CHAPTER 9
BINOMIAL INTEREST RATE TREES
AND THE EVALUATION OF BONDS WITH EMBEDDED OPTIONS
9.1 INTRODUCTION
An option is a right to buy or sell a security at a specific price on or possibly before a specified
date. As we saw in Chapter 7, many bonds have a call feature giving the issuer the right to buy
back the bond from the bondholder. In addition to callable bonds, there are putable bonds, giving
the bondholder the right to sell the bond back to the issuer, sinking fund bonds in which the issuer
has the right to call the bond or buy it back in the market, and convertible bonds that give the
bondholder the right to convert the bond into specified number of shares of stock.
The inclusion of option features in a bond contract makes the evaluation of such bonds
more difficult. A 10-year, 10% callable bond issued when interest rate are relatively high may be
more like a 3-year bond given that a likely interest rate decrease would lead the issuer to buy the
bond back. Determining the value of such a bond requires taking into account not only the value
of the bond’s cash flow, but also the value of the call option embedded in the bond. One way to
capture the impact of a bond’s option feature on its value is to construct a model that incorporates
the random paths that interest rates follow over time. Such a model allows one to value a bond’s
option at different interest rate levels.
One such model is the binomial interest rate tree.
Patterned after the binomial option pricing model (BOPM), this model assumes that interest rates
follow a binomial process in which in each period the rate is either higher or lower. In this
chapter, we examine how to evaluate bonds with option features using a binomial interest rate tree
approach. We begin by defining a binomial tree for one-period spot rates and then showing how
the tree can be used to value a callable bond. After examining the valuation of a callable bond,
we then show how the binomial tree can be extended to the valuation of putable bonds, bonds
1
with sinking funds, and convertible bonds. In this chapter, we focus on defining the binomial tree
and explaining how it can be used to value bonds with embedded options; in the next chapter, we
take up the more technical subject of how the tree can be estimated.
9.2 BINOMIAL INTEREST RATE MODEL
A binomial model of interest rates assumes a spot rate of a given maturity follows a binomial
process where in each period it has either a higher or lower rate. For example, assume that a oneperiod, riskless spot rate (S) follows a process in which in each period the rate is equal to a
proportion u times its beginning-of-the-period value or a proportion d times its initial value,
where u is greater than d. After one period, there would be two possible one-period spot rates: Su
= uS0 and Sd = dS0. If the proportions u and d are constant over different periods, then after two
periods there would be three possible rate. That is, as shown in Figure 9.2-1, after two periods the
one-period spot rate can either equal: Suu = u2S0, Sud = udS0, or Sdd = d2S0. Similarly, after three
periods, the spot rate could take on four possible values: Suuu = u3S0, Suud = u2dS0, Sudd = ud2S0,
and Sddd = d3S0.
To illustrate, suppose the current one-period spot rate is 10%, the upward parameter u is
1.1 and the downward parameter d is .95. As shown in Figure 9.2-2, the two possible one-period
rates after one period are 11% and 9.5%, the three possible one-period rates after two periods are
12.1%, 10.45%, and 9.025%, and the four possible rates are three periods are 13.31%, 11.495%,
9.927, and 8.574%.
9.2.1 Valuing a Two-Period Bond
Given the possible one-period spot rates, suppose we wanted to value a bond that matures in two
2
periods. Assume that the bond has no default risk or embedded option features and that it pays an
8% coupon each period and a $100 principal at maturity. Since there is no default or call risk, the
only risk an investor assumes in buying this bond is market risk. This risk occurs at time period
one. At that time, the original two-period bond will have one period to maturity where there is a
certain payoff of $108. We don’t know, though, whether the one-period rate will be 11% or
9.5%. If the rate is 11%, then the bond would be worth Bu = 108/1.11 = 97.297; if the rate is
9.5%, the bond would be worth Bd = 108/1.095 = 98.630. Given these two possible values in
Period 1, the current value of the two-period bond can be found by calculating the present value
of the bond’s expected cash flows in Period 1. If we assume that there is an equal probability (q)
of the one-period spot rate being higher (q = .5) or lower (1-q = .5), then the current value of the
two-period bond (B0) would be 96.330 (see Figure 9.2-3).
B0 
q[ Bu  C]  (1  q )[ Bd  C]
1  S0
B0 
.5[97.297  8]  .5[98.630  8]
 96.330
110
.
Now suppose that the two-period, 8% bond has a call feature that allows the issuer to buy
back the bond at a call price (CP) of 98. Using the binomial tree approach, this call option can be
incorporated into the valuation of the bond by determining at each node whether or not the issuer
would exercise his right to call. The issuer will find it profitable to exercise whenever the bond
price is above the call price (assuming no transaction or holding costs). This is the case when the
one-period spot rate is 9.5% in Period 1 and the bond is priced at 98.630. 1 The price of the bond
In this case, the issuer could buy the bond back at 98 financed by issuing a one-year bond at
9.5% interest. One period later the issuer would owe 98(1.095) = 107.31; this represents a savings
of 108-107.31 = 0.69. Note, the value of that savings in period one is .69/1.095 = 0.63 which is
equal to the difference between the bond price and the call price: 98.630 - 98 = .63.
1
3
in this case would be the call price of 98. It is not profitable, however, for the issuer to exercise
the call at the spot rate of 11% when the bond is worth 97.297; the value of the bond in this case
remains at 97.297. In general, since the bond is only exercised when the call price is less than the
bond value, the value of the callable bond in Period 1 is therefore the minimum of its call price or
its value as an otherwise noncallable bond:
BCt  Min[ BNC
t , CP].
Rolling the two callable bond values in Period 1 of 97.297 and 98 to the present, we obtain a
current price of 96.044.
BCt 
.5[97.297  8]  .5[98  8]
 96.044.
110
.
As we should expect, the bond’s embedded call option lowers the value of the bond from 96.330
to 96.044. The value of the callable bond in term of the binomial tree is shown in Figure 9.2-4a.
Note, at each of the nodes in Period 1, the value of the callable bond is determined by selecting
the minimum of the otherwise noncallable bond or the call price, and then rolling the callable
bond value to the current period.
Instead of using a price constraint at each node, the price of the callable bond can
alternatively be found by determining the value of call option at each node, VtC, and then
subtracting that value from the noncallable bond value (BtC = Btnc - VtC). In this two-period
case, the value of the call is the maximum of either its intrinsic value (exercise value), Btnc - VtC,
or zero:
4
VtC  Max[ Bt  CP, 0 ].
This approach yields a current value of the call option of .2864. Subtracting this call value from
the noncallable bond value of 96.330, we obtain the same callable bond value of 96.044 (see
Figure 9.2-4b).
9.2.3 Valuing a Three-Period Bond
The binomial approach to valuing a two-period bond requires only a one period binomial tree of
one-period spot rates. If we want to value a three-period bond, we in turn need a two-period
interest rate tree. For example, suppose we wanted to value a three-period, 9% coupon bond with
no default risk or option features. In this case, market risk exist in two periods: Period 3, where
there are three possible spot rates and Period 2, where there are two possible rates. To value the
bond, we first determine the three possible values of the bond in Period 2 given the three possible
spot rates and the bond’s certain cash flow next period (maturity). As shown in Figure 9.2-5, the
three possible values in Period 2 are Buu = 109/1.121 = 97.2346, Bud = 109/1.1045 = 98.6872, and
Bdd = 99.977. Given these values, we next roll the tree to the first period and determine the two
possible values there. Note, in this period the values are equal to the present values of the
expected cash flows in Period 2; that is:
.5[97.2346  9]  .5[98.6872  9]
 96.3612
111
.
.5[98.6872  9]  .5[99.977  9]
Bd 
 98.9334.
1095
.
Bu 
5
Finally, using the bond values in Period 1, we roll the tree to the current period where we
determine the value of the bond to be 96.9521:
B0 
.5[96.3612  9]  .5[98.9334  9]
 96.9521.
110
.
If the bond is callable, we can determine its value by first comparing each of the
noncallable bond values with the call price in Period 2 (one period from maturity) and taking the
minimum of the two as the callable bond value. We next roll the callable bond values from
Period 2 to Period 1 where we determine the two bond values at each node as the present value of
the expected cash flows, and then for each case we select the minimum of the value we calculated
or the call price. Finally, we roll those two callable bond values to the current period and
determine the callable bond’s price as the present value of Period 1's expected cash flows.
Figure 9.2-6a shows the binomial tree value of the three-period, 9% bond given a call
feature with a CP = 98. Note, at the two lower nodes in Period 2, the bond would be called at 98
and therefore the callable bond price would be 98; at the top node, the bond price of 97.2346
would prevail. Rolling these prices to Period 1, the present values of the expected cash flows are
96.0516 at the 11% spot rate and 97.7169 at the 9.5% rate. Since neither of these values are less
than the CP of 98, each represents the callable bond value at that node. Rolling these two values
to the current period, we obtain a value of 97.258 for the three-period callable bond.
The alternative approach to valuing the callable bond is to determine the value of the call
option at each node and then subtract that value from the noncallable value to obtain the callable
bond’s price. However, different from our previous two-period case, when there are three periods
or more, we need to take into account that prior to maturity the bond issuer has two choices: she
6
can either exercise the option or hold it for another period. The exercising value, IV, is
IV  Max[ Bnc
t  CP, 0 ],
while the value of holding, VH , is the present value of the expected exercise value next period:
VH 
qIVu  (1  q )IVd
.
1 S
If the value of holding exceeds IV, the issuer will hold the option another period and the
value of the call in this case will be the holding value, VH. In contrast, if IV is greater than the
holding value, then the issuer will exercise the call immediately and the value of the option will
be IV. Thus, the value of the call option is equal to the maximum of IV or VH:
VC  Max[IV, 0 ].
Figure 9.2-6b shows this valuation approach applied to the three-period callable bond.
Note, in Period 2 the value of holding is zero at all three nodes since next period is maturity
where it is too late to call. The issuer, though, would find it profitable to exercise in two of the
three cases where the call price is lower than the bond values. The three possible callable bond
values in Period 2 are:
C
BCuu  Bnc
uu  Vuu  97.2346  Max[ 97.2346  98, 0 ]  97.2346
C
BCud  Bnc
ud  Vud  98.6872  Max[ 98.6872  98, 0 ]  98
nc
BCdd  Bdd
 VddC  99.977  Max[99.977  98, 0 ]  98.
7
In Period 1, the noncallable bond price is lower than the call price at the lower node. In this case,
the IV is 98.8395 - 98 = .8394. The value of holding the call, though, is 1.2165:
VH 
.5[ Max[98.6872  98, 0]]  .5[ Max[99.977  98, 0]]
 12165
.
.
1095
.
Thus, the issuer would find it more valuable to defer the exercise one period. As a result, the
value of the call option is Max[IV, VH] = Max[.8394, 1.2165] = 1.2165 and the value of the
callable bond is 97.7169 (the same value we obtained using the price constraint approach):
BCd  Bdnc  VdC  98.9334  12165
.
 97.7169.
At the upper node in Period 1 where the price of the noncallable is 96.3612, the exercise value is
zero. The value of the call option in this case is equal to its holding value of .3095:
VH 
.5[ Max[97.2364  98, 0]]  .5[ Max[98.6872  98, 0]]
 .3095,
111
.
and the value of the callable bond is 96.0517:
C
BCu  Bnc
u  Vu  96.3612  .3095  96.0517.
Finally, rolling the two possible option values of .3095 and 1.2165 in Period 1 to the current
period, we obtain the current value of the option of .6936 and the same callable bond value of
96.258 that we obtained using the first approach:
8
.5[.3095]  .5[12165
.
]
 .6936
110
.
BC0  B0nc  V0C  96.9521  .6936  96.258.
V0C 
9.2.4 Alternative Binomial Valuation Approach
In valuing an option-free bond with the binomial approach, we started at the bond’s maturity and
rolled the tree to the current period. An alternative but equivalent approach is to calculate the
weighted average value of each possible paths defined by the binomial process. This value is
known as the theoretical value.
To see this approach, consider again the three-period, 9% option-free bond valued with a
two-period interest rate tree. For a two-period interest rate tree, there are four possible interest
rate paths. That is, to get to the second-period spot rate of 9.025%, there is one path (spot rate
decreasing two consecutive periods); to get to 10.45%, there are two paths (decrease in the first
period and increase in the second and increase in the first and decrease in the second); to get to
12.1% there is one path (increase two consecutive periods). Given the three-period bond’s cash
flows of 9, 9, and 109, the value or equilibrium price of each path is obtained by discounting each
of the cash flows by their appropriate 1-, 2-, and 3-period spot rates:
CF1
CF2
CF3


.
Path i
Path i 2
(1  S1 ) (1  S2 )
(1  S3Path i ) 3
1
S1Path
S0  .10spot rate is equal to the geometric average of the current and expect oneThe t-period
S2Path 1  [(1  S0 )(1  Sd )]1/ 2  1  [(110
. )(1095
. )]1/ 2  1  .0974972
period spot
rates. For example, for path 1 (path with two consecutive1/ 3decreases in rates), its oneS3Path 1  [(1  S0 )(1  Sd )(1  Sdd )]1/ 3  1  [(110
. )(1095
. )(109025
.
)]  1  .0950761.
B0Path i 
period rate is St = S1 = 10%, its two-period rate is S2 = 9.74972 (geometric average of S0 = 10%
and Sd = 9.5%), and its three-period rate is S3 = 9.50761 (geometric average of S0 = 10%, Sd =
9
9.5%, and Sdd = 9.025%).
Discounting the three-period bond’s cash flows by these rates yields a value for path 1 of
98.65676:
B0Path1 
9
9
109


 98.65676.
2
110
.
(10974971
.
)
(10950761
.
)3
The periodic spot rates and bond values for each of the four paths are shown at the bottom
of Figure 9.2-5. Given the path values, the bond’s weighted average value is obtained by
summing the weighting values of each path with the weights being the probability of attaining
that path. For a two-period interest rate tree with a probability of the rate increasing in one period
being q = .5, the probability of attaining each path is .25. Using these probabilities, the threeperiod bond’s weighted average value or theoretical value is equal to 96.9521 – the same value
we obtained earlier by rolling the bond’s value from maturity to the current period.
9.3 VALUING BONDS WITH OTHER OPTION FEATURES
In addition to call features, bonds can have other embedded options such as a put option, a stock
convertibility clause, or a sinking fund arrangement in which the issuer has the option to buy
some of the bonds back either at their market price or at a call price. The binomial tree can be
easily extended to the valuation of bonds with these embedded option features.
9.3.1 Putable Bond
A putable bond, or put bond, gives the holder the right to sell the bond back to the issuer at a
10
specified exercise price (or put price), PP. In contrast to callable bonds, putable bonds benefit the
holder: If the price of the bond decreases below the exercise price, then the bondholder can sell
the bond back to the issuer at the exercise price. From the bondholder’s perspective, a put option
provides a hedge against a decrease in the bond price. If rates decrease in the market, then the
bondholder benefits from the resulting higher bond prices, and if rates increase, then the
bondholder can exercise, giving her downside protection. Given that the bondholder has the right
to exercise, the price of a putable bond will be equal to the price of an otherwise identical
nonputable bond plus the value of the put option (V0P ):
B0P  B0np  V0P .
Since the bondholder will find it profitable to exercise whenever the put price exceeds the
bond price, the value of a putable bonds can be found using the binomial approach by comparing
bond prices at each node with the put price and selecting the maximum of the two, Max[Bt, PP].
The same binomial value can also be found by determining the value of the put option at each
node and then pricing the putable bond as the value of an otherwise identical nonputable bond
plus the value of the put option. In using the second approach, the value of the put option, like the
call option, will be the maximum of either its intrinsic value (or exercising value), IV = Max[PPBt,0], or its holding value (the present value of the expected exercising value next period). In
most cases, though, the put’s intrinsic value will be greater than its holding value.2
To illustrate, suppose the three-period, 9% option-free bond in our previous example had a
put option giving the bondholder the right to sell the bond back to the issuer at an exercise price
Since the price of the bond approaches its face value as it approach maturity, the bond’s price
will be increasing if rates do not change. As a result, the put option tends to lose its time value as
the bond approaches maturity and therefore holding value tends to decrease over time.
2
11
of PP = 97. Using the two-period tree of one-period spot rates and the corresponding bond values
for the option-free bond (Figure 9.2-5), we start, as we did with the callable bond, at Period 2 and
investigate each of the nodes to determine if there is an advantage for the holder to exercise. In all
three of the cases in Period 2, the bond price exceeds the exercise price (see Figure 9.3-1a); thus,
there are no exercise advantages in this period and each of the possible prices of the putable bond
are equal to their nonputable values and the values of each of the put options are zero.3 In Period
1, though, it is profitable for the holder to exercise when the spot rate is 11%. At that node, the
value of the nonputable bond is 96.3612, compared to PP = 97; thus the value of putable bond is
its exercise price of 97:
BPu  Max[96.3612, 97]  97.
The putable bond price of 97 can also be found by subtracting the value of the put option from the
price of the non-putable bond (see Figure 9.3-1b). The value of the put option at this node is
.6388,
VuP  Max[IV, VH ]  Max[97  96.3612, 0 ]  .6388;
thus, the value of the putable bond is
Note, even at the relatively high spot rate of 12.5%, the bond price exceeds the put price of 97,
reflecting the fact that the bond price is approaching its face value with one period left to
maturity.
3
12
P
BPu  Bnp
u  Vu
BPu  96.3612  .6388  97.
At the lower node in Period 1, it is not profitable to exercise nor is there any holding value of the
put option since there is no exercise advantage in Period 2. Thus at the lower node, the nonputable bond price prevails. Rolling the two putable bond values in Period 1 to the present, we
obtain a current value of the putable bond of 97.2425:
B0P 
.5[97  9]  .5[98.9334  9]
 97.2425.
110
.
This value also can be obtained using the alternative approach by computing the present value of
the expected put option value in Period 1 and then adding that to the current value of the
nonputable bond. With possible exercise values of .6388 and 0 in Period 1, the current option
value is .2904:
V0P 
.5[.6388]  .5[ 0 ]
 .2904.
110
.
Thus:
B0P  B0np  V0P
B0P  96.9521  .2904  97.2425.
The inclusion of the put option in this example causes the bond price to increase from
96.9521 to 97.2425, reflecting the value of the put option to the bondholder.
13
9.3.3 Sinking-Fund Bonds
Many bonds have sinking fund clauses specified in their indenture requiring that the issuer make
scheduled payments into a fund. The sinking fund agreement may also include a provision
requiring an orderly retirement in which the issuer is to buy up a certain proportion of the bond
issue each period. Often when the sinking fund agreement includes this provision, the issuer is
given an option of either purchasing the bonds in the market or calling the bonds at a specified
call price. This option makes the sinking fund valuable to the issuer. If interest rates are
relatively high, then the issuer will be able to buy back the requisite amount of bonds at a
relatively low market price; if rates are low and the bond price high, though, then the issuer will
be able to buy back the bond on the call option at the call price. Thus, a sinking fund bond with
this type of call provision should trade at a lower price than an otherwise identical non-sinking
fund bond.
Similar to callable bonds, a sinking fund bond can be valued using the binomial tree
approach. To illustrate, suppose a company issues a $15M, three-period bond with a sinking fund
obligation requiring that the issuer sink $5M of face value after the first period and $5M after the
second, with the issuer having an option of either buying the bonds in the market or calling them
at a call price of 98. Assume the same interest rate tree and bond values characterizing the threeperiod, 9% noncallable described in Figure 9.2-5 apply to this bond without its sinking fund
agreement. With the sinking fund, the issuer has two options: At the end of Period 1, the issuer
can buy $5M worth of the bond either at 98 or at the bond’s market price, and at the end of Period
2, the issuer has another option to buy $5M worth of the bond either at 98 or the market price.
The value of the Period 1 option (in terms of $100 face value) is .4243:
14
Page 15 Insert
and the value of the Period 2 call option is .6936:
Page 15 Insert
Note, since the sinking fund arrangement requires an immediate exercise or bond purchase at the
specified sinking fund dates, the possible values of the sinking fund’s call features at those dates
are equal to the intrinsic values. This differs from the valuation of a standard callable bond where
a holding value is also considered in determining the value of the call option.
Since each option represents 1/3 of the issue, the value of the bond’s sinking fund option
is
V0SF  (1 / 3) (.4243)  (1 / 3) (.6936)  .3726,
and the value of the sinking fund bond is 96.645:
NSF
BSF
 V0SF
0  B0
BSF
0  96.9521  .3726  96.5795.
Thus, the total value of the $15M face value issue is $14.4869M:
15
Value of the Issue 
96.5795I
F
G
H100 J
K$15M  $14.4869M.
Or
Value of the Issue 
96.9521I
F
F.3726I$15M  $14.4869M.
$15M  G J
G
J
H100 K H100 K
Like a standard callable bond, a sinking fund provision with a call feature lowers the value of an
otherwise identical non-sinking fund bond.
9.3.3 Convertible Bond
A convertible bond gives the holder the right to convert the bond into a specified number of
shares of stock. Convertibles are often sold as a subordinate issue, with the conversion feature
serving as a bond sweetner. To the investor, convertible bonds offer the potential for a high rate of
return if the company does well and its stock price increases, while providing some downside
protection as a bond if the stock declines. Convertibles are usually callable, with the convertible
bondholder usually having the right to convert the bond to stock if the issuer does call.
Convertible Bond Terms
Suppose our illustrative three-period, 9% bond were convertible into four shares of the underlying
company’s stock. The conversion features of this bond include its conversion ratio, conversion
value, and straight debt value. The conversion ratio (CR) is the number of shares of stock that
can be converted when the bond is tendered for conversion. The conversion ratio for this bond is
four. The conversion value (CV) is the convertible bond’s value as a stock. At a given point in
time, the conversion value is equal to the conversion ratio times the market price of the stock
16
(PtS):
CVt  (CR) PtS .
If the current price of the stock were 92, then the bond’s conversion value would be CV =
(4)($92) = $368.4 Finally, the straight debt value (SDV) is the convertible bond’s value as a
nonconvertible bond. This value is obtained by discounting the convertible’s cash flows by the
discount rate on a comparable non-convertible bond.
Minimum and Maximum Convertible Bond Prices
Arbitrage ensures that the minimum price of a convertible bond, is the greater of either its straight
debt value or its conversion value:
Min BCB
 Max[CVt SDVt ].
t
If a convertible bond is priced below its conversion value, arbitrageurs could buy it, convert it to
stock, and then sell the stock in the market to earn a riskless profit. Arbitrageurs seeking such
opportunities would push the price of the convertible up until it is at least equal to its CV.
Similarly, if a convertible is selling below its SDV, then arbitrageurs could profit by buying the
convertible and selling it as a regular bond.
In addition to a minimum price, if the convertible is callable, the call price at which the
issuer can redeem the bond places a maximum limit on the convertible. That is, the issuer will
find it profitable to buy back the convertible bond once its price is equal to the call price. Buying
Another convertible bond term is its conversion price. The conversion price is the bond’s par
value divided by the conversion ratio: F/CR. This bond has a conversion price of 25. The
conversion price is only applicable when the bond is trading at par. Many convertible bond
contracts, though, specify terms (e.g., the conversion ratio) in terms of the conversion price.
4
17
back the bond, in turn, frees the company to sell new stock or bonds at prices higher than the
stock or straight debt values associated with the convertible. Thus, the maximum price of a
convertible is the call price. The actual price of the convertible will be at a premium above its
minimum value but below the maximum.
Valuation of Convertibles Using Binomial Trees
The valuation of a convertible bond with an embedded call is more difficult than the valuation of
a bond with just one option feature.
In the case of a callable convertible bond, one has to
consider not only the uncertainty of future interest rates, but also the uncertainty of stock prices.
A rate decrease, for example, may not only increase the convertible’s SDV and the chance the
bond could be called, but if the rate decrease is also associated with an increase in the stock price,
it may also increase the conversion value of the convertible and the chance of conversion. The
valuation of convertibles therefore needs to take into the random patterns of interest rates, stock
prices, and the correlation between them.
To illustrate the valuation of convertibles, consider a three-period, 10% convertible bond
with a face value 1000, which can be converted to 10 shares of the underlying company’s stock
(CR = 10). To simplify the analysis, assume the bond has no call option and no default risk, that
the current yield curve is flat at 5%, and that the yield curve will stay at 5% for the duration of the
three periods (i.e., no market risk). In this simplified world, the only uncertainty is the future
stock price. Like interest rates, suppose the convertible bond’s underlying stock price follows a
binomial process in which each period it can either increase to equal u times its initial value or
decrease to equal d times the initial value, where u = 1.1, d= 1/1.1 = .9091, and the current stock
price is 92. The possible stock prices resulting from this binomial process are shown in Figure
18
9.3-2, along with the convertible bond’s conversion value.
Since spot rates are assumed constant, the value of the convertible bond will only depend
on the stock price. To value the convertible bond, we start at the maturity date of the bond. At that
date, the bondholder will have a coupon worth 100 and will either convert the bond to stock or
receive the principal of 1000. At the top stock price of 122.45, the convertible bondholder would
exercise her option, converting the bond to ten shares of stock. The value of the convertible bond
at the top node in Period 3 would therefore be equal to its conversion value of 1224.50 plus the
$100 coupon:
BCB
uuu  Max[ CV, F]  C
 Max[1224.50,1000]  100
 1324.50.
Similarly, at the next stock price of 101.20, the bondholder would also find it profitable to
convert; thus, the value of the convertible in this case would be its conversion value of 1012 plus
the $100 coupon. At the lower two stock prices in Period 3 of 83.6366 and 69.12, conversion is
worthless; thus, the value of the convertible bond is equal to the principal plus the coupon: 1100.
In Period 2, at each node the value of the convertible bond is equal to either the present
value of the bond’s expected value at maturity or its conversion value. At all three stock prices,
the present values are greater than the bond’s conversion values, including at the highest stock
price; that is, at PSuu = 111.32, CV is 1113.20 compared to the convertible bond value of 1160.24:
BCB
uu 
.5[1324.50]  .5[1112]
 1160.24.
105
.
19
Thus, in all three cases, the values of holding the convertible bond option are greater than the
exercising values. Similarly, the two possible convertible bond values in Period 1 also exceed
their conversion values. Rolling the tree to the current period, we obtain a convertible bond value
of 1164.29. As we would expect, this value exceeds both the convertible bond’s current
conversion value of 920 and its SDV of 1136.16:
SDV 
100
100
1100


 113616
. .
2
105
.
(105
. )
(105
. )3
As noted, the valuation of a convertible become more complex when the bond is callable.
With callable convertible bonds, the issuer will find it profitable to call the convertible prior to
maturity whenever the price of the convertible is greater than the call price. However, whenever
the bondholder is faced with a call, she usually has the choice of either tendering the bond at the
call price or converting it to stock. Since the issuer will call whenever the call price exceeds the
convertible bond price, he is in effect forcing the holder to convert. By doing this, the issuer takes
away the bondholder’s value of holding the convertible, forcing the convertible bond price to
equal its conversion value.
To see this, suppose the convertible bond is callable at CP = 1100. At the top stock price
of 111.32 in Period 2, the conversion value is 1113.20 (see Figure 9.3-3). In this case, the issuer
can force the bondholder to convert by calling the bond. The call option therefore reduces the
value of the convertible from 1160.24 to 1113.20. In period 1, the call price of $1100 is below
both the conversion value (1012) and the bond value (1126.92). In this case, the issuer would call
the bond and the holder would take the call instead of converting. The value of the callable
convertible bond in this case would be the call price of 1100. Rolling this value and period 1’s
lower node value to the current period, we obtain a value for the callable convertible bond of
20
1140.80, which is less than the noncallable convertible bond value of 1164.29 and greater than the
straight debt value of a noncallable bond of 1136.16.
In the above two cases, we assumed for simplicity that the yield curve remained constant
at 5% for the period. As noted, the complexity of valuing convertibles is taking into account the
uncertainty of two variables – stock prices and interest rates. Looking historically at stock prices
and general interest rate, there is some evidence to suggest that for many stocks, there is a
negative correlation between their prices and interest rates. Often in high interest rate periods
(e.g., late 1970s, early 1980s, and the early1990s), stock prices tends to be relatively low (bear
market), and in periods of relatively low rates, stock prices tend to be comparatively high (bull
market). Such historical trends suggest that for many stocks there is an inverse relation between
their prices and interest rates.
A simple way to model such behavior is to use correlation or regression analysis to first
estimate the relationship between a stock’s price and the spot rate, and then either with a binomial
model of spot rates identify the corresponding stock prices or with a binomial model of stock
prices identify the corresponding spot rates. For example, suppose using regression analysis, we
estimated the following relationship between the stock in our above example and the one-period
spot rate:
St  .16  .001PtS .
Using this equation, the corresponding spot rates associated with the stock prices from the threeperiod tree would be
21
PtS
122.45
111.32
101.20
92.00
83.64
78.76
69.12
St
3.800%
4.870%
5.900%
6.800%
7.636%
8.124%
9.090%
Figure 9.3-4 shows the binomial tree of stock prices along with their corresponding spot
rates. Given the rates and stock prices, the methodology for valuing the convertible bond is
identical to our previous analysis. Again, we start at maturity where we value the bond at each
node as the maximum of either its conversion value or face value. Given these values, we then
move to Period 2, where we first determine the present value of Period 3's expected bond values
using the spot rates we’ve estimated. We then compare each of those values with the call price
and the conversion value. As we noted in the previous case, if the call price exceeds the
convertible bond value, then the issuer will call the bond, and depending on which is more
profitable, the bondholder will either accept the call and receive the call price or will convert to
stock. As in our previous case, at the top node in Period 2, the call price of 1100 is below the
convertible bond value of 1161.67. In this case, the issuer will call the bond and the holder will
find it more profitable to convert; that is, the conversion value of 1113.20 exceeds the call price of
1110.
Thus, the convertible bond would be equal to the conversion value. The other two
convertible bond prices in Period 2 are less than the call price, implying the issuer would not
exercise; the prices also are greater than the conversion value, implying the holder would not
convert. Thus, the values of the convertibles in these two cases are equal to the present values of
their expected cash flows for the next period. In period 1, the call price is less than the bond value
22
(1108.96) and conversion value (1012) at the top node. In this case the issuer would call and the
bondholder would find it better to accept the call instead of converting; thus, the convertible bond
price at this node would be the call price of 1100. Rolling the tree to the current period with this
value and the lower node value, we obtain a convertible bond value of 1098.57. This value is
lower than the previous case in which we assumed a constant yield curve at 5%. The price is also
lower than the SDV. That is, using the tree the value of the bond as a nonconvertible, callable
bond is 1086.09 (these calculations are left to the reader).
It should be noted that modeling a bond with multiple option features and influenced by
the random patterns of more than one factor is more complex in practice than the simple model
described above. The above model is intended only to provide some insight into the dynamics
involved in valuing a bond with embedded convertible and call options given different interest
rate and stock price scenarios.
9.4 CONCLUSION
In 1985, Merrill Lynch introduced the liquid yield option note (LYON). The LYON is a zerocoupon bond, convertible into the issuer’s stock, callable, with the call price increasing over time,
and putable with the put price increasing over time. The LYON is a good example of how
innovative the investment community can be in structuring debt instruments with option clauses.
This type of innovation has also been quite extensive in the construction of mortgage-backed
securities. Faced with the problem of prepayment risk, various types of mortgage securities were
created in the 1980s with different claims that addressed such risk. Mortgage-backed securities,
LYONs, and other securities with embedded option features all can be evaluated, though, using a
binomial tree. (Some of the end-of-the-chapter problems call for the valuation of bonds with
23
multiple option features.) In Chapter11 we will examine mortgage-backed securities and many of
their derivative securities which have option features, and in Part 4 we will examine options and
futures contracts on debt securities and interest rates offered by exchanges and financial
institutions. Before examining these securities, though, we first need to address a more
fundamental question of how we estimate the binomial tree. This is the subject of the next
chapter.
24
Fi g ure 9 .2 -1 :B in om i al T ree o f O n ePeri od Sp o t R a tes
S uu u  u 3 S 0
Suu  u 2 S0
S u  uS 0
S u u d  u 2 dS 0
S u d  udS 0
S0
S u d d  ud 2 S 0
S d  dS 0
Sdd  d S0
2
Sddd  d S 0
3
25
F ig u re 9 .2 -2 : B in o m ia l T ree
u = 1 .1 , d = .9 5 , S o = .1 0
S u u u  .1 33 1
S u u  .121
S u  .11
S0  .10
S u u d  .11495
S u d  .1045
S u d d  .09927
Sd  .095
Sdd  .09025
S d d d  .08 57
26
Figure 9.2-3: Value of TwoPeriod Option-Free Bond:
C = 8 and F =100
Buu  108
Bu 
.5[97.297  8].5[98.630  8]
110
.
B0  96330
.
B0 
108
 97.297
111
.
B ud  108
Bd 
108
 98.630
1.095
Bdd  108
27
Figure 9.2-4a
Figure 9.2  4a: Value of
2  Period , 9% Callable
B uu  108
bond with CP  98
BuC  Min[ BuNC , CP ]
 Min[97.297, 98]
 97.297
.597297
[ . 8].598
[ 8]
110
.
B0 96044
.
B0 
B uu  1 0 8
BdC  Min[ BdNC , CP ]
 Min[98.630, 98]
 98
B uu  108
Figure 9.2-4b
Figure 9.2-4b
Value of the Call Option and
Callable Bond
VuC  Max[ BuNC  CP, 0]
 Max[97.297  98, 0]  0
BuC  Bunc  VuC
.5[0].5[.630]
110
.
 .2864
 97.297  0  97.297
V0C 
B0C  B0nc  V0C
VdC  Max[ BdNC  CP , 0 ]
 96.330  .2864  96.044
 Max[ 98.630  98, 0 ]
 .630
BdC  Bdnc  VdC
 98.630  .630  98
28
Figure 9.2-5
Value of Three-Period Option-Free Bond
C = 9, F = 100
1 0 9
B
u u

1 0 9
1 .1 2 1
 9 7 .2 3 4 6
Bu 
.5[ 9 7 .2 34 6  9 ]  .5[ 9 8 .6 87 2  9 ]
1 0 9
1.1 1
 9 6. 3 61 2
B0 
.5 [ 9 6 .3 61 2  9 ]  .5 [ 9 8 .9 33 4  9 ]
B
1 .1 0
u d
 9 6 .9 52 1

1 0 9
1 .1 0 4 5
 9 8 .6 8 7 2
Bd 
.5[ 9 6 .6 8 7 2  9 ]  .5[ 9 9 .9 7 7  9 ]
1 .0 9 5
1 0 9
 9 8.9 33 4
B
d d

1 0 9
1 .0 9 0 2 5

9 9 .9 7 7
1 0 9
Path 1
10%
9.50%
9.03%
Path 3
10%
11.00%
10.45%
Period t
1
2
3
yt0
0.1
0.09749715
0.0950761
CF
9
9
109
PV
Path 2
Period t
yt0
8.181818
10%
1
0.1
7.47198
9.50%
2
0.09749715
83.00296 10.45%
3
0.09982649
98.65676
Period t
yt0
CF
PV
Path 4
Period t
yt0
1
0.1
9
8.181818
10%
1
0.1
2
0.10498869
9
7.371007 11.00%
2
0.10498869
3
0.10482577
109
80.82489 12.10%
3
0.11030022
96.37771
Theoretical Value = (98.65676 + 97.58588 + 96.37771 + 95.18805)/4 = 96.9521
29
CF
9
9
109
CF
9
9
109
PV
8.181818
7.47198
81.93208
97.58588
PV
8.181818
7.371007
79.63523
95.18805
Figure 9.2-6a
Value of Three-Period Callable Bond
Exhibit 3a: Value of Callable
B
C

uu
M i n [ 9 7.2 3 4 6 , 9 8]

Bond: C = 9, F = 100
9 7.2 3 4 6
.5[ 9 7.234 6  9 ]  .5 [ 9 8  9 ]
Bu 
11
. 1
 9 6.051 6
C
Bu  Min[ 9 6.051 6 , 9 8]  9 6.051 6
B0 
C
.5[ 9 6. 0 5 1 6  9 ]  . 5[ 9 7. 7 1 6 9  9 ]
C
B
ud
1.1 0

M i n [ 9 8 .6 8 7 2 , 9 8 ]

9 8
 9 6. 2 5 8
Bd 
. 5[ 9 8  9 ]  .5 [9 8  9 ]
1. 0 9 5
 9 7 .7 1 6 9
Bd  M in [ 9 7 .7 1 6 9 , 9 8 ]  9 7 .7 1 6 9
C
C
B
dd

M i n [ 9 9 .9 7 7 , 9 8 ]

9 8
Figure 9.2-6b
NC
F i gu re 8. 2 - 7: V al ue of C a llab le
Buu  97 .2346
B on d : C = 9, F = 10 0:
Vuu  IV  0
B
C
t

nc
Bt
 Vt
C
C
nc
BuuC  Buu
 V uuC  97 .2346
NC
Bu
 9 6 .3 6 1 2
IV  0
VH 
C
Bu
NC
B0
VH 
.5 ( 0 )  .5 ( .6 8 7 2 )
 .3 0 9 5
1 .1 1
 9 6 .3 6 1 2  .3 0 9 5  96 .0 5 1 6
 9 6 .9 5 2 1
.5 (. 3 0 9 5 )  .5 (1 .2 1 6 5 )
NC
B ud  9 8 .6 8 7 2
C
 I V  .6 8 7 2
V ud
 .6 9 3 6
1 .1 0
B
B 0  9 6 .9 5 2 1  .6 9 3 6  9 6 .2 5 8
C
NC
Bd
C
ud
nc
C
 B ud  V ud  9 8
 9 8 .9 3 3 4
I V  .9 3 3 4
VH 
.5 ( .6 8 7 2 )  .5 ( 1.9 7 7 )
 1 .2 1 6 5
1.0 9 5
C
B d  9 8 .9 3 3 4  1.2 1 6 5  9 7 .7 1 6 9
NC
Bd d  9 9.97 7
V dd  IV  1 .97 7
C
C
nc
C
Bd d  B d d  V d d  9 8
30
Figure 9.3-1a
Value of Putable Bond
V a lu e o f P u ta b le B o nd :
S u u  12.1%, B unup  97.2346
C = 9 , F = 1 0 0 , P P = 97
Bu u  Max[ 97.2346, 97]
P
Su  11%
Bu 
 97.2346
.5 [ 9 7 .2 34 6  9 ]  .5[ 9 8 .6 87 2  9 ]
1.1 1
 9 6 .3 61 2
P
Bu  M a x [ 9 6.3 61 2 , 9 7 ]  9 7
S u d  10.45%, B u d  98.6872
np
S0  10 %
.5 [ 9 7  9 ]  .5 [9 8 .9 33 4  9 ]
B0 
P
BuPd  Max[98.6872, 97]
1 .1 0
 98.6872
 9 7 .2 42 5
S d  9.5%
Bd 
.5[ 98.6872  9 ]  .5[ 99.977  9 ]
111
.
 98.9334
B
P
d
S d d  9.025%, B dndp  99.977
 Max [ 98.9334 , 97 ]  98.9334
BdPd  Max[99.977, 97]
 99.977
Figure 9.3-1b
F ig ure 8. 3 - 1b: V al ue o f P ut ab le
np
Su u  12 .1%, B u u  97 .2346
B o nd: C = 9 , F = 1 00:
V u u  IV
P
B Pt  B tnp  V t P
 Max [97  97 .2346, 0]  0
S u  11%, B u  96 .3612
np
IV  M ax [ 97  96 .3612 , 0 ]  .6388
B  B u u  V u u  97 .2346
P
uu
np
P
VH  0
V uP  M ax [ IV , V H ]  .6388
P
np
P
B u  B u  V u  96 .3612  .6388  97
V u d  IV
P
S 0  1 0 % , B 0  9 6 .9 5 2 1
np
VH 
P
.5 (. 6 3 8 8 )  .5 ( 0 )
 Max [97  98 .6872 , 0]  0
 .2 9 0 4
1 .1 0
np
B0  B0  V0
np
Su d  10. 45%, B u d  98 .6872
B
P
P
ud
 B u d  Vu d  98.6872
np
P
 9 6 .9 5 2 1  .2 9 0 4  9 7 .2 4 2 5
S d  9.5%, B dnp  98 .9334
IV  M ax [ 97  98.9334 , 0 ]  0
VH  0
V dd  IV
P
P
V d  M ax [ IV , V H ]  0
P
np
S dd  9.025%, B dd  99 . 977
np
 Max [ 97  99 .977 , 0]  0
P
B d  B d  V d  98.9334  0  98.9334
31
B
P
dd
 B dd  V dd  99 .977
np
P
Page 15 Insert
Value of Sinking Fund Call
Su  11%
nsf
Bu
Value of Sinking Fund
Call in period 1
Vu
SF
 96 . 3 6 12
 IV  M ax [ 96 . 3 6 12  98 , 0 ]  0
Su  11%
V0
SF

. 5 ( 0 )  .5 ( . 9 3 3 4 )
 .4 2 4 3
1 .1 0
S d  9 .5%
 98.9334
n sf
Bd
Vd
SF
 IV  M ax[ 98. 9334  98 , 0 ]  .9334
Page 15 Insert
S uu  1 2 .1 %
n sf
Value of Sinking Fund
Call in period 2
B uu
S u  1 1%
SF
Vu

.5 ( 0 )  .5 ( . 6 8 7 2 )
V
SF
uu
 9 7. 2 3 4 6
 M a x [ 9 7. 2 3 4 6  9 8 , 0 ]
 0
1.1 1
 .3 0 9 5
S ud  1 0 .4 5 %
S0  1 0 %
V 0S F 
ns f
B ud
 9 8 .6 8 7 2
.5 ( .3 0 9 5 )  .5 (1 .2 1 6 5 )
V udSF  M a x [ 9 8 .6 8 7 2  9 8 , 0 ]
1.1 0
 .6 9 3 6
 .6 8 7 2
S d  1 1%
Vd
SF

.5 (. 6 8 7 2 )  .5 ( 1 .9 7 7 )
1 .0 9 5
 1.2 1 6 5
S dd
 9 .0 2 5 %
B ddns f  9 9 . 9 7 7
V ddSF  M a x [ 9 9 . 9 7 7  9 8 , 0 ]
 1 .9 7 7
32
F ig u re 9 .3  2
Convertible Bond
C  100, F  1000
S
Puuu
 122.45
CV  1224.50
P  11132
.
CV  111320
.
.
5
(
1324
.50) .5(1112)
BuuCB 
105
.
 1160.24
S
uu
For Stock: u  11
. , d  .9091
Convertible: CR  10
PuS  10120
.
B
 1000
C
 100
CB
Buuu
 Max[CV , F ]  C
 1324.50
S
Puud
 10120
.
CV  1012
CV  1012
CB
u
F
.5(1160.24  100) .5(1053.33  100)

105
.
 1149.64
F
 1000
C
 100
CB
uud
B
PudS  92
CV  920
P0S  92
CV  920
.5(1112 )  .5(1100)
1.05
 1053.33
CB
Bud

.5(1149.32  100) .5(1095.69  100)
105
.
 1164.29
B0CB 
PdS  8364
.
CV  836.40
.5(105333
.  100) .5(1047.62  100)
105
.
 109569
.
S
Pudd
 83.64
CV  836.40
F
 1000
C
 100
CB
udd
B
PdSd  7 8 .7 6
1100
1.0 5
 1 0 4 7 .6 2
B dCdB 
S
Pddd
 69.12
CV  69120
.
F  1000
C
CB
ddd
B
33
 Max[CV , F ]  C
 1100
BuCB 
C V  7 8 7 .6 0
 Max[CV , F ]  C
 1112
 100
 Max[CV , F ]  C
 1100
Figure 9.3  3
Callable Convertible Bond
PuuS  111.32, Suu  5%
CP  1100, C  100, F  1000
CV  1113.20, CP  1100
For Stock : u  11
. , d  .9091
.5[1324.50].5[1112]
1.05
 1160.238
BuuCB / NC 
Convertible: CR  10
P  10120
. , Su  5%
S
u
BuuCB / Call  1113.20
CV  1012, CP  1100
.5[1113.20  100]
P  92, S0  5%
S
0
CV  920, CP  1100
.5[1100  100]
CB / C
0
B
.5[1095.69  100]

105
.
 1140.80
CV  122350
.
F  1000, C  100
CB
Buuu
 Max[CV , F ]  C
 1324.50
S
Puud
 10120
.
CV  1012
.5[1035.58  100]
105
.
 1126.92
BuCB / NC 
PudS  92, Sud  5%
CV  920, CP  1100
BuCB / Call  1100
BudCB / NC
P  83.64, Sd  5%
CV  836.40, CP  1100
S
d
BdCB / NC
S
Puuu
 122.45
.5[1112].5[1100]

105
.
 1053.33
BudCB / Call  1053.33
.5[1053.33  100]
.5[1047.62  100]

105
.
 1095.69
PddS  78.76, Sdd  5%
F  1000, C  100
CB
Buud
 Max[CV , F ]  C
 1112
S
Pudd
 83.64
CV  836.40
F  1000, C  100
CB
Budd
 Max[CV , F ]  C
 1100
CV  786.60, CP  1100
BdCB / Call  1095.69
1110
105
.
 1047.62
BddCB / NC 
BddCB / Call  1047.62
S
Pddd
 69.12
CV  69120
.
F  1000, C  100
CB
Bddd
 Max[CV , F ]  C
 1100
34
Figure 9.3  4
Callable Convertible Bond
CP  1100, C  100, F  1000
PuuS  111.32, Suu  4.87%
CV  1113.20, CP  1100
For Stock : u  11
. , d  .9091
S  .16  .001P S
Convertible: CR  10
S
Puuu
 122.45
CB / NC
uu
B
P  10120
. , Su  5.9%
S
u
.5[1324.50].5[1112]

10487
.
 1161.67
BuuCB / Call  1113.20
CV  1012, CP  1100
.5[1113.20  100]
.5[103558
.  100]
1.059
 1108.96
BuCB / NC 
P0S  92, S0  6.8%
CV  920, CP  1100
.5[1100  100]
B0CB / C
.5[1046.55  100]

1068
.
 1098.57
CB / Call
u
B
 1100
CV  920, CP  1100
CB / NC
ud
B
.5[1112].5[1100]

1068
.
 1035.58
BudCB / Call  1035.58
PdS  83.64, Sd  7.636%
CV  836.40, CP  1100
BdCB / NC
P  92, Sud  6.8%
S
ud
.5[103558
.  100]
.5[1017.35  100]

107636
.
 1046.55
BdCB / Call  1046.55
CV  1223.50
F  1000, C  100
CB
Buuu
 Max[CV , F ]  C
 1324.50
S
Puud
 10120
.
CV  1012
F  1000, C  100
CB
Buud
 Max[CV , F ]  C
 1112
S
Pudd
 83.64
CV  836.40
F  1000, C  100
CB
Budd
 Max[CV , F ]  C
PddS  78.76, Sdd  8124%
.
 1100
CV  786.60, CP  1100
1110
108124
.
 1017.35
BddCB / NC 
BddCB / Call  1017.35
S
Pddd
 69.12
CV  69120
.
F  1000, C  100
CB
Bddd
 Max[CV , F ]  C
 1100
35