# Equal Probability Assumption in Binomial Trees

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.

Thanks.

I’m not sure why log-normality enters into the discussion; these are trees of interest rates, not prices.

I find it simpler to think of the “probabilities” as weights.

When creating a binomial tree, you have essentially two options:

• Decide on the weights and calculate the values (prices, interest rates, whatever) at each node
• Decide on the values at each node and calculate the weights

The risk-neutrality comes from calibrating the tree to price risk-free bonds correctly, using risk-free rates.

The one-period forward rates are one-period risk-free rates; they’re the one-period risk-free forward rates implied by the risk-free spot curve.

Risk-neutrality is a product of using risk-free rates and calibrating the tree to price risk-free bonds correctly.

I hope that I have.