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.


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.

As the lawyers would say, “Asked and answered.”

I hope that I have.

@S2000magician Thank you very much for your informative and kind reply.

I understand your point on the fact that the forward rate the model uses is risk-free, which would imply that the binomial model assumes risk-neutrality.

In light of the above, from my knowledge I recall that the risk-neutral probability could be found in the way shown above.

So, when the CFA notes state the fact that the probabilities are equal in the lognormal model as shown on the previous attached photo, are they referring to the real (physical) probabilities or risk-neutral probabilities? I understand that there is a material difference between the two. And from my knowledge (if I am not mistaken), risk-neutral probabilities, rather than physical probabilities, should be used when discounting cash flows with risk-free rates.

Thank you very much once again and sorry if my reply may seem bewildering.

The formula on your paper deals with prices, not interest rates.

I’m pretty sure that you can’t extrapolate it to interest rates (i.e., discount factors); in general, the average of reciprocals doesn’t equal the reciprocal of the average.