muRisQ Advisory

LIBOR Fallback

and Quantitative Finance

Planning for the end of LIBORCass Business School – 19 June 2019

Marc Henrard

muRisQ Advisory and University College London

muRisQ Advisory

LIBOR Fallback – Quantitative Finance

1 From basics to fallback [Pay-off]

2 Fallback implementation – Adjusted RFR [Measurability]

3 Spread Adjustment and Value transfer [Filtration]

4 Conclusions

5 Annexes

A Quant Perspective on IBOR Fallback Consultation Resultshttps://ssrn.com/abstract=3308766

Fallback transformershttps://murisq.blogspot.com/2018/10/

libor-fallback-transformers.html

Planning for the end of LIBOR, Cass Business School – 19 June 2019 2

muRisQ Advisory

LIBOR Fallback – Quantitative Finance

1 From basics to fallback [Pay-off]

2 Fallback implementation – Adjusted RFR [Measurability]

3 Spread Adjustment and Value transfer [Filtration]

4 Conclusions

5 Annexes

Planning for the end of LIBOR, Cass Business School – 19 June 2019 3

muRisQ Advisory

Basics: LIBOR derivatives datesBenchmark dates

Fixing date (θ)

Underlying deposit effective date (σ) – T+2 for IBOR

Underlying deposit maturity date (τ)

θ σ τ

Underlying deposit

Derivative dates

Start accrual date (u)

End accrual date (v)

Payment date (tP)

u v tP

Accrual period

Planning for the end of LIBOR, Cass Business School – 19 June 2019 4

muRisQ Advisory

Basics: Vanilla IRS and OIS

LIBOR IRS

θ σ = u v = tp =?τ

Deposit = Accrual

OIS

t0 tn tP

t0 t1 t2 ti tn−1tn

Accrual/compounding period

Typically: tP = last fixing publication + 2 business days

Cmp(IO , (ti )i=0,··· ,n) =

(n−1∏i=0

(1 + δOi I

O(ti ))− 1

).

Planning for the end of LIBOR, Cass Business School – 19 June 2019 5

muRisQ Advisory

Basics: pricing derivatives

LIBOR coupon pay-off in tP :

Lj(θ)

Planning for the end of LIBOR, Cass Business School – 19 June 2019 6

muRisQ Advisory

Basics: pricing derivatives

LIBOR coupon pay-off in tP :

Lj(θ)

Present value in s < tP (textbook):

Ncs EX [(Nc

tP)−1Lj(θ)

∣∣Fs

]

Planning for the end of LIBOR, Cass Business School – 19 June 2019 7

muRisQ Advisory

Basics: pricing derivatives

LIBOR coupon pay-off in tP :

Lj(θ)

Present value in s < tP (textbook):

Ncs EX [(Nc

tP)−1Lj(θ)

∣∣Fs

]Present value in s < t (reality):

Ncs EX [(Nc

tP)−1(1{d > θ}Lj(θ) + 1{d ≤ θ}?

)∣∣Fs

]with d the discontinuation date of the LIBOR.The question mark in the formula is not a typo, it is the fallback!

Pricing LIBOR derivatives is not basic anymore!

Planning for the end of LIBOR, Cass Business School – 19 June 2019 8

muRisQ Advisory

Fallback

What is fallback?T

od

ay

An

nou

nce

men

t

Dis

con

tin

uat

ion

Jan

uar

y20

22?

Replacement of a discontinued benchmark by an adjusted RFR andan spread adjustment (ISDA master agreement).

Lj(θ) = FRj(θ) + Spread

θ u v

Lj(θ)

FRj(θ)

Spread

Planning for the end of LIBOR, Cass Business School – 19 June 2019 9

muRisQ Advisory

Fallback and quant finance

After the fallback the pricing formula will become

Ncs EX [(Nc

v )−1(1{d > θ}I j(θ) + 1{d ≤ θ}(FRj(θ) + S(X , [a− l , a]))

) ∣∣Fs

].

where d is the discontinuation date, a the announcement date andS the spread. The notations X and l will be explained later.The dates d and a are stopping times.

Planning for the end of LIBOR, Cass Business School – 19 June 2019 10

muRisQ Advisory

Fallback

Criteria

Consistency

Absence Value Transfer

Absence Manipulation (RFR and spread)

Similarity with existing instruments

Valuation simplicity

Risk management

Achievable!

An ISDA consultation took place from 12 July to 23 October 2018(GBP, CHF, JPY, AUD). The results of the consultation have beenpublished on 27 November 2018.USD fallback consultation in progress. Consultation expectedrelated EUR. Document on the technical details and the legalwording expected Q3 2019.

Planning for the end of LIBOR, Cass Business School – 19 June 2019 11

muRisQ Advisory

LIBOR Fallback – Quantitative Finance

1 From basics to fallback [Pay-off]

2 Fallback implementation – Adjusted RFR [Measurability]

3 Spread Adjustment and Value transfer [Filtration]

4 Conclusions

5 Annexes

Planning for the end of LIBOR, Cass Business School – 19 June 2019 12

muRisQ Advisory

Fallback – Adjusted RFR options

Option 3: Compounded Setting in Arrears RateConsultation text: “The fallback could be to the relevant RFRobserved over the relevant IBOR tenor and compounded dailyduring that period.”

FRj(t0) =1

δj(t0)

(n∏

i=1

(1 + δOi I

O(si−1))− 1

).

Pro: Interest rate, same term, similar to OIS, available?Con: Available late, achievable?, wrong period? FRA?

θ u=?tP? v =?τ tP?

t0 t1 t2 ti tn−1tn

Accrual period

Planning for the end of LIBOR, Cass Business School – 19 June 2019 13

muRisQ Advisory

Fallback – Adjusted RFR options

The adjusted rate is computed between t0 = σ and tn = τ ;

FRj(θ) = Cmp(IO , (ti )i=0,··· ,n).

is not Fθ-measurable anymore, it is only Ftn -measurable.In many case tp < τ . The new pay-off is not measurable on itspayment date!The measurability requirement may appear as a technical term ofno practical importance, but is it not; it is only the precisedescription of the very practical requirement that before you areable to pay an amount, you need to know what amount should bepaid.

Planning for the end of LIBOR, Cass Business School – 19 June 2019 14

muRisQ Advisory

Fallback – Adjusted RFR options

Option X1: Term rate / OIS BenchmarkOption not proposed in the ISDA consultation.In t0 the rate of the OIS associated to the IBOR start and maturitydeposit dates is measured to create an OIS benchmark.Pro: Term interest rate, correct period, similar to OISCon: Not in ISDA consultation. No term rate benchmark yet.FSB OSSGs message.

ICE swap rate. ARRC recommendations for bond, loan,securitisation include term rates. Term SOFR expected by end2021. Consultation Euro term rate as EURIBOR-fallback. Workinggroup on Term SONIA Rate (TSRR) + 3 term rate proposals.FCA’Bailey comment at BoE panel (June 2019): enough liquidityin Sonia-linked instruments for a forward-looking term rate

[...] very large legacy [...] various ways to deal with thisand our view on this is that nothing is off the table.

Planning for the end of LIBOR, Cass Business School – 19 June 2019 15

muRisQ Advisory

Fallback – Forward-looking v backward-looking

OIS benchmark and compounded in arrears are the same beforethe fixing date in term of valuation and risk management.

The floating leg amount paid is the spot OIS rate as measured atthe fixing date t0 for the period [u, v ] and it is paid in v .

EX [(Ncv )−1NδF c(t0; u, v)

]= N EX

[(Nc

t0)−1δNct0 EX [(Nc

v )−1F c(t0; u, v)∣∣Ft0

]]= N EX

[(Nc

t0)−1

(Pc(t0, u)

Pc(t0, v)− 1

)Pc(t0, v)

]= N EX [(Nc

t0)−1 (Pc(t0, u)− Pc(t0, v))]

= N (Pc(0, u)− Pc(0, v))

(1) tower property of conditional expectation. (2) F c(t0; u, v) isFt0-measurable, the formula of the forward and the definition ofPc(t0, v). (3) martingale property of Pc(., x).

Planning for the end of LIBOR, Cass Business School – 19 June 2019 16

muRisQ Advisory

Fallback – Forward-looking v backward-looking

The cash flow in the backward-looking option is also paid in v butbased on the composition of the daily rates ri compounded overthe period. This is similar to the floating payment of an OIS. Thepresent value is given by

EX

[(Nc

v )−1N

(n∏

i=1

(1 + δi ri )− 1

)]= NPc(0, v)

(Pc(0, u)

Pc(0, v)− 1

)= N (Pc(0, u)− Pc(0, v))

Planning for the end of LIBOR, Cass Business School – 19 June 2019 17

muRisQ Advisory

LIBOR Fallback – Quantitative Finance

1 From basics to fallback [Pay-off]

2 Fallback implementation – Adjusted RFR [Measurability]

3 Spread Adjustment and Value transfer [Filtration]

4 Conclusions

5 Annexes

Planning for the end of LIBOR, Cass Business School – 19 June 2019 18

muRisQ Advisory

Fallback - spread options

When?Values computed just before the discontinuation’s announcement.

Option 1: Forward ApproachThe spread adjustment [is] calculated based on observed marketprices for the forward spread between the relevant IBOR and theadjusted RFR in the relevant tenor at the time the fallback istriggered.

Option 2: Historical Mean/Median ApproachThe spread adjustment [is] based on the mean or median spotspread between the IBOR and the adjusted RFR calculated over asignificant, static lookback period (e.g., 5 years, 10 years) prior tothe relevant announcement or publication triggering the fallbackprovisions.Pro: No manipulation, easy to compute and verify.Con: Value transfer. Not related to current market situation.

Planning for the end of LIBOR, Cass Business School – 19 June 2019 19

muRisQ Advisory

Adjustment Spread Computation

The adjustment spread is denoted S(X , [a− l , a]).

It depends on methodology parameters X still to be decided(mean/median, data trimming, transition period). It also dependson market data measured in the period [a− l , a]. Theannouncement date is unknown (stopping time) and the look-backperiod l is an element of the methodology parameters. All theLIBOR derivatives are now path dependent up to thediscontinuation’s announcement date.

Ncs EX [(Nc

v )−1(1{d > θ}I j(θ) + 1{d ≤ θ}(FRj(θ) + S(X , [a− l , a]))

) ∣∣Fs

].

When is the value transfer taking place?

Planning for the end of LIBOR, Cass Business School – 19 June 2019 20

muRisQ Advisory

Has the value transfer started?

Jan-

12

Jan-

13

Jan-

14

Jan-

15

Jan-

16

Jan-

17

Jan-

18

Jan-

19

Jan-

20

Date

0

0.5

1

1.5

2

2.5

3R

ate

(in %

)

Overnight compounding in arrears fallback - past rates

USD-LIBOR-3MEFFR-CMP-3MSpreadRunning Measures 1-JunRunning Measures 1-JulRunning Measures 1-AugRunning Measures 1-SepRunning Measures 1-OctRunning Measures 1-NovRunning Measures 1-DecRunning Measures 1-Jan

Planning for the end of LIBOR, Cass Business School – 19 June 2019 21

muRisQ Advisory

Has the value transfer started?

Jan-

17

Jan-

18

Jan-

19

Date

0

0.1

0.2

0.3

0.4

0.5

0.6R

ate

(in %

)

Overnight compounding in arrears fallback - past rates

USD-LIBOR-3MEFFR-CMP-3MSpreadRunning Measures 1-JunRunning Measures 1-JulRunning Measures 1-AugRunning Measures 1-SepRunning Measures 1-OctRunning Measures 1-NovRunning Measures 1-DecRunning Measures 1-Jan

Planning for the end of LIBOR, Cass Business School – 19 June 2019 22

muRisQ Advisory

Has the value transfer started?

Jul-2

018

Oct-20

18

Jan-

2019

Apr-2

019

Date

15

20

25

30

35

40

45

50

Spr

ead

(in b

ps)

LIBOR3M-EFFR basis 30YRunning Measures 5YRunning Measures 7YRunning Measures 10Y

See blog for more graphs:http://multi-curve-framework.blogspot.com/2019/05/

making-money-on-libor-fallback-5.htmlPlanning for the end of LIBOR, Cass Business School – 19 June 2019 23

muRisQ Advisory

Fallback in the language of quants – Spread

The spread S(X , [a− l , a]) is Fa-mesurable.

Two filtrations (or maybe many): the pure market one, that wehave denoted Fs , and a second one, containing the fallbackmethodology with material non-public information Gs : Fs ⊂ Gs .

What is the difference between

Ncs EX [(Nc

w )−1X (FRj(θ) + S(X , [a− l , a]))∣∣F s

]and

Ncs EX [(Nc

w )−1X (FRj(θ) + S(X , [a− l , a]))∣∣Gs] .

The historical spreads used to compute S cannot be manipulated.

Is it possible to manipulate the methodology and G?

Is it possible to know (part of) G before the general public thatknowns only F?

Planning for the end of LIBOR, Cass Business School – 19 June 2019 24

muRisQ Advisory

LIBOR Fallback – Quantitative Finance

1 From basics to fallback [Pay-off]

2 Fallback implementation – Adjusted RFR [Measurability]

3 Spread Adjustment and Value transfer [Filtration]

4 Conclusions

5 Annexes

Planning for the end of LIBOR, Cass Business School – 19 June 2019 25

muRisQ Advisory

Benchmarks – Timetable

Jul 2014 FSB’s paper Reforming Interest Rate BenchmarksApr 2017 GBP preferred RFR: Reformed SONIAJun 2017 USD preferred RFR: SOFRApr 2018 First SOFR rate publishedSep 2018 EUR preferred RFR: ESTEROct 2018 First ISDA consultation on fallbackNov 2018 Results of first ISDA consultation on fallbackMay 2019 Second ISDA consultation on fallback??? 2019 Brexit???? 2019 New ISDA definitions including fallback provisionsOct 2019 First ESTER rate publishedOct 2019 EONIA recalibration??? 20?? SOFR/ESTER as collateral rateEnd 2021 End LIBOR agreement FCA/LIBOR panel banksDec 2021 End transition period for EU Benchmark RegulationDec 2021 End of EONIA

Planning for the end of LIBOR, Cass Business School – 19 June 2019 26

muRisQ Advisory

Conclusions

A clear fallback procedure is required. The ISDA consultationand its results have left many details open, including theachievability of the option selected. A substantial work is stillrequired to obtain a non-ambiguous fallback procedure.[Pay-off and Measurability]

Spread methodology selected responsible for value transfer.Some transfer may have taken place already and some maytake place later. [Filtration]

The selected transition and fallback solutions require asubstantial work from a quant finance and systemsperspective.

Planning for the end of LIBOR, Cass Business School – 19 June 2019 27

muRisQ Advisory

Contacts

muRisQ AdvisoryWeb: murisq.com

Email: info@murisq.com

Quantitative financeDerivatives marketModel validation

Market infrastructure

Planning for the end of LIBOR, Cass Business School – 19 June 2019 28

muRisQ Advisory

LIBOR Fallback – Quantitative Finance

1 From basics to fallback [Pay-off]

2 Fallback implementation – Adjusted RFR [Measurability]

3 Spread Adjustment and Value transfer [Filtration]

4 Conclusions

5 Annexes

Planning for the end of LIBOR, Cass Business School – 19 June 2019 29

muRisQ Advisory

Risk Transition through the Fallback

Oct-18

Jan-

19

Apr-1

9

Jul-1

9

Oct-19

Jan-

20

Apr-2

0

Jul-2

0

Oct-20

Date

0

2

4

6

8

10

12

14

16

18

20

PV

01 (

KU

SD

/bp)

Ann

ounc

emen

t Dat

e

Dis

cont

inua

tion

Dat

e

Legacy LIBOR3MLegacy DSC-ONFallback LIBOR3MOIS Benchmark DSC-ONSpot DSC-ONIn Advance DSC-ONIn Arrears DSC-ON

Planning for the end of LIBOR, Cass Business School – 19 June 2019 30

muRisQ Advisory

Risk – mixed delta

Delta risk/bucketed PV01. Risk as of December 2018 under thehypothesis that discontinuation will take place on 1 Jan 2022.

Produced by scalable tools based on production-grade libraries.

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muRisQ Advisory

Estimating the value transfer

Exact methodologies (spreads) are not known yet. Nevertheless itis possible to estimate the value impacts for different scenarios.

Spot/historical spread impact Discontinuation date impact

Example: USD 1000 swap portfolio, change adjustment spreadand discontinuation date (extract from POC with test portfolio).

Planning for the end of LIBOR, Cass Business School – 19 June 2019 32

muRisQ Advisory

Vanilla becoming exotics? Cap/floor

Vanilla IBOR caplet with strike K̄ , expiry t0 and paid-off in v :

δj(t0)(I j(t0)− K̄ )+.

Amount in the case of compounding setting in arrears? Amountstill paid in v and is given, for a spread S , by

δj(t0)

(1

δj(t0)

(n∏

i=1

(1 + δ0i I

O(si−1))− 1

)+ S − K̄

)+

The important difference is the the expiry date, which is nowdelayed to sn−1 = v − 1d . The pay-off can be writen as

max

(n−1∏i=0

(1 + δiF

O(si , si , si+1)), 1 + δK

)− 1

The vanilla IBOR cap/floor are becoming Asian options usingcompounding as averaging method on rates.

Planning for the end of LIBOR, Cass Business School – 19 June 2019 33

muRisQ Advisory

Is the GBP spread curve flat?

0 5 10 15 20 25Maturity (in years)

9

10

11

12

13

14

15

16

17

18

19S

prea

d (in

bps

)

Planning for the end of LIBOR, Cass Business School – 19 June 2019 34