Adjacent nodes in an interest rate tree

CFA Level II
#1

I understand from the lesson that adjacent forward rates at any nodal period are 2 standard deviations appart. In an exercice, we are looking at the value of the lower one period rate, and we know the following :

  • the upper rate : 5,8365%
  • the volatility = 25%

The solution is : lower one period rate = 5,8365% * e(-2 * 0.25)

Could someone explain to me why we use exponential and not just 5,8365% * (-2 * 0.25) ?

Should we systematically use exponential to compute adjacent nodes, or is it specific to this exercice?

#2

There’s no good reason to choose the exponential treatment over simply multiplying (though your formula for multiplying is wrong in either case). Someone a long time ago decided that we would treat the volatility as continuous rather than discrete, so that’s the way it’s done. Always.

#3

Thank you S2000 magician, it is clearer! But then, why do you say that the formulas are wrong? I should not multiply? (if not mistaken, it is the solution I read in the material)

#4

If you do simple multiplication, you will not find the solution in the answer choices.

#5

You would multiply by (1 + 2×25%) to go to the next higher node, or divide by (1 + 2×25%) to go to the next lower node.