# Binomial int tree

Tim Brospack is generating a binomial interest rate tree assuming a volatility of 15%. Current 1-year spot rate is 5%. The 1-year forward rate in the second year is either a low estimate of 5.250% or a high estimate of 7.087%. The middle 1-year forward rate in year three is estimated at 6.25%. The lower node 1-year forward rate in year three is closest to:
ANS- 4.63%

CAN SOMEONE EXPLAIN HOW

There is a formula for this.
The rule with interest rate trees is that each node on the same year is two 2 volatilities apart from each other. We would raise Euler’s number by 2*volatility, and then multiply by the middle node.

So in this question we are given the rate for the middle node. Now what we have to do is multiply that middle number by e^(-vol*2), and then that should give us the rate for the lower node.

This also works for the upper node too! If we were to instead multiply the middle number by e^(vol*2), then we would get the rate for the upper node.

This also works when you are trying to find the upper node from the lower node. Try it out! If you had the lower node of 4.630%, and you wanted to find the upper node’s rate, you would multiply that by e^(vol*4) and you get the same upper node value.

As long as you have the volatility of the tree, and one of the rates in a given year, you can solve for the rest of the nodes for that same year!