Chapter 14
Interest Rate Options:
Fundamentals
Spot Option: Definition
• An option is a security that gives the holder the
right (but not the obligation) to buy an asset at a
specific price (exercise price: X) on or possibly
before a specific date (expiration date: T).
– A Call option is an option that gives the holder
the right to buy a specific asset or security.
– A Put option is an option that gives the holder
the right to sell a specific asset or security.
Terms
– Option Holder: Buyer of the option; has the
right to exercise; long position.
– Option Writer: Seller of the option; has the
responsibility to fulfill the terms of the option if
the holder exercises; short position.
– Option Premium: Price of the option:
• C = call premium
• P = put premium
Terms
• American Option: Option that can be
exercised at any time on or before the
expiration date.
• European Option: Option that can be
exercised only on the expiration date.
Symbols
• Symbols:
• T = Expiration or when the option is exercised.
• 0 = current period
• t = any time between current and expiration
• Example:
• CT = Call price at expiration
• C0 = Current call price
Terms
• Spot Option: Options contracts on stocks,
debt securities, foreign currencies, and
indices are sometimes referred to as spot
options or options on actuals.
• This reference is to distinguish them from
options on futures contracts (also called
options on futures, futures options, and
commodity options).
Futures Options
• A futures option gives the holder the right to
take a position in a futures contract.
– Call option on a futures contract gives the
holder the right to take a long position in the
underlying futures contract.
– Put option on a futures option gives the holder
the right to take a short position in the
underlying futures contract.
Futures Options
• Like all option positions, the futures option
buyer pays an option premium for the right
to exercise, and the writer, in turn, receives
a premium when he sells the option and is
subject to initial and maintenance margin
requirements on the option position.
Futures Call Option
• A call option on a futures contract gives the holder
the right to take a long position in the underlying
futures contract when she exercises, and requires
the writer to take the corresponding short position
in the futures.
• Upon exercise, the holder of a futures call option
in effect takes a long position in the futures
contract at the current futures price and the writer
takes the short position and pays the holder via the
clearinghouse the difference between the current
futures price and the exercise price.
Futures Put Option
• A put option on a futures option entitles the holder
to take a short futures position and the writer the
long position.
• Upon exercise, the put holder in effect takes a
short futures position at the current futures price
and the writer takes the long position and pays the
holder via the clearinghouse the difference
between the exercise price and the current futures
price.
Exercising Futures Call Options
• In practice, when the holder of a futures call option
exercises, the futures clearinghouse will establish for the
exercising option holder a long futures position at the
futures price equal to the exercise price and a short futures
position for the assigned writer.
• Once this is done, margins on both positions will be
required and the position will be marked to market at the
current settlement price on the futures.
• When the positions are marked to market, the exercising
call holder’s margin account on his long position will be
equal to the difference between the futures price and the
exercise price, ft-X, while the assigned writer will have to
deposit funds or near monies worth ft-X to satisfy her
maintenance margin on her short futures position.
Exercising Futures Call Options
• Thus, when a futures call is exercised, the holder
takes a long position at ft with a margin account
worth ft-X; if he were to immediately close the
futures he would receive cash worth ft-X from the
clearinghouse.
• The assigned writer, in turn, is assigned a short
position at ft and must deposit ft-X to meet her
margin.
Exercising Futures Put Options
• If the futures option is a put, the same procedure
applies except that holder takes a short position at
ft (when the exercised position is marked to
market), with a margin account worth X-ft, and the
writer is assigned a long position at ft and must
deposit X-ft to meet her margin.
Differences Between
Futures Options and Spot Options
• Spot options and futures options are
equivalent if
– The options and the futures contracts expire at
the same time
– The carrying-costs model holds
– The options are European.
Differences Between
Futures Options and Spot Options
•
There are, though, several factors that serve to differentiate the
two contracts:
1. Since many futures contracts are relatively more liquid than their
corresponding spot security, it is usually easier to form hedging or
arbitrage strategies with futures options than with spot options.
2. Futures options often are easier to exercise than their corresponding
spot. For example, to exercise an option on a T-bond futures, one
simply assumes the futures position, while exercising a spot T-bond
option requires an actual purchase or delivery.
3. Most futures options are traded on the same exchange as their
underlying futures contract, while most spot options are traded on
exchanges different from their underlying securities. This, in turn,
makes it easier for futures options traders to implement arbitrage and
hedging strategies than spot options traders.
Markets for Interest Rate Options
• Many different types of interest rate options
are available on the
– Organized futures and options exchanges, and
– OTC market
Markets for Interest Rate Options
• Exchange-traded interest rate options include both
futures options and spot options.
– On the U.S. exchanges, the most heavily traded options are
the CME’s and CBOT’s futures options on T-bonds, T-notes,
T-bills, and Eurodollar contracts.
– The CBOE, AMEX, and PHLX have offered options on
actual Treasury securities and Eurodollar deposits. These
spot options, however, proved to be less popular than futures
options and have been delisted.
– A number of non-U.S. exchanges, though, do list options on
actual debt securities, typically government securities.
Markets for Interest Rate Options
• There is a large OTC market in debt and interestsensitive securities and products in the U.S. and a
growing OTC market outside the U.S..
• Currently, security regulations in the U.S. prohibit
off-exchange trading in options on futures.
• All U.S. OTC options are therefore options on
actuals.
Markets for Interest Rate Options
• The OTC markets in and outside the U.S. consists
primarily of dealers who make markets in the
underlying spot security, investment banking
firms, and commercial banks.
• OTC options are primarily used by financial
institutions and non-financial corporations to
hedge their interest rate positions.
Markets for Interest Rate Options
• The option contracts offered in the OTC
market include:
– Spot options on Treasury securities
– LIBOR-related securities
– Special types of interest rate products, such as:
•
•
•
•
Interest rate calls and puts
Caps
Floors
Collars
Types: CBOT’s Futures Options
The CBOT offers trading on interest rate
futures options on T-bonds, T-notes with
maturities of 10 years, 5 years, and 2 years,
the Municipal Bond Index, and the
Mortgage-Backed bond contract.
Types: CBOT’s Futures Options
Futures Options on T-Bonds and Notes
• The call and put contracts on the T-bond and Tnote futures are set with exercise prices that are
one point apart (104, 105, 106, etc.; ½ point
intervals for other T-notes) and with expiration
months following the March, June, September,
and December cycle, with one expiration month
being the one in front of the month with the
current quarter.
• The premiums on the options are quoted as a
percentage of the face value of the underlying
bond or note.
Types: CBOT’s Futures Options
Futures Options on T-Bonds and Notes
• A buyer of an April 104 T-bond futures call trading
at 2 –11 (or 2 11/64 = 2.171875) would pay
$2,171.87 for the option to take a long position in
the April T-bond futures at an exercise price of
$104,000.
Types: CBOT’s Futures Options
Futures Options on T-Bonds and Notes
• If long-term rates were to subsequently drop, causing the
April T-bond futures price to increase to ft = 108, then the
holder, upon exercising, would have a long position in the
April T-bond futures contract and a margin account worth
$4,000.
• If she closed her contract at 108, she would have a profit of
$1,828.13:
ft  X
 108  104 
F  
$100,000  $4,000

100
 100 
  $4,000  $2,171.87  $1,828.13
Value of M arg in 
Types: CBOT’s Futures Options
Futures Options on T-Bonds and Notes
• By contrast, if long-term rates were to stay the
same or increase, then the call would be
worthless and the holder would simply allow it to
expire, losing the $2,171.87 premium.
Types: CME’s Futures Options
Futures Options on Eurodollars and T-Bills
• The CME offers trading on futures options on Tbills, Eurodollar deposits, and 30-day LIBOR
contracts.
• The maturities of the options correspond to the
maturities on the underlying futures contracts.
• The exercise quotes are based on the system used
for quoting the futures contracts.
Types: CME’s Futures Options
Futures Options on Eurodollars and T-Bills
• The exercise prices on the Eurodollar and T-bill futures
contracts are quoted in terms of an index equal to 100
minus the annual discount yield: 100 – RD. The
formula for X:
100  R D (.25)
X
($1M)
100
Types: CME’s Futures Options
Futures Options on Eurodollars and T-Bills
• The option premiums are quoted in terms of an index
point system.
• For T-bill and Eurodollars, the dollar value of an option
quote is based on a $25 value for each basis point
underlying a $1M T-bill or Eurodollar.
• The actual quotes are in percents; thus a 1.25 quote
would imply a price of $25 times 12.5 basis points:
($25)(12.5) = $312.50
Or simply:
($250)(1.25) = $312.50
Types: CME’s Futures Options
Note:
• For the closest maturing month, the options
are quoted to the nearest quarter of a basis
point.
• For other months, they are quoted to the
nearest half of a basis point.
Types: CME’s Futures Options
• Example: The actual price on a March Eurodollar call with an
exercise price of 94.5 quoted at 5.92 is $1,481.25. The price is
obtained by rounding the 5.92 quoted price to 5.925, converting
the quote to basis points (multiply by 10), and multiplying by
$25:
(5.925)(100)($25) = $1,481.25
Or
(5.925)($250) = $1,481.25
• A 10.30 quote on a 94.5 April call indicates a call price of $2,575:
(10.30)(10)($25) = $2,575
Or
(10.30)($250) = $2,575
Types: CME’s Futures Options
An investor buying the 94.5 March call would
therefore pay $1,481.25 for the right to take a long
position in the CME’s $1M March Eurodollar
futures contract at an exercise price of $986,250:
100  (100  94.5)(90 / 360)
X 
$1,000,000  $986,250
100
Types: CME’s Futures Options
If short-term rates were to subsequently drop, causing the March
Eurodollar futures price to increase to an index price of 95.5 (RD =
4.5 and ft = [[100 – 4.5(90/360)]/100]($1,000,000) = $988,750), the
holder, upon exercising, would have a long position in the CME
March Eurodollar futures contract and a futures margin account
worth $2,500. If she closed the position at 95.5, she would realize a
profit of $1,018.75:
M arg in Value  f t  X  $988,750  $986,250  $2,500
M arg in Value  $25 (Futures Index  Exercsie Index )
 $25[95.5  94.5](100)  $2,500
  $2,500  $1,481.25  $1,018.75
Types: CME’s Futures Options
• If short-term rates were at RD = 5.5% and stayed
there or increased, then the call would be worthless
and the holder would simply allow it to expire,
losing her $1,018.75 premium.
Types: OTC Options
• OTC options can be structured on almost any
interest-sensitive position an investor or borrower
may wish to hedge.
• U.S. Treasuries, LIBOR-related instruments, and
Mortgage-backed securities are often the most
common underlying security.
• When spot options are structured on securities, terms
such as the specific underlying security, its maturity
and size, the option’s expiration, and the delivery are
all negotiated.
Types: OTC Options
• For an OTC option on a Treasury, the underlying
security is often a recently auctioned Treasury (onthe-run bond), although some selected existing
securities (off-the-run securities) are used.
• The bid-ask spreads on OTC Treasury options tend
to be larger than exchange-trade ones.
• The option maturities on OTC contracts can range
from one day to several years, with many of the
options being European.
Types: OTC Options
• In the case of OTC spot T-bond or T-note options,
OTC dealers either offer or will negotiate contracts
giving the holder to right to purchase or sell a
specific T-bond or T-note.
• Example: A dealer might offer a T-bond call option
to a fixed income manager giving him the right to
buy a specific T-bond maturing in year 2016 and
paying a 6% coupon with a face value of $100,000.
Types: OTC Options
• Note: Because the option contract specifies a
particular underlying bond, the maturity of the
bond, as well as its value, will be changing during
the option's expiration period.
• Example: a one-year call option on the 15-year
bond, if held to expiration, would be a call option
to buy a 14-year bond.
Types: OTC Options
• Note: A spot T-bill option contract offered by a
dealer on the OTC market usually calls for the
delivery of a T-bill meeting the specified criteria
(e.g., principal = $1 million; maturity = 91 days).
• With this clause, a T-bill option is referred to as a
fixed deliverable bond, and unlike
specific-security T-bond options, T-bill options can
have expiration dates that exceed the T-bill's
maturity.
Types: OTC Options
• A second feature of a spot T-bond or T-note options
offered or contracted on the OTC market is that the
underlying bond or note can pay coupon interest
during the option period.
• As a result, if the option holder exercises on a
non-coupon paying date, the accrued interest on the
underlying bond must be accounted for. For a
T-bond or T-Note option, this is done by including
the accrued interest as part of the exercise price.
Types: OTC Options
• Like futures options, the exercise price on a spot
T-bond or T-note option is quoted as an index equal
to a proportion of a bond with a face value of $100
(e.g., 95).
• If the underlying bond or note has a face value of
$100,000, then the exercise price would be:
 Index 
X 
($100,000)  Accrued Interest

 100 
Types: OTC Options
• The prices on spot T-bond and T-note options are
typically quoted like futures T-bond options in terms
of points and 32nds of a point. Thus, the price of a
call option on a $100,000 T-Bond quoted at 1 5/32 is
$1,156.25 = (1.15623/100)($100,000)
Types: OTC Options
Interest Rate Call and Interest Rate Put
• In addition to option contracts on specific
securities, the OTC market also offers a number of
interest-rate option products.
• These products are usually offered by commercial
or investment banks to their clients.
• Two products of note are interest rate calls and
interest rate puts.
Types: OTC Options
Interest Rate Call
• An interest rate call, also called a caplet, gives the buyer a
payoff on a specified payoff date if a designated interest rate,
such as the LIBOR, rises above a certain exercise rate, Rx.
• On the payoff date:
– If the designated rate is less than Rx, the interest rate call
expires worthless.
– If the rate exceeds Rx, the call pays off the difference
between the actual rate and Rx, times a notional principal,
NP, times the fraction of the year specified in the contract.
Types: OTC Options
Interest Rate Call
• Example: Given an interest rate call with a
designated rate of LIBOR, Rx = 6%, NP = $1M,
time period of 180 days, and day-count convention
of actual/360, the buyer would receive a $5,000
payoff on the payoff date if the LIBOR were 7%:
(.07-.06)(180/360)($1M) = $5,000
Types: OTC Options
Interest Rate Call
• Interest rate call options are often written by
commercial banks in conjunction with futures
loans they plan to provide to their customers.
• The exercise rate on the option usually is set
near the current spot rate, with that rate often
being tied to the LIBOR.
Types: OTC Options
Interest Rate Call
• Example: A company planning to finance a future $10M
inventory 60 days from the present by borrowing from a
bank at a rate equal to the LIBOR + 100 BP at the start of
the loan could buy from the bank an interest rate call option
with an exercise rate equal to say 8%, expiration of 60 days,
and notional principal of $10M.
• At expiation, 60 days later, the company would be entitled
to a payoff if rates are higher than 8%. Thus, if rates on the
loan increase, the company would receive a payoff that
would offset the higher interest on the loan.
Types: OTC Options
Interest Rate Put
• An interest rate put, also called a floorlet, gives the
buyer a payoff on a specified payoff date if a
designated interest rate is below the exercise rate,
Rx.
• On the payoff date:
– If the designated rate is more than Rx, the interest rate
put expires worthless.
– If the rate is less than Rx, the put pays off the difference
between Rx and the actual rate times a notional principal,
NP, times the fraction of the year specified in the
contract.
Types: OTC Options
Interest Rate Put
• Example: Given an interest rate put with a
designated rate of LIBOR, Rx = 6%, NP = $1M,
time period of 180 days, and day-count convention
of actual/360, the buyer would receive a $5,000
payoff on the payoff date if the LIBOR were 5%:
(.06-.05)(180/360)($1M) = $5,000
Types: OTC Options
Interest Rate Put
• A financial or non-financial corporation that is
planning to make an investment at some future date
could hedge that investment against interest rate
decreases by purchasing an interest rate put from a
commercial bank, investment banking firm, or
dealer.
Types: OTC Options
Interest Rate Put
• Example: suppose that instead of needing to borrow $10M,
the previous company was expecting a net cash inflow of
$10M in 60 days from its operations and was planning to
invest the funds in a 90-day bank CD paying the LIBOR.
• To hedge against any interest rate decreases, the company
could purchase an interest rate put (corresponding to the
bank's CD it plans to buy) from the bank with the put having
an exercise rate of say 7%, expiration of 60 days, and
notional principal of $10M.
• The interest rate put would provide a payoff for the company
if the LIBOR were less than 7%, giving the company a hedge
against interest rate decreases.
Types: OTC Options
Cap
• A popular option offered by financial
institutions in the OTC market is the cap.
• A plain-vanilla cap is a series of European
interest rate call options – a portfolio of
caplets.
Types: OTC Options
Cap
• Example: A 7%, two-year cap on a three-month
LIBOR, with a NP of $100M, provides, for the next
two years, a payoff every three months of (LIBOR .07)(.25)($100M) if the LIBOR on the reset date
exceeds 7% and nothing if the LIBOR equals or is
less than 7%.
• Note: Typically, the payoff does not occur on the
reset date, but rather on the next reset date (three
months later).
Types: OTC Options
Cap
• Caps are often written by financial
institutions in conjunction with a floatingrate loan and are used by buyers as a hedge
against interest rate risk.
Types: OTC Options
Cap
• Example:
– A company with a floating-rate loan tied to the LIBOR
could lock in a maximum rate on the loan by buying a cap
corresponding to its loan.
– At each reset date, the company would receive a payoff
from the caplet if the LIBOR exceeded the cap rate,
offsetting the higher interest paid on the floating-rate loan;
on the other hand, if rates decrease, the company would pay
a lower rate on its loan while its losses on the caplet would
be limited to the cost of the option.
• Thus, with a cap, the company is able to lock in a
maximum rate each quarter, while still benefiting with
lower interest costs if rates decrease.
Types: OTC Options
Floor
• A plain-vanilla floor is a series of European
interest rate put options – a portfolio of
floorlets.
Types: OTC Options
Floor
• Example: A 7%, two-year floor on a three-month
LIBOR, with a NP of $100M, provides, for the next
two years, a payoff every three months of (.07 LIBOR)(.25)($100M) if the LIBOR on the reset
date is less than 7% and nothing if the LIBOR
equals or exceeds 7%.
Types: OTC Options
Floor
• Floors are often purchased by investors as a tool to
hedge their floating-rate investment against interest
rate declines.
• Thus, with a floor, an investor with a floating-rate
security is able to lock in a minimum rate each
period, while still benefiting with higher yields if
rates increase.
Fundamental Strategies
• Fundamental Strategies:
–
–
–
–
Call Purchase
Naked Call Write
Put Purchase
Naked Put Write
Profit Graph
• Option Strategies can be evaluated in terms
of a profit graph.
ST

• A profit graph is a plot of the option
position’s profit, π, and underlying spot
price, S, or futures price, f, relation at
expiration or when the option is exercised.
Fundamental Spot Option Positions
• To see the major characteristics of option positions,
consider spot call and put options on an OTC 6% Tbond with face value of $100,000, a maturity at the
option’s expiration of 15 year, no accrued interest at
the option’s expiration date, and currently selling at
par.
ST

• Suppose the T-bond’s exercise prices for both call
and put options (X) are $100,000 (quoted at 100)
and an investor can buy the options at a call or put
premium of $1,000 (quoted at 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
Naked Call Write
• Sell T-bond call for: X= 100,000, C=1000.

1,000
100,000
4,000
ST
105,000 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
Put Purchase
• Buy T-bond put: X=100,000, P = 1000

4,000
95,000
 1000
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
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
Fundamental Futures
Option Positions
• The important characteristics of futures options can also be seen
by examining the profit relationships for the fundamental call and
put positions formed with these options.
• The next two exhibits show the profit and futures price
relationship at expiration for a long call and put positions on a Tbill futures.
ST

• The call and put options each have exercise prices equal to 90
(index) or X = $975,000, are priced at $1,250 (quote of 5:
(5)($250) = $1,250) and it is assumed the T-bill futures option
expires at the same time as the underlying T-bill futures contract.
Fundamental Futures Option Positions
Call Purchase
• The numbers shown in the call exhibit reflect a case in
which the holder exercises the call at expiration, if
profitable, when the spot price is equal to the price on the
expiring futures contract.
ST

• Example: At ST = fT = $980,000, the holder of the 90 T-bill
futures call would receive a cash flow of $5,000 for a profit
of $3,750 = $5,000-$1,250.
– That is, upon exercising the holder would assume a long position in
the expiring T-bill futures priced at $980,000 and a futures margin
account worth $5,000 ((fT - X) = $980,000 - $975,000) = $5,000).
– Given we are at expiration, the holder would therefore receive
$5,000 from the expired futures position, leaving her with a profit of
$3,750.
Futures Options on Treasury Securities
Call on T-bill Futures:
Exercise at 980,000 : Holder goes
• X = IMM 90 or X = $975,000
long at f T  980,000 and then closes
• PT = 5 or C = $1,250
by going short at f T  980,000,
• Futures and options futures have same expiration.and receives f T X  980,000  975,000 :
  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
Fundamental Futures Option Positions
Naked Call Write
• The opposite profit and futures price relation is
attained for a naked call write position.
• In this case:
– If the T-Bill futures is at $975,00 or less, the writer of
the futures call would earn the premium of $1,250.
– If fT > $975,000, he, upon the exercise by the holder (or
assignment by the clearinghouse), would assume a
short position at fT and would have to pay fT -X to bring
the margin on his expiring short position into balance.
ST

Fundamental Futures Option Positions
Put Purchase
• The next exhibit shows a long put position on the 90 T-bill
futures purchased at $1,250.
• In the case of a put purchase, if the holder exercises when fT is
less than X, then he will have a margin account worth X-fT on
an expiring short futures position.
ST

• Example: If ST = fT = $970,000 at expiration, then the put
holder upon exercising would receive $5,000 from the
expiring short futures (X - fT = $975,000 - $970,000) yielding
a profit from her futures option of $3,750.
• The put writer's position would be the opposite.
Futures Options on Treasury Securities
Put on T-bill Futures:
• X = IMM 90 or X = $975,000
• PT = 5 or P = $1,250
• Futures and options futures have same expiration.
Exercise at 970,000 : Holder goes
short at f T  970,000 and then
closes by going long at f T  980,000,
and receives Xf T  975,000  970,000 :
  975000  97000012503750.
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
Fundamental Interest Rate Option Positions
• The next exhibit shows the profit graph and table
for an interest rate call with the following terms:
–
–
–
–
–
Exercise rate = 6%
Reference rate = LIBOR
NP = $10M
Time period as proportion of a year = .25
Cost of the option = $12,500.
ST

Fundamental Interest Rate Option Positions
• If the LIBOR reaches 7.5% at expiration, the
holder would realize a payoff of (.075.06)($10M)(.25) = $37,500 and a profit of
$25,000.
ST

• If the LIBOR is 6.5%, the holder would breakeven
with the $12,500 payoff equal to the option’s cost.
• If the LIBOR is 6% or less, there would be no
payoff and the holder would incur a loss equal to
the call premium of $12,500.
Fundamental Interest Rate Option Positions
LIBOR
4.0%
5.0%
6.0%
6.5%
7.0%
7.5%
8.0%

 $12,500
 $12,500
 $12,500
0
$12,500
$25,000
$37,500
$

37500
2
12500
  Max[LIBOR  .07,0](.25)($10M)  $12,500
4
6
6.5
8
LIBOR %
Fundamental Interest Rate Option Positions
• The next exhibit shows the profit graph and table
for an interest rate put with terms similar to those
of the interest rate call:
– Exercise rate = 6%
– Reference rate = LIBOR
– NP = $10M
– Time period as proportion of a year = .25
– Cost of the option = $12,500.
ST

Fundamental Interest Rate Option Positions
$
LIBOR
4.0%
4.5%
5.0%
5.5%
6.0%
7.0%
8.0%

$37,500
$25,000
$12,500
0
 $12,500
 $12,500
 $12,500
  Max[LIBOR  .07,0](.25)($10M)  $12,500
37500

4
12500
5 5.5
6
7
8
LIBOR %
Other Strategies
• One of the important features of an option is that it
can be combined with positions in the underlying
security and other options to generate a number of
different investment strategies.
ST

• Two well-known strategies formed by combining
option positions are straddles and spreads.
Straddle
• A straddle purchase is formed by buying both a
call and put with the same terms – the same
underlying security, exercise price, and expiration
date.
ST

• A straddle write, in contrast, is constructed by
selling a call and a put with the same terms.
Straddle Purchase
•
Buy 100 T-bond put for 1 and buy 100 T-bond call for 1:

4,000
3,000
95,000
100,000
105,000
ST
1,000
2,000
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
Spread
• A spread is the purchase of one option and
the sale of another on the same underlying
security but with different terms:
– Money Spread: Different exercise prices
– Time Spread: Different expirations
– Diagonal Spread: Different exercice prices and
expirations
ST

Spread
• Two of the most popular time spread positions are
the bull spread and the bear spread:
– A bull call spread is formed by buying a call with a
certain exercise price and selling another call with a
higher exercise price, but with the same expiration date.
– A bear call spread is the reversal of the bull spread; it
consists of buying a call with a certain exercise price
and selling another with a lower exercise price.
ST

• The same spreads also can be formed with puts.
Bull Spread
•
Buy 100 T-bond call for 1 and sell 101 T-bond call for .75:


1,000
750
100,000
101,000
102,000
ST

250
Bull Spread
1,000
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
Other Strategies
1. Bull Call Spread: Long in call with low X and short in
call with high X.
2. Bull Put Spread: Long in put with low X and short in
put with high X
3. Bear Call Spread: Long in call with high X and short in
call with low X
4. Bear Put Spread: Long in put with high X and short in
put with low X
5. Long Butterfly Spread: Long in call with low X, short
in 2 calls with middle X, and long in call with high X
(similar position can be formed with puts)
6. Short Butterfly Spread: Short in call with low X, long
in 2 calls with middle X, and short in call with high X
(similar position can be formed with puts)
ST

Other Strategies
7. Straddle Purchase: Long call and put with similar
terms
8. Strip Purchase: Straddle with additional puts (e.g.,
long call and long 2 puts)
9. Strap Purchase: Straddle with additional calls (e.g.,
long 2 calls and long put)
10. Straddle Sale: Short call and put with similar terms
(strip and strap sales have additional calls and puts)
11. Money Combination Purchase: Long call and put
with different exercise prices
12. Money Combination Sale: Short call and put with
different exercise prices
ST

Microstructure
• Many options are traded on organized
exchanges.
• The purpose of any exchange is to provided
marketability.
Microstructure
• Option Exchanges provide marketabilty by:
– Listings
– Standardization
• Set: X, T, and Size
– Market Makers or Specialist
– Option Clearing Corporation
Standardization
• Similar to the futures exchanges, the option
exchanges standardize contracts by setting
expiration dates, exercise prices, and
contract sizes on options.
Standardization
• The derivative exchanges also impose two limits on option
trading: exercise limits and position limits. These limits are
intended to prevent an investor or groups of investors from
having a dominant impact on a particular option.
• An exercise limit specifies the maximum number of option
contracts that can be exercised on a specified number of
consecutive business days (e.g., 5 days) by any investor or
investor group.
• A position limit sets the maximum number of options an
investor can buy and sell on one side of the market.
– A side of the market is either a bullish or bearish position. An investor
who is bullish could profit by buying calls or selling puts, while an
investor with a bearish position could profit by buying puts and selling
calls.
Continuous Trading
• As noted in Chapter 12, on the futures exchanges
such as the CBOT, CME, and LIFFE, continuous
trading is provided through locals who are willing
to take temporary positions to make a market.
• Many of the option exchanges, though, use market
makers and specialist to ensure a continuous
market.
Option Clearing Corporation, OCC
• Derivative exchanges have a clearinghouse (CH) or option
clearing corporation (OCC), as it is referred to on the
option exchange.
• In the case of options, the CH:
– Intermediates each transaction that takes place on the
exchange
– Guarantees that all option writers fulfill the terms of
their options if they are assigned
– Manages option exercises, receiving notices and
assigning corresponding positions to clearing members.
Option Clearing Corporation, OCC
• As an intermediary, the OCC functions by
breaking up each option trade.
• After a buyer and seller complete an option trade,
the OCC steps in and becomes the effective buyer
to the option seller and the effective seller to the
option buyer.
• At that point, there is no longer any relationship
between the original buyer and seller.
Option Clearing Corporation, OCC
• By breaking up each option contract, the OCC
makes it possible for option investors to close
their positions before expiration.
• If a buyer of an option later becomes a seller of
the same option, or vice versa, the OCC computer
will note the offsetting position in the option
investor's account and will therefore cancel both
entries.
OCC Example
• Suppose A buys a 100 T-bond futures call
from B for 3:
– A is long; B is short.
• After this contract is established, the OCC
breaks it up:
– A’s right to exercise is now with the OCC.
– B’s responsibility is when the OCC exercises;
this could be when someone (not just A)
notifies the OCC they want to exercise.
OCC Record: Entry 1
• A: Right to exercise
• B: Responsibility
Offsetting Transaction for ‘A’
• Suppose the price of T-bonds increase to 110,
pushing the price of the 100 T-bond futures call up
to 12.
• Seeing the opportunity to profit, suppose A sells a
100 T-bond futures call to C for 12.
– OCC breaks up this contract.
– OCC’s new entry of ‘responsibility for A’
cancels ‘A’s right entry’; thus A’s position is
closed.
OCC Record Entry 2
•
•
•
•
A: Right to exercise
B: Responsibility
A: Responsibility
C: Right to Exercise
Closed
Offsetting Transaction for B
• Suppose when the price of the T-bond is at 110
and the price of the 100 T-bond call is at 12, B
begins to worry and decides to close his short
position. He can do this by going long in the 100
T-bond Call.
• Suppose B buys a 100 T-bond futures call from D.
– OCC breaks up the contract.
– OCC’s new entry of ‘right for B’ cancels ‘B’s
responsibility’ entry; thus B’s position is closed.
OCC Record Entry 3
•
•
•
•
B: Responsibility
C: Right
B: Right
D: Responsibility
Closed
Importance of the OCC
• By acting as an intermediary, the OCC makes it
possible for option investors to close their
positions.
• In this example:
– ‘A’ was able to buy a 100 T-Bond futures call for 3,
then later close her position by selling another 100 TBond call for 12: Profit = 9.
– ‘B’ was able to sell a 100 T-Bond futures call for 3,
then later close his position by buying another 100 TBond call for 12: loss = 9.
Margin Requirements and Costs
• Margin Requirements:
– Initial Margin: Option writers are required to
deposit cash or risk-free securities with their
brokers to secure their positions.
– Maintenance Margin: Writers are required to
post additional cash or risk-free securities when
the underlying security or futures prices move
against them.
• Other Costs: Commissions; bid-ask spread.
Margin Requirements
• For futures options, there are two sets of margins:
– Margin requirement for the option writer
– Futures margin requirement that must be met if
the futures option is exercised.
• If the futures option is exercised, both the holder
and writer must establish and maintain the futures
margin positions, with the writer’s margin position
on the option now being replaced by his new
futures position.
Types of Transactions
• The CH provides marketability by making it
possible for option investors to close their
positions instead of exercising.
• In general, there are four types of trades an
investors of an exchange-traded option can make:
• Opening
• Exercising
• Expiring
• Closing (or Offsetting)
Types of Transactions
• Opening transaction occurs when investors initially buy
or sell an option.
• Expiring transactions is allowing the option to expire; that
is, doing nothing when the expiration date arrives because
the option is worthless (out of the money).
• Exercising transaction: If it is profitable, a holder can
exercise.
• Offsetting or closing transactions: Holders or writers of
options can close their positions with offsetting or closing
transactions or orders.
Types of Transactions
• As a general rule, option holders should close their
positions rather than exercise.
• If there is some time to expiration, an option
holder who sells her option will receive a price
that exceeds the exercise value.
• Because of this, many exchange-traded options are
closed.
OTC Options
• In the OTC option market, interest rate option
contracts are negotiable, with buyers and sellers
entering directly into an agreement.
• Thus, the dealer's market provides option contracts
that are tailor-made to meet the holder's or writer's
specific needs.
• The OTC market does not have a clearinghouse to
intermediate and guarantee the fulfillment of the
terms of the option contract, nor market makers or
specialists to ensure continuous markets; the
options, therefore, lack marketability.
OTC Options
• Since each OTC option has unique features, the
secondary market is limited.
• Prior to expiration, holders of OTC options who
want to close their position may be able to do so
by selling their positions back to the original
option writers or possibly to an OTC dealer who is
making a market in the option.
• Because of this inherent lack of marketability, the
premium on OTC options are higher than
comparable exchange-trade ones.
Option Price Relations: Calls
• The price of any option is constrained by certain boundary
conditions.
• One of those boundary conditions is the intrinsic value.
• By definition, the intrinsic value (IV) of a call at a time prior to
expiration (let t signify any time prior to expiration), or at
expiration (T again signifies expiration) is the maximum of the
difference between the price of the underlying security or
futures (St or ft) and the exercise price or zero:
IV = Max[ft-X,0] or Max[St-X,0]
Option Price Relations: Calls
• The intrinsic value can be used as a reference to
define in-the-money, on-the-money and
out-of-the-money calls.
Type
Spot Call
In-the-Money
St > X => IV > 0
On-the-Money
St = X => IV = 0
Out-of-the-Money St < X => IV = 0
Futures Call
ft > X => IV > 0
ft = X => IV = 0
ft < X => IV = 0
Option Price Relations: Calls
• For an American futures option, the IV defines a
boundary condition in which the price of a call has
to trade at a value at least equal to its IV:
Ct  Max[ft-X,0]
• If this condition does not hold (Ct< Max[ft-X,0]),
an arbitrageur could buy the call, exercise, and
close the futures position.
Option Price Relations: Calls
•
Example: Suppose a T-bill futures contract expiring in 182
days were trading at $987,862 (index = 95.1448) and a 95 Tbill futures call expiring in 182 days (X = $987,500) were
trading at $100, below its IV of $362.
•
Arbitrageurs could realize risk-free profits by
1. Buying the call at $100
2. Exercising the call to obtain a margin account worth ft - X
= $987,862 - $987,500 = $362 plus a long position in the
T-bill futures contract priced $987,862
3. Immediately closing the long futures position by taking an
offsetting short position at $987,862.
Option Price Relations: Calls
•
Example: Doing this, arbitrageurs would realize a risk-free
profit of $262.
•
By pursuing this strategy, though, arbitrageurs would push
the call premium up until it is at least equal to its IV of
ft-X = $362 and the arbitrage profit is zero.
•
Note: This arbitrage strategy requires that the option be
exercised immediately. Thus, the condition applies only to
an American futures option.
•
The boundary conditions for European futures, American
spot, and European spot interest rate options are explained
in Chapter 14, Exhibit 14.14.
Option Price Relations: Calls
•
The other component of the value of an option is the time
value premium (TVP).
•
By definition, the TVP of a call is the difference between
the price of the call and its IV:
TVP = Ct - IV
•
Example: If the 95 T-bill futures call expiring in 182 days
(X = $987,500) were trading at $562 when the T-bill
futures contract expiring in 182 days were trading at
$987,862 (index = 95.1448), the IV would be $362 and the
TVP would be $200. It should be noted that the TVP
decreases as the time remaining to expiration decreases.
Option Price Relations: Calls
• Graphically, the relationship between Ct, TVP, and
IV is depicted in the next exhibit. In the figure,
graphs plotting the call price and the IV (on the
vertical axis) against the futures price (on the
horizontal axis) are shown for the American 95 Tbill futures call option.
Call and Futures Price Relation
C = IV + TVP
• 95 T-bill Futures Call Price Curve
C t , IV
Call Curve
C IV
1200
1000
500
100
C
986,000
X  987,500
988,500
ft
Option Price Relations: Calls
• The IV line shows the linear relationship between
the IV and the futures price.
• The IV line, in turn, serves as a reference for the
call price curve (CC).
• The noted arbitrage condition dictates that the
price of the call cannot trade (for long) at a value
below its IV.
• Graphically, this means that the call price curve
cannot go below the IV line.
Option Price Relations: Calls
• The call price curve (CC) in the exhibit shows the
positive relationship between Ct and ft.
• The vertical distance between the CC curve and
the IV line, in turn, measures the TVP.
• The CC curve for a comparable call with a greater
time to expiration would be above the CC curve,
reflecting the fact that the call premium increases
as the time to expiration increases.
Option Price Relations: Calls
• Note: The slopes of the CC curves approach the
slope of the IV line when the security price is
relatively high
– known as a deep-in-the-money-call
• The slopes of CC curves approach zero (flat)
when the price of the futures is relatively low
– known as a deep out-of-the-money call
Call Function
• The call price curve illustrates the positive relation
between a call price and the underlying security or
futures price and the time to expiration. An
option’s price also depends on the volatility of the
underlying security or futures contract.
• Call Function:
C t  f (S or f , X, T, )




Price and Variability Relation
• Since a long call position is characterized by
unlimited profit potential if the security or futures
increases but limited losses if it decreases, a call
holder would prefer more volatility rather than
less.
Price and Variability Relation
• Greater variability suggests:
– On the one hand, a given likelihood that the
security will increase substantially in price,
causing the call to be more valuable.
– On the other hand, greater volatility also
suggests a given likelihood of the security price
decreasing substantially.
• Given that a call’s losses are limited to just the
premium when the security price is equal to the
exercise price or less, the extent of the price
decrease would be inconsequential to the call
holder.
Price and Variability Relation
• Thus, the market will value a call option on a
volatile security or contract more than a call on
one with lower variability.
Price and Variability Relation
• The positive relationship between a call’s premium
and its underlying security’s volatility is illustrated
in the next exhibit. The exhibit shows two call
options:
1. A call option on Bond A with an exercise price of X =
100 in which the underlying bond is trading at 100 and
has a variability characterized by an equal chance of
Bond A either increasing by 10% or decreasing by 10%
by the end of the period (assume theses are the only
possibilities).
2. A call option on Bond B with an exercise price of X =
100 in which the underlying bond is trading at 100 but
has a greater variability characterized by an equal chance
of Bond B increasing or decreasing by 20% by the end
of the period.
Price and Variability Relation
Bond A  110
Bond A Call : IV  10
Bond A  100
Bond A Call : X  100
Variabilit y  10%
Bond A  90
Bond A Call : IV  0
Bond B  120
Bond B Call : IV  20
Bond B  100
Bond B Call : X  100
Variabilit y  20%
Bond B  80
Bond B Call : IV  0
Price and Variability Relation
•
Given the variability of the underlying bonds, the IV on the
call for Bond B would be either 20 or 0 at the end of the
period, compared to value of only 10 and 0 for the call on
Bond A.
•
Since, Bond B’s call cannot perform worse than Bond A’s
call, and can do better, it follows there would be a higher
demand and therefore price for the Bond B call than the
Bond A call.
•
Thus, given the limited loss characteristic of an option, the
more volatile the underlying security, the more valuable the
option, all other factors being equal.
Option Price Relations: Puts
• The price of a put at a given point in time prior to
expiration (Pt) also can be explained by reference to
its IV, boundary conditions, and TVP.
• In the case of puts, the IV is defined as the maximum
of the difference between the exercise price and the
security or futures price or zero:
IV = Max[X-ft,0] or Max[X-St,0]
Option Price Relations: Puts
• In-the-money, on-the-money, and
out-of-the-money puts are defined as:
Type
Spot Put
In-the-Money
X > St => IV > 0
On-the-Money
X = St => IV = 0
Out-of-the-Money X < St => IV = 0
Futures Put
X > ft => IV > 0
X = ft => IV = 0
X < ft => IV = 0
Option Price Relations: Puts
• For an American futures option, the IV defines a
boundary condition in which the price of the put
has to trade at a price at least equal to its IV:
Pt  Max[X-ft,0]
• If this condition does not hold, an arbitrageur could
buy the put, exercise, and close the futures position.
Option Price Relations: Puts
•
•
Example: Suppose a T-bill futures contract expiring in 182
days were trading at $987,200 and a 95 T-bill futures put
expiring in 182 days (X = $987,500) were trading at $100,
below its IV of $300.
Arbitrageurs could realize risk-free profits by
1. Buying the put at $100
2. Exercising the put to obtain a margin account worth X ft = $987,500 - $987,200 = $300 plus a short position
in the T-bill futures contract priced $987,200
3. Immediately closing the short futures position by taking
an offsetting short position at $987,200.
Option Price Relations: Puts
• Doing this, the arbitrageur would realize a risk-free
profit of $200.
• By pursuing this strategy, though, arbitrageurs
would push the put premium up until it is at least
equal to its IV of X-ft = $300 and the arbitrage
profit is zero.
• Exhibit 14.16 in the text explains with examples the
arbitrage strategies governing the boundary
conditions for the European futures put options and
the American and European spot put options.
Option Price Relations: Puts
• The TVP for a put is defined as
TVP = Pt – IV
Option Price Relations: Puts
Put Price Curve
• Graphically, the IV and TVP can be seen for an American
futures options in the next exhibit.
• The figure shows a negatively sloped put-price curve (PP)
and a negatively sloped IV line going from the horizontal
intercept (where ft = X) to the vertical intercept where the
IV is equal to the exercise price when the futures is trading
at zero (i.e., IV = X, when ft = 0).
• The slope of the PP curve approaches the slope of the IV
line for relatively low futures prices (deep in-the-money
puts) and approaches zero for relatively large futures
prices (deep-out-of-the money puts).
Put and Futures Price Relation
P = IV + TVP
• Put Price Curve:
Pt
IV
Put Curve
P
P
X
ft
Put Function
• The price of a put option depends not only on the
underlying security or futures price and time to
expiration, but also on the volatility of the
underlying security or futures contract.
• Since put losses are limited to the premium when
the price of the underlying security or futures is
greater than or equal to the exercise price, put
buyers, like call buyers, will value puts on
securities or futures with greater variability more
than those with lower variability.
Put Function
• Put Function:
Pt  f (S or f , X T, )

  
Closing Instead of Exercising
• As noted earlier, if there is some time to
expiration, an option holder who sells her option
will receive a price that exceeds the exercise
value; that is, if she sell the option, she will
receive a price that is equal to an IV plus a TVP; if
she exercises, though, her exercise value is only
equal to the IV.
• Thus, by exercising instead of closing she loses
the TVP.
Closing Instead of Exercising
• Thus, an option holder in most cases should close
her position instead of exercising.
• There are some exceptions to the general rule of
closing instead of exercising.
• For example, if an American option on a security
that was to pay a high coupon that exceeded the
TVP on the option, then it would be advantageous
to exercise.
Put-Call Parity Model
• Since the call and put derive their values from the
underlying security, the prices of the call, put, and
security are related. The relation governing their
prices is know as the Put-Call Parity Model.
• The put-call parity relation can be seen in terms of
a conversion position.
Put-Call Parity Model
•
For European spot options on debt securities with
no coupons (e.g., T-bills and Eurodollar deposits),
the conversion consist of
1. A long position in an underlying security that will have
a maturity equal to the maturity of the option’s
underlying security (e.g., a T-bill that will have
maturity of 91 days at the expiration on the spot T-bill
option)
2. A short position in a call and a long position in a put
with the same exercise price and time to expiration.
•
As shown in the next slide, the conversion yields
a certain cash flow at expiration equal to the
exercise price.
CF of Conversion at T
Closing Position
Long Bond
Long Put
Short Call
Net
ST < X
ST
X- ST
0
X
ST = X
ST
0
0
X
ST > X
ST
0
-(ST-X)
X
Put-Call Parity Model
•
•
To preclude arbitrage, the risk-free conversion portfolio
must be worth the same as a risk-free pure discount bond
with a face value of X maturing at the same time as the
option’s expiration.
Thus, in equilibrium:
X
P  C  S0 
T
(1  R f )
e
0
e
0
Put-Call Parity Model
•
•
•
The put-call parity condition on options on T-bonds, Tnotes, and other debt securities paying interest is similar to
options on zero coupon bonds except that the accrued
interest on the underlying bond is included.
That is, at expiration the conversion will yield a risk-free
cash flow equal to the exercise price plus the accrued
interest.
Thus, the equilibrium value of the conversion will equal
the value of a risk-free bond with a face value of X plus
the accrued interest:
X  Accrued Interest
P  C  S0 
(1  R f ) T
e
0
e
0
Put-Call Futures Parity Model
• For European futures options, the conversion is
formed with:
– Long position in the futures contract
– Long position in a put
– Short position in a call on the futures contract.
• As shown in the next slide, if the options and the
futures contracts expire at the same time, then the
conversion would be worth X – f0 at expiration,
regardless of the price on the futures contract.
Put-Call Futures Parity Model
Closing Position
Long Futures
Long Futures Put
Short Futures Call
Net
fT < X
fT- f0
X- fT
0
X- f0
fT = X
fT- f0
0
0
X- f0
fT > X
fT- f0
0
-(fT-X)
X- f0
Put-Call Futures Parity Model
• Since this position yields a risk-free return, in
equilibrium its value would be equal to the present
value of a risk-free bond with a face value of X-f0
(remember the futures contract has no initial
value).
• Thus:
P C
e
0
e
0
X  f0

T
(1  R f )
Put-Call Futures Parity Model
• Note: If the carrying-cost model holds and the
futures and options expire at the same time, then
the equilibrium relation defining put-call parity for
European futures options will be equal to the putcall parity for European spot options.
• This can be seen algebraically, by substituting the
carrying cost equation S0(1+Rf)T for f0 in the
above equation.
• Also note that put-call parity is defined in terms of
European options, not American.
Websites
• Information on the CBOE: www.cboe.com
• For market information and prices on futures options go to
www.cme.com and click on “Market Data,” and go to
www.cbt.com and click on “Quotes and Trades.”
• For general information and other links:
www.optioncentral.com
• For more information on options go to www.isda.org