Explain how one-sided and key rate durations help assess the rate sensitivity of bonds that contain embedded options

One-sided durations measure sensitivity to a fall and to a rise in rates separately: down-duration = (PV₋ − PV₀) ÷ (ΔCurve × PV₀) and up-duration = (PV₀ − PV₊) ÷ (ΔCurve × PV₀). They reveal asymmetry that a single effective duration averages away. For a callable bond near its call price, the down-duration is smaller, because the call caps gains when rates fall. For a putable bond near its put price, the up-duration is smaller, because the put limits losses when rates rise. Key rate durations show which points on the curve matter: a callable bond deep in the money is most sensitive to the rate at its call date.