Geometrically, the model is no longer flat but curved, and the degree of curvature is called the convexity.
Formally, the convexity adjustment arises from the Jensen inequality in probability theory: the expected value of a convex function is greater than or equal to the function of the expected value: Geometrically, if the model price curves up on both sides of the present value (the payoff function is convex up, and is above a tangent line at that point), then if the price of the underlying changes, the price of the output is greater than is modeled using only the first derivative.
This value is isolated via a straddle – purchasing an at-the-money straddle (whose value increases if the price of the underlying increases or decreases) has (initially) no delta: one is simply purchasing convexity (optionality), without taking a position on the underlying asset – one benefits from the degree of movement, not the direction.
From the point of view of risk management, being long convexity (having positive Gamma and hence (ignoring interest rates and Delta) negative Theta) means that one benefits from volatility (positive Gamma), but loses money over time (negative Theta) – one net profits if prices move more than expected, and net loses if prices move less than expected.
Applying Girsanov's theorem[1] allows expressing the dynamics of the modeled financial variables under the pricing measure and therefore estimating this convexity adjustment.