Regarding the binomial interest rate tree model, I struggle to understand the assumption of equal probabilities under log-normality.
To the best of my knowledge, interest rate tree models are based on risk-neutral probabilities, which in general the risk-neutral probabilities for the lattice tree would not equal.
However, I do notice that the textbook is discounting the expected bond cashflow with the one-year FORWARD rate, not the one-period risk-free rate conventional in risk-neutral valuation.
Am I getting confused and missing the fact that risk-neutrality is not assumed in the binomial tree model for interest rates?
Moreover, to be precise, why are the “probabilities” equal under log-normality in the binomial tree model?
It would be great if anyone could shed some light in this.