Recently, spectrum sharing has been considered as a promising solution to improve the spectrum utilization. It however may be vulnerable to security problems as the primary and secondary network access the same resource. Therefore, in this paper, we focus on the performance analysis of a cognitive radio network in the presence of an eavesdropper (EAV) who illegally listens to the primary user (PU) communication in which the transmit power of the secondary transmitter (SU-Tx) is subject to the joint constraint of peak transmit power of the SU-Tx and outage probability of the PU. Accordingly, an adaptive transmit power policy and an analytical expression of symbol error probability are derived for the SU. Most importantly, security evaluations of primary network in terms of the probability of existence of non-zero secrecy capacity and outage probability of secrecy capacity are obtained. Numerical results reveal a fact that the security of the primary network does not only depends on the channel mean powers between primary and secondary networks, but also strongly depends on the channel condition of the SU-Tx to EAV link and transmit power policy of the SU-Tx.