Snowballs

Snowballs come up when studying how the coupons of structured notes work, and when learning how rate exotics are valued and their risks managed. What sets a snowball apart from a floating-rate note is that its coupon depends on the previous period's coupon.

How the coupon is set

A snowball sets each period's coupon from the previous period's coupon and a floating rate¹. The first period has no previous period, so its coupon is a fixed rate agreed in advance. From the second period on, the coupon is the previous period's coupon plus an add-on, minus that period's floating rate².

Let $C_n$ be the rate for period $n$ ($C_1$ is the fixed rate for period 1), $A_n$ the add-on for period $n$, and $r_n$ the floating rate for period $n$. From period 2 on ($n \ge 2$), the rate is:

$$C_n = \max(C_{n-1} + A_n - r_n,\ 0)$$

$\max(\cdot,\ 0)$ replaces a negative value inside the parentheses with 0, and this is the 0% floor on the coupon. The form of the formula follows the example in the 2005 paper by Bender et al. That example uses LIBOR as the floating rate and sets the coupon as $(\text{previous coupon} + A - \text{LIBOR})^+$, which keeps only the positive part². The LIBOR used in the example is a benchmark that ceased publication in September 2024.

In the example by Bender et al., the add-on $A_n$ is a value set in the contract and grows by a fixed step each year. The first year pays a fixed rate, and coupons are paid every six months².

The floating rate is the rate observed at the start of the period, and the coupon is paid at the end of that period². The right-hand side of the formula contains the previous period's coupon, so every coupon is determined by a chain that runs back to the first period.

An example of the payments

Consider a hypothetical example in which Company J holds a snowball note with a face value of 10 billion yen, a term of 5 years, and annual coupons. The rate for period 1 is a fixed 3.0%, and the add-on is 2.0% each period. The rate from period 2 on is set by the formula above. Two cases are assumed, one in which the floating rate falls and one in which it rises.

The first table is the case where the floating rate falls (1.0%, 0.8%, 0.5%, then 0.5%).

The second table is the case where the floating rate rises (2.0%, 3.0%, 3.5%, then 4.0%).

In the falling case, the rate for year 5 is 8.2%: the previous rate of 6.7%, plus 2.0%, minus the floating rate of 0.5%. In the rising case, the previous rate of 0.5% plus 2.0% minus the floating rate of 4.0% gives −1.5%, which the floor turns into 0%.

For comparison, consider a note that also has a fixed 3.0% in year 1 but pays each period's floating rate as is from year 2. Its coupons over 5 years total 300 + 100 + 80 + 50 + 50 = 580 million yen in the falling case and 300 + 200 + 300 + 350 + 400 = 1,550 million yen in the rising case. In this example, the snowball's coupons total 2,130 million yen more than the floating-rate note's in the falling case and 700 million yen less in the rising case. The comparison covers only the total coupons. It excludes the redemption of face value, present value, and the issuer's credit.

Figure description

Coupons received by Company J when the floating rate falls. The coupon grows every period: 300, 400, 520, 670, and 820 million yen in years 1 to 5. Redemption of face value is not included. The length of each arrow is a rough guide to the amount.

Figure description

Coupons received by Company J when the floating rate rises. The coupons are 300, 300, 200, 50, and 0 million yen in years 1 to 5. They fall from year 3, and in year 5 nothing is received because of the 0% floor. Redemption of face value is not included. The length of each arrow is a rough guide to the amount.

How path dependence works

The coupon on a floating-rate note depends only on that period's floating rate. A snowball's coupon includes the previous period's coupon, so it depends on how rates have moved up to that point. This is called path dependence. The general properties of path-dependent products are covered in Overview of Exotic Options.

As long as the floor does not bind, each period's rate equals the period-1 rate $C_1$ plus the differences for each period (the add-on minus the floating rate, $A_k - r_k$).

Deep diveWriting each period's rate as a sum of differences

Without the floor, $C_n = C_{n-1} + A_n - r_n$. Substituting this in turn from period 2 through period $n$ gives:

$$C_n = C_1 + \sum_{k=2}^{n} (A_k - r_k)$$

In the falling case of the example, the rate for year 5 is $C_1 = 3.0\%$ plus the differences of 1.0%, 1.2%, 1.5%, and 1.5%, which is 8.2%.

What follows applies while the floor does not bind. A period's difference feeds into the coupons of all later periods, not just its own. A period in which the floating rate is below the add-on pushes up later coupons, and a period in which it is above pushes them down. Compare two paths under the same contract, where the floating rate in a single period is 0.1 percentage points higher. Later rates stay 0.1 percentage points lower and the gap does not widen. If the floating rate in every period is 0.1 percentage points higher, the differences add up, and the gap in rates grows in later periods.

In the falling case of the example, the 1.0% difference in year 2 is also part of the 8.2% rate in year 5. The snowball's rate rises from 4.0% to 8.2%, while the floating-rate note's rate falls from 1.0% to 0.5%.

The 0% floor cuts this build-up short. In the rising case of the example, the formula gives −1.5% for year 5, but the coupon stops at 0%, and the −1.5% is not carried into the next period. The next period's rate is counted again from 0%.

The version with a cancellation right

Some snowballs carry a right to end the trade before maturity. The example by Bender et al. is a cancelable snowball swap that can be canceled after the first year³. Cancellation removes all later payments.

With this right, deciding whether to cancel requires an expected present value of the later payments (a conditional expectation)⁴. Because a snowball's later coupons depend on past rates, estimating this expectation requires a calculation along each path the rates may take.

Valuation and risk

In a LIBOR market model, the expectation needed to value the cancelable snowball in the example by Bender et al. has no closed-form expression. Bender et al. value it with nested Monte Carlo simulation⁵.

Part of an approximation to the coupon of a cancelable snowball can be valued in closed form with the Black model. The exercise policy derived from that approximation (the rule for deciding when to cancel) serves as the starting point of the valuation⁶. A general explanation of how valuation depends on model assumptions is in Model risk.

Prices in actual markets, hedging practice, liquidity, and the issuer's credit are not covered on this page.

Notes

1. Haruyasu Ota, "マルコフ関数モデルによる金利オプションの価格付けの実用化" (Putting the Markov-functional model into practice for pricing interest rate options), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2010), page 136, footnote 9 (in Japanese). ↩

2. Christian Bender, Anastasia Kolodko, and John Schoenmakers, "Iterating Snowballs and related path dependent callables in a multi-factor Libor model" (WIAS Preprint No. 1061, October 2005), page 6, section 3.1 (page 8 of the PDF), where the formula for K_i appears. ↩ ↩² ↩³ ↩⁴

3. Bender et al. (2005), page 6, section 3.1 (page 8 of the PDF), and page 2, section 2.2 (page 4 of the PDF). ↩

4. Bender et al. (2005), the Abstract (page 3 of the PDF) and page 2, section 2.2 (page 4 of the PDF). ↩

5. Bender et al. (2005), page 6, section 3.1 (page 8 of the PDF), and page 5, section 2.4 (page 7 of the PDF). ↩

6. Bender et al. (2005), page 7 (page 9 of the PDF), around equations (14) and (15). ↩