Interest rate caps and floors

A borrower paying a floating rate on a term loan faces one exposure repeated many times: the rate resets each period, and each reset is a separate chance to be hurt. The instrument that matches that exposure is a strip of separate options.

An interest-rate cap is a portfolio of caplets. Each caplet covers one interest period, and it pays

payoff = Notional × max(reference rate - K, 0) × tau

where tau is the period's accrual factor. The payoff is determined on the fixing date at the start of the period and paid at the end, which is exactly how the underlying loan works.

Three details decide whether an implementation is right.

The caplet's expiry is the fixing date, and its payment is one period later. The Black formula therefore uses T = time to fixing inside d1 and d2, and the discount factor to the payment date outside the bracket. Using one date for both is the commonest error in a first implementation. Discounting at the fixing date applies less discounting than the contract calls for, so the premium comes out too high by roughly one period's worth.

The underlying is the forward rate for that period, read off the discount curve as F = (D(T1)/D(T2) - 1) / tau. Not the spot rate, not the current fixing.

The first period is not an option. A one-year cap on a quarterly loan has four interest periods, but the first period's rate was set at the start, before the cap existed. There is nothing left to be uncertain about, so a standard one-year quarterly cap contains three caplets, expiring at three, six and nine months. Counting four is a real and expensive error.

A floor is the mirror: a strip of floorlets, each a put on the same forward rate, paying Notional × max(K - r, 0) × tau. A lender with floating-rate assets buys floors for the same reason a borrower buys caps.

Why are caps quoted at a flat volatility?

One more market convention has to be stated, because it explains a puzzle every learner meets. Caps are quoted at a single flat volatility, one number applied to every caplet in the strip, chosen so that the resulting premium matches the market price. That flat number is not the volatility of any individual forward rate. The real market has a different volatility for each expiry, and reconstructing it from cap quotes is called stripping the caplet volatilities. To see the difference concretely: caplet volatilities of 26%, 22% and 19% on the curve in Worked example 1 give a cap premium of $74,356.85, and the single flat volatility that reproduces that same total is 21.786675%. The total agrees by construction. The individual caplets do not: the first is $2,429.47 too cheap at the flat volatility and the third is $2,597.86 too dear. Price a cap with a flat volatility and you have the right cap and the wrong caplets, which matters the moment you unwind part of it.

Cap minus floor is a payer swap

The identity that ties the rate-option world back to the swap world is worth deriving rather than memorising, because the derivation is two lines and it never leaves you.

Take one period. The caplet is N·tau·D·[F·N(d1) - K·N(d2)] and the floorlet is N·tau·D·[K·N(-d2) - F·N(-d1)]. Subtract:

caplet - floorlet = N·tau·D·[ F·(N(d1) + N(-d1)) - K·(N(d2) + N(-d2)) ]
                  = N·tau·D·(F - K)

because N(x) + N(-x) = 1 for any x. The volatility vanishes. Sum over the periods and the right-hand side is the present value of a payer swap on the same schedule at the same fixed rate K.

cap(K) - floor(K) = payer swap at K

Three things follow, and all three are used in practice.

The identity is model-free in the volatility. It holds whatever sigma is, and it holds if the two legs use different volatilities per caplet, provided the same volatility is used for the caplet and the floorlet of a given period. That makes it the best available test of a cap-pricing implementation: price both legs, subtract, and compare against a swap valued off the curve alone. In Worked example 1 the residual is $2.9 × 10⁻¹¹ on a $38,492.86 swap value, which is machine noise.

The at-the-money-forward strike falls straight out. Set K equal to the par forward-starting swap rate and the payer swap is worth zero, so the cap and the floor must cost the same. On the module's curve that strike is 4.962799%, at which the cap and the floor are each worth $55,005.22.

The zero-cost collar is the same identity read as a construction. Buy a cap at a strike above the forward, sell a floor at a strike below it, and choose the floor strike so that the two premiums cancel. The borrower pays nothing, is protected above the cap strike and gives up the benefit of any fall below the floor strike. Worked example 7 solves for that floor strike on an Indian rupee curve and finds a band of 43.79 basis points around a forward of 7.0084%, which is narrower than most people expect and is the honest answer to what "free protection" buys.

The instructive failure is also worth carrying. If the cap strike is set below the forward, no floor strike below the forward can fund it, because the cap is then more valuable than any such floor. Practice item P13 shows a 7.00% cap on a 7.0084% forward that no zero-cost collar can finance. Free protection has a boundary, and the boundary is the forward.