Debt OPTIONS
Options on Treasury Securities:
T-Bill Options
• Options on T-Bills give the holder the right to buy a T-Bill
with a face value of $1M and maturity of 91 days.
• Exercise price is quoted in terms of the IMM index and
the following formula can be used to determine X:
100  RD (.25)
X 
($1 M )
100
• The option premium is quoted in terms of annual discount
points (PT). The actual premium is
PT
C or P 
(.25) ($1 M )
100
Options on Treasury Securities:
T- Bond Options
• Options on T-Bonds give the holder the right to buy a
specified T-Bond with a face value of $100,000.
• Exercise price is quoted as a percentage of par (e.g. IN =
90). If the holder exercises, she pays the exercise price
plus the accrued interest:
IN
X 
($100,000)  Acc Int
100
• The option premium is quoted in terms of points (PT).
The actual premium is
PT
C or P 
($100,000)
100
Fundamental Strategies
• There are six fundamental strategies:
–
–
–
–
–
–
Call Purchase
Naked Call Write
Covered Call Write
Put Purchase
Naked Put Write
Covered Put Write
Profit Graph
• Option Strategies can be evaluated in terms
of a profit graph.
• A profit graph is a plot of the option
position’s profit and security price relation
at expiration or when the option is
exercised.

ST
ST

Figure 17.3-1: Call Purchase
• Buy T-Bond call: X = $100,000, C = $1000

4,000
100,000
1,000
105,000
ST
Call Purchase
Spot Price at T
90000
95000
100000
101000
102000
103000
104000
105000
106000
Profit/Loss
-1000
-1000
-1000
0
1000
2000
3000
4000
5000
Figure 17.3-2: Naked Call Write
• Sell T-Bond call for: X= 100,000, C=1000.

1,000
100,000
4,000
105,000
ST
Naked Call Write
Spot Price at T
90000
95000
100000
101000
102000
103000
104000
105000
106000
Profit/Loss
1000
1000
1000
0
-1000
-2000
-3000
-4000
-5000
Figure 17.3-3: Covered Call Write
•
Long T-Bond at 100,000, short 100 T-Bond call at 1.

1,000
95,000
4,000
100,000
ST
Covered Call Write
Spot Price at T
90000
95000
100000
101000
102000
103000
104000
105000
106000
Short Call
Profit/Loss
1000
1000
1000
0
-1000
-2000
-3000
-4000
-5000
Long T-Bond
Total
Profit/Loss Profit/Loss
-10000
-9000
-5000
-4000
0
1000
1000
1000
2000
1000
3000
1000
4000
1000
5000
1000
6000
1000
Figure 17.3-4: Put Purchase
• Buy T-Bond put: X=100,000, P = 1000

4,000
95,000
1,000
100,000
ST
Put Purchase
Spot Price at T
90000
95000
100000
101000
102000
103000
104000
105000
106000
Profit/Loss
9000
4000
-1000
-1000
-1000
-1000
-1000
-1000
-1000
Figure 17.3-5: Naked Put Write
• Sell T-Bond put: X =100,000, P = 1000

1,000
95,000
4,000
100,000
ST
Naked Put Write
Spot Price at T
90000
95000
100000
101000
102000
103000
104000
105000
106000
Profit/Loss
-9000
-4000
1000
1000
1000
1000
1000
1000
1000
Figure 17.3-6: Covered Put Write
•
Short T-Bond at 100,000, short 100 T-Bond put at 1.

1,000
100,000
4,000
105,000
ST
Covered Put Write
Spot Price at T
90000
95000
100000
101000
102000
103000
104000
105000
106000
Short Put
Profit/Loss
-9000
-4000
1000
1000
1000
1000
1000
1000
1000
Short T-Bond
Profit/Loss
10000
5000
0
-1000
-2000
-3000
-4000
-5000
-6000
Total
Profit/Loss
1000
1000
1000
0
-1000
-2000
-3000
-4000
-5000
Other Strategies
Figure 17.4-1: Straddle Purchase
•
Buy 100 T-Bond put for 1 and buy 100 T-Bond call for 1:

4,000
3,000
95,000
1,000
2,000
100,000
105,000
ST
Straddle Purchase
Spot Price at T
94000
97000
98000
100000
102000
103000
106000
Call Purchase
Profit/Loss
-1000
-1000
-1000
-1000
1000
2000
5000
Put Purchase
Profit/Loss
5000
2000
1000
-1000
-1000
-1000
-1000
Total
Profit/Loss
4000
1000
0
-2000
0
1000
4000
Figure 17.4-2: Bull Spread
•
Buy 100 T-Bond call for 1 and sell 101 T-Bond call for .75:


1,000
750
100,000
250
1,000
101,000
102,000

ST
Bull Spread
Spot Price at T
94000
97000
98000
100000
100250
101000
102000
103000
106000
100 Call Purchase at 1 101 Call Sale at .75
Profit/Loss
Profit/Loss
-1000
750
-1000
750
-1000
750
-1000
750
-750
750
0
750
1000
-250
2000
-1250
5000
-4250
Total
Profit/Loss
-250
-250
-250
-250
0
750
750
750
750
Hedging
Table 17.8-4: Hedging the Cost of a September T-Bill
Purchase with a T-Bill Call
Call: X = 94 (985,000), C = 1 ($2,500)
1
2
1
Spot Rate: R
Spot Price
7.5
981250
7.25
981875
6.75
983125
6.5
983750
6.25
984375
6
985000
5.75
985625
5.5
986250
5.25
986875
5
987500
4.75
988125
100  R (.25)
Spot price  ST 
$1 M
100
100  (100  94)(.25)
X
$1 M  $985,000
100
1
90
C0 
($1 M )  $2,500
100 360
F
IJF
IJ
G
G
H KH K
L$1,000,000 O
Hedged YTM  M
NEffective Cost P
Q
365/ 91
1
3
Profit/Loss
-2500
-2500
-2500
-2500
-2500
-2500
-1875
-1250
-625
0
625
4
Effective Costs
col 2 - col 3
983750
984375
985625
986250
986875
987500
987500
987500
987500
987500
987500
5
Hedged
YTM
0.0679212
0.065204167
0.059795959
0.05710472
0.054422013
0.051747806
0.051747806
0.051747806
0.051747806
0.051747806
0.051747806
Table 17.8-5: Hedging a Future T-Bond Sale with a
T-Bond Put
T-Bond: M = 15yrs at T; Coupon = 6%
1
Spot Index
91
91.5
92
92.5
93
93.5
94
94.5
95
95.5
96
96.5
Estimated YTM 
Put: X = 94,000, P = 1000
2
3
4
Long Put
Spot Price
Estimated YTM
Profit/Loss
91000
0.069109948
2000
91500
0.068581375
1500
92000
0.068055556
1000
92500
0.067532468
500
93000
0.06701209
0
93500
0.066494401
-500
94000
0.065979381
-1000
94500
0.065467009
-1000
95000
0.064957265
-1000
95500
0.064450128
-1000
96000
0.063945578
-1000
96500
0.063443596
-1000
$6000  ($100,000  Spot price) / 15
($100,000  Spot price) / 2
5
Revenue
Col 2 + Col 4
93000
93000
93000
93000
93000
93000
93000
93500
94000
94500
95000
95500
Futures Options on
Treasury Securities
• Futures options give the holder the right to take a
futures position:
– Futures Call Option gives the holder the right to go long.
When the holder exercises, she obtains a long position in the
futures at the current price, ft, and the assigned writer takes
the short position and pays the holder ft - X.
– Futures Put Option gives the holder the right to go short.
When the holder exercises, she obtains a short position at the
current futures price, ft, and the assigned writer takes the long
position and pays put holder X - ft.
• Futures option on Treasuries: Options on T-Bill
Futures, T-Bond Futures, and T-Note Futures.
Exhibit 17.9-1: Futures Options on
Treasury Securities
Call on T-Bill Futures:
Exercise at 980,000: Holder goes
long at f T  980,000 and then closes
• X = IMM 90 or X = $975,000
by going short at f T  980,000,
and receives f T  X  980,000  975,000:
• PT = .5 or C = $1,250
• Futures and options futures have same expiration.   980000  975000  1250  3750.
ST  f T
C  ( f T  975,000)
10.5
973,750
 $1250
 1250
10.0
9.5
9.0
8.5
975,000
976,250
977,500
978,750
 1250
0
1250
2500
8.0
980,000
3750
RD


3750
975000
1250
980000
ST  f T
Exhibit 17.9-2: Futures Options on
Treasury Securities
Exercise at 970,000: Holder goes
Put on T-Bill Futures:
short at f T  970,000 and then
closes by going long at f T  980,000,
• X = IMM 90 or X = $975,000
and receives X  f T  975,000  970,000:
• PT = .5 or P = $1,250
  975000  970000  1250  3750.
• Futures and options futures have same expiration.
RD
ST  f T
 P  (975,000  f T
12.0
970,000
 $1250
3750
115
.
110
.
971,250
972,500
2500
1250
10.5
973,750
0
10.0
9.5
975,000
976,250
 1250
 1250
9.0
8.5
977,500
978,750
 1250
 1250
8.0
980,000
 1250

3750

970000
1250
972250
ST  f T
Table 17.9-1: Put-Call-Futures Parity
Expiration Cash Flow
Position
Long Futures
Long Put
Short Call
Total
Investment
0
P0
C0
P0  C0
fT  X
fT  f0
X  fT
fT  X
ST  f 0
0
fT  X
fT  f0
0
0
X  f0
0
X  f0
fT  X
X  f0
Value of the conversion:
P0  C0  0  ( X  f 0 )(1  R f )  T
Hedging Cases
Exhibit 18.2-2: Hedging $5M CF in June with
June T-Bill Futures Call
Call: X = 90 (975,000), C = 1.25 ($3,125), n = 5.1282051
1
2
3
4
Spot Rate: R
Spot Price Profit/Loss nTB
8
980000
9615.38456 5.112
8.5
978750
3205.12819 5.112
9
977500
-3205.1282 5.112
9.5
976250
-9615.3846 5.112
9.75
975625
-12820.513 5.112
10
975000
-16025.641 5.112
10.25
974375
-16025.641 5.115
10.5
973750
-16025.641 5.118
10.75
973125
-16025.641 5.122
11
972500
-16025.641 5.125
11.25
971875
-16025.641 5.128
 C  5128205
.
[ Max ( ST  $975,000, 0]  $3,125]
nTB 
$5 M   C
ST
YTM
Ln ($1M ) O
M
N$5 M P
Q
365/ 91
TB
1
5
YTM
0.093
0.093
0.093
0.093
0.093
0.093
0.096
0.098
0.101
0.104
0.107
Managing the Maturity
Gap with T-Bill Put
• Case: In June, a bank makes a $1M loan for 180 days which it
plans to finance by selling a 90-day CD now at the LIBOR of
8.258% and a 90-day CD ninety days later (in September) at the
LIBOR prevailing at that time. To minimize its exposure to market
risk, the bank buys a T-Bill put at X = IMM = 90 for $$1250.
X  IMM  90 or RD  10%
100  (10)(.25)
X 
($1 M )  $975,000
100
Bank sells $1 M CD now ( June) at 8.258%.
At the September maturity , the bank will owe
$1.019758 M . To hedge this liability , the bank
would need to buy 10459056
.
puts:
CFT $1.019758 M

 10459056
.
puts
X
$975,000
Cost  (10459056
.
)($1250)  $1307
nP 
Maturity Gap Hedged with T-Bill Puts
(1)
( 2)
(3)
( 4)
(5)
RD
ST
RateCD
p
Debt on CD
(6)
( 7)
Funds Needed Debt 90 days later
(5)  (4)
(6)[1  (3)]90/ 365
(8)
Rate
[(7) / $1 M ]365/180
7%
8%
$982,500
$980,000
.07588
.08690
 1307
 1307
1,019,750
1,019,750
1,021,065
1,021,065
1,039,646
1,042,262
8.203%
8.756%
9%
$977,500
.09807
 1307
1,019,750
1,021,065
1,044,893
9.313%
10% $975,000
11% $972,500
.10940
.12080
 1307
1307
1,019,750
1,019,750
1,021,065
1,018,451
1,047,500
1,047,500
9.867%
9.867%
12% $970,000
.13240
3922
1,019,750
1,015,836
1,047,500
9.867%
Assume 90  day CD rate is .25% greater than T  Bill rate:
365/ 91
 $1 M 
RateCD  
 .0025  1

S
 T 
 P  14059056
.
[ Max[$975,000  ST , 0]  $1307
Hedging future T-Bond Sale
With T-Bond Puts
• Case: Three months from the present (.25 of year), a
bond manager plans to sell a T-Bond with maturity of
15.25 years, F = $100,000, and coupon rate = 10%.
• Manager hedges the sale against interest rate increases
by buying one put option on a T-Bond with a current
maturity of 15.25 years and face value of $100,000.
The put has an expiration of T = .25 years, exercise
price of X = IN = 95 or X = $95,000, and is trading at
P = 1 - 5 or P = [1.15625/100]($100,000) = $1156.
Hedging future T-Bond Sale
With T-Bond Puts
• Hedge T-Bond Sale:
ST
91
92
93
93844
.
94
95
96
97
Yield
1110%
.
10.97%
10.85%
10.74%
10.72%
10.60%
10.48%
10.35%
P
$2,844
$1,844
$844
0
 $156
 $1,156
 $1,156
 $1,156
Hedged revenue
$93,844
$93,844
$93,844
$93,844
$93,844
$93,844
$94,844
$95,844
$10,000  [$100,000  [( ST / 100)($100,000)] / 15]
$100,000  ( ST / 100)($100,000)
2
Hedged revenue  ( ST / 100)($100,000)   P
Yield  ARTM 
Hedging Future Bond Portfolio
Sale With T-Bond Puts
• Case: Three months from the present (.25 of year), a bond manager
plans to liquidate a bond portfolio consisting of AAA, AA, and A
bonds. The portfolio currently has a WAM of 15.25 years, F =
$10M, WAC = 10%, and has tended to yield a rate 1% above TBond rates.
• Manager hedges the sale against interest rate increases by buying
put options on a T-Bond with a current maturity of 15.25 years and
face value of $100,000. The put has an expiration of T = .25 years,
exercise price of X = IN = 95 or X = $95,000, and is trading at P =
1 - 5 or P = [1.15625/100]($100,000) = $1156.
• To hedge, the manager buys 105.26316 T-Bond puts for $121,684:
F
$10 M

 105.26316
X
$95,000
Cost  (105,26316) ($1156)  $121,684
nP 
Hedging Future Bond Portfolio
Sale With T-Bond Puts
• Hedge Bond Portfolio Sale:
ST
Yield
93
94
95
96
97
98
10.85%
10.72%
10.60%
10.48%
10.35%
10.23%
P
Bond revenue
Hedged revenue
$88,842
$16,421
 $121,684
 $121,684
 $121,684
 $123,684
$8.7298 M
$8.8108 M
$8.8865 M
$8.9633 M
$9.0477 M
$9.1266 M
$8.80 M
$8.80 M
$8.80 M
$8.84 M
$8.92 M
$9.00 M
15
$10 M
(1  yield )15
t 1
 P  105.26316[ Max[$95,000  ( ST / 100)($100,000), 0 ]  $121,684
Hedged revenue  Bond revenue   P
Bond revenue 
$1 M
 (1  yield )
t

Interest Rate Options
Interest Rate Options
• Interest rate call option gives the holder the
right to a payoff if an interest rate (e.g.,
LIBOR) exceeds a specified exercise rate;
interest rate put option gives the holder the
right to a payoff if an interest rate is less
than the exercise rate.
• Interest rate options are written by
commercial banks in conjunction with a
future loan or CD investment.
Interest Rate Call Option
Case:
• A company plans to borrow $10M in sixty days from
Sun Bank. The loan is for 90 days with the rate equal to
LIBOR in 60 days plus 100 BP.
• Worried that rates could increase in the next 60 days,
the company buys an interest rate call from the bank for
$20,000.
• Terms: Exercise Rate = 7%; call premium plus interest
will be paid at the maturity of the loan; any interest rate
payoff will be paid at the loan’s maturity.
• See Chapter 17.
Interest Rate Put Option
Case:
• A company plans to invest $10M in sixty days in a Sun
Bank 90-day CD. The CD will pay the LIBBER.
• Worried that rates could decrease in the next 60 days,
the company buys an interest rate put from the bank for
$15,000.
• Terms: Exercise Rate = 7%; put premium plus interest
will be paid at the maturity of the CD; any interest rate
payoff will be paid at the CD’s maturity.
• See Chapter 17
Caps: Series of Interest
Rate Call Options
• A Cap is a series of interest rate calls that expire at or near
the interest rate payment dates on a loan. They are written by
financial institutions in conjunction with a variable rate loan.
Case:
• A company borrow $50M from Commerce Bank to finance its yearly
construction projects. The loan starts on March 1 at 8% and is reset
every three months at the prevailing LIBOR.
• Cap: In order to obtain a maximum rate while still being able to obtain
lower rates if the LIBOR falls, the company buys a Cap from the bank for
$100,000 with exercise Rate = 8%.
• See Chapter 17
Floor: Series of Interest
Rate Put Options
• A floor is a series of interest rate puts that expire at or
near the payment dates on a loan. They are purchased
by financial institutions in conjunction with a variable
rate loan they are providing.
Case:
• Commerce Bank purchases a floor with an exercise rate
of 8% for $70,000 from another institution to protect
the variable rate loan it made.
• See Chapter 17
Table 17.8-1: Profit and Interest Rate Relation from
Closing a Long 94 T-Bill Call Purchased at 1
Call: X = 94 (985,000), C = 1 ($2,500)
Spot Rate: R
6.5
6.25
6
5.75
5.5
5.25
5
4.75
4.5
4.25
4
100  R(.25)
$1 M
100
100  (100  94)(.25)
X
$1 M
100
1
90
C0 
($1 M )  $2,500
100 360
Spot price  ST 
F
IJF
IJ
G
G
H KH K
Spot Index = 100-R
93.5
93.75
94
94.25
94.5
94.75
95
95.25
95.5
95.75
96
Spot Price
983750
984375
985000
985625
986250
986875
987500
988125
988750
989375
990000
Long Call
Profit/Loss
-2500
-2500
-2500
-1875
-1250
-625
0
625
1250
1875
2500
Table 17.8-3: Profit and Interest Rate Relation from
Closing a Long 94 T-Bond Put Purchased at $1000
T-Bond: M = 15yrs at T; Coupon = 6%;
Spot Index
90
90.5
91
91.5
92
92.5
93
93.5
94
94.5
95
Estimated YTM 
Put: X = 94,000, P = 1000
Spot Price
90000
90500
91000
91500
92000
92500
93000
93500
94000
94500
95000
$6000  ($100,000  Spot price) / 15
($100,000  Spot price) / 2
Estimated YTM
0.070175439
0.069641295
0.069109948
0.068581375
0.068055556
0.067532468
0.06701209
0.066494401
0.065979381
0.065467009
0.064957265
Long Put
Profit/Loss
3000
2500
2000
1500
1000
500
0
-500
-1000
-1000
-1000
Table 17.8-2: Profit and Interest Rate Relation from
Closing a Long 94 T-Bill Put Purchased at 1
Put: X = 94 (985,000), C = 1 ($2,500)
Spot Rate: R
8
7.75
7.5
7.25
7
6.75
6.5
6.25
6
5.75
5
100  R(.25)
$1 M
100
100  (100  94)(.25)
X
$1 M
100
1
90
P0 
($1 M )  $2,500
100 360
Spot price  ST 
F
IJF
IJ
G
G
H KH K
Spot Index = 100-R
92
92.25
92.5
92.75
93
93.25
93.5
93.75
94
94.25
95
Spot Price
980000
980625
981250
981875
982500
983125
983750
984375
985000
985625
987500
Long Put
Profit/Loss
2500
1875
1250
625
0
-625
-1250
-1875
-2500
-2500
-2500