Convexity measures the rate of change of duration with respect to yield. While duration provides a linear approximation of price sensitivity, convexity captures the curvature: the fact that bond prices rise more when yields fall than they decline when yields rise by the same amount.
For a standard non-callable bond, convexity is always positive. This means the duration estimate understates price gains when yields fall and overstates price losses when yields rise. The price change approximation including convexity is:
Price change = -modified duration x yield change + 0.5 x convexity x (yield change)^2
Convexity scales roughly with the square of maturity. A 30-year Treasury has substantially more convexity than a 10-year, which is one reason long-duration bonds carry a convexity premium: investors value the asymmetric payoff.
Convexity matters most for large yield changes. For small moves (a few basis points), the linear duration approximation is sufficient. For large moves (50+ bps), ignoring convexity produces meaningful pricing errors, especially for long-dated instruments.
Traders who are long convexity benefit from volatility because they gain more from favorable moves than they lose from unfavorable ones. This is why convexity is sometimes called "free insurance" for bondholders, though the market prices it through lower yields on high-convexity instruments.