estimating the binomial tree

8/10/2019 ESTIMATING THE BINOMIAL TREE

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CHAPTER 10

ESTIMATING THE BINOMIAL TREE

10.1 INTRODUCTION

With most bonds having some option clause, the binomial approach to valuing bonds is bynecessity becoming the standard pricing model for bonds. In the last chapter, weve examinedhow the tree can be used to price bonds with call and put options, sinking fund agreements, andconvertible clauses. We have not addressed, though, the more fundamental problem of how toestimate the tree. In this chapter, we examine two models that have been used for estimatingbinomial interest rate trees. Before we describe these model, though, we first need to look athow we can subdivide the tree so that the assumed binomial process is defined in terms of arealistic lengths of time, with a sufficient number of possible rates at maturity.

10.2 SUBDIVIDING THE BINOMIAL TREE

The binomial model is more realistic when we subdivide the periods to maturity into a numberof subperiods. That is, as the number of subperiods increases, the length of each period becomessmaller, making the assumption that the spot rate will either increase or decrease more plausible,and the number of possible rates at maturity increases, which again adds realism to the model.or example, suppose the three!period bond in our illustrative example in "hapter # were athree!year bond. Instead of using a three period binomial tree, where the length of each periodis a year, suppose we evaluate the bond using a six!period tree with the length of each periodbeing half a year. If we do this, we need to divide the one!year spot rates and the annual couponby two, ad$ust the u and d parameters to reflect changes over a six!month period instead of one

year %this ad$ustment will be discussed in the next section&, and define the binomial tree of spotrates for five periods, each with a length of six months. igure '(.)!' shows a five!periodbinomial tree for one!year spot rates with the length of each period being six months, u *'.(+##,and d * .-+-.

iven the spot rates in igure '(.)!', the value of a three!year, / option!free bondand callable bond with "0 of # are shown in igures '(.)!). 1ote, in valuing the bonds at eachnode, we use a 2+3 coupon %reflecting the accrued interest& and discount at a six!month rate%annual spot rate divided by )&. In examining the tree for the callable bond, observe that inperiod 3 it is advantageous for the issuer to call the bond at all five interest rate cases, includingthe higher rates. This reflects the fact that the bond prices are approaching their face value. 4lsonote, that except at the lowest node in 0eriod +, the holding values of the call e5ual or exceedthe intrinsic values at all of the earlier nodes.

In this example, we valued a three!year bond with a five!period tree of one!year spotrates, with the length of each period being six months. If we wanted to value the bond every5uarter, then we would need a ''!period tree of spot rates with the length of each period beingthree months and with the annual rates divided by + and u and d ad$usted to reflect movementsover three months. In general, let6

'


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h * length of the period in years7n * number of periods of length h defining the maturity of the bond,

where n * %maturity in years&8h.

Thus, a three!year bond evaluated over 5uarterly periods %h * '8+ of a year&, would have a

maturity of n * %9 years&8%'8+ years& * ') periods and would re5uire a binomial interest rates treewith n!' * ')!' * '' periods. To evaluate the bond over monthly periods %h * '8') of a year&,the bonds maturity would be n * %9 years&8%'8') years& * 9: periods and would re5uire a 93!period binomial tree of spot rates7 for weekly periods %h * '83)&, the bonds maturity would be'3: periods of length one week, and we would need a '33!period tree of spot rates. Thus, bysubdividing we make the lengths of each period smaller, which makes the assumption of onlytwo rates at the end of one period more plausible, and we increase the number of possible ratesat maturity.

10.3 ESTIMATING THE BINOMIAL TREE

In practice, determining the value of a bond using a binomial tree re5uires that we be able toestimate the random process that spot rates follow. There are two general approaches toestimating binomial interest rate movements. The first %models derived by ;endleman andBartter %'#(& and "ox, Ingersoll, and ;oss %%'#3&& is to estimate the u and d parameters basedon estimates of the spot ratess mean and variability. The estimating formula for u and d areobtained by solving for the u and d values which make the mean and variance of spot ratesresulting from a binomial process e5ual to their estimated values. Bond values obtained usingthis approach can then be compared to actual bond prices to order to determine if the bond ismispriced. The second approach %models derived by Black,


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e5ual to a proportion d times the initial rate, with probability of the increase in one period being5 * .3. 4t the end of n periods, this binomial process yields a distribution of nC' possible spotrates %e.g., for n * 9, there are four possible rates6 A uuu* u9A(, Auud* u)dA(, Audd* ud) A(, and Addd* d9A(&. This distribution, though, is not normally distributed since spot rates cannot be negative%i.e., we normally do not have negative interest rates&. =owever, the distribution of spot rates can

be converted into a distribution of logarithmic returns, gn, where6

This distribution can take on negative values and will be normally distributed if 5 *.3. igure'(.9!' shows the binomial distributions of spot rates for n * ', ), and 9 periods and theircorresponding logarithmic returns for the case in which u * '.', d * .3, A ( * '(/, and 5 * .3.4s shown in the figure, when n * ', there are two possible spot rates of ''/ .3/, withrespective logarithmic returns of .(39 and !.(3'96

when n * ), there are three possible spot rates of ').'/, '(.+3/, and .()3/ withcorresponding logarithmic returns of

9

g S

Sn

n= ln .(

g uS

Su

g dS

Sd

u

d

= HG KJ = = =

=FHG

IKJ = = =

ln ln ln . .

ln ln ln . . 7

(

(

(

(

'' (39

3 (3'9

bg bg

bg bg

g u S

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g udS

Sud

g d S

Sd

uu

ud

dd

= HG KJ = = =

=FHG

IKJ = = =

=FHG

IKJ = = =

ln ln ln . .

ln ln ln % . &%. & .

ln ln ln . . 7

)

(

(

) )

(

(

)(

(

) )

'' '(:

'' 3 (++

3 '():

ch c h

bg b g

ch c h


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when n * 9, there are four possible spot rates of '9.9'/, ''.+3/, .)-3/, and #.3-+/, withlogarithmic returns of

The probability of attaining any one of these rates is e5ual to the probability of the spot rateincreasing $ times in n periods, pn$. That is, the probability of attaining spot rate '(.++/ in0eriod ) is e5ual to the probability of the spot rate increasing one time %$ * '& in two periods %n* )&, p)'. In a binomial process this probability can be found using the following formula6'

Thus after two periods, the probability of the spot rate e5ualing '.(:/ is p))* .)3, '(.+3/ isp)'*.3, and .()3/ is p)(*.)3. Dsing these probabilities, the expected value and the variance ofthe distribution of logarithmic returns after two periods would be e5ual to E%g n& * +.+/ andF%gn& * .('(#6

The means and variances for the three distributions are shown at the bottom of igure'(.9!'. In examining each distributions mean and variance, note that as the number of periodsincreases, the expected value and variance increase by a multiplicative factor, such that E%rn& *nE%r'& and F%rn& * nF%r'&. 4lso, note that the expected value and the variance are also e5ual to

'nG %;ead as n factorial& is the product of all numbers from n to '7 also (G * '.

+

g u S

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g u dS

Su d

g ud S

Sud

g d S

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uuu

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= HG KJ = = =

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IKJ = = =

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IKJ = = =

ln ln ln . .

ln ln ln % . &%. & .

ln ln ln % . &%. & .

ln ln ln . . .

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ch c h

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j n j=

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% &G G% & .'

E g

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% & . %. & . %. & . %. & . .

% & . H. . . H. . . H. . . .

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)3 '(: (++ 3 (++ (++ )3 '(): (++ ('(#

= + + =

= + + =


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S"(n+ ,"$ u and d

iven the features of a binomial distribution, the formulas for estimating u and d are found bysolving for the u and d values which make the expected value and the variance of the binomialdistribution of the logarithmic return of spot rates e5ual to their respective estimated parametervalues, under the assumption that 5 * .3 %or e5uivalently that the distribution is normal&. If we

let and eand Fe be the estimated mean and variance of the logarithmic return of spot rates for a

period e5ual in length to n periods, then our ob$ective is to solve for the u and d whichsimultaneously satisfy the following e5uations6

If 5 * .3, then the values for u and d %formulas for the values& which satisfy the two e5uationsare

%Aee the appendix at the end of his chapter for the algebraic derivation.&In terms of our example, if the estimated expected value and variance of the logarithmic

return were .(++ and .('(#, respectively, for a period e5ual in length to n * ), then using theabove e5uations, u would be '.' and d would be .36

Annua(*d M*an and Va$an%*

In order to facilitate the estimation of u and d for a number of bonds with different maturates, it

is helpful to use an annualied mean and variance %4eand F4e & &. 4nnualied parameters are

3

E g nE g n q u q d

V g nV g n q q u d

n

n

% & % & H ln % & ln

% & % & % & ln% 8 & .

= = +

= =

'

'

)

'

'

nE g n q u q d

nV g n q q u d V

e

e

% & H ln % & ln

% & % & ln% 8 & .

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)

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'

= + =

= =

u e

d e

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e e

=

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8 8

8 8

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= =

= =

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+

. 8 . 8

. 8 . 8

.

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('(# ) (++ )

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3


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obtained by simply multiplying the estimated parameters of a given length by the number ofperiods of that length that make up a year. or example, if 5uarterly data is used to estimate the

mean and variance %5eand F5e&, then we simply multiply those estimates by four to obtain the

annualied parameters %4e* +5eand %F4e* +F5e&. Thus, if the estimated 5uarterly mean and

variance were .()) and .((3+, then the annualied mean and variance would be .(## and .()':,

respectively)

. 1ote, when the annualied mean and variance are used, then these parameters mustby multiplied by the proportion h, defined earlier as the time of the period being analyedexpressed as a proportion of a year and n is not needed since h define the length of trees periodtree6

If the annualied mean and variance of the logarithmic return of one!year spot rates were .(++and .('(#, and we wanted to evaluate a three!year bond with six!month periods %h * J of a

year&, then we would use a six!period tree to value the bond %n * %9 years&8%J& * : periods& and uand d would be '.' and .36

If we make the length of the period monthly %h*'8')&, then we would value the three!year bondwith 9:!period tree and u and d would be e5ual to '.(9+)+ and .-+(69

Est!atn+ A*and VA

*Usn+ Hst"$%a( Data

To estimate u and d re5uires estimating the mean and variance6 eand Fe.The simplest way to do

this is to estimate the parameters using the average mean and variance from an historical sample

)1ote, the annualied standard deviation cannot be obtained simply by multiplying the 5uarterlystandard deviation by four. ;ather, one must first multiply the 5uarterly variance by four andthen take the s5uare root of the resulting annualied variance.

91ote, in the e5uations for u and d, as n increases the mean term in the exponent goes to ero

5uicker than the s5uare root term. 4s a result, for large n %n * 9(&, the mean terms impact on uand d is negligible and u and d can be estimates as

u e and d eu

V n V ne e= = =8 8

.'

.

:

u e

d e

hV h

hV h

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eA

eA

eA

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u e

d e

= =

= =

+

+

% 8 &%. & % 8 &%. &

% 8 &%. & % 8 &%. &

.

. .

' ) ('(# ' ) (++

' ) ('(# ' ) (++

''(

3

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= =

= =

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+

% 8 &%. & % 8 &%. &

% 8 &%. & % 8 &%. &

.

. .

' ') ('(# ' ') (++

' ') ('(# ' ') (++

'(9+)+

-+(

g g

b gb g


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of spot rates %such as the rates on T!Bills&. 4s an example, historical 5uarterly one!year spot ratesover '9 5uarters are shown in Table '(.9!'. The ') logarithmic returns are calculated by takingthe natural log of the ratio of spot rates in one period to the rate in the previous period %A t8At!'&.rom this data, the historical 5uarterly logarithmic mean return and variance are6

@ultiplying the historical 5uarterly mean and variance by +, we obtain an annualied mean andvariance, respectively, of ( and .(':#9:.

iven the estimated annualied mean and variance, u and d can be estimated once wedetermine the number of periods to subdivide %see bottom of Table '(.9!'&.

/*atu$*s

The u and d formulas derived here assume an interest rate process in which the variance andmean are stable and where the end!of!the!period distribution is symmetrical. Kther models canbe used to address cases in which these assumptions do not hold. @ertons mixed diffusion!$ump model, for example, accounts for the possibilities of infre5uent $umps in the underlyingprice or interest rate, and "ox and ;osss constant elasticity of variance model is applicable forcases in which the variance is inversely related to the underlying price or rate. ?ohnson,0awlukiewic, and @ehta, in turn, have developed a skewness!ad$usted binomial model forpricing securities when the distribution of logarithmic returns is characteried by skewness.

4 binomial interest rates tree generated using the u and d estimation approach is

constrained to have an end!of!the!period distribution with a mean and variance that matches theanalysts estimated mean and variance. The tree is not constrained, however, to yield a bondprice that matches its e5uilibrium value. 4s a result, analysts using such models need to makeadditional assumptions about the risk premium in order to explain the bonds e5uilibrium price.In contrast, calibration models are constrained to match the current term structure of spot ratesand therefore yield bond prices that are e5ual to their e5uilibrium values.

10.3-2 Ca('$at"n M"d*(

The calibration model generates a binomial tree by first finding spot rates which satisfy a

variability condition between the upper and lower rates. iven the variability relation, the modelthen solves for the lower spot rate which satisfies a price condition where the bond valueobtained from the tree is consistent with the e5uilibrium bond price given the current yieldcurve.

Va$a'(t) C"ndt"n

In our derivation of the formulas for u and d we assumed that the distribution of the logarithmic

-

et

t

e

t e

t

g

Vg

= = =

=

= =

=

=

')(

')(

''

(+:):

''((+)(

'

')

)

'

') b g .. .


8/28

return of spot rates was normal. This assumption also implies the following relationship betweenthe upper and lower spot rate6

That is, from the binomial process we know

Therefore6

Aubstituting the e5uations for u and d, we obtain6

or in terms of the annualied variance6

Thus, given a lower rate of .3/ and an annualied variance of .((3+, the upper rate for aone!period binomial tree of length one year %h * '& would be ''/.6

If the current one!year spot rate were '(/, then these upper and lower rates would beconsistent with the upward and downward parameters of u * '.' and d * .3. This variabilitycondition would therefore result in a binomial tree identical to the one shown in igure '(.9!'.6

P$%* C"ndt"n

Kne of the problems with using $ust a variability condition %or e5uivalently $ust u and destimates& is that it does not incorporate all of the information. 4s we discussed in "hapter 3, theyield curve reflects not only the supply and demand for bonds of different maturity segments, butalso the expectations of investors about future interest rates. Thus, in addition to the variabilityrelation between upper and lower spot rates, the calibration model also tries to generate a

#

S S eu dV ne=

) 8.

S uS

S dS

u

d

=

=

(

(.

S

uS

S

d

S S u

d

u d

d

= =

=

(

( .

S S e

e

S S e

u d

V n n

V n n

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V n

e e

e e

e

=

=

+

+

8 8

8 8

8,

)

S S eu dhVe

A

=)

.

S eu = = 3/ ''/.) ((3+. .


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binomial tree that is consistent with the current yield curves spot rates. This is done by solvingfor a lower spot rates which satisfies the variability relation and also yields a bond price that ise5ual to the e5uilibrium bond price. To see this, suppose the current yield curve has one!, two!,and three!year spot rates of y' * '(/, y)* '(.'))9#/, and y9 * '(.)++##/, respectively.urthermore, suppose that we estimate the annualied logarithmic mean and variance to be .

(+#':- and .((3+, respectively. Dsing the u and d approach, a one!period tree of length oneyear would have up and down parameters of u * '.')9: and d * .-3. iven the current one!period spot rate of '(/, the trees possible spot rate would be Au* ''.)9:/ and Ad* .-3/.These rates, though, are not consistent with the existing term structure. That is, if we value atwo!year pure discount bond %0


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Aolving the above e5uation for Ad, yields a rate of .3/. 4t Ad * .3/, we have a binomial treeof one!year spot rates of Au * .3/ and Ad* .3/ that simultaneously satisfies our variabilitycondition and price condition7 that is, the rate is consistent with the estimated volatility of .((3+

and the current yield curve with one!year and two!year spot rates of '(/ and '(.'))9#/ %seeigure '(.9!)&.It should be noted that the lower rate of .3/ represents a decline from the current rate

of '(/, which is what we tend to expect in a binomial process. This is because we havecalibrated the binomial tree to a relatively flat yield curve. If we had calibrated the tree to apositively sloped yield curve, then it is possible that both rates next period could be greater thanthe current rate7 although the upper rate will be greater than the lower. or example, if the currenttwo!year spot rate were '(.3/ instead of '(.'())9#3, then the e5uilibrium price of a two!yearbond would be .#'# and the Adand Auvalues which calibrate the tree to this price and variabilityof .((3+ would be '(.)((::/ and ''.#'3:/. By contrast, if we had calibrated the tree to anegatively sloped curve, then it is possible that both rates next period could be lower than the

current one.

T"-P*$"d Bn"!a( T$**

iven our estimated one!year spot rates after one period of .3/ and ''/, we can now move tothe second period and determine the trees three possible spot rates using a similarmethodology.The variability condition follows the same form as the one period7 that is6

Aimilarly, the price condition re5uires that the binomial value of a three!year 0


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Dsing an iterative %trial and error& approach, we find that a lower rate of Add* .()3/ yields abinomial value that is e5ual to the e5uilibrium price of the three!year bond of .-9' %see igure'(.9!9&6

The two!period binomial tree is obtained by combining the upper and lower rates foundfor the first period with the three rates found for the second period %igure '(.9!+&. This yields atree that is consistent with the estimated variability condition and with the current term structureof spot rates. To grow the tree, we continue with this same process. or example, to obtain thefour rates in 0eriod 9, we solve for the Addd which along with the spot rates found previously forperiods one and two and the variability relations, yields a value for a four!year 0


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portfolio6 a portfolio constructed so that it has the same cash flows. The replicating portfolio ofa coupon bond, in turn, is the portfolio of ero discount bonds. Thus, one of the importantfeatures of the calibration model is that it yields prices on option!free bonds that are arbitragefree.

In addition to satisfying an arbitrage!free condition on option!free bonds, the calibration

model also values a bonds embedded options as arbitrage!free prices. To see this, consider athree!year, / callable bond priced in terms of the calibrated binomial tree shown in igure'(.9!+. The values of the bond and its call option are the same as the one shown in igure '(.9!:4s shown, the value of the call option in period ' is .9(3 when the rate is at ''/ and '.)':3when the rate is at .3/, and the value of the option in the current period is .:9:. Each of thesevalues were determined by calculating the present value of each options expected value. Thesevalues are also e5ual to the values of their replicating portfolios. or example, the current callprice of .:9: is e5ual to the value of a portfolio consisting of a one!year, / option!free bondand a two!year, / option!free bond constructed so that next year the portfolio is worth .9(3 ifthe spot rate is ''/ and '.)':3 if the rate is at .3/. Apecifically, given the possible cash flowson the two!year bond of BuC " * %'(8'.''& C * '(-.'#'# and Bd C " * %'(8'.(3& C *

'(#.3+99-, and the cash flow on the one!year bond of '(, the replicating portfolio is formedby solving for the number of one!year bonds, n', and the number of two!year bonds, n), where6

Aolving for n'and n), we obtain n'* !.::()-9: and n)* .:-+)3. Thus, a portfolio formed bybuying .:-+)3 issues of a two!year bond and shorting .::()-9: issues of a one!year bond willyield possible cash flows next year of .9(3 if the spot rate is at ''/ and '.)':3 if the rate is at

.3/. @oreover, given the current one!year and two!year bond prices of .(' and #.(:+9, thevalue of this replicating portfolio is .:9:6

Aince the replicating portfolio and the call option have the same cash flows, by the law of oneprice they must be e5ually priced. Thus, in the absence of arbitrage, the price of the call is e5ualto .:9:, which is the same price we obtained by discounting the options expected value.

The two call values in period ' of .9(3 and '.)':3 are likewise e5ual to the values oftheir replicating portfolios. That is, at the ''/ rate a replicating portfolio consisting of

')

n n

n n

' )

' )

'( '(-'#'# 9(3

'( '(#3+99- ')':3

% & % . & .

% & % . & . .

+ =

+ =

B

B

VRP

'

) )

(

'(

''( ('

''(

'(

''('))9## (:+9

::()-9: (' :-+)3 # (:+9 :9:

= =

= + =

= + =

..

. % . &.

% . &% . & %. &% . & .


13/28

!.+:'(#()- issues of a one!year, / bond %original two!year bond& and .+-9(#)- of the two!year, / bond %original three!year& will yield cash flows in period ) e5ual to ( if the spot rate is').'/ and .:#-) if the rate is '(.+3/. 4t that node, the price on the one!year bond is '(8'.''* #.'#'# and the price of the two!year is :.9:') %see igure '(.9!:&. 4t these prices, thevalue of the replicating portfolio is .9(3: * %!.+:'(#(9&%#.'#'#& C %.+-9(#)-&%:.9:')&,

which matches the value of the call. 4t the lower node, the replicating portfolio consists of!.#':3'+ one!year, / bonds priced at .3+99# * '(8'.(3 and ' two!year, / bond pricedat #.99+ %see igure '(.9!:&. This portfolios possible cash flows in period ) match thepossible call values of .:#-) and '.-- and its period ' value is e5ual to the call value of'.)':3.

With the option values e5ual to their replicating portfolio values, the calibration modelhas the feature of pricing embedded options e5ual to their arbitrage!free prices. Because of thisfeature and the feature of pricing option!free bonds e5ual to their e5uilibrium prices, thecalibration model is referred to as an arbitrage!free model. Atudents of option pricing may recallthat arbitrage!free models can alternatively be priced using a risk!neutral pricing approach.When applied to bond pricing, this approach re5uires finding the pseudo probabilities that make

a binomial tree of bond prices e5ual to the e5uilibrium price. The risk!neutral pricing approachis e5uivalent to the calibration approach.The calibration model presented here is the Black!ee model, for example, assumes interest rates each period are determined by the previousrate plus or minus an additive rather than multiplicative random shock. The Black!Lurasinskimodel, in turn, is characteried by a mean reversion process in which short!term rates revert toa central tendency. Each of these models, though, is characteried by the property that if theirassumption about the evolution of rates is correct, the models bond and embedded optionprices are supported by arbitrage. This arbitrage!free feature of the calibration model is one ofthe main reasons that many practitioners favor this model over the e5uilibrium model.

10. OPTION-ADUSTED SPREAD

Knce weve derived a binomial tree, we can then use it to value any bond with embedded optionfeatures. In addition to valuation, the tree also can be used to estimate the option spread %thedifference in yields between a bond with option features and an otherwise identical option!freebond&, as well as the duration and convexity of bonds with embedded option features.

The simplest way to estimate the option spread is to estimate the MT@ for a bond with anoption given the bonds values as determined by the binomial model, then subtract that rate fromthe MT@ of an otherwise identical option!free bond. or example, in the previous example thevalue of the three!year, / callable was :.)3#, while the e5uilibrium price of the noncallablewas :.3)'. Dsing these price, the MT@ on the callable is '(.+)3:/ and the MT@ on thenoncallable is '(.'-, yielding an option spread of .)33:/6

'9


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Kne of the problems with using this approach to estimating the spread is that not all ofthe possible cash flows of the callable bond are considered. In three of the four interest ratescenarios, for example, the bond could be called, changing the cash flow pattern. from threeperiods of , , and '( to two periods of , and '(-. 4n alternative approach that addressesthis problem is the option!ad$usted spread %K4A& analysis.

K4A analysis solves for the option spread, k, which makes the average of the presentvalues of the bonds cash flows from all of the possible interest rate paths e5ual to the bondsmarket price. The first step in this approach is to specify the cash flows and spot rates for each

path. In the case of the three!year bond valued with a two!period binomial interest rate tree,there are four possible paths6

0ath ' 0ath ) 0ath 9 0ath +

Time A' " A' " A' " A' "

(')9

.'(.(3

.(()3!

!

'(-

.'(.(3.'(+3

!

!

'(-

.'(

.''.'(+3

!

!

'(-

.'(

.''.')'

!

!

'(

iven the four paths, we next determine the appropriate two!year spot rates %y)& and three!yearrates %y9& to discount the cash flows. These rates can be found using the geometric mean and theone!year spot rates from the tree7 that is6

'+

NoncallableYTM YTM YTM

YTM

CallableYTM YTM YTM

YTM

Option Spread YTM YTMC nc

6 .% & % &

.

6 .% & % &

.

. . .

:3)'

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'

'(

''('-/

:)3#

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'( +)3:/ '( '-/ )33:/.

) 9

) 9

=+

++

++

=

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++

++

=

= = =

Path

y

y

y

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'(

''( '(3 ' (-+-)

''( '(3 '(()3 ' (3(-3'

'

)' )

9

' 9

=

= =

= =

.

H% . &% . & .

H% . &% . &% . & .

8

8

Path

y

y

y

)

'(

''( '(3 ' (-+-)

''( '(3 ''(+3 ' (#':

'

)' )

9

' 9

=

= =

= =

.

H% . &% . & .

H% . &% . &% . & .

8

8

Path

y

y

y

9

'(

''( ''' ' '(+##-''( ''' ''(+3 ' '

'

)' )

9

' 9

=

= == =

.

H% . &% . & .H% . &% . &% . & .

8

8

Path

y

y

y

+

'(

''( ''' ' '(+##-''( ''' '')' ' '

'

)' )

9

' 9

=

= == =

.

H% . &% . & .H% . &% . &% . & .

8

8


15/28

iven a discount rate e5ual to the spot rate plus the spread, k, the final step is to solve for the kwhich makes the average present values of the paths e5ual to the callable bonds market price,B@(7 that is6

1ote, if the market price is e5ual to the binomial value we obtained using the calibrationmodel, then the option spread, k, is e5ual to ero. This reflects the fact that we have calibratedthe tree to the yield curve and have consider all of the possibilities. In practice, though, we donot expect the market price to e5ual the binomial value. If the market price is below thebinomial value, then k will be positive. @any analyst in trying to identify mispriced bonds usethe K4A approach to estimate k instead of comparing the market price with the binomial value.

10. DURATION AND CONVE4IT5

>ike our earlier measure of bond value, we defined a bonds duration %modified and @acauley&without factoring in its embedded option features. In a period of low interest rates, when thelikelihood of a call is high, we would expect differences in the cash flows of a callable andotherwise identical noncallable and therefore differences in their durations. The inclusion ofoption features also changes the convexity of a bond. Bonds with call options are sometimessaid to have negative convexity, meaning that the bonds duration moves inversely with ratechanges. That is, an option!free bonds duration move in the opposite direction as interest rates,increasing in absolute value as rates decrease and decreasing as rates increase. When a bond has

a call option, though, a rate decrease can lead to an early call, which shortens the life of the bondand lowers its duration, while an interest rate increase tends to length the expected life of thebond causing its duration to increase. Thus, the option features of a bond can have a significantimpact on the bonds duration and convexity.

4s first noted in "hapter -, the duration and convexity of bonds with embedded optionfeatures can be estimated using a binomial tree and what is referred to as the effective durationand convexity measures. Effective duration and convexity are defined as

'3

B

M

(

)

)

)

) 9

''(

#

'(-+-)

''(

'(

'(-+-)

''(

'(

''(+##-

''(

''(+###-

'(

'''()##:

=

++

+

+NM QP

++

++

LNM

OQP

++

++

LNM

OQP

+ + + + + +

LNM PH

GGGGGGGGG

GG

JJJJJJJJJ

JJ

. % . &

. % . &

. % . &

. % . & % . &

E!!ecti"e #uration B B

B y

E!!ecti"e Con"e$ity B B BB y

=

= +

+

+ +

)

)

(

(

(

)

% &% &

% &% &,


16/28

where6 B!* price associated with a small decrease in rates7BC* price associated with small increase in rates.

The binomial tree calibrated to the yield curve can be used to estimate B(, B! and BC.irst, the current yield curve and calibrated tree can be used to determine B (7 next, B! can be

estimated by allowing for a small e5ual decrease in each of the yield curve rates %e.g. '( basispoints& and then using the tree calibrated to the new rates to find the price7 finally, B Ccan beestimated in a similar way by allowing for a small e5ual increase in the yield curves rates andthen estimating the bond price using the tree calibrated to these higher rates.

10.6 C"n%(us"n

In the last two chapters, weve examined how binomial interest rate trees can be constructed andused to value bonds with call and put options, sinking fund arrangements, and convertibilityclauses. iven that many bonds, as well as other debt contract, have embedded option feature,

the binomial tree represents both a useful and practical approach to the valuation of debtsecurities. In this "hapter weve focused on introducing how the tree can be estimated. Bondswith multiple option features and ones that are sub$ect to the uncertainties of more than onefactor, such as callable convertible bonds or >MK1A, are more difficult to evaluate and mayre5uire more complex binomial models. Each, though, can be evaluated by extending the basicbinomial tree presented here.

':


17/28

APPENDI4 A

ALGEBRAIC DERIVATION O/ THE u AND d /ORMULAS

A. 1 DERIVATION O/ THE E7UATIONS /OR E8+n9 and V8+n9

@athematically, the random variable gncan be defined in terms of u, d, n, the number of upwardmoves in n periods, $, and the number of downward moves, n!$. That is6

By applying the rules of logarithms, gncan also be expressed as

1ote that in E5uation %4.'&, the only random variable is $, the number of upward moves in nperiods. Thus, the expected value of $ is

iven that the probability of an upward move in one period is 5, the expected number of upwardmoves in n periods therefore is E%$& * n5. Aubstituting n5 for E%$& in E5uation %4.)& andrearranging the resulting e5uation, the expected value of gncan be expressed alternatively asfollows6

The bracket expression on the right!hand side of E5uation %4.9& is the expected logarithmicreturn for one period. Thus, E5uation %4.9& shows that E%gn& is e5ual to the expected logarithmicreturn for one period times n.

The mathematical expression for F%gn& is found first by substituting the expressions for gnand E%gn& into the general e5uation for F%gn&, then rearranging as follows to obtain E5uation%4.+&&6

'-

g S S u dn nj n j= = ln% 8 & ln .(

g j u n j d

A g j u d n d

n

n

= +

= +

ln % & ln

% . & ln% 8 & ln .'

% . & % & % & ln% 8 & ln .A E g E j u d n dn) = +

E g nq u n d

A E g n q u q d

n

n

% & ln ln

% . & % & H ln % & ln .

= +

= + 9 '


18/28

or one period, the number of upward moves %$& is either ' or (, with the expected valueof the number of upward moves, E%$&, e5ual to 5. With the probability of $ * ' being 5 and theprobability of $ * ( being '!5, the variance of upward moves for one period, F%$'&, is therefore5%'!5&. That is6

In a binomial process the variance of $ for n periods, F%$&, is e5ual to the variance ofupward moves in one period times n7 thus6

Aubstituting E5uation %4.3& into %4.+&, we obtain the alternative expression for F%gn&6

A. 2 ALGEBRAIC SOLUTIONS /OR u AND d

If a spot rates possible logarithmic returns in the future are characteried by a symmetricdistribution %ero skewness& in which the parameter values increase with the length of thedifferencing period, then the binomial distribution for logarithmic returns is a goodapproximation of the way a securityNs return performs over time. If this is the case, then we canestimate u and d by finding the values which make E%gn& and F%gn& for the distribution e5ual totheir respective empirical %estimating& values. @athematically, this is done by setting E5uation

%4.9& for the mean e5ual to its estimated value, e, and E5uation %4.:& for the variance e5ual to

its estimated value, Fe, to obtain the following e5uation system6

'#

V g E g E g

V g E j u d n d E j u d n d

V g E j E j u d

V g E j E j u d

A V g V j u d

n n n

n

n

n

n

% & H % &

% & H ln% 8 & ln % & ln% 8 & ln

% & H% % &%ln% 8 &

% & H % & Hln% 8 &

% . & % & % &Hln% 8 & .

=

= +

=

=

=

)

)

)

) )

)+

V j q q q q

V j q q q q q

V j q q q q q q q

% & % & % &% &

% & % &% &

% & % &.

') )

'

) 9

') 9 ) 9

' ' (

' '

) '

= +

= +

= + + =

% . & % & % & % &.A V j nV j nq q3 ''

= =

% . & % & % &Hln% 8 & .A V g nq q u dn: ')=


19/28

The u and d formulas are found by solving these e5uations simultaneously for the unknowns.

This e5uation system, however, consist of two e5uations and three unknowns, u, d, and 5, andtherefore cannot be solved without additional information about one of the unknowns. If we

assume that the distribution of gnis symmetric, then we can set 5 * .3 in E5uations %4.-& and%4.#&.


20/28

inally, express E5uations %4.'9& and %4.'+& as exponents to obtain the e5uations for the u and dvalues which satisfy E5uations %4.& and %4.'(&6

E5uations %4.'3& and %4.':& are the familiar Binomial Kption 0ricing @odel e5uations forestimating u and d. The e5uations determine the values of u and d, given the estimated varianceand the mean of the spot rates logarithmic return for the period and the assumption that spotrate changes follow a binomial process.

)(

ln % 8 & 8 % 8 &

% . & ln 8 % 8 & .

d n V n n

A d V n n

e e e

e e

=

= +

)

'+

% . &

% . & .

8 % 8 &

8 % 8 &

A u e

A d e

V n n

V n n

e e

e e

'3

':

=

=

+

+


21/28

FIGURE 10.2-1SIX-PERIOD TREE

igure .)!'6 ive!0eriod Binomial Tree of Apot ;ates

Ao * '(/, u * '.(+##, d * .-+- ') :(/.

'' -+/.

'(:(/.

'( '#:3/.

+::3/.

# -#,/.

'( +3/.

, -'))/.

(':/.

') '(/.

'' )+3/.

'( -)'-/.

:+9/.

)3/.

''39-/.

''((/.

'( ))9/.

, 3/.

'( +##/.

-+-/.

'(/

)'


22/28

FIGURE 10.2-2BOND VALUE WITH SIX-PERIOD TREE

igure .9!)6 Three!Mear, /

Bond Falued with :!0eriod Tree') :/, # ) :3

):3

#

. .

.

B

V %V

B B V

nc

C

C n c C

=

= =

= =

''-+/, #:#'

:#'

#

. .

.

B

V %V

B B V

nc

C

C nc C

=

= =

= =

'( :/ , (-'

'(-'

#

. .

.

B

V %V

B B V

nc

C

C nc C

=

= =

= =

+ ::3/, - --

'---

#

. .

.

B

V %V

B B V

nc

C

C nc C

=

= =

= =

#-#/, '(('(

)'(

#

. .

.

B

V %V

B B V

nc

C

C nc C

=

= =

= =

'('#3:/, +93

'+93

#

. .

.

B

V %V

B B V

nc

C

C nc C

=

= =

= =

') '(/, - (#3

( ++:(

: :3)3

. .

, .

.

B

%V V

B B V

nc

&

C nc C

=

= =

= =

'')+3/, -#-9

( #)+

- (+9:

. .

, .

.

B

%V V

B B V

nc

&

C nc C

=

= =

= =

'(+3/, #:('3

:(3' ''')

-+'(9

. .

. , .

.

B

%V V

B B V

nc

&

C nc C

=

= =

= =

-'))/, )#3'

' )#3' ' 39'--3+

. .

. , ..

B

%V VB B V

nc

&

C nc C

=

= =

= =

(':/, 9+

' 9+ '#3+

#

. .

. , .

B

%V V

B B V

nc

&

C nc C

=

= =

= =

:+9/, #39+)

39 +) ' ) :3

- )9--

. .

. , .

.

B

%V V

B B V

nc

&

C n c C

=

= =

= =

) 3/, 3( -9

' 3( -9 ' :3 3

-#3'+

. .

. , .

.

B

%V V

B B V

nc

&

C n c C

=

= =

= =

''39-/, :+)93( :((-

3#))-

. ., .

.

B%V V

B B V

nc

&

C nc C

=

= =

= =

'(-)'-/, -3(#

( 3:-

:339'

. .

, .

.

B

%V V

B B V

nc

&

C nc C

=

= =

= =

''/, :'-:

( -9#'

3+9##

B

%V V

B B V

nc

&

C nc C

=

= =

= =

.

, .

.

'())9/, -39:+

( '(-'#

:+:)#

. .

, .

.

B

%V V

B B V

nc

&

C nc C

=

= =

= =

3/, # # ):3

# ):3 ' +( ):

- +'-9

. .

. , .

.

B

%V V

B B V

nc

&

C n c C

=

= =

= =

'( +##/, : 9(:9

( #3#:

3++:+

. .

, .

.

B

%V V

B B V

nc

&

C nc C

=

= =

= =

-+-/, - (#

( ''#)

: -):

. .

, .

.

B

%V V

B B V

nc

&

C nc C

=

= =

= =

'(/, : -::

( -)-

3-:

B

%V V

B B V

nc

&

C nc C

=

= =

= =

.

, .

.

Per iod ' C: ' (( + 3 ' (+ 3(6 . .+ = + =

))


23/28

/IGURE 10.3-1: BINOMIL DISTRIBUTION O/ LOGARITHMIC RETURNS

S( '(= .

S p

g

d

d

=

= =

. ,

ln%. & .

(3

3 (3'9

'(

S p

g

u

u

= =

= =

. , .

ln% . & .

'' 3

'' (39

''

S p

g

uu

uu

= =

= =

. , .

ln% . & .

')' )3

'' '(:

))

)

S p

g

ud

ud

= =

= =

. , .

ln % . &%. & .

'(+3 3

'' 3 (++(

)'

b g

S pg

dd

dd

= =

= =

. , .ln%. & .(()3 )3

3 '():)(

)

S p

g

uud

uud

= =

= =

. , .

ln % . & %. & .

''+3 9-3

'' 3 '99

9)

)c h

S p

g

uuu

uuu

= =

= =

. , .

ln . .

'99' ')3

'' )#3

99

9c h

S p

g

udd

udd

= =

= =

. , .

ln % . &%. & .

()-3 9-3

'' 3 ((-9

9'

)c h

S p

g

ddd

ddd

= =

= =

. , .

ln %. & . .

(#3-+ ')3

3 '39

9(

9c hE g

V g

n

n

% &

% &

.

.

())

((3+

.

.

(++

('(#

.

.

(::

(':)

Bino(ial #i)tribution

o! arith(ic return)

u d q

log

. , . , .= = ='' 3 3

)9


24/28

TABLE 10.3-1Est!atn+ M*an and Va$an%* t& Hst"$%a( Data

Quarter Spot RateAt At8At!' g t * %At8At!'& %gt! e&)

98.198.98.!98."99.199.99.!99."##.1##.##.!##."#1.1

1#.$%1#.#%9."%8.8%9."%

1#.#%1#.$%1#.#%9."%8.8%9."%

1#.#%1#.$%

&.9"!".9"##.9!$1.#$81.#$!81.#$##.9"!".9"##.9!$1.#$81.#$!81.#$##

'''.#(!8'.#$19'.#$(9.#$$#.#$18.#(8!'.#(8!'.#$19'.#$(9.#$$#.#$18.#(8!

e) #

&.##!!99.##!!8.##"!"!.##"!($.##!819.##!!99.##!!99.##!8!.##"!"!.##"!($.##!819.##!!99

*+e ) .#"$9$

+e) ",#- * (7 *

e ) "*+e) ",.#"$9$- * .(':#9:

Le/gth h / 0or !'ear2o/3

D

estimating the binomial tree

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