Emotions that somebody develops based on an argument do not only depend on the argument itself - they are also influenced by a subjective evaluation of the argument's potential impact on the self. For instance, an argument to ban plastic bottles might cause fear of losing a job for a bottle industry worker, which lowers the convincingness - presumably independent of its content. While binary emotionality of arguments has been studied, such cognitive appraisal models have only been proposed in other subtasks of emotion analysis, but not in the context of arguments and their convincingness. To fill this research gap, we propose the Contextualized Argument Appraisal Framework to model the interplay between the sender, receiver, and argument. We adapt established appraisal models from psychology to argument mining, including argument pleasantness, familiarity, response urgency, and expected effort, as well as convincingness variables. To evaluate the framework and pave the way for computational modeling, we develop a novel role-playing-based annotation setup, mimicking real-world exposure to arguments. Participants disclose their emotion, explain the main cause, the argument appraisal, and the perceived convincingness. To consider the subjective nature of such annotations, we also collect demographic data and personality traits of both the participants and ask them to disclose the same variables for their perception of the argument sender. The analysis of the resulting ContArgA corpus of 4000 annotations reveals that convincingness is positively correlated with positive emotions (e.g., trust) and negatively correlated with negative emotions (e.g., anger). The appraisal variables particularly point to the importance of the annotator's familiarity with the argument.