The presence of obstinate types in bargaining has been shown to alter dramatically the bargaining equilibrium strategies and outcomes. This paper shows that outside options may cancel out the effect of obstinacy in bargaining. When parties have access to stationary outside options, we show that when opting out is preferable to accepting the inflexible demand of the other party, there is a unique Perfect Bayesian Equilibrium in which each party reveals himself as rational as soon as possible. A similar conclusion holds when outside options may only be available at a later date or when only one party has access to an outside option.