Caps and Floors

This trade comes up when a company that borrows at a floating rate wants to set a ceiling on its interest payments, or when a financial institution that handles floating-rate structured deposits hedges the upper and lower limits built into them.

Receiving the difference from the strike rate

An interest rate cap (cap) is an option under which the seller pays the buyer the difference when a rate is above a rate agreed in advance. An interest rate floor (floor) pays the buyer the difference when the rate is below the agreed rate¹. This page calls the agreed rate the strike rate. The rate referenced is a benchmark rate for floating rates, such as TIBOR (Tokyo Interbank Offered Rate). Both caps and floors are traded as OTC derivatives¹. Calls, puts, and other option basics are covered in Option Basics: Calls and Puts.

Consider a hypothetical example in which Company A buys a cap from Bank B. The terms are a notional amount of 10 billion yen, a term of 5 years, annual interest payments, and a strike rate of 1.50% a year. Company A pays Bank B a price (the premium) in full at the trade date. In each year in which the floating rate is above 1.50%, Bank B pays Company A the difference multiplied by 10 billion yen and by the length of the period (1 year). The notional amount of 10 billion yen itself is not exchanged.

With a cap, however high the floating rate rises, the buyer receives the part above the strike rate from the seller. A cap sets an upper limit on changes in the rate, and a floor sets a lower limit¹.

Payments caplet by caplet

The payments on a cap are set separately for each interest period. If the floating rate $L$ set at the start of a period is above the strike rate $K$, the cap buyer receives the following amount at the end of the period².

$$\text{Payment received} = M \times \delta \times \max(L - K, 0)$$

$M$ is the notional amount and $\delta$ is the length of the period (in years). As with other interest rate trades, the way days are converted into years follows the day count convention³. An option that covers the payment for just one period is called a caplet, and one period of a floor is called a floorlet. A cap can be seen as a bundle of caplets, and a floor as a bundle of floorlets⁴. The floorlet pays $\max(K - L, 0)$ multiplied by the notional amount and the length of the period².

This formula assumes a benchmark rate, like TIBOR, that is set at the start of the period. For benchmark rates set at the end of the period, such as TONA (Tokyo Overnight Average Rate) compounded in arrears, the rate is not determined the way this formula assumes⁵. The terms of caps and floors that reference a benchmark rate set in arrears are not explained here.

In this example, every period is 1 year long, and business-day adjustments to the fixing and payment dates are ignored. Suppose that for Company A's cap, the floating rate is set at 1.0% in 1 year, 1.5% in 2 years, 2.0% in 3 years, and 2.5% in 4 years. The rate for the first year is already set at the trade date, so the example covers the 4 caplets from year 1 to year 5.

For the period from year 3 to year 4, Company A receives 10 billion yen × 1 year × (2.0% − 1.5%) = 50 million yen. For the periods set at 1.0% and 1.5%, the difference is zero or below, so nothing is received. The premium is taken as about 62 million yen, the price under the Black model found in the section on the Black and Bachelier models below.

Figure description

Payments seen by Company A, the cap buyer. Money moves only now, when the premium is paid, and on the payment dates of the periods in which the floating rate was above the strike rate (years 4 and 5). Nothing happens in years 2 and 3. The amounts received are the same as in the table in the text. Whether the total received exceeds the premium depends on how rates move.

In this example, the total received of 150 million yen exceeds the premium. If the floating rate stayed at 1.5% or below for all 5 years, however, Company A would receive nothing, and the premium of 62 million yen would simply be a cost. The profit and loss of Bank B, the seller, is the opposite of Company A's: if rates do not rise, Bank B gains the premium, and the higher rates rise, the more it pays.

Using a cap to limit interest payments on a loan

A company borrowing at a floating rate can put a ceiling on its interest payments by buying a cap⁶. This is because the amount by which the floating rate exceeds the strike rate is received from the cap and can go toward the interest on the loan. In exchange, the cost is higher than a floating-rate loan without a cap by the amount of the premium⁶.

Suppose Company A has borrowed 10 billion yen at a floating rate for 5 years and has bought the cap from the previous section. Its interest payments for years 2 to 5 are as follows.

Even in the years when the floating rate rises to 2.0% or 2.5%, the net interest stops at 150 million yen (the equivalent of 1.50%). In years when the floating rate falls, the interest goes down, just as it would on a loan without a cap. If rates do not rise, the premium of 62 million yen is an extra cost compared with not buying the cap. Some floating-rate loans come with a cap built in, and their rate is higher than that of an ordinary floating-rate loan by the amount of the premium⁶.

A floor is used as a lower limit by the party that receives a floating rate. Some structured deposits put a lower limit on their floating rate, and the floating-rate Japanese government bonds for retail investors are designed so that their rate does not fall below 0%. These products can be seen as having a floor built in⁷. Financial institutions that offer structured deposits hedge the floors and caps built into them by trading caps and floors⁷.

The Black and Bachelier models

The price of a cap is found by pricing each caplet and adding the prices up⁹. A caplet is an option whose underlying asset is the floating rate set at the start of the period. The valuation uses the forward rate $F$ for that period, derived from the yield curve. The market uses two models that make different assumptions about how the forward rate moves¹⁰.

The Black model assumes that the rate of change of the forward rate follows a normal distribution. The forward rate then follows a lognormal distribution and does not take values at or below zero¹⁰. Let $T$ be the number of years until the floating rate is set, $P(0, T+\delta)$ the discount factor to the payment date, and $\sigma_B$ the Black volatility. The price of a caplet is then⁹:

$$\text{Caplet} = M \delta P(0, T+\delta) \left[ F \Phi(d_1) - K \Phi(d_2) \right]$$ $$d_1 = \frac{\ln(F/K) + \sigma_B^2 T/2}{\sigma_B \sqrt{T}}$$ $$d_2 = d_1 - \sigma_B \sqrt{T}$$

$\Phi$ is the cumulative distribution function of the standard normal distribution. The price of a floorlet is $M \delta P(0, T+\delta) \left[ K \Phi(-d_2) - F \Phi(-d_1) \right]$¹¹. The market convention is to quote the price of a cap as the sum of Black model prices using a single volatility common to all the caplets⁹.

The Bachelier model (normal model) assumes that the forward rate itself follows a normal distribution. Because it treats rates as equally likely to move up or down, the forward rate can also take values at or below zero. The model was proposed by Louis Bachelier in 1900, and in practice it seems to be most often called the normal model¹². The Black model cannot produce a price when the forward rate is at or below zero, so when rates were negative, information vendors stopped providing Black volatilities, and the Bachelier model came into wide use¹⁰. The Bachelier model volatility (normal volatility) $\sigma_N$ is expressed as a change in the rate and quoted in bp. The price of a caplet is then¹³:

$$\text{Caplet} = M \delta P(0, T+\delta) \left[ (F - K) \Phi(d) + \sigma_N \sqrt{T} \varphi(d) \right]$$ $$d = \frac{F - K}{\sigma_N \sqrt{T}}$$

$\varphi$ is the probability density function of the standard normal distribution. In practice, the volatilities of the two models are linked by the approximation $\sigma_N \approx F \sigma_B$¹⁴.

Now value Company A's cap from the first example using the same yield curve as in the example on the swap valuation page. The Black volatility is 30% and the normal volatility is 40bp. With a forward rate of 1.34%, 30% × 1.34% ≈ 0.40%, so the two volatilities roughly correspond.

The total is 62,499,186 yen under the Black model and 59,061,714 yen under the Bachelier model¹⁵. For the caplet covering years 1 to 2, the forward rate is low at 0.90%, and 30% × 0.90% ≈ 27bp is smaller than 40bp, so the Bachelier model gives the higher price.

Deep diveNegative rates and the shifted Black model

The Black model formula contains $\ln(F/K)$. When $F$ is zero or below, the logarithm cannot be taken, so no price can be calculated¹³. Before rates turned negative, the Black model was also the standard for swaptions¹⁰.

Another approach adds a fixed amount to the forward rate and the strike rate to make them positive before applying the Black model. This is called the shifted lognormal model (shifted Black). How much to add is said to depend largely on market convention. As of October 2021, practitioners use both the Bachelier model and the shifted Black model¹⁶.

How the value changes with rates and volatility

The value of a cap rises as the forward rate rises and as volatility increases. The table below revalues Company A's cap from the previous section, changing one condition at a time. To move the yield curve up or down, all the market rates it is built from (the overnight rate and the fixed rates of OIS) are moved by the same amount and the curve is rebuilt. As a result, the forward rates move by slightly different amounts than the rates that were shifted.

As in the previous section, the amounts are values calculated with the simulator¹⁵.

Raising the yield curve and raising volatility both benefit Company A, the buyer, and cost Bank B, the seller, the same amount. If rates fall or volatility decreases, the buyer loses and the seller gains. The value of a floor rises as the forward rate falls. When the yield curve is lowered by 1.5 percentage points, the forward rates fall below zero, so the Black model cannot calculate a price, but the Bachelier model gives 763,819 yen.

Cap minus floor is an interest rate swap

If the strike rate, term, payment dates, and notional amount are the same, the price of a cap equals the price of a floor plus the value of an interest rate swap⁸. The interest rate swap here pays the strike rate as the fixed rate and receives the floating rate. This is because buying a cap and selling a floor means that, in every period, you receive the difference if the floating rate is above the strike rate and pay the difference if it is below.

$$\max(L - K, 0) - \max(K - L, 0) = L - K$$

This relationship is the put-call parity between caps and floors. It follows from the payments themselves, so it holds whichever model is used. In Company A's first example, a floor with the same terms is priced at 167,321,889 yen under the Black model and 163,884,417 yen under the Bachelier model. Subtracting these from the cap prices gives −104,822,703 yen under both models, which matches the value of an interest rate swap paying a fixed rate of 1.50% from year 1 to year 5¹⁵. The forward rates (0.90% to 1.34%) are below 1.50%, so the value is negative for the party paying the fixed rate of 1.50%.

Risks and trading practice

The cap buyer's loss is limited to the premium paid. The seller pays more the further the floating rate rises above the strike rate, so its loss can far exceed the premium received. As the section "How the value changes with rates and volatility" shows, the market value moves with the yield curve and volatility.

When the premium is paid at the trade date, as in this example, the cap buyer pays the premium first and receives the difference from the seller at the end of each period. The risk that the seller becomes unable to pay is the counterparty's credit risk. Contract documentation and whether a trade goes through a central counterparty are not explained here.

Caps and floors are traded over the counter, and buyers ask dealers for prices after setting the terms. For example, there is an order that reads: "I would like a bid on a floor with a notional amount of 100 million yen, a term of 10 years, semiannual frequency, a strike price of 0%, and DTIBOR (6-month) as the reference rate" (our translation)⁷.

As of August 2021, floors are used to hedge structured deposits and similar products, so floors with a strike rate of 0% tend to be traded, while caps are traded at a variety of strike rates⁷. As of the same date, for analysis of interest rate options, Swaptions are used more than caps and floors. Swaptions are relatively more liquid, and their data is easier to obtain¹⁷.

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: swaptions), Ministry of Finance, "ファイナンス" (Finance), August 2021 issue (in Japanese), "６．キャップとフロア" (6. Caps and floors), pages 58–59. ↩ ↩² ↩³

2. Ernesto Mordecki, "24. Pricing Fixed Income Derivatives through Black's Formula," slides 18–19 ("24e. Interest rate options"). ↩ ↩²

3. European Banking Federation, interest rate transactions annex (Edition 2020), Sections 3(3) and 4(2), page 2. The floating amount is calculated with a day count fraction. ↩

4. 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: swaptions), Ministry of Finance, "ファイナンス" (Finance), August 2021 issue (in Japanese), "６．３ キャップレットとフロアレット" (6.3 Caplets and floorlets), page 59. ↩

5. Bank of Japan Financial Markets Department and Financial System and Bank Examination Department, and Financial Services Agency Strategy Development and Management Bureau and Supervision Bureau, "円LIBOR移行対応の振り返りと今後の取り組み" (Bank of Japan Review 2022-J-8, May 2022, in Japanese), page 3, Chart 2 "代替金利指標の３つの選択肢" (Three options for alternative benchmarks), the row "金利決定のタイミング" (Timing of rate fixing). It lists TIBOR as "前決め" (fixed in advance) and TONA compounded in arrears as "後決め" (fixed in arrears). An English version is available: "Review of JPY LIBOR Transition and Future Initiatives" (Bank of Japan Review 2022-E-4, May 2022), page 3, Chart 2. ↩

6. Central Council for Financial Services Information, "3-5 オプション取引活用の例（やさしいデリバティブ）" (Examples of using option trading, Easy Derivatives, in Japanese), "キャップ付ローン" (Loans with a cap). ↩ ↩² ↩³

7. 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: swaptions), Ministry of Finance, "ファイナンス" (Finance), August 2021 issue (in Japanese), "６．キャップとフロア" (6. Caps and floors), page 59. ↩ ↩² ↩³ ↩⁴

8. Ernesto Mordecki, "24. Pricing Fixed Income Derivatives through Black's Formula," slide 20 ("Remarks"). ↩ ↩²

9. Martin Haugh, "Market Models" (Term Structure Models: IEOR E4710, 2010), page 2, formula (3) under "Black's Formula for Caplets" and the paragraph after it. The document's $\delta Z_t^{T+\delta}$ corresponds to $\delta P(0, T+\delta)$ on this page, and its $L(t, T)$ to $F$ (the formula is per unit of notional). ↩ ↩² ↩³

10. Takahiro Hattori, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to swaptions (models): the normal (Black) model and the SABR model), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), "２．１ 正規分布と対数正規分布" (2.1 Normal and lognormal distributions) and "２．２ 数式を用いたモデルの概要" (2.2 Overview of the models using formulas), pages 56–57. The document explains the models using the swap rate of a swaption. ↩ ↩² ↩³ ↩⁴

11. This formula is derived from the caplet formula and the relationship caplet − floorlet = $M \delta P(0, T+\delta)(F - K)$ (see the section "Cap minus floor is an interest rate swap"). Page 2 of Haugh's "Market Models" says that similar formulas exist for floorlets. ↩

12. Takahiro Hattori, "スワップション入門（モデル編）" (Introduction to swaptions (models)), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), pages 57–58 and note 5 on page 58. ↩

13. Takahiro Hattori, "スワップション入門（モデル編）" (Introduction to swaptions (models)), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), "３．２ ノーマルおよびブラック・モデルにおけるオプションの公式（価格式）" (3.2 Option pricing formulas under the normal and Black models), pages 60–61, formula (4), note 12, and formula (5). The document's formulas are for a payer swaption; replacing the annuity $A$ with the one-period $\delta P(0, T+\delta)$ gives the caplet formula. ↩ ↩²

14. Takahiro Hattori, "スワップション入門（モデル編）" (Introduction to swaptions (models)), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), "３．１ データでみるノーマル・ボルとブラック・ボルの違い" (3.1 The difference between normal vol and Black vol in the data), pages 59–60, formula (3). ↩

15. Amounts are the simulator's results rounded to the nearest yen. ↩ ↩² ↩³

16. Takahiro Hattori, "スワップション入門（モデル編）" (Introduction to swaptions (models)), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), "２．４ シフティッド・ログノーマル・モデル" (2.4 The shifted lognormal model), pages 58–59. ↩

17. 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: swaptions), Ministry of Finance, "ファイナンス" (Finance), August 2021 issue (in Japanese), "６．キャップとフロア" (6. Caps and floors), page 58. ↩