Butterfly Trades
Butterfly Trades
A butterfly is a sophisticated three-leg yield curve trade that isolates the "hump" or curvature of the curve, allowing traders to profit from curve shape without taking directional (up/down yields) risk.
Key takeaways
- A butterfly uses three maturity segments (typically 2-year, 5-year, and 10-year) in a specific ratio designed to be duration-neutral
- The position profits from changes in curve curvature — the degree to which the middle of the curve bulges relative to a straight line connecting the short and long ends
- Butterflies are duration-neutral but require careful execution and monitoring
- Profits are typically small on a percentage basis (a few basis points), so butterflies rely on leverage and tight transaction costs
- Butterflies are primarily professional/institutional strategies due to complexity and execution costs
The Butterfly Structure
The standard butterfly trade is constructed as:
Long 2-year bond
Short (2x) 5-year bond
Long 10-year bond
The position can also be structured in reverse (short the wings, long the body).
The key is the weighting. A butterfly is not just any three-leg position; it is specifically sized so that:
- It is duration-neutral: The total duration (weighted by position size) is zero or near-zero, making the position insensitive to parallel shifts in yields (all yields up or down equally).
- It isolates curvature: The position profits (or loses) based on whether the middle of the curve (the 5-year) is rich or cheap relative to the line connecting the short and long ends.
Duration-Neutral Sizing
To illustrate, suppose:
- 2-year duration: 2.0 years
- 5-year duration: 4.5 years
- 10-year duration: 9.0 years
A duration-neutral butterfly might be:
- Long $4.5M 2-year (duration: 2.0 × $4.5M = $9M)
- Short $2M 5-year (duration: 4.5 × $2M = -$9M)
- Long $4.5M 10-year (duration: 9.0 × $4.5M = $40.5M)
Wait, that does not balance. Let me recalculate:
- Long $9M 2-year (duration: 2.0 × $9M = $18M)
- Short $4M 5-year (duration: 4.5 × $4M = -$18M)
- Long $2M 10-year (duration: 9.0 × $2M = $18M)
Net duration: $18M - $18M + $18M = $18M. That is still not neutral.
The correct sizing is:
- Long $9M 2-year (duration: 2.0 × $9M = $18M)
- Short $4M 5-year (duration: 4.5 × $4M = -$18M)
- Long $2M 10-year (duration: 9.0 × $2M = $18M)
Net duration: $18 - $18 + $18 = $18M notional duration. But we want neutral duration.
Actually, the issue is I should think in terms of dollar duration (DV01), not notional. Let me reconsider.
If each $1M of bonds has durations of 2, 4.5, and 9 years respectively:
For duration neutrality: $X_2 × 2 + $X_5 × 4.5 + $X_10 × 9 = 0
If we set $X_5 = -$2M (short $2M 5-year), then: $X_2 × 2 + (-$2M) × 4.5 + $X_10 × 9 = 0 $X_2 × 2 - $9M + $X_10 × 9 = 0
If we set $X_2 = $X_10$ (to be symmetric), then: $X × 2 - $9M + $X × 9 = 0 $X × 11 = $9M $X = $0.82M
So a duration-neutral butterfly might be:
- Long $0.82M 2-year
- Short $2M 5-year
- Long $0.82M 10-year
This is a small position, with the shorts being larger than the longs. The position is duration-neutral but has positive convexity (benefits from non-parallel curve moves).
Butterfly PnL: How It Works
With the position sized as above, the butterfly profits from changes in the 5-year yield relative to the 2-year and 10-year.
Scenario 1: Curve flattens uniformly (short-end yields rise, long-end yields fall)
- All yields shift by the same amount... wait, that is not a parallel shift. Let me clarify.
Scenario 1: All yields rise by 10 bps (parallel shift)
- 2-year yield rises from 2.0% to 2.1% → bond price falls by approximately 2.0% × 0.1% = 0.02%.
- 5-year yield rises from 2.5% to 2.6% → bond price falls by approximately 4.5% × 0.1% = 0.045%.
- 10-year yield rises from 3.0% to 3.1% → bond price falls by approximately 9.0% × 0.1% = 0.09%.
Now apply the position:
- Long $0.82M 2-year: loses $0.82M × 0.02% = $164.
- Short $2M 5-year: gains $2M × 0.045% = $900 (shorts profit when prices fall).
- Long $0.82M 10-year: loses $0.82M × 0.09% = $737.
Net PnL: -$164 + $900 - $737 = -$1. (This should be nearly zero due to duration neutrality.)
Scenario 2: Non-parallel shift; 5-year yields fall by 10 bps while 2-year and 10-year yields are unchanged
- 2-year yield unchanged: no price change.
- 5-year yield falls from 2.5% to 2.4% → bond price rises by approximately 4.5% × 0.1% = 0.045%.
- 10-year yield unchanged: no price change.
Apply the position:
- Long $0.82M 2-year: no change.
- Short $2M 5-year: loses $2M × 0.045% = $900 (shorts lose when prices rise).
- Long $0.82M 10-year: no change.
Net PnL: -$900. (The 5-year became richer, and we were short, so we lose.)
Scenario 3: Curve becomes more humped; 5-year yields fall relative to 2-year and 10-year Suppose the 2-10 spread stays the same (both rise 5 bps), but the 5-year only rises 0 bps:
- 2-year yield rises from 2.0% to 2.05% → price falls by ≈ 2.0 × 0.05% = 0.001%, loss on long $0.82M.
- 5-year yield unchanged: no price change.
- 10-year yield rises from 3.0% to 3.05% → price falls by ≈ 9.0 × 0.05% = 0.0045%, loss on long $0.82M.
Apply the position:
- Long $0.82M 2-year: loses ≈ $0.82M × 0.001% = $8.
- Short $2M 5-year: no change.
- Long $0.82M 10-year: loses ≈ $0.82M × 0.0045% = $37.
Net PnL: -$45. (Both wings lost, but the short middle position broke even.)
Actually, this scenario does not show the butterfly profit. Let me reconsider.
A butterfly profits when:
- The curve becomes less humped (the 5-year yield falls relative to the 2-10 line), meaning the 5-year becomes richer (more expensive), or
- The curve becomes more humped (the 5-year yield rises relative to the 2-10 line), meaning the 5-year becomes cheaper.
Given that we are short the 5-year in the butterfly:
- We profit if the 5-year becomes richer (prices rise, yields fall).
- We lose if the 5-year becomes cheaper (prices fall, yields rise).
Corrected Scenario 3: Curve becomes less humped (5-year becomes relatively richer)
- 2-year yield rises from 2.0% to 2.1% (long, so lose).
- 5-year yield falls from 2.5% to 2.4% (short, so gain!).
- 10-year yield rises from 3.0% to 3.1% (long, so lose).
The short position in the 5-year gains more (0.045% gain on $2M) than the long positions lose (0.02% + 0.09% = 0.11% combined on $1.64M), so the net is a gain.
The butterfly is profitable when the 5-year is rich relative to the straight-line interpolation between the 2-year and 10-year.
When Butterflies Are Attractive
Butterflies are most attractive when:
1. The middle of the curve is visibly overvalued or undervalued
Traders use interpolation models to estimate the "fair" 5-year yield given the 2-year and 10-year. If the actual 5-year yield is significantly below (rich) or above (cheap) the interpolated level, a butterfly becomes attractive.
For example, if the 2-year yields 2.0%, the 10-year yields 3.0%, and the interpolated 5-year should yield 2.5%, but the actual 5-year yields 2.3% (rich), then shorting the 5-year and going long the wings becomes attractive.
2. Curve-fitting arbitrage
Corporate bond investors often buy corporate bonds in the 5-7 year zone because this segment has strong natural demand (pension funds, life insurers). If corporate demand is temporarily very high, corporate 5-year spreads compress, potentially making the Treasury 5-year rich. A butterfly exploits this by being short the rich Treasury 5-year.
3. Supply-demand imbalances
If the Treasury auction calendar is heavy in 5-year issuance (a new supply influx depresses 5-year prices, making yields high), a butterfly would long the cheap 2-year and 10-year and short the now-cheap 5-year, betting for normalization.
4. Volatility and dealer dynamics
During periods of high volatility or when major dealers are unwinding positions, the 5-year can become dislocated from its "fair" value relative to the 2-10 line. Butterflies can exploit these temporary dislocations.
Challenges and Risks
Butterfly trades have significant challenges:
1. Small profit margins
A typical butterfly profit is 2-5 basis points, which is tiny. For a $10M position, a 3 bps profit amounts to $3,000 gross. Transaction costs (bid-ask spreads on three bonds) can be $5K-$10K, so the trade needs to be sized much larger or executed with tight spreads to be profitable.
2. Execution difficulty
Implementing a butterfly requires buying and selling three different bonds simultaneously. In times of market stress, the bids and offers on 2-year, 5-year, and 10-year bonds might not be available, or they might move significantly while the trader is executing the legs.
3. Model risk
The "fair" 5-year yield (the interpolation between 2 and 10) depends on the interpolation model. If the model is wrong, the butterfly setup is wrong. Different traders use different models, which can be a source of consistent divergence or a sign that the model-based arbitrage is weak.
4. Holding period and mark-to-market risk
Even if the butterfly is theoretically profitable, the market might move against it before the trade captures its profit. If the entire curve (2, 5, and 10) rise sharply, all legs lose (despite duration neutrality, there can be second-order effects from convexity). The trader must be willing to hold the position through temporary mark-to-market losses.
5. Leverage required
Because butterfly profits are small, most butterfly trades use leverage (repo financing) to amplify returns. This introduces financing risk: if repo rates spike or funding dries up, the position becomes untenable.
Variations on the Butterfly
While the 2-5-10 butterfly is most common, traders also use:
- 3-5-7 butterfly: Short the 5-year, long the 3-year and 7-year. Useful for focusing on the middle segment.
- Reverse butterfly: Long the 5-year, short the 2-year and 10-year. Profitable when the 5-year becomes cheap (the reverse of the normal butterfly).
- Generalized butterfly: Any three-leg position with duration neutrality, tailored to specific curve segments.
Real-World Example
In September 2019, a trader noticed:
- 2-year Treasury: 1.60% yield.
- 5-year Treasury: 1.40% yield (interpolation from 2 and 10 suggests ≈1.70%, so the 5-year is cheap).
- 10-year Treasury: 1.80% yield.
The 5-year was trading cheap to the 2-10 line (the interpolation suggested 1.70%, but it yielded only 1.40%), suggesting a butterfly long the wings (long 2 and 10) and short the 5-year would profit if the 5-year normalized.
A trader might have sized:
- Long $10M 2-year.
- Short $25M 5-year.
- Long $15M 10-year.
(Sizing adjusted for duration neutrality.)
Over the following months, if the 5-year yield normalized to 1.65%, the position would profit on the short 5-year leg while the long wings have less downside (due to duration-neutral construction).
(Note: These 2019 yields are illustrative; actual yields differed.)
Butterflies vs. Steepeners and Flatteners
| Aspect | Butterfly | Steepener | Flattener |
|---|---|---|---|
| Legs | 3 (2Y, 5Y, 10Y) | 2 (short-end, long-end) | 2 (short-end, long-end) |
| Duration neutral | Yes | Optional | Optional |
| Risk | Curvature risk | Directional + curve slope risk | Directional + curve slope risk |
| Profit driver | Middle of curve richness | Curve steepening | Curve flattening |
| Complexity | High | Moderate | Moderate |
| Execution difficulty | High | Moderate | Moderate |
Who Uses Butterflies
Butterflies are primarily used by:
- Professional fixed-income traders.
- Bond dealers making markets.
- Quantitative hedge funds with sophisticated curve models.
- Large asset managers (pension funds, insurance) with expert bond trading operations.
Retail investors rarely use butterflies due to execution difficulty, leverage requirements, and small profit margins.
Next
Butterflies are complex relative-value trades that isolate curve curvature. For most investors, the most practical yield curve trade is the simpler carry trade: buy bonds, hold for carry and price appreciation, and manage duration risk along the way. The final article of this chapter provides a practical one-page reference for all yield curve concepts, spreads, and strategies.