# If short rates $r(t)$ do not determine the bond prices $P(t, T)$, then what is the basis for short rate models?

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?

• They DO NOT determine bond prices under the objective measure, but they DO under the risk neutral one. – ab94 Sep 7 '19 at 14:04

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.