Floor
From Open Ideas
Definition
A floorlet is a contract in which one party receives the positive difference between a fixed interest rate level $L$ and a variable interest rate $r$ over a certain period $\Delta$ on a notional amount $F$. The counterparty is obliged to pay the positive difference, but receives a premium in return.
A floor is contract that can be interpreted as a portfolio of floorlets.
The payoff function
The time $T_{i}$ value of a floorlet is, by definition, given by
$\displaystyle{\left(L - r_{s}(T_{i-1},T_{i}) \right)^{+} \Delta \cdot F}$ .
Since the payoff is known in $T_{i-1}$, the value in $T_{i-1}$ can easily determined by discounting, i.e.
$\displaystyle{P(T_{i-1}, T_{i})\left(L - r_{s}(T_{i-1},T_{i}) \right)^{+} \Delta \cdot F}$
or
$\displaystyle{P(T_{i-1}, T_{i})\left((1 + \Delta L \right) - (1 + \Delta r_{s}(T_{i-1},T_{i})))^{+} \cdot F}$
In combination with
$\displaystyle{P(t,T) := \frac{1}{1+\Delta r_{s}} }$
one obtains the expression
$\displaystyle{F \:(1 + \Delta L) \: \left(P(T_{i-1}, T_{i}) - \frac{1}{1 + \Delta L} \right)^{+} } \,.$
It follows that the value of a floorlet must be equal to the value of $F \cdot (1+\Delta L)$ European call options with strike price $K = \frac{1}{1+\Delta L}$ on a zerobond maturing at $T_{i}$.
