Affective personality traits have been associated with a risk of developing mental and cognitive disorders and can be informative for early detection and management of such disorders. However, conventional personality trait detection is often biased and unreliable, as it depends on the honesty of the subjects when filling out the lengthy questionnaires. In this paper, we propose a method for objective detection of personality traits using physiological signals. Subjects are shown affective images and videos to evoke a range of emotions. The electrical activity of the brain is captured using EEG during this process and the multi-channel EEG data is processed to compute the inter-hemispheric asynchrony of the brainwaves. The most discriminative features are selected and then used to build a machine learning classifier, which is trained to predict 16 personality traits. Our results show high predictive accuracy for both image and video stimuli individually, and an improvement when the two stimuli are combined, achieving a 95.49% accuracy. Most of the selected discriminative features were found to be extracted from the alpha frequency band. Our work shows that personality traits can be accurately detected with EEG data, suggesting possible use in practical applications for early detection of mental and cognitive disorders.