Chapter 9: 3/23 Lecture
Valuing Bonds with Embedded Options
Copyright 2006, Jeffrey M. Mercer, Ph.D.
1
Relative Value Analysis
 Relative value analysis is used to help
identify bonds that are underpriced
(“cheap”), overpriced (“rich”), or fairly
priced.
 We use yield spread measures to
accomplish this.
 Yield spreads have to measured relative to
interest rates from some benchmark.
 Our interpretation of relative value depends
upon the benchmark choice.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Benchmarks
 Benchmark interest rates come from:



Treasury securities, or
a specific bond sector with similar credit risk,
liquidity, and maturity characteristics, or
the issuer’s own (other) bonds.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Interpretation of Spread Measures
 the nominal spread is a spread measured relative to
the yield curve.

YTMGM – YTMTbond = nominal spread.
 the zero-volatility spread (or static spread) is a
spread relative to the spot rate curve.

Spot rates on GM bond = spot rates on Tbond + 500
bps.
 the option adjusted spread is a spread relative to
the spot rate curve.

The OAS adjusts GM’s z-spread for the “option value”
(in bps). If option value = 100 bps, OAS = 500-100 =
400 bps. In this case, the 400 bps would compensate
for default and liquidity risks.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Interpretation of Spread Measures:
Treasury Market Benchmark
 If we compare the rates on a callable corporate bond
with U.S. Treasury rates, we can summarize as
follows:
Spread measure
Benchmark
Spread reflects
compensation for risks:
Nominal
yield curve
credit, option, liquidity
Zero-volatility
spot rate curve
credit, option, liquidity
OAS
spot rate curve
credit, liquidity
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Interpretation of Spread Measures:
Specific Bond Sector Benchmark
 If we compare the rates on a callable corporate
bond with rates from a specific sector, we can
summarize as follows:
Spread measure
Benchmark
Reflects compensation
for risks (see next
slide):
Nominal
sector yield curve
credit, option, liquidity
Zero-volatility
sector spot rate curve
credit, option, liquidity
OAS
sector spot rate curve
credit, liquidity
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Risks from prior slide
 Where credit risk in this case means:

Credit risk of a security under consideration
relative to the credit risk of the sector used as
the benchmark.
 Where liquidity risk in this case means:

liquidity risk of a security under consideration
relative to the liquidity risk of the sector used
as the benchmark.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Interpretation of Spread Measures:
Issuer-Specific Benchmark
 If we compare a callable bond’s rates with rates on
other bonds from the same issuer, we can
summarize as follows:
Spread measure
Benchmark
Reflects compensation
for risks:
Nominal
issuer yield curve
option, liquidity
Zero-volatility
issuer spot rate curve
option, liquidity
OAS
issuer spot rate curve
liquidity
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Relative Value Analysis and Spreads:
Example
 When a bond has an embedded option, we
must use the option-adjusted spread in
relative value analysis.
 But the OAS has to be measured relative to a
benchmark, and the choice of benchmark
determines how we interpret “relative value.”
 Example 1: (next slide)
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Example 1
 Benchmark: Treasury market rates
 The bond under consideration is a triple B rated





corporate bond with an embedded option.
Nominal spread between our bond and the
benchmark = 170 bps
Nominal spread between option-free BBB’s and the
benchmark = 145 bps
Z-spread = 160 bps
OAS = 125 bps
Is the bond cheap, rich, or fair?
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Example 1: continued
 The bond has an embedded option, so the
OAS should be used.
 The only comparison that can be made is the
OAS of 125 bps to the nominal spread for
option-free BBB’s of 145.
 Based on this comparison, the bond is rich
(i.e., overvalued).

Recall that the z-spread and the nominal
spread are typically close, given a reasonably
sloped yield curve.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Example 2
 Benchmark: AA-rated callable corporate bonds
 The bond under consideration is a triple B rated





corporate bond with an embedded option.
Nominal spread between our bond and the
benchmark = 110 bps
Nominal spread between the benchmark and optionfree AA’s = 90 bps
Z-spread = 100 bps
OAS = 80 bps
Is the bond cheap, rich, or fair?
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Example 2: continued
 The bond has an embedded option, so the
OAS should be used.
 The only comparison that can be made is the
OAS of 80 bps to the nominal spread for
option-free AA’s of 90.
 Based on this comparison, the bond is rich
(i.e., overvalued).

Recall that the z-spread and the nominal
spread are typically close, given a reasonably
sloped yield curve.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Example 3
 Benchmark: Rates on other issues from the






specific issuer of our bond.
The bond under consideration is a triple B rated
corporate bond with an embedded option.
Nominal spread between our bond and the
benchmark = 30 bps
Nominal spread between the benchmark and optionfree AA’s = 90 bps
Z-spread = 20 bps
OAS = -25 bps
Is the bond cheap, rich, or fair?
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Example 3: continued
 The bond has an embedded option, so the
OAS should be used.
 The only measure we need to look at is the
OAS.
 Sine OAS is negative, the bond is rich (i.e.,
overvalued).
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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“Required OAS” vs. “Security OAS”
 Define required OAS as the OAS available
on comparable securities (i.e., same credit
risk, liquidity risk, and maturity).
 Define security OAS as the OAS on the
security under consideration.
 Then,



“Cheap” is security OAS > required OAS.
“”Rich” is security OAS < required OAS.
“Fair” is security OAS = required OAS.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 and pp. 312 - 313
 Exhibit 8 demonstrates that the theoretical
price of the four year, 6.5% callable bond,
with volatility = 10%, is 102.899.
 No suppose the market price is 102.218.
 The bond is underpriced (relative to our
model) by $0.681.
 The OAS equals the constant spread that,
when added to the rates in the binomial tree,
will make the model value equal to the market
value.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 and pp. 312 - 313
 Exhibit 11 demonstrates that the OAS is 35
bps.
 Thus, a positive OAS (in this case) is
consistent with the bond being undervalued.

Note: “In this case” because the benchmark
rates are from this specific issuer, so credit
risk and liquidity risk differences are controlled
for (see top of page 302 where this issuer’s
bonds are first introduced).
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 and pp. 312 - 313
 Again assume that the market price is
102.218.
 But now, assume that our volatility forecast is
20% (rather than 10%).
 At 20% volatility, the model value of the bond
would decrease, and the option value would
increase.
 But since the market price didn’t change, the
OAS must decrease.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 and pp. 312 - 313
 Thus, our OAS estimate, and our relative
value analysis for bonds with embedded
options, depend heavily on our volatility
estimate!
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Can you solve the following problem?
 Calculate the bond’s price in Exhibit 11 if the
OAS is 50 bps instead of 35 bps.
 Draw the binomial tree diagram, as in Exhibit
11, showing the computed value and call
price if exercised, and the adjusted interest
rates.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Original Exhibit 11
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NHHH will be:
97.084
97.084
6.5
9.6987%
 9.6987% = 9.5487% + 0.15%
 97.084 = 106.5/(1.096987)
 Will not be called.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NHHL will be:
98.327
98.327
6.5
8.0312%
 8.0312% = 7.8812% + 0.15%
 98.327 = 106.5/(1.08312)
 Will not be called.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NHLL will be:
99.844
99.844
6.5
5.5483%
 6.6660% = 6.5160% + 0.15%
 99.844 = 106.5/(1.06666)
 Will not be called.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NLLL will be:
100.902
100.000
6.5
5.5483%
 5.5483% = 5.3983% + 0.15%
 100.902 = 106.5/(1.055483)
 Will be called at 100.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NHH will be:
96.931
96.931
6.5
7.5053%
 7.5053% = 7.3553% + 0.15%

1  97.084  6.5 98.327  6.5 
96.931  

2  1.075053
1.075053 
 Will not be called.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NHL will be:
99.388
99.388
6.5
6.2354%
 6.2354% = 6.0854% + 0.15%

1  98.327  6.5 99.844  6.5 
99.388  

2  1.062354
1.062354 
 Will not be called.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NLL will be:
101.166
100.000
6.5
5.1958%
 5.1958% = 5.0458% + 0.15%

1  99.844  6.5 100  6.5 
101.166  

2  1.051958
1.051958 
 Will be called at 100.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NH will be:
98.874
98.874
6.5
5.9289%
 5.9289% = 5.7789% + 0.15%

1  97.084  6.5 99.388  6.5 
98.874  

2  1.059289
1.059289 
 Will not be called.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node NL will be:
101.190
100.000
6.5
4.9448%
 4.9448% = 4.7948% + 0.15%

1  99.388  6.5 100  6.5 
101.190  

2  1.049448 1.049448 
 Will be called at 100.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 11 Problem
 Node N will be:
101.863
4.0000%
 4.0000% = 3.850% + 0.15%

1  98.874  6.5 100  6.5 
101.863  

2  1.04000
1.04000 
 Will be called at 100.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Effective Duration and Convexity
 You will not have to calculate effective
duration or effective convexity on the exam
(pp. 314-315).
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Valuing Putable Bonds
 Recall that with a putable bond the owner has
the right to force the issuer to redeem the
bond and pay it off.
 Using the binomial model, the process we
follow is the same as with callable bonds,
except that we assume the bond will be “put”
if the price falls below some level.
 Let’s use the same interest rate tree as in
Exhibit 5, and consider a 4 year 6.5% bond
that is putable in one year at a price of 100.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 14: Putable Bond
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 15
Computed price is
below 100.
Computed price is
below 100.
It will be put.
It will be put.
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Exhibit 15
 From Node NHH:
1  100  6.5 100  6.5 
99.528  

2 1.070053 1.070053 
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Value of the Put Option
 We know from earlier coverage that this bond, if it is
option-free, is priced at $104.643.
 Exhibit 14 shows the putable bond is priced at
$105.327.
 Since:
Value of Put Option  Value of putable bond  Value of option - free bond
the option value is $105.327 - $104.643 = $0.684
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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Value of the Put Option
 We know that the option’s value increases
with increases in volatility.
 Therefore, if we increase our volatility
estimate, our model will produce a higher
price estimate for the putable bond.
 This makes sense since the bondholder owns
the option (which is increasing in value).
Copyright 2006, Jeffrey M. Mercer, Ph.D.
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