Bonds Callable

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Chapter 14

Bonds with Embedded

Options and Options onBonds

FIXED-INCOME SECURITIES



Outline• Callable and Putable Bonds

– Institutional Aspects

– Valuation

• Convertible Bonds – Institutional Aspects

– Valuation

• Options on Bonds – Institutional Aspects

– Valuation

– Uses



Callable Bonds and Putable BondsBond with Embedded Options

• Callable bonds – Issuer may repurchase at a pre-specified call price

– Typically called if interest rates fall

• A callable bond has two disadvantages for an investor – If it is effectively called, the investor will have to invest in another bond

yielding a lower rate

– A callable bond has the unpleasant property for an investor to appreciateless than a normal similar bond when interest rates fall

– Therefore, an investor will be willing to buy such a bond at a lower pricethan a comparable option-free bond

• Examples – The UK Treasury bond with coupon 5.5% and maturity date 09/10/2012 can

be called in full or part from 09/10/2008 on at a price of pounds 100

– The US Treasury bond with coupon 7.625% and maturity date 02/15/2007can be called on coupon dates only, at a price of $100, from 02/15/2002 on

– Such a bond is said to be discretely callable



Callable and Putable BondsInstitutional Aspects

• Putable bond holder may retire at a pre-specified

price• A putable bond allows its holder to sell the bond at

par value prior to maturity in case interest ratesexceed the coupon rate of the issue

• So, he will have the opportunity to buy a new bond ata higher coupon rate

• The issuer of this bond will have to issue another

bond at a higher coupon rate if the put option isexercised

• Hence a putable bond trades at a higher price than a

comparable option-free bond



Callable and Putable BondsYield-to-Worst

• Let us consider a bond with an embedded call option

trading over its par value• This bond can be redeemed by its issuer prior to

maturity, from its first call date on

– One can compute a yield-to-call on all possible call dates – The yield-to-worst is the lowest of the yield-to-maturity and all yields-to-call

• Example – 10-year bond bearing an interest coupon of 5%, discretely callable after 5

years and trading at 102 – There are 5 possible call dates before maturity

– Yield-to-worst is 4.54%

Yield-to-call

year 5 4.54%

year 6 4.61%

year 7 4.66%

year 8 4.69%

year 9 4.72%

Yield-to-maturity

year 10 4.74%



Callable and Putable BondsValuation in a Binomial Model

• Let us assume that a binomial tree has been already

built and calibrated as explained in Chapter 12• Recursive procedure

– Price cash-flow to be discounted on period n-1 is the minimum value of theprice computed on period n and call price on period n

– And so on until we get the price P of the callable bond• Example

– We consider a callable bond with maturity two years, annual coupon 5%,callable in one year at 100

– r 0

= 4%, r u

= 4.66% and r l

= 4.57% (cf. example in Chapter 12)

– We have Pu = 105/1.0466 = 100.32 and Pl = 105/1.0457 = 100.41

– Finally, price of the callable bond

( ) ( )96.100%41

541.100,100min

%41

532.100,100min

2

1

=⎟ ⎠

⎞

⎜⎝

⎛

+

+

++

+

= P



Callable and Putable BondsMonte Carlo Approach

• Step 1: generate a large number of short-term interest ratepaths using some dynamic model (see Chapter 12)

• Step 2: along each interest rate path, the price P of the bondwith embedded option is recursively determined

• The price of the bond is computed as the average of its prices

along all interest rate pathsPeriod Path1 Path2 Path3 Path4 Path5 Path6

1 4,00% 4,00% 4,00% 4,00% 4,00% 4,00%2 4,08% 4,14% 4,29% 4,24% 4,28% 4,28%

3 3,83% 4,02% 4,35% 4,27% 4,24% 4,23%

4 4,15% 3,88% 4,25% 3,87% 4,17% 4,30%5 4,27% 4,26% 4,68% 4,58% 4,29% 3,99%6 4,69% 4,49% 4,33% 4,29% 4,47% 4,32%

7 4,88% 5,10% 5,24% 5,08% 5,27% 4,70%8 5,14% 4,94% 4,75% 5,54% 5,25% 5,08%

9 5,24% 5,47% 5,15% 5,26% 5,43% 5,64%10 5,59% 5,04% 5,29% 5,58% 5,38% 5,02%



Callable and Putable BondsMonte Carlo Approach - Example

• Price a callable bond with annual coupon 4.57%,maturity 10 years, redemption value 100 and callableat 100 after 5 years

• Prices of the bond under each scenario

Path1 Path2 Path3 Path4 Path5 Path6Price of the callable bond 100.43 100.55 99.90 99.76 99.68 100.55

• Price of the bond is average over all paths

P=1/6(100.43+100.55+99.9+99.76+99.68+100.55)=100.14• The Monte Carlo pricing methodology can also be

applied to the valuation of all kinds of interest rates

derivatives



Convertible Bonds

Definition

• Convertible securities are usually either convertible bonds orconvertible preferred shares which are most often

exchangeable into the common stock of the company issuingthe convertible security

• Being debt or preferred instruments, they have an advantage to

the common stock in case of distress or bankruptcy• Convertible bonds offer the investor the safety of a fixed incomeinstrument coupled with participation in the upside of the equitymarkets

• Essentially, convertible bonds are bonds that, at the holder'soption, are convertible into a specified number of shares



Convertible Bonds

Terminology

• Convertible bonds – Bondholder has a right to covert bond for pre-specified number of

share of common stock

• Terminology – Convertible price is the price of the convertible bond

– Bond floor or investment value is the price of the bond if there is noconversion option

– Conversion ratio is the number of shares that is exchanged for abond

– Conversion value = current share price x conversion ratio

– Conversion premium = (convertible price – conversion value) /conversion value

– Income pickup is the amount by which the yield to maturity of theconvertible bond exceeds the dividend yield of the share



Convertible Bonds

Examples

• Example 1: – Current bond price = $930

– Conversion ratio: 1 bond = 30 shares common

– Current stock price = $25/share

– Market Conversion Value = (30 shares)x(25) = $750

– Conversion Premium = (930 – 750) / 750 = 180 / 750 = 24%

• Example 2: AXA Convertible Bond – AXA has issued in the € zone a convertible bond paying a 2.5% coupon

rate and maturing on 01/01/2014; the conversion ratio is 4.04

– On 12/13/2001, the current share price was €24.12 and the bid-askconvertible price was 156.5971/157.5971

– The conversion value was equal to €97.44 = 4.04 x 24.12

– The conversion premium calculated with the ask price 157.5971 was61.73% = (157.5791 - 97.44)/ 97.44

– The conversion of the bond into 4.04 shares can be executed on any date

before the maturity date



Convertible Bonds

Bloomberg Description



Convertible Bonds

Uses

• For the issuer – Issuing convertible bonds enables a firm to obtain better financial

conditions – Coupon rate of such a bond is always lower to that of a bullet bond with the

same characteristics in terms of maturity and coupon frequency

– This comes directly from the conversion advantage which is attached tothis product

– Besides the exchange of bonds for shares diminishes the liabilities of thefirm issuer and increases in the same time its equity so that its debtcapacity is improved

• For the convertible bondholder

– The convertible bond is a defensive security, very sensitive to a rise in theshare price and protective when the share price decreases

– If the share price increases, the convertible price will also increase

– When share price decreases, price of convertible never gets below thebond floor, i.e., the price of an otherwise identical bullet bond with no

conversion option



Convertible Bonds

Determinants of Convertible Bond Prices

• Convertible bond is similar to a normal coupon bondplus a call option on the underlying stock – With an important difference: the effective strike price of the call option will

vary with the price of the bond

• Convertible securities are priced as a function of – The price of the underlying stock

– Expected future volatility of equity returns

– Risk free interest rates

– Call provisions

– Supply and demand for specific issues

– Issue-specific corporate/Treasury yield spread – Expected volatility of interest rates and spreads

• Thus, there is large room for relative mis-valuations



Convertible BondsConvertible Bond Price as a Function of Stock Price

P a r i t y

B o n d

P r i c e

S tra i g h t B o n d

S to c k P r ic e

C o n v e r t i b l eB o n d



Convertible Bonds

Convertible Bond Pricing Model

• A popular method for pricing convertible bonds is thecomponent model – The convertible bond is divided into a straight bond component and a call

option on the conversion price, with strike price equal to the value of thestraight bond component

– The fair value of the two components can be calculated with standardformulas, such as the famous Black-Scholes valuation formula.

• This pricing approach, however, has severaldrawbacks – First, separating the convertible into a bond component and an option

component relies on restrictive assumptions, such as the absence ofembedded options (callability and putability, for instance, are convertiblebond features that cannot be considered in the above separation)

– Second, convertible bonds contain an option component with a stochasticstrike price equal to the bond price



Convertible Bonds

Convertible Bond Pricing Models

• Theoretical research on convertible bond pricing wasinitiated by Ingersoll (1977a) and Brennan andSchwartz (1977), who both applied the contingentclaims approach to the valuation of convertiblebonds.

• In their valuation models, the convertible bond pricedepends on the firm value as the underlying variable.Brennan and Schwartz (1980) extend their model byincluding stochastic interest rates.

• These models rely heavily on the theory of stochasticprocesses and require a relatively high level ofmathematical sophistication



• The price of the stock only can go up to agiven value or down to a given value

S

uS

dS

• Besides, there is a bond (bank account) thatwill pay interest of r

Convertible Bonds

Binomial Model



• We assume u (up) > d (down)

• For Black and Scholes we will need d = 1/u• For consistency we also need u > (1+r) > d

• Example: u = 1.25; d = 0.80; r = 10%

S=100

S = 125

S = 80

Convertible Bonds

Binomial Model



• Basic model that describes a simple world.

• As the number of steps increases, it becomesmore realistic

• We will price and hedge an option: it appliesto any other derivative security

• Key: we have the same number of states and

securities (complete markets)⇒ Basis for arbitrage pricing

Convertible Bonds

Binomial Model



• Introduce an European call option: – K = 110 – It matures at the end of the period

S=100

uS = 125

dS = 80

S C (K=110)

Cu = 15

Cd = 0

Convertible Bonds

Binomial Model



• We can replicate the option with the stock

and the bond• Construct a portfolio that pays Cu in state u

and Cd

in state d

• The price of that portfolio has to be the sameas the price of the option

• Otherwise there will be an arbitrageopportunity

Convertible Bonds

Binomial Model



• We buy Δ shares and invest B in the bank

• They can be positive (buy or deposit) or negative(shortsell or borrow)

• We want then,

⎭⎬⎫

=++Δ=++Δ

d

u

C r BdS

C r BuS

)1(

)1(

• With solution,

)1)((

;

)( r d u

C d C u B

d uS

C C ud d u

+−

×−×=

−

−=Δ

Convertible Bonds

Binomial Model



• In our example, we get for stock:

3

1

80251100

015=

−×−=

−−=

)..( d)S(u

C C Δ

d u

• And, for bonds:

24.24

)1.1()8.025.1(

158.0025.1

)1)((

−=

×−

×−×=

+−

×−×=

r d u

C d C u B ud

• The cost of the portfolio is,

09.924.2410031 =−×=+Δ BS

Convertible Bonds

Binomial Model





• Remember that

)1)((;

)( r d u

C d C u B

d uS

C C ud d u

+−

×−×=

−

−=Δ

• And BS C +Δ=

• Substituting,

)1)(()( r d u

C d C u

d u

C C C ud d u

+−

×−×+

−

−=

Convertible Bonds

Binomial Model



• After some algebra,

⎥⎦

⎤⎢⎣

⎡

−

+−+

−

−+×

+= d u C

d u

r uC

d u

d r

r C

)(

)1(

)(

1

1

1

• Observe the coefficients,

)(

)1(,

)(

1

d u

r u

d u

d r

−

+−

−

−+

• Positive

• Smaller than one

• Add up to one⇒Like a probability.

Convertible Bonds

Binomial Model



• Rewrite

[ ]d u C pC pr

C ×−+××+

= )1(1

1

• Where

)(

)1(1,

)(

1

d u

r u p

d u

d r p

−

+−=−

−

−+=

• This would be the pricing of: – A risk neutral investor

– With subjective probabilities p and (1-p)

Convertible Bonds

Binomial Model



• Suppose the following economy,

S

uS

dS

u2S

udS

d 2S

• We introduce an European call with strike price K that

matures in the second period

Convertible Bonds

Binomial Model



• The price of the option will be:

)],0max()1(2

),0max()1(

),0max([)1(

1

22

222

K udS p p

K S d p

K S u pr

C

−×−××+

−×−+

−××+

=

• There are “two paths” that lead to the intermediate

state (that explains the “2”)

• Suppose we know the volatility σ and the time tomaturity t, we can retrieve u and d (see B&S)

Convertible Bonds

Binomial Model

ud eu nt /1;/ == σ



Convertible Bond

Valuation Methodology

• Given that a convertible bond is nothing but anoption on the underlying stock, we expect to be ableto use the binomial model to price it

• At each node, we test – a. whether conversion is optimal

– b. whether the position of the issuer can be improved by calling the bonds

• It is a dynamic procedure: max(min(Q1,Q2),Q3)),where

– Q1 = value given by the rollback (neither converted nor called back) – Q2 = call price

– Q3 = value of stocks if conversion takes place



Convertible Bond

Example

• Example – We assume that the underlying stock price trades at $50.00 with a

30% annual volatility

– We consider a convertible bond with a 9 months maturity, aconversion ratio of 20

– The convertible bond has a $1,000.00 face value, a 4% annualcoupon

– We further assume that the risk-free rate is a (continuouslycompounded) 10%, while the yield to maturity on straight bondsissued by the same company is a (continuously compounded) 15%

– We also assume that the call price is $1,100.00

– Use a 3 periods binomial model (t/n=3 months, or ¼ year)



Convertible Bond

Example

• We have

d ud r p

ud

eu

−

−+=

==

==

1

.86071

1618.14/13.

• Actually (continuously compounded rate)

.5478607.1618.1

8607.4

%10exp

=−

−⎟ ⎠

⎞⎜⎝

⎛

= p



Convertible Bond

Example

$78.42

G 10.00% looks like a stock: use risk-free rate

$1,568.31 conversion: 78.42>1040/20=52

$67.49

D 10.00% calling or converting does not change the bond value because it is already essentially equity

$1,349.86

$58.09 $58.09

B 11.03% H 10.00% looks like a stock: use risk-free rate

$1,191.13 $1,161.83 conversion: 58.09>1040/20=52

$50.00 $50.00

A 12.15% E 12.27% bond should not be converted because 1,073.18>50*20=1,000

$1,115.41 $1,073.18

$43.04 $43.04

C 13.51% I 15.00% looks like a risky bond: use risky rate

$1,006.23 $1,040.00 no conversion: 43.04<1040/20=52

$37.04

F 15.00% bond should not be converted because 1,001.72>50*20=1,000

$1,001.72

$31.88

J 15.00% looks like a risky bond: use risky rate

$1,040.00 conversion: 31.88<1040/20=52

Bond is Called



Convertible Bond

Example

• At node G, the bondholder optimally choose toconvert since what is obtained under conversion($1,568.31), is higher than the payoff under theassumption of no conversion ($1,040.00)

• The same applies to node H

• On the other hand, at nodes I and J, the value underthe assumption of conversion is lower than if thebond is not converted to equity – Therefore, bondholders optimally choose not to convert, and the payoff is

simply the nominal value of the bond, plus the interest payments, that is$1,040.00



Convertible Bond

Example

• Working our way backward the tree, we obtain atnode D the value of the convertible bond as thediscounted expected value, using risk-neutralprobabilities of the payoffs at nodes G and H

( )( )83.161,1131.568,1e=$1,349.86%10

12

3-

×−+××

p p

• At node F, the same principle applies, except that it

can be regarded as a standard bond• We therefore use the rate of return on a non

convertible bond issued by the same company, 15%

( )( )040,11040,1e=$1,001.72%15

123-

×−+××

p p



Convertible Bond

Example

• At node E, the situation is more interesting because theconvertible bond will end up as a stock in case of an up move

(conversion), and as a bond in case of a down move (noconversion)

• As an approximate rule of thumb, one may use a weighted

average of the riskfree and risky interest rate in thecomputation, where the weighting is performed according to the(risk-neutral) probability of an up versus a down move

px10% + (1-p)x15% = 12.27%

• Then the value is computed as

( )( )040,1183.161,1e=$1,073.18

%27.1212

3-

×−+×

×

p p



Convertible Bond

Example

• Note that at nodes D, E and F, calling or converting is notrelevant because it does not change the bond value since the

bond is already essentially equity• At node B, it can be shown that the issuer finds it optimal to call

the bond

• If the bond is indeed called by the issuer, bondholders are left

with the choice between not converting and getting the callprice ($1,100), or converting and getting $20x58.09=1,161.8$,which is what they optimally choose

• This is less than $1,191.13, the value of the convertible bond if

it were not called, and this is precisely why it is called by theissuer

• Eventually, the value at node A, i.e., the present fair value of the convertible bond, is computed as $1,115.41



Convertible Bond

Allowing for Stochastic Interest Rates

4.0%

4.5%

5.0%

3.6%

4.0%

3.2%

Interest Rate

Tree

$10

$12$9

$8 $11 $14

Common Stock

Price Tree



Convertible Bonds

Convertible Arbitrage

• Convertible arbitrage strategies attempt to exploitanomalies in prices of corporate securities that areconvertible into common stocks

• Roughly speaking, if the issuer does well, theconvertible bond behaves like a stock, if the issuerdoes poorly, the convertible bond behaves likedistressed debt

• Convertible bonds tends to be under-priced because

of market segmentation: investors discount securitiesthat are likely to change types



Convertible Bonds

Convertible Arbitrage

• Convertible arbitrage hedge fund managers typicallybuy (or sometimes sell) these securities and then

hedge part or all of the associated risks by shortingthe stock

• Take for example Internet company AOL's zero

coupon converts due Dec. 6, 2019 – These bonds are convertible into 5.8338 shares of AOL stock

– With AOL common stock trading at $34.80 on Dec. 31, 2000, theconversion value was $203 (=5.8338 x 34.80)

– As the conversion value is significantly below the investment value

(calculated at $450.20), the investment value dominated and theconvertible traded at $474.10

– When, or if, the stock trades above $77.15, the conversion value willdominate the pricing of the convertible because it will be in excess of theinvestment value



Convertible Bonds

Mechanism

• In a typical convertible bond arbitrage position, the hedge fundis not only long the convertible bond position, but also short an

appropriate amount of the underlying common stock• The number of shares shorted by the hedge fund manager is

designed to match or offset the sensitivity of the convertiblebond to common stock price changes

• As the stock price decreases, the amount lost on the longconvertible position is countered by the amount gained on theshort stock position, theoretically creating a stable net positionvalue

• As the stock price increases, the amount gained on the longconvertible position is countered by the amount lost on theshort stock position, theoretically creating a stable net positionvalue

• This is known as delta hedging

C



Convertible Bonds

Mechanism

Parity =

ConvertibleBond

ConvertibleBondPrice

Stock Price

Delta =Change in Price of Conv Bond

Change in Price of Stock

Stock Price

Conversion Ratio

C tibl B d



Convertible Bonds

Mechanism

• In the AOL example, the delta for the convertible isapproximately 50%

• This means that for every $1 change in the conversion value,the convertible bond price changes by 50 cents

• To delta hedge the equity exposure in this bond we need toshort half the number of shares that the bond converts into, for example 2.9 shares (5.8338\2)

• The combined long convertible bond/short stock position shouldbe relatively insensitive to small changes in the price of AOL's

stock• Over-hedging is sometimes appropriate when there is concern

about default, as the excess short position may partially hedgeagainst a reduction in credit quality

C tibl B d



Convertible Bonds

Risks Involved

• Because a convertible bond is essentially a bond plus an optionto switch so that these strategies will typically

– make money if expected volatility increases (long vega) – make money if the stock price increases rapidly (long gamma)

– pay time-decay (short theta)

– make money if the credit quality of the issuer improves (short the credit differential)

• The risks involved relate to

– changes in the price of the underlying stock (equity market risk) – changes in the interest rate level (fixed income market risk)

– changes in the expected volatility of the stock (volatility risk)

– changes in the credit standing of the issuer (credit risk)

• The convertible bond market as a whole is also prone to

liquidity risk as demand can dry up periodically, and bid/askspreads on bonds can widen significantly

• There is also the risk that the HF manager will be unable tosustain the short position in the underlying common shares

• In addition, convertible arbitrage hedge funds use varyingdegrees of leverage, which can magnify both risks and returns

O ti B d



Options on Bonds

Terminology• An option is a contract in which the seller (writer)

grants the buyer the right to purchase from, or sell to,

the seller an underlying asset (here a bond) at aspecified price within a specified period of time

• The seller grants this right to the buyer in exchangefor a certain sum of money called the option price oroption premium

• The price at which the instrument may be bought orsold is called the exercise or strike price

• The date after which an option is void is called theexpiration date – An American option may be exercised any time up to and including the

expiration date

– A European option may be exercised only on the expiration date

O ti B d



Options on Bonds

Factors that Influence Option Prices• Current price of underlying security

– As the price of the underlying bond increases, the value of a call optionrises and the value of a put option falls

• Strike price – Call (put) options become more (less) valuable as the exercise price

decreases

• Time to expiration – For American options, the longer the time to expiration, the higher theoption price because all exercise opportunities open to the holder of theshort-life option are also open to the holder of the long-life option

• Short-term risk-free interest rate – Price of call option on bond increases and price of put option on bond

decreases as short-term interest rate rises (through impact on bond price)

• Expected volatility of yields (or prices) – As the expected volatility of yields over the life of the option increases, the

price of the option will also increase

Options on Bonds



Options on Bonds

Pricing

• Options on long-term bonds – Interest payments are similar to dividends.

– Otherwise, long-term bonds are like options on stock: – We can use Black-Scholes as in options on dividend-paying equity

• Options on short-term bonds – They do not pay dividends

– Problem: they are not like a stock because they quickly converge to par

– We cannot directly apply Black-Scholes

• Other shortcomings of standard option pricing

models – Assumption of a constant short-term rate is inappropriate for bond options

– Assumption of a constant volatility is also inappropriate: as a bond movescloser to maturity, its price volatility decline

Options on Bonds



Options on Bonds

Pricing

• A solution to avoid the problem is to consider aninterest rate model, as described in Chapter 12 – The following figure shows a tree for the 1-year rate of interest (calibrated

to the current TS)

– The figure also shows the values for a discount bond (par = 100) at eachnode in the tree

6%

6.5%

5.5%

6.5%

5.5%

7%

6%

5%

7%

6%

5%

7.5%

6.5%

5.5%

4.5%

7.5%

6.5%

5.5%

4.5%

83.97

88.2

89.8

93

94

95

100

100

100

100

83.97

88.2

89.8

88.2

89.8

93

94

95

93

94

95

100

100

100

100

100

100

100

100

Interest rates

Bond prices

Options on Bonds



Options on Bonds

Pricing

• Consider a 2-year European call on this 3-year bondstruck at 93.5

• Start by computing the value at the end of the tree – If by the end of the 2nd year the short-term rate has risen to 7% and the

bond is trading at 93, the option will expire worthless

– If the bond is trading at 94 (corresponding to a short-term rate of 6%) the

call option is worth 0.5

– If the bond is trading at 95 (short-term rate = 5%), the call is worth 1.5

• Working our way backward the tree

( )

( )

( ) 5573.5.5.%61

1

9479.5.15.5.5.%5.51

1

2347.5.5.05.%5.61

1

0 =×+×+

=

=×+×+

=

=×+×+=

d u

l

u

C C C

C

C

Options on Bonds



Options on Bonds

Put-Call Parity• Assumption no coupon payments and no premature exercise

• Consider a portfolio where we purchase one zero coupon bond,

one put European option, and sell (write) one European calloption (same time to maturity T and the same strike price X)

• Payoff at date T

T < X :

You hold the bond: BT The call option is worthless: 0

The put option is worth: X - BT Thus, your net position is: X

T ≥ X :

You hold the bond: BT The call option is worth: -(BT - X)

The put option is worthless: 0 Thus, your net position is: X

Options on Bonds



Options on Bonds

Put-Call Parity – Con’t

• No matter what state of the world obtains at theexpiration date, the portfolio will be worth X

• Thus, the payoff from the portfolio is risk-free, andwe can discount its value at the risk-free rate r

• We obtain the call-put relationship

rT rT XeC B P Xe P C B −− −+=⇒=−+ 000000

• For coupon bonds

)(000 Coupons PV XeC B P rT −−+= −

Documents

Bonds Callable