# what's the difference between instantaneous short rate and instantaneous forward rate?

In the short rate models, sometimes it models the instantaneous short rate and sometimes it models the instantaneous forward rate. Does instantaneous short rate = F(0, t + tau) and instantaneous forward rate = F(0, t, t + tau)?

In more standard notation the instantaneous forward rate is written as $$f(t,T)$$, that is, the continuously compounded interest rate seen at $$t$$ for the infinitesimal interest period $$[T,T+dt]\,.$$ Heath Jarrow & Morton decided to model those forward rates. Likewise, one can model the constant maturity forward rates $$f(t,t+\tau)\,.$$ These models are very general (almost too general to be useful in practice) and need further specifications. For example: when the volatility of the forward rate $$f(t,T)$$ is chosen to be $$\sigma(t)\exp(-\int_t^T\lambda(s)\,ds)$$ you will get the Hull-White model of the short rate $$r(t)=f(t,t)$$ which satisfies the SDE $$dr(t)=\lambda(t)(\theta(t)-r(t))\,dt+\sigma(t)\,dW_t\,.$$ HJM's work showed that the no arbitrage properties of many interest rate models can be studied under a common framework.