We tackle in this work the problem for a player to efficiently bid in Simultaneous Ascending Auctions (SAA). Although the success of SAA partially comes from its relative simplicity, bidding efficiently in such an auction is complicated as it presents a number of complex strategical problems. No generic algorithm or analytical solution has yet been able to compute the optimal bidding strategy in face of such complexities. By modelling the auction as a turn-based deterministic game with complete information, we propose the first algorithm which tackles simultaneously two of its main issues: exposure and own price effect. Our bidding strategy is computed by Monte Carlo Tree Search (MCTS) which relies on a new method for the prediction of closing prices. We show that our algorithm significantly outperforms state-of-the-art existing bidding methods. More precisely, our algorithm achieves a higher expected utility by taking lower risks than existing strategies.