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Gross Basis for treasury futures = clean price - future price* conversion factor

Is there a way to estimate the gross basis, say 10 days from now, given that you know what the forward bond price is?

Conversion factor is constant as well.

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Gross basis today is as your formula: $$GB_{today} = P_{today} - F*C_f$$ Forward gross basis depends on the future price of the bond: $$GB_{forward} = P_{forward} - F*C_f$$ The important part being the futures price is not expected to change. You stated you know what forward price is but for clarity it is: $$P_{forward} = P_{today} * (1 + dayfrac * reporate)$$

Net basis is the difference between the two prices measured at settlement, so the forward net basis does not equal the forward gross basis unless your forward price is the delivery settlement price.

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