Calculating interest rate Caps and Floors using a Binomial

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Transcript Calculating interest rate Caps and Floors using a Binomial 

Calculating interest rate Caps and
Floors using a Binomial Tree
Galabe Sampid Marius, Keskus Eren,
& Celestin Kamta
Abstract
In this paper, we give a brief description of the meaning
of an interest rate Caps and Floors in the LIBOR
market; and then try to create an application in
MATLAB to value Caps and Floors at different levels
using a binomial tree.
We do not go into details of interest rate Caps and
Floors since our task is to create an application to value
Caps and Floors using a binomial tree.
Interest Rate Caps and Floor
What is an interest rate Cap?
An interest rate cap is a series of European call
options (also known as caplets) on a specified
interest rate, usually the LIBOR interest rate.
The buyer of the cap receives money if on the
maturity of any of the caplets, the reference or the
underlying rate exceeds the agreed strike price of
the cap.
Interest rate Cap protects an investor who has borrowed funds
on a floating interest rate basis against the risk of paying very
high interest rates.
At time 𝑑𝑖 the payoff of the ith caplet is given by:
𝐢𝑑𝑖𝑖 = π‘π›Ώπ‘šπ‘Žπ‘₯ 𝑙 𝑑𝑖 , 𝑑𝑖 βˆ’ 𝛿 – 𝐾, 0
With no other payments,
where
N = face value,
𝑑𝑖 = respective payment dates ( i = 1,2.3,….,n:
where n = number of caplets )
𝛿 = 𝑑𝑖+1 βˆ’ 𝑑𝑖
K = cap rate
The payoff diagram is as seen in figure 1 bellow
Floating Interest Rate
No Cap
Cap Rate
Fig 1
gain
Reference Rate
What is an interest rate Floor?
An Interest rate floor is a series of European put
options (also known as "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 fixed below the agreed strike price of the floor.
At time 𝑑𝑖 the payoff of is given by:
πΉπ‘‘π‘–π‘–βˆ’π›Ώ = π‘π›Ώπ‘šπ‘Žπ‘₯ 𝐾 βˆ’ 𝑙 𝑑𝑖 , 𝑑𝑖 βˆ’ 𝛿 , 0
Interest Rate Floor protects an investor who has lent funds
on a floating rate basis against receiving very low interest
rate
The payoff diagram is as seen in figure 2 bellow
Floating interest Rate
Floor
No Floor
gain
Floor Rate
Fig 2
Reference Rate
In the Black-Scholes world, the value of the Cap and Floor
are respectively
𝑁𝛿 π‘˜ 𝑃 0, π‘‘π‘˜+1 πΉπ‘˜ 𝑁 𝑑1 βˆ’ π‘…π‘˜ 𝑁 𝑑2
and
Where
and
𝑁𝛿 π‘˜ 𝑃 0, π‘‘π‘˜+1 [π‘…π‘˜ 𝑁 𝑑2 βˆ’ πΉπ‘˜ 𝑁 𝑑1 ] respectively
𝑑1 =
ln(πΉπ‘˜ /π‘…π‘˜ )+(πœŽπ‘˜2 𝑑 π‘˜ )/2
πœŽπ‘˜ π‘‘π‘˜
𝑑2 = 𝑑1 βˆ’ πœŽπ‘˜ π‘‘π‘˜
πΉπ‘˜ = is the forward interest rate at time 0 for the period between time π‘‘π‘˜
and π‘‘π‘˜+1 .
πœŽπ‘˜ = volatility of the forward interest rate
There also exist put-call parity relations between the Caps and
Floors. That is,
Value of Cap = Value of Floor + Value of Swap
Here, the Cap and Floor have the same strike price. The Swap is
an agreement to receive LIBOR and pay a fixed rate of strike
with no exchange of payments on the first reset day.
One big advantage with caps is that the buyer limits his potential
loss to the premium paid, but retains the right to benefit from
favorable rate movements. On the other hand, the borrower
buying a cap limits exposure to rising interest rates while
retaining the potential to benefit from falling rates. However, the
premium is a non refundable fund which is paid upfront by the
buyer. They can also lose all their values if the expire out of the
money or when approaching maturity.
Because Caps and Floors allow an investor to benefit from
changes in interest rates; while at the same time limiting any
downside losses, they act as some sort of insurance to the
investor.
Tack så Mycket