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