Transcript PowerPoint Version of Binomial Tree Pricing Notes

Bonds with embedded options
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Callable Bonds
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A problem with traditional pricing is that it
ignores options imbedded in the bond such as
a call option.
The call feature increases reinvestment rate
risk since the bond will be called if rates are
low, especially if lower than the coupon rate.
As the yield decreases the price increase
lessens because there is a higher probability
of the bond being called (price compression)
Valuing a Callable Bond
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You can think of a callable bond as having
two components: A noncallable bond and a
call option. The price of the callable bond
would then be equal to the noncallable bond
price minus the call option price.
Valuation Model Review
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Remember from before that the appropriate
rate to use is not a single rate, but the zero
spot rate or the forward rates (example on
next slide)
The value of the callable bond will be tied
directly to the volatility of interest rates. To
price the bond we will use a binomial tree
model.
5.25% coupon bond, 3 years to
maturity, yearly payments
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Assume you have the following observed yield
curve, spot rates, and forward rates.
Maturity
Rate
1 yr
2 yr
3 yr
YTM Market Value Spot Rate
3.5%
4.0%
4.5%
5.25

100
3.5%
100
4.01%
4.531%
5.25 100 105.25
2

3
 102.075
Forward
3.5%
4.5225%
5.5792%
(1.035) (1.0401) (1.04531)
5.25
5.25
105.25


 102.075
(1.035) (1.035)(1.04523) (1.035)(1.04523)(1.055792)
Binomial Tree Model
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We are going to represent the two possible
paths of interest rates in a tree structure.
Let each time be denoted as a decision Node N
with a subscript denoting whether it represent
the higher or lower interest rate environment.
The current level of interest rates is r*, it may
increase to state H or decrease to state L If
the level of interest rates increases to H the
bond will have a value of VH in the next period.
Likewise if rates decrease to L the value will be
Binomial Tree Model
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l VH
NH
V
l
r*
N
l VL
NL
Time 0
Time 1
Value at Time 0
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The value of the bond at time 0 can be
calculated assuming that there is an equal
chance of obtaining either state.
The expected value at time 0 is then equal to
the PV of total amount you would receive in
each state multiplied by the probability of the
state (in this case 1/2)
The total amount you would receive is the
value or the bond plus any other cash flows
received (For example the coupon payment)
Binomial Tree Model
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VH
l
C
NH H
V0
l
r*
N
1  VH  CH VL  CL 
V0  


2  1 r *
1 r * 
Time 0
VL
l
NL CL
Time 1
Simple Example
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Assume that the current level of interest rates
is 3.5% as in our previous example.
If the higher interest rate price is $98 with a
coupon of $5 and the lower interest rate price
is $102 with the same $5 of coupon the value
at time 0 would be
1  98  5 102  5 


  101.449
2  1.035
1.035 
Binomial Tree Model continued
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Starting today we want to think about the
future path of interest rates.
Start with the time 0 (today) and think about
the future path of interest rates. We will
assume that over the next year there are two
possible outcomes for the one year rate at
time 1.
Let the lower rate be r1L and the higher rate
be r1H
Binomial Tree Method continued
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Assuming the lower rate in the next period
the higher rate can be found from the
equation.
r1H=e2sr1L
where:
e = natural logarithm 2.71828
(x = lny y=ex)
For example let r1L=4.5% and s=10%
then r1H = .045e2(.10)=.054963
Binomial Tree Model
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VH
l CH
NH r1H=.05496
V0
l
r*
N
Time 0
VL
l CL
NL r =.045
1L
Time 1
Extending the model
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It is easy to extend the model to add a
second year.
From each of the decision nodes NH and NL
you can just repeat the same tree.
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Binomial Tree Model
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V0 l
N
r=.035
Time 0
VH
C
l H
r1H=.05496
NH
VL
l
NL CL
r1L=.045
Time 1
VHH
l CHH
NHH r2HH
VHL
l CHL
NHL r2HL
VLL
l CLL
NLL r2LL
Time 2
The Two Year Bond
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Using the observed yield curve from before,
the two year bond would have a 4% coupon
rate implying $4 coupon payments each year.
At maturity the bond will have a value of $100
Substitute the value in for VHH, VHL,and VLL.
Let the coupon be $4 in each period.
Binomial Tree Model
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V0 l
N
r=.035
Time 0
VH
C =4
l H
r1H=.05496
NH
VL
l
NL CL = 4
r1L=.045
Time 1
VHH = 100
l CHH = 4
NHH r2HH
VHL = 100
l CHL = 4
NHL r2HL
VLL= 100
l CLL= 4
NLL r2LL
Time 2
Finding VH and VL
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The values of the bond can be found at time
1 by applying the earlier formula
1  VH  CH VL  CL 
V0  


2  1 r *
1 r * 
1  VHH  CHH VHL  CHL  1  100  4 100  4 
  
VH  


  98.582
2  1  r1H
1  r1H  2  1.05496 1.05496 
1  VLL  CLL VHL  CHL  1  100  4 100  4 
  
VL  


  99.522
2  1  r1L
1  r1L  2  1.045 1.045 
Binomial Tree Model
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V0 l
N
r=.035
Time 0
VH=98.852
C =4
l H
r1H=.05496
NH
VL=99.522
l
NL CL = 4
r1L=.045
Time 1
VHH = 100
l CHH = 4
NHH r2HH
VHL = 100
l CHL = 4
NHL r2HL
VLL= 100
l CLL= 4
NLL r2LL
Time 2
Iterative Procedure
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We assumed an interest rate of 4.5% for r1L
this is the correct rate IF V0 can be found
given the current values it the tree and V0 is
equal to the market price of 100
1  V  CH VL  CL  1  98.582  4 99.522  4 
V0   H


 
  99.567
2  1 r *
1  r *  2  1.035
1.035 
Since the price from the tree is too low, the
rate r1L must be lower to increase the price.
Try a new price and repeat until the correct
price 4.074% is found
Iterative procedure
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Given the rate of 4.074 the expected value of
the two possible changes in interest rates is
equal to the current value, in other words it
is “fairly priced.”
The change in rates requires finding new
values for r1H and for VH and VL
Binomial Tree Model
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V0 l
N
r=.035
Time 0
VH=99.071
C =4
l H
r1H=.04976
NH
VL=99.929
l
NL CL = 4
r1L=.04074
Time 1
VHH = 100
l CHH = 4
NHH r2HH
VHL = 100
l CHL = 4
NHL r2HL
VLL= 100
l CLL= 4
NLL r2LL
Time 2
Interpretations
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r1H and r1Lare a set of forward rates from time
1 to time 2 or 1f1 as it was previously called.
Notice that since the change in rates makes a
difference in the value of the bond, for each
forward rate we will have a different value of
the bond.
The next step, time 3
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The model could be extended again to include
the next year. The YTM for the three year
treasury was 4.5% so the coupons at every
time period become 4.50.
The goal is to find a value for r2LL that will
allow us to move from right to left through
the tree to produce a value of 100 again at V0
r2HL=r2LLe2s as before and r2HH=r2HLe2sr2LLe4s
the correct rate is then 4.53%
Binomial Tree Model
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V0 l
N
r=.035
Time 0
VH =98.074
C = 4.50
l H
r1H=.04976
NH
VL=99.926
l
NL CL = 4.50
r1L=.04074
Time 1
VHH = 97.886
l CHH = 4.50
NHH r2HH=.06757
VHL = 99.022
l CHL = 4.50
NHL r2HL =.05532
VLL= 100
l CLL= 4.50
NLL r2LL=.0453
Time 2
Valuing an Option Free Bond
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The rates in the binomial tree now represent
the correct rates for the on the run treasury
yield curve that we started with. It is now
possible to use it to value the three year
5.25% coupon bond.
Starting with year three the values VHH, VHL,
and VLL can be found then we can work right
to left through the tree
Binomial Tree Model
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V0 l
N
r=.035
Time 0
VH =99.461
C = 5.25
l H
r1H=.04976
NH
VL=101.333
l
NL CL = 5.25
r1L=.04074
Time 1
VHH = 98.588
l CHH = 5.25
NHH r2HH=.06757
VHL = 99.732
l CHL = 5.25
NHL r2HL =.05532
VLL= 100.689
l CLL= 5.25
NLL r2LL=.0453
Time 2
Value of the bond
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1  VH  C H VL  CL 
V0  


2  1 r *
1 r * 
1  99.461  5.25 101.333  5.25 
 

  102.075
2
1.035
1.035

The same value as we calculated before!!!
Valuing a Call Option
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Assume that the the bond can be called at the
end of the first year or later for a call price of
$100.
If the value at a node is greater than $100
then the bond will be called (the yield is less
than the coupon) and the firm can refinance
at a lower rate. Starting on the right, if the
value exceeds 100 it needs to be replaced,
then the tree is worked right to left again.
Binomial Tree Model
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V0 l
N
r=.035
Time 0
VH =99.461
C = 5.25
l H
r1H=.04976
NH
VL=101.001
l
NL VL=100
CL = 5.25
r1L=.04074
Time 1
VHH = 98.588
l CHH = 5.25
NHH r2HH=.06757
VHL = 99.732
l CHL = 5.25
NHL r2HL =.05532
VLL= 100.689
VLL =100
l C = 5.25
LL
NLL
r2LL=.0453
Time 2
Value of callable bond
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1  VH  C H VL  C L 
V0  


2  1 r *
1 r * 
1  99.461  5.25 100  5.25 
 

  101.4302
2
1.035
1.035 
The value has decreased because of the call
option
The value of the call option is then
102.075-101.4302=0.6448
Put option
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The same model could be used to value a put
option.
Now, you look at increases in rates that lower
the price.
If the value of the bond at the node is less
than the puttable value then the option would
be exercised and the value of the bond
becomes the put value.
Assume that our bond has a put option after
year one with the puttable value being $100
Binomial Tree Model (Put)
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V0 l
N
r=.035
Time 0
VH =100.261
C = 5.25
l H
r1H=.04976
NH
VL=101.461
l
NL CL = 5.25
r1L=.04074
Time 1
VHH = 98.588
VHH=100
l
C = 5.25
NHH HH
r2HH=.06757
VHL = 99.732
l VHL=100
NHL CHL = 5.25
r2HL =.05532
VLL= 100.689
l CLL= 5.25
NLL r =.0453
2LL
Time 2
Value of puttable bond
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1  VH  C H VL  C L 
V0  


2  1 r *
1 r * 
1  100.261  5.25 101.461  5.25 
 

  102.523
2
1.035
1.035

The value has increased because of the put
option.
The value of the put option is
102.075 – 102.523= -0.448
Modeling Risk
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Modeling risk is the risk that the valuation
model has produced an incorrect result due to
assumptions used in the model.
Higher volatility lowers the value of the call
option (raises value of put)
Lower volatility raises the value of the a call
option (lower the value of a put)
Option Adjusted Spread
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The constant spread that when added to all
the forward rates in the binomial tree will
make the theoretical value equal to the
market price.
OAS Intuition
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Converts the difference between the
valuation and the market price into a spread
measure.
The key is the inputs in the model
If the tree uses the treasury spot curve, the
OAS represents the richness or cheapness of
the security plus a credit spread
If the tree uses issuer’s spot rate curve, then
the credit risk is already incorporated.
OAS and total yield spreads
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The OAS is attempting to separate the
amount of the nominal spread that is the
result of option risk. Therefore it reports a
spread that is adjusted for the option.
Example: Assume you have calculated the
OAS of a BBB callable corporate bond
compared to non callable treasuries to be 120
Bp. would imply that the BBB pays 120 Bp
more because of the liquidity and credit risk
etc. the spread has removed the portion of
the spread attributable to the option.
OAS and benchmarks
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In the previous example the OAS was a
representation of credit and liquidity risks. If
instead of using the on the run treasury as a
benchmark we used the on the run issues for
the same issuer (the issuer of the BBB). Then
credit risk is also not part of the spread, only
liquidity and other factors.
OAS is the spread after adjusting for options,
what it actually represents however depends
upon the benchmark being used…
OAS in our example
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Assume that the 5.25% callable three year
coupon bond is currently selling for $101.17
Previously we found the price to be
$101.4302
The OAS would be the additional yield added
to the binomial interest rate tree at every
yield that produced a value for the bond of
$101.17, in this case the OAS is 45 Bp
Binomial Tree Model
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V0 l
N
r=.035
Time 0
VH =99.833
C = 5.25
l H
r1H=.005426
NH
VL=100.6946
l
NL VL=100
CL = 5.25
r1L=.04524
Time 1
VHH = 102.6309
VHH = 100
l
CHH = 5.25
NHH r =.07208
2HH
VHL = 103.817
VHL = 100
l C = 5.25
HL
NHL r2HL =.0598
VLL= 104.809
VLL =100
l C = 5.25
LL
NLL
r2LL=.0498
Time 2
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Value of callable bond (with OAS)
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1  VH  C H VL  C L 
V0  


2  1 r *
1 r * 
1  99.83306  5.25 100  5.25 
 

  101.1703
2
1.035
1.035 
Funding Cost as a Benchmark
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Often the on the run issues of the LIBOR is
used as the benchmark.
LIBOR is used as a benchmark borrowing rate
that the institution pays to obtain funds. It
can then compare its cost of funding to LIBOR
by looking at the spread above LIBOR it pays
to obtain funds.
As long as the assets spread relative to LIBOR
is greater than the spread the institution must
pay to obtain funding, it is covering the
funding cost.
Effective Duration and Convexity
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Effective Duration (option adjusted duration)
allows a yield change to change the expected
future cash flows.
Quick approximation of duration
and convexity
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P- = the price if yield is decreased by x Bp
P+ = the price if yield is increased by x Bp
P0 = the initial price Dy=change in rate (x Bp in
dec
form)
P P
duration 
-

2(P0 )( Dy)
P  P  2( P0 )
convexity 
2
(P0 )( Dy)
Calculating P+ and P-
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Calculate the OAS
2) Shift the on the run yield curve by a small
basis points
3) Construct a binomial interest rate tree based
on the new yield curve
4) To each of the short rates add the OAS to
adjust the tree
5) Use the adjusted tree to determine the value
of P.
1)
Valuing a Step Up Callable Note
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The Binomial model can be expanded to cover
other types of options.
One possibility is a note whose coupon rte
changes over the life of the note. In this case
the coupon rate may increase in the future.
Initially, the procedure is the same as before.
After developing the interest rate tree, the
bond is valued using the coupon rates that
correspond to what the bond will pay.
Valuing a Floating Rate Note
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On a floating rate bond the payment at the
end of the year is determined by the rate at
the beginning of the year.
Therefore the coupon payment for each node
will be based off of the interest rate for that
node (the rate at the beginning of the period
determines the coupon at the end of the
period).
The valuation therefore uses the coupon for
that node as the payment at the next point in
A Capped Floater
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If the floating rate bond has a cap, then
whenever the coupon is above the cap the
value of the coupon will be based off of the
cap.
Analysis of Convertible Bonds
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A convertible bond can allows the holder to
convert the bond into a predetermined
number of shares of common stock of the
issuer.
It may also be callable and puttable.
Exchangeable securities allow conversion to
the stock of another firm.
The conversion privilege may extend over the
entire life of the of the issue or a portion of
the issue
Conversion Ratio
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The conversion ratio is the number of shares the
holder will receive if the conversion option is
exercised.
Assume that the conversion ratio is 25, this would
imply that you would receive 25 shares for each
$1,000 of par value.
The conversion price is the price per share implied by
the conversion ratio. In the example above this
would be $1000/25 = $40
If not issued at par, the conversion price is found by
dividing the issue price per 1000 by the conversion
Other embedded options
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Often the convertible is also callable, usually
with a non callable period at the beginning of
the period of the issue
The issue may also be puttable. Hard Put –
the issuer must redeem with cash Soft Put –
the issuer may use common stock, cash, or
subordinated notes, or a combination of the
three
Minimum Value
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The conversion (or parity) value is the value
of the security if it is converted immediately.
The minimum value is then the greater of 1)
the conversion price and 2) its value as a
security without the conversion option (also
called the straight value or investment value).
If the security does not sell a the greater of
the two, then there are arbitrage possibilities.
Market Conversion Price
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If the convertible bond is bought then
converted immediately into stock, the
buyer is effectively paying a share price
based on the value of the security.
The market conversion price is
marketprice of
market
convertible security
conversion
conversionratio
price
Market Conversion Price
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If the actual market price increases above the
market conversion price the value of the
convertible bond should increase by the same
percentage.
Buying the convertible bond rather than the
underlying stock results in basically paying a
premium for the stock an can be expressed as
a ratio based on the market price.
Why pay a premium?
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The investor may be willing to pay the
premium if there is an expectation of
receiving a higher current income from the
coupon payments than from possible
dividends on the stock.
One way to address this is looking at the
amount of time it takes to recover the
premium paid (ignoring the time value of
money)
Premium Payback Period
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The amount of time to recover the premium is
marketconversion
premuin
premiumper share
payback 
favorableincome
period
differential per share

marketconversionprice - current marketprice
coponinterest- (conversion ratio x commonstock dividend per share)
conversionratio
Downside Risk
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It is often assumed that the price of the
convertible security cannot fall below the
straight value, therefore some participants
look at the ratio of the market price to the
straight value as a measure of downside risk.
However the straight value changes as
interest rates change so this does not truly
provide a good measure of downside risk .
Up side potential
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Drake
Drake University
The up side depends upon the valuation of
the common stock and the potential for gain
the stock price.
If the straight value is significantly higher
than the value implied by the common stock
price then it is referred to as a fixed income
equivalent or busted convertible.
If the conversion value from the stock price is
higher than the straight value it is a common
stock equivalent.
Option based value
UNIVERSITE
D’AUVERGNE
Drake
Drake University
We have ignored the true value of the
security
The convertible security value should equal
the straight value plus the value of a call
option of the stock of the firm.
IF the bond has a call feature then the value
becomes: the straight value plus the value of
the cal option on the stock minus the value of
the call option on the bond.