Cap & Floor

An interest rate cap is a ceiling on a floating rate. A borrower with a floating loan buys a cap struck at, say, 4%: every period the market rate fixes above 4%, the cap pays them the difference. Below 4%, nothing happens and they simply enjoy the lower rate.

It's insurance, priced like insurance: an upfront premium buys protection for the life of the deal. Unlike a swap, which locks a fixed rate and gives up all benefit from falling rates, a cap protects the downside while keeping the upside — that asymmetry is what the premium pays for.

A floor is the mirror image: a minimum rate, bought by floating-rate receivers (lenders, FRN investors) fearing rate collapses.

A strip of little options

A 5-year quarterly cap is really 20 independent options ("caplets"), one per reset: each pays \(\max(r_i - K, 0)\times\delta\times N\) at its period's end. The cap premium is simply the sum of caplet values — periods further out cost more (more time for rates to wander).

Structures built from caps and floors

• Collar: buy a cap, sell a floor — cheaper (often zero-cost) protection, rate confined to a band.
• Corridor: buy a cap, sell a higher-strike cap — capped protection at lower premium.
• Embedded: floored FRNs, capped mortgages and structured notes all contain these options implicitly.

Market colour

Caps became front-page finance in 2022–23: US commercial real-estate loans typically require borrowers to hold caps, and when rates jumped, replacement caps that had cost $100k suddenly cost millions — a squeeze that materially affected property refinancing.

Caplet pricing

Each caplet is a call on its forward rate \(f_i\), a martingale under its own forward measure. Bachelier (normal) pricing, today's standard:

Under RFR benchmarks the "rate" is compounded-in-arrears, so the option's effective observation extends into the accrual period — vol accrues partly during the period itself, handled by an adjusted variance term (\(T_i \to T_i + \tfrac{\delta}{3}\)-style corrections in simple models).

Vol surfaces and stripping

Markets quote flat vols — one vol repricing the whole cap — per maturity and strike. Desks strip these into forward (spot) caplet vols, the genuinely informative objects, via bootstrap: each maturity's cap minus the previous cap's caplets reveals marginal caplet value. The caplet surface and the swaption cube must cohere; their misalignment is traded (cap/swaption arbitrage) and is a standard calibration tension in term-structure models.

Caps vs. swaptions — the correlation wedge

A cap is a basket of options on individual forwards (no correlation dependence); a swaption is an option on a basket (average) of forwards — worth less when forwards decorrelate. The price gap between a cap and the matching swaption straddle strip prices inter-forward correlation, one of the few places it is directly observable.