What is the T-forward measure and why is it useful?

Question

AcadiFi · Accepted Answer

The T-forward measure uses the zero-coupon bond P(t,T) as numeraire instead of the money-market account. Under this measure, the process X(t)/P(t,T) is a martingale for any tradeable asset X.

Magic property: under Q^T, the forward price of ANY asset delivered at T equals its expected spot price at T. That is, F(t,T) = E^T[S(T) | F_t]. No discounting inside the expectation — discounting is handled by the bond price outside.

This simplifies pricing European-style payoffs tremendously. A call on a bond with maturity T has price: C_0 = P(0,T) * E^T[max(S(T) - K, 0)]. The discount factor is deterministic (today's bond price), and the expectation is under Q^T where the underlying has nice distribution properties.

```mermaid
graph LR
    A[Risk-Neutral Q] --> B[Change Numeraire]
    B --> C[T-Forward Q^T]
    C --> D[Forward = Expected Future Spot]
```

Practical use at a firm like Halcyon Rates: pricing bond options under Hull-White becomes a closed-form Black-like formula because under Q^T the bond yield is Gaussian. The LMM uses different forward measures for each LIBOR rate. FX forwards, inflation-linked bonds, quanto options — all cleaner under forward measure.

For CFA, key takeaway: changing numeraire from cash to a T-maturity bond makes the forward price a martingale and decouples discounting from expectation.

What is the T-forward measure and why is it useful?

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