Caps And Floors

A borrower with a floating-rate loan is exposed to rates rising indefinitely, the same way a homeowner without insurance is exposed to their house burning down. A cap is exactly that insurance: for a premium paid up front, the borrower is reimbursed whenever the floating rate resets above some agreed level, and pays nothing extra (just keeps paying the market floating rate) whenever it doesn't. A floor is the mirror image, protecting a lender against rates falling too low. Because a loan resets many times over its life, the insurance has to cover every single reset — so a cap isn't one option, it's a bundle of them, one per period.

One cap, many caplets

A cap on a rate over $N$ periods is a portfolio of $N$ independent options, called caplets, each paying off only if the rate observed for its own period exceeds the strike:

$$\text{Caplet payoff}_i = \tau_i \times N_\text{notional} \times \max(L_i - K,\ 0),$$

where $L_i$ is the floating rate that resets for period $i$, $K$ is the cap's strike rate, $\tau_i$ the period's year fraction, and $N_\text{notional}$ the notional amount. In plain English: each caplet only pays out for its own reset, ignoring every other period, and the cap's total value is just the sum of pricing each caplet separately (typically with the Black formula, treating each forward rate as lognormal) and discounting each back to today.

Worked example 1 — pricing one caplet

A caplet covers a 3-month period starting in 1 year, notional $10,000,000, strike $K = 5.00\%$. The forward rate for that period is $F = 5.20\%$, forward volatility $\sigma = 20\%$, and the discount factor to the payment date is $D = 0.945$. Using Black's formula with $d_1 = \frac{\ln(F/K) + 0.5\sigma^2 t}{\sigma\sqrt{t}}$ at $t=1$: $\ln(5.20/5.00) = 0.0392$, $0.5 \times 0.04 \times 1 = 0.02$, sum $= 0.0592$, divided by $\sigma\sqrt{t}=0.20$ gives $d_1 = 0.296$, and $d_2 = 0.296 - 0.20 = 0.096$. Using $N(0.296) \approx 0.6163$ and $N(0.096) \approx 0.5382$: caplet value = D \times \tau \times N_\text{notional} \times [F\,N(d_1) - K\,N(d_2)] = 0.945 \times 0.25 \times 10{,}000{,}000 \times [0.052 \times 0.6163 - 0.05 \times 0.5382] = 2{,}362{,}500 \times [0.03205 - 0.02691] = 2{,}362{,}500 \times 0.00514 \approx \12{,}143$.

Worked example 2 — summing the strip

If a 2-year cap has 8 quarterly caplets and (for simplicity) each one is worth roughly $10,000 given its own forward rate and volatility, the cap's value is approximately 8 \times 10{,}000 = \80{,}000$ — but in practice each caplet uses a different forward rate (from the curve) and often a different implied volatility (from the volatility term structure), so caplets further out are rarely priced identically; a real desk sums 8 distinct numbers, not one number multiplied by 8.

The $N(d_1)$ and $N(d_2)$ terms above are areas under exactly this curve — drag the mean and see how a caplet's forward rate sitting further above its strike shifts probability mass, and value, further in its favor.

What this means in practice

Corporate treasurers buy caps to put a ceiling on floating-rate borrowing costs without giving up the benefit of rates falling, and banks selling caps hedge each caplet largely independently since each depends on a different forward rate and vol point. A collar (buy a cap, sell a floor) is a common cost-reducing structure, financing the cap's premium by giving up some downside protection.