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HJM framework uses the instantaneous forward rate $f(t,T)$ in the resulting dynamics and pricing formulas (like in Hull-White or Ho-Lee model).

But clearly market does not have an $f(t,T)$ formula, so I guess you can create one of your own (or create a $P(t,T)$ function and transform it to $f(t,T)$). So far I've thought about some alternatives, but I haven't found much literature about it.

Any thoughts on these alternatives or is there something better? My thoughts:

  1. Make a cubic spline interpolation of observable $P(t,T)^*$ to get a continuous form of $P(t,T)$.
  2. Same as 1, but with a high order polynomial.
  3. Fit a Nelson-Siegel-Svenson model to zero coupon rates, then get $P(t,T)$ and continue.
  4. Fit an equilibrium short rate model like CIR to observable $P(t,T)^*$, then get a $P(t,T)$ function and continue.

My intuition tells me that 1 or 2 might be better as they fit exactly to market data. With 3 and 4, models lose its arbitrage-free condition which is its essence.

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