Calibrating binominal tree

CFA Level II Fixed Income
#1

I can’t quite understand can we calibrate a binominal tree without excel? If we have estimated f(1,1), f(2,1) etc. and the volatility is given, can we build the binominal tree without Solver? If not, why?

My logic is as follows: at t=1 if I have estimated f(1,1):

i(1,high) shall be equal to f(1,1) multiplied by e power sigma;

i(1,low) shall be equal to f(1,1) multiplied by e power minus sigma

At t=2 for estimated f(2,1):

i(2, high, low) = f(2,1)

i(2, high,high) = f(2,1) multiplied by e power 2*sigma

i(2, low, low) = f(2,1) multiplied by e power -2*sigma etc.

When I try to build a binominal tree in such a way however I’m not achieving the interest rate tree that is given by CFAI / Wiley. I haven’t found in Reading 43 where I’m wrong. Any idea?

#2

if such a question comes up, just take the middle rate from the answers and plug it into the tree and see if it works.

#3

I’ll do what you suggested during the exam but now I’m just trying to understand why the abovementioned formulas work only approximately.

#4

Because calibrating a binomial tree involves solving polynomial equations.

You can solve for the rates in period 1 (solving a quadratic equation), in period 2 (solving a cubic equation) and in period 3 (solving a quartic equation) analytically, but you cannot solve for the rates in later periods analytically (quintic and higher-order polynomial equations cannot be solved using radicals, a theorem proven initially by Évariste Galois in the late 1820s).