Debt Instruments and Markets
Professor Carpenter
American Options and
Callable Bonds
Concepts and Buzzwords
ƒ American Options
ƒ Valuing an American
Call on a Coupon
Bond
ƒ Valuing a Callable
Bond
ƒ Interest Rate
Sensitivity of a
Callable Bond
•
exercise policy,
value-maximization,
redeem
Reading
Tuckman, chapter 19.
American Options and Callable Bonds
1
Debt Instruments and Markets
Professor Carpenter
American Options
¾An
American option can be exercised before
its expiration date.
¾For options on zero-coupon bonds (or on any
asset that pays no cash flows before the
option expiration date)
• it would never be optimal to exercise a call early,
• it might be optimal to exercise a put early.
No Early Exercise of Calls on Assets that
Have No Payments Prior to Expiration:
American call value
≥ European call value
= European put value + V − dT K
> V − K = exercise value
Prior to expiration, an American call on a "nonpaying" asset is worth more than its exercise value.
American Options and Callable Bonds
2
Debt Instruments and Markets
Professor Carpenter
American Options on Assets with
Intervening Payments:
Early Exercise
For call options on assets that make
payments prior to option expiration, such as
coupon bonds, early exercise is optimal if
the call gets deep enough in the money.
Exercise or Wait?
ƒ Advantage to early
exercise:
– Receive cash flows before
the expiration date, if the
underlying asset pays any
American Options and Callable Bonds
ƒ Advantages to waiting:
– Keep the option not to
exercise (the put)
– Do not pay any money
now (get interest on K)
3
Debt Instruments and Markets
Professor Carpenter
American Call on the Coupon Bond
¾ Consider
an American call option on $100 par of
the 2-year, 5.5%-coupon bond.
¾ The strike price of the call is $100.
¾ The call expires at time 2.
Underlying 5.5%-Coupon Bond
Each node in the tree below lists the current ex-coupon price of the coupon
bond, and the current price of a zero with 6 months to maturity. At each
node, the bond is simply priced as a package of zeroes .
Time 0
Time 0.5
Time 1
= 0.970857x2.75+0.941787x2.75
+0.91318x102.75
100.0019
0.973047
0.5
0.5
99.0890
0.970857
100.9548
0.976941
0.5
0.5
0.5
0.5
= 0.973047x2.75+0.947649x2.75+
0.922242x2.75+0.897166x102.75
American Options and Callable Bonds
98.6063
0.966581
100.0207
0.973533
101.1547
0.979071
0.5
0.5
0.5
0.5
0.5
0.5
Time 1.5
98.8626
0.962167
99.6684
0.970009
100.3113
0.976266
100.8227
0.981243
4
Debt Instruments and Markets
Professor Carpenter
Exercise Policy for the American Call
¾ To
value the call, we need to know its future cash
flows at each time and state.
¾ The future cash flows of the option depend on the
option holder's exercise policy.
¾ At each time and state the option holder must
decide whether or not to exercise the option.
¾ The exercise policy is a rule that specifies the
option holder's action at each time and state.
Value-Maximizing Exercise Policy
¾ We
assume that whomever holds the option will
choose, at each time and state, to exercise it or not
depending on which action maximizes the option's
market value.
¾ This assumption is reasonable as long as the option
holder can freely sell the option.
¾ Then, if ever the option holder wants to get out of his
position when the option is worth more than its
exercise value, the option holder just sells the
option instead of exercising it.
American Options and Callable Bonds
5
Debt Instruments and Markets
Professor Carpenter
Valuing the American Call
¾ The
value of the call depends on the exercise policy,
which determines its future cash flows.
¾ At any point in time, the exercise decision involves
comparing the exercise value of the call and the
market value of the call if left unexercised.
¾ The unexercised call value depends on its future
possible market values.
¾ Therefore, the exercise policy and the call value must
be determined together, working backward from the
expiration date.
Call Value at the Bond's Maturity Date
¾ The
stated expiration date of the call is time 2.
¾ However, at time 2, the underlying coupon bond
matures and is worth par, which is the strike price of
the option.
¾ Therefore, at time 2, the option is worthless.
¾ Consequently, its effective expiration date is really
one period prior to the maturity date of the coupon
bond, time 1.5.
American Options and Callable Bonds
6
Debt Instruments and Markets
Professor Carpenter
Call at Time 1.5
Coupon Bond
Call
98.8626
0
99.6684
0
100.3113
0.3113
100.8227
0.8227
Call at Time 1
Coupon Bond and
Discount Factor
Time 1
98.6063
0.966581
100.0207
0.973533
101.1547
0.979071
American Options and Callable Bonds
Call
Time 1
Time 1.5
exercise value < 0
wait value = 0
Î call value = 0
0
exercise value = 0.0207
wait value = 0.973533x
0.5(0+0.3113)=0.1515
Î call value=0.1515
0
exercise value=1.1547
wait value=0.979071x
0.5(0.3113+0.8227)=0.5552
Î call value=1.1547
0.3113
0.8227
7
Debt Instruments and Markets
Professor Carpenter
Call at Time 0.5
Call
Coupon Bond and
Discount Factor
Time 0.5
Time 0.5
exercise value<0
wait value=0.970857x
0.5(0+0.1515)=0.0736
Î call value=0.0736
99.0890
0.970857
100.9548
0.976941
exercise value=0.9548
wait value=0.976941x
0.5(0.1515+1.1547)=0.6380
Î call value=0.9548
Time 1
0
0.1515
1.1547
Call at Time 0
Coupon Bond and
Discount Factor
Call
Time 0
Time 0
100.0019
0.973047
exercise value=0.0019
wait value=0.973047x
0.5(0.0736+0.9548)=0.5003
Î call value=0.5003
Time 0.5
0.0736
American Options and Callable Bonds
0.9548
8
Debt Instruments and Markets
Professor Carpenter
Callable Bond
¾ Consider
a firm that issues a 2-year, 5.5%coupon bond that is callable at par.
¾ This means that the issuer has the option to call
(or ‘redeem’, or buy back) the bond for par at any
time for par.
¾ By selling a callable bond to investors, the issuer
has effectively
ƒ sold a noncallable bond to the investors and
ƒ simultaneously bought a call option on the bond from
those investors.
Valuing the Callable Bond
¾ Callable
Bond = Noncallable Bond - Call Option
¾ To value the callable bond, assume that the issuer
follows a strategy for exercising the call that
ƒ maximizes the value of the call, or equivalently,
ƒ minimizes the value of the callable bond, the issuer's liability.
¾ Since
we have already valued the noncallable and the
option, we already know the value of the callable bond
at each time and state.
ƒ For example, at time 0, the callable is worth
100.0019 - 0.5003 = 99.5016
¾ The
next slides verify this.
American Options and Callable Bonds
9
Debt Instruments and Markets
Professor Carpenter
Callable Bond at Time 1.5
Noncallable Bond
Callable Bond
98.8626
98.8626
99.6684
99.6684
100.3113
100 (called)
100.8227
100 (called)
Callable Bond at Time 1
Noncallable Bond and
Discount Factor
Time 1
98.6063
0.966581
100.0207
0.973533
101.1547
0.979071
American Options and Callable Bonds
Callable Bond
Time 1
called value=100
wait value = 0.966581x
(0.5(98.8626+99.6684)+ 2.75)= 98.6063
Î callable bond value = 98.6063
called value=100
wait value = 0.973533x
(0.5(99.6684+100)+2.75)=99.8692
Î callable bond value=99.8692
called value=100
wait value=0.979071x102.75= 100.5995
Î callable bond value=100
Time 1.5
98.8626
99.6684
100
(called)
100
(called)
10
Debt Instruments and Markets
Professor Carpenter
Callable Bond at Time 0.5
Callable Bond
Noncallable Bond
and Discount Factor
Time 0.5
Time 0.5
called value=100
wait value=0.970857x
(0.5(98.6063+99.8692)+2.75)=
99.0155
Î callable bond value=99.0155
99.0890
0.970857
100.9548
0.976941
called value=100
wait value=0.976941x
(0.5(99.8692+100)+2.75)=
100.3168
Î callable bond value=100
Time 1
0.5
98.6063
0.5
0.5
99.8692
0.5
100
(called)
Callable Bond at Time 0
Coupon Bond and
Discount Factor
Time 0
Call
Time 0
Time 0.5
99.0155
100.0019
0.973047
American Options and Callable Bonds
called value=100
wait value=0.973047x
(0.5(99.0155+100)+2.75)=99.5016
Î call value=99.5016
100
(called)
11
Debt Instruments and Markets
Professor Carpenter
Price Tree for Callable and Noncallable Bonds
Each node in the tree lists the current ex-coupon price of the noncallable
bond, the price of the option, and the ex-coupon price of the callable bond.
Time 0
100.0019
0.5003
99.5016
Time 0.5
99.0890
0.0736
99.0155
100.9548
0.9548
100 (called)
Time 1
98.6063
0
98.6063
100.0207
0.1515
99.8692
101.1547
1.1547
100 (called)
Time 1.5
98.8626
0
98.8626
99.6684
0
99.6684
100.3113
0.3113
100 (called)
100.8227
0.8227
100 (called)
Another Way to See Callable = Noncallable – Option:
See Refunding Profit as Option Exercise Value
ƒ Suppose that the way the firm finances the call is by
selling new noncallable bonds with the same coupon
and maturity as the old debt, a “refunding.”
ƒ The profit from refunding
= proceeds of sale of new debt - call price
= value of the noncallable -100
=option exercise value
ƒ With this plan in place, a firm that has issued a callable
bond plans to pay the noncallable cash flows until
maturity but gets to do a refunding along the way.
ƒ The firms’ net position = short a callable bond = short a
noncallable bond, long an option.
American Options and Callable Bonds
12
Debt Instruments and Markets
Professor Carpenter
Interest Rate Delta of the Callable Bond
Time 0.5
Noncallable
Option
Callable
Short Rate
99.0890
0.0736
99.0155
6.004%
100.9548
0.9548
100
4.721%
ir∆ = −
Bu − Bd
ru − rd
99.0890 − 100.9548
= 145.43
0.06004 − 0.04721
0.0736 − 0.9548
ir∆ call = −
= 68.69
0.06004 − 0.04721
99.0155 − 100
ir∆ callable = −
= 76.74
0.06004 − 0.04721
ir∆ noncallable = −
The relationship that holds for prices also holds for deltas:
Callable (77) = Noncallable (145) - Call Option (68)
In particular, the callable has less interest rate
sensitivity than the noncallable.
American Options and Callable Bonds
13