In this paper, we examine the security issue of unmanned aerial vehicle (UAV)-aided communication systems. We consider the scenario where a source communicates with a legitimate UAV in the presence of a full-duplex eavesdropper. We assume that the full-duplex eavesdropper performs both eavesdropping and malicious jamming simultaneously. In order to safeguard data transmissions against such an eavesdropper, the source transmits artificial noise (AN) signals together with information signals. We adopt the hybrid outage probability as the performance metric, which combines the transmission outage probability and the secrecy outage probability. We derive a compact expression for the hybrid outage probability. Using the derived expression, we determine the optimal power allocation factor between information signals and AN signals that minimizes the hybrid outage probability. In addition, we examine the impact of the operating height of UAV on the secrecy performance, showing that the optimal secrecy performance can be achieved by judiciously controlling the height of UAV.