An interest rate cap has several exercise dates, so it can be seen as a combination of several options. One of these options taken out alone is a caplet, and a cap is a portfolio of caplets. Once the caplet price is known, the cap price follows, so pricing theory is discussed in terms of caplets1.

Let δ\delta be the length of the period, LL the floating rate fixed at the start of the period, and KK the strike rate. The payoff per unit of notional is δ max⁡(L−K, 0)\delta\,\max(L - K,\,0) at the end of the period2.

Consider a hypothetical example in which a caplet has a notional of 10 billion yen, a strike rate of 2.0%, and a period of 0.5 years. If the floating rate is fixed at 2.5%, the payoff is 10 billion yen × 0.5 × (2.5% − 2.0%) = 0.025 billion yen, that is, 25 million yen. If the floating rate is fixed at 2.0% or lower, the payoff is 0, and the buyer's cost is only the premium paid.

In the market, the price of a cap is quoted by adding up caplet prices from the Black formula, one caplet at a time3. In the LIBOR market model, each caplet is priced under the forward measure of its own payment date, and the result agrees with the Black model formula. The corresponding option on a floor is a floorlet1.

Notes

  1. Takahiro Hattori, "債券（金利）オプション入門―スワップションについて―" (Introduction to bond (interest rate) options: on swaptions), Ministry of Finance, "Finance", August 2021 (in Japanese), section 6.3, "Caplets and floorlets" (PDF page 11). The source has one spelling variant ("キャプレット"), which is aligned with this site's headword. ↩ ↩2

  2. Martin Haugh, "Market Models" (Columbia University, Term Structure Models: IEOR E4710, 2010), page 2, "Black's Formula for Caplets". The payoff at time T+δT + \delta is δ(L(T,T)−K)+\delta(L(T,T) - K)^+. The text uses the form for a notional of 1 with the symbols replaced. The example is calculated from this formula. ↩

  3. Haugh, "Market Models", page 2 (a cap price is the sum of caplet prices, and the market convention is to quote prices with Black's formula) and page 8, equations (25) and (26) (Black's formula for each caplet is derived in the LMM). ↩