In future wireless networks, the availability of information on the position of mobile agents and the propagation environment can enable new services and increase the throughput and robustness of communications. Multipath-based simultaneous localization and mapping (SLAM) aims at estimating the position of agents and reflecting features in the environment by exploiting the relationship between the local geometry and multipath components (MPCs) in received radio signals. Existing multipath-based SLAM methods preprocess received radio signals using a channel estimator. The channel estimator lowers the data rate by extracting a set of dispersion parameters for each MPC. These parameters are then used as measurements for SLAM. Bayesian estimation for multipath-based SLAM is facilitated by the lower data rate. However, due to finite resolution capabilities limited by signal bandwidth, channel estimation is prone to errors and MPC parameters may be extracted incorrectly and lead to a reduced SLAM performance. We propose a multipath-based SLAM approach that directly uses received radio signals as inputs. A new statistical model that can effectively be represented by a factor graph is introduced. The factor graph is the starting point for the development of an efficient belief propagation (BP) method for multipath-based SLAM that avoids data preprocessing by a channel estimator. Numerical results based on synthetic and real data in challenging single-input, single-output (SISO) scenarios demonstrate that the proposed method outperforms conventional methods in terms of localization and mapping accuracy.