A Simple Multi-Curve Model for Pricing SOFR
Futures and Other Derivatives
Fabio Mercurio∗
Abstract
In this note we propose a simple two-factor multi-curve model where Fed-fund,
SOFR and LIBOR rates are modeled jointly. The model is used to price the newly
quoted SOFR futures as well as Eurodollar futures. We then derive pricing formulas
for SOFR-based swaps, and show how the valuations of LIBOR-based swaps as well
as LIBOR-SOFR basis swaps are impacted by the introduction of a new LIBOR
fallback.
1
Introduction
With the purpose of proposing a new, IOSCO compliant, interest rate benchmark, the US
Alternative Reference Rates Committee (ARRC) identified a treasuries repo financing rate,
which they called Secured Overnight Funding Rate (SOFR), as the best replacement for
LIBOR. Similar decisions were made by regulators in other countries. In the UK, LIBOR
will be replaced with reformed SONIA, while CHF OIS and swap markets have transitioned
from TOIS to SARON, a secured overnight rate based on repo trades.
Each business day, starting April 3, 2018, the New York Fed has been publishing the
SOFR on the New York Fed website. The SOFR includes all trades in the Broad General
Collateral Rate plus bilateral Treasury repurchase agreement (repo) transactions cleared
through the Delivery-versus-Payment (DVP) service offered by the Fixed Income Clearing
Corporation (FICC). It is calculated as a volume-weighted median of transaction-level
tri-party repo data collected from the Bank of New York Mellon as well as GCF Repo
transaction data and data on bilateral Treasury repo transactions cleared through FICC’s
DVP service.
On May 7, 2018, CME launched 1-month and 3-month SOFR futures contracts. The
contract listings of the 1-month futures comprise the nearest 7 calendar months. The
contract listings of the 3-month futures comprise the 20 March quarterly months starting
∗
The views and opinions expressed in this article are my own and do not represent the opinions of any
firm or institution. Bloomberg LP and Bloomberg Professional service are trademarks and service marks
of Bloomberg Finance L.P., a Delaware limited partnership, or its subsidiaries. All rights reserved.
1
Electronic copy available at: https://ssrn.com/abstract=3225872
with June 2018. CME announced that they are targeting Q3-2018 for clearing OTC SOFRbased swaps, while LCH cleared its first SOFR swap and SOFR-Fed-fund basis swap on
July 16, 2018.
The existing data on SOFR and SOFR-based derivatives leads to the creation of a
SOFR interest-rate curve that can be stripped and extrapolated using available quotes.
In this paper, we introduce a simple multi-curve model that allows for the consistent
calculation of the convexity adjustments of SOFR futures as well as of Eurodollar futures.
We also introduce pricing formulas for SOFR-based swaps and show how the valuations
of LIBOR-based swaps as well as LIBOR-SOFR basis swaps change because of the new
LIBOR fallback that will be introduced by ISDA.
2
Main assumptions, definitions and notation
We assume a continuous-time framework where both the Fed-fund and SOFR rates are
approximated by instantaneous rates, whose time-t values are denoted by r(t) and s(t),
respectively. A plot of Fed-fund and SOFR rates is provided in Figure 1. Rates r(t) and
s(t) have associated money-market accounts and discount curves. We denote by P (t, T ),
resp. Ps (t, T ), the price at time t of the OIS, resp. SOFR, zero-coupon bond with maturity
T.
We assume the existence of a risk-neutral measure Q, and that numeraires are defined
by the OIS curve. We denote by E the expectation with respect to Q, and by ET the
expectation with respect to the T -forward measure whose associated numeraire is the OIS
zero-coupon bond P (t, T ). An equivalent set up can be considered where numeraires are
defined by the SOFR curve.
Cleared derivatives whose Price Alignment Interest (PAI) is the Fed-fund rate, are
priced using OIS discounting. That is, the time-t value of a payoff H received at time T is
given by:
h RT
i
V (t) = E e− t r(u) du H|Ft
(1)
= P (t, T )ET [H|Ft ]
where Ft is the sigma-algebra generated by the modeled risk factors up to time t. Instead,
cleared derivatives whose PAI is the SOFR rate, can be priced using SOFR discounting.
In this case, see Fuji, Shimada and Takahashi (2010) or Piterbarg (2010, 2012), the time-t
value of payoff H due at time T is
h RT
i
V (t) = E e− t s(u) du H|Ft
(2)
Given a quarterly time structure T0 , T1 , . . . , Tn , we define the forward LIBOR at time
t for the interval [Tj−1 , Tj ) as in Mercurio (2009), that is we set:
Lj (t) = ETj L(Tj−1 , Tj )|Ft
where L(t, T ) is the time-t spot LIBOR with maturity T . We denote by τj the year fraction
for the interval [Tj−1 , Tj ), and by Qj the Tj -forward measure.
2
Electronic copy available at: https://ssrn.com/abstract=3225872
Figure 1: Daily fixings of SOFR and Fed-fund rates (in %) from April 2, 2018 to July 27,
2018. Source: Bloomberg.
3
Modeling OIS, SOFR and LIBOR dynamics
We consider a multi-curve model that extends the parsimonious multi-curve model of
Moreni and Pallavicini (2010). We model the joint evolution of Fed-fund (that is, OIS),
SOFR and LIBOR rates assuming that: i) forward LIBORs follow a shifted-lognormal
LMM; ii) the Fed-fund and SOFR rates evolve according to a Hull-White one-factor (1990)
model; iii) the SOFR-OIS basis is deterministic.
As is clear from Figure 1, the SOFR shows a more diffusive behavior than the Fedfund rate, whose historical evolution is closer to that of a point process. The reason for
using Hull-White (1990) dynamics for both SOFR and Fed-fund rates is purely driven by
convenience as well as by the consideration that, in this paper, we model OIS rates only
for discounting payoffs and for defining pricing measures.1
3.1
OIS rate dynamics
The instantaneous OIS short rate r(t) is assumed to follow, under Q, a Hull-White onefactor (1990) model:
r(t) = x(t) + α(t)
where α is a deterministic function, and
dx(t) = −ax(t) dt + σ(t) dZ(t),
x(0) = 0
(3)
where a is a positive constant, σ is deterministic, and Z is a standard Brownian motion
under Q.
1
We also notice that the longer the tenor of an OIS rate, the more diffusive its historical behavior is.
3
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As is well known, see also Appendix A, the price at time t of the OIS zero-coupon bond
with maturity T is given by:
h RT
i
P (t, T ) = E e− t r(u) du |Ft
(4)
P (0, T ) A(0, t)A(t, T ) −B(t,T )x(t)
e
=
P (0, t)
A(0, T )
where
Z T
1
2
2
A(t, T ) = exp
σ (u)B (u, T ) du
2 t
1
B(t, T ) =
1 − e−a(T −t)
a
Matching OIS zero-coupon bond prices at time 0, see again Appendix A, leads to
Z t
α(t) = f (0, t) +
σ 2 (u) e−a(t−u) B(u, t) du
0
where f (0, t) is the instantaneous OIS forward rate at time 0 with maturity t.
The simply-compounded OIS forward rate for the interval [Tj−1 , Tj ) is defined, as in
the classic single-curve case, by
1 P (t, Tj−1 )
Fj (t) =
−1
τj P (t, Tj )
This is the correct definition also under SOFR discounting thanks to the assumption of
deterministic OIS-SOFR basis.
The forward rate Fj (t) is a martingale under the (OIS) Tj -forward measure Qj . Application of Ito’s lemma and formula (4) lead to the following Qj -dynamics:
1
dFj (t) = Fj (t) +
[B(t, Tj ) − B(t, Tj−1 )]σ(t) dZj (t)
(5)
τj
where Zj is a standard Qj -Brownian motion.
3.2
SOFR dynamics
We model the instantaneous SOFR short rate s(t) by assuming that
s(t) = x(t) + β(t)
where x is the process defined in (3) and β is a deterministic function to be used to calibrate
the time-0 SOFR curve.
This implies that
s(t) − r(t) = β(t) − α(t) = γ(t)
4
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which is consistent with our assumption of deterministic SOFR-Fed-fund basis.
The SOFR zero-coupon bond at time t with maturity T is defined by:
h RT
i
Ps (t, T ) = E e− t s(u) du |Ft
h RT
i
R
− t r(u) du − tT γ(u) du
=E e
e
|Ft
= e−
RT
t
γ(u) du
(6)
P (t, T )
Contrary to P (t, T ), which is a tradable asset, bond Ps (t, T ) is not tradable in this paper
set up. Still, it can be stripped from market data of SOFR-based instruments.
Basis γ(t) can be inferred from OIS and SOFR zero-coupon bond prices at time 0. In
fact:
Z T
P (0, T )
γ(u) du = ln
(7)
Ps (0, T )
0
so
γ(t) = fs (0, t) − f (0, t)
(8)
and
Z
β(t) = fs (0, t) +
t
σ 2 (u) e−a(t−u) B(u, t) du
0
where fs (0, t) is the instantaneous SOFR forward rate at time 0 with maturity t. Moreover,
we can write:
Ps (0, T ) A(0, t)A(t, T ) −B(t,T )x(t)
e
Ps (t, T ) =
Ps (0, t)
A(0, T )
(9)
P (0, t) Ps (0, T )
= P (t, T )
P (0, T ) Ps (0, t)
Assuming continuous compounding instead of daily, the simply-compounded SOFR
forward rate Fjs (t) for the interval [Tj−1 , Tj ) is defined by
R Tj
s
Tj
Tj−1 s(u) du
1 + τj Fj (t) = E e
|Ft
(10)
and can be expressed as
1+
τj Fjs (t)
Tj
=E
=
=
=
=
R Tj
e
Tj−1
s(u) du
|Ft
R
R Tj
T
1
− t j r(u) du Tj−1 s(u) du
E e
e
|Ft
P (t, Tj )
R
R Tj
T
1
− t j−1 r(u) du Tj−1 γ(u) du
E e
e
|Ft
P (t, Tj )
j
P (t, Tj−1 ) RTTj−1
γ(u) du
e
P (t, Tj )
Ps (t, Tj−1 )
Ps (t, Tj )
5
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(11)
Therefore, also SOFR forward rates can be written as ratios of bond prices (minus 1 and
divided by the year fraction). However, this is only true in the deterministic basis case.
We refer to Appendix B for the formulas in the general case.
3.3
Forward LIBOR dynamics
Forward LIBORs Lj , j = 1, . . . , n, are assumed to evolve under their corresponding forward
measures Qj according to:
dLj (t) = σj (t)[Lj (t) + αj ] dWj (t)
(12)
where the σj ’s are deterministic, αj ’s are constant, and Wj is a standard Qj -Brownian
motion, j = 1, . . . , n.
We assume a one-factor model for simplicity, that is we set dWi (t) dWj (t) = dt for all
i, j.
Explicit bond pricing (4) allows us to derive explicit dynamics of forward LIBORs under
Q as well. The drift of Lj under Q can be obtained as follows:
dLj (t) d ln P (t, Tj )
dt
= ρσj (t)σ(t)B(t, Tj )[Lj (t) + αj ]
Drift(Lj ; Q) = −
where ρ denotes the instantaneous correlation between each Lj and r, that is dWj (t) dZ(t) =
ρj dt, which is constant because of the one-factor nature of our forward LIBOR dynamics.
The Q-dynamics of Lj is then given by:
dLj (t) = ρσj (t)σ(t)B(t, Tj )[Lj (t) + αj ] dt + σj (t)[Lj (t) + αj ] dW (t)
(13)
where W is a standard Q-Brownian motion such that dW (t) dZ(t) = ρ dt.
3.4
LIBOR model calibration
We assume that the ATM caplet lognormal volatility σjATM for Lj , j = 1, . . . , n, is quoted
in the market, and define the average volatility of Lj as
s
Z Tj−1
1
V̄j :=
σj2 (t) dt
(14)
Tj−1 0
If σj is constant, then V̄j = σj .
For any given αj , the ATM caplet volatility can be calibrated exactly if and only if we
have:
2
Lj (0)
1 ATM p
1
−1
V̄j = p
Φ
2Φ
σ
Tj−1 − 1 +
(15)
2[Lj (0) + αj ]
2 j
2
Tj−1
where Φ denotes the standard normal distribution function, see for instance Mercurio
(2017). Equivalent formulas can be presented based on an ATM caplet normal volatility.
The shift parameter αj can be either set to a predefined value, for instance αj = 1/τj ,
or calibrated to the ATM slope of the caplet smile.
6
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3.5
The LIBOR-OIS basis
Given the OIS rate Fj and LIBOR Lj , we define the associated basis spread Bj in a
multiplicative way:
[1 + τj Fj (t)][1 + τj Bj (t)] = 1 + τj Lj (t)
so that
Lj (t) − Fj (t)
1
Bj (t) :=
=
1 + τj Fj (t)
τj
"
Lj (t) +
Fj (t) +
1
τj
1
τj
#
−1
(16)
By definition Bj is Qj−1 -martingale. Using (5) and (12), the Qj -dynamics of Bj is:
Lj (t) + αj
1
σj (t)
dWj (t) − (B(t, Tj ) − B(t, Tj−1 ))σ(t) dZj (t)
dBj (t) = · · · dt + Bj (t) +
τj
Lj (t) + τ1j
(17)
3.6
The OIS model calibration
In the absence of OIS option data, we fix a value of a, for instance a = 0.03, and calibrate
σ(t) by minimizing the volatility of basis, for a given n, following Mercurio (2017, 2018).
Assuming a constant σ, minimizing the volatility of Bn in (17) at time 0 yields:
σ=
Ln (0) + αj
ρσn (0)
B(0, Tn ) − B(0, Tn−1 ) Ln (0) + τ1n
(18)
An alternative σ can be obtained by minimizing the standard deviation of the log return
of Bn + 1/τn from time 0 to time Tn−1 .
If αn = 1/τn , then there exists a function σ(t) that minimizes the volatility of Bn at
each time t, that is:
ρσn (t)
(19)
σ(t) =
B(t, Tn ) − B(t, Tn−1 )
Alternatively, σ(t) can be chosen so as to maximize the smoothness of the SOFR curve
constructed using existing quotes of both SOFR futures and SOFR swap rates.
4
The pricing of Eurodollar futures
The Eurodollar futures rate at time t for the same interval [Tj−1 , Tj ) is defined by
fj (t) = E[L(Tj−1 , Tj )|Ft ]
(20)
and is associated with the Eurodollar-futures contract, with unit notional, that pays out
1 − L(Tj−1 , Tj ) at time Tj−1 .
A futures convexity adjustment is defined to be the difference between the corresponding
futures and forward LIBOR rates, that is:
Cj (t) = fj (t) − Lj (t)
7
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(21)
Under our previous assumptions, and thanks to (13), Eurodollar-futures convexity adjustments can be calculated exactly and in closed form, see Henrard (2014) or Mercurio
(2017, 2018). We have:
Z Tj−1
Cj (0) = [Lj (0) + αj ] exp ρ
σj (t)σ(t)B(t, Tj ) dt − 1
(22)
0
5
The pricing of CME 1m-SOFR futures
We consider a 1m-SOFR futures contract with maturity T , and whose delivery month is
represented by the interval [T − δ, T ), where δ is approximately one month, and where
we assume T − δ ≥ 0. We approximate the arithmetic average of daily SOFR during the
delivery month by:2
Z
1 T
s(u) du
(23)
δ T −δ
so the corresponding 1m-SOFR futures rate at time 0, fs1m (0; T − δ, T ), is calculated as
follows:
Z
i
1 h T
1m
fs (0; T − δ, T ) = E
s(u) du
δ
T −δ
Z
i 1Z T
1 h T
(24)
x(u) du +
β(u) du
= E
δ
δ T −δ
T −δ
Z
1 T
=
β(u) du
δ T −δ
since E[x(t)] = 0 for each t.
Using (48), we then get:
1 Ps (0, T − δ) 1
A(0, T )
ln
+ ln
δ
Ps (0, T )
δ A(0, T − δ)
1
1
A(0, T )
= ln[1 + δFs (0; T − δ, T )] + ln
δ
δ A(0, T − δ)
1m
= Rs (0; T − δ, T ) + Cs (0; T − δ, T )
fs1m (0; T − δ, T ) =
(25)
where Fs (0; T − δ, T ) and Rs (0; T − δ, T ) denote, respectively, the simply-compounded and
continuously-compounded SOFR forward rates for the interval [T − δ, T ), and Cs1m (0; T −
δ, T ) is the 1m-SOFR futures convexity adjustment, that is:
Cs1m (0; T − δ, T ) =
2
1
A(0, T )
ln
δ A(0, T − δ)
The case of the exact discrete-time payoff is addressed in Harris (2018).
8
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(26)
In the case of a constant σ(t) ≡ σ, this convexity adjustment is explicitly given by
σ2
2 −aT
1 −2aT
1m
aδ
2aδ
Cs (0; T − δ, T ) =
δ + e (1 − e ) − e
(1 − e )
2δa2
a
2a
σ2
= [3T 2 − 3T δ + δ 2 ] + O(a)
6
(27)
Since δ ≈ 1/12, the maximum T ≈ 7/12 and σ typically below 1%, then Cs1m (0; T − δ, T )
is likely to be a fraction of a basis point even for the longest quoted maturity. So, to a
high degree of accuracy, fs1m (0; T − δ, T ) ≈ Rs (0; T − δ, T ).
As per the prompt futures contract, that is when T − δ < 0, part of the integral in (23)
has already been accumulated. In this case, we have:
Z
Z
1 T
1 0
1m
fs (0; T − δ, T ) =
s(u) du +
β(u) du
δ T −δ
δ 0
(28)
Z
1 0
1 A(0, T )
=
s(u) du + ln
δ T −δ
δ Ps (0, T )
6
The pricing of CME 3m-SOFR futures
We consider a 3m-SOFR futures contract with maturity Tj (corresponding to the 3rd
Wednesday of the futures delivery month), and whose reference quarter is represented by
the interval [Tj−1 , Tj ). We approximate the compounded daily SOFR interest rate during
the reference quarter by:3
j
1 RTTj−1
s(u) du
−1
(29)
e
τj
3m
(0), is
So, assuming Tj−1 ≥ 0, the corresponding 3m-SOFR futures rate at time 0, fs,j
calculated as follows:
R Tj
3m
Tj−1 s(u) du
1 + τj fs,j (0) = E e
R Tj
R Tj
β(u) du
Tj−1 x(u) du
=E e
e Tj−1
(30)
R Tj
Ps (0, Tj−1 ) A(0, Tj )
Tj−1 x(u) du
=
E e
Ps (0, Tj ) A(0, Tj−1 )
where we used again (48).
3
Also in this case, the exact discrete-time payoff is addressed in Harris (2018).
9
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By the tower property of conditional expectations, we can write
R Tj
R Tj
x(u)
du
x(u)
du
E e Tj−1
= E E e Tj−1
|FTj−1
B(Tj−1 ,Tj )x(Tj−1 )
= E A(Tj−1 , Tj ) e
1
= A(Tj−1 , Tj ) e 2 B
R Tj−1
2 (T
j−1 ,Tj ) 0
σ 2 (u)e−2a(Tj−1 −u) du
where we used (42) and (40). This leads to:
R Tj−1 2
−2a(Tj−1 −u)
1 2
Ps (0, Tj−1 ) A(0, Tj )
du
A(Tj−1 , Tj ) e 2 B (Tj−1 ,Tj ) 0 σ (u)e
Ps (0, Tj ) A(0, Tj−1 )
(31)
Ps (0, Tj−1 ) Uj
=
e
Ps (0, Tj )
3m
1 + τj fs,j
(0) =
where the last equality defines Uj . In the case of a constant σ(t) ≡ σ, Uj is explicitly given
by:
σ 2 −a(Tj +Tj−1 )
Uj = 3 e
− e−2aTj + e−a(Tj −Tj−1 ) + 2a(Tj − Tj−1 ) − 1 + 2e−aTj − 2e−aTj−1
2a
σ2 3
2
3
=
2Tj − 3Tj Tj−1
+ Tj−1
+ O(a)
6
σ2
= (Tj − Tj−1 ) 3Tj2 − (Tj − Tj−1 )2 + O(a)
6
(32)
As per the prompt futures contract, that is when Tj−1 < 0, part of the integral in (29)
has already been accumulated. In this case we have:
R
R
R0
Tj
Tj
3m
x(u)
du
Tj−1 s(u) du
1 + τj fs,j (0) = e
E e0
e 0 β(u) du
R
R0
Tj
A(0, Tj )
x(u)
du
Tj−1 s(u) du
E e0
=e
(33)
Ps (0, Tj )
R0
s(u) du A(0, Tj )
= e Tj−1
A(0, Tj )
Ps (0, Tj )
Recalling (11), the SOFR forward rate Fjs (0) can be obtained from the quoted futures
3m
rate fs,j
(0) as follows:
h1
i
1
3m
+ fs,j
(0) e−Uj −
Fjs (0) =
τj
τj
3m
3m
so, the 3m SOFR futures convexity adjustment, Cs,j
(0) := fs,j
(0) − Fjs (0), is given by
h1
i
3m
3m
Cs,j
(0) =
+ fs,j
(0) 1 − e−Uj
τj
(34)
h1
i
s
Uj
=
+ Fj (0) e − 1
τj
10
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Since
write
1
τj
3m
>> fs,j
(0) and Uj is small, when σ(t) ≡ σ, using the last equality in (32), we can
3m
Cs,j
(0) ≈
σ2
1
Uj ≈ Tj2
τj
2
which gives a simple formula for assessing the size of the convexity adjustment.
7
Stripping discount factors from futures
Let us denote by T11m , . . . , TN1m the maturities of the N quoted 1m-SOFR futures rates,
3m
and by T13m , . . . , TM
the maturities of the M quoted 3m-SOFR futures rates, where N ≤ 7
and M ≤ 20. Let us then denote by T01m and T03m the start dates of the delivery periods
for the prompt 1m- and 3m- SOFR futures contracts, respectively. The 1m- and 3m-SOFR
futures maturities overlap. For instance, on June 27, 2018, their order was:
3m
T11m < T21m < T31m < T13m < T41m < T51m < T61m < T23m < T71m < T33m < T43m < . . . < T20
SOFR discount factors can be stripped from the available 1m- and 3m-SOFR futures
3m
as follows.
quotes for maturities T11m , . . . , TN1m and T13m , . . . , TM
1m
Denoting by δj the year fraction for the interval [Tj−1 , Tj1m ), the discount factor for T11m
is obtained using (28), that is:
Ps (0, T11m )
=
R0
1m
1m 1m
1m s(u) du−δ1 fs (0;T0 ,T1 )
1m
A(0, T1 ) e T0
The subsequent discount factors, for j = 2, . . . , N , are then obtained recursively using (25),
that is:
A(0, Tj1m ) −δj f 1m (0;T 1m ,T 1m )
1m
s
j−1 j
e
(35)
Ps (0, Tj1m ) = Ps (0, Tj−1
)
1m
A(0, Tj−1
)
3m
As per 3m-SOFR futures, we denote by τj the year fraction for the interval [Tj−1
, Tj3m ),
3m
and calculate the discount factor for T1 using (33), that is:
R0
Ps (0, T13m ) = e
T03m
s(u) du
A2 (0, T13m )
3m
1 + τ1 fs,1
(0)
The subsequent discount factors, for j = 2, . . . , M , are then obtained recursively using
(31), that is:
3m
Ps (0, Tj−1
) Uj
Ps (0, Tj3m ) =
e
3m
1 + τj fs,j (0)
In the absence of SOFR swap quotes, a SOFR curve can then be constructed by assuming, for instance, a deterministic basis between OIS swap or forward rates and the
corresponding synthetic SOFR rates. This is the approached followed by Bloomberg. A
snapshot of the Bloomberg SOFR curve on July 30, 2018 is presented in Figure 2.
If SOFR swap quotes are available, one can then use them to construct a SOFR curve
in the mid to long end, see the following section for the pricing formulas.
11
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Figure 2: The SOFR curve on July 30, 2018, based on the daily fixing and quoted futures
rates. SOFR swaps rates are created synthetically. Source: Bloomberg.
8
The valuation of a SOFR fixed-floating swap
Consider a swap where the floating leg pays at each time Tj , j = a + 1, . . . , b, a rate that
is obtained by compounding the daily fixings of the SOFR from Tj−1 to Tj , and where the
0
fixed leg pays the fixed rate K on dates Tc+1
, . . . , Td0 , with Tc0 = Ta and Td0 = Tb .
The floating-leg payment at time Tj is approximately given by (29) times the corresponding year fraction, that is
R Tj
e
Tj−1
s(u) du
−1
By (1) and (10), the value of this payment at time t ≤ Tj−1 is
R Tj
Ps (t, Tj−1 )
s
Tj
Tj−1 s(u) du
− 1|Ft = τj P (t, Tj )Fj (t) = P (t, Tj )
−1
P (t, Tj ) E e
Ps (t, Tj )
Therefore, at time t ≤ Ta , the SOFR swap value to the fixed-rate payer is given by
b
X
τj P (t, Tj )Fjs (t)
−K
j=a+1
d
X
τj0 P (t, Tj0 )
j=c+1
0
where τj0 denotes the year fraction for the fixed-leg interval [Tj−1
, Tj0 ).
The corresponding forward swap rate is then defined as the fixed rate K that makes
the swap value equal to zero at time t, that is:
Pb
s
j=a+1 τj P (t, Tj )Fj (t)
S(t) =
Pd
0
0
j=c+1 τj P (t, Tj )
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0
When Ta < t < Ta+1 ≤ Tc+1
, the SOFR swap value to the fixed-rate payer and the
corresponding swap rate become
R
b
d
X
X
t
1
s(u)
du
s
T
P (t, Ta+1 ) e a
−1 +
τj P (t, Tj )Fj (t) − K
τj0 P (t, Tj0 ) (36)
Ps (t, Ta+1 )
j=a+2
j=c+1
and
R
t
Pb
s(u)
du
1
s
P (t, Ta+1 ) e Ta
−
1
+
j=a+2 τj P (t, Tj )Fj (t)
Ps (t,Ta+1 )
S(t) =
Pd
0
0
j=c+1 τj P (t, Tj )
Formulas simplify when SOFR is the chosen PAI. In this case, by (2) and thanks to the
assumption of a deterministic basis γ, the OIS discount factors in the swap and swap-rate
formulas must be replaced with the corresponding SOFR ones. Therefore, under SOFR
discounting, the formulas for a SOFR forward-start swap and the associated forward swap
rates are equal to the corresponding LIBOR-based ones in a single-curve set up. The exact
same result can be obtained, more naturally and directly, by assuming that SOFR, and
not Fed-fund, is the risk-free rate.
9
The new valuation of a LIBOR fixed-floating swap
Consider a standard LIBOR-based swap where the floating leg pays at each time Tj , j =
a + 1, . . . , b, the LIBOR L(Tj−1 , Tj ) times the associated year fraction τj , and where the
0
fixed leg pays the fixed rate K on dates Tc+1
, . . . , Td0 , with associated year fractions τj0 . We
set Tc0 = Ta and Td0 = Tb .
Assuming OIS discounting, by the pricing formula (1), the swap value to the fixed-rate
payer at time t < Ta+1 is given by
b
X
τj P (t, Tj )Lj (t) − K
j=a+1
d
X
τj0 P (t, Tj0 )
j=c+1
where we set Lj (t) = L(Ta , Ta+1 ) if Ta ≤ t < Ta+1 . This valuation relies on LIBOR being
published at least until the last LIBOR fixing date Tb−1 , so that forwards Lj (t) can be
defined accordingly. However, soon enough this may no longer be the case. LIBOR is
in fact very likely to be discontinued before the end of 2021, and ISDA already started
consultations with the industry on the definition of a new LIBOR fallback. This means
that swaps like the above are standard up to some payment time Tk (included), and from
Tk (excluded) on become swaps written on a new interest rate index. Assuming Tk > Ta ,
the valuation of the above swap must then be modified as follows:4
k
X
j=a+1
τj P (t, Tj )Lj (t) +
b
X
τj P (t, Tj )L̂j (t) − K
d
X
τj0 P (t, Tj0 )
(37)
j=c+1
j=k+1
4
It Tk ≤ Ta , then the (forward-start) swap will pay the LIBOR fallback on each payment date of the
floating leg, leading to a simpler valuation formula.
13
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where L̂j (t) denotes the forward at time t of the new LIBOR fallback L̂(Tj−1 , Tj ), that is:
L̂j (t) = ETj L̂(Tj−1 , Tj )|Ft
The methodology for the new LIBOR fallback L̂(Tj−1 , Tj ) has not been decided yet. However, the consensus is that it will be defined as the sum of a SOFR-based term rate
R(Tj−1 , Tj ) applied to the interval [Tj−1 , Tj ], and a LIBOR-SOFR basis spread S(T ∗ ) calculated at the time T ∗ < Tk when an official announcement of LIBOR discontinuation will
be given:
L̂(Tj−1 , Tj ) = R(Tj−1 , Tj ) + S(T ∗ )
where S(T ∗ ) is the difference between the LIBOR and the SOFR-based term rate calculated
at T ∗ .5 Therefore, we can write:
L̂j (t) = Rj (t) + ETj S(T ∗ )|Ft
where Rj (t) is the time-t forward of R(Tj−1 , Tj ), that is: Rj (t) = ETj R(Tj−1 , Tj )|Ft .
Forwards L̂j (t) can be calculated using a multi-curve model, like the one we propose
in this paper, where SOFR and LIBOR (and possibly OIS) rates are jointly modeled.
This allows us to calculate SOFR-based forwards Rj (t), should the choice of term rate
R(Tj−1 , Tj ) generate a convexity adjustment for Rj (t), as is the case for instance when
we set R(Tj−1 , Tj ) = s(Tj−1 ). It will also allow us to calculate expected values of S(T ∗ ),
should it be modeled as stochastic. In fact, there is no a-priori reason why SOFR-based
and LIBOR-based swaps will be valued in a way consistent with the the assumption of a
deterministic basis S(T ∗ ).
10
The valuation of a LIBOR-SOFR basis swap
A LIBOR-SOFR basis swap is a swap with two floating legs, one being the floating leg of
a LIBOR fixed-floating swap, the other being the floating leg of a SOFR-based swap with
the same maturity and payment frequency. Let us denote by Ta the (forward) start date
of the swap, and by Tj , j = a + 1, . . . , b its payment dates. Assuming, for ease of notation,
the same day count convention, and hence year fractions, for the two legs, the value at
time t of the basis swap to the LIBOR payer is:
b
X
τj P (t, Tj )Fjs (t)
j=a+1
−
k
X
τj P (t, Tj )Lj (t) −
j=a+1
b
X
τj P (t, Tj )L̂j (t)
j=k+1
where, for simplicity, we also assume t ≤ Ta < Tk . The case when Ta < t < Ta+1 , which
we omit for brevity, is based on the formula for the floating-leg value in (36).
5
An alternative definition is based on using the forward spread at T ∗ for the interval [Tj−1 , Tj ].
14
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11
Conclusions
We have introduced a simple multi-curve model to price SOFR futures, as well as SOFR
swaps, with the purpose of building a SOFR curve. Then, we have shown how the valuations of standard LIBOR-based swaps and LIBOR-SOFR basis swaps change because of
LIBOR discontinuation and the introduction of the new LIBOR fallback.
At the time of writing, there are still many more questions than answers. For instance:
• How will a risk-free term rate be calculated?
• How will LIBOR fallbacks be defined?
• Will there be a LIBOR fallback basis between cash instruments and derivatives?
• Will there be a “zombie” LIBOR, that is a surviving LIBOR based on just a few
contributions?
• Will there be a new LIBOR proxy calculated off existing contracts or new ones that
may be created for this purpose?
• When will the market start to trade SOFR-based caps, swaptions and then nonvanilla interest-rate derivatives?
• How to transition from a LIBOR-based contract to a SOFR-based one?
• etc
Some of these questions will hopefully be addressed soon. In the meantime, multi-curve
models can be used to quantify the impact of LIBOR discontinuation and the introduction
of a new risk-free rate on the valuation of derivatives.
References
[1] Fujii, M., Shimada, Y. and A. Takahashi (2010). A note on construction of multiple swap curves with and without collateral. Available online at
https://www.fsa.go.jp/frtc/seika/discussion/2009/20100203-1.pdf
[2] Harris, J. (2018) Modeling Interest Rate Futures Convexity. Working paper,
Bloomberg, London.
[3] Henrard, M. (2014) Interest Rate Modelling in the Multi-Curve Framework: Foundations, Evolution and Implementation. Palgrave Macmillan.
[4] Hull, J., White, A. (1990) Pricing Interest-Rate-Derivative Securities. Review of financial studies 3(4), 573-592.
[5] Mercurio, F. (2009) Interest Rates and The Credit Crunch: New Formulas and Market
Models. Available online at:
http://papers.ssrn.com/sol3/papers.cfm?abstract id=1332205
[6] Mercurio, F. (2017) The Present of Futures: Valuing Eurodollar-Futures Convexity
Adjustments in a Multi-Curve World. Available online at:
http://papers.ssrn.com/sol3/papers.cfm?abstract id=2987832
[7] Mercurio, F. (2018) The Present of Futures, r isk.net March, 1-6
15
Electronic copy available at: https://ssrn.com/abstract=3225872
[8] Moreni, N. and Pallavicini, A. (2010) Parsimonious HJM Modelling for Multiple YieldCurve Dynamics. Available online at: https://arxiv.org/abs/1011.0828
[9] Piterbarg, V. (2010) Funding beyond discounting: collateral agreements and derivatives pricing, Risk February, 97-102.
[10] Piterbarg, V. (2012). Cooking with collateral, Risk August, 58-63.
Appendix A: Some analytical results
For the reader’s convenience, we here report some standard and well-known results for the
short-rate model driven by the SDE (3).
Solution of the SDE (3)
The SDE (3) can be solved in closed form. We have:
Z t
σ(u) e−a(t−u) dZ(u)
x(t) =
(38)
0
Equivalently, for any t < T , we can write:
x(T ) = x(t) e
−a(T −t)
T
Z
σ(u) e−a(T −u) dZ(u)
+
(39)
t
so, conditional on x(t), x(T ) is normally distributed with
E x(T )|x(t) = x(t) e−a(T −t)
Z T
Var x(T )|x(t) =
σ 2 (u) e−2a(T −u) du
(40)
t
The integral of x and bond pricing
The integral of x can also be expressed in closed form:
Z T
Z
x(T ) − x(t) 1 T
x(u) du = −
+
σ(u) dZ(u)
a
a t
t
Z T
= B(t, T )x(t) +
σ(u)B(u, T ) dZ(u)
(41)
t
where we used (39) and set B(t, T ) = [1 − e−a(T −t) ]/a. This shows that also
Gaussian with mean and variance given by
hZ T
i
E
x(u)|Ft = B(t, T )x(t)
t
hZ T
i Z T
Var
x(u)|Ft =
σ 2 (u)B 2 (u, T ) du
t
t
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RT
t
x(u) du is
(42)
which, using the explicit formula for the moment generating function of a normally distributed random variable, leads to
i
h RT
(43)
E e− t x(u) du |Ft = A(t, T ) e−B(t,T )x(t)
where
Z T
1
2
2
A(t, T ) = exp
σ (u)B (u, T ) du
2 t
Therefore, the price at time t of the OIS zero-coupon bond with maturity T is given by:
R
− tT r(u) du
|Ft
P (t, T ) = E e
R
R
T
(44)
− tT x(u) du
=E e
|Ft e− t α(u) du
= e−
RT
t
α(u) du
A(t, T )e−B(t,T )x(t)
Matching OIS zero-coupon bond prices at time 0 gives
Z T
A(0, T )
α(u) du = ln
P (0, T )
0
(45)
which, taking the time-T derivative of both sides, leads to
Z t
σ 2 (u) e−a(t−u) B(u, t) du
α(t) = f (0, t) +
0
Using (45), the OIS zero-coupon bond prices become:
P (t, T ) =
P (0, T ) A(0, t)A(t, T ) −B(t,T )x(t)
e
P (0, t)
A(0, T )
(46)
The integrals of α, β and γ
From (45), we immediately get, for any t < T ,
Z T
A(0, T )
P (0, t)
α(u) du = ln
+ ln
A(0, t)
P (0, T )
t
(47)
By the same token, the integral of β is given by the same formula (45), provided that OIS
bond prices are replaced with the corresponding SOFR ones, that is
Z T
A(0, T )
Ps (0, t)
β(u) du = ln
+ ln
(48)
A(0, t)
Ps (0, T )
t
Finally, the integral of γ can be obtained by difference:
Z T
Z T
Z T
Ps (0, t)
P (0, t)
γ(u) du =
β(u) du −
α(u) du = ln
− ln
Ps (0, T )
P (0, T )
t
t
t
which agrees with (7).
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(49)
Appendix B: the general case of a stochastic γ(t)
We derive general formulas for SOFR zero-coupon bonds and forward rates under the
assumption of a stochastic SOFR-Fed-fund basis.
The SOFR zero-coupon bond at time t with maturity T is defined by the first line of
(6). We have:
i
h RT
− t s(u) du
|Ft
Ps (t, T ) = E e
h RT
i
RT
(50)
= E e− t r(u) du e− t γ(u) du |Ft
i
h RT
= P (t, T ) ET e− t γ(u) du |Ft
Therefore, the ratio of SOFR and OIS discount factors is given by the expected value, in
the corresponding forward measure, of the stochastic discount factor calculated using an
instantaneous rate equal to γ(t).
The simply-compounded SOFR forward rate Fjs (t) for the interval [Tj−1 , Tj ) is defined
by (10). We have:
R Tj
s
Tj
Tj−1 s(u) du
1 + τj Fj (t) = E e
|Ft
R
R Tj
T
1
− t j r(u) du Tj−1 s(u) du
=
E e
|Ft
e
P (t, Tj )
R
R Tj
T
1
− t j−1 r(u) du Tj−1 γ(u) du
=
E e
|Ft
e
P (t, Tj )
(51)
R Tj
P (t, Tj−1 ) Tj−1 Tj−1 γ(u) du
=
E
e
|Ft
P (t, Tj )
h R Tj
i
R Tj
Tj
− t γ(u) du
|Ft
Ps (t, Tj−1 ) E e
γ(u)
du
Tj−1
h
iE
|Ft
e Tj−1
=
Ps (t, Tj ) ETj−1 e− RtTj−1 γ(u) du |F
t
In general, therefore, 1 + τj Fjs (t) is equal to the corresponding ratio of SOFR discount
factors, provided it is multiplied by a correction term, which is identically equal to 1 when
γ(t) is deterministic, and model dependent when γ is stochastic.
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