Trusting in others and reciprocating that trust with trustworthy actions are crucial to successful and prosperous societies. The trust game has been widely used to quantitatively study trust and trustworthiness, involving a sequential exchange between an investor and a trustee. Deterministic evolutionary game theory predicts no trust and no trustworthiness, whereas the behavioral experiments with the one-shot anonymous trust game show that people substantially trust and respond trustworthily. To explain these discrepancies, previous works often turn to additional mechanisms, which are borrowed from other games such as the prisoner's dilemma. Although these mechanisms lead to the evolution of trust and trustworthiness to an extent, the optimal or the most common strategy often involves no trustworthiness. In this paper, we study the impact of asymmetric demographic parameters (e.g., different population sizes) on game dynamics of the trust game. We show that, in a weak-mutation limit, stochastic evolutionary dynamics with the asymmetric parameters can lead to the evolution of high trust and high trustworthiness without any additional mechanisms in well-mixed finite populations. Even full trust and near full trustworthiness can be the most common strategies. These results are qualitatively different from those of the previous works. Our results thereby demonstrate rich evolutionary dynamics of the asymmetric trust game.