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From my understanding:

FwdPx= SpotPx - Accruals + Financing

Assume that the yield curve is flat/or that the bond yield stays the same the next day, i.e. that the market is unchanged and that the only force the bond is subject to is the Pull to Par.

Then if it is trading above 100 its price the next day will by slightly lower and the opposite if it is trading below 100. Therefore we in T+1 you actually have a change in price (call it “dP”)

My assumption is that the initial FwdPx (I.e. the FwdPx with the same starting date and end date of the previous day, not shifted by one day), will also have to move by dP + (1day Accrual - 1day Financing), due to the changes in both the spot price and the fact that the “carry” component is one day less. Is this correct?

The original question is to understand what would happen to a Basis position (long CTD, short Future), from T to T+1 if nothing at all happens, or in other words if only the deterministic components of the bond make their effect on the bond, assuming that there is a net basis equal to zero.

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No the forward price of a bond on a fixed date does not pull to par. If the forward yield stays the same, so does the forward price. In a scenario where yields don’t move , it is the spot price that gradually moves toward the forward price. Does it answer your question ?

Edit to address some comments: Consider the following example : a completely flat yield curve, all yields are 5%. A 5 year zero coupon bond has a spot price of $ 100/1.05^5 $. The forward price of this bond one year from now is $ 100/1.05^4 $. Now allow one day to pass. Assuming all yields are still at 5%, the spot price of the bond will increase slightly through the P2P to $ 100/1.05^{4+364/365} $. The forward price to the same exact date as before doesn’t change , but the forward price for the date which is one year from the new spot date increases to $ 100/1.05^{3+364/365} $.

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  • $\begingroup$ The forward price does not pull to par IF the forward yield stays the same, but the pull to par assumes that the spot yield stays the same, and thus the spot price must change, which in turn must mean the forward must change. The pull to par of the spot price is not something that can be excluded, because it is not determined by a movement in the spot curve and is independent from it. Therefore the forward price must change, why wouldn’t it? $\endgroup$
    – Giuseppe
    Commented Jan 12 at 21:06
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    $\begingroup$ The "pull to par" is by definition the natural movement of the Spot when there is no new information coming in about future rates. The Forward Price is determined mathematicallly from the available information and does not change in this scenario, the FP is constant (recalculation gives the same result as before). In real life, i.e. in a world with uncertainty, on the Pull to Par are superimposed other movements up or down in SP and FP caused by changed information about rates. $\endgroup$
    – nbbo2
    Commented Jan 13 at 9:04
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    $\begingroup$ In other words we have decomposed bond price movements into 2 parts (pull to par, or predictable movements) and non-pull to par (stochastic, unpredictable) movements. The SP has both types, the FP only unpredictable movements. The FP has all the properties of an optimal forecast conditional on known information, it can't change if the information does not change. $\endgroup$
    – nbbo2
    Commented Jan 13 at 9:36
  • $\begingroup$ @nbbo2, in the case of a deep discount bond, if you had to sell a bond forward for date X on date t, I'd assume that price would be different from the price on date t+1, for the same forward expiring on date X. 1d less of carry and in particular the bond will have pulled higher to par (even if yields are unchanged)..that seems like it is just mathematical from the fwd(t) = spot(t) - carry(t)..if I reevaluate with t=t+1, this price will have changed even if yields are unchanged.. $\endgroup$
    – user68819
    Commented Jan 13 at 11:02
  • $\begingroup$ dm63 and @nbbo2 have provided very clear and accurate explanations here. $\endgroup$
    – Attack68 ♦
    Commented Jan 13 at 11:14

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