Range Accruals

The coupon on a range accrual depends on how many days a rate stays inside a range. The party receiving the coupon weighs its outlook on that count. The issuer values and hedges the digital options inside the note.

How the coupon is set by days inside the range

A range accrual sets its coupon by counting the days on which a reference rate is inside a range. The manual that the Board of Governors of the Federal Reserve System wrote for its examiners (February 1998, the "manual" below) covers this kind of note in its section on structured notes¹. The note is also called a range note or accrual note. The coupon accrues daily, at a rate set by a reference index. The note has two rates. It accrues the higher rate for the period when the index is inside the range, and the lower rate, which may be 0, when the index is outside. Notes have also been issued with indexes other than interest rates, such as currencies, commodities, and equities.

Four items need to be set.

The number of days inside the range divided by the number of observation days gives the share of days inside the range. The coupon for one period is the coupon for the days inside and the coupon for the days outside, weighted by this share. How far the rate falls outside the range does not matter. Only whether it is inside or outside sets the coupon¹. Some notes observe monthly or quarterly, or once during the term. With quarterly observation, only the position of the quarterly observation day affects the coupon¹.

There is also an example of the same idea using a stock index. The product pays the notional amount multiplied by the share of days over three months on which the Nikkei 225 is inside the range of 13,000 to 14,000. With a notional amount of 10 million US dollars and 70% of the days inside the range, the payoff is 7 million US dollars³.

A worked example of the coupon

The example uses a bond on which the issuer pays and the holder receives, with an interest rate as the reference. Consider a hypothetical example in which Company I holds a range accrual note with a notional amount of 10 billion yen for one year. Company I receives coupon every three months. The range for the reference rate is 0.50% to 2.00%. The rate on days inside the range is 3.0% a year, and the rate on days outside is 0%. Each period has 60 observation days, and each period is calculated as one quarter of a year. If no day is inside the range, the coupon for that period is 0.

The coupon for one period is set by the following formula.

$$\text{Coupon} = \text{Notional amount} \times \text{Rate inside the range} \times \frac{1}{4} \times \frac{\text{Days inside the range}}{\text{Observation days}}$$

A notional amount of 10 billion yen times 3.0% a year times one quarter is 75 million yen. This is the coupon for one period if the rate stays inside the range throughout.

Figure description

Company I receives a coupon of 75 million yen after 3 months, 56.25 million yen after 6 months and 37.5 million yen after 9 months. After 12 months there is no coupon to receive, because the reference rate was never inside the range. The length of each arrow is a guide to the size of the amount. Principal is not included.

The total coupon for the year is 168.75 million yen, or 1.6875% a year on the notional amount. For comparison, a bond with the same notional amount pays a fixed coupon of 1.2% a year. Its coupon is 120 million yen for the year.

The note matches this 120 million yen when the share of days inside the range is 40% (300 million yen × 40%). Under these conditions, a share above 40% beats the fixed bond and a share below 40% falls short of it. The next section sets out the difference for each rate path in a table.

How the coupon changes with rate moves

The manual summarizes the note this way. In a stable environment in which the reference index stays inside the range, the party receiving the coupon gets more than from other floating-rate notes. In a volatile environment in which the index moves outside the range, it gets less². The more days inside the range, the more coupon the note pays. The narrower the range, the larger the coupon enhancement over similar products². As in the example in the previous section, a note with a narrower range and a higher rate inside the range pays more if the rate stays inside the range, and less if it moves outside.

For the example note, the results for each rate path are as follows.

The loss when the rate moves outside the range is the same whether the reference rate rises above the top or falls below the bottom.

A combination of digital options

The party receiving the coupon is in the same position as one who has sold, on each observation day, a series of digital option calls struck at the top of the range and puts struck at the bottom¹. The definition and valuation of digital options are covered in Digital Options.

Rewritten as a formula for one period's coupon, this explanation becomes the following. Let $r_{\text{in}}$ be the rate inside the range, $r_{\text{out}}$ the rate outside, and $N$ the number of observation days.

$$\text{Coupon} = \text{Notional amount} \times \frac{1}{4} \times \left[\, r_{\text{in}} - (r_{\text{in}} - r_{\text{out}}) \times \frac{\text{Days above the top} + \text{Days below the bottom}}{N} \,\right]$$

The base is the coupon if the rate had been inside the range every day, and the difference between the two rates is subtracted only for the days outside the range. The amount subtracted corresponds to what the digital call and digital put that were sold pay from the seller to the buyer for each day. The payment for a day occurs only when the rate is either above the top or below the bottom. When the rate outside the range is 0, the amount paid is one day's share of the coupon at the rate inside the range (one period's coupon divided by the number of observation days). This formula restates the manual's explanation in terms of days, and it is not a formula that appears in the manual.

Each observation day has one pair of digital options, so the price is the sum of as many digitals as there are observation days. A product that pays by the share of days inside a range can be thought of as a set of digitals, one for each day of the period (the stock index example above)³. What the holder receives for selling the digitals shows up in the coupon as the enhancement to the rate inside the range.

There is another way to set the range, which is to sell barrier options instead of digitals. A down-and-out put sits at the bottom of the range and an up-and-out call at the top, and if the index touches a barrier on even one day, the coupon for that period is 0. The manual calls this form quite rare⁴.

Valuation and hedging

As the previous section showed, valuation starts from the price of a digital for each observation day. The first point is delta at the edge of the range. A digital option's payoff jumps from 0 to a fixed amount when the price crosses the strike. The delta of an FX digital option is proportional to 1 divided by the product of the square root of the time to expiry and the volatility. Approximating a digital with a call spread (a long and a short vanilla call with nearby strikes) smooths the payoff and avoids very large (potentially infinite) delta numbers⁵. This formula is for an FX digital.

Applying this property to the digital for each observation day, the delta of that day's digital becomes large when the observation day is near and the reference rate is near the top or bottom of the range. The seller of the digitals is the bond holder, and the issuer holds the opposite position. If the issuer adjusts its hedging position to match delta, a large rebalance would be needed each time the rate crosses an edge. This is an inference from the property above, and it is not a practice described in the sources.

The second point is dependence on the model. The price of a digital call is given by the change in the vanilla call price when the strike moves slightly, so the difference in volatility across strikes (the skew) enters the price. Consider a hypothetical example in which a stock has an expiry of 1 year, a current price equal to the strike, and volatility of 25%. Volatility moves 2.5 percentage points for each 10% move in the strike. The price with the skew is 55% of the payout. The price from a constant volatility that ignores the skew is 45%⁶. Because the price of the digital for each observation day changes with the volatility at each strike, the price of a range accrual also appears to depend on how the volatility smile is treated. This is an inference from the stock example. Differences in treatment can be one source of model risk.

For forms with an early redemption provision, a 2023 paper shows how to value a callable range accrual linked to the CMS spread using the least-squares Monte Carlo method⁷. The CMS spread is covered in CMS Spread Options.

Risks

The risk for the party receiving the coupon is that the reference rate stays outside the range and the coupon falls. The manual explains that because of the leverage in the sold digitals, notes of this type often move sharply in price and carry large price risk⁸.

The form of the risk changes with how the range is set. In a form observed every day, the coupon falls as more days land outside the range. In the barrier form described in the previous section, one day outside the range sets the coupon for the whole period to 0. The previous section also noted that some forms carry early redemption provisions.

Differences from similar products

What this article does not cover

The valuation model and delta formula for a note that references interest rates, differences by type of reference rate (a short-term rate or a swap rate, for example), and the contents of early redemption provisions are not covered on this page.

Notes

1. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual" (February 1998), Section 4040.1 "Structured Notes," pp. 4–5, "Range Notes." ↩ ↩² ↩³ ↩⁴ ↩⁵

2. The same Section 4040.1, p. 5, "Range Notes." ↩ ↩² ↩³

3. Martin Haugh, Columbia University, "Black-Scholes and the Volatility Surface" (IEOR E4707, 2013), Section 6 "Pricing Exotics," Example 4 "Pricing a Range Accrual" (including the hint). ↩ ↩²

4. The same Section 4040.1, p. 5, "Range Notes." The barrier form and the "quite rare" assessment. ↩

5. OpenGamma, "Digital Forex Options," Section 2 (the delta formula) and Section 3, "Call spread approximation," "avoiding huge (potentially infinite) delta numbers" (pp. 1–3). ↩

6. The same Haugh lecture note, Section 6 "Pricing a Digital Option" and Example 3 "Pricing a digital." ↩

7. He, Hsieh, Huang, and Lin, "Valuation of callable range accrual linked to CMS Spread under generalized swap market model," International Review of Financial Analysis, vol. 90 (2023), abstract. ↩

8. The same Section 4040.1, p. 5, "Range Notes," final paragraph. ↩