Bond Convexity

\[ \frac{ \Delta P } { P } = - D_E \Delta y + \frac{1}{2} C_E \delta y ^ 2 \]

where \(D_E, C_E \) are the effective duration and effective convexity of the bond

Dollar Convexity \[ C = \frac{ \partial^2 P } { \partial y^2 } \]

Effective Convexity \[ C_E = \frac{C}{P} \]

Approximate convexity \[ C_E \approx - \frac{ P ( y + \Delta y ) - 2 P(y) + P ( y - \Delta y ) } {P(y) \Delta y ^ 2 } \]