(instantenous) fwd rate

I believe fwd rate refers to an interest rate from a future start date (like next Aug) to a future maturity date (next Nov). We are talking about the market rate to transpire on that start date. That yet-unknown rate could be inferred (in a risk-neutral sense) today, using the live market rates.

The basic calc is documented in my blog ...

When the loan tenor becomes overnight (or, theoretically, shorter than a nanosec), we call it the instantaneous fwd rate. This rate, again, can be estimated. Given observation time is today, we can estimate the fwd rate for different "fwd start dates", denoted tau. We can plot this fwd rate as a function of tau.