# Why is the movement in a binomial tree 50% up or down?

In valuing a bond with a binominal tree, why is the movement up or down 50%? Why not 40% / 60%?

You could make it 40/60 if you wanted to.

Why would you want to?

This is because the binomial model uses a lognormal distribution which assumes that the probability are equal for an up and down movement

I believe that you’re incorrect about that assumption.

I have a 10-period binomial tree model I build in Excel in which I assumed equal probabilities for up and down movements. I just changed it so that I can specify the probability of the up and down movements. I calibrated it for some arbitrary par curve with 50/50 probabilities and priced a 10-year bond with a weird coupon. Then I changed the probabilities to 40/60, recalibrated it, and priced the same 10-year bond. The prices differ by less than one penny.

For giggles, I changed P(up) to 10%, then to 90%. The prices were all within one penny of each other.

The choice of probability for the up movement does, however, have an effect on the estimated price of bonds with embedded options, assuming that the rule for exercising the option is that it is always exercised when it is in the money.

Interesting.

The question was ‘why’ we assume 50% prob not if it has an effect; (of course if we changed the probabilities the value would change).

The binomial method for bond price estimation uses a lognormal distribution ( Reading 35, page 86 of the 2019 CFA institute FI and Derivatives book). This is because modeling interest rates follows a lognormal random walk.

For an option-free bond, the value _ doesn’t _ change. Reread what I wrote, above.

I’m aware of that.

The definition of a lognormal random walk, as far as I know, does not depend on the assumption that the probability of an up movement is 50%. And, as I mentioned above, for an option-free bond it doesn’t have to, as you get the same price no matter what you assume the probability (technically, the weight) is.

The section in the textbook I noted above explains clearly why. I hope that helps.

If you disagree with the CFA notes, I’m not sure what else I can add of value to this question.

It doesn’t explain why, clearly or otherwise.

What the text says is, “This expected value is the average of the value for the forward rate being higher, to be denoted below by VH, and the value for the forward rate being lower, VL. It is a simple average because in the lognormal model the probabilities are equal for the rate going up or down.”

So they say that the probabilities are equal (which isn’t remotely an explanation of _ why _), but they certainly don’t say that they have to be equal. And, as I mentioned earlier, they don’t have to be equal for an option-free bond: you get the same price whether they’re 50/50 or 40/60 or 90/10 or 10/90.

Great. Thanks for clarifying.

Thanks S2000magician as always, and appreciate the replies too, Tolu Dipo.

My pleasure.

You’re quite welcome.