In finance, an interest rate cap is a type of interest rate derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for each month the LIBOR rate exceeds 2.5%.

Similarly, an interest rate floor is a derivative contract in which the buyer receives payments at the end of each period in which the interest rate is below the agreed strike price.

Caps and floors can be used to hedge against interest rate fluctuations. For example, a borrower who is paying the LIBOR rate on a loan can protect himself against a rise in rates by buying a cap at 2.5%. If the interest rate exceeds 2.5% in a given period the payment received from the derivative can be used to help make the interest payment for that period, thus the interest payments are effectively "capped" at 2.5% from the borrowers' point of view.

Interest rate cap

An interest rate cap is a derivative in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. An example of a cap would be an agreement to receive a payment for each month the LIBOR rate exceeds 2.5%. They are most frequently taken out for periods of between 2 and 5 years, although this can vary considerably. Since the strike price reflects the maximum interest rate payable by the purchaser of the cap, it is frequently a whole number integer, for example 5% or 7%.

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-expiry 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.

Valuation of CMS Caps

Caps based on an underlying rate (like a Constant Maturity Swap Rate) cannot be valued using simple techniques described above. The methodology for valuation of CMS Caps and Floors can be referenced in more advanced papers.

Implied Volatilities

  • An important consideration is cap and floor (so called Black) volatilities. Caps consist of caplets with volatilities dependent on the corresponding forward LIBOR rate. But caps can also be represented by a "flat volatility", a single number which if plugged in the formula for valuing each caplet recovers the price of the cap i.e. the net of the caplets still comes out to be the same. To illustrate: (Black Volatilities) → (Flat Volatilities) : (15%,20%,....,12%) → (16.5%,16.5%,....,16.5%)
  • Therefore, one cap can be priced at one vol. This is extremely useful for market practitioners as it reduces greatly the dimensionality of the problem: instead of tracking n caplet Black volatilities, you need to track just one: the flat volatility.
  • Another important relationship is that if the fixed swap rate is equal to the strike of the caps and floors, then we have the following put–call parity: Cap-Floor = Swap.
  • Caps and floors have the same implied vol too for a given strike.
  • Imagine a cap with 20% vol and floor with 30% vol. Long cap, short floor gives a swap with no vol. Now, interchange the vols. Cap price goes up, floor price goes down. But the net price of the swap is unchanged. So, if a cap has x vol, floor is forced to have x vol else you have arbitrage.
  • Assuming rates can't be negative, a Cap at strike 0% equals the price of a floating leg (just as a call at strike 0 is equivalent to holding a stock) regardless of volatility cap.

Compare

  • Interest rate swap

Notes

References

  • Basic Fixed Income Derivative Hedging - Article on Financial-edu.com.
  • Convexity Conundrums by Patrick Hagan
  • Martingales and Measures: Black's Model Dr. Jacqueline Henn-Overbeck, University of Basel
  • Bond Options, Caps and the Black Model Dr. Milica Cudina, University of Texas at Austin
  • Online Caplet And Floorlet Calculator Dr. Shing Hing Man, Thomson Reuters Risk Management