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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 $$ϕ_F= -\textrm{DV01}_B / \textrm{DV01}_F = -(\textrm{DV01}_B / \textrm{DV01}_{CTD} )\cdot \textrm{CF}_{CTD}$$

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2 Answers 2

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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.

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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.

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  • $\begingroup$ I understand that the invoce price (so the price the buyer of the futures has to pay at the seller at delivery) is: contract size*(futures price * conversion factor + accrued interest underlying bond). This has to be equal (at delivery) to contract size * price CTD (dirty). So we have: price (clean) CTD = Price futures * CF. So Price futures = Price (clean) CTD / CF. So DV01 Futures = DV01 (clean) CTD /CF ? $\endgroup$
    – luca dibo
    Commented Dec 4, 2023 at 15:44
  • $\begingroup$ At expiry...yes..us futures have a whole month to 'expire' though $\endgroup$
    – user68819
    Commented Dec 4, 2023 at 22:38
  • $\begingroup$ What i don’t understand is why should i use DV01 of the CTD computed without taking info account the accrued interest of the CTD. I mean, the DV01 of a bond is computed mathematically using the dirty price of the bond * modified duration of the bond / 10000. Here it seems that in order to compute DV01 of the CTD i should do: clean price CTD * modified duration CTD / 10000. Why is that? If you can explain that i give you the bounty $\endgroup$
    – luca dibo
    Commented Dec 5, 2023 at 22:38
  • $\begingroup$ accrued interest is a static value. It has no dependency to yields. Including or excluding it yields the same risk sensitivity. $\endgroup$
    – Attack68 ♦
    Commented Jan 22 at 6:37

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