Interest rate cap and floor

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.[1] 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%.[1] By comparison the underlying index for a cap is frequently a LIBOR rate, or a national interest rate.[1] The extent of the cap is known as its notional profile and can change over the lifetime of a cap, for example, to reflect amounts borrowed under an amortizing loan.[1] The purchase price of a cap is a one-off cost and is known as the premium.[1]

The purchaser of a cap will continue to benefit from any fall in interest rates below the strike price, which makes the cap a popular means of hedging a floating rate loan.[1]

The interest rate cap can be analyzed as a series of European call options, known as caplets, which exist for each period the cap agreement is in existence. Unlike other types of option, it is generally not necessary for the purchaser of a cap to notify the seller in order to exercise it, as this will happen automatically if the interest rate exceeds the strike price.[1] Each caplet is settled in cash at the end of the period to which it relates.[1]

In mathematical terms, a caplet payoff on a rate L struck at K is

where N is the notional value exchanged and 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 1 February 2007 struck at 2.5% with a notional of 1 million dollars. Then if the USD LIBOR rate sets at 3% on 1 February you receive

Customarily the payment is made at the end of the rate period, in this case on 1 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 is below the agreed strike price of the floor.

Interest rate collars and reverse collars

An interest rate collar is the simultaneous purchase of an interest rate cap and sale of an interest rate floor on the same index for the same maturity and notional principal amount.

A reverse interest rate collar is the simultaneous purchase of an interest rate floor and simultaneously selling an interest rate cap.

Valuation of interest rate caps

Black model

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 . Under this model, a caplet on a LIBOR expiring at t and paying at T has present value

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
N is the standard normal CDF.

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.

As negative interest rates became a possibility and then reality in many countries at around the time of Quantitative Easing, so the Black model became increasingly inappropriate (as it implies a zero probability of negative interest rates). Many substitute methodologies have been proposed, including shifted log-normal, normal and Markov-Functional, though no new standard is yet to emerge.[2]

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..

The size of cap and floor premiums are determined by a wide range of factors

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

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Notes

References

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