interest rate cap is a derivative in which the buyer receives money
at the end of each period in which an interest rate exceeds
the agreed strike price. An example of a cap would be an agreement
to receive money for each month the LIBOR rate exceeds 2.5%.
rate cap can be analyzed as a series of European call options
or caplets which exists for each period the cap agreement
is in existence.
a caplet payoff on a rate L struck at K is
is the notional value exchanged and Î± 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
2007 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.
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 is below the agreed strike price of
of interest rate caps
and most common valuation of interest rate caplets is via the
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
purchase of an interest rate cap and sale of an interest rate
floor on the same index for the same maturity and notional principal
cap rate is set above the floor rate.
- The objective
of the buyer of a collar is to protect against rising interest
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 buyerâ€™s effective interest
an interest rate floor and simultaneously selling an interest
- 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.
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.
of cap and floor premiums are determined by a wide range of
- The relationship
between the strike rate and the prevailing 3-month LIBOR
are highest for in the money options and lower for at
the money and out of the money options
increase with maturity.
option seller must be compensated more for committing
to a fixed-rate for a longer period of time.
economic conditions, the shape of the yield curve, and the
volatility of interest rates.
yield curve -- caps will be more expensive than floors.
steeper is the slope of the yield curve, ceteris paribus,
the greater are the cap premiums.
premiums reveal the opposite relationship. ]
- An important
consideration is cap and floor volatilities. Caps consist
of caplets with volatilities dependent on the corresponding
forward LIBOR rate. But caps can 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%)
one cap can be priced at one vol.
important intuition is that caps and floors are duals.
Cap-Floor = Swap.
and floors have the same implied vol too for a given strike.
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.