Why a Forward Rate Is Not a Martingale Under Every Forward Measure
Summary
The document poses a fixed-income pricing puzzle: a one-period swap rate can be treated as a forward rate, while its payment date may differ from the accrual period’s end date. The question compares valuation under the swap annuity measure with valuation under a forward measure, and asks how delayed payment affects a forward rate agreement and a single-period constant-maturity swap.
The text lays out the apparent equality between the two products when their rate payoff and payment date match, then contrasts it with the familiar result that a forward rate is a martingale under the measure associated with its accrual end date. It provides the setup and pricing expressions but no answer resolving the paradox. Consequently, it is useful as a statement of a measure-change and payment-delay question, not as a complete derivation or pricing result; the treatment also assumes the stated single-period structure.
Key ideas
- A swap rate is described as a martingale under its annuity measure.
- For a single-period swap, the floating rate corresponds to a forward rate over the accrual interval.
- The payment date can differ from the accrual end date, raising a measure-selection issue.
- The document frames, but does not resolve, the relationship between delayed-payment FRA and CMS valuation.
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Full text
# Single-Period Annuity vs Forward Measure
# Single-Period Annuity vs Forward Measure
Swap rate is a martingale under its annuity measure. For a single-period swap, its rate is effectively a forward rate starting on $T_s$ and ending on $T_e$. The swap fixed and float leg payments are made on $T_p$. Can it be argued that since swap annuity is effectively a zero-coupon bond $P(T, T_p)$ in this case -- strictly speaking up to a factor which can be taken to be unity without a loss of generality -- the swap/forward rate is then a martingale in the forward measure $T_p$? Since $T_p$ can be pretty much arbitrary -- and is not necessarily equal to $T_e$ -- this leads to a paradoxical conclusion that a forward rate from $T_s$ to $T_e$ is a martingale in any $T$-forward measure. However, it is known that a forward rate is only a martingale in $T_e$-forward measure, since it is inversely proportional to $P(T, T_e)$. How is the paradox resolved?
Restating the paradox differently, in terms of FRA and CMS pricing. If we consider a single-period CMS leg that pays the swap rate discussed above on $T_p$, its value at time $t$ should thus be $V_{A}(t) = A(t) E^A[ \frac{P(T_r, T_p)}{A(T_r)} S(T_r)] = A(t) E^{A}[S(T_r)] = A(t) S(t) = P(t,T_p) F(t, T_s, T_e)$. If we consider a simple FRA agreement paying on $T_e$, its value at time $t$ should be $V_{T_e}(t) = P(t, T_e) E^{T_e}[F(T_r, T_s, T_e)] = P(t, T_e) F(t, T_s, T_e)$. If we consider a FRA paying on $T_p$ instead, that FRA is now undisguisable from the CMS leg above, and its value at time $t$ is thus $V_{T_p}(t) = V_{A}(t) = P(t,T_p) F(t, T_s, T_e)$. However, one expects a convexity adjustment associated with FRA payment delay from $T_e$ to $T_p$ that is not reduced to switching discounting factor in front of the expected rate from $P(t, T_e)$ to $P(t, T_p)$.
To restate yet again. A FRA on the rate between $T_s$ and $T_e$ paying on $T_p$ has the same payoff as a single-period CMS paying on $T_p$ with the CMS rate being a one-period swap with the floating rate between $T_s$ and $T_e$ paying on $T_p$. Since payoffs of the FRA (which has a pay delay) and CMS are the same, their present values are the same. That present value (computed as the present value of the CMS) is (again) $V_{T_p}(t) = V_{A}(t) = A(t) E^A[ \frac{P(T_r, T_p)}{A(T_r)} S(T_r)] = A(t) E^{A}[S(T_r)] = A(t) S(t) = P(t,T_p) F(t, T_s, T_e)$. Where is the catch?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.