In earlier posts, we devoted considerable effort to understanding the Eurodollar spot market and forward contracts on the Eurodollar rate or LIBOR, called FRAs. In this post, we cover options on LIBOR. Although these are not the only interest rate options, their characteristics are sufficiently general to capture most of what we need to know about options on other interest rates. First recall that a Eurodollar is a dollar deposited outside of the United States. The primary Eurodollar rate is LIBOR, and it is considered the best measure of an interest rate paid in dollars on a nongovernmental borrower. These Eurodollars represent dollar-denominated time deposits issued by banks in London borrowing from other banks in London.
Before looking at the characteristics of interest rate options, let us set the perspective by recalling that FRAs are forward contracts that pay off based on the difference between the underlying rate and the fixed rate embedded in the contract when it is constructed. For example, consider a 3 X 9 FRA. This contract expires in three months. The underlying rate is six-month LIBOR. Hence, when the contract is constructed, the underlying Eurodollar instrument matures in nine months. When the contract expires, the payoff is made immediately, but the rate on which it is based, 180-day LIBOR, is set in the spot market, where it is assumed that interest will be paid 180 days later. Hence, the payoff on an FRA is discounted by the spot rate on 180-day LIBOR to give a present value for the payoff as of the expiration date.
Just as an FRA is a forward contract in which the underlying is an interest rate, an interest rate option is an option in which the underlying is an interest rate. Instead of an exercise price, it has an exercise rate (or strike rate), which is expressed on an order of magnitude of an interest rate. At expiration, the option payoff is based on the difference between the underlying rate in the market and the exercise rate. Whereas an FRA is a commitment to make one interest payment and receive another at a future date, an interest rate option is the right to make one interest payment and receive another. And just as there are call and put options, there is also an interest rate call and an interest rate put.
An interest rate call is an option in which the holder has the right to make u known interest payment and receive an unknown interest payment. The underlying is the unknown interest rate. If the unknown underlying rate turns out to be higher than the exercise rate at expiration, the option is in-the-money and is exercised; otherwise, the option simply expires. An interest rate put is an option in which the holder has the right to make an unknown interest payment and receive a known interest payment. If the unknown underlying rate turns out to be lower than the exercise rate at expiration, the option is in-the-money and is exercised; otherwise, the option simply expires. All interest rate option contracts have a specified size, which, as in FRAs, is called the notional principal. An interest rate option can be European or American style, but most tend to be European style. Interest rate options are settled in cash. .
As with FRAs, these options are offered for purchase and sale by dealers, which are financial institutions, usually the same ones who offer FRAs. These dealers quote rates for options of various exercise prices and expirations. When a dealer takes an option position, it usually then offsets the risk with other transactions, often Eurodollar futures.
To use the same example we used in introducing FRAs, consider options expiring in 90 days on 180-day LIBOR. The option buyer specifies whatever exercise rate he desires. Let us say he chooses an exercise rate of 5.5 percent and a notional principal of $10 million. Now let us move to the expiration day. Suppose that 180-day LIBOR is 6 percent. Then the call option is in-the-money. The payoff to the holder of the option is This money is not paid at expiration, however; it is paid 180 days later. There is no reason why the payoff could not be made at expiration, as is done with an FRA. The delay of payment associated with interest rate options actually makes more sense, because these instruments are commonly used to hedge floating-rate loans in which the rate is set on a given day but the interest is paid later.
Note that the difference between the underlying rate and the exercise rate is multiplied by 1801360 to reflect the fact that the rate quoted is a 180-day rate but is stated as an annual rate. Also, the interest calculation is multiplied by the notional principal.
In general, the payoff of an interest rate call is
Days in underlying rate
(Notional Principal)Max(O,Underlying rate at expiration – Exercise rate) 360 ) (4-1)
The expression Max(0,Underlying rate at expiration – Exercise rate) is similar to a form that we shall commonly see throughout this post for all options. The payoff of a call option at expiration is based on the maximum of zero or the underlying minus the exercise rate. If the option expires out-of-the-money, then “Underlying rate at expiration – Exercise rate” is negative; consequently, zero is greater. Thus, the option expires with no value. If the option expires in-the-money, “Underlying rate at expiration – Exercise rate” is positive. Thus, the option expires worth this difference (multiplied by the notional principal and the Days1360 adjustment). The expression “Days in underlying rate,” which we used in earlier posts, refers to the fact that the rate is specified as the rate on an instrument of a specific number of days to maturity, such as a 90-day or 180-day rate, thereby requiring that we multiply by 901360 or 1801360 or some similar adjustment.
For an interest rate put option, the general formula is Days in underlying rate
(Notional Principal)Max(O,Exercise rate – Underlying rate at expiration)
For an exercise rate of 5.5 percent and an underlying rate at expiration of 6 percent, an interest rate put expires out-of-the-money. Only if the underlying rate is less than the exer- cise rate does the put option expire in-the-money.
As noted above, borrowers often use interest rate call options to hedge the risk of rising rates on floating-rate loans. Lenders often use interest rate put options to hedge the risk of falling rates on floating-rate loans. The form we have seen here, in which the option expires with a single payoff, is not the more commonly used variety of interest rate option. Floating-rate loans usually involve multiple interest payments. Each of those payments is set on a given date. To hedge the risk of interest rates increasing, the borrower would need options expiring on each rate reset date. Thus, the borrower would require a combination of interest rate call options. Likewise, a lender needing to hedge the risk of falling rates on a multiple-payment floating-rate loan would need a combination of interest rate put options.
A combination of interest rate calls is referred to as an interest rate cap or sometimes just a cap. A combination of interest rate puts is called an interest rate floor or sometimes just a floor.8 Specifically, an interest rate cap is a series of call options on an interest rate, with each option expiring at the date on which the floating loan rate will be reset, and with each option having the same exercise rate.9 Each option is independent of the others; thus, exercise of one option does not affect the right to exercise any of the others. Each component call option is called a caplet. An interest ratefloor is a series of put options on an interest rate, with each option expiring at the date on which thefloating loan rate will be reset, and with each option having the same exercise rate. Each component put option is called a floorlet. The price of an interest rate cap or floor is the sum of the prices of the options that make up the cap or floor.
A special combination of caps and floors is called an interest rate collar. An interest rate collar is a combination of a long cap and a short floor or a short cap and a long floor. Consider a borrower in a floating rate loan who wants to hedge the risk of rising interest rates but is concerned about the requirement that this hedge must have a cash outlay up front: the option premium. A collar, which adds a short floor to a long cap, is a way of reducing and even eliminating the up-front cost of the cap. The sale of the floor brings in cash that reduces the cost of the cap. It is possible to set the exercise rates such that the price received for the sale of the floor precisely offsets the price paid for the cap, thereby completely eliminating the up-front cost. This transaction is sometimes called a zero-cost collar. The term is a bit misleading, however, and brings to mind the importance of noting the true cost of a collar. Although the cap allows the borrower to be paid from the call options when rates are high, the sale of the floor requires the borrower to pay the counterparty when rates are low. Thus, the cost of protection against rising rates is the loss of the advantage of falling rates. Caps, floors, and collars are popular instruments in the interest rate markets.
Although interest rate options are primarily written on such rates as LIBOR, Euribor, and Euroyen, the underlying can be any interest rate.

Options on bonds, usually called bond options, are primarily traded in the over-the-counter markets. Options exchanges have attempted to generate interest in options on bonds, but have not been very successful. Corporate bonds are not very actively traded; most are purchased and held to expiration. Government bonds, however, are very actively traded; nevertheless, options on them have not gained widespread acceptance on options exchanges. Options exchanges generate much of their trading volume from individual investors, who have far more interest in and understanding of stocks than bonds. Thus, bond options are found almost exclusively in the over-the-counter market and are almost always options on government bonds. Consider, for example, a U.S. Treasury bond maturing in 27 years. The bond has a coupon of 5.50 percent, a yield of 5.75 percent, and is selling for $0.9659 per $1 par. An over-the-counter options dealer might sell a put or call option on the bond with an exercise price of $0.98 per $1.00 par. The option could be European or American. Its expiration day must be significantly before the maturity date of the bond. Otherwise, as the bond approaches maturity, its price will move toward par, thereby removing much of the uncertainty in its price. The option could be specified to settle with actual delivery of the bond or with a cash settlement. The parties would also specify that the contract covered a given notional principal, expressed in terms of a face value of the underlying bond.
Continuing our example, let us assume that the contract covers $5 million face value of bonds and is cash settled. Suppose the buyer exercises a call option when the bond priceis at $0.995. Then the option is in-the-money by $0.995 – $0.98 = $0.015 per $1 par. The seller pays the buyer 0.015($5,000,000) = $75,000. If instead the contract called for delivery, the seller would deliver $5 million face value of bonds, which would be worth $5,000,000($0.995) = $4,975,000. The buyer would pay $5,000,000($0.98) = $4,900,000. Because the option is created in the over-the-counter market, the option buyer would assume the risk of the seller defaulting.
Even though bond options are not very widely traded, another type of related option is widely used, especially by corporations. This family of options is called interest rate options. These are quite different from the options we have previously discussed, because the underlying is not a particular financial instrument