# Bonds with embedded options

Document Sample

```					Bonds with embedded options

UNIVERSITE
D’AUVERGNE
Drake
DRAKE UNIVERSITY
UNIVERSITE
D’AUVERGNE

Callable Bonds                   Drake
Drake University

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)
UNIVERSITE
D’AUVERGNE

Valuing a Callable Bond              Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Valuation Model Review                    Drake
Drake University

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.
UNIVERSITE

5.25% coupon bond, 3 years to                             D’AUVERGNE

Drake
maturity, yearly payments                               Drake University

Assume you have the following observed yield
curve, spot rates, and forward rates.
Maturity   YTM Market Value Spot Rate                  Forward
Rate
1 yr       3.5%            100     3.5%                3.5%
2 yr       4.0%            100     4.01%               4.5225%
3 yr       4.5%
5.25       5.25 100 105.4.531%
25                  5.5792%
        2
       3
 102.075
(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)
UNIVERSITE
D’AUVERGNE

Binomial Tree Model                    Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Binomial Tree Model       Drake
Drake University

l VH
NH

V
l
r*
N

l VL
NL
Time 0          Time 1
UNIVERSITE
D’AUVERGNE

Value at Time 0                    Drake
Drake University

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)
UNIVERSITE
D’AUVERGNE

Binomial Tree Model           Drake
Drake University

VH
l
C
NH H

V0
l
r*
N
VL
1  VH  CH VL  CL    l
V0                      NL CL
2  1 r *   1 r * 
Time 0             Time 1
UNIVERSITE
D’AUVERGNE

Simple Example                    Drake
Drake University

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 
UNIVERSITE
D’AUVERGNE

Binomial Tree Model continued            Drake
Drake University

Starting today we want to think about the
future path of interest rates.
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
UNIVERSITE
D’AUVERGNE

Binomial Tree Method continued              Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Binomial Tree Model             Drake
Drake University

VH
l CH
NH r1H=.05496

V0
l
r*
N
VL
l CL
NL r =.045
1L

Time 0         Time 1
UNIVERSITE
D’AUVERGNE

Extending the model               Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Binomial Tree Model               Drake
Drake University

VHH
l CHH
VH               NHH r2HH
C
l H
r1H=.05496           VHL
NH
V0 l                              l CHL
N
r=.035                           NHL r2HL
VL
l
NL CL
r1L=.045            VLL
l CLL
NLL r2LL
Time 0            Time 1                Time 2
UNIVERSITE
D’AUVERGNE

The Two Year Bond                   Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Binomial Tree Model              Drake
Drake University

VHH = 100
l CHH = 4
VH               NHH r2HH
C =4
l H
r1H=.05496           VHL = 100
NH
V0 l                              l CHL = 4
N
r=.035                           NHL r2HL
VL
l
NL CL = 4
r1L=.045            VLL= 100
l CLL= 4
NLL r2LL
Time 0            Time 1               Time 2
UNIVERSITE
D’AUVERGNE

Finding VH and VL                             Drake
Drake University

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   1 r                
 2 1.05496  1.05496   98.582
2       1H    1  r1H                      

1  VLL  CLL VHL  CHL  1  100  4 100  4 
VL   1 r                
 2 1.045  1.045   99.522
2        1L   1  r1L                      
UNIVERSITE
D’AUVERGNE

Binomial Tree Model              Drake
Drake University

VHH = 100
l CHH = 4
VH=98.852        NHH r2HH
C =4
l H
r1H=.05496           VHL = 100
NH
V0 l                              l CHL = 4
N
r=.035                           NHL r2HL
VL=99.522
l
NL CL = 4
r1L=.045            VLL= 100
l CLL= 4
NLL r2LL
Time 0            Time 1               Time 2
UNIVERSITE
D’AUVERGNE

Iterative Procedure                             Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Iterative procedure                Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Binomial Tree Model              Drake
Drake University

VHH = 100
l CHH = 4
VH=99.071        NHH r2HH
C =4
l H
r1H=.04976           VHL = 100
NH
V0 l                              l CHL = 4
N
r=.035                           NHL r2HL
VL=99.929
l
NL CL = 4
r1L=.04074          VLL= 100
l CLL= 4
NLL r2LL
Time 0            Time 1               Time 2
UNIVERSITE
D’AUVERGNE

Interpretations                  Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

The next step, time 3                 Drake
Drake University

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 V 0
r2HL=r2LLe2s as before and r2HH=r2HLe2sr2LLe4s
the correct rate is then 4.53%
UNIVERSITE
D’AUVERGNE

Binomial Tree Model               Drake
Drake University

VHH = 97.886
l CHH = 4.50
VH =98.074       NHH r2HH=.06757
C = 4.50
l H
r1H=.04976           VHL = 99.022
NH
V0 l                              l CHL = 4.50
N
r=.035                           NHL r2HL =.05532
VL=99.926
l
NL CL = 4.50
r1L=.04074          VLL= 100
l CLL= 4.50
NLL r2LL=.0453
Time 0            Time 1                Time 2
UNIVERSITE
D’AUVERGNE

Valuing an Option Free Bond               Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Binomial Tree Model              Drake
Drake University

VHH = 98.588
l CHH = 5.25
VH =99.461       NHH r2HH=.06757
C = 5.25
l H
r1H=.04976           VHL = 99.732
NH
V0 l                              l CHL = 5.25
N
r=.035                           NHL r2HL =.05532
VL=101.333
l
NL CL = 5.25
r1L=.04074          VLL= 100.689
l CLL= 5.25
NLL r2LL=.0453
Time 0            Time 1                Time 2
UNIVERSITE
D’AUVERGNE

Value of the bond                  Drake
Drake University

1  VH  C H VL  C L 
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!!!
UNIVERSITE
D’AUVERGNE

Valuing a Call Option                   Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Binomial Tree Model              Drake
Drake University

VHH = 98.588
l CHH = 5.25
VH =99.461       NHH r2HH=.06757
C = 5.25
l H
r1H=.04976           VHL = 99.732
NH
V0 l                              l CHL = 5.25
N
r=.035                           NHL r2HL =.05532
VL=101.001
l
NL VL=100              VLL= 100.689
CL = 5.25           VLL =100
r1L=.04074        l C = 5.25
LL
NLL
r2LL=.0453
Time 0            Time 1                  Time 2
UNIVERSITE
D’AUVERGNE

Value of callable bond                 Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Put option                    Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Binomial Tree Model (Put)         Drake
Drake University

VHH = 98.588
VHH=100
l
VH =100.261       C = 5.25
NHH HH
C = 5.25          r2HH=.06757
l H
r1H=.04976         VHL = 99.732
NH
V0 l                         l VHL=100
N
r=.035                      NHL CHL = 5.25
VL=101.461
l                   r2HL =.05532
NL CL = 5.25
r1L=.04074        VLL= 100.689
l CLL= 5.25
NLL r =.0453
2LL
Time 0          Time 1              Time 2
UNIVERSITE
D’AUVERGNE

Value of puttable bond                 Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Modeling Risk                      Drake
Drake University

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)
UNIVERSITE
D’AUVERGNE

Drake University

the forward rates in the binomial tree will
make the theoretical value equal to the
market price.
UNIVERSITE
D’AUVERGNE

OAS Intuition                       Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

OAS and total yield spreads             Drake
Drake University

The OAS is attempting to separate the
amount of the nominal spread that is the
result of option risk. Therefore it reports a
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.
UNIVERSITE
D’AUVERGNE

OAS and benchmarks                    Drake
Drake University

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.
what it actually represents however depends
upon the benchmark being used…
UNIVERSITE
D’AUVERGNE

OAS in our example                  Drake
Drake University

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
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
UNIVERSITE
D’AUVERGNE

Binomial Tree Model                Drake
Drake University

VHH = 102.6309
VHH = 100
l
CHH = 5.25
VH =99.833       NHH r =.07208
2HH
C = 5.25
l H                   VHL = 103.817
r1H=.005426
NH                      VHL = 100
V0 l                              l CHL = 5.25
N
r=.035                           NHL r2HL =.0598
VL=100.6946
l
NL VL=100              VLL= 104.809
CL = 5.25           VLL =100
r1L=.04524        l C = 5.25
LL
NLL
r2LL=.0498
Time 0            Time 1                  Time 2
UNIVERSITE
D’AUVERGNE

Value of callable bond (with OAS)         Drake
Drake University

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 
UNIVERSITE
D’AUVERGNE

Funding Cost as a Benchmark               Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Effective Duration and Convexity          Drake
Drake University

allows a yield change to change the expected
future cash flows.
UNIVERSITE

Quick approximation of duration            D’AUVERGNE

Drake
and convexity                       Drake University

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
P P           dec form)
duration       -    
2(P0 )( Dy)

P  P  2( P0 )
convexity 
(P0 )( Dy) 2
UNIVERSITE
D’AUVERGNE

Calculating P+ and P-               Drake
Drake University

1)   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
5)   Use the adjusted tree to determine the value
of P.
UNIVERSITE
D’AUVERGNE

Valuing a Step Up Callable Note            Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Valuing a Floating Rate Note               Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

A Capped Floater                   Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Analysis of Convertible Bonds              Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Conversion Ratio                        Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Other embedded options                   Drake
Drake University

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
UNIVERSITE
D’AUVERGNE

Minimum Value                      Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Market Conversion Price                   Drake
Drake University

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
market price of
market
convertible security
conversion 
conversion ratio
price
UNIVERSITE
D’AUVERGNE

Market Conversion Price               Drake
Drake University

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.
UNIVERSITE
D’AUVERGNE

Drake University

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)
UNIVERSITE
D’AUVERGNE

Drake University

The amount of time to recover the premium is

market conversion
premuin
payback 
favorable income
period
differential per share
market conversion price - current market price

n
copon interest - (conversio ratio x commonstock dividend per share)
conversion ratio
UNIVERSITE
D’AUVERGNE

Downside Risk                   Drake
Drake University

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 .
UNIVERSITE
D’AUVERGNE

Up side potential                   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.
UNIVERSITE
D’AUVERGNE

Option based value                  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.

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 7 posted: 2/18/2012 language: English pages: 60
How are you planning on using Docstoc?