for domestic issuance of
convertible bonds is three years and longest is five years; the conversion is available at
least after half year after issuance. However, in market abroad, the lasting period is
comparatively longer for convertible bonds, which is averagely 15 years or so. The
restriction on lasting period lead to a larger possibility of loss for investors and of
deviation from actual value than that in foreign market. What’s more, the reading
system in national capital market is not fully consistent with one of assumption of
Black-Scholes model, that is, short security. Despite the inconsistency between our
trading system and prerequisite of Black-Scholes model, the theory value is still close
to market value, which, I think, chances are that securities margin trading provide an
effective way for short securities. It can be forecasted that, with the continuous
completion of trading system in domestic market, the market value of convertible bonds
will convergent to actual value.
(3) Irrational investment of investors
In domestic security market, speculation widely exists, especially in stock market,
where investors have tendency to put the capital in stock with strong fluctuation, no
matter institution or private. Such circumstances easily cause difference between theory
value and market value.
(4) Volatility in stock price
In the Black-Scholes model, the estimation of convertible bonds depends on value of
underlying stock owned by issuers. The estimation of volatility have great influence in
stock price.
Apart from all the reasons mentioned above, market value of convertible bond are
38. 36
influenced by call provision, sell provision and downredressal provision; however, all
of those are not included in the Black-Scholes model, which may also cause the
deviation in the price.
39. 37
5 Conclusion
The paper mainly discussed the pricing of convertible bond based on Black-Scholes
model. First three chapters provide some prerequisite about introduction of paper,
information of convertible bond and Black-Scholes model. In the last chapter, empirical
study, the result show Black-Scholes model can be applied to the pricing of the
convertible bond in Chinese financial market. But the difference between the theoretical
value and market value also illustrates some problems involved in market for
convertible bond, such as the insufficient knowledge about convertible bonds among
investors, deficient regulation in the market, investors’ irrational behaviors and strong
volatility in stock market.
40. 38
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