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I'm self-studying BDT and I'm having some difficulty with what is meant by the "short-rate volatility parameter for the first year" and "the short-rate volatility parameter for the second year," as in the below problem (taken from an actuarial practice exam on models for financial economics):

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The confusion is because, intuitively, I would think "first year" refers to time $t \in [0, 1)$, "second year" refers to $t \in [1, 2)$, etc. This is because I always think about time as measured starting from $t = 0$.

So I would think that if the author intended for $\sigma_1 = 0.5\ln{(r_u/r_d)}$ to be the short rate volatility for $t \in [1, 2)$, it would have been stated "the short term volatility at the beginning of the second year."

The below quote is taken from my other study manual on models for financial economics by Abraham Weishaus:

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This seems to confirm my thoughts, or I may just be confused. Am I thinking about this incorrectly?

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To start the numbering of years with 0 is weird, but from the answer its clear, that the next three years are meant. Since we discount the bond with $\frac{1}{(1+r_{0})}$ it's clear, that $r_{0}$ is the rate for the timeperiod starting now and ending in one year.

Than we discount with all possible combinations of $r_{u}$, $r_{d}$ and $r_{uu}$, $r_{ud}$, $r_{dd}$ which means these are the rates for the following two years

That the author calls the first year $y_{0}$, the second year $y_{1}$ and the third year $y_{2}$ seems unusal. If he has a reason for it, it's not apparent from the excerpt you posted.

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