Exercise: Rho vs DV01 for a long-dated swaption

Consider a 10-year European receiver swaption on a 10-year USD swap. The swaption value $V$ depends on the swap rate and on the entire discount curve.

A junior quant computes

\rho_V = \partial V / \partial r

by bumping a single flat risk-free rate used in a Black-76 style pricing formula — treating the entire curve as one number — and reports "rho =

+150{,}000

per

1\%

." The senior trader responds: "That's not the hedge ratio I need. Give me DV01 instead."

Tasks

State in one sentence why $\rho$ is inadequate for a product whose value depends on the entire yield curve.
Define DV01 and key-rate DV01 (KRD) formally.
For a 10-year swaption whose underlying swap has cash flows at years $10, 11, \ldots, 20$ , sketch how you would compute KRD at the $10$ -year, $15$ -year, and $20$ -year curve pillars. Which KRD is likely largest in absolute value?
Give one scenario in which a positive $\rho$ and a zero DV01 at every key rate can coexist. Why is this possible, and what does it tell you about the flat-rate shift?

Hint

A flat-rate parallel shift is a special linear combination of key-rate shifts. DV01 vs KRD is like a single-number approximation vs a vector.