Convexity is the curvature of the price/yield relationship. A straight bond has positive convexity: each successive rate decline produces a larger price gain, each successive rise a smaller loss. It is a free good: gains accelerate, losses decelerate. A callable gives it away. Here is the price of each instrument at constant OAS across a ±200 bp parallel shift, with the change from the previous row in the "step" columns:
| Curve shift | Straight | step | Callable | step | Putable | step |
|---|---|---|---|---|---|---|
| −200 bp | 107.5498 | 102.8646 | 107.5979 | |||
| −150 bp | 105.6333 | −1.917 | 102.3656 | −0.499 | 105.7601 | −1.838 |
| −100 bp | 103.7613 | −1.872 | 101.7280 | −0.638 | 104.2014 | −1.559 |
| −50 bp | 101.9327 | −1.829 | 100.6655 | −1.062 | 102.7583 | −1.443 |
| 0 | 100.1462 | −1.787 | 99.5000 | −1.166 | 101.6914 | −1.067 |
| +50 bp | 98.4006 | −1.746 | 98.1481 | −1.352 | 100.8681 | −0.823 |
| +100 bp | 96.6949 | −1.706 | 96.5997 | −1.548 | 100.0841 | −0.784 |
| +150 bp | 95.0278 | −1.667 | 94.9745 | −1.625 | 99.4701 | −0.614 |
| +200 bp | 93.3984 | −1.629 | 93.3774 | −1.597 | 99.0034 | −0.467 |
How do you see negative convexity in the numbers?
Read the straight column from the bottom up (as rates fall): the gain per 50 bp step grows monotonically: 1.629, 1.667, 1.706, 1.746, 1.787, 1.829, 1.872, 1.917. That acceleration is positive convexity, in points. Read the callable column the same way: 1.597, 1.625, 1.548, 1.352, 1.166, 1.062, 0.638, 0.499. The gains shrink; 200 bp into the rally an extra 50 bp is worth half a point instead of one and a half. That deceleration is negative convexity, and it is the most important price behaviour in the callable and mortgage markets.
The statistic practitioners quote is up/down capture. On a 100 bp parallel move from base:
| Instrument | Rally (−100 bp) | Sell-off (+100 bp) |
|---|---|---|
| Straight | +3.615 pts (+3.610%) | −3.451 pts (−3.446%) |
| Callable | +2.228 pts (+2.239%) | −2.900 pts (−2.915%) |
| Putable | +2.510 pts (+2.468%) | −1.607 pts (−1.581%) |
The callable captures 61.6% of the straight bond's rally but 84.0% of its sell-off, most of the pain, well under two-thirds of the gain. That asymmetry is the price of the option you sold, and it is why callables must out-yield bullets to be worth owning. The putable is the mirror: 69.4% of the rally, only 46.6% of the sell-off, paid for in lower yield.
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Does duration plus convexity predict a callable's rally?
One more diagnostic. Estimate the 100 bp rally with duration alone, add the convexity term, compare to the truth. Straight bond: duration-only +3.5281%, with convexity +3.6099%, actual +3.6099%, a +8 bp correction, almost perfectly accurate. Callable: duration-only +2.5770%, with convexity +2.2392%, actual +2.2392%, a negative −34 bp correction. A risk system running on modified duration and assuming positive convexity will systematically overstate what a callable book earns in a rally and understate what it loses in a sell-off, in the same direction every time.