Deriving the Bond Futures Hedge Ratio from CTD DV01

# mathematical proof of the hedge ratio formula for bond futures


# mathematical proof of the hedge ratio formula for bond futures












We know that the hedge ratio ϕ_F that we should use in order to to the duration-hedging through bond futures is:

$$ϕ_F= -(DV01_B / DV01_{CTD} )\cdot CF_{CTD}$$

Where $\textrm{DV01}_B$ is the dollar duration of the bond I want to hedge divided by 10000, i.e. it is equal to: $$(\textrm{modified duration B} \cdot \textrm{dirty ctv B}) /10000$$

$\textrm{DV01}_{CTD}$ is the dollar duration of the CTD bond divided by 10000, i.e. it is equal to: $$(\textrm{modified duration CTD} \cdot \textrm{dirty ctv CTD}) /10000$$

$\textrm{CF}_{CTD}$ is the conversion factor of the CTD bond

How can I proof this formula?

The part I don't get is why I can write: $\textrm{DV01}_F = \textrm{DV01}_{CTD} / \textrm{CF}_{CTD}$,

So, I can proof that $ϕ_F= -\textrm{DV01}_B / \textrm{DV01}_F$

What I cannot proof is that $$ϕ_F= -\textrm{DV01}_B / \textrm{DV01}_F = -(\textrm{DV01}_B / \textrm{DV01}_{CTD} )\cdot \textrm{CF}_{CTD}$$

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/77897

You want to be a little careful with respect to what your sensitivities (DV01s) are measuring here and with respect to what settlement time.

Let,

$$ DV01_B^t = \text{risk sensitivity of price of bond, B, relative to yield at settlement, t} = \frac{\partial P^t_B}{\partial y^t_B} $$

Suppose a future has one CTD, so there is no optionality then the immediate futures price, F, is determined from the price of the CTD bond at delivery time, d, and the conversion factor (and the delivery price is a function of the spot price and the repo rate to delivery):

$$ F = \frac{1}{cf} P_{ctd}^{delivery} $$

Then the future price's sensitivity to the yield on the CTD at delivery is:

$$ \frac{\partial F}{\partial y_{ctd}^{delivery}} = \frac{1}{cf} DV01_{ctd}^{delivery} $$

Thus, if you suppose you had a position in a bond, B, that is not the CTD you might choose to hedge that bond with the CTD bond, in which case you would look to match their spot DV01s, multiplied by the notional trading:

$$ N_B DV01_B^{spot} + N_{ctd} DV01_{ctd}^{spot} = 0 $$

In this case your hedge ratio, $\phi = - \frac{DV01_{ctd}^{spot}}{DV01_B^{spot}}$

At this point you need to think about how you want to incorporate the repo rate or how you want to relate yields at delivery with spot yields. If you make the assumption that an appropriate hedge ratio can be obtained by using forward DV01s you can write:

$$ \phi = - \frac{DV01_{ctd}^{delivery}}{DV01_B^{delivery}} = - \frac{cf}{DV01_B^{delivery}} \frac{\partial F}{\partial y^{delivery}_{ctd}} $$

This is essentially the same answer as previously, labelled with settlements.

## Answer by user68819 (score 0)

https://quant.stackexchange.com/a/77575

Well you need to account for carry too which is easy

Very simply..

Bond future price ~ ctd bond forward / cf

This implied dv01 of the bond fut ~ dv01 of ctd forward/cf

Obviously I'm missing the value of and risk of optionality here.