Inflation Caps and Floors

Companies and pension funds with debts or income linked to prices use this trade when they want to hedge only one side, either rising or falling prices.

A right that puts a ceiling and a floor on inflation

An inflation cap is a right to receive from the seller an amount equal to the notional amount times the part by which the rate of inflation exceeds the strike. An inflation floor is the reverse: a right to receive the notional amount times the part by which the rate falls below the strike¹. The buyer pays a premium at the start in exchange for the right¹. Inflation is measured with a price index such as the consumer price index (CPI). A trade that, like a swap, exchanges a fixed rate for the rate of inflation is covered under Inflation Swaps.

The strike is set as a rate of inflation. When the rate of inflation is below the strike of a cap, or above the strike of a floor, there is no payoff, and the buyer's cost is the premium alone.

Consider a hypothetical example in which Company U buys the following inflation cap from Bank I. Company U owes, in five years, an amount equal to 10 billion yen times the change in a price index.

One payment at maturity, or a payment each year

Inflation caps and floors come in two forms that differ in how the payoff is determined.

One payment at maturity

In the form that pays once at maturity (the zero-coupon form), the payoff is determined by the change in the price index from the start to maturity. At maturity, the seller of a cap pays the notional amount times the amount by which the ratio of index levels exceeds the strike compounded over the term. The seller of a floor pays in the opposite direction, the amount by which the compounded strike exceeds the ratio of index levels². The compounded strike is the annual strike compounded over the number of years; in the example it is 1.03 multiplied five times. The source uses this form because it makes the pricing formulas easier to build².

$$\text{Cap payoff} = N \times \max\left(\frac{I_T}{I_0} - (1 + k)^T,\ 0\right)$$ $$\text{Floor payoff} = N \times \max\left((1 + k)^T - \frac{I_T}{I_0},\ 0\right)$$

$N$ is the notional amount, $k$ is the annual strike, $T$ is the term in years, and $I_0$ and $I_T$ are the price index at the start and at maturity. $\max(a, 0)$ is the larger of $a$ and 0.

For the example cap, $(1 + k)^T$ is about 1.1593. If the index rises to 120 at maturity, the ratio of index levels is 1.20, and Company U receives the 10 billion yen notional times the difference between 1.20 and about 1.1593, which is about 407.26 million yen³. If the index rises only to 110, the ratio is 1.10, which does not reach about 1.1593, and there is no payoff. Suppose Company U also buys a floor with the same notional amount of 10 billion yen, a term of 5 years and a strike of 0%. If the index falls to 95 it receives 500 million yen, and if the index rises to 105 there is no payoff.

A payment each year

The annual form bundles one-year options lined up at each yearly expiry. A one-year option is called a caplet (a floorlet for a floor), and a cap (floor) is a bundle of them with the same strike and notional amount whose expiries are one year apart¹. A caplet pays the notional amount times the part by which that year's rate of inflation exceeds the strike. The rate for the year is the ratio of the price index at the end of the year to the index at the start, minus 1⁴. Figure 1 shows the payoff of a caplet with a strike of 3% a year.

Figure description

The payoff, before the premium is paid, of one year (a caplet) of an inflation cap with a strike of 3% a year. The horizontal axis is inflation over one year, and the vertical axis is the payoff as a percentage of the notional amount. If inflation is 3% or lower, the payoff is 0, and above 3% it grows by the amount inflation exceeds 3%. At 5% the payoff is 2% of the notional amount, which is 200 million yen on a notional amount of 10 billion yen. The payoff of a floor has the mirror-image shape around the strike.

Consider a hypothetical example in which Company U buys an annual cap with a notional amount of 10 billion yen, a term of 3 years and a strike of 3% a year, for a premium of 60 million yen. Suppose inflation in the three years is 2%, 5% and 4%. In year 1 it does not reach 3% and there is no payoff. In year 2 Company U receives 2% (5% − 3%), or 200 million yen, and in year 3 it receives 1% (4% − 3%), or 100 million yen (Figure 2).

Figure description

The cash flows of Company U, which bought an annual inflation cap with a notional amount of 10 billion yen, a term of 3 years and a strike of 3% a year. At the start it pays a premium of 60 million yen. If inflation in the three years is 2%, 5% and 4%, there is no payoff after 1 year because inflation does not reach the strike. After 2 years it receives 200 million yen, and after 3 years it receives 100 million yen. The length of each arrow is a rough guide to the size of the amount. The cash flows of the seller, Bank I, are in the opposite direction to those of Company U.

In the actual market, caps and floors are priced as bundles of caplets and floorlets with different expiries. The values of individual caplets are not quoted directly; they are obtained by interpolating from the values of caps with a limited set of maturities⁵.

Reference lag and the payoff index

An inflation cap refers to a rate of inflation calculated from a price index. The features are as follows.

Because compiling a price index takes time, the index on the maturity date cannot be used to calculate the payoff⁸. This delay (the reference lag) arises for the same reason as in inflation swaps.

Strike levels, and what they reveal

The strike is set as a level of annual inflation, and several levels are traded for each maturity. In US zero-coupon quotes from October 2009 to March 2012, caps had strikes of 1% to 6% a year and floors had strikes of −2% to 3% a year, with quotes for maturities of 1, 3, 5, 7 and 10 years⁷. Floors have negative strikes, so rights can be placed on the side where prices fall as well. The data come from BGC Partners, which provided quotes for this period. There are also anecdotal reports that the most heavily traded contracts are the 0% floor and the 5% cap⁷.

The method for finding the distribution approximates inflation as taking only whole-number values. A combination of caps with strikes one percentage point apart (buy one, sell two, buy one) is equivalent to a product that pays one unit of notional amount only when inflation ends up exactly at that level. If the values of such combinations are collected for each strike at the same maturity, the distribution of inflation that the market implies can be obtained. The method does not assume a normal distribution or any similar form⁹.

What changes the value

The value of a cap rises as inflation becomes more likely to exceed the strike, and the value of a floor rises as inflation becomes more likely to fall below it. The mean of the distribution of inflation obtained from option prices equals the forward inflation swap rate¹⁰. If that rate rises, a cap with the same strike gains in value and a floor loses value.

Another factor is the size of the fluctuations in inflation, that is, volatility. The higher the volatility, the more often inflation ends far above the strike for a cap, or far below it for a floor. Because the payoff never goes below zero, both gain in value. The premium of the floor embedded in the new type of Japanese inflation-linked bond, which guarantees the principal, rose in periods of high volatility such as the COVID-19 pandemic¹¹.

Handling volatility, and models

A zero-coupon caplet can be valued with a formula similar to the Black model, using the volatility of the forward price index. The volatility in the formula is a market value for each strike and maturity¹². Because it is backed out of market quotes, it takes a shape in which the value differs by strike (a volatility smile). To reproduce this shape in a single model, models with stochastic volatility, or with a leverage function that captures differences across strikes, have been proposed¹³. There is no single established valuation method.

An annual caplet deals with the movement of the ratio to the previous year's index, so its price depends on the volatilities of two forward price indexes and their correlation, the volatility of nominal interest rates, and other inputs¹⁴. For a market model that assumes forward price indexes move lognormally, formulas using an approximation have been derived¹⁴.

Deep diveThe formulas for zero-coupon caps and floors

If the forward ratio of index levels $I_T / I_0$ is assumed to move lognormally, the values of the cap and the floor can be written as follows¹².

$$\text{Cap} = N \cdot P \cdot \left\{F\,\Phi(d_1) - K\,\Phi(d_2)\right\}$$ $$\text{Floor} = N \cdot P \cdot \left\{K\,\Phi(-d_2) - F\,\Phi(-d_1)\right\}$$ $$d_1 = \frac{\ln(F / K) + \sigma^2 \tau / 2}{\sigma\sqrt{\tau}}, \quad d_2 = d_1 - \sigma\sqrt{\tau}$$

Here, $F$ is the current value of the forward ratio $I_T / I_0$ of index levels. The strike $K$ is $(1 + k)^T$ compounded over the term. The discount factor $P$ runs to the payment date. The market volatility is $\sigma$, and $\tau$ is the number of years until the price index is fixed. The cumulative distribution function of the standard normal distribution is $\Phi$. The symbols are adapted to those in the main text by normalizing the notional amount and the base index in the source.

What they are used for

Caps and floors suit anyone with obligations or income linked to prices who wants to fix only one side of that exposure. Consider Company U's obligation and what happens if it buys the cap. The obligation is 10 billion yen times the ratio of the price index over the 5 years. If the index rises to 120, the obligation is 12 billion yen, but Company U receives about 407 million yen from the cap, and adding the 100 million yen premium gives an effective burden of about 11.693 billion yen, against 12 billion yen without the cap. If the index rises only to 110, the obligation is 11 billion yen, the cap pays nothing, and the burden including the premium is 11.1 billion yen. Without the cap the burden would have been 11 billion yen, so Company U pays the premium on top.

If instead Company U holds a contract under which it will receive 10 billion yen, linked to the price index, in 5 years, it can buy a floor to limit the drop in its receipt if the index falls. With a floor with a strike of 0% and a premium of 50 million yen, if the index falls to 95 the contract pays 9.5 billion yen, the floor pays 500 million yen, and after deducting the premium the total is 9.95 billion yen. If the index rises to 105, the contract pays 10.5 billion yen, the floor pays nothing, and after deducting the premium the total is 10.45 billion yen. Without the floor the receipt would be 10.5 billion yen, so the premium reduces it.

Pensions are one example of a user. According to a 2012 Bank of England article, the United Kingdom has pensions whose liabilities grow with the rise in retail prices, up to a maximum of 5%, and do not fall (limited price indexation). The article says these give an incentive to buy a 0% RPI floor and sell a 5% cap¹⁵. The long maturities of 20 to 30 years in the UK market were heavily affected by trades of this kind by pension funds¹⁶. In the United States around 2012, mutual funds and insurance companies were buyers and sellers of caps and floors¹⁷. Some users also used them to hedge structured products¹⁸.

Floors embedded in bonds and contracts are another form of use. Lecture materials from January 2006 report that inflation-linked bonds issued by banks carried a floor with a strike of 0%. The floor sets a lower limit of zero on inflation so that the coupon stays positive even when prices fall, and the materials say that a floor with a low strike is the most heavily traded inflation option¹⁹. A 2012 paper explains that US Treasury Inflation-Protected Securities (TIPS) do not repay principal below par even if prices fall, and that this feature creates an option-like shape²⁰. The principal guarantee (floor) on Japanese inflation-linked bonds is covered in Inflation-Indexed JGBs and BEI.

Risks

The buyer's loss is limited to the premium. The seller is obliged to pay the amount the buyer receives. For a cap, that amount grows as prices rise.

The trades are over-the-counter derivatives that do not go through an exchange, and liquidity is thin. A 2012 Bank of England article says that the inflation option markets in the UK, the US and the euro area are relatively young and small, that trading is sporadic, and that quotes may be slow to react to changes in the market¹⁶. Few values are available by maturity, so interpolation is needed, and there is no single valuation method.

The reference lag of the price index is also a risk. The payoff is determined by the index some months before maturity, so price movements close to maturity are not reflected in the payoff.

The buyer also bears the risk that the seller cannot pay (counterparty credit risk). In the US zero-coupon trades used in the 2012 paper, the buyer received collateral to guard against this risk²¹.

Notes

1. Tom Smith, "Option-implied probability distributions for future inflation" (Bank of England, Quarterly Bulletin 2012 Q3), the box "Caps, caplets, floorlets and floors" on page 226. ↩ ↩² ↩³

2. Kitsul and Wright, "The Economics of Options-Implied Inflation Probability Density Functions" (NBER Working Paper 18195, June 2012), page 3, "2 Inflation Caps and Floors". ↩ ↩²

3. Multiplying 1.03 five times gives about 1.159274, and the payoff is 10 billion yen times (1.20 − 1.159274), which is 407,259,257 yen. The text and the table round to the nearest 10,000 yen. ↩

4. Fabio Mercurio, "Pricing inflation-indexed options with stochastic volatility" (Winter School lecture, January 2006), slides 18 and 24. In the source's formula, the payoff is multiplied by the length of the year's period as well as the notional amount. The text omits this by treating the period as one year. ↩

5. Smith (2012 Q3), page 225. ↩

6. Smith (2012 Q3), page 225. As of 2012, the lag is 2 months for the UK and 3 months for the US and the euro area. Kitsul and Wright (June 2012), footnote 6 on page 10, says about 2 months for the US. ↩

7. Kitsul and Wright (June 2012), page 4 and its footnote 2. The data are daily quotes provided by BGC Partners from 6 October 2009 to 27 March 2012. The statement in footnote 2 is described as "anecdotal reports". ↩ ↩² ↩³

8. Kitsul and Wright (June 2012), footnote 6 on page 10. The source gives "delays in compiling CPI data" as the reason. ↩

9. Kitsul and Wright (June 2012), page 5. The source assumes the approximation that inflation takes only whole-number values. ↩

10. Smith (2012 Q3), page 227. ↩

11. Takahiro Hattori, "物価連動国債入札－発展編（フロア・オプション、流動性等）－" (Inflation-linked JGB auctions: advanced edition (floor option, liquidity, etc.)), Ministry of Finance, "ファイナンス" (Finance), March 2024 issue (in Japanese), "２.３　オプションという観点でみた元本保証" (2.3 The principal guarantee seen as an option). ↩

12. Orcan Ögetbil and Bernhard Hientzsch, "Inflation Models with Correlation and Skew" (arXiv:2405.05101, May 2024), section 3.1 (pages 5–6). The source's formulas write the base index and the notional amount with separate symbols; the text expresses them with the ratio $I_T / I_0$. ↩ ↩²

13. Ögetbil and Hientzsch (May 2024), the abstract and section 1 (pages 1–2). The source is on the side that proposes these models. ↩

14. Mercurio (January 2006), slides 19 to 23. ↩ ↩²

15. Smith (2012 Q3), footnote 5 on page 225. ↩

16. Smith (2012 Q3), page 225. ↩ ↩²

17. Kitsul and Wright (June 2012), page 4. The source says that filings with the US Securities and Exchange Commission (SEC) show that mutual funds and insurance companies take part. ↩

18. Smith (2012 Q3), "Table A" on page 225. ↩

19. Mercurio (January 2006), slide 4. ↩

20. Kitsul and Wright (June 2012), page 7, "2.1 The Embedded Deflation Option in TIPS". ↩

21. Kitsul and Wright (June 2012), page 3. This describes the US dollar quote data that the source deals with. ↩