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Modeling EURIBOR and RFR Exposure in a Multi-Factor Rates Framework

Article Quant Q&A · Author: Olórin
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Summary

The author describes a rates-pricing problem in which a multi-factor short-rate model for SOFR or ESTR, calibrated to swaption volatility, must also price exotic instruments linked to EURIBOR. The products include callable bonds, range accruals, CMS spread structures, and target redemption notes. The challenge is that EURIBOR-linked instruments coexist with ESTR-based swaps and swaptions used for calibration, so the model’s risk-free-rate dynamics do not directly match the exotics’ floating reference rates.

Earlier work used market volatility cubes for vanilla swaptions and historical correlation to approximate mid-curve swaption spread options. For the expanded request, the author considers adding a stochastic volatility factor and reviews multi-factor models, including the forward market model. The post does not choose a final approach or present pricing results. It highlights unresolved questions about modeling the basis between EURIBOR and compounded ESTR, and whether forward-looking rates in the cited framework correspond to EURIBOR term fixings.

Key ideas

  • A model calibrated to ESTR or SOFR instruments may not directly price exotics linked to EURIBOR.
  • The proposed framework uses multiple short-rate factors and may include stochastic volatility.
  • Earlier mid-curve spread options were approximated using historical correlation.
  • The key open modeling choice is how to represent the basis between term rates and compounded overnight rates.
  • The document raises, but does not resolve, whether forward-looking rates in the reviewed model match EURIBOR fixings.

Tags

Fixed incomeDerivatives pricingVolatilityOptions
Full text
# A rates model for EUR and USD pricing in different underlyings (EURIBOR (yes) or ESTR, and SOFR)


# A rates model for EUR and USD pricing in different underlyings (EURIBOR (yes) or ESTR, and SOFR)












Being a house mainly focused on almost everything else that rates products we never had a "rates pricer", no surprise. The best connected to rates thing we have is an equity/fx/what have you / rates hybrid model where you can only have one rates underlying for which you diffuse the short rate in an Ho-Lee manner and we never used it.

Several years ago I had a request to price (mainly USD) swaptions (see The "I want to price swaptions" request), the swaptions being vanialla or mid-curve, request that I solved by using Bloomberg's volatility cubes for the vanilla swaptions and by writing mid-curve swaptions as options on the spread of two different forward swap rates for which I used historical correlation (the market on real spread options being inexistant). So nothing fancy but it did the job of allowing execution of simple rates volatility strats.

Now, I have a new rates request, of different nature and specification.

I was first asked to provide a rates pricer to price USD swaptions/cap/floors but also other products like : callable bonds, range accruals, products payment short term/long term CMS spreads, TARNs. The products were supposed to be freshly issued so that all involded forward rates where compounded SOFRs and I made my choice for a multi-factor short rate model (with 3 to 5 factor) with an hesitation regarding wheter or not to add and stochastic volatility factor. Basically everything that I have to price will depend on (forward) zero-coupons that I can explicitely calculate, sometimes tediously (think for instance about adapting swap rates approximate dynamics calculations from chapter 12 or 13 from Piterbarg and Andersen to the SOFR case), so eveything is fine.

But the request specification changed : now it will mainly be EUR instruments and still but just a tiny bit of USD instruments. And I've be sent term sheets of EUR instruments (range accs, TARN, callable bonds) where EURIBOR is still used even if the instruments were issued at the beginning of 2023 ! While EUR swaps and swaptions (on which I want to calibrate my model) are not EURIBOR but (compounded) ESTR linked. (The forward rates are calculated roughly as explained in p.1-3 from Mingyang Xu's "SOFR Derivative Pricing Using a Short Rate Model", in the same way as it is done in the USD case.)

So I have a 3 to 5 factors short rate model possibly with a stochastic volatility which diffuse a RFR (risk free rate), SOFR or ESTR, calibrated on swpations, that will have to price in the USD case SOFR linked light exotics (ok) and EURIBOR linked light exotics (not ok) in the EUR case.

Conclusion : either I change the model, or I make hypothesis on the basis between EURIBORs I will encouter and ESTR, or on the basis bewteen for instance EURIBOR3M and 3M ESTR forward rate deduced from daily compound of ESRT on the 3M period.

So I started to look the recent/semi recent litterature a bit, came across the two papers from Mercurio and Lyashenko on the FMM, which could do the job. (Hum, in these papers "forward-looking" forward rates are defined as would OIS swap rates be defined, and I don't see why this would be (for 3M periods for instance) the same as an EURIBOR 3M ... I posted a related question here.)

I still can't make up my mind, so that advises are welcome.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.