In conventional cognitive radio ad hoc networks, both primary users (PUs) and secondary users (SUs) are typically equipped with omni-directional antennas, which can cause high interference. As a result, SUs usually suffer from the poor network connectivity due to the high interference caused by PUs. Different from omni-directional antennas, directional antennas can concentrate the transmission on some desired directions and reduce interference on other undesired directions. In this paper, we investigate the network connectivity of a novel cognitive radio ad hoc network with directional antennas (DIR-CRAHNs), in which both PUs and SUs are equipped with directional antennas. We derive the local connectivity (i.e., probability of node isolation) of such DIR-CRAHNs in closed form and numerical results show that the proposed DIR-CRAHNs have higher local connectivity than conventional cognitive radio ad hoc networks.