Duration tells you approximately how much a bond's price will change for a given change in interest rates. But duration is only a linear approximation of a curved relationship — and the curvature itself contains information that matters enormously for bond investors, mortgage markets, and options traders. That curvature is convexity: one of the most important and most underappreciated concepts in fixed income.
Duration: the linear approximation
Duration estimates that a 1% rise in yields will cause a bond's price to fall by approximately [duration] percent. A 10-year bond with duration of 8 years falls roughly 8% when yields rise 1%. But this is a linear approximation of a non-linear relationship. The true price/yield curve is curved (convex toward the origin) — meaning small changes in yield are approximated well by duration, but large changes in yield cause the bond to perform better than duration alone predicts. Convexity measures how curved this relationship is, and allows a more precise second-order approximation: ΔPrice ≈ −Duration × ΔYield + (1/2) × Convexity × (ΔYield)².
Negative convexity: the mortgage market problem
Not all bonds have positive convexity. Callable bonds and mortgage-backed securities (MBS) exhibit negative convexity in certain yield environments. With a callable bond, the issuer can redeem it when rates fall — exactly when the bondholder would benefit most from holding it. This option caps the bond's price appreciation: as yields fall toward the call price, the bond's price rise is truncated because the market prices in the probability of early redemption. The same effect appears in MBS: when rates fall, homeowners refinance their mortgages at lower rates, prepaying the MBS principal and shortening the bond's effective maturity. This is prepayment risk — the bond shortens in duration just when duration was most valuable — creating negative convexity.
Convexity in options: the key insight
Convexity appears in a different form in options markets, where it is captured by gamma. An option is inherently convex — its payoff profile is non-linear: it benefits asymmetrically from moves in the underlying (unlimited upside on a call, fixed maximum loss equal to premium paid). Buying options buys positive convexity; selling options sells positive convexity (or equivalently, takes on negative convexity). This is why long options positions benefit from large moves regardless of direction — their convex payoff means they gain more from large moves than they lose from small ones, all else equal. This is the same mathematical principle that makes a positively convex bond perform better than duration predicts in large rate moves.