Volatility adjustment for SOFR/OIS caplet referencing LIBOR vol

Suppose that today the price of a 3m LIBOR caplet with 6m expiry has been calibrated with a particular implied volatility.

How would one go about thinking about an adjustment to that volatility to account for the price of a caplet which is not based on 3m LIBOR but on 3m compounded OIS/SOFR. There seems to be a number of practical issues:

• the OIS caplet must have an expiry 3m greater than that of the LIBOR caplet to account for all possible interim fixings that are compounded over the period.
• the OIS caplet's volatility will be affected by the volatility of each contributing OIS rate within the 3m period, which occur on different dates.
• the OIS and LIBOR may have a basis so that strike may be further/nearer the underlying expected rate.

The adjustment is required since we have existing LIBOR framework and systems which will temporarily adopt OIS caplets (proxied by LIBOR) with that adjustment to get closer to true value.

Estimated or more specific quantitatively theoretic answers all appreciated.

• Have you checked recent papers by Lyashenko & Mercurio (2019, 2020) and Piterbarg (2020) on adapting rate models to RFRs? For the market data feed, you can always use the OIS surface which from experience should not be very far away from RFR (at least in the case of FedF and SOFR, they don't seem to be). Then current practice seems to be for RFR caplets to be in-arrears i.e. the accrual period $[T_S,T_E]$ is the same as a LIBOR caplet, except that the RFR rate is known at $T_E$ instead of $T_S$ as in LIBOR (therefore the exercise decisions is shifted 3m/6m forward, see Piterbarg's paper). Nov 23 '20 at 15:15
• Finally, in both these papers there is an assumption that volatility decays linearly throughout the compound accrual period $[T_S,T_E]$. The authors then find you can basically apply the Black normal formula where the expiry is set to be $T^\star:=(T_S-t)^++(T_E-\max(T_S,t))^3/(3(T_E-T_S)^2)$ and the implied volatility from the appropriate surface with expiry $T_E$ and tenor $T_E-T_S$. Nov 23 '20 at 15:17
• See Equations (5), (7), (13), (14) and Section 6.3 in papers.ssrn.com/sol3/papers.cfm?abstract_id=3330240. Nov 23 '20 at 15:21
• those look really useful links, thanks Ill take a look.
– Attack68
Nov 23 '20 at 16:55
• This might of interest to you, SABR for RFR caplets/floorlets: dx.doi.org/10.2139/ssrn.3567655 Jan 30 at 11:53

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:

1. 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.
2. 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: $$\text{d}r_i(t)=\pmb{1}_{t where the indicator function is in the spirit of [1], i.e. no more vol when the rate is fixed. We assume the Brownian Motions have pairwise correlation $$\rho$$. Similarly for the LIBOR rate (or whatever forward-looking rate of interest): $$\text{d}L_t=\pmb{1}_{t

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 $$t the overnight rate is not yet fixed and the value of $$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 $$\omega:=\delta/\Delta$$. 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 $$t. The variance of the RFR caplet rate can be approximated by the variance of $$\widetilde{R}_{T+\Delta}$$: \begin{align} V(R_{T+\Delta}|\mathscr{F}_t) &\approx E(\widetilde{R}_{T+\Delta}^2|\mathscr{F}_t) \\[4pt] &=\sum_{i=1}^n\sum_{j=0}^{n-1}w^2 E(r_i(t\wedge t_{i-1})r_j(t\wedge t_{j-1})|\mathscr{F}_t) \\ &=w^2\sigma_{T,\Delta}^2\rho\sum_{i=1}^n\sum_{j=1}^n (t_{i-1}\wedge t_{j-1}-t) \end{align} Note that: $$\sum_{j=1}^n(t_{i-1}\wedge t_{j-1}-t) =\sum_{j=1}^i(t_{j-1}-t)+(n-i)(t_{i-1}-t)$$ Therefore: \begin{align} \sum_{i=1}^n\sum_{j=1}^n(t_{i-1}\wedge t_{j-1}-t) %&=\sum_{i=1}^n\sum_{j=1}^i(t_{j-1}-t) %+\sum_{i=1}^n(n-i)(t_{i-1}-t) %\\ %&=\sum_{j=1}^n\sum_{i=j}^n(t_{j-1}-t) %+\sum_{i=1}^n(n-i)(t_{i-1}-t) %\\ &=\sum_{i=1}^n(2(n-i)+1)t_{i-1} \end{align}

Using the fact that $$t_i=t_0+i\delta$$ and after some manipulations of the sums, we find that: $$E(\widetilde{R}_{T+\Delta}^2|\mathscr{F}_t) =\sigma_{T,\Delta}^2\rho\left((T-t)+\frac{n\delta}{3}+\frac{\delta}{3n^2}-\frac{\delta}{n}\right)$$ Note 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. On the other hand, we have that $$n\delta=\Delta$$. Hence ignoring the last two terms, we have the following approximation to the total variance of the RFR caplet rate: $$V(R_{T+\Delta}|\mathscr{F}_t) \approx \sigma_{T,\Delta}^2\rho\left((T-t)+\frac{\Delta}{3}\right)$$ This is the same expression found in [2] and [3] under a continuous compounding assumption if $$\rho=1$$. 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 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 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: $$\sqrt{V(R_{T+\Delta}|\mathscr{F}_t)} \approx \sqrt{\alpha V(L_T|\mathscr{F}_t)}$$ where: $$\alpha:=1+\frac{\Delta}{3(T-t)}$$ 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 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.
• Could you please define what $t_i \wedge t_j$ means, I am unfamiliar with this notation. Aug 26 at 11:35
• @BrownianBread $x\wedge y:=\min\{x,y\}$ and $x\vee y:=\max\{x,y\}$. Aug 26 at 13:02
• Already a superb answer, +1 for that. Could you maybe briefly extend on the case where $t\geq t_0$, as in your notation I somewhere lost the index $j$ due to assuming the opposite. Might be a very simple confusion of mine using your notation vs. Mercurio/Lyashenko at the same time; but if I computed it correctly (under $\rho=1$ assumption), the variance of the ongoing caplet should read $\sigma_{T, \Delta}^2 \left[ \frac{(T_j - t)^3}{3(T_j - T_{j-1})^2} \right]$. Could you "translate" this somehow into your notation as I actually find your derivation better than in the paper :-) Aug 26 at 16:01
• Moreover, thinking about GBP SONIA Options a little longer: would you by chance know whether brokers actually quote the "forward looking" vols $\sigma^2_{T, \Delta}$ or already the backward-looking adjusted one? Else one might "double count" , no ... ? If there is such a straightforward transform, why bother to quote fwd looking at all, or put differently, why bother "for liquidity in RFR options to pick up", when we simply can use the existing surfaces as of today? Aug 27 at 11:43