When modelling stocks we specify the model in terms of the dynamics of the stock itself (e.g. in Black-Scholes, Heston and SABR - often denoted $S$).
However, as I am reading about Term Structure Models many sources start with the dynamics of the short rate (often denoted $r$) or state price vector (often denoted $X$).
Why are we using this the approach as opposed to just model the bond prices directly? And what is the difference between modelling short rates and state prices?
If you specify separate price processes for bonds with different maturities, the resulting model is typically not arbitrage free. Solving the bond prices by instead specifying a short rate process under the risk neutral measure is a convenient way to guarantee that there are no arbitrage opportunities.
How about the difference between modelling short rates under the risk neutral measure and specifying a process for state prices or the stochastic discount factor? Typically there is not much difference as the same short rate models could be derived either way. It is more about which approach is more convenient.
Note that specifying a short rate process, however, is not the only way to model bonds. For example the so called Market Model instead specifies a separate process for different forward rates and assumes these processes are correlated.
