Interest-rate cap

A cap limits the floating rate on a notional principal by paying the excess over a strike for successive accrual periods. It is a sum of caplets. For accrual fraction $\delta$ and reset at $T$ with payment at $U$, the unit-notional payoff is $\delta (L(T;T,U)-K{)}^{+}$ at $U$, where $L(T;T,U)=[P(T,U{)}^{-1}-1]/\delta$ and $P$ is a zero-coupon bond price.

One payment period of an interest-rate cap. For $1+\delta K>0$, its value at the reset date is $(1-(1+\delta K)P(T,U){)}^{+}$. Thus the caplet is equivalent before reset to $1+\delta K$ units of a put expiring at $T$ on the maturity-$U$ zero-coupon bond, with strike $(1+\delta K{)}^{-1}$.

For a Gaussian forward-rate field with deterministic volatility, put $F=P(t,U)/P(t,T)$, $K_b=(1+\delta K{)}^{-1}$, and $v^2={\int }_t^T\Vert \Sigma (s,U)-\Sigma (s,T){\Vert }^{2}ds$. Under the T-forward measure, $F$ is lognormal with that integrated variance. With $d_1=\log (F/K_b)/v+v/2$ and $d_2=d_1-v$, the displayed expression is the caplet price. At $v=0$ use the discounted intrinsic bond-put value. Gaussian instantaneous rates do not make the reset floating rate Gaussian; the exponential bond relation is what yields this bond-option formula.