Why a LIBOR Floating Leg Is Not at Par Under OIS Discounting
Summary
The document examines whether a standard LIBOR floating-rate leg remains worth par when its cash flows are discounted using the OIS curve. It introduces the single-curve bond and forward-rate identity that would imply par value, then asks whether that identity remains valid when OIS discount factors are used. The discussion also sketches a change of measure using an OIS zero-coupon bond as numeraire and an expectation involving the future LIBOR fixing.
The answer distinguishes the discount curve from the projection curve in a dual-curve framework. Discount factors come from OIS, while forward LIBOR rates are derived from a separate LIBOR curve; consequently, the single-curve identity generally does not hold, and the floating leg need not have par value. The stated exception is the absence of an OIS-LIBOR basis. The document gives a conceptual distinction and a rate formula, but no numerical example or detailed derivation, so it does not quantify the basis effect.
Key ideas
- A LIBOR floating leg is not generally worth par when discounted using OIS rates.
- In a dual-curve setup, OIS determines discounting while a LIBOR curve supplies projected forward rates.
- The single-curve bond identity depends on consistent discount and projection curves.
- Par value can hold when there is no basis between OIS and LIBOR.
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Full text
# Floating leg of a standard swap still has a value at par when we use the OIS as discount factor?
# Floating leg of a standard swap still has a value at par when we use the OIS as discount factor?
Does a bond paying floating coupon `LIBOR`, still has the value at par when we use the `OIS` as discount factor? It seems only when the Identity: $$B(t,T_2)(1+(T_2-T_1)F(t,T_1,T_2))=B(t,T_1)$$ still holds, the proposition above will be true. Here $B(t,T)$ is the value of zero coupon bond, $F(t,T_1,T_2)$ is the `forward LIBOR.`
In John Hull's book `Options, Futures and Other Derivatives 9th` `page 205` ,shows the way to calculate the forward LIBOR implied in `Swap rate` under `OIS discounting.` But it's the case we know $B(t,T_1),$ but don't know $B(t,T_2).$
If we know both $B(t,T_1)$ and $B(t,T_2).$ Can we calculate the forward LIBOR still as above identity?
Denote
$D_{ois}(t):$ the discounted factor of OIS
$B(t,T):$ Bond price
$E_t[]:$ Conditional expectation at time $t$ under OIS-risk neutral measure which makes $D_{ois}(t)B(t,T)$ martingale for all $T.$
Use $N(t) = D_{ois}(t)B(t,T_1)$ as a numeraire to change the measure into OIS $T_1$-forward measure $E^{T_1}_t[]$(simply use expectation represent new measure).
Then $$ \dfrac{B(t,T)}{B(t,T_1)} = \dfrac{D_{ois}(t)B(t,T)}{D_{ois}(t)B(t,T_1)}$$ should be martingale under $E^{T_1}_t[].$ Then use the definition of the forward LIBOR $F(t, T, T_{1})$ we can prove that $$\dfrac{1}{D_{ois}(T)}E_{T}\left[D_{ois}(T_1)\Big((T_1-T) \cdot F(T, T, T_{1})+1\Big)\right] = 1.$$
## Answer by Antoine Conze (score 1)
https://quant.stackexchange.com/a/36506
No, the Libor floater is not worth par when discounting at OIS (unless there is no basis between OIS and Libor).
In a dual curve settings discounting is done at $B_{OIS}(t, T)$, whereas the forward libors are computed on the projection curve as $F(t, T_1, T_2) = (B_{libor}(t, T_1)/B_{libor}(t, T_2) - 1)/(T_2 - T_1)$, where $B_{OIS}(t, T)$ is the discount factor on the OIS curve and $B_{libor}(t, T)$ is the discount factor on the libor curve.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.