# Forward interest rate curve family parametrization

There are many academic sources, books and articles, introducing forward interest rate curve. For example, those authors define $$f(\tau)=f(\tau;\beta_0,\beta_1,\beta_2,\lambda)$$ as a function of time to maturity $$\tau$$, dependent on parameters to be estimated. Such approach is use while introducing, e.g., Nelson-Siegel or Svensson model.

However, given the spot interst rate structure $$R(\tau)$$, the forward rate $$f(\tau)$$, estimated right now, should also have one more argument $$t$$, i.e. $$f(\tau)$$ is in fact $$f(t;\tau)$$, since it's the rate for the period $$[t;t+\tau]$$, implied from the spot term structure $$R$$.

Please, let me know, if I'm missing something. Where is $$t$$ in the forward rate curve definition?

I would personally take the view that the parameters are where the "observation time" index comes into the picture. In practical terms, this could mean that at time $$t$$ we calibrate $$\Theta(t) = \beta_{0; t}, \beta_{1, t}, \beta_{2, t}, \lambda_t$$ so that $$f$$, which is otherwise not varying with $$t$$, is "close to" the observed forward curve.
Those are instantaneous forwards, spanning $$[τ,τ+dτ]$$. They are a complete description of the rates economy since all other forwards can be implied from these.