We introduce two new notions called the Daugavet constant and Δ-constant of a point, which measure quantitatively how far the point is from being Daugavet point and Δ-point and allow us to study Daugavet and Δ-points in Banach spaces from a quantitative viewpoint. We show that these notions can be viewed as a localized version of certain global estimations of Daugavet and diametral local diameter two properties such as Daugavet indices of thickness. As an intriguing example, we present the existence of a Banach space X in which all points on the unit sphere have positive Daugavet constants despite the Daugavet indices of thickness of X being zero. Moreover, using the Daugavet and Δ-constants of points in the unit sphere, we describe the existence of almost Daugavet and Δ-points, as well as the set of denting points of the unit ball. We also present exact values of the Daugavet and Δ-constant on several classical Banach spaces, as well as Lipschitz-free spaces. In particular, it is shown that there is a Lipschitz-free space with a Δ-point, which is the furthest away from being a Daugavet point. Finally, we provide some related stability results concerning the Daugavet and Δ-constant.