In order to ensure reliable autonomous driving, the system must be able to detect future dangers in sufficient time to avoid or mitigate collisions. In this paper, we propose a collision risk assessment algorithm that can quantitatively assess collision risks for a set of local path candidates via the lane-based probabilistic motion prediction of surrounding vehicles. First, we compute target lane probabilities, which represent how likely a driver is to drive or move toward each lane, based on lateral position and lateral velocity in curvilinear coordinates. And then, collision risks are computed by incorporating both model probability distribution of lanes and a time-to-collision between a pair of predicted trajectories. Finally, collision risks are plotted on a trajectory plane that represents each set of the tangential acceleration and the final lateral offset of local path candidates. This collision risk map provides intuitive risk measures, and can also be utilized to determine a control strategy for a collision avoidance maneuver. Validation of the model is conducted by comparing the model probabilities with the maneuver probabilities derived from the next generation simulation database. Furthermore, the effectiveness of the proposed algorithm is verified in two driving scenarios, preceding vehicle braking and cut-in, on a curved highway with multiple vehicles.