chapter 14 interest rate options: fundamentals. spot option: definition an option is a security that...

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


• 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.


• There is a large OTC market in debt and interest-sensitive 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.


• 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.


• 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 OptionsTypes: 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.

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.


Futures Options on T-Bonds and Notes• The call and put contracts on the T-bond and T-

note 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.

Futures Options on T-Bonds and Notes• The call and put contracts on the T-bond and T-

note 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.


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.

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.


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:

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:

13.828,1$87.171,2$000,4$

000,4$000,100$100

104108F

100

XfinargMofValue t


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.

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 OptionsTypes: CME’s Futures Options

Futures Options on Eurodollars and T-Bills

• The CME offers trading on futures options on T-bills, 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.

Futures Options on Eurodollars and T-Bills

• The CME offers trading on futures options on T-bills, 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.


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:

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:

)M1($100

)25(.R100X D


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:

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


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.

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.


• 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:

• A 10.30 quote on a 94.5 April call indicates a call price of $2,575:

• 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:

• A 10.30 quote on a 94.5 April call indicates a call price of $2,575:

(5.925)(100)($25) = $1,481.25

Or

(5.925)($250) = $1,481.25

(10.30)(10)($25) = $2,575Or(10.30)($250) = $2,575


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:

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:

250,986$000,000,1$100

)360/90)(5.94100(100X


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:

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:

75.018,1$25.481,1$500,2$

500,2$)100](5.945.95[25$

)IndexExercsieIndexFutures(25$ValueinargM

500,2$250,986$750,988$XfValueinargM t


• 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.

• 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 OptionsTypes: 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.

• 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.


• For an OTC option on a Treasury, the underlying security is often a recently auctioned Treasury (on-the-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.

• For an OTC option on a Treasury, the underlying security is often a recently auctioned Treasury (on-the-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.


• 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.

• 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.


• 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.

• 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.


• 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.

• 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.


• 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.

• 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.


• 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:

• 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:

InterestAccrued)000,100($100

IndexX


• 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

• 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)


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.

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.


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.

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.


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%:

Interest Rate Call• Example: Given an interest rate call with a



(.07-.06)(180/360)($1M) = $5,000


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.

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.


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.

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.


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.

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.


Interest Rate Put• Example: Given an interest rate put with a



Interest Rate Put• Example: Given an interest rate put with a



(.06-.05)(180/360)($1M) = $5,000


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.

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.


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.

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.


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.

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.


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).

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).


Cap• Caps are often written by financial

institutions in conjunction with a floating-rate loan and are used by buyers as a hedge against interest rate risk.

Cap• Caps are often written by financial

institutions in conjunction with a floating-rate loan and are used by buyers as a hedge against interest rate risk.


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.

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.


Floor• A plain-vanilla floor is a series of European

interest rate put options – a portfolio of floorlets.

Floor• A plain-vanilla floor is a series of European

interest rate put options – a portfolio of floorlets.


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%.

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%.


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.

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 StrategiesFundamental Strategies

• Fundamental Strategies:– Call Purchase– Naked Call Write– Put Purchase– Naked Put Write

• Fundamental Strategies:– Call Purchase– Naked Call Write– Put Purchase– Naked Put Write

Profit GraphProfit Graph

• Option Strategies can be evaluated in terms of a profit graph.

• 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.

• Option Strategies can be evaluated in terms of a profit graph.

• 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.

ST

Fundamental Spot Option PositionsFundamental Spot Option Positions

• To see the major characteristics of option positions, consider spot call and put options on an OTC 6% T-bond 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.

• 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).

• To see the major characteristics of option positions, consider spot call and put options on an OTC 6% T-bond 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.

• 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).

ST

Call Purchase

• Buy T-bond call: X = $100,000, C = $1000

TS100 000, 105 000,

4 000,

1 000,Call PurchaseSpot Price at T Profit/Loss

90000 -100095000 -1000100000 -1000101000 0102000 1000103000 2000104000 3000105000 4000106000 5000

Naked Call Write

• Sell T-bond call for: X= 100,000, C=1000.

ST105 000,

1 000,

4 000,

100 000,Naked Call WriteSpot Price at T Profit/Loss

90000 100095000 1000100000 1000101000 0102000 -1000103000 -2000104000 -3000105000 -4000106000 -5000

Put Purchase

• Buy T-bond put: X=100,000, P = 1000

ST95 000,

4 000,

1000

100 000,Put Purchase

Spot Price at T Profit/Loss90000 900095000 4000

100000 -1000101000 -1000102000 -1000103000 -1000104000 -1000105000 -1000106000 -1000

Naked Put Write

• Sell T-bond put: X =100,000, P = 1000

ST95 000,

4 000,

1 000,

100 000,Naked Put WriteSpot Price at T Profit/Loss

90000 -900095000 -4000

100000 1000101000 1000102000 1000103000 1000104000 1000105000 1000106000 1000

Fundamental Futures Option Positions

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 T-bill futures.

• 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.

• 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 T-bill futures.

• 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.

ST

Fundamental Futures Option PositionsFundamental 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.

• 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.

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.

• 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.

ST

Futures Options on Treasury Securities Call on T-bill Futures:• X = IMM 90 or X = $975,000

• PT = 5 or C = $1,250

• Futures and options futures have same expiration.

R S f fD T T C T

( , )

$1250

. ,

. ,

. ,

. ,

. ,

. ,

975 000

105 973 750 1250

10 0 975 000 1250

9 5 976 250 0

9 0 977 500 1250

85 978 750 2500

8 0 980 000 3750

TT fS 975000

1250

980000

3750

.37501250975000980000

:000,975000,980Xfreceivesand

,000,980fatshortgoingby

closesthenand000,980fatlong

goesHolder:000,980atExercise

T

T

T


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.

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


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.

• 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.

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.

• 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.

ST

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.

R S f fD T T P T

( ,

$1250

. ,

. ,

. ,

. ,

. ,

. ,

. ,

. ,

. ,

975 000

12 0 970 000 3750

115 971 250 2500

110 972 500 1250

10 5 973 750 0

10 0 975 000 1250

9 5 976 250 1250

9 0 977 500 1250

8 5 978 750 1250

8 0 980 000 1250

TT fS 972250

1250

970000

3750

.37501250970000975000

:000,970000,975fXreceivesand

,000,980fatlonggoingbycloses

thenand000,970fatshort

goesHolder:000,970atExercise

T

T

T

Fundamental Interest Rate Option PositionsFundamental 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


• 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.

• 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.

ST

500,12$)M10)($25](.0,07.LIBOR[Max

500,37$%0.8

000,25$%5.7

500,12$%0.7

0%5.6

500,12$%0.6

500,12$%0.5

500,12$%0.4

LIBOR

%LIBOR

$

62 4 5.6 8

12500

37500



• 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.

• 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

500,12$)M10)($25](.0,07.LIBOR[Max

500,12$%0.8

500,12$%0.7

500,12$%0.6

0%5.5

500,12$%0.5

000,25$%5.4

500,37$%0.4

LIBOR

%LIBOR

$

654 5.5 8

12500

37500

7


Other StrategiesOther 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.

• Two well‑known strategies formed by combining option positions are straddles and spreads.

• 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.

• Two well‑known strategies formed by combining option positions are straddles and spreads.

ST

StraddleStraddle

• 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.

• A straddle write, in contrast, is constructed by selling a call and a put with the same terms.

• 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.

• A straddle write, in contrast, is constructed by selling a call and a put with the same terms.

ST

Straddle Purchase

• Buy 100 T-bond put for 1 and buy 100 T-bond call for 1:

ST95 000, 100 000, 105 000,

2 000,

1 000,

3 000,

4 000,

Straddle PurchaseCall Purchase Put Purchase Total

Spot Price at T Profit/Loss Profit/Loss Profit/Loss94000 -1000 5000 400097000 -1000 2000 100098000 -1000 1000 0

100000 -1000 -1000 -2000102000 1000 -1000 0103000 2000 -1000 1000106000 5000 -1000 4000

SpreadSpread

• 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

• 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

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SpreadSpread

• 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.

• The same spreads also can be formed with puts.

• 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.

• The same spreads also can be formed with puts.

ST

Bull Spread

• Buy 100 T-bond call for 1 and sell 101 T-bond call for .75:

TS100 000, 101 000,

250

750

1 000,

1 000,

102 000,

Bull Spread

100 Call Purchase at 1 101 Call Sale at .75 TotalSpot Price at T Profit/Loss Profit/Loss Profit/Loss

94000 -1000 750 -25097000 -1000 750 -25098000 -1000 750 -250

100000 -1000 750 -250100250 -750 750 0101000 0 750 750102000 1000 -250 750103000 2000 -1250 750106000 5000 -4250 750


ST

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 X3. Bear Call Spread: Long in call with high X and short in call with low X4. Bear Put Spread: Long in put with high X and short in put with low X5. 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

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 prices12. Money Combination Sale: Short call and put with different exercise prices

MicrostructureMicrostructure

• Many options are traded on organized exchanges.

• The purpose of any exchange is to provided marketability.

• Many options are traded on organized exchanges.

• The purpose of any exchange is to provided marketability.

MicrostructureMicrostructure

• Option Exchanges provide marketabilty by:– Listings– Standardization

• Set: X, T, and Size

– Market Makers or Specialist– Option Clearing Corporation

• Option Exchanges provide marketabilty by:– Listings– Standardization

• Set: X, T, and Size

– Market Makers or Specialist– Option Clearing Corporation

StandardizationStandardization

• Similar to the futures exchanges, the option exchanges standardize contracts by setting expiration dates, exercise prices, and contract sizes on options.

• Similar to the futures exchanges, the option exchanges standardize contracts by setting expiration dates, exercise prices, and contract sizes on options.

StandardizationStandardization

• 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.

• 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 TradingContinuous 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.

• 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, OCCOption 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.

• 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.


• 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.

• 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.


• 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.

• 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 ExampleOCC 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.

• 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 1OCC Record: Entry 1

• A: Right to exercise

• B: Responsibility



Offsetting Transaction for ‘A’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.

• 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 2OCC Record Entry 2



• A: Responsibility

• C: Right to Exercise

Closed

Offsetting Transaction for BOffsetting 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.

• 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 3OCC Record Entry 3


• C: Right

• B: Right

• D: Responsibility

Closed

Importance of the OCCImportance 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 T-Bond 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 T-Bond call for 12: loss = 9.

• 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 T-Bond 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 T-Bond call for 12: loss = 9.

Margin Requirements and CostsMargin 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: – 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 RequirementsMargin 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.

• 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 TransactionsTypes 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)

• 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)


• 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.

• 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.


• 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.

• 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 OptionsOTC 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.

• 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 OptionsOTC 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.

• 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: CallsOption 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:

• 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]


• The intrinsic value can be used as a reference to define in‑the‑money, on‑the‑money and out‑of‑the‑money 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 Futures Call

In‑the‑Money On-the-Money Out-of-the-Money

St > X => IV > 0

St = X => IV = 0

St < X => IV = 0

ft > X => IV > 0

ft = X => IV = 0

ft < X => IV = 0


• 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:

• If this condition does not hold (Ct< Max[ft‑X,0]), an arbitrageur could buy the call, exercise, and close the futures position.

• 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:

• If this condition does not hold (Ct< Max[ft‑X,0]), an arbitrageur could buy the call, exercise, and close the futures position.

Ct Max[ft‑X,0]


• Example: Suppose a T-bill futures contract expiring in 182 days were trading at $987,862 (index = 95.1448) and a 95 T-bill 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.

• Example: Suppose a T-bill futures contract expiring in 182 days were trading at $987,862 (index = 95.1448) and a 95 T-bill 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.


• 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.

• 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.


• 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:

• 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.

• 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:

• 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.

TVP = Ct ‑ IV

• 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 T-bill futures call option.

• 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 T-bill futures call option.


Call and Futures Price RelationC = IV + TVP

• 95 T-bill Futures Call Price Curve

IV,C t

IV

tf

CurveCall

X 987 500, 988 500,986 000,

500

100

12001000

C

C

• 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.

• 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.


• 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.

• 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.


• 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

• 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 FunctionCall 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:

• 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:

),T,X,forS(fCt ),T,X,forS(fCt

Price and Variability RelationPrice 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.

• 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.


• 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.

• 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.


• Thus, the market will value a call option on a volatile security or contract more than a call on one with lower variability.

• Thus, the market will value a call option on a volatile security or contract more than a call on one with lower variability.


• 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.

• 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.

100X:CallABond

100ABond

10IV:CallABond

110ABond

0IV:CallABond

90ABond

100X:CallBBond

100BBond

20IV:CallBBond

120BBond

0IV:CallBBond

80BBond

%10yVariabilit

%20yVariabilit



• 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.

• 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: PutsOption 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:

• 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]


• In‑the‑money, on‑the‑money, and out‑of‑the‑money puts are defined as:

• In‑the‑money, on‑the‑money, and out‑of‑the‑money puts are defined as:

Type Spot Put Futures Put

In‑the‑Money On-the-Money Out-of-the-Money

X > St => IV > 0

X = St => IV = 0

X < St => IV = 0

X > ft => IV > 0

X = ft => IV = 0

X < ft => IV = 0


• 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:

• If this condition does not hold, an arbitrageur could buy the put, exercise, and close the futures position.

• 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:

• If this condition does not hold, an arbitrageur could buy the put, exercise, and close the futures position.

Pt Max[X-ft,0]


• 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.

• 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.


• 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.

• 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.


• The TVP for a put is defined as• The TVP for a put is defined as

TVP = Pt – IV


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 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

tfX

CurvePutP

P

Put FunctionPut 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.

• 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 FunctionPut Function

• Put Function:• Put Function:

),TX,forS(fPt ),TX,forS(fPt

Closing Instead of ExercisingClosing 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.

• 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 ExercisingClosing 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.

• 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 ModelPut-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.

• 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.


• 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.

• 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 TCF of Conversion at T

Closing Position ST < X ST = X ST > X

Long BondLong PutShort Call

ST

X- ST

0

ST

00

ST

0-(ST-X)

Net X X X


• 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:

• 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:

Tf

0e0

e0 )R1(

XSCP


• The put-call parity condition on options on T-bonds, T-notes, 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:

• The put-call parity condition on options on T-bonds, T-notes, 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:

Tf

0e0

e0 )R1(

InterestAccruedXSCP

Put-Call Futures Parity ModelPut-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.

• 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.


Closing Position fT < X fT = X fT > X

Long FuturesLong Futures PutShort Futures Call

fT- f0

X- fT

0

fT- f0

00

fT- f0

0-(fT-X)

Net X- f0 X- f0 X- f0


• 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:

• 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:

Tf

0e0

e0 )R1(

fXCP


• 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 put-call 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.

• 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 put-call 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.

WebsitesWebsites

• 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

• 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

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