# Interest rate cap and interest rate floor

Interest rate caps and interest rate floors are interest rate derivatives with an optional character. With an interest rate cap, the buyer receives a payment at the end of each period in which the agreed reference interest rate is above the agreed base value . With an interest floor, the buyer receives a payment at the end of each period in which the agreed reference interest rate is below the agreed base value.

## functionality

When concluding a cap or floor, the following points in particular are agreed between the two contracting parties:

• the term of the contract
• the face value of the deal
• the base value . This is generally a liquid money market rate . For example, Euribor rates are common here for the euro and Libor rates for US dollars , pounds sterling and Swiss francs .
• the exercise price . This is called the cap rate for caps and floor rate for floors .
• the length of the interest rate adjustment periods. The often multi-year term of the contract is divided into individual interest rate adjustment periods, called caplets or floorlets , which are usually three or six months, less often one or twelve months (other variants are theoretically possible, but very rare). A fixing date and a payment date are assigned to each cap or floorlet. Usually the date of the fixing is two bank working days before the start of a cap or floorlet and the date of payment is the last day of a cap or floorlet.

On the date of the fixing , the current value of the corresponding money market rate is determined ( fixing ) and, depending on the type of cap or floor, it is determined whether and in what amount a payment will be made.

The nominal of the cap or floor is denoted by , the cap rate by and the floor rate by . The interest rate observed on the market is denoted by, the fraction of the length of the current interest period in days and the number of days in a year by . The payment of the cap is that of the floor . ${\ displaystyle N}$${\ displaystyle r ^ {\ text {cap}}}$${\ displaystyle r ^ {\ text {floor}}}$${\ displaystyle r ^ {\ text {fixing}}}$${\ displaystyle \ alpha}$${\ displaystyle P ^ {\ text {cap}}}$${\ displaystyle P ^ {\ text {floor}}}$

Among other things, there are the following types of caps and floors.

### Plain vanilla

If the observed money market rate exceeds the cap rate, the cap pays the product of the nominal value and the difference between the money market rate and the cap rate on the date of payment. Conversely, a floor pays the product of the nominal value and the difference between the floor rate and the money market rate. In both cases, based on the days of the period as a fraction of the year, since the interest rate is a p. a.-interest.

• Withdrawal function cap: ${\ displaystyle P ^ {\ text {cap}} = N \ cdot \ alpha \ cdot \ max (r ^ {\ text {fixing}} - r ^ {\ text {cap}}, 0)}$
• Payout function Floor: ${\ displaystyle P ^ {\ text {floor}} = N \ cdot \ alpha \ cdot \ max (r ^ {\ text {floor}} - r ^ {\ text {fixing}}, 0)}$

### Cap and floor as a chooser option

This variant is described in more detail in the article Chooser Option . Here the cap or floor only has an effect over a limited number of the available periods that the buyer can choose.

### Cap and floor with in-area fixing

This variant differs from the plain vanilla variant in that the date of the fixing is not one or two bank working days before the start of the interest period, but rather before the end of the interest period.

### Digital cap and digital floor

If the fixing exceeds the cap rate, the cap pays a fixed amount regardless of the extent of the excess . Similarly, a fixed amount is also paid for a floor if the fixing was below the floor rate. ${\ displaystyle d}$

### Cap and floor with index factor

In addition to the contractually fixed points described above, an index factor is agreed. If the -fold of the observed money market rate exceeds the cap rate, the cap pays the product of the nominal value and the difference of -fold the money market rate and the cap rate on the date of payment . Conversely, a floor pays the product of the nominal value and the difference between the floor rate and times the money market rate and money market rate. ${\ displaystyle i> 0}$${\ displaystyle i}$${\ displaystyle i}$${\ displaystyle i}$

• Withdrawal function cap: ${\ displaystyle P ^ {\ text {cap}} = N \ cdot \ alpha \ cdot \ max (i \ cdot r ^ {\ text {fixing}} - r ^ {\ text {cap}}, 0)}$
• Payout function Floor: ${\ displaystyle P ^ {\ text {floor}} = N \ cdot \ alpha \ cdot \ max (r ^ {\ text {floor}} - i \ cdot r ^ {\ text {fixing}}, 0)}$

### Collar

A collar is the combination of the payout profiles from a cap and a floor. It corresponds to buying a cap and selling a floor with . The buyer of a collar is thus paid if the interest rate fixing is above the cap rate. If the fixing is below the floor rate, the buyer pays. ${\ displaystyle r ^ {\ text {cap}}> r ^ {\ text {floor}}}$

• Withdrawal function collar: ${\ displaystyle P ^ {\ text {collar}} = {\ begin {cases} N \ cdot \ alpha \ cdot (r ^ {\ text {fixing}} - r ^ {\ text {cap}}) & {\ text {if}} r ^ {\ text {cap}}

## example

For example, if interest caps were bought for the equivalent of 100,000 Swiss francs with a maximum interest rate of 3.5% and the underlying interest rate in the Swiss franc currency is 4%, the difference is 0.5% of the equivalent of 100,000 francs credited based on the interest period.

With the help of caps, the interest for a floating rate bond can be fixed up to a maximum amount for the issuer . If the maximum amount of the interest rate is exceeded, the cap pays out the difference between the actual interest rate and the upper limit. Loans with a variable interest rate with a cap are also often sold as so-called cap loans .

## rating

The evaluation is discussed on a cap. The evaluation of a floor can be derived from this.

A caplet is an option on a swap , a swaption , with only one interest period. Since the individual caplets are independent of each other, each caplet can be assessed individually. The cash value of the cap then corresponds to the sum of the cash values ​​of the individual caplets.

### Black

The usual rating for caps is based on the Fischer Black model . The model assumes that the underlying interest rate is log-normally distributed with volatility . In this model, the caplet has a Euribor interest rate that is fixed at the time and pays at the time${\ displaystyle \ sigma}$${\ displaystyle t}$${\ displaystyle T}$

${\ displaystyle V = P (0, T) \ left (FN (d_ {1}) - r ^ {\ text {cap}} N (d_ {2}) \ right)}$,

With

${\ displaystyle P (0, T)}$is the discount factor from today.${\ displaystyle T}$
${\ displaystyle F}$is the forward interest rate. thus corresponds${\ displaystyle F}$${\ displaystyle {\ frac {1} {\ alpha}} \ left ({\ frac {P (0, t)} {P (0, T)}} - 1 \ right)}$
${\ displaystyle d_ {1} = {\ frac {\ ln (F / r ^ {\ text {cap}}) + 0.5 \ sigma ^ {2} t} {\ sigma {\ sqrt {t}}}}}$

and

${\ displaystyle d_ {2} = d_ {1} - \ sigma {\ sqrt {t}}}$.

There is thus an equivalence relation between the volatility and the present value of the cap. Because all other variables contain undisputed values, it makes no difference whether the price of a cap or the volatility is mentioned. Accordingly, the price of the caps and floors is not negotiated on the market, but rather the volatility. This volatility is also called Black Vola or Implied Volatility .

### Bond put

It can be shown that a caplet is similar to a bond option , so that an assessment using bond options is also possible.

## literature

• John C. Hull : Options, Futures, and Other Derivatives. 6th edition. Pearson Studium, Munich et al. 2006, ISBN 3-8273-7142-2 , pp. 740f. ( Economics. Business Administration, Stock Exchange & Finance ).