8 BONDSWITHEMBEDDEDOPTIONS
ANALYSIS OF BONDS
WITH
EMBEDDED OPTIONS
A BOND WITH AN EMBEDDED OPTION IS ONE IN
WHICH EITHER THE ISSUER OR THE BONDHOLDER
HAS THE OPTION TO CHANGE A BOND’S CASH FLOWS
MOST COMMON EMBEDDED OPTION IS :
CALL OPTION
Issue
Price
YTM
(%)
Treasury
C=8.8%
96.61
9.15
Corporate
C=8.8%
87.07
10.24
Yield Spread =
109 BP
This simple analysis does not take into consideration
•
The term structure of interest rate
•
Volatility of interest rate
STATIC SPREAD
Will the cash flow analysis be the same for :
•a zero coupoun 25-year corporate bond
•a 8.8% coupon, 25-year corporate bond
NO
?
Risk asociated with holding a corporate over a Treasury
STATIC SPREAD
Zero Volatility spread
Z-spread
Spread that will make the PV of the cash flows from the corporate
bond, when discounted at the Swap zero-rates + spread , equal to
the corporate’s bond price
Static spread in our example would therefore be 120BP and not 109BP
The shorter the maturity of the bond, the less the static spread will
differ from the traditional yield spread.
On Bloomberg, when hitting YAS on a bond,
Z-spread is the static spread and OAS, the Option Adjutsted spread).
The option price is Z-spread- OAS
CALLABLE BONDS
•The holder of a callable bond has given the issuer
the right…to call (buy back) the issue prior to expiration .
Disadvantage for the bondholder :
Reinvestment
Risk
Lack of price
Appreciation potential
Lack of price
appreciation potential
PRICE COMPRESSION
PRICE/YIELD RELATIONSHIP FOR A CALLABLE BOND
PRICE
a’
P’
Bullet Bond (convexity shape)
b
Callable
Bond (a-b)
a
Price Compression
Y’
YIELD
PRICE/YIELD RELATIONSHIP FOR A CALLABLE BOND
PRICE
Price Compression
P’
b
Callable
Bond (a-b)
a
Y’
YIELD
A bond with an embedded option (call) can be considered as a portfolio
of : bond + Option
A bullet bond
A call option
YTMcallable<
/ >
YTMnoncallable
YTMcallable >
YTMnoncallable
CALLABLE BOND
NONCALLABLE BOND - CALL OPTION PRICE
PRICE/YIELD RELATIONSHIP FOR A CALLABLE BOND
PRICE
What is the option’s price ?
a’
PNC - PC = Call Option Price
PNC
Non callable Bond
b
PC
Callable
Bond (a-b)
a
Y’’
Y’
YIELD
VALUATION MODEL
CALLABLE BOND
NONCALLABLE BOND - CALL OPTION PRICE
VALUATION MODEL
PUTTABLE BOND
NONCALLABLE BOND + PUT OPTION PRICE
The price of an option free bond is the present value of the cash
flows discounted at the spot rates. What is the bond price ?
YEAR
ZERO
RATES
COUPON
RATE
(yearly)
Mkt VALUE
1
3.5%
5.25%
100
2
4.01%
5.25%
100
3
4.541%
5.25%
100
Will this bond trade at a premium/discount ? Premium because all zero rates < coupon rates
5.25/1.035 +
5.25/(1.0401)2
+
105.25/(1.451)3 = 102.047
When analysing embedded options, consideration must be
given to :
INTEREST RATE VOLATILITY
We are trying to determine how the 1-period forward rate can
vary over time based on some assumption about interest rate
volatility
We do this by introducing a Binomial
interest-rate Tree
OBJECTIVE
Determine whether the forward rates are
correctly reflected in the price of a bond
An interest rate model makes assumptions
about the relationship between the level of
short term interest rates and interest rate
volatility
NODE (time period)
r0
N
TODAY
r1H
NH
r1L
NL
1 year
r3HHH
NHHH
r2HH
NHH
r2HL
NHL
r2LL
NLL
2 years
r3HHL
NHHL
r3HLL
NHLL
r3LLL
NLLL
3 years
H : higher of the
two forward rates
r0
N
r1H
NH
r1L
NL
L : lower of the
two forward rates
TODAY
1 year
r3HHH
NHHH
r2HH
NHH
r2HL
NHL
r2LL
NLL
2 years
r3HHL
NHHL
r3HLL
NHLL
r3LLL
NLLL
3 years
H : the higher 1-year rate
one year from now
r0
N
r1H
NH
r1L
NL
L : the lower 1-year rate
one year from now
TODAY
1 year
r3HHH
NHHH
r2HH
NHH
r2HL
NHL
r2LL
NLL
2 years
r3HHL
NHHL
r3HLL
NHLL
r3LLL
NLLL
3 years
H : the higher 1-year rate
two year from now
r2HH
NHH
r0
N
r1H
NH
r1L
NL
r2HL
NHL
r2LL
NLL
L : the lower 1-year rate
two year from now
TODAY
1 year
2 years
r3HHH
NHHH
r3HHL
NHHL
r3HLL
NHLL
r3LLL
NLLL
3 years
N is the root of the tree and is nothing more than the current
1-year forward rate which is denoted by r0
The next year 1-year forward rate can take 2 possible values
of equal probability of occuring. One rate will be higher than the other.
It is assumed that the 1-year rate can evolve over time based on
a random process called Lognormal Random Walk with a certain
volatility.
= assumed volatility of the 1-year forward rate
r1,H = the higher 1-year rate one year from now
r1,L = the lower 1-year rate one year from now
r1,H = r1,L (e )
2
r1,H = r1,L (e )
2
If
r1,L = 4.074% with a 10% volatility…
Then
r1,H = 4.976%
YEAR 2
•3 different outcomes in the second year for the 1-year rate.
R2,LL = 1-year rate in the second year assuming the lower rate in
the first year and the lower rate in the second year
R2,HH = 1-year rate in the second year assuming the higher rate in
the first year and the higher rate in the second year
R2,HL = 1-year rate in the second year assuming the higher rate in
the first year and the lower rate in the second year(or vice versa)
r2,HH= r2,LL (e )
4
r2,HL= r2,LL (e )
2
r3e6
NHHH
r2e4
NHH
r0
N
TODAY
r1e2
NH
r1
NL
1 year
r2e2
NHL
r2
NLL
2 years
r3e4
NHHL
r3e2
NHLL
r3
NLLL
3 years
DETERMINING THE
VALUE AT A NODE
Components to price a bond ?
•Coupon (C)
•Forward rate ( r )
•Maturity ( t )
The value of the bond at each node depends on the future cash flow
r3e6
NHHH
•The appropriate rate to use is
the 1-year forward rate at the node
r2e4
NHH
r0
N
TODAY
r1e2
NH
r1
NL
1 year
r2e2
NHL
r2
NLL
2 years
r3e4
NHHL
r3e2
NHLL
r3
NLLL
3 years
•The appropriate rate to use is
the 1-year forward rate at the node
r2e4
NHH
r1e2
NH
r2e2
NHL
VH = Bond’s value for the higher rate
VL = Bond’s value for the lower rate
C = Coupon rate of the bond
The cash flow at each node is either :
• VH + C for the higher rate
• VL + C for the lower rate
What is the present value of VH + C ?
What is the present value of VL + C ?
VH + C
1 + r
VL + C
1 + r
VALUE AT NODE
VH + C
1 + r
+
VL + C
1 + r
--------------------------------------------------------------------------
2
EXAMPLE
The goal here is to determine wether the 1-year low rate in 1 year r 1,L
used to price the bond is correct
• 2 YEAR BOND
•TRADING AT 100 TODAY
•VOLATILITY = = 10%
•ANNUAL COUPON = 4%
Step by step process….
Step 1 :
Select a value for r1 , lowest 1-year rate one year from now
Let’s select r1 arbitrarily = 4.5%
Step 2 : Determine the corresponding value for the higher 1-year
forward rate.
(2 *0.10)
r 1,H = 0.045e
= 5.496%
Step 3 :
Compute the bond’s value one year from now
(at maturity for us, therefore 100 + 4 = 104)
Step 4 :
Calculate the bond’s value in step3 using the higher rate
V H = 104/1+0.05496 = 98.585
Step 5 = Calculate the bond’s value in step3 using the lower rate
V L = 104/1+0.045 = 99.522
Step 6 =
Add the coupon to V H and V L to get the cash flow
at N H and N L
V H + C = 102.582
V L + C = 103.522
Step 7 =
Calculate the PV of those 2 values using the root
rate of 3.5%
102.582 / 1.035 = 99.13
103.522 / 1.035 = 100.021
Step 8 =
Calculate the average of the two PV
(99.13 + 100.021 ) / 2 = 99.567
WHAT WAS THE PRICE OF OUR BOND TODAY ?
100
Remember step 1 : lowest 1-year rate one year from now
let’s select r1 = 4.5%
What is needed is to find the exact 1-year forward rate,
one year from now, so that our bond price becomes 100 instead
of 99.567
Will r 1 have to be higher or lower ?
Lower (4.074%)
Next step is to determine the low 1-year rate two years from now.
It needs to be done by trail and error on Excel.
For this, we analyse a 3-year 4 ½ coupon bond that trades at par.
We know from previous calculations that the 1-year, one year from now,
is at 4,074% and that the 1-year rate today is 3,50%.
R1,0 = 3,5%
R1,1 = 4,074%
R1,2 = ?
Vol
Face Value
Coupon
10%
Year
0
1
2
3
100
4,50%
V
100
C
4,50%
Data based on the market
V
97,88497
C
4,50%
YTM
Check
100,00
6,758%
V
98,07298
V
100
C
4,50%
C
4,50%
R
4,976%
V
102,075
V
99,0212
C
4,50%
C
4,50%
R
3,50%
YTM
5,533%
V
99,92529
V
100
C
4,50%
C
4,50%
R
4,074%
V
100
C
4,50%
V
99,9713
C
4,50%
YTM
4,530%
Now that we have all three low rates,
R1,0 = 3,5%
R1,1 = 4,074%
R1,2 = 4,53%
…..it is easy to determine the other rates on the binomial tree
with the formula :
R1,H = R1,Le2∞
And complete the process to determine any 3-year bond price at t0
Valuing a Callable Corporate Bond
Same process as an option free bond except :
•When the call option may be exercised by the issuer
the bond value at the node must be changed to reflect the
lower of its value if it is not called and call price.
The price of an option free bond is the present value of the cash
flows discounted at the spot rates. What is the bond price ?
5.25/1.035
YEAR
ZERO
RATES
COUPON
RATE
(yearly)
Mkt VALUE
1
3.5%
5.25%
100
2
4.01%
5.25%
100
3
4.54%
5.25%
100
+
5.25/(1.0401)2
+
105.25/(1.0454)3 = 102.075
Suppose this same bond is callable at 100 in year 2…..
Any bond valuation above 100 (node NL an NLL) must be called at 100.
Call option = non callable bond – callable bond
102.075 -
101.432 =
0.643
On Bloomberg, when hitting YAS on a bond,
Z-spread is the static spread and OAS, the Option Adjutsted spread).
The option price is Z-spread- OAS
Hope you enjoyed the class !
WITH
EMBEDDED OPTIONS
A BOND WITH AN EMBEDDED OPTION IS ONE IN
WHICH EITHER THE ISSUER OR THE BONDHOLDER
HAS THE OPTION TO CHANGE A BOND’S CASH FLOWS
MOST COMMON EMBEDDED OPTION IS :
CALL OPTION
Issue
Price
YTM
(%)
Treasury
C=8.8%
96.61
9.15
Corporate
C=8.8%
87.07
10.24
Yield Spread =
109 BP
This simple analysis does not take into consideration
•
The term structure of interest rate
•
Volatility of interest rate
STATIC SPREAD
Will the cash flow analysis be the same for :
•a zero coupoun 25-year corporate bond
•a 8.8% coupon, 25-year corporate bond
NO
?
Risk asociated with holding a corporate over a Treasury
STATIC SPREAD
Zero Volatility spread
Z-spread
Spread that will make the PV of the cash flows from the corporate
bond, when discounted at the Swap zero-rates + spread , equal to
the corporate’s bond price
Static spread in our example would therefore be 120BP and not 109BP
The shorter the maturity of the bond, the less the static spread will
differ from the traditional yield spread.
On Bloomberg, when hitting YAS on a bond,
Z-spread is the static spread and OAS, the Option Adjutsted spread).
The option price is Z-spread- OAS
CALLABLE BONDS
•The holder of a callable bond has given the issuer
the right…to call (buy back) the issue prior to expiration .
Disadvantage for the bondholder :
Reinvestment
Risk
Lack of price
Appreciation potential
Lack of price
appreciation potential
PRICE COMPRESSION
PRICE/YIELD RELATIONSHIP FOR A CALLABLE BOND
PRICE
a’
P’
Bullet Bond (convexity shape)
b
Callable
Bond (a-b)
a
Price Compression
Y’
YIELD
PRICE/YIELD RELATIONSHIP FOR A CALLABLE BOND
PRICE
Price Compression
P’
b
Callable
Bond (a-b)
a
Y’
YIELD
A bond with an embedded option (call) can be considered as a portfolio
of : bond + Option
A bullet bond
A call option
YTMcallable<
/ >
YTMnoncallable
YTMcallable >
YTMnoncallable
CALLABLE BOND
NONCALLABLE BOND - CALL OPTION PRICE
PRICE/YIELD RELATIONSHIP FOR A CALLABLE BOND
PRICE
What is the option’s price ?
a’
PNC - PC = Call Option Price
PNC
Non callable Bond
b
PC
Callable
Bond (a-b)
a
Y’’
Y’
YIELD
VALUATION MODEL
CALLABLE BOND
NONCALLABLE BOND - CALL OPTION PRICE
VALUATION MODEL
PUTTABLE BOND
NONCALLABLE BOND + PUT OPTION PRICE
The price of an option free bond is the present value of the cash
flows discounted at the spot rates. What is the bond price ?
YEAR
ZERO
RATES
COUPON
RATE
(yearly)
Mkt VALUE
1
3.5%
5.25%
100
2
4.01%
5.25%
100
3
4.541%
5.25%
100
Will this bond trade at a premium/discount ? Premium because all zero rates < coupon rates
5.25/1.035 +
5.25/(1.0401)2
+
105.25/(1.451)3 = 102.047
When analysing embedded options, consideration must be
given to :
INTEREST RATE VOLATILITY
We are trying to determine how the 1-period forward rate can
vary over time based on some assumption about interest rate
volatility
We do this by introducing a Binomial
interest-rate Tree
OBJECTIVE
Determine whether the forward rates are
correctly reflected in the price of a bond
An interest rate model makes assumptions
about the relationship between the level of
short term interest rates and interest rate
volatility
NODE (time period)
r0
N
TODAY
r1H
NH
r1L
NL
1 year
r3HHH
NHHH
r2HH
NHH
r2HL
NHL
r2LL
NLL
2 years
r3HHL
NHHL
r3HLL
NHLL
r3LLL
NLLL
3 years
H : higher of the
two forward rates
r0
N
r1H
NH
r1L
NL
L : lower of the
two forward rates
TODAY
1 year
r3HHH
NHHH
r2HH
NHH
r2HL
NHL
r2LL
NLL
2 years
r3HHL
NHHL
r3HLL
NHLL
r3LLL
NLLL
3 years
H : the higher 1-year rate
one year from now
r0
N
r1H
NH
r1L
NL
L : the lower 1-year rate
one year from now
TODAY
1 year
r3HHH
NHHH
r2HH
NHH
r2HL
NHL
r2LL
NLL
2 years
r3HHL
NHHL
r3HLL
NHLL
r3LLL
NLLL
3 years
H : the higher 1-year rate
two year from now
r2HH
NHH
r0
N
r1H
NH
r1L
NL
r2HL
NHL
r2LL
NLL
L : the lower 1-year rate
two year from now
TODAY
1 year
2 years
r3HHH
NHHH
r3HHL
NHHL
r3HLL
NHLL
r3LLL
NLLL
3 years
N is the root of the tree and is nothing more than the current
1-year forward rate which is denoted by r0
The next year 1-year forward rate can take 2 possible values
of equal probability of occuring. One rate will be higher than the other.
It is assumed that the 1-year rate can evolve over time based on
a random process called Lognormal Random Walk with a certain
volatility.
= assumed volatility of the 1-year forward rate
r1,H = the higher 1-year rate one year from now
r1,L = the lower 1-year rate one year from now
r1,H = r1,L (e )
2
r1,H = r1,L (e )
2
If
r1,L = 4.074% with a 10% volatility…
Then
r1,H = 4.976%
YEAR 2
•3 different outcomes in the second year for the 1-year rate.
R2,LL = 1-year rate in the second year assuming the lower rate in
the first year and the lower rate in the second year
R2,HH = 1-year rate in the second year assuming the higher rate in
the first year and the higher rate in the second year
R2,HL = 1-year rate in the second year assuming the higher rate in
the first year and the lower rate in the second year(or vice versa)
r2,HH= r2,LL (e )
4
r2,HL= r2,LL (e )
2
r3e6
NHHH
r2e4
NHH
r0
N
TODAY
r1e2
NH
r1
NL
1 year
r2e2
NHL
r2
NLL
2 years
r3e4
NHHL
r3e2
NHLL
r3
NLLL
3 years
DETERMINING THE
VALUE AT A NODE
Components to price a bond ?
•Coupon (C)
•Forward rate ( r )
•Maturity ( t )
The value of the bond at each node depends on the future cash flow
r3e6
NHHH
•The appropriate rate to use is
the 1-year forward rate at the node
r2e4
NHH
r0
N
TODAY
r1e2
NH
r1
NL
1 year
r2e2
NHL
r2
NLL
2 years
r3e4
NHHL
r3e2
NHLL
r3
NLLL
3 years
•The appropriate rate to use is
the 1-year forward rate at the node
r2e4
NHH
r1e2
NH
r2e2
NHL
VH = Bond’s value for the higher rate
VL = Bond’s value for the lower rate
C = Coupon rate of the bond
The cash flow at each node is either :
• VH + C for the higher rate
• VL + C for the lower rate
What is the present value of VH + C ?
What is the present value of VL + C ?
VH + C
1 + r
VL + C
1 + r
VALUE AT NODE
VH + C
1 + r
+
VL + C
1 + r
--------------------------------------------------------------------------
2
EXAMPLE
The goal here is to determine wether the 1-year low rate in 1 year r 1,L
used to price the bond is correct
• 2 YEAR BOND
•TRADING AT 100 TODAY
•VOLATILITY = = 10%
•ANNUAL COUPON = 4%
Step by step process….
Step 1 :
Select a value for r1 , lowest 1-year rate one year from now
Let’s select r1 arbitrarily = 4.5%
Step 2 : Determine the corresponding value for the higher 1-year
forward rate.
(2 *0.10)
r 1,H = 0.045e
= 5.496%
Step 3 :
Compute the bond’s value one year from now
(at maturity for us, therefore 100 + 4 = 104)
Step 4 :
Calculate the bond’s value in step3 using the higher rate
V H = 104/1+0.05496 = 98.585
Step 5 = Calculate the bond’s value in step3 using the lower rate
V L = 104/1+0.045 = 99.522
Step 6 =
Add the coupon to V H and V L to get the cash flow
at N H and N L
V H + C = 102.582
V L + C = 103.522
Step 7 =
Calculate the PV of those 2 values using the root
rate of 3.5%
102.582 / 1.035 = 99.13
103.522 / 1.035 = 100.021
Step 8 =
Calculate the average of the two PV
(99.13 + 100.021 ) / 2 = 99.567
WHAT WAS THE PRICE OF OUR BOND TODAY ?
100
Remember step 1 : lowest 1-year rate one year from now
let’s select r1 = 4.5%
What is needed is to find the exact 1-year forward rate,
one year from now, so that our bond price becomes 100 instead
of 99.567
Will r 1 have to be higher or lower ?
Lower (4.074%)
Next step is to determine the low 1-year rate two years from now.
It needs to be done by trail and error on Excel.
For this, we analyse a 3-year 4 ½ coupon bond that trades at par.
We know from previous calculations that the 1-year, one year from now,
is at 4,074% and that the 1-year rate today is 3,50%.
R1,0 = 3,5%
R1,1 = 4,074%
R1,2 = ?
Vol
Face Value
Coupon
10%
Year
0
1
2
3
100
4,50%
V
100
C
4,50%
Data based on the market
V
97,88497
C
4,50%
YTM
Check
100,00
6,758%
V
98,07298
V
100
C
4,50%
C
4,50%
R
4,976%
V
102,075
V
99,0212
C
4,50%
C
4,50%
R
3,50%
YTM
5,533%
V
99,92529
V
100
C
4,50%
C
4,50%
R
4,074%
V
100
C
4,50%
V
99,9713
C
4,50%
YTM
4,530%
Now that we have all three low rates,
R1,0 = 3,5%
R1,1 = 4,074%
R1,2 = 4,53%
…..it is easy to determine the other rates on the binomial tree
with the formula :
R1,H = R1,Le2∞
And complete the process to determine any 3-year bond price at t0
Valuing a Callable Corporate Bond
Same process as an option free bond except :
•When the call option may be exercised by the issuer
the bond value at the node must be changed to reflect the
lower of its value if it is not called and call price.
The price of an option free bond is the present value of the cash
flows discounted at the spot rates. What is the bond price ?
5.25/1.035
YEAR
ZERO
RATES
COUPON
RATE
(yearly)
Mkt VALUE
1
3.5%
5.25%
100
2
4.01%
5.25%
100
3
4.54%
5.25%
100
+
5.25/(1.0401)2
+
105.25/(1.0454)3 = 102.075
Suppose this same bond is callable at 100 in year 2…..
Any bond valuation above 100 (node NL an NLL) must be called at 100.
Call option = non callable bond – callable bond
102.075 -
101.432 =
0.643
On Bloomberg, when hitting YAS on a bond,
Z-spread is the static spread and OAS, the Option Adjutsted spread).
The option price is Z-spread- OAS
Hope you enjoyed the class !