Reading Year Labels and Short Rates in the Black-Derman-Toy Model
Summary
The note clarifies how to interpret year labels and short-rate nodes in a Black-Derman-Toy bond tree. It explains that the initial rate, r₀, discounts the bond over the period from now to one year, while the up and down rates at the next tree level apply over the following year, with later nodes covering subsequent periods. This resolves the apparent mismatch between counting the first year from time zero and labels that start at y₀.
The explanation uses the discounting sequence in the example as its evidence: the initial bond discount uses r₀, and later discounting uses the rates at subsequent tree levels. It concludes that calling the periods y₀, y₁, and y₂ is unusual, but the excerpt gives no reason for that convention. The note is a notation clarification rather than a derivation of the model or a general account of how to calibrate it.
Key ideas
- The initial short rate r₀ applies to the period from the present to one year later.
- Rates at the next tree level apply to the following year.
- The tree’s discounting sequence can help resolve ambiguous year labels.
- Starting year labels at y₀ is unusual, and the excerpt does not explain why it is done.
Tags
Full text
# Clarification on the Black-Derman-Toy model regarding measuring time and notation
# Clarification on the Black-Derman-Toy model regarding measuring time and notation
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):
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:
This seems to confirm my thoughts, or I may just be confused. Am I thinking about this incorrectly?
## Answer by Ami44 (score 1, accepted)
https://quant.stackexchange.com/a/32370
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.