I'm relatively new to working with interest rate models and I am having some conceptual difficultly considering how a short-rate model can be utilized when calculating OAS of a mortgage-based bond, whose prepayment behavior will be based on longer term rates in any given month ($t$). For example, as a simple model assumption, a mortgage borrower might elect to prepay when 10 yr rates are 100 bps lower than their current coupon rate on the loan. A prepayment penalty calculation within the prepay model might also be based on current long term rates (such as the US10yr or US7yr).
In the context of a short-rate model this is my question specifically: if I calibrate a short-rate tree (such as BDT or Ho-Lee) then generate sample paths from this tree, can I infer a term structure of interest rates given a short-rate $r$ in a month $t$ at any point in the model/simulation?
