Constructing a Binomial interest rate tree

[Tural]

Hi all,

Why do we need to use forward rate multiplied by e⁽⁺¹⁾ and e^(-1) standard deviation fot Time 1; and by e⁽⁺²⁾ and e^(-2) standard deviations for Time 2? In my understanding, since Time 1 is accommodating one standard deviation already, would not it be logical to multiply by e⁽⁺¹⁾& e^(-1) at all consecutive steps?

Thanks

[S2000magician]

No, it would not be logical, nor consistent.

If the rate lower rate (r_L) at time 1 is the forward rate times e^−1×σ and the upper rate (r_U) at time 1 is the forward rate times e^+1×σ, then the upper rate satisfies:

r_U = r_L × e^2×σ.