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毕 业 论 文

论文名称： Empirical Study of Pricing of Convertible

Bonds Based on Black- Scholes Model

学院： 金融管理学院

专业： 财务管理（中加合作）

学号： 1212049

学生姓名： 董思哲

指导教师： 张玮倩

2016 年 3 月

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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

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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.

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Key Words: Convertible Bond, Pricing, Black-Scholes Model

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中 文 摘 要

作为金融衍生品之一的可转换债券十分复杂，因为它同时具有债权和期权的

两方面特性。作为债券，可转换债券是融资人的融资渠道，同时对于投资者来说

可转换债券避免了金融市场上的一定的风险，因为债权在到期日会偿还本金，按

期支付利息。然而，可转换债券又不是普通的公司债券。在某种特定的情况下，

可转换债券可以进行转换，赎回和回售等等。这也进一步认证了为什么可转换债

券比一般普通的债券更为复杂，同时这两者之间的定价方式也大为不同。可转换

债券自从上世纪八十年代就风靡美国，中国也在 2006 年开始建立起较为规范的

可转换债券市场。但是，尽管如此，可转换债券在中国的流行时间还是很短，还

很难被认定为成熟的投资产品，所以广大的投资者对可转换债券并非十分的了解。

为了进一步加深对这一复杂衍生品的认识，研究其定价是非常好的研究角度。

在这样的研究背景下，本文主要解决中国可转换债券的定价问题，并基于学

术界经典的 Black-Scholes模型进行了实证分析。

在本文的结论部分展示了实证分析的结果，并且基于此提出了若干可以解释

市场价值和理论价值的差异的原因。

关键词：可转换债券，定价，Black-Scholes 模型

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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

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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-

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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

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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.

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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.

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the valuation model used in the valuation of convertible bond.

B = ∑𝐼𝑡

(1+𝑖)𝑡𝑛𝑡=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.

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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.

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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

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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

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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

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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

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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.

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(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；

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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）

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,

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

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

22

113008 Dianqi CB 10/20,90% 4-Jan-16 Untriggered

128009 GoerTek CB 15/30,90% 5-Jan-16 Untriggered

110032 Sanyi CB 10/20,85% 15-Jan-16 Untriggered

110033 Guomao CB 10/20,90% 26-Feb-16 Untriggered

110034 Jiuzhou CB 10/20,85% 2-Feb-15 Untriggered

110035 Baiyun CB 15/30,90% 22-Jan-16 Untriggered

113009 Guangqi CB 10/20,85% 18-Dec-15 Untriggered

123001 Lanbiao CB 15/30,90% 12-Dec-14 Untriggered

128010 Shunchang CB 20/30,85% 22-Jan-16 Untriggered

Data Source: Wind Information

Table 2 Basic Information of 10 Convertible Bond (Cont.)

Code Name Term Conversion

Price

Conversion

Start Date

Conversion

End Date

110030 Gree CB 5 20.9 30-Jun-15 24-Dec-19

113008 Dianqi CB 6 10.66 3-Aug-15 1-Feb-21

128009 GoerTek CB 6 26.33 19-Jun-15 11-Dec-20

110032 Sanyi CB 6 7.5 4-Jul-16 3-Jan-22

110033 Guomao CB 6 9.03 5-Jul-16 5-Jan-22

110034 Jiuzhou CB 6 18.78 21-Jul-16 14-Jan-22

110035 Baiyun CB 5 12.88 5-Sep-16 25-Feb-21

113009 Guangqi CB 6 21.99 22-Jul-16 21-Jan-22

123001 Lanbiao CB 6 15.3 27-Jun-16 17-Dec-21

128010 Shunchang CB 6 9.44 29-Jul-16 21-Jan-22

Data Source: Wind Information

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.

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

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 𝜇𝑡 as continuously compounded daily returns.

26

𝜇𝑡 = ln(𝑆𝑡𝑆𝑡−1

)

Second, calculate the standard deviation of 𝜇𝑡. Define𝜎𝜇 as daily return standard

deviation.

𝜎𝜇 = √∑（𝜇𝑡 − �̅�）

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

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%

28

19-Oct-15 20.19 0.75%

20-Oct-15 19.79 -2.00%

21-Oct-15 17.81 -10.54%

22-Oct-15 18.35 2.99%

23-Oct-15 19.32 5.15%

26-Oct-15 19.5 0.93%

27-Oct-15 19.29 -1.08%

28-Oct-15 18.54 -3.97%

29-Oct-15 18.9 1.92%

30-Oct-15 18.83 -0.37%

2-Nov-15 18.09 -4.01%

3-Nov-15 17.73 -2.01%

4-Nov-15 18.82 5.97%

5-Nov-15 19.35 2.78%

6-Nov-15 20.2 4.30%

9-Nov-15 20.43 1.13%

10-Nov-15 20.23 -0.98%

11-Nov-15 21.33 5.29%

12-Nov-15 20.8 -2.52%

13-Nov-15 19.98 -4.02%

16-Nov-15 20.36 1.88%

17-Nov-15 20.35 -0.05%

18-Nov-15 21.47 5.36%

19-Nov-15 21.24 -1.08%

20-Nov-15 21.36 0.56%

23-Nov-15 21.73 1.72%

24-Nov-15 21.29 -2.05%

25-Nov-15 21.24 -0.24%

26-Nov-15 21.37 0.61%

27-Nov-15 19.48 -9.26%

30-Nov-15 19.93 2.28%

1-Dec-15 21.3 6.65%

2-Dec-15 22.52 5.57%

3-Dec-15 22.54 0.09%

4-Dec-15 22.44 -0.44%

7-Dec-15 21.89 -2.48%

8-Dec-15 20.9 -4.63%

9-Dec-15 21.41 2.41%

10-Dec-15 20.46 -4.54%

11-Dec-15 20.22 -1.18%

14-Dec-15 20.71 2.39%

15-Dec-15 21.08 1.77%

16-Dec-15 20.86 -1.05%

17-Dec-15 21.79 4.36%

18-Dec-15 21.74 -0.23%

21-Dec-15 22.54 3.61%

22-Dec-15 22.66 0.53%

23-Dec-15 22.8 0.62%

24-Dec-15 22.03 -3.44%

25-Dec-15 22.23 0.90%

28-Dec-15 21.07 -5.36%

29-Dec-15 21.34 1.27%

30-Dec-15 21.57 1.07%

31-Dec-15 21.24 -1.54%

4-Jan-16 19.12 -10.52%

5-Jan-16 18.84 -1.48%

6-Jan-16 19.23 2.05%

7-Jan-16 17.31 -10.52%

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

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

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%

32

128009 GoerTek

CB 90.77 14.95 105.73 126.45 20.73 19.60%

110032 Sanyi CB 87.95 2.04 90.00 108.06 18.06 20.07%

110033 Guomao

CB 87.96 3.08 91.03 112.46 21.42 23.54%

110034 Jiuzhou CB 86.88 8.24 95.12 128.58 33.46 35.17%

110035 Baiyun CB 88.75 4.51 93.26 100.00 6.74 7.23%

113009 Guangqi

CB 87.54 10.19 97.73 118.96 21.23 21.72%

123001 Lanbiao CB 88.60 4.98 93.58 111.30 17.72 18.94%

128010 Shunchang

CB 88.19 3.06 91.25 125.05 33.80 37.04%

Table 7 Modified Theoretical Value of Convertible Bond

Code Name Coefficient Option

Value

Theory

Value

Closing

Price

Differ

ence

Deviation

Degree

110030 Gree CB 0.92 6.83 99.72 120.68 20.96 21.02%

113008 Dianqi

CB

0.95 4.03 94.43 117.55 23.12 24.48%

128009 GoerTek

CB

0.94 14.08 104.86 126.45 21.60 20.60%

110032 Sanyi

CB

0.93 2.01 89.96 108.06 18.10 20.12%

110033 Guomao

CB

0.46 1.96 89.92 112.46 22.54 25.07%

110034 Jiuzhou

CB

0.85 7.29 94.17 128.58 34.40 36.53%

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

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.

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

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.

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.

38

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3. Jian Liu, Mengxian Tao, Chaoqun Ma, Fenghuawen, ‘Utility indifference pricing of

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