Convexity is a measure of the degree of curvature of the price/yield relationship, to indicate the error in the estimated change in bond's price based on duration.
Effective convexity = (V_ + V+ - 2 x Vo) / (2 x Vo x dy2).
It is the 2nd derivative of price function wrt yield. For callable bond, V is capped at call price.
Contribution of convexity = Convexity x dy2.
For noncallable bond, convexity effect is always +ve no matter which direction interest rate move but it can be –ve if the bond has embedded options.
Thus, approximate bond price change (using duration and convexity), i.e.
Approx. price change = -1 x Duration x dy + Convexity x dy2
Modified Convexity vs. Effective Convexity
With modified convexity the cash flows do not change due to a change in interest rates.
Effective Convexity, on the other hand, assumes that cash flow does change due to a change in interest rates.
When bonds have options, it is best to use effective convexity just like you should use effective duration. For option-free bonds, either convexity measure will be a positive value, whereas when it comes to bonds with options, the effective convexity could be negative even if the modified convexity is positive.
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