A butterfly buys the belly and sells the wings, or the reverse, and what it is long is curvature: the middle of the curve against the average of the two ends. Everything about it turns on the weighting convention, so state the convention before quoting the trade.
How is a butterfly sized, and what is it exposed to?
Take a body of $100m of the five-year, whose DV01 is $44,539. Split the hedge evenly across the wings and each wing needs $22,270 of DV01, which is $117,041,635 of the two-year and $28,060,396 of the ten-year. The whole structure is DV01-neutral, and it raises $45,102,031 of cash because the wings sell more notional than the body buys. That financing is real and belongs in the profit and loss.
Under a 50/50 weighting the trade's exposure is exactly one quantity:
curvature = (change in the body's yield) - 0.5 x (change in the two-year) - 0.5 x (change in the ten-year)
Everything else nets out. Compute it for each of the three scenarios and the trade's behaviour follows without further arithmetic.
| Scenario | 2y | 5y | 10y | Curvature | Butterfly P&L |
|---|---|---|---|---|---|
| Bull steepener | -30 bp | -20 bp | -5 bp | -2.50 bp | +$113,305 |
| Bear steepener | +5 bp | +20 bp | +30 bp | +2.50 bp | -$116,237 |
| Parallel | +25 bp | +25 bp | +25 bp | 0.00 bp | -$1,036 |
The parallel move is the one the butterfly is insulated from, and the insulation is structural: any move in which the body's yield changes by the average of the two wing changes produces zero curvature and therefore no first-order profit and loss. The residual $1,036 on a $100m body is the convexity mismatch again, since the two wings' convexities of 4.615632 and 75.580012 do not average to the body's 23.210605.
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How do weightings and condors change the trade?
Both steepeners are curvature events under this weighting and they point in opposite directions, which surprises people who expect a butterfly to be slope-neutral. A 50/50 butterfly is neutral to slope only when the body sits at the DV01-weighted midpoint of the wings, and on this curve the five-year does not. Managers who want slope neutrality use a regression weighting, in which the wing DV01s are split according to each wing's historical beta to the body rather than evenly, and the trade then expresses residual curvature against a fitted relationship rather than raw curvature. Both are defensible. Quoting one and computing the other is not.
A condor is the same idea with four points instead of three, buying two adjacent belly maturities against two wings. It trades a smoother, less lumpy exposure for a smaller signal, and everything above transfers unchanged.