With the rapid adoption of intermittent energy sources, HVDC transmission plays an increasingly significant role in power systems. To optimize the utilization of AC/DC grids with a high share of RES, there is a critical need for an accurate quantification tool that can effectively represent the uncertain operational characteristics of RES while incorporating the control capabilities of HVDC converter stations. This paper addresses this need by introducing a general polynomial chaos based non-sampling, one-shot mathematical framework capable of solving the stochastic optimal power flow problem in hybrid AC/DC grids. The proposed framework not only incorporates HVDC controllability and accounts for uncertainties caused by demand and renewable generation, but it also offers the additional capability of assessing the hosting capacity in meshed AC/DC grids. To show the effectiveness of the framework, three case studies are conducted on 5-bus and 67-bus AC/DC test systems. The 5-bus system is used for comparison with Monte Carlo simulations, while the studies on 67-bus system emphasize the enhanced flexibility of HVDC converters and explore the effects of various confidence levels. The findings of this research contribute to optimizing grid operation in RES-rich systems, offering guidance for the optimal utilization of hybrid AC/DC grids.