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Introduction to Pricing Approach

An interest rate cap is designed to provide insurance against the interest expense from a floating rate loan exceeding a certain range. That is, a cap will have a positive payoff when the reference interest rate ends up above the fixed rate specified in the contract. The borrowing entity still pays an amount of interest based on periodic fixing of the floating rate, but when that exceeds the interest 'cap' the net cost of borrowing is partially offset by the payoff from the option.

An interest rate floor on the other hand is analogous to a standard put option. The holder of the floor will receive a payoff when the reference interest rate falls below the benchmark rate, and this structure can therefore be used to guarantee a minimum return for a lender. A collar is simply the combination of a long position in a cap and a short position in a floor, and is typically used to ensure that the cost of floating-rate debt does not fall outside of a certain range.

Because all of these instruments are structured with reference to a particular floating-rate debt issue, the caps and floors actually consist of a series of caplets or floorlets with expirations that match the rate-set dates of the bond. Consider a cap that is written to insure that the cost of a particular floating-rate note does not exceed 7%. The note has a face value of $100,000,000, is reset on a semi-annual basis and matures in 3 years. A cap associated with this debt will consist of 6 caplets, one with an expiration date that matches each of the six rate reset dates, but with payoffs that occur six months later to coincide with the actual interest payment dates. For example, assume that the relevant reference rate has reached 8% at one of the reset dates. The payoff (six months later) on the corresponding caplet will be:

 

Equation Template

If general, the payoff function for the caplet that matures at time ti with payment to be made at ti+1 can be written as:

 

Equation Template

 

Ri = the value of the reference floating rate at reset date i

Rx = the fixed strike (or cap) rate for the deal

ri = the time (in years) between the i'th reset date and payment date

FV = the face value of the deal

 

The payoff structure is the standard payoff for a call option. If the reference interest rate (Ri) is assumed to have a lognormal distribution with volatility then the caplet can be valued using the following equation:

 

Equation Template

where,

discount factor for the period between time 0 and time ti+1

Fi = the forward interest rate for the period between time ti and ti+1

d1 =

d2 =

Note that the modification affects the discounting procedure. For most standard options the payoff occurs on the same date as the comparison between the spot and strike values, and the appropriate discount factor therefore has the same maturity as the expiration date. However for the caplet, the payoff occurs at the end of the period after the rate-set has been made. This is consistent with the fact that bond interest payments are normally made in arrears.

Using exactly the same logic, the Black model for floorlets is defined as:

 

Equation Template

Note that in the definition of d1 and d2 given above the volatility measure for each caplet or floorlet has a subscript. This implies that the volatility measure can potentially differ for each of the optlets that make up caps or floors. It is however standard practise to simplify the volatility input by assuming that the rate is constant across all of the maturity dates of the optlets. That is:

 

Equation Template

These volatilities are referred to as flat volatilities.

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