Predicting the Maximum Lead from Final Scores in Basketball: A Diffusion Model

We present an analysis of a time-dependent stochastic model of the real-time progress of the difference between the home and the away team's scores in sports contests. The model was originally proposed by Stern (1994) and treats the evolution of the home team lead (or deficit) as a Wiener diffusion process. We derive the distribution, the mean, and the variance, of the maximum home team lead during one match, both unconditionally, and also conditional on the final score. We present estimates of the model's parameters, and apply the model to the joint distribution of the maximum home team lead and the final results of one complete season (2010/2011) of the top German basketball league. The diffusion model predicts the maximum home team lead from the final results fairly accurately which suggests that static result-oriented accounts may profitably be complemented by dynamic accounts which model the real-time events leading up to the final score.