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As I see people trading 10-year notes are often quoting implied volatilities in terms of daily (or yearly) basis points. While at first I thought it is simply a translation from % to the actual spot change, it seems to be not the case and is somehow to be calculated through the volatility of the yield. However, I can't find anywhere a good step-by-step representation of how it's done and numbers I get manually are different from what I am supposed to get. I assume it is a pretty basic question, but I'd be very interested to know how is it actually done and what this measure is supposed to represent?

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Basis point implied volatility is calculated as follows:

$\sigma_{bp} = \frac{100F\sigma}{DV01}$

where $F$ is the price of underlying TY future, $\sigma$ is implied volatility, and $DV01$ is a dollar duration of the cheapest to deliver bond (it is published daily by the exchange, or can be calculated manually).

The calculation of $DV01$ using modified duration is well presented here, and the details follow. Say, the face value of the cheapest to deliver bond is $\$V$, total number of payment periods is $n$, payment frequency is $f$ (reciprocal of number of payments per year), and interest paid is $R$. The cashflow for each of the payment periods is

\begin{align*} C_{i<n} &= VRf \\ C_n &= VRf + V \end{align*}

The yield to maturity $y$ can be approximated by numerically solving the following equation, expressing the fair future value: $$F = \sum_{i=1}^n FV_i = \sum_{i=1}^n \frac{C_i}{(1+fy)^i}$$

Finally , the dollar value duration DV01 is:

$$DV01 = \frac{f}{F(1+fy)}\sum_{i=1}^n \frac{iC_i}{(1+fy)^i}$$

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