Modified duration and analytical convexity are closed-form derivatives of a price function whose cash flows are fixed. When the cash flows themselves move with rates, those formulas are not approximations, they are answers to a different question. The replacements are effective measures, defined by brute-force revaluation:
Effective duration = (V₋ − V₊) / (2 × V₀ × Δy) Effective convexity = (V₊ + V₋ − 2V₀) / (V₀ × Δy²)
where V₋ and V₊ are the values after shifting the benchmark curve down and up by Δy, recalibrating the tree, and revaluing at the same OAS. Every clause matters: shift the benchmark curve, not the bond's own yield; recalibrate, so the tree stays arbitrage-free; hold OAS constant, because letting it float would contaminate rate sensitivity with credit moves. The three instruments of this module's US example, at a constant OAS of 56.20 bp:
| Instrument | Shock | V₋ | V₀ | V₊ | Effective duration | Effective convexity |
|---|---|---|---|---|---|---|
| Straight | ±100 bp | 103.7613 | 100.1462 | 96.6949 | 3.528 | +16.4 |
| Callable | ±100 bp | 101.7280 | 99.5000 | 96.5997 | 2.577 | −67.6 |
| Putable | ±100 bp | 104.2014 | 101.6914 | 100.0841 | 2.024 | +88.8 |
Three readings. The callable's duration is materially shorter (2.58 versus 3.53), the call truncates the long end of the cash flows in exactly the scenarios where duration would otherwise be longest. The putable's is shorter still (2.02), because the put converts a four-year bond into something closer to a two-year bond when rates rise. And the callable's convexity is negative while the putable's is strongly positive.
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How big a shock should you use for effective convexity?
A crown-level caveat on shock size. Effective convexity for an option-embedded bond is shock-size dependent, because the exercise boundary is a kink a small shock may not straddle. The same callable at the same OAS gives +5.1 at ±25 bp, −74.9 at ±50 bp and −67.6 at ±100 bp. That is not an arithmetic bug; it is what a Bermudan kink does to a finite-difference second derivative on a coarse lattice. Refining the lattice confirms it: rebuilding the identical bond on the identical zero curve gives ±25 bp convexity of +5.1, +12.5, +3.9 and −127.1 at 1, 2, 4 and 12 steps per year, while the ±100 bp measure is stable and negative throughout (−67.6, −76.8, −80.6, −86.4). The rule: use a shock large enough to move the exercise decision (±50 to ±100 bp), report the shock size, and never compare a ±25 bp convexity to a ±100 bp one.