CMS Spread Options

It comes up when an investor, or the holder of a bond paying interest linked to the slope of the yield curve, holds a view on changes in that slope. On the pricing side, how the correlation between the two swap rates is handled is central to valuation.

An option on the difference of two rates

A CMS spread option compares the difference between two swap rates of different tenors, such as 10 years and 2 years, with a strike. An option whose payoff is set by comparing the difference between the prices of two underlying assets with a strike is called a spread option, and examples of underlyings include the share prices of different companies or long-term and short-term interest rates¹. A CMS spread option is a spread option whose underlyings are the swap rates referenced by a CMS.

The difference is the long-term swap rate minus the short-term swap rate. For example, "10-year swap rate − 2-year swap rate" reflects the slope of the yield curve. How a swap rate is referenced is covered in Constant Maturity Swaps (CMS).

The payoff of a call is set by the following value times the notional amount, where $S_{\text{long}}$ is the long-term swap rate on the observation date, $S_{\text{short}}$ is the short-term swap rate, and $K$ is the strike².

$$\max\left(S_{\text{long}} - S_{\text{short}} - K,\ 0\right)$$

The payoff of a put reverses the comparison of the difference and the strike: $\max(K - S_{\text{long}} + S_{\text{short}},\ 0)$.

Figure description

This chart shows the payoff on the observation date of the buyer of a call with a strike of 0.5%, as a percentage of the notional amount. The premium is not included. The horizontal axis is the difference between the 10-year and 2-year swap rates. The payoff is 0% when the difference is 0.5% or lower, and rises by the amount by which the difference exceeds 0.5%. At a difference of 1.5%, the payoff is 1.0%.

How the payoff is set, with a numerical example

The call pays the amount by which the difference exceeds the strike, and the put pays the amount by which it falls below. Consider a fictional example with a notional amount of 10 billion yen, a period of one year, and a strike of 0.50%. Company F buys the call, and the payoff of a put with the same strike is shown alongside. The payoff is "notional amount × the amount by which the difference exceeds the strike (for the put, falls below it)" (the period is one year, so the period factor is 1). The premium of the call is 0.20% of the notional amount (20 million yen), and the time value of interest is ignored.

If the difference is 0.50% or lower, Company F receives nothing from the call, and the premium of 20 million yen is a straight loss. Above 0.50% the payoff rises, and once the difference exceeds 0.70% (the strike plus the premium of 0.20%), the position is profitable even after deducting the premium. At a difference of 1.00%, the payoff is 50 million yen and the profit is 30 million yen. The put works the other way: the further the difference falls below 0.50%, the larger the payoff. It is 50 million yen at a difference of 0.00% and 100 million yen at −0.50% (an inverted curve).

The seller's position is the opposite. A call seller receives only the premium, and when the difference exceeds the strike, pays the excess to the buyer. The more the difference widens, the more the seller pays, so the loss grows as the difference widens.

A call is called a caplet and a put a floorlet. The numerical experiments in the paper by Belomestny and others also use a caplet and a floorlet on the difference between the 10-year and 2-year swap rates³. Standard caps and floors have a single rate as the underlying, while this option has the difference between two rates.

The slope of the curve and the payoff

What moves the payoff is not the level of each swap rate but the difference. The wider the gap between the long-term and short-term rates (the steeper the curve), the larger the difference, and the call payoff rises by the amount by which the difference exceeds the strike. When the gap between long-term and short-term rates narrows (the curve flattens), the difference approaches 0. When short-term rates rise above long-term rates (the curve inverts), the difference becomes negative.

A call with a strike of 0.50% pays nothing when the difference is 0.50% or lower. When the curve flattens or inverts, the difference is more likely to fall below this level. A call with a strike of 0% pays as long as the curve is not inverted (the difference is positive). The flatter the slope, the larger the put payoff. If the 10-year and 2-year rates move in the same direction by the same amount, the difference does not change. For this reason, the payoff may not change even when the level of interest rates moves a lot.

The effect of correlation on the premium

The lower the correlation between the two swap rates, the more the difference varies and the higher the option premium. The standard deviation of the difference, $\sigma_{\text{diff}}$, is set by the standard deviations of the two swap rates, $\sigma_1$ and $\sigma_2$, and the correlation coefficient $\rho$, as follows.

$$\sigma_{\text{diff}}^2 = \sigma_1^2 + \sigma_2^2 - 2\rho\,\sigma_1\sigma_2$$

When the correlation coefficient is close to 1, the two rates move alike and the difference barely moves. As the correlation coefficient falls, the difference moves more. Consider a fictional example in which the standard deviation of each swap rate is 100bp a year, the period is one year, and the strike equals the expected value of the difference. The greater the variation of the difference, the more both the call and the put are worth. Under these assumptions, the price is the standard deviation of the difference times √(T/(2π)), where T is 1 year⁴.

With a notional amount of 10 billion yen, 1bp is 1 million yen, so the price at a correlation coefficient of 0.9 is about 17.8 million yen, and at 0.1 about 53.5 million yen. In this comparison, the volatility of each swap rate is held constant.

Valuation requires an estimate of the correlation between the logarithms of the swap rates⁵. The price of a product with several underlying assets is affected by the correlation between the underlyings, and this correlation risk is a risk that a product with a single underlying does not have⁶.

Valuation methods and model differences

The payoff depends on two swap rates, so valuation needs their joint distribution. In a practitioner's presentation (Ramesh, 2022), the distribution of each swap rate is set by a model fitted to swaption prices (such as the SABR model)⁷. There is not necessarily only one method in practice, and the approximation formula in the paper by Belomestny and others (2008) uses a lognormal approximation. The function that builds a joint distribution from the two distributions is called a copula⁸.

The Gaussian copula, one kind of copula, links two distributions with a single correlation coefficient. The correlation needed to reproduce a given price with the Gaussian copula is called the implied correlation. With the Gaussian copula, the implied correlation is constant regardless of the strike. To represent prices whose implied correlation differs by strike (that is, prices with a smile), copulas with extra parameters, such as the power-Gaussian copula, are used⁹. Even with the same individual distributions, changing the shape of the copula changes the shape of the implied correlation smile. Because the price changes with the model chosen, it is subject to model risk.

Deep diveThe formula of the Gaussian copula

Let $F_X$ and $F_Y$ be the cumulative distribution functions of the two swap rates. The joint distribution function can then be written with a copula $C$ as $F_{X,Y}(x, y) = C(F_X(x), F_Y(y))$. The Gaussian copula is the following formula¹⁰.

$$C_{\rho}(u, v) = \Phi_{\rho}\left(\Phi^{-1}(u),\ \Phi^{-1}(v)\right)$$

$\Phi_{\rho}$ is the cumulative distribution function of the bivariate standard normal distribution with correlation coefficient $\rho$, and $\Phi$ is the cumulative distribution function of the standard normal distribution.

There are also ways to approximate the price with a formula. If the two swap rates can be approximated by a lognormal distribution, a spread option with a strike of 0 can be approximated with Margrabe's formula¹¹. The paper by Belomestny and others (2008) calculates the price in a measure that discounts the payoff to match the payment date (the forward measure)¹². The swap rate seen as an expected value at the time of the payment generally does not equal the forward swap rate derived from today's market, so a convexity adjustment is needed¹³. The details of the adjustment are covered in Constant Maturity Swaps (CMS).

The Monte Carlo method is used both to calculate prices and as a benchmark to check the accuracy of approximation formulas. The approximation formula was derived under a lognormal assumption. In the numerical experiments of the paper, the difference between the approximation formula's price and the Monte Carlo price was a few bp in the example with a 20-year maturity¹⁴.

Uses and risks

A bond that pays interest linked to the difference between two swap rates is traded by parties with a view on how the slope of the curve will change. Some products whose interest rises as the curve steepens are called steepeners. The paper by Belomestny and others (2008) states that from the start of 2006, the US and European yield curves flattened sharply and CMS spreads (the difference between two swap rates) narrowed. Interest fell, investors holding this kind of product suffered large losses, and the liquidity of the products had also fallen, it says¹⁵.

Consider a fictional example in which Company F holds a bond with a notional amount of 10 billion yen, receives interest linked to the difference, and hedges with an option. The interest is "notional amount × difference" (considered only where the difference is 0% or higher). Company F buys a put with a strike of 0.50% for a premium of 15 million yen. The put pays the amount by which the difference falls below 0.50%.

When the difference falls to 0.50% or lower, the total stops at 35 million yen, which limits the decline in interest. When the difference exceeds 0.50%, the put pays nothing, and the interest received is reduced by the premium of 15 million yen. In this example, Company F pays the premium in return for putting a floor under what it receives when the difference falls.

The risks can be viewed in three parts: the seller's loss, liquidity, and uncertainty in valuation. The loss of a call seller grows as the difference widens, and liquidity can become low when the products fall out of favor, as after the curve flattening in 2006. In valuation, the price changes with the model.

Differences from other products

A CMS is an interest rate swap that references a swap rate as its floating rate, and it references one rate. A CMS spread option puts an option form on the difference between two rates, and it is close to a basket option in that it is affected by correlation.

Notes

1. 中村一岳 (Institute for Monetary and Economic Studies, Bank of Japan), 「相関デリバティブのリスク特性やリスク管理手法に関する考察」 (A study of the risk characteristics and risk management of correlation derivatives; English title is the writer's translation), Discussion Paper No. 2003-J-7, page 3, "スプレッド･オプション" (spread options) (in Japanese). ↩

2. Belomestny, Kolodko, Schoenmakers, "Pricing CMS spreads in the Libor market model" (WIAS Preprint No. 1386, December 2008), page 3, equation (3) in "2 Main setup". ↩

3. The same paper, page 11, the explanation before "Tables 2-4" and "Table 2. CMS spread caplet, strike 0.5%". ↩

4. Calculation on this page. Each swap rate is assumed to follow a normal distribution, the strike equals the expected value of the difference, and discounting is ignored. ↩

5. Belomestny, Kolodko, Schoenmakers, "Pricing CMS spreads in the Libor market model," page 4, the explanation after equation (6) ((ii), estimation of the correlation between the logarithms of the two swap rates). ↩

6. 中村一岳 (Institute for Monetary and Economic Studies, Bank of Japan), 「相関デリバティブのリスク特性やリスク管理手法に関する考察」 (A study of the risk characteristics and risk management of correlation derivatives; English title is the writer's translation), page 1, "１．はじめに" (Introduction) (in Japanese). ↩

7. Prithvi Ramesh, "Generalized Power-Gaussian Copula & CMS Spread Option Smiles" (UBS Quantitative Analytics, May 2022), slide 3, "Marginal-distributions in the T-forward measure". ↩

8. The same presentation, slide 3, "Copula function" and "Joint-distribution". ↩

9. The same presentation, slide 4 (Gaussian copula and implied correlation) and slides 5 and 6 (power-Gaussian copula and the implied correlation smile). ↩

10. The same presentation, slide 4, "Gaussian Copula & Implied Correlations". ↩

11. Belomestny, Kolodko, Schoenmakers, "Pricing CMS spreads in the Libor market model," page 3, equations (4) and (5) in "2 Main setup". ↩

12. The same paper, page 1, "1 Introduction". ↩

13. 田中敬一, 山田健 and 渡部敏明, 「金利派生商品の効率的な価格付け：確率密度関数の近似を用いて」 (Efficient pricing of interest rate derivatives: using approximations of probability density functions; English title is the writer's translation), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2006), pages 2 and 10 (in Japanese). ↩

14. Belomestny, Kolodko, Schoenmakers, "Pricing CMS spreads in the Libor market model," page 11, the explanation before "Tables 2-4," and Table 4 on page 12. ↩

15. Belomestny, Kolodko, Schoenmakers, "Pricing CMS spreads in the Libor market model," page 1, "1 Introduction". ↩

16. Prithvi Ramesh, "Generalized Power-Gaussian Copula & CMS Spread Option Smiles," slides 3 to 5. ↩