In this paper we first present a geometric approach to option bounds. We show that if two risk neutral probability density functions intersect for certain number of times, then comparing the fatness of their tails we can tell which of them gives higher option prices. Thus we can derive option bounds by identifying the risk neutral probability density function which intersects all admissible ones for certain number of times. Applying this approach we tighten the first order stochastic dominance option bounds from concurrently expiring options when the maximum value of the risk neutral density are known.