Sanaa Khan K1306336

FACULTY OF SCIENCE, ENGINEERING AND COMPUTING

School of Computer Science and

Mathematics

BSc (Hons) DEGREE

IN

Financial Mathematics with Business Management

Name: Sanaa Khan

ID Number: K1306336

Project Title: Valuation of Convertible Bonds

Date: 11/04/16

Supervisor: Luluwah Al-Fagih

WARRANTY STATEMENT

This is a student project. Therefore, neither the student nor Kingston University

makes any warranty, express or implied, as to the accuracy of the data or conclusion

of the work performed in the project and will not be held responsible for any

consequences arising out of any inaccuracies or omissions therein.

Abstract

In this paper, we will be discussing methods of pricing a European style convertible bond

(CB), i.e. where conversion can only take place at maturity. Pricing methods include using

the Black-Scholes model to price the bond by splitting components to help simplify the

procedure. Furthermore, contract features will be looked upon, to give a better perspective

as to what is said between the issuer and the bondholder, as well as how the CB is formed

and the features within it. The paper will also be looking at the analysis of price sensitivities

and how different features affect the price of a CB and the impact they have on a portfolio

containing a CB.

Sanaa Khan K1306336

Contents

Abstract i

1. Introduction 1

2. Payoff Profiles 4

2.1. Notation 4

2.2. Bondholder’s Perspective 5

2.3. Bond Issuer’s Perspective 5

2.4. Payoff 6

2.5. A Zero-Sum Game 7

2.5.1. Example 7

3. Contract Features of a Convertible Bond 9

3.1. Convertible Bond Financing 9

3.2. Maturity 9

3.3. Principle 9

3.4. Conversion Ratio 10

3.5. Call Provisions 10

3.6. Put Provisions 10

3.7. Coupon Payments 11

3.8. Refix Clause 11

3.9. Other Non-Standard Clauses 12

3.10. Termination 12

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4. Properties of a Convertible Bond 13

4.1. Conversion Price 13

4.2. Parity 13

4.3. Premium to Parity 14

4.4. Investment Premium 14

4.5. Bond Floor 15

4.6. Price Sensitivities 16

4.7. Upper and Lower Bounds 19

5. Pricing Methods 21

5.1. Mathematical Background 21

5.2. Monte Carlo Simulation 21

5.3. Lattice based Method 22

5.4. Reduced Form Approach 22

5.5. Tsiveriotis-Fernandes Method 23

5.6. Black-Scholes Method 23

6. Black-Scholes Model 26

6.1. The Bond Price 26

6.2. Example 29

6.3. Margrabe Formula 31

7. Conclusion 34

8. References 35

1

1. Introduction Convertible bonds (CB) were first used during the 1960s. Convertible bonds are hybrid

securities; they use both equity and debt. A convertible bond is a bond such that the holder

of the bond; that being the investor is able to convert it into cash or equity when they feel it

would be beneficial to them [7 - pg 58]. Ingersoll’s (1977) research suggests that the general

valuation procedure would be to set up the price of the convertible and equate it to the

maximum value of a straight bond, or the value it holds within the common stock (after

conversion) given that at some point in the near future. The value found from this, would

then be discounted back to the present value. Yan, Yi, Yang and Liang (2015) state they wish

to keep hold of the bond, in which case they will receive interest payments; or they could

convert it into the company’s stocks. The bondholder would ideally pick a strategy in which

they would be able to maximise the CB value.

The issuers of convertible bonds are usually smaller firms. Smaller firms who are looking into

getting finances. The reason for this is because smaller firms are not as well-known and need

financing when their credit is low [20]. It is found that when a weaker firm wishes to issue a

CB, it shows they have faith in their project. This enhances their chances of gaining investors

for their company. However, a larger firm would not need to issue convertible bonds as they

would easily be able to get funding and or loans as they are more known within the industry.

If a larger firm wanted to issue a bond, they would not have enough buyers.

The motivation behind the smaller firms issuing the convertible bonds is due to the fact they

lack stable credit histories. This means they would have to pay higher interest payments;

also known as coupons - to their debt holders. The size of a firm usually is a reason as to why

there is an issuance of convertible bonds [12]. Firm size is associated with bankruptcy costs;

since smaller firms are more vulnerable to failure and are risk averse. Smaller firms face

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higher degree information asymmetry, this could increase the cost of the debt. It could also

lead to having more restrictive contracts – also known as covenants, had they wanted to

issue a straight bond. This is a reason why larger firms just offer straight bonds. A convertible

bond is more flexible the way it works, matters are stated within the contract, as well as

being set out if the firm breaks the contract they (the bondholder) will receive a premium

[12]. The motivation behind issuing CBs is the fact firms will have interest rate-cost savings,

in comparison to issuing straight corporate1 bonds [20].

Another reason why firms issue CBs is to ensure the investor has no entitlement in the

running of the business. This would mean having the ability to vote for the directors that

would only be in control of the common stockholders. This makes it attractive to firms, as

they know their positions will not be endangered nor questioned. Kwok (2014) suggests that

convertible bonds are chosen by firms over straight bonds due to the lower coupon rate.

CBs have a callable feature which means it can be redeemed by the issuer prior to the

contractual date, this paper will follow a European styled CB. At this point, a price – in the

form of a penalty, would be paid to the bondholder, as the company is forcing them to

either convert or surrender the bond [ref 7 page 58].

Owning a convertible bond is like playing a game. The bondholder is allowed to convert the

bond when they see it is beneficial for them. Suppose the bondholder converts before the

call date set within the covenant; it would mean the shareholders were not able to call the

bond when they thought it would be beneficial for them [7]. According to Yan, Yi, Yang and

Liang (2015) when the coupon rate is bounded above by the interest rate multiplied by the

strike price, that is when the bondholder will convert the CB. The conversion for the issuer

will take place when the coupon rate is lower than the dividend rate multiplied by the strike

1 Information Asymmetry: a party within a transaction has more information on the other party that

they are dealing with. Due to this, a party is likely to take advantage of the other party’s lack of knowledge.

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price; though this paper will not be discussing dividends used within CBs. The contract is

terminated when the coupon rate lies in between the two bounds, at that point both parties

will terminate the contract.

The bondholder will be receiving coupon payments, over the life of the CB, up until the

contract has reached its expiry (maturity). Prior to maturity, the bondholder has the right to

convert their bond into the company’s shares. Close to the end of the contract the company

have the right to call the bond back and force the bondholder to capitulate the bond to the

company.

In this paper, we will be discussing methods of pricing a European style convertible bond

(CB), i.e. where conversion can only take place at maturity. Pricing methods include using

the Black-Scholes model to price the bond by splitting components to help simplify the

procedure. Furthermore, contract features will be looked upon, to give a better perspective

as to what is said between the issuer and the bondholder, as well as how the CB is formed

and the features within it. The paper will also be looking at the analysis of price sensitivities

and how different features affect the price of a CB and the impact they have on a portfolio

containing a CB.

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2. Payoff Profiles

A payoff is what is received by the bondholder during the lifetime of the bond. The

bondholder has two options, (i) to receive the face value, or (ii) the share price multiplied by

the conversion ratio – the one with the greater value is what the bondholder will receive.

First we introduce some notation that will be used throughout the paper.

2.1 Notation

• 𝑁 – Face value

• 𝐶𝑟 – Conversion ratio

• 𝑆𝑡 – Share price at time 𝑡

• 𝐶 – Coupon payment

• 𝑇 – Maturity

• 𝐶𝑃 – Conversion price

• 𝐾 – Strike price

• 𝐵𝐹 – Bond floor

• 𝑃 – Price of CB. 𝑃 = 𝑃(𝑡) – price of CB at time 𝑡

• 𝑃𝑎 – Parity

• 𝑟 – interest rate

• 𝑐𝑡 – European call option

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2.2 Bondholder’s Perspective:

The bondholder’s perspective is very important during the issuance of the bond, all the way

up until maturity. In order to show their perspective the diagram below shows the true

picture of what they expect in return.

The four scenarios show what the payoff would be when the bond is converted, the

bondholder would receive their payoff; however, they would receive the maximum out of

the two. The next is the issuer calling the bond earlier, that way they end up paying them

the exercise price. When the bond is not exercised from either side the face value is given to

the bondholder. The default value would be received when the value is below the bond

floor; knowing the bond floor is the lowest boundary, below that the bondholder would then

receive zero.

2.3 Bond Issuer’s Perspective:

When the bondholder converts, the issuer pays the bondholder the maximum value out of

the (𝑆𝑡𝐶𝑟, 𝑁). When the issuer voluntarily calls the bond they pay the strike to the

bondholder. When the bond is not exercised by either party the face value is paid back to

Conversion: max(𝑆𝑡𝐶𝑟, 𝑁)

Not exercised: 𝑁

Default: 0

Issuer calls bond: 𝐾

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the holder at maturity. Default takes place when the value of the CB has fallen below the

bond floor.

Both parties have the opportunity to gain and to lose, but the total of the sum of their

choices will equal to zero. Following these scenarios we can see, how a zero sum game is

easily associated with CBs, regardless of the decision or the option which is put in front of

the two parties (See section 2.5.1 below).

2.4 Payoff

The payoff is a function [26]: Payoff:𝑚𝑎𝑥(𝑁, 𝐶𝑟𝑆𝑇); that tells the bondholder how much

they will receive from the bond issuer. When the bondholder decides to convert the bond

they will be receiving the maximum amount out of the two values. This means that the share

price is low the bondholder will not be converting and would prefer to receive the face value

𝑁. However, they will receive amount 𝐶𝑟𝑆𝑇 when the share price is high and the bondholder

chooses to convert. However, if there is a final coupon payment which is to be made then

the function changes to: 𝑚𝑎𝑥(𝑁 + 𝐶, 𝐶𝑟𝑆𝑇).Again, the same rule applies; whichever value

is higher is what the bondholder will receive at conversion.

Therefore, the value of the CB at maturity is given by:

𝑃(𝑇) = 𝐶𝑟𝑆𝑇𝑖𝑓𝐶𝑟𝑆𝑇 ≥ 𝑁

𝑃(𝑇) = 𝑁 + 𝐶𝑖𝑓𝐶𝑟𝑆𝑇 < 𝑁 + 𝐶

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2.5 A Zero-sum Game

A zero-sum game is a game where if one player gains then the other makes a loss. A CB can

be seen as a zero-sum game since the payoff of the CB follows the same ideology. Only one

party will be gaining something out of the game and one will lose, making the sum of the

game equal to zero. Each player uses a strategy to ensure that they are reducing their

opponent’s payoff.

Nash equilibrium (NE) is a term used within the theory of games; it is the solution concept of

a competitive game between two or more players. It is a way in which strategies are used to

make a profit. The game consists of: a set of actions and the choices of the set actions and

the impact they have on each player. The participants of the game are known to be in NE

when making strategic decisions, whilst considering their opponents decisions too. It is seen

that NE does not mean there will be a larger payoff amount necessarily for all players within

the game; it could be the case where a player receives a smaller amount due to the choice

they make [9].

2.5.1 Example of a zero-sum game:

To illustrate this in more detail, we look at a general example:

Two players within the game (the investor and the issuer) are both playing for the higher

payoff. The first player (purple) picks one of the two actions, either 1 or 2, without sharing it

with the other party. The second player (green) then picks an option out of the three

available choices. Again, player 2 chooses without player 1 knowing their decision. This then

leads to their choices being revealed, and the players are able to see their points and the

impact it has had on the payoff due to their choice. In this case, if purple picked option 2 and

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green decided to pick B, it would come to be that purple has gained 20 points and green has

now lost 20 points.

𝑨 𝑩 𝑪

𝟏 30, − 30 −10, 10 20, − 20

𝟐 10, − 10 20, − 20 −20, 20

The next step they would be taking is to ensure they are able to maximise their payoff.

Purple in this case could then say “With the second option, I could lose 20 and only win 20,

but with option 1 I could lose 10 but gain 30, which mean option 1 is more beneficial.”

Having the same strategy, Green would pick option C, that way they could gain 20, only if

Purple has picked option 2. On the other hand, if Purple were to pick option 2, only to know

that Green is more likely to pick option B. The strategy behind this would mean that Green

would then have already chosen option B, and player Purple would pick option 2; which

would lead to Purple gaining 20 and Green losing 20. Both players would be playing in order

to gain the highest payoff possible, that too with the intention of knowing what the other

player has chosen.

The probability within this example reveals that Purple should choose option 1, which has

the probability of 4/7 and option 2 which has the probability of 3/7. Whereas Green should

set the probabilities: 0, 4/7 and 3/7 to the three options A, B and C. In this case Purple will

then be gaining 20/7 on an average per game the two players are involved in.

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3. Contract features of a Convertible Bond (CB)

In this section, we look at the contract features in a CB.

3.1. CB Financing

A feature within a covenant is CB financing, this is the feature which meets the needs of the

issuer and the borrower [11].

3.2. Maturity: 𝑻

Maturity is the end of the contractual date set for the bond, also known as the expiration

date. This is when the firm has to pay back the entire amount back to the investor. Chan and

Chen (2004) state that maturity is usually between the years “…2, 3, 5, 7 and 10…” [6-pg 6]

however, it is possible to have some which last longer than 10 years. Brennan and Schwartz

(1980) found that the value of the CB depends on the maturity, as it has an impact on the

underlying asset risk of the company issuing the bond.

3.3. Principal or Face Value: 𝑵

A principal is the face value of the CB, the amount that the bond can be redeemed at

maturity. There have been occasions where the bond has been redeemed at maturity for a

larger contracted price, than the principal of the bond [11]. This is usually due to the change

in share price, when there is an increase that is when the investor receives a larger amount

at conversion.

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3.4. Conversion Ratio: 𝑪𝒓

Conversion ratio is mentioned within the covenant, it is highly common for there to be a

scheduled timing as to when the conversion of the ratio takes place. It is usually adjusted

over the life of the CB. The conversion ratio determines how many shares the bondholder

will receive at conversion. The way the conversion ratio works is the par value of the

convertible bond, would be divided by conversion price, which would then all be multiplied

by the price per share. This would be a method in which the value would be evaluated, as

well as the firm knowing how much they would have to pay [11]. An example to follow this

would be: If the company has the par value set to £1000 and the conversion ratio has been

set to 25 shares, using this information we can find that the conversion price would be £40.

3.5. Call Provisions: 𝒄𝒕

CBs tend to have a call feature, this allows the investor to know the CB would be called back.

This is when the firm decides to purchase the CB back at a particular date and time. The firm

can force the investor to convert/surrender the bond to them within a brief period of time

[11]. When a bond is called before maturity, the firm pay a penalty, which is pre-set. Lau and

Kwok (2004) suggest that the firm should call back the bond when it reaches the call price.

3.6. Put Provisions: 𝒑𝒕

Put provisions is a least common factor used within the contract. This is when the

bondholder is able to sell the CB back to the firm at a particular price and date [11]. Usually,

the firm will set a date when the put provision can start, as the contract would contain a

statement which states that the bondholder would have to keep the bond for a certain

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period of time before they convert the bond. However, a put provision has specified dates; it

is not continued throughout the life of the CB. A put provision is in place to increase the

protection of the bondholder, which leads to the increase in value of the CB [30].

3.7. Coupon Payments: 𝑪

Coupon payments are for the investor to gain over the period of the bond. It is agreed upon

how often they are to be received, i.e. monthly, annually or half-yearly. Amiram, Kalay, Kalay

and Ozel (2014) have suggested firms that face a higher information asymmetry are more

likely to issue bonds with a higher coupon value. An influential factor for coupon payments is

agency conflicts. This is when the shareholder wishes to gain an increase in the share value,

but the management are not cooperating. The coupon payment is a contractual term which

is able to reduce the agency conflicts. Lastly, the firms which face intense agency conflicts

tend to issue CBs with higher coupon rates [7].

3.8. Refix Clause

A refix clause is a feature which is used in the contract to make it more desirable to the

investor. It alternates the conversion ratio or the conversion price, which is subjected to

share price level between the issuance of the bond, up until maturity. The refix clause adds

additional value to the investor, which means an increase in the premium price paid for the

CB. The refix ensures the bondholder is protected against the decrease in share price [11].

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3.9. Other Non-Standard Clauses

The conversion segment usually states that the bondholder will receive a combination of

assets, both shares and cash. It is not necessary that they receive just shares when they

convert the CB [11].

3.10. Termination

Termination is when the contract is cancelled, which means the investor will no longer be

receiving any coupon payments. There are three occasions where the contract is terminated:

if the issuer calls back the bond before maturity, if the bondholder chooses to convert any

time up until maturity or if both the issuer and the bondholder choose to stop altogether.

The second scenario will mean the investor will be receiving 𝐶𝑟𝑆𝑡 at time 𝑡, with the

prearranged conversion rate 𝐶𝑟. The last scenario is if neither party exercises the CB from

the issuance up until the maturity. The bondholder is then expected to sell the CB back to

the issuer at maturity. However, when the bondholder sells the CB back, it is expected to be

sold according to the pre-set amount or the other option is to convert it into equity at the

conversion rate 𝐶𝑟 [29].

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4. Properties of a Convertible Bond (CB)

In this section we will be looking at the properties of a CB and how they are derived and

simplified.

4.1 Conversion Price: CP

The conversion price (𝐶𝑃) is the price per share at which the CB can be converted to stock.

The strike of each call option is equal to the conversion price: 𝐶𝑃. So, 𝐾 = 𝐶𝑃. The formula

shows how the 𝐶𝑃 is found for the bond. The formula for the conversion price is equal to the

face value divided by the conversion ratio [26].

4.2 Parity: Pa

The parity is the value of the shares that a party would receive if the CB were to be

converted immediately. The parity is usually a percentage of the face value of the CB. It uses

the share price which is multiplied by the conversion ratio, which is then divided by the face

value. The value of 𝐶𝑃 is the reciprocal of the face value and the conversion ratio, which is

why it is easily substituted within the formula for the parity. When the face value is at par or

1, then the parity is equal to the share price multiplied by the conversion ratio [26].

𝑃𝑎 =𝑆 × 𝐶𝑟𝑁

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=𝑆

𝐶𝑃

= 𝑆 × 𝐶𝑟

4.3 Premium to Parity (%)

Parity is presented in the form of a percentage; it is the percentage the investor is willing to

spend above the market price of the share for the CB. The fact the CB will be paying coupons

to the bondholder, which in turn could be higher than the dividends paid to the

shareholders. Such a yield would increase the value of the premium, which would be of an

advantage for the firm [26]. Parity is known as the lower boundary for the CB’s speculative

value.

4.4 Investment Premium

This is used to allow the firm to know how much the investor is willing to pay for the option

to convert embedded in the CB. This is also known as the premium to the bond floor, this is

usually the percentage of the bond floor. This means the lowest price the bond could fall to

is worked out as a percentage [26].

𝑃 − 𝐵𝐹𝐵𝐹

𝑃 − 𝑃𝑎𝑃𝑎

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4.5 Bond Floor: 𝑩𝑭

The bond floor is the value of the discounted cash flows of the CB. The bond floor is known

as the lower set boundary for the CB, it is usually given as a theoretical price for low share

prices of the bond. When the CB falls onto the bond floor it is looked upon as being

worthless, as it has hit the lowest price it can be if converted. As the share price increases,

the CB moves away from the bond floor and towards the conversion value [26].

The bond floor is given by:

𝐵𝐹 =∑𝐶𝑡𝑖𝑒(−𝑟𝑡𝑖) +𝑁𝑒(−𝑟𝑇)

𝑁𝑐

𝑖=1

Where 𝐶𝑡𝑖 is the value of each coupon payment at the time 𝑡𝑖, r is the interest rate and 𝑁𝑐 is

the number of upcoming coupon payments [26].

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The graph is an illustration of the price of a CB up until year 5. The dotted line represents the

bond floor value and the parity value of the bond. As mentioned, the bond floor is the

lowest value the CB can take. The graph gives an indication as to how the bondholder would

be looking into converting this CB. The convertible’s value will be driven by the value of the

underlying shares received when the bond is converted; this is known as the parity. When

the share price is low, it is unlikely the bondholder will want to convert the bond, as they will

receive a lower amount from the issuer [26].

4.6 Price Sensitivities

The delta tells us how sensitive the price of the CB is to the economic changes to the share

price, and the exact effect it has on the CB. Delta should be closer to zero if the investor

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does not want the price of the CB to change. This means as the share price changes over

time it would have an impact on the face value of the bond, which is a disadvantage for the

bond issuer. Hence why it is likely they would want to ensure delta is closer to zero; meaning

if the portfolio is near zero it would not be affected by the share price changes over time.

However, if the issuer does not ensure delta is closer to zero, it would mean the value of the

convertible will be increasing and decreasing over time up until maturity.

In particular, we can see below:

∆ = 𝜕𝑃

𝜕𝑆

lim𝑆→∞

∆ = lim𝑆→∞

𝑑𝑃

𝑑𝑆

=𝑑

𝑑𝑆lim𝑆→∞

𝑃

=𝑑

𝑑𝑆

𝑆 × 𝐶𝑟𝑁

=𝑑

𝑑𝑆(𝑆) ×

𝐶𝑟𝑁

=𝐶𝑟𝑁

lim𝑆→0

∆ = 0(𝑎𝑠𝑠𝑢𝑚𝑖𝑛𝑔𝑛𝑜𝑑𝑒𝑓𝑎𝑢𝑙𝑡)

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The last line where delta is equal to zero can also be seen in the graph above, where the

price of the CB does not change much for low share price. If the share price is falling, then it

would mean the CB value would be shifting to the bond floor [26].

𝜕2𝑃

𝜕𝑆2> 0

𝜕∆

𝜕𝑆> 0

The gamma, on the other hand, is the differential of the delta; it is looked at to ensure the

portfolio is insensitive to price movements. An increase in the gamma is usually caused

when the share price is increasing, as well as decreasing when the share price drops. The

larger the value of Γ the more likely it is affected by the change in share price [26].

The gamma tells us how often we need to rebalance the portfolio to make it ‘delta-neutral’,

i.e. how many shares need to be bought or sold in order to rebalance a portfolio of

convertible bonds. This can be very expensive for the firm to continuously rebalance, which

would mean having delta being closer to zero would be beneficial for the investor.

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4.7 Upper and Lower bounds

The above graph shows the price of the CB, with the upper and lower bounds. The green

dashed line which is going through the bond shows how the upper bound is formed; it is

produced with the conversion value and the share price at time t. As 𝑆𝑡 → ∞, the value of

the CB, 𝑃 ⟼ 𝐶𝑟𝑆𝑡.

At maturity 𝑇 or before, the bondholder could exercise the bond, which would then lead to

them receiving their payoff of:

𝑃(𝑇) = max(𝑁, 𝐶𝑟𝑆𝑡)

St

CB

CrSt

𝑁 + 𝐶𝑟𝑆𝑡

𝑁

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On the other hand, the lower bound is seen as being the parity. If the price of the CB falls

below the parity then there would be a possibility for arbitrage2 profits through purchasing a

CB in the market, as well as that selling it and then replacing the borrowed shares through

conversion [30]. It has been found that it may not always be possible to short sell, however

the possibility still remains. Hence:

𝑃(𝑡) ≥ 𝐶𝑟(𝑡)𝑆(𝑡)

2 Arbitrage: The trade that makes a profit by exploiting the price differences of identical or similar

financial instruments, on different markets or in different forms. Arbitrage exists due to the result of market inefficiencies, this provides mechanism to ensure prices do not deviate substantially from the fair value for a long period of time.

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5. Pricing Methods

In this section we will be looking at pricing methods of CBs and review some of the

mathematical background needed.

5.1 Monte Carlo Simulation (MCS)

There are several pricing methods to price a CB. One method to price CBs is using the Monte

Carlo Simulation (MCS) method; this would mean using differential equations to price the

CB. The method would be suitable to find the prices of the coupon payments and the

dividend payments. Coupon payments are usually solved using a continuous method rather

than the discrete method. The Monte Carlo Simulation uses computational algorithms;

which is mainly used in three different cases: optimisation, numerical integration and from

generating draws from probability distribution. Ulam (1949) has stated how this method

uses integration, as well as probabilities to value the price of the CB. An example of the MCS

method being used would be tossing a fair coin to see if a party wins a pound coin, the

winner would gain a pound extra. In order to play the game, both players must have a pound

coin. This would mean the investor and the bond issuer are both risk seeking, as they prefer

to gamble their initial wealth.

5.2 Lattice-based method

Another way in which a bond is priced is the lattice-based method. This method consists of

producing a binomial tree, in which it would show independent and equal paths. Lattice

models are useful as they are easy to compute and they reduce computing time. Lattice

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methods lose accuracy when they are dealing with two or more variables; as well as losing

efficiency when discrete payments are being dealt with and early-exercise options [28].

The disadvantage of the method would be the number of nodes can increase quickly as the

number of time steps increase. [32]

5.3 Reduced Form Approach

Another method to valuate a CB is the reduced form approach, which uses the Poisson

distribution model. The reduced form simplifies the valuation of the bond; it makes an

empirical analysis of the possibility of the bonds. An advantage of this method is that it

avoids the need to determine the firms optimal call policy, also it does not require any other

source of information about the firms’ financial state. This pricing method produces a closer

fit for CB prices, as well as producing low pricing errors. This method is known to be more

appealing than other methods of valuing CBs. When valuing the CB with the reduced form

approach, the factors taken into consideration are both call and default intensities under the

risk. This specific method captures the differences between the features call and default

decisions [14].

5.4 Tsiveriotis-Fernandes (TF) Method

The method proposes to split the CB into two segments: a cash-only and equity. The cash-

only is subjected to credit risk, whereas the equity is not. The method is found to being

popular to price the CB, all because it is very simple and it holds the ability to incorporate

vital traits of CBs that have limited market data [30]. There are three methods in which this

is used to ensure the PDEs are solved efficiently, ensuring the boundary condition and

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discontinuities are controlled well within the calculations. The methods are: explicit method,

implicit method and the Crank-Nicolson method.

5.5 Black-Scholes (B-S) Method

B-S is a method which analyses the theory of the corporate pricing formula. In order to

derive this formula there are assumptions which the B-S model follows. The assumptions

are:

Price of the underlying asset follows Geometric Brownian motion (GBM).

No arbitrage opportunities.

Unlimited short-selling.

The risk-free interest rate is constant, and the same for borrowing and lending.

No taxes or transaction costs.

Underlying asset can be traded continuously and in infinitesimally small numbers of

units.

In the next section B-S will be used to model the price of the CB.

5.6 Mathematical background:

Before we proceed, we must state some mathematical definitions.

Ω (omega) is the set of all possible outcomes, known as sample space [15].

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A filtration (ℱ𝑡 ) represents the history of a process up to time t. It is a model of

information. The filtration is a collection of subsets known as ‘events’. We can assign

a probability to each of those events [15].

ℙ is a probability measure, shows how likely an event is to happen, giving a number

𝜖[0,1] [15].

Brownian motion (BM): A BM (also known as the Wiener process) is a stochastic

process,

𝑊 = (𝑊𝑡)t≥0 that satisfies the following:

(i) 𝑊0 = 0

(ii) 𝑊 has continuous sample paths

(iii) 𝑊 has stationary increments: 𝑊𝑡 − 𝑊𝑠 ~ 𝑊𝑡−𝑠for any 0 ≤𝑠 < 𝑡.

(iv) 𝑊 has independent increments

(v) 𝑊𝑡 ~ 𝑁(𝜇𝑡, 𝜎2𝑡) for any 𝑡>0.

Brownian motion with drift:

𝑊0 + 𝜇𝑡 + 𝜎𝑊𝑡

Where𝜇𝑡 is the drift and 𝜎 is known as the diffusion coefficient or the volatility.

Geometric Brownian motion (GBM):

A BM with drift can take negative values which is not very useful. Therefore, we use

a GBM to model the price of a stock.

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25

𝑆𝑡 = 𝑆0𝑒(𝜇−

𝜎2

2)𝑡+𝜎𝑊𝑡

𝑆𝑡 denotes the stock price at time t and 𝑊𝑡 is a standard Brownian motion (SBM).

The stock price follows the GBM with stochastic differential equation [30]:

𝑑𝑆𝑡 = 𝜇𝑆𝑡𝑑𝑡 + 𝜎𝑆𝑡𝑑𝑊𝑡

The drift within the equation is µ, which represents the average growth of the asset

price. The future asset price - is found from the random changes within the price;

this is seen with a random variable which is drawn from normal distribution with the

mean being zero. The σ represents the volatility in the equation, it is known as the

dimension of the standard deviation of the returns within the portfolio [24].

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6. Black-Scholes Model

The pricing model uses Black-Scholes (BS) to model the price of an option, an option such as

a European call option. The model follows the Geometric Brownian Motion (GBM) with a

constant drift and volatility. The method when applied to a stock option, the model itself

includes the constant price, the strike price and the time to maturity, as well as the value of

the money.

A call option is purchasing the right to buy an asset at an agreed price, on or before the

particular set date; the right within the option is only received by the investor when they pay

a premium for the option [16].

6.1. The Bond Price

The bond price is under the Black-Scholes (BS) model satisfies the BS partial differential

equation:

𝜎2

2𝑆𝑡2𝑓𝑆𝑡𝑆𝑡 + (𝑟𝑆𝑡 − 𝐶)𝑓𝑆𝑡 − 𝑟𝑓 − 𝑓(𝑇−𝑡) + 𝑐 = 0

𝐶 denotes the coupon payments, 𝑐 being the amount of coupons paid out. 𝑟 is the interest

rate, 𝑆𝑡 is the share price and 𝜎2 is the variance return, 𝑓 in this case is: 𝑓 = (𝑆𝑡 , (𝑇 − 𝑡)).

(Ingersoll 1977)

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However, the price of the bond can be seen as a combination of a straight bond with a call

option. This means the formula for the straight bond will be applied, as well as the formula

for the call option, if the bond is callable.

Straight bond and a call option [30]:

𝑃𝑟𝑖𝑐𝑒𝑜𝑓𝐶𝐵 = 𝑃𝑟𝑖𝑐𝑒𝑜𝑓𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡𝑏𝑜𝑛𝑑 + 𝑃𝑟𝑖𝑐𝑒𝑜𝑓𝐸𝑢𝑟𝑜𝑝𝑒𝑎𝑛𝑐𝑎𝑙𝑙𝑜𝑝𝑡𝑖𝑜𝑛

𝑋(𝑡) = 𝑁∑𝐶𝑒−𝑟(𝑡𝑖−𝑡) +𝑁𝑒−𝑟(𝑇−𝑡)𝑛

𝑖=1

Where 𝑡𝑖 represents the dates at which the coupons are paid.

Price of a European call option [30]:

𝑉(𝑡) = 𝑆𝑡Φ(𝑑1) − 𝐾𝑒−𝑟(𝑇−𝑡)Φ(𝑑2)

Where 𝑑1 and 𝑑2 are [30]:

𝑑1 =log (

𝑆𝑡𝐾) + (𝑟 +

𝜎2

2 )(𝑇 − 𝑡)

𝜎√𝑇 − 𝑡

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𝑑2 = 𝑑1 − 𝜎√𝑇 − 𝑡

The value of Φ(𝑑1) and Φ(𝑑2) is found to be the cumulative probability distribution

function for a standard normal distribution.

This then leads to:

𝑃(𝑡) = 𝑋(𝑡) + 𝑉(𝑡)

This is finding the final value of the price of the CB, finding the sum of the two components.

However, splitting the two components relies on restrictive assumptions, an example being

the embedded options. Splitting the components mean not being able to call the bond, nor

being able to sell it back to the issuer; which are the features available within a CB. However,

these cannot be taken into consideration with the BS equation above [30]. As a convertible

usually is American styled, it becomes hard to value it, in terms of the closed-form approach

of the BS model. The closed-form approach is computing the value of the option; the

method gives the issuer an idea of the pricing and the behavior of the CB. The method

becomes complicated to use when continuous time intervals are used within the option; it

becomes difficult to find the price if early exercise is available within the option. Hence, for

CBs it is not appropriate, unless it is European in nature [27].

The payoff for a call option is: 𝑚𝑎𝑥(𝑆𝑇 − 𝐾, 0), the bondholder has two options from which

they will receive one, (i) if 𝑆𝑇 − 𝐾 is negative then the payoff value is 0 and (ii) if 𝑆𝑇 − 𝐾 is

positive then the payoff value is an integer that the bondholder will receive.

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The fair price of a CB is defined using the replicating strategy, using units of stock and cash;

this continues to follow the BS model of valuation.

6.2. Example

Initial Stock (S0) 300

Time (T) 5

Strike Price (K) 350

Risk-free rate (r ) 0.02

Volatility 0.1

Face Value (N) 500

Coupon Payments (C ) 0

d1 0.18305321

d2 -0.04055359

Call option V(t) 18.56223972

Bond Value Bf 0.455940983

Convertible Bond (CB) 19.018180713

Table 1 shows the current data for a CB and the price of it when the components have been

split into a call option and a straight bond. Using these parameters two graphs will be

plotted, (i) which will contain the increase of the strike price, whilst the other parameters

remain the same and (ii) where the interest rate will be increasing, again whilst the rest of

3 Table 1

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the parameters remain the same. This will help to show the impact of changing a parameter

and how this affects the price of the CB.

4

The graph shows the impact of the change in the conversion price, it shows how the price of

the CB price has decreased as the value of 𝐶𝑃 has increased. The other parameters remain

the same, however it is seen how the value of the CB changes with the change in a

parameter value. As the conversion price increases, the value of the call option is out-of-the-

money, which means the payoff would be equal to zero as the conversion price continues to

increase. If the conversion price is high, the bondholder would be able to purchase shares in

the market, more shares than which they would receive at conversion. In this case, the

bondholder would not be exercising the CB and would see benefit in buying the shares in the

market.

4 Graph 1: change in conversion price

0

50

100

150

200

250

150 200 250 300 350

Price of CB with increasing conversion price

P(t)

Conversion Price (𝐶𝑃)

Co

nve

rtib

le B

on

d P

rice

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5

The graph shows the impact of the change in the interest rate on the price of a CB. The

graphs shows as the rate of interest increases the CB is worth more, the same can be said for

the call option; an increase in the interest rate, increases the value of the option and the CB.

6.3. Margrabe Formula

The Margrabe formula generalises the BS pricing model to price options, which gives the

holder of the option the right to exchange but not the obligation to exchange ‘S’ units of one

asset into ‘P’ units of another [25]. In this case, a CB can be viewed as a risky straight bond

and the option to exchange the straight bond for a specific amount of shares. The Margrabe

model assumes that the assets follow the GBM with the correlation 𝜌 [30]. In this case,

5 Graph 2: change in interest rate

0

10

20

30

40

50

60

0.02 0.03 0.04 0.05 0.06

Price of CB with increasing interest rate

P(t)

Risk-free Interest rate (𝑟)

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however, there are contradictions to applying the GBM to the CBs; this is seen with the

results, as the results we obtain are sensible.

The equation for the replicating portfolio (exchange option) is shown by [25]:

𝐸(𝑡) = 𝑄1𝑆1𝑒((𝑎1−𝑟)(𝑇−𝑡))Φ(𝑑1) − 𝑄2𝑆2𝑒

((𝑎2−𝑟)(𝑇−𝑡))Φ(𝑑2)

Where [25]:

𝑑1,2 =ln (

𝑆1𝑆2) + (𝑎1 − 𝑎2 ±

�̂�2

2)(𝑇 − 𝑡)

�̂�√𝑇 − 𝑡

�̂� = √𝜎12𝜎2

2 − 2𝜌𝜎1𝜎2

Where 𝑎1 = 𝑎2 = 𝑟, 𝑆1, 𝑆2 are assets which are chosen to be exchanged and 𝑄1and 𝑄2 are

the quantities of the assets. There are no dividend payments in this case; therefore, early

exercise of the CB will not be optimal for the bondholder [25]. The definition of Φ(𝑑1) and

Φ(𝑑2) can be found in section 6.1.

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An exchange option is seen to be European in nature, which means it cannot be called by the

issuers and there are no coupon payments made [30]. In this case, under the assumption of

the Margrabe model, 𝑆1can be seen as our share price 𝑆𝑡 and 𝑄1 following to be the

conversion ratio 𝐶𝑟, 𝑆2 is the price of the bond 𝑃(𝑡). Since there is only one unit of the bond

this means 𝑄2 = 1. The price of the bond 𝑃(𝑡), at time 𝑡 = 0 is, 𝑃(0) = 𝑁𝑒−𝛿𝑇. Where 𝛿 is

the continuous compounded yield rate.

The replicating portfolio, as seen as above, consists of 𝐶𝑟𝑒((𝑎1−𝑟)(𝑇−𝑡))Φ(𝑑1) amount of

shares and 𝑒((𝑎2−𝑟)(𝑇−𝑡))Φ(𝑑2) of loaned cash. When there is a change in the share price,

i.e: 𝑆𝑡 → ∞ then, Φ(𝑑1);Φ(𝑑2) → 1 this then leaves the replicating strategy being a long

position in share value: 𝑆𝑡𝐶𝑟𝑒((𝑎1−𝑟)(𝑇−𝑡)) and a short position with the cash amount:

𝑁𝑒−𝛿𝑇+((𝑎1−𝑟)(𝑇−𝑡)). This gets balanced out due to the risky long position, which then leads

to: as 𝑆𝑡 → ∞, 𝑃(𝑡) = 𝑆𝑡𝐶𝑟𝑒((𝑎1−𝑟)(𝑇−𝑡)) [30].

Using the value of 𝑑𝑆𝑡 (section 5.6), following the BS model the value of the call option

becomes [30]:

𝑉(𝑡) = 𝑆𝑡Φ(𝑑1) − 𝐾𝑡𝑒−𝑟(𝑇−𝑡)Φ(𝑑2)

Where in this case 𝑑1 and 𝑑2 are equal to [30]:

𝑑1,2 =ln (

𝑆𝑡𝐾) + (𝑟 ±

𝜎2

2 )(𝑇 − 𝑡)

𝜎√𝑇 − 𝑡

A more detailed discussion of the margrabe formula is left for future work.

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7. Conclusion

Convertible bonds (CB) are interesting as they combine two financial instruments. We see

that CBs can be modelled as a zero-sum game between the bondholder and the bond issuer.

This paper focuses on the pricing of a CB, going through stages of the bond before finally

breaking down the price of the bond, as well as discussing the methods used to price the

bond. An example of these is Black-Scholes model, it is commonly used to help price the

bond, however we also find that using the lattice-based method also works well. Under the

Black-Scholes model, we are able to split the components of the CB into a straight bond and

a call option.

The interesting factor is that the issuer and the bondholder are able to change the features

within the bond according to their needs, which means both parties have a fair advantage. A

feature such as the premium to pay the issuer would be discussed and can be changed

before signing for the bond. This means the CB are attractive for both the issuer and the

bondholder, as it helps the issuer with financing their needs and gives the bondholder the

right to convert when they feel it is beneficial for them.

The paper also looks at an example of a CB for a specific set of parameters. We see how that

affects the overall price of the CB and whether it increases or decreases with changes in

interest rate and conversion price.

CBs offer the investor a greater right than the issuer of the bond – the right for which they

have paid for, thus it is acceptable. However, the instruments used within the contract must

be carefully understood by both parties in order to be able to make the investment.

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