Bond Price Martingales Under the T-Forward Measure
Summary
The document clarifies what is a martingale under a forward measure and whether clean or dirty bond prices qualify. A bond price alone is not generally a martingale under the T-forward measure; the relevant martingale is the asset price expressed relative to the numeraire, here the zero-coupon bond maturing at T. The measure is commonly used in modeling forward rates such as LIBOR.
For a coupon-bearing bond, clean price excludes accrued interest while dirty price includes it. The answers state that this distinction does not change the central martingale principle because the coupon is known. They also describe pricing a coupon bond as its principal repayment and each coupon valued separately, then summing the components. The discussion is conceptual and offers no worked stochastic model; the exact martingale statement depends on specifying the asset, numeraire, and cash flows consistently.
Key ideas
- Martingale statements require expressing an asset relative to a chosen numeraire.
- Under the T-forward measure, the natural numeraire is the zero-coupon bond maturing at T.
- A coupon bond's standalone price is not generally a martingale under that measure.
- Coupon bond valuation can be decomposed into principal and individual coupon cash flows.
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Full text
# Is a bond expiring at $T$ clean or dirty price a martingale under the $T$-Forward measure? # Is a bond expiring at $T$ clean or dirty price a martingale under the $T$-Forward measure? 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. ## Answer by Gordon (score 4, accepted) https://quant.stackexchange.com/a/21347 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. ## Answer by zsljulius (score 1) https://quant.stackexchange.com/a/21358 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. ## Answer by SmallChess (score 1) https://quant.stackexchange.com/a/21360 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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