Yield Curve Curvature, Butterflies and Structured Products
This is a retelling of Chapter 6 (Part 2) from “Trading the Fixed Income, Inflation and Credit Markets: A Relative Value Guide” by Neil C. Schofield and Troy Bowler (Wiley, 2011, ISBN: 978-0-470-74229-7).
In the first half of Chapter 6 we covered yield curve basics, short-end trades, and slope strategies. Now we move to curvature. This is where butterfly trades live. We’ll also look at constant-maturity products and structured products like range accruals.
Butterfly Spreads: The Idea
A butterfly trade is how you bet on a kink in the yield curve. Say the 5-year swap rate looks too high compared to the 2-year and 10-year rates. You could just receive fixed on a 5-year swap and wait. But that’s a big directional bet.
Instead, you isolate the 5-year rate by taking offsetting positions at adjacent maturities. You receive fixed at 5 years (the “belly”) and pay fixed at 2 and 10 years (the “wings”). If the kink corrects itself, you profit. If the curve moves in parallel, the wings and belly roughly cancel. If the curve pivots around the 5-year point, the wings cancel each other.
Common butterfly combos include 1-2-3, 2-3-5, 2-5-10, and 5-10-15 year maturities. You can also trade them on a forward-starting basis.
Weighting the Butterfly
There’s no single “right” way to size the legs. The book covers four approaches.
Cash and duration neutral means you invest zero net cash and the total DV01 of the wings equals the DV01 of the belly. This protects against small parallel shifts but not against slope changes. The math involves solving two simultaneous equations to get the nominal amounts for each wing.
50/50 weighting splits the DV01 risk equally between the two wings. Each wing carries half the belly’s DV01. This protects against both parallel moves and curve rotations (pivots around the belly). The trade isn’t cash neutral, though.
For a 2s5s10s butterfly using 50/50 weights, if you have $10 million in the 5-year with a DV01 of $4,696 per $10m, then each wing needs $2,348 of DV01 exposure. That means a bigger nominal in the 2-year (lower DV01 per dollar) and a smaller nominal in the 10-year (higher DV01 per dollar).
Regression weighting recognizes that yields at different maturities don’t always move by the same amount. You run a regression of changes in the belly rate on changes in the wing rates. The resulting coefficients become your hedge ratios. If regression gives you 0.61 for the 2-year and 0.62 for the 10-year, then 61% of the belly’s DV01 goes into the 2-year wing and 62% into the 10-year wing.
This hedges against typical curve movements rather than perfectly parallel ones. The downside is that regression coefficients aren’t stable over time.
PCA (Principal Component Analysis) uses factor analysis to determine weightings. The third principal component captures curvature. When factor 3 hits extreme values, the curve is unusually kinked. You use the factor loadings to weight the butterfly so it’s hedged against factors 1 and 2 (level and slope) and only exposed to factor 3 (curvature).
Reading a Butterfly Quote
Butterfly spreads are quoted on a bid/offer basis. The 50/50 spread formula is: (2 x belly rate - short wing rate - long wing rate) / 2. So a 2s5s10s butterfly spread equals 5-year minus half the 2-year minus half the 10-year.
The language can feel backwards. A high spread means the belly is “cheap” (rate too high, expected to fall). A low spread means the belly is “rich” (rate too low, expected to rise). But smart traders don’t rely on gut feel. They use Z-scores to evaluate whether the current spread is historically unusual.
If a trader thinks the 5-year rate will fall relative to the wings (the spread will decrease), they “sell the spread.” They receive the belly rate and pay the wings. In market speak, the 5-year is expected to “outperform” because from a bond perspective, a falling rate means rising prices.
Carry on Butterfly Trades
Just like steepeners and flatteners, butterfly carry is revealed by comparing spot and forward spreads.
If the forward butterfly spread is less than the spot spread, a long butterfly position carries positively. The spread can decrease by the difference before you lose money.
If the forward spread is higher than spot, the position carries negatively. The spread needs to increase beyond the negative carry for profit.
The book’s example: spot butterfly spread at 24.5 basis points, 12-month forward spread at 4.5 basis points. The long butterfly only loses money if the spread drops by more than 20 basis points. That’s a comfortable cushion.
Volatility, Slope, and Curvature
These three things are connected through the level of interest rates.
We already know that low rates tend to come with high lognormal vol and steep curves. Now add curvature to the picture. When rates are low and curves are steep, the curve typically becomes more concave: very steep at the short end, flat at the long end. This means the 2s10s30s butterfly spread (measuring curvature) tends to increase.
So there’s a positive relationship between implied volatility and curvature. When vol is high, curvature is usually pronounced. When vol is low, the curve tends to be flatter and smoother.
These relationships don’t always hold perfectly. The period from late 2008 onward was unusual. Central banks pinned short-term rates near zero and ran quantitative easing. The result was that the long end became more volatile than the short end, which is the opposite of normal behavior. Actual vol fell because short rates barely moved, and implied vol followed it down.
Volatility, Curvature, and Skew
There’s also a link between curve shape and the volatility skew. When the curve is very concave, traders tend to buy out-of-the-money payers on long-dated swap tenors. They’re betting the long end will steepen, driven by rising long-term rates. When the curve is convex (short end flat or inverted, long end flat), traders buy out-of-the-money receivers expecting rates to fall.
Bottom line: skew is more pronounced when the curve has extreme curvature, and milder when the shape is less dramatic.
Constant-Maturity Products
A constant-maturity swap (CMS) has one leg tied to a swap rate of a fixed maturity that never changes. Unlike a regular swap where you’re exposed to a rate that rolls down the curve over time, a CMS always references, say, the 5-year rate, even after two years have passed.
In its basic form, a CMS swap is floating vs floating. One leg resets to the constant-maturity rate, the other pays LIBOR plus or minus a spread. The spread is set so the deal is fair at inception.
If you’re a “CMS receiver” (getting the CMS rate, paying LIBOR plus spread), you’re betting the curve stays steep. A “CMS payer” is betting on flattening.
CMS products can also be quoted using a “participation” instead of a spread. Rather than adjusting the LIBOR leg, you adjust the CMS leg. For example, receiving 92% of the CMS 10-year rate and paying 12-month EURIBOR flat.
CMS caps and floors work like regular caps and floors but reference a swap rate instead of LIBOR. CMS spread options let you bet on the difference between two swap rates, which is really a bet on the curve’s slope.
Pricing CMS Products
Pricing gets technical, but the intuition is about replication. A CMS caplet can be replicated by buying a strip of payer swaptions at increasing strikes. A CMS floorlet is replicated by buying a receiver swaption at the floor strike and selling receivers at lower strikes.
The catch is that the swaption payoff is concave (the present value of the payoff depends on discounting over the swap’s remaining life), while the CMS option payoff is linear. To bridge this gap, practitioners apply a “convexity adjustment” that shifts the forward swap rate upward.
One practical consequence: changes in demand for CMS products affect long-dated swaption volatility, because banks need to hedge their CMS exposure using swaptions.
Range Accruals
Range accruals are structured products where you earn a coupon only on days when a reference rate stays within a specified range. The coupon formula is something like: Fixed Rate x (n/N), where n is the number of days the rate is in range and N is total days in the period.
The investor is essentially selling a strip of daily digital options. That’s where the yield enhancement comes from.
These can reference CMS rates too. The book gives the example of the “Atlantic Range Accrual note”: a 10-year note paying CMS 10-year + 0.10% times the fraction of days that 3-month USD LIBOR stays at or below 7.50%.
The structures come in fixed and floating coupon flavors. Fixed might pay 5.30% for each qualifying day. Floating might pay EURIBOR plus a spread on qualifying days. The issuer usually has the right to call the note on coupon dates, which gives the investor another source of yield (they’re short the call option).
Where the barriers sit matters. The further out-of-the-money the barrier relative to forward rates, the less yield enhancement you get. Makes sense: the less likely the rate is to breach the barrier, the cheaper the embedded option.
That wraps up Chapter 6. From basic curve terminology through butterfly trades to structured products, the progression shows how traders can isolate increasingly specific views on the yield curve. Next, we apply relative value thinking to credit markets.
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