I will refer to Risk-Free Rates (RFR) for greater generality, instead of OIS or SOFR. There are two dimensions to your question, I will treat them separately.
How to adjust a LIBOR vol surface to price RFR options?
When lacking RFR option quotes, I think there is no evident solution. Some pointers I would consider:
- From anecdotal evidence I've seen for USD LIBOR vs SOFR as well as GBP LIBOR vs SONIA, we should probably expect vol surfaces for LIBOR and RFR to be roughly similar, with differences in the order of $\pm$5bps. This is not surprising, in that RFR will replace LIBOR hence we can reasonably expect the market will trade at similar levels. Hence you might just pick the LIBOR vol surface as it is to price RFR products: in any case, this solution is temporary until liquidity in RFR options develops.
- For GBP, the replacement RFR rate is (reformed) SONIA, which has been around much longer than USD SOFR or EUR ESTER. Evidence does seem to show the market for SONIA options is more developed than other RFR. For example, data from ISDA shows there were \$46.7Bn of notional traded in GBP SONIA options or cap/floors during Q1 2021, against \$14.7Bn for USD SOFR though the trade count is larger for USD (401 against 315 trades for GBP SONIA), see [4] (1). So you might compare GBP LIBOR and SONIA surfaces, infer a basis between these vols, and apply it to other currencies such as USD.
How to adjust a forward-looking vol to price a backward-looking option?
For this question, more concrete answers are available. In a normal framework where Black's formula can be used, results in [1], [2] and [3] show a backward-looking RFR caplet requires an adjustment to its total variance. This is because volatility of the compounded rate decreases progressively during the accrual period as more and more fixings are set, whereas for a forward-looking rate (such as LIBOR) the fixing is set once and for all at the beginning of the accrual period.
While the papers mentioned provide proofs under the assumption of continuous compounding, I will derive a similar result under daily compounding, which corresponds to the actual market practice.
Let us assume that we have a term structure of forward overnight rates $r_i(t):=r(t,t_{i-1},t_i)$ with fixing dates $T=t_0,\dots,t_{n-1}$ and pay dates $t_1,\dots,t_n=T+\Delta$ over the accrual (and compounding) period $[T,T+\Delta]$. Ignoring weekends and holidays, we assume each rate has the same accrual period $\delta=t_i-t_{i-1}$ for all $i$, where $\delta$ is one day. We also assume each overnight rate follows a Brownian Motion with a certain volatility. Finally, there is also a term structure of LIBOR rates $L_t:=L(t,T,T+\Delta)$, which we also assume follow a Brownian Motion.
If you have a forward-looking, term-based volatility surface (e.g. the LIBOR vol surface), a sensible assumption is to set the vols of the forward overnight rates between $T$ and $T+\Delta$ to be equal to the LIBOR implied volatility $\sigma_{T,\Delta}$ for expiry $T$ and tenor $\Delta$ (this would correspond to pointer 1 above). Then each overnight rate is distributed as:
$$\text{d}r_i(t)=\pmb{1}_{t<t_{i-1}}\sigma_{T,\Delta}\text{d}W_i(t)$$
where the indicator function is in the spirit of [1], i.e. no more vol when the rate is fixed. We assume the Brownian Motions have pairwise correlation $\rho$. Similarly for the LIBOR rate (or whatever forward-looking rate of interest):
$$\text{d}L_t=\pmb{1}_{t<T}\sigma_{T,\Delta}\text{d}W(t)$$
We now define the forward RFR compounded rate $R$ for tenor $\Delta$ as follows:
$$
R_t
:=R(t,T,T+\Delta)
:=\frac{1}{\Delta}\left(\prod_{i=1}^n
(1+\delta r_i(t\wedge t_{i-1}))-1\right)
$$
Note that for $t<t_{i-1}$ the overnight rate is not yet fixed and the value of $R$ is based on the forward overnights, whereas for $t\geq t_{i-1}$ the rate is now determined.
Now, consider a LIBOR caplet, this product has the following payoff at time $T+\Delta$:
$$V_{\text{Libor}}(T+\Delta)=(L_{T}-K)^+$$
On the other hand, a backward-looking RFR caplet pays at that same date:
$$V_{\text{Rfr}}(T+\Delta)=(R_{T+\Delta}-K)^+$$
Note a fundamental difference here: the LIBOR caplet $V_{\text{Libor}}$ is $\mathscr{F}_T-$measurable, while on the other hand the RFR payoff $V_{\text{Rfr}}$ is $\mathscr{F}_{T+\Delta}-$measurable. This is where the adjustment will come from.
We now introduce the following Taylor approximation to $R$:
$$
\widetilde{R}_t
:=\widetilde{R}(t,T,T+\Delta)
:=\frac{1}{\Delta}\sum_{i=1}^n\delta r_i(t\wedge t_{i-1})
=\sum_{i=1}^n\omega r_i(t\wedge t_{i-1})
$$
where $\omega:=\delta/\Delta=1/n$. The variance of the LIBOR caplet rate is straightforward to obtain:
\begin{align}
V(L_T|\mathscr{F}_t)
=E(L_T^2|\mathscr{F}_t)
=\sigma_{T,\Delta}^2(T-t)
\end{align}
For ease of exposition, let us assume that $t<t_0$ so that $\pmb{1}_{t<t_{i-1}}=1$ for all $i$. The variance of the RFR caplet rate can be approximated by the variance of $\widetilde{R}_{T+\Delta}$:
\begin{align}
V(R_{T+\Delta}|\mathscr{F}_t)
\approx V(\widetilde{R}_{T+\Delta}|\mathscr{F}_t)
&=(w\sigma_{T,\Delta})^2V\left(\left.\sum_{i=1}^nW_i((T+\Delta)\wedge t_{i-1})\right|\mathscr{F}_t\right)\\[3pt]
&=(w\sigma_{T,\Delta})^2V\left(\left.\sum_{i=1}^nW_i(t_{i-1})\right|\mathscr{F}_t\right)
\end{align}
Each variable $W_i(t_{i-1})|_{t}$ is normal with zero mean and variance $(t_{i-1}-t)=(T-t)+i\delta$. Using Cholesky decomposition, the pairwise covariance between the summands is:
\begin{align}\text{Cov}(W_i(t_{i-1}),W_j(t_{j-1}))
=\rho((t_{i-1}\wedge t_{j-1})-t)
=\rho((T-t)+(i\wedge j)\delta)
\end{align}
The vector $(W_1,\dots,W_n)$ is Gaussian and $\widetilde{R}$ can be expressed as a linear combination of $W_1,\dots,W_n$ therefore is itself Gaussian. Through standard calculations (see e.g. this post) we deduce that:
\begin{align}
\tag{1}
V(\widetilde{R}_{T+\Delta}|\mathscr{F}_t)
=\sigma_{T,\Delta}^2\left(
(T-t)\left(\rho+\frac{1-\rho}{n}\right)
+\rho\frac{\Delta}{3}+\delta\left(\frac{1}{2}+\frac{3-2\rho}{6n}\right)
\right)
\end{align}
Note that $\lim_{n\rightarrow\infty}\delta(n)=0$ hence:
$$\tag{2}\lim_{n\rightarrow\infty} V(\widetilde{R}_{T+\Delta}^2|\mathscr{F}_t)
=\sigma_{T,\Delta}^2\rho\left((T-t)+\frac{\Delta}{3}\right)$$
This is the same expression found in [2] and [3] under a continuous compounding assumption if $\rho=1$. Note for a flat correlation, you are restricted to positive or close to zero values (see this post) however in practice we expect consecutive daily rates to be tightly (positively) correlated so this should not be much of a restriction.
In practice, observe that the third and fourth term above will be very small: $\delta$ is an overnight accrual so around 0.004 or 0.003 (depending on your day count convention), whereas $n$ is the number of fixings for the accrual period, which should be between 60 and 90 for a 3m tenor. Hence we can ignore the last two terms in $(1)$ and use the approximation $(2)$; because $\widetilde{R}$ is normally distributed, you can use Black's formula with the variance above to price your backward-looking RFR caplet.
Some remarks:
- The vanishing volatility throughout the accrual period is reflected in the "adjusted" expiry $(T-t)+\Delta/3$ which is obviously higher than that of a forward-looking caplet where the rate is fixed at $T$. In particular, assuming $\rho=1$, we have:
$$
\sqrt{V(R_{T+\Delta}|\mathscr{F}_t)}
\approx \sqrt{\alpha_1V(L_T|\mathscr{F}_t)}
$$
where:
$$\alpha_1:=1+\frac{\Delta}{3(T-t)}$$
For pricing backward-looking caplets, you might want to adjust your LIBOR vol surface by the factor $\alpha$ then feed the adjusted vols to Black's formula.
- The correlation parameter $\rho$ allows you some flexibility when marking these caplets, especially when the market is still developing and there is no RFR vol surfaces. In particular, if the vol $\sigma_{T,\Delta}$ comes straight out from a LIBOR surface, this correlation parameter might be used to mark the basis between LIBOR and RFR vols. The adjustment factor becomes:
$$\alpha_\rho:=\rho\alpha_1$$
The above setting can be subject to refinements. For example, you might want to specify different vols for each overnight rate, as well as different pairwise correlations: while maybe unwarranted for short tenors such as 1m, for a 1y-6m RFR caplet, you could set the overnight vols between 1y and 1y3m to the 1y-3m LIBOR vol, whereas for the rest you use the 1y3m-3m LIBOR vol and set the correlation to match the correlation between the 1y-3m and 1y3m-3m LIBORs.
References
[1] Lyashenko, Andrei and Mercurio, Fabio. "Looking Forward to Backward-Looking Rates: A Modeling Framework for Term Rates Replacing LIBOR". February 2019. Available at SSRN.
[2] Piterbarg, Vladimir. "Interest Rates Benchmark Reform and Options Markets". March 2020. Available at SSRN.
[3] Piterbarg, Vladimir. "Benchmark reform goes non-linear". Risk Magazine, September 2020.
[4] ISDA. "Transition to RFRs Review: First Quarter of 2021". April 2021. Available at ISDA SwapsInfo.
(1) Trade data is reported by the Depository Trust & Clearing Corporation (DTCC) and only covers trades that are required to be disclosed under US regulations and includes cleared and non-cleared OTC IRD transactions.
