Chapter14-Bonds with embeded Options

Chapter 14

Bonds with Embedded Options and Options on

Bonds

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 appreciate

less than a normal similar bond when interest rates fall – Therefore, an investor will be willing to buy such a bond at a lower price

than 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 02/15/2007 can 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 rates exceed the coupon rate of the issue

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

• The issuer of this bond will have to issue another bond at a higher coupon rate if the put option is exercised

• 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-callyear 5 4.54%year 6 4.61%year 7 4.66%year 8 4.69%year 9 4.72%

Yield-to-maturityyear 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 the price 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– r0 = 4%, ru = 4.66% and rl = 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

532.100,100min21

P

Callable and Putable BondsMonte Carlo Approach

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

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

• The price of the bond is computed as the average of its prices along all interest rate paths

Period Path1 Path2 Path3 Path4 Path5 Path61 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 callable at 100 after 5 years

• Prices of the bond under each scenarioPath1 Path2 Path3 Path4 Path5 Path6

Price of the callable bond 100.43 100.55 99.90 99.76 99.68 100.55

• Price of the bond is average over all pathsP=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 BondsDefinition

• Convertible securities are usually either convertible bonds or convertible preferred shares which are most often exchangeable into the common stock of the company issuing the 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 income instrument coupled with participation in the upside of the equity markets

• Essentially, convertible bonds are bonds that, at the holder's option, 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 no

conversion option– Conversion ratio is the number of shares that is exchanged for a

bond– 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 the

convertible 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-ask

convertible 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 was

61.73% = (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 to this

product– Besides the exchange of bonds for shares diminishes the liabilities of

the firm issuer and increases in the same time its equity so that its debt capacity is improved

• For the convertible bondholder– The convertible bond is a defensive security, very sensitive to a rise in the

share 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 the bond

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 bond plus 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 Bonds Convertible Bond Price as a Function of Stock Price

Parity

BondPrice

Straight Bond

Stock Price

ConvertibleBond

Convertible BondsConvertible Bond Pricing Model

• A popular method for pricing convertible bonds is the component 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 the straight bond component

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

• This pricing approach, however, has several drawbacks

– First, separating the convertible into a bond component and an option component relies on restrictive assumptions, such as the absence of embedded options (callability and putability, for instance, are convertible bond features that cannot be considered in the above separation)

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

Convertible BondsConvertible Bond Pricing Models

• Theoretical research on convertible bond pricing was initiated by Ingersoll (1977a) and Brennan and Schwartz (1977), who both applied the contingent claims approach to the valuation of convertible bonds.

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

• These models rely heavily on the theory of stochastic processes and require a relatively high level of mathematical sophistication

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

S

uS

dS

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

Convertible BondsBinomial 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


• Basic model that describes a simple world.• As the number of steps increases, it becomes

more realistic• We will price and hedge an option: it applies

to any other derivative security• Key: we have the same number of states and

securities (complete markets) Basis for arbitrage pricing


• 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


• 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 same as the price of the option

• Otherwise there will be an arbitrage opportunity


• We buy shares and invest B in the bank• They can be positive (buy or deposit) or negative

(shortsell or borrow)• We want then,

d

u

CrBdSCrBuS

)1()1(

• With solution,

)1)((;

)( rduCdCuB

duSCC uddu


• In our example, we get for stock:

31

80251100015

)..(d)S(u

CCΔ du

• And, for bonds:

24.24)1.1()8.025.1(

158.0025.1)1)((

rduCdCu

B ud

• The cost of the portfolio is,

09.924.2410031

BS


• The price of the European call must be 9.09.• Otherwise, there is an arbitrage opportunity.• If the price is lower than 9.09 we would buy

the call and shortsell the portfolio• If higher, the opposite• We have computed the price and the hedge

simultaneously:– We can construct a call by buying the stock and borrowing– Short call: the opposite


• Remember that

)1)((;

)( rduCdCu

BduS

CC uddu

• And

BSC • Substituting,

)1)(()( rduCdCu

duCC

C uddu


• After some algebra,

du Cdu

ruCdu

drr

C)(

)1()(

11

1

• Observe the coefficients,

)()1(,

)(1

duru

dudr

• Positive• Smaller than one• Add up to one

Like a probability.


• Rewrite

du CpCpr

C

)1(1

1

• Where

)()1(1,

)(1

durup

dudrp

• This would be the pricing of:– A risk neutral investor– With subjective probabilities p and (1-p)


• Suppose the following economy,

S

uS

dS

u2S

udS

d2S

• We introduce an European call with strike price K that matures in the second period


• The price of the option will be:

)],0max()1(2),0max()1(

),0max([)1(

1

22

222

KudSppKSdp

KSupr

C

• There are “two paths” that lead to the intermediate state (that explains the “2”)

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


udeu nt /1;/

Convertible BondValuation Methodology

• Given that a convertible bond is nothing but an option on the underlying stock, we expect to be able to 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 BondExample

• 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, a

conversion ratio of 20– The convertible bond has a $1,000.00 face value, a 4% annual

coupon– We further assume that the risk-free rate is a (continuously

compounded) 10%, while the yield to maturity on straight bonds issued 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

dudrp

ud

eu

1

.860711618.14/13.

• Actually (continuously compounded rate)

.5478607.1618.1

8607.4%10exp

p


$78.42G 10.00% looks like a stock: use risk-free rate

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

$67.49D 10.00% calling or converting does not change the bond value because it is already essentially equity

$1,349.86

$58.09 $58.09B 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 $50.00A 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 $43.04C 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

$37.04F 15.00% bond should not be converted because 1,001.72>50*20=1,000

$1,001.72

$31.88J 15.00% looks like a risky bond: use risky rate

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

Bond is Called


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

• The same applies to node H• On the other hand, at nodes I and J, the value under

the assumption of conversion is lower than if the bond 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


• Working our way backward the tree, we obtain at node D the value of the convertible bond as the discounted expected value, using risk-neutral probabilities of the payoffs at nodes G and H

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

123-

pp

• 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-

pp


• At node E, the situation is more interesting because the convertible bond will end up as a stock in case of an up move (conversion), and as a bond in case of a down move (no conversion)

• As an approximate rule of thumb, one may use a weighted average of the riskfree and risky interest rate in the computation, 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%27.12

123-

pp


• Note that at nodes D, E and F, calling or converting is not relevant 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 call price ($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 the issuer

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

Convertible BondAllowing for Stochastic Interest Rates

4.0%

4.5%

5.0%

3.6%

4.0%

3.2%

Interest RateTree

$10

$12$9

$8 $11 $14

Common StockPrice Tree

Convertible Bonds Convertible Arbitrage

• Convertible arbitrage strategies attempt to exploit anomalies in prices of corporate securities that are convertible into common stocks

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

• Convertible bonds tends to be under-priced because of market segmentation: investors discount securities that are likely to change types

Convertible BondsConvertible Arbitrage

• Convertible arbitrage hedge fund managers typically buy (or sometimes sell) these securities and then hedge part or all of the associated risks by shorting the 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, the

conversion 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 the convertible traded at $474.10

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

Convertible BondsMechanism

• In a typical convertible bond arbitrage position, the hedge fund is 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 convertible bond to common stock price changes

• As the stock price decreases, the amount lost on the long convertible position is countered by the amount gained on the short stock position, theoretically creating a stable net position value

• As the stock price increases, the amount gained on the long convertible position is countered by the amount lost on the short stock position, theoretically creating a stable net position value

• This is known as delta hedging


Parity =

ConvertibleBond

ConvertibleBondPrice

Stock Price

Delta = Change in Price of Conv BondChange in Price of Stock

Stock PriceConversion Ratio


• In the AOL example, the delta for the convertible is approximately 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 to short 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 should be 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 hedge against a reduction in credit quality

Convertible BondsRisks Involved

• Because a convertible bond is essentially a bond plus an option to 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/ask spreads on bonds can widen significantly

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

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

Options on BondsTerminology

• 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 a specified price within a specified period of time

• The seller grants this right to the buyer in exchange for a certain sum of money called the option price or option premium

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

• The date after which an option is void is called the expiration 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

Options on BondsFactors that Influence Option Prices

• Current price of underlying security– As the price of the underlying bond increases, the value of a call option rises

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 the option

price because all exercise opportunities open to the holder of the short-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 BondsPricing

• 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 moves

closer to maturity, its price volatility decline


• A solution to avoid the problem is to consider an interest 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 each node 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%

83.9788.2

89.8

93

94

95

100

100

100

100

83.9788.2

89.8

88.2

89.8

93

94

95

93

94

95

100

100

100

100

100

100

100

100

Interest rates

Bond prices


• Consider a 2-year European call on this 3-year bond struck 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

du

l

u

CCC

C

C

Options on BondsPut-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 call option (same time to maturity T and the same strike price X)

• Payoff at date T

BT < 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 BT 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 BondsPut-Call Parity – Con’t

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

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

• We obtain the call-put relationshiprTrT XeCBPXePCB 000000

• For coupon bonds

)(000 CouponsPVXeCBP rT

Chapter14-Bonds with embeded Options

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