## IBOR Transition: Modelling RFR term rates to price IR derivatives

## Perspectives

# IBOR Transition: Modelling RFR term rates to price IR derivatives

Thu 21 Jan 2021

One of the anticipated challenges in the transition from IBOR rates to risk-free rates (RFRs) is the management of its impact on quantitative models. The ones currently used for pricing IBOR-linked financial instruments account for term rates which are “forward-looking”. The RFRs replacing the IBORs are all overnight rates. This means that a term rate must be derived from the overnight rates before being used in a contract.

Regulators consider two main term rates associated with a given RFR, namely:

- the “backward-looking” compounded setting-in-arrears rate[1], which is only fixed on the last day of the application period, or
- the “forward-looking” market-implied prediction of what the above-compounded rate will be.

The former is the preferred choice on the derivatives market. Yet, its growing momentum brings into question how to efficiently update the existing interest rate modelling framework to include “backward-looking” rates.

One potential answer is the generalised Forward Market Model (FMM), proposed by Lyashenko and Mecurio[2] in 2019. They show that by extending the definition of the Zero-Coupon Bond (ZCB) price process to beyond its maturity[3], one could extend the popular LIBOR Market Model (LMM) to describe the dynamics of the forward rate between the term rate fixing and payment times. One of the most important new risk-free rates is the Secured Overnight Funding Rate (SOFR), set to replace the USD LIBOR. Using the authors’ proposed set-up, Mazars’ Quantitative Solutions team illustrates with real data how the FMM can be used to bootstrap a SOFR curve from SOFR futures[4].

### 1. The FMM model

The forward ratemodelled under the FMM is defined by

whererepresents the valuation date, and represent the start (fixing) and end (payment) dates of the period under consideration, respectively, denotes the timeprice of ZCB with maturity, and represents the year fraction for the interval . By the definition of the extended ZCB, this forward ratehas the following desirable properties:

- before the fixing time, it coincides with the classical forward rate for this interval,
- after the payment time, it coincides with a continuous approximation of the compounded setting-in-arrears rate,
- during the application period it aggregates these two rates.

This rate is a martingale under the *extended* -forward measure and conveniently allows one to model both “forward- and backward-looking” term rates under a common framework. Under the FMM, we assume the dynamics of are given by:

whereis a standard Brownian motion under the extended forward measure, is an adapted process and is a monotone decreasing, piece-wise differentiable function such thatfor, andfor. The functionensures that the volatility of decreases to zero as we move through the accrual period and more of the fixings that contribute to the final value of become known. After the end date, is zero and is fixed and equal to the realised “backward-looking” rate at time . This model can be used to price several derivatives, including futures, caps, swaps and swaptions.

### 2. Pricing SOFR Futures

SOFR futures are one of the most liquid instruments based on the new RFRs, and they can also be used to strip SOFR discount curves, so the implementation will be restricted to the three-month SOFR futures[5] contract. This work can easily be extended to pricing additional vanilla instruments as more liquidity becomes available.

The timevalue of a SOFR futures contract that pays at time is given by , where [6] represents the “backward-looking” three-month continuously compounded rate. The futures contract is subject to margining, so profits can be reinvested daily at a higher rate and losses can be financed at a lower rate. This means that a futures contract has a higher value than the corresponding forward contract (FRA), and this difference in values is the *convexity adjustment*. The convexity adjustment at time for the futures contract with fixing date and payment date is denoted by and given by

where is the expectation under the risk-neutral measure. Clearly, this value depends on the model used to describe and, given the market observed forward rate , it specifies the futures price. The problem of pricing SOFR3M futures is therefore reduced to the issue of calculating convexity adjustments for each application period .

In this implementation, it is assumed – as proposed by the original authors – that the SOFR short rate can be modelled according to the Ho and Lee model (which falls into the Heath, Jarrow, and Morton (HJM) framework) with constant volatility . It is also assumed that the function in equation 2 is the *linear decay function:* . This HJM representation of the SOFR short rate allows one to get the following FMM dynamics for the forward term rate [7].

Using these dynamics[8] makes it possible to solve equation 3, and the convexity adjustment for the period can be approximated by

### 3. Implementation

#### 3.1 SOFR zero-curve benchmark with Bloomberg using a simple SOFR volatility assumption

We use Bloomberg Market Data and calculate the convexity adjustments for a range of listed SOFR 3M futures with start dates ranging from June 2020 to March 2025. The valuations were performed as of 30 June, 21 August, 1 September and 15 September 2020 to test the validity of the model over a three-month period (equivalent to the application period of the first SOFR futures contract under consideration).

We are unable to calibrate to swaptions due to the current illiquidity of the options market on SOFR. An analysis of historical SOFR volatility[9] indicates that a value is the best available proxy to the implied SOFR volatility. We discuss the volatility assumption in Section 4.2.

The implementation was performed in Python, and convexity adjustments were computed for each SOFR futures contract. Figures 1, 2, and 3 compare the convexity adjustments calculated as of 30 June, 21 August, and 1 September 2020, respectively, with Bloomberg’s data provided on these dates. Clearly, the model yields values for the convexity adjustments very close to those calculated by Bloomberg. While the model seems to provide the best fit for convexity adjustments on 21 August 2020, these values are sensitive to the calibration of the parameter and a more precise calibration of this parameter will remedy this behaviour. Based on sensitivity analysis we performed, it was observed that a change in leads to a better fit to Bloomberg values for September 2020 data. Since our goal here was to demonstrate the applicability of the model rather than perfectly fit Bloomberg data, we have not attempted to do so in these results. We also observe that our calculated values diverge from the Bloomberg values at longer maturities (greater than three years). This is because the difference between the FMM and the model used by Bloomberg to calculate convexity adjustments becomes more pronounced at longer maturities[10].

Figure1: Convexity Adjustments on 30 June 2020

Figure 2: Convexity Adjustments on 21 August 2020

Figure 3: Convexity Adjustments on 1 September 2020

As previously discussed, SOFR futures contracts will form a key part of the SOFR curve stripping methodology. Therefore, we also use our calculated convexities to bootstrap the SOFR discount factors from SOFR futures and compare these against Bloomberg. Table 1 presents discount factors calculated by stripping the SOFR futures as of 21 August 2020 and bootstrapping the discount curve from 16 September 2020[11] to 15 September 2021 (we use a period of one year spanning four futures contracts).

Payment Date | Bloomberg Discount Factor | Bootstrapped Discount Factor | Difference (bps) |

16/09/2020 | 0.999960938 | 0.999960938 | 0.0000 |

16/12/2020 | 0.999778211 | 0.999778670 | 0.0046 |

17/03/2021 | 0.999615796 | 0.999617915 | 0.0212 |

16/06/2021 | 0.999531241 | 0.999536918 | 0.0568 |

15/09/2021 | 0.999474704 | 0.999486526 | 0.1182 |

Table 1: Comparison of bootstrapped discount factors from SOFR futures with Bloomberg

Typically, SOFR futures are only used in the short end of curve construction and would not be used to strip curves for maturities greater than one year. With a difference of just 0.12 bps at the maturity of one year, we have demonstrated that this implementation enables us to obtain reasonable values for both convexity adjustments and the stripped discount curve.

#### 3.2 The volatility assumption

The purpose of this analysis is to assess the robustness of the assumption of constant volatility we made in our implementation of the FMM. In the implementation above, we used a constant bps and quoted SOFR futures prices[12] to strip the FRA curve (using equation 4). We now check the implied volatility in equation if we were to use the FRAs derived from a market SOFR discount curve built using non-futures instruments (from here on referred to as the SOFR swaps curve). We use real market data from Bloomberg. Thus, our test aims to assess what volatility for the SOFR is being implied by the market quoted SOFR futures and SOFR swaps curve and whether it agrees with our assumption of a constant bps.

One of this test’s underlying assumptions is that the market SOFR swaps curve should be aligned to the SOFR futures curve. To check this, we compare the forward rates (3-month FRAs) implied by each of these curves (these curves are calculated using equation 1). Figure 4 illustrates the two forward curves calculated as of 30 June 2020. Discount factors for both curves are sourced from Bloomberg.

Figure 4: Comparison of forward curves implied by discount curves built using different instruments.

While the two curves are close and show similar trends, there is a difference between the FRAs implied from the curves, large enough to cause significant differences in the implied volatilities. One major issue we encounter is that using the FRAs calculated from the swaps curve yields negative convexities, i.e. the implied FRA is greater than 100 – the futures price for the particular contract. The same behaviour is observed for the other valuation dates considered (21 August 2020, 1 September 2020, and 15 September 2020).

Table 2 shows the volatilities implied using the market data for SOFR futures and SOFR swaps curve. The columns in this table represent valuation dates. The rows represent the futures contract under consideration (for example, the Jun20 + 3 futures contract is the 3M SOFR Future with accrual period beginning on 17 June 2020 and ending on 16 September 2020). Cases where the implied volatility is “NA” correspond to a negative implied convexity, as mentioned above. Further, the implied volatility varies substantially and does not support our assumption of a constant bps. This behaviour might result from differences in the futures and swaps markets or a result of data quotes from different times and different data providers. While it shows that the SOFR futures and swaps market data are not consistent, we believe it implies that care must be taken when assuming bps. This assumption should not be made lightly and without an analysis of its potential impact.

Futures Contract | 30-Jun-20 | 21-Aug-20 | 01-Sep-20 | 15-Sep-20 |

Jun20 + 3 | NA | NA | NA | NA |

Sep20 + 3 | NA | 264 | NA | NA |

Dec20 + 3 | 21 | 102 | NA | NA |

Mar21 + 3 | 45 | NA | NA | NA |

Jun21 + 3 | 9 | 34 | NA | NA |

Table 2: SOFR Volatility (in bps) implied by swap curve

The IBOR transition has made it necessary to review the current interest rate modelling framework. Lyashenko and Mecurio provided a natural and elegant way to extend the existing framework to address the significant differences in the way term rates will be considered when IBOR rates are no longer referenced.

In this article, Mazars’ Quantitative Solutions team investigated the impact of the volatility assumption that needs to be made to bootstrap a SOFR zero-curve from futures quotes.

This article contributes to the research in RFR modelling and the quantitative impacts of the IBOR transition[13] that Mazars Quantitative Solutions team has been carrying out.

**This article was written by Mariem Bouchaala, Akhilesh Bansal** **and Hannah Maidment.**

[1] This rate, denoted by is defined by . The product is over the business days in , is the year fraction for this interval, and is the RFR ﬁxing on date with associated day-count fraction . [2] Mecurio, F., & Lyashenko, A. (2019, March 5). Looking Forward to Backward-Looking Rates: A Modelling Framework for Term Rates Replacing LIBOR. Retrieved from SSRN. [3] The extended zero-coupon bond price is defined by . [4] Link. [5] Currently, both 1-month and 3-month futures are traded on the CME. [6] . [7] See Libor replacement II: completing the generalised forward market model for more details. [8] Note that the above dynamics are under the FMM measure, we need to do a change of measure to get the dynamics of R(t) under the risk neutral measure (a big advantage of the FMM over the LMM).

[9] The volatility is calculated as the standard deviation of absolute daily returns of the SOFR and calibrated over a two-year period. This is consistent with the normal volatility assumptions in the context of negative rates. Note that we filtered out some spikes from the SOFR data as they skew the standard deviation and tested over smaller calibration periods as well. Following our analysis, we choose heuristically as this is also consistent with the value used by M. Henrard in Overnight Futures: Convexity Adjustments (2018). [10] Convexity adjustments scale up with the square of maturity. [11] Note that we have avoided complications for bootstrapping from 31 August 2020 to 16 September 2020 by letting the first bootstrapped value be the discount factor on 16 December 2020 and letting the value of the bootstrapped curve on the first date (16 September 2020) be equal to the market observed value. [12] Practically we use (100-futures price). [13] For more information on modelling of SOFR risk factors, please visit Mazars’ Financial service blog.

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