Interest rate cap and floor
Encyclopedia
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%.
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.
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%.
The interest rate cap can be analyzed as a series of European call options or caplets which exist for each period the cap agreement is in existence.
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 the 1st of August.
, 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.
. 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
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
.
s, such as the Hull-White model
have this degree of tractability. Thus we can value caps and floors in those models..
What about Collars?
Interest rate collar
…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.
And Reverse Collars?
…buying an interest rate floor and simultaneously selling an interest rate cap.
The size of cap and floor premiums are determined by a wide range of factors
Derivative (finance)
A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...
in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price
Strike price
In options, the strike price is a key variable in a derivatives contract between two parties. Where the contract requires delivery of the underlying instrument, the trade will be at the strike price, regardless of the spot price of the underlying instrument at that time.Formally, the strike...
. 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
Hedge (finance)
A hedge is an investment position intended to offset potential losses that may be incurred by a companion investment.A hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance, forward contracts, swaps, options, many types of...
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 derivativeDerivative (finance)
A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...
in which the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price
Strike price
In options, the strike price is a key variable in a derivatives contract between two parties. Where the contract requires delivery of the underlying instrument, the trade will be at the strike price, regardless of the spot price of the underlying instrument at that time.Formally, the strike...
. An example of a cap would be an agreement to receive a payment 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 exist for each period the cap agreement is in existence.
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 receiveCustomarily the payment is made at the end of the rate period, in this case on the 1st of August.
Interest rate floor
An interest rate floor is a series of European put options or floorlets on a specified reference rateReference rate
A reference rate is a rate that determines pay-offs in a financial contract and that is outside the control of the parties to the contract. It is often some form of LIBOR rate, but it can take many forms, such as a consumer price index, a house price index or an unemployment 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
Strike price
In options, the strike price is a key variable in a derivatives contract between two parties. Where the contract requires delivery of the underlying instrument, the trade will be at the strike price, regardless of the spot price of the underlying instrument at that time.Formally, the strike...
of the floor.
Black model
The simplest and most common valuation of interest rate caplets is via the Black modelBlack model
The Black model is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing bond options, interest rate caps / floors, and swaptions...
. Under this model we assume that the underlying rate is distributed log-normally with volatility
Volatility (finance)
In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...
. Under this model, a caplet on a LIBOR expiring at t and paying at T has present valuewhere
- P(0,T) is today's discount factor for T
- F is the forward priceForward priceThe forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...
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
Implied volatility
In financial mathematics, the implied volatility of an option contract is the volatility of the price of the underlying security that is implied by the market price of the option based on an option pricing model. In other words, it is the volatility that, when used in a particular pricing model,...
.
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 modelShort rate model
In the context of interest rate derivatives, a short-rate model is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.-The short rate:...
s, such as the Hull-White model
Hull-White model
In financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates...
have this degree of tractability. Thus we can value caps and floors in those models..
What about Collars?
Interest rate collar
…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.
- The cap rate is set above the floor rate.
- The objective of the buyer of a collar is to protect against rising interest rates (while agreeing to give up some of the benefit from lower interest rates).
- The purchase of the cap protects against rising rates while the sale of the floor generates premium income.
- A collar creates a band within which the buyer’s effective interest rate fluctuates
And Reverse Collars?
…buying an interest rate floor and simultaneously selling an interest rate cap.
- The objective is to protect the bank from falling interest rates.
- The buyer selects the index rate and matches the maturity and notional principal amounts for the floor and cap.
- Buyers can construct zero cost reverse collars when it is possible to find floor and cap rates with the same premiums that provide an acceptable band.
The size of cap and floor premiums are determined by a wide range of factors
- The relationship between the strike rate and the prevailing 3-month LIBOR
- premiums are highest for in the money options and lower for at the money and out of the money options
- Premiums increase with maturity.
- The option seller must be compensated more for committing to a fixed-rate for a longer period of time.
- Prevailing economic conditions, the shape of the yield curveYield curveIn finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
, and the volatility of interest rates.- upsloping yield curveYield curveIn finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
—caps will be more expensive than floors. - the steeper is the slope of the yield curveYield curveIn finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
, ceteris paribus, the greater are the cap premiums. - floor premiums reveal the opposite relationship.
- upsloping yield curve
Valuation of CMS Caps
Caps based on an underlying rateLIBOR (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 volatilities. Caps consist of caplets with volatilities dependent on the corresponding forward LIBOR rate. But caps can also be represented by a "flat volatility", so the net of the caplets still comes out to be the same. (15%,20%,....,12%) → (16.5%,16.5%,....,16.5%)
- So one cap can be priced at one vol.
- 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% and floor with 30%. 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.
- 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.
External links
- 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 BaselUniversity of BaselThe University of Basel is located in Basel, Switzerland, and is considered to be one of leading universities in the country...
- Bond Options, Caps and the Black Model Dr. Milica Cudina, University of Texas at AustinUniversity of Texas at AustinThe University of Texas at Austin is a state research university located in Austin, Texas, USA, and is the flagship institution of the The University of Texas System. Founded in 1883, its campus is located approximately from the Texas State Capitol in Austin...
- Online Caplet And Floorlet Calculator Dr. Shing Hing Man, Thomson Reuters Risk Management
- Introduction to Caps, Floors, Collars and Swaptions