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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)?

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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.

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