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When we say Bond prices are martingale under T-Forward measure, do we mean their Clean Price is a martingale or is it their dirty price.

I guess it should be dirty price, as clean price is just a convenient representation while dirty price is the actual price of the bond. But it will be great if somebody could please validate/invalidate my thoughts.

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Your question is not really clear. Why do you need the bond price to be a martingale under the $T$-forward measure? The $T$-forward measure is used mainly for modelling the LIBOR rate. Note that, the bond price itself is not a martingale; instead, the bond price relative to the $T$-maturity zero coupon bond price is a martingale.

For a coupon bond, it does not matter whether it is a clean price or dirty price, as the coupon is a known quantity in either case.

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Gordon has the correct answer. When you talk about a certain measure, it is associated with a certain numeraire. In this case, it is the zero coupon bond that matures at T. So the bond price is denoted in terms of this numeraire. To price a bond with coupon, you should price it as two parts: (a) the principal to be repaid at maturity, this is equivalent to a zero coupon bond. (b) Every single coupons up to maturity, you should price them separately. At the end, when you sum these together, you will get the time 0 price of the bond.

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The answer is neither. An asset by itself should not be a martingale, it's the asset relative to some other tradable asset a martingale. Martingale is a mathematical concept to compare an asset relative to something else; for example, you would be comparing against the cash account in risk-neutral.

Furthermore, the T-forward measure is usually used for LIBOR rates because by definition each LIBOR rate can be easily be turned into a T-forward measure (check the definition yourself and you will see).

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