Pricing Caps and Floors with Backward-Looking Overnight Rates
Summary
The document asks how to adapt cap and floor pricing from the LIBOR framework to compounded overnight reference rates. Under LIBOR, a spot-starting zero-coupon bond can be expressed using a known forward rate, supporting a forward-measure martingale argument. With a backward-looking compounded rate, the realized rate is not known at the start of its accrual period, so the same bond construction does not directly apply.
The author considers using the fixed rate on a single-period overnight indexed swap as a forward-looking proxy, then deriving bond prices and a martingale property from those swap rates. This is presented as a proposed line of reasoning, not an established method or a verified pricing result. The question highlights the modeling challenge of connecting realized compounded rates to tradable term structures; it provides no derivation, empirical evidence, or definitive answer, and the proposed equalities and measure-change argument require validation.
Key ideas
- LIBOR caplet pricing can use a forward rate that is known at the start of its accrual period.
- Compounded overnight rates are backward-looking and are only determined over the accrual period.
- The document proposes using single-period overnight indexed swap rates to relate bond prices to overnight rates.
- The proposed bond-price and martingale relationships are posed as questions rather than established results.
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Full text
# Post-LIBOR pricing of Caps and Floors
# Post-LIBOR pricing of Caps and Floors
Many posts on this page are dedicated to pricing rates options within the old LIBOR framework, which basically relies on the fact that the forward Libor rate $L(t, T_1,T_2)$ is a martingale under the $T_1$ forward measure (i.e. this post here for example)
This in turn relies on the assumption that a zero-coupon bond can be written as an inverse of the Libor rate, i.e. a spot-starting zero coupon bond can be written as:
$$P(t_0,T_1):=\frac{1}{1+\tau L(t_0, t_0, T_1)}$$
With the new ARR rates, how does this work? I have read the paper "Looking forward to backward looking rates" and they propose using a hybrid numeraire which is basically a zero-coupon bond that turns into a money-market numeraire after maturity, but I fail to see how this allows us to construct a zero-coupon bond from the ARR rates?
A zero coupon bond $P(t_0,T_1)$ in the ARR world would have to be written as:
$$\frac{1}{1+\tau R_{ARR}(t_0, t_0, T_1)}$$
With:
$$R_{ARR}(t_0, t_0, T_1)=\prod_{t_0}^{T_1}\left(1+\frac{\delta(t)}{360}r_i\right)-1$$
The rate $R_{ARR}$ is obviously not known till $T_1$ (unlike the old Libor rate $L(t, T_1,T_2)$), so as of time $t_0$, we cannot simply write the zero coupon bond in terms of the $R_{ARR}$ like we used to be able to do with $L(t_0, t_0,T_1)$.
So do we just write the zero coupon bond in terms of single-period OIS swaps? Say an OIS SOFR swap expiring in 3 months has the fixed rate basically equal to the expected value of the $R_{ARR}(t_0, t_0, T_1)$ with $T_1$ being equal to 3 months. Calling this fixed rate $K(t_0, t_0, T_1)$, we could then write:
$$P(t_0,T_1)=\frac{1}{1+\tau K(t_0, t_0, T_1)}$$
Then, we could argue that:
$$\frac{P(t_0,T_1)}{P(t_0,T_{2})}=1+\tau K(t_0, T_1, T_{2})$$
And then, recycling the logic of the old Libor world, we could argue that in general, $K(t, T_1, T_{2})$ is a martingale under the $T_1$ forward measure.
Pricing a caplet would then work via rewriting the ARR rate in the pay-off in terms of the forward-looking OIS rate $K$.
But I am not sure if this is how things work nowadays?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.