Butterfly Trades
Understanding butterfly spreads and relative value trading
Butterfly Trades
A butterfly trade is a sophisticated relative value strategy that involves three legs: buying two “wing” positions and selling a “body” position (or vice versa). Unlike the two-legged curve trades we covered in Section 4.3, butterflies are designed to profit from changes in the curvature of the yield curve at a specific point, while remaining neutral to both parallel shifts and overall steepening/flattening.
What is a Butterfly Trade?
A butterfly consists of:
- Two wing positions: Long (or short) bonds at the short and long ends
- One body position: Short (or long) bond in the middle maturity
The classic butterfly structure is:
- Buy the wings (e.g., 5Y and 20Y JGBs)
- Sell the body (e.g., 10Y JGB)
- Weight positions to be duration-neutral and curve-neutral
Butterfly Value Formula
The butterfly value measures the curvature of the yield curve at a specific point:
\[\text{Butterfly} = Y_{middle} - \frac{1}{2}(Y_{short} + Y_{long})\]Where:
- $Y_{middle}$ = Yield of the body bond (e.g., 10Y)
- $Y_{short}$ = Yield of the short wing (e.g., 5Y)
- $Y_{long}$ = Yield of the long wing (e.g., 20Y)
Interpretation:
- Positive butterfly: Middle yield is higher than the average of the wings (curve is humped)
- Negative butterfly: Middle yield is lower than the average of the wings (curve is flat or inverted at that point)
Example:
- 5Y yield: 0.50%
- 10Y yield: 0.90%
- 20Y yield: 1.20%
This positive 5bp butterfly indicates the 10Y is “rich” relative to the wings.
Why Trade Butterflies?
Butterfly trades offer several advantages:
- Isolate curvature risk: Profit from local curve shape changes without exposure to parallel shifts or overall steepening/flattening
- Mean reversion: Butterfly values tend to fluctuate around historical averages, creating profitable mean-reversion opportunities
- Lower volatility: Three-legged structure provides multiple hedges, reducing overall position risk
- Leverage opportunities: Low volatility allows use of higher leverage
- Rich/cheap identification: Statistical analysis identifies mispriced points on the curve
Duration-Neutral and Curve-Neutral Construction
A properly constructed butterfly must be:
- Duration-neutral: No exposure to parallel yield curve shifts
- Curve-neutral: No exposure to simple steepening or flattening
To achieve this, we need to carefully weight the three legs. The general approach:
\[W_{wings} = \frac{DV01_{body}}{DV01_{wing1} + DV01_{wing2}}\]Where:
- $W_{wings}$ = Weight (notional ratio) for each wing position
- $DV01_{body}$ = DV01 of the body bond
- $DV01{wing1}$, $DV01{wing2}$ = DV01 of the two wing bonds
Worked Example: 5s-10s-20s Butterfly
Let’s construct a ¥10 billion 5s-10s-20s butterfly trade.
Market Data:
- 5Y JGB #505: Yield 0.50%, DV01 = ¥4,800 per ¥1bn
- 10Y JGB #362: Yield 0.90%, DV01 = ¥9,200 per ¥1bn
- 20Y JGB #187: Yield 1.20%, DV01 = ¥17,500 per ¥1bn
Current butterfly value: \(\text{Butterfly} = 0.90\% - \frac{1}{2}(0.50\% + 1.20\%) = 0.05\% = 5bp\)
Trade View: We believe 5bp is rich (above historical average of 2bp). We expect the butterfly to narrow (10Y yield falls relative to wings).
Trade Structure:
- Sell body (sell 10Y)
- Buy wings (buy 5Y and 20Y)
Step 1: Determine Position Sizes
Starting with ¥10 billion notional in the body:
- Sell ¥10 billion of 10Y JGB
For the wings, we need: \(W_{wings} = \frac{DV01_{10Y}}{DV01_{5Y} + DV01_{20Y}} = \frac{9,200}{4,800 + 17,500} = \frac{9,200}{22,300} = 0.413\)
Total wings notional: ¥10bn ÷ 0.413 = ¥24.2 billion
Now distribute between the two wings proportionally to their DV01s:
\[\text{5Y notional} = 24.2bn \times \frac{4,800}{22,300} = ¥5.2bn\] \[\text{20Y notional} = 24.2bn \times \frac{17,500}{22,300} = ¥19.0bn\]Final Position:
- Buy ¥5.2 billion of 5Y JGB
- Sell ¥10.0 billion of 10Y JGB
- Buy ¥19.0 billion of 20Y JGB
Step 2: Verify Duration Neutrality
- 5Y DV01: ¥5.2bn × ¥4,800 = ¥24.96 million per bp
- 10Y DV01: ¥10.0bn × ¥9,200 = ¥92.00 million per bp
- 20Y DV01: ¥19.0bn × ¥17,500 = ¥332.50 million per bp
Net DV01: +¥24.96m - ¥92.00m + ¥332.50m = ≈ ¥265m per bp
Wait—this isn’t duration-neutral! This reveals an important point about butterflies.
The Trade-Off:
In practice, there are two common approaches to butterfly construction:
Approach 1: Regression-Weighted Butterfly Use historical regression analysis to determine weights that minimize residual curve exposure. This typically results in approximately equal DV01s on the short and long wings, with the body’s DV01 equal to the sum.
Approach 2: Equal Wing Notionals (Simplified) For a rough butterfly, some traders use equal notionals in both wings, then adjust the body to achieve approximate duration neutrality. This is simpler but less precise.
Revised Calculation (Regression-Based):
For a duration-neutral 5s-10s-20s butterfly, typical weights are:
- Buy ¥10 billion of 5Y JGB (DV01 = ¥48 million)
- Sell ¥10 billion of 10Y JGB (DV01 = ¥92 million)
- Buy ¥5 billion of 20Y JGB (DV01 = ¥87.5 million)
Net DV01 ≈ ¥48m - ¥92m + ¥87.5m ≈ ¥43.5m per bp
Still not perfect, but much closer. The residual can be hedged with a small futures position if needed.
Step 3: Calculate P&L
Assume the butterfly narrows from 5bp to 2bp (our target):
New yields:
- 5Y: 0.50% (unchanged for simplicity)
- 10Y: 0.87% (falls 3bp)
- 20Y: 1.20% (unchanged)
P&L approximation:
The body position (short 10Y) profits when 10Y yield falls:
- 10Y P&L: ¥10bn × ¥9,200 × 3bp = +¥276 million
The wings experience minimal change in this scenario.
Net P&L ≈ +¥276 million from a 3bp butterfly narrowing.
This simplified calculation shows the profit potential, though in practice all three yields would adjust.
Common JGB Butterfly Structures
| Butterfly | Typical Use | Characteristic |
|---|---|---|
| 2s-5s-10s | Short-end positioning | Sensitive to BOJ policy expectations |
| 5s-10s-20s | Most liquid belly trade | Balanced exposure across curve |
| 10s-20s-30s | Long-end positioning | Pension/insurance demand dynamics |
| 5s-10s-30s | Wide butterfly | Larger curvature exposure |
Statistical Analysis: Rich/Cheap Identification
Professional butterfly traders use statistical methods to identify when a butterfly is “rich” or “cheap”:
Step 1: Calculate Historical Butterfly Values
Track the butterfly spread over time (e.g., past 2 years of daily data).
Step 2: Calculate Z-Score
\[\text{Z-Score} = \frac{\text{Current Butterfly} - \text{Mean Butterfly}}{\text{Standard Deviation}}\]Interpretation:
- Z > +1.5: Butterfly is rich → Consider selling (buy body, sell wings)
- Z < -1.5: Butterfly is cheap → Consider buying (sell body, buy wings)
-
** Z < 1.0**: Butterfly is fair value → No clear trade
Example:
- Current 5s-10s-20s butterfly: 5bp
- 2-year mean: 2bp
- 2-year standard deviation: 1.5bp
This +2.0 Z-score suggests the butterfly is significantly rich, presenting a selling opportunity.
Grid Point Sensitivity (GPS)
Butterfly trades are closely related to Grid Point Sensitivity analysis (covered in detail in Section 5.5). GPS measures how a portfolio’s value changes with yield movements at specific maturity points (e.g., 2Y, 5Y, 10Y, 20Y, 30Y).
A butterfly trade essentially creates targeted GPS exposure:
- Positive GPS at the wing maturities (5Y and 20Y)
- Negative GPS at the body maturity (10Y)
- Near-zero GPS at other maturities
This precision allows traders to isolate specific curve risks without affecting the rest of the portfolio.
Risks in Butterfly Trading
1. Execution Risk
Butterflies require simultaneous execution of three legs. Challenges include:
- Leg risk: One or two legs execute, market moves before completing all three
- Liquidity: Less liquid maturities (e.g., 20Y, 30Y) have wider bid-offer spreads
- Market impact: Large butterflies can move prices, especially in off-the-run bonds
2. Curvature Shifts in Unexpected Ways
Historical patterns may not hold during:
- BOJ policy regime changes
- Market stress events
- Structural demand shifts (e.g., pension rebalancing)
3. Basis Risk Between Bonds
The calculation assumes specific bonds represent their maturity points, but:
- On-the-run vs off-the-run: Liquidity premiums can distort relationships
- Coupon effects: Different coupons trade at different yields even for same maturity
- Roll risk: As bonds age, they may no longer represent the intended maturity point
4. Model Risk
Regression-based weights assume past relationships persist. During structural breaks (like YCC exit), historical correlations may fail.
When Butterfly Trades Work Best
Butterflies are most effective when:
- Mean reversion is strong: Curve curvature fluctuates around stable averages
- Volatility is moderate: Too low = small profit opportunities; too high = unpredictable moves
- Liquidity is good: Tight bid-offer spreads minimize execution costs
- No structural regime change: BOJ policy remains stable
Avoid butterflies during:
- Major BOJ policy transitions
- Market stress with flight-to-quality flows
- Heavy issuance concentrated at one maturity
Practical Implementation Tips
- Start with liquid structures: 5s-10s-20s is most liquid; avoid exotic combinations
- Use on-the-run bonds: Tighter spreads reduce execution costs
- Monitor continuously: Butterfly values can move quickly; set alerts on Z-scores
- Scale gradually: Build positions over time to minimize market impact
- Hedge residual duration: Use small futures positions to neutralize any remaining DV01
- Track financing costs: Repo rates on different maturities affect carry
Relationship to Other Strategies
Butterfly trades build on concepts from earlier strategies:
- Curve trades (4.3): Butterflies are three-legged extensions of two-legged curve trades
- Carry trades (4.2): Butterflies have carry implications depending on yield levels
- Grid Point Sensitivity (3.5): Butterflies create targeted GPS exposures
Understanding butterflies prepares you for even more complex relative value strategies used by quantitative trading desks.