The question title says it all: We know that in general, specifying the short rate $r(t)$ does not specify the bond prices $P(t, T)$. So how can a model for short rates—for example the Vasicek model—be powerful enough to price interest rate derivatives?
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2$\begingroup$ They DO NOT determine bond prices under the objective measure, but they DO under the risk neutral one. $\endgroup$– ab94Commented Sep 7, 2019 at 14:04
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1 Answer
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Let $r(s)$ be the process of a short rate. Then, by risk neutral pricing, $$ P(t,T) = \mathbb{E}^\mathbb{Q}\left[ \exp\left( -\int_t^T r(s)\mathrm{d}s\right) \Bigg| \mathcal{F}_t\right].$$ Thus, the zero-coupon bond is determined completely by the short rate process. Here, $P(t,T)$ denotes the time $t$ price of a zero-coupon bond maturing at time $T$. You just take the risk-neutral expectation of the discounted payoff. The payoff is $1$ for almost all states of the world $\omega\in\Omega$ (assuming no default risk). Thus, the price of the bond is the conditional expectation of the discount factor. The risk-neutral measure $\mathbb{Q}$ uses a bank account $(B_t)$ as numeraire with $\mathrm{d}B_t=r(t)B_t\mathrm{d}t$.
Short rate models (such as Vasicek, Hull-White, CIR, etc.) specify a stochastic model for $r(s)$, typically a (perhaps multidimensional) SDE and then, you can find (sometimes analytical) prices for bonds, bond options, swaptions etc.
The easiest case is a deterministic and constant short rate $r(s)\equiv r$. Then, $$P(t,T)=e^{-r(T-t)}$$ and clearly the short rate $r$ gives you the bond price.
