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I am reading The Treasury Bond Basis by Burghardt. In chapter 2, it states that the bond with lowest converted price net of Carry is the CTD which I am a little confused. $$\mathrm{Converted\ Price}=\frac{\mathrm{Bond\ Price}-\mathrm{Carry}}{CF}$$ The PNL of the buying the basis and deliver the underlying bond at delivery date is $$PNL = FuturePrice*CF - BondPrice + Carry $$ $$= CF * (FuturePrice - \frac{BondPrice-Carry}{CF}) $$ $$= CF*(FuturePrice-ConvertedPrice)$$

It seems to me the bond with the highest PNL here is the CTD, and even if it has lower Converted price net of carry, it doesn't guarantee a larger PNL here.

Could anyone assist? Thank you so much!

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There exist many ways to rank bonds for CTD, some of which are

  1. Highest IRR
  2. Lowest converted price (net of carry)
  3. Highest PNL (aka net basis)
  4. Highest hedge ratio

Which one is the best? All and none at the same time. As soon as they disagree, this means there is some optionality and so they are all wrong.

The correct way is to realise that there is optionality and some (simple) model is needed.

But traders still talk about CTD? Yes, because with "low" rates the optionality is little, because they are used to, because their IT systems only support a single CTD and because they don't like when it switches between 2 bonds (and so it gets overridden in the system).

An observation about methods 1 & 3 (which are the most popular): they are circular in the sense that one cannot compute them without already having a Future Price available. How would you even price the future if you needs its price to start?

But maybe the most fundamental question is: what will you use the CTD for? Most often it is used to compute the DV01 of the futures position in the book.

This becomes a bad idea when the ranking changes. Imagine an option desk where the unitary delta of calls can only be 1 or 0? They won't like it. Here it is exactly the same.

Anyway, from an option pricing point of view, my preferred ranking is 2, akin to intrinsic value.

EDIT: What about the bond future (quoted) price? It can be used to calibrate the model (for instance fitting a repo spread), which can then be used to price the bond future in different scenarios.

Imagine you use methods 1 and 3: would you be able to apply any shift to the curve and see how the ranking changes? If the ranking is price dependent, then you will find yourself in a circular dependency, unable to do it.

Or, if you want to check the impact of a shadow bond? You will realise the importance of an independent way to price the future contract (and so to rank the bonds), which does not need a price to start with.

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  • $\begingroup$ Thank you for your answer Andrea. Why we dont have future price for methods 1 & 3? Thought we have the realtime price for it. Unless you are talking about the Future price at expiration? In this case, I am not sure why would we need to use the future price at expiration since the current future price is the one that we need to calculate the PNL that we locked in when we enter the basis trade at time 0. $\endgroup$ Commented Feb 18 at 19:55
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The pnl to the short is:

invoice price - spot price at expiry.

Prior to this she will choose a bond for forward delivery which will maximize:

invoice price - fwd price = invoice price - (spot price - carry) = -basis net of carry,

or minimise basis net of carry if you flip the choice around.

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  • $\begingroup$ Yes I agree with this. And I think this is the number we should be looking at when ranking CTD. But I am not sure why converted price can work when getting the CTD. $\endgroup$ Commented Feb 18 at 18:43
  • $\begingroup$ He's just dividing through by the conversion factor. It's the same thing. $\endgroup$ Commented Feb 19 at 4:05
  • $\begingroup$ The bond with the lowest converted price will also minimise the net basis $\endgroup$ Commented Feb 19 at 5:41
  • $\begingroup$ Thank you so much for your reply.. I must be missing something here. Isnt net basis = CF*((spot price-carry)/CF - FuturePrice)? If we only minimize the converted price which is the first term (spot price-carry)/CF, how can we make sure after multiply by CF it is still the smallest across the deliverable universe? $\endgroup$ Commented Feb 19 at 12:45
  • $\begingroup$ You are right, the 4 ranking methods in my answer are all "qualitatively" similar, but mathematically different. $\endgroup$ Commented Feb 19 at 18:50

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