HJM, briefly

* HJM uses (inst) fwd rate, which is continuously compounded. Some alternative term structure models use the “short rate” i.e. the extreme version of spot overnight rate. Yet other models [1] use the conventional “fwd rate” (i.e. compounded 3M loan rate, X months forward.)

[1] the Libor Mkt Model

* HJM is mostly under RN measure. The physical measure is used a bit in the initial SDE…

* Under RN measure, the fwd rate follows a BM (not a GBM) with instantaneous drift rate and instantaneous variance both time-dependent but slow-moving. Since it’s not GBM, the N@T is Normal, not LG
** However, to use the market-standard Black’s formula, the discrete fwd rate has to be LN

* HJM is the 2nd generation term-structure model and one of the earliest arbitrage free model. In contrast, the Black formula is not even an interest rate model.

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