# Interest rate cap and floor

 Contents

## Interest rate cap

An interest rate cap is a series of European call options or caplets on a specified interest rate, usually the LIBOR interest rate. The underlying rate is known as the reference rate. The buyer of the cap receives money if on the maturity of any of the caplets, the reference rate exceeds the agreed strike price of the cap.

In formulas a caplet payoff on a rate L struck at K is

[itex] N\alpha max(L-K,0)[itex]

where N is the notional value exchanged and [itex]\alpha[itex] 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 2005 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.

## 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 below the agreed strike price of the floor.

## Valuation of interest rate caps

### 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 [itex]\sigma[itex]. Under this model, a caplet on a LIBOR expiring at t and paying at T has present value

[itex] V = P(0,T)(FN(d_1) - KN(d_2))[itex]

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 [itex] (\frac{P(0,t)}{P(0,T)} - 1)/\alpha [itex]
K is the strike
[itex]d_1 = \frac{log(F/K) + 0.5 \sigma^2t)}{\sigma\sqrt{t}}[itex]

and

[itex]d_2 = d_1 - \sigma\sqrt{t}[itex]

Notice that there is a 1-to-1 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.

### 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.

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