Interest rate cap and floor
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[edit] Interest rate cap
An interest rate cap is a derivative in which the buyer receives money at the end of each period in which an interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive money for each month the LIBOR rate exceeds 2.5%.
The interest rate cap can be analyzed as a series of European call options or caplets which exists for each period the cap agreement is in existence.
In formulas a caplet payoff on a rate L struck at K is
where N is the notional value exchanged and α is the day count fraction corresponding to the period to which L applies. For example suppose you own a caplet on the six month USD LIBOR rate with an expiry of 1st February 2007 struck at 2.5% with a notional of 1 million dollars. Then if the USD LIBOR rate sets at 3% on 1st February you receive 1m*0.5*max(0.03-0.025,0) = $2500. Customarily the payment is made at the end of the rate period, in this case on 1st August.
[edit] Interest rate floor
An interest rate floor is a series of European put options or "floorlets" on a specified reference rate, usually LIBOR. The buyer of the floor receives money if on the maturity of any of the floorlets, the reference rate fixed is below the agreed strike price of the floor.
[edit] Valuation of interest rate caps
[edit] Black
The simplest and most common valuation of interest rate caplets is via the Black model. Under this model we assume that the underlying rate is distributed log-normally with volatility σ. Under this model, a caplet on a LIBOR expiring at t and paying at T has present value
- V = P(0,T)(FN(d1) − KN(d2))
where
- P(0,T) is today's discount factor for T
- F is the forward price of the rate. For LIBOR rates this is equal to
- K is the strike
and
Notice that there is a one-to-one mapping between the volatility and the present value of the option. Because all the other terms arising in the equation are indisputable, there is no ambiguity in quoting the price of a caplet simply by quoting its volatility. This is what happens in the market. The volatility is known as the "Black vol" or implied vol.
[edit] As a bond put
It can be shown that a cap on a LIBOR from t to T is equivalent to a multiple of a t-maturity put on a T-maturity bond. Thus if we have an interest rate model in which we are able to value bond puts, we can value interest rate caps. Similarly a floor is equivalent to a certain bond call. Several popular short rate models, such as the Hull-White model have this degree of tractability. Thus we can value caps and floors in those models..
