think of it as a callable bond, the convexity is bond covexity minus call option convexity

second, as rate drop, duration of MBS decrease, as rate drop, price increase for a normal bond, in aggregate, dv01 decrease in magnitude (but still negative), so dv01 increase, leading to negative convexity(sensitivity of dv01)

third, as rates drop, more prepayment, as a result, MBS increase in price not as fast as a normal bond, the curvature is concave, meaning negative convexity

mortgage bond is a portfolio of long position in bond and short position in option.