In institutional fixed income markets, directional bets on whether interest rates will rise or fall are notoriously volatile and exposed to macroeconomic surprises. Rates trading desks, primary dealers, and relative value macro hedge funds systematically hedge parallel rate risk to trade the shape of the sovereign yield curve.
While steepeners and flatteners trade the curve's first-order slope (e.g., 2s/10s or 5s/30s spreads), the Treasury Butterfly Spread isolates the curve's second-order derivative: curvature (the hump or belly dip). By pairing a position in an intermediate maturity bond (the "belly") against offsetting positions in shorter and longer maturity bonds (the "wings"), traders isolate pure curvature changes while neutralizing directional level and slope moves.
01. Anatomy of the Butterfly Spread
A butterfly spread consists of three sovereign Treasury issues along the constant maturity curve:
- Short Wing ($T_{short}$): The shorter maturity bond (e.g., 2-Year or 5-Year Treasury note).
- Belly ($T_{belly}$): The intermediate maturity bond (e.g., 5-Year or 10-Year Treasury note).
- Long Wing ($T_{long}$): The longer maturity bond (e.g., 10-Year or 30-Year Treasury bond).
The two most liquid benchmark butterflies traded in the global rates market are:
- The 2s/5s/10s Butterfly: 2Y Wing, 5Y Belly, 10Y Wing. Reflects intermediate monetary policy expectations and cyclical growth outlook.
- The 5s/10s/30s Butterfly: 5Y Wing, 10Y Belly, 30Y Wing. Reflects long-end term premium shifts, Treasury auction concession absorption, and liability-driven investment (LDI) pension hedging.
02. Curvature Spread Formulations
The conventional market metric for butterfly curvature is expressed in basis points ($bps$):
Equivalently, the butterfly spread represents the difference between the two adjacent curve slopes:
When the Fly Spread widens (increases in value), the belly yield rises relative to the linear interpolation of the wings. This is known as a belly cheapening or curve humping. Conversely, when the Fly Spread narrows, the belly yield falls relative to the wings, known as a belly richening.
03. Duration-Neutral & DV01 Weighting
A common retail misconception is that a butterfly is structured with equal $50\%$ wing notionals. In institutional trading, a 50/50 allocation creates severe duration mismatch: because the 10-year note has approximately 4.5 times the duration of a 2-year note, a parallel curve shift will dominate the trade's P&L.
To achieve true curvature isolation, traders solve for DV01 Neutrality, where DV01 represents the Dollar Value of a Basis Point ($DV01 = \text{Notional} \times \text{Modified Duration} \times 0.0001$).
Under exact DV01-neutral weights, a $+50\text{ bps}$ parallel shift in benchmark yields produces a net P&L of exactly $0, leaving the position exposed purely to relative curvature and twist shifts.
04. Carry & Roll-Down Dynamics
Because relative value trades may take weeks or months to converge, carry and roll-down dictate the holding return of the position. A fly with positive expected curvature may still lose money if the financing carry drag exceeds the curvature convergence.
Where $R$ is the general collateral repo financing rate (e.g., SOFR). Roll-down represents the capital gain achieved as each security ages into a shorter, lower-yielding tenor along an upward-sloping yield curve:
05. Curve Twist Regimes & Convexity
Beyond parallel shifts, the yield curve experiences non-linear deformations known as curve twists:
| Twist Regime | Short Wing (2Y) | Belly (5Y) | Long Wing (10Y) | Fly P&L Impact |
|---|---|---|---|---|
| Bull Steepening | Sharp Yield Drop | Moderate Drop | Unchanged / Minor Drop | Belly richens; sell fly gains |
| Bear Flattening | Sharp Yield Rise | Moderate Rise | Mild Rise | Belly cheapens; buy fly gains |
| Curvature Hump | Unchanged | Yield Spike (+20 bps) | Unchanged | Long Barbell / Short Belly maximum profit |
| Curvature Dip | Unchanged | Yield Plunge (-20 bps) | Unchanged | Long Belly / Short Barbell maximum profit |
06. Execution, Repo Drag & Trade Sizing
When shorting the belly (e.g., 5-year note), traders must borrow the security in the bilateral repo market. If the 5-year is on "special" (trading at a repo rate significantly below general collateral SOFR, e.g., 1.50% vs 4.50%), the financing drag of maintaining the short position increases dramatically, eroding theoretical relative value alpha.