Hi @sm23 There is a lot to say about your brief questions, which are very good, but much of the substance of the answers is thematic in Tuckman's Chapter 9 and 10. Further, it's hard to just talk about these concepts, they are best understood in the spreadsheets. But briefly,
We want to distinguish between the recombining short-term rate tree and the rate term structure. The recombining tree plots possible (one standard deviation) changes in the short-term (in Tuckman, either six month or one year) interest rate; so the tree says (eg) "today the one-year rate is 10.0% and next year it might jump up to a one-year rate of 12.0% or down to a one-year rate of 8.0%." The term structure reveals the sequence of rates for a zero-coupon bond: the one-year rate is 10.0%, the two-year rate is 9.95%, the three-year rate is 9.90% etc. In Model 1, if the short-term rate recombines to the same value (e.g., 10.0%), then it is because of the discussed convexity that the term structure will decline (although the decline will be offset by any risk premium, which acts in the opposite direction); this convexity is due to the volatility (aka, uncertainty) in the future rate, which shifts the price/yield away from its value if the rate were certain.
This is advanced (even GARP has misapplied this concept), please see https://www.bionicturtle.com/forum/threads/vasicek-model-question.10477/ . As Tuckman explain, the basic Vasicek model employs mean reversion with the theta parameter, Θ. For most practical purposes, we can stop there and view this long-run value of the short-rate as analogous to the long-run variance in GARCH(1,1); i.e., the rate toward which the short-rate feels a "gravitational pull" or central tendency. Going deeper, this theta is parsed into a "true long term rate" and a risk premium.
The mean reversion parameter naturally dampens the term structure of volatility, which is different than the term structure of the interest rate. If we think about scaling volatility when the variable is independent, that's a dispersion process, right? 2-year volatility is sqrt(2) times greater than 1-year volatility. But mean reversion is "pulling the volatility" back with central tendency. I hope that's helpful. Thanks!
I just read them off our learning spreadsheet at https://www.bionicturtle.com/topic/learning-spreadsheet-tuckman-chapter-7/ . I was just trying to overcome the conceptual issue. You can see the XLS for the exact calcs; also, our notes show the calculation, and Tuckman the source explains very carefully, maybe you can take a look at all of that pre-existing work? if you still have a specific question, let me know!