Hull–White

dr = (θ(t) − a r) dt + σ dW

r is today’s overnight rate. θ(t) is the pull toward a level, and it can change with t so the model can sit on a starting curve. a is how strongly a gap from that level is closed. σ is the size of the random shock. dW is that shock over a small step. The shock is a model input. It is not a forecast of the next fixing.

One classroom step: the short rate is 6%, and θ(t) is pulling toward 4%. a is 0.05, so the next small step is drawn back toward 4%. σ is 0.01, so that same step is then shocked. The next point on the path is the pull and the shock together.

Vasicek uses the same pull, with the target level held constant instead of θ(t). CIR ties the size of the shock to the level of the rate, which is one way the model keeps the rate from going negative. Heath–Jarrow–Morton writes the shock on every forward rate along the curve, rather than on a single overnight rate. Hull–White is the short-rate model this track uses when a Bermudan has to compare exercise with waiting.

A European swaption can be priced from one forward swap rate and one volatility, because there is one exercise date. A Bermudan needs a rate on each exercise date, and a value for still holding the option. That comparison is the Bermudan exercise lesson.