This paper investigates the day-ahead optimal reserve management problem of electric vehicle (EV) aggregator. Geographically dispersed EVs are coordinated by the aggregator to participate in the day-ahead reserve market. A bilevel model is proposed to formulate the interaction between the aggregator and the EV owners. In the upper level, the EV aggregator aggregates the reserve capacity provided by the EV owners and then bids in the reserve market. In the lower level, the EV owners decide their optimal energy charging/discharging and reserve capacity based on the reserve price released by the aggregator. Compared with existing works, our proposed bilevel model is more practical. To be specific, we take into account the exclusive right constraint for accessing EV battery, i.e., the battery cannot be simultaneously accessed by the EV owner and the aggregator. This practical model leads to a bilevel mixed integer nonlinear program, which is difficult to solve because the lower level problem incorporates nonconvex integer variables. A novel exact algorithm is developed to solve it and the finite convergence is proved. Comprehensive case studies demonstrate the economic merits of our proposed model to both the aggregator and the EV owners. We also compare the solution optimality with the state-of-the-art approach and thus validate the effectiveness of our proposed exact algorithm.