Numerix Support Papers
Liang Wu
Executive Director of
Financial Engineering, Pricing
RFR Cap/Floor Formulae
and Content
lwu@numerix.com
September 2022
Numerix LLC
Corporate Headquarters
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5th Floor
New York, NY 10016
Tel +1 212 302 2220
In this paper we provide the details for pricing RFR Caps/Floors with both forward­
looking and backward­looking payoffs. We consider rate dynamics based on the
Black and Bachelier models.
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RFR Cap/Floor Formulae | September 2022
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Definitions
The money market account B(t) is defined to grow at the risk­free rate r(t), such that B(0) = 1
and dB(t) = r(t) B(t) dt. This gives
B(t) = e
∫t
0
r(u) du
.
A stochastic discount factor is defined as
D(t, T ) = e−
∫T
t
r(u) du
=
B(t)
.
B(T )
With these the definitions, we can introduce the concept of extended zero­coupon bond price
P (t, T ) which can have a definition both before and after time T as described below (also see
[1] for details):
• When t ≤ T , P (t, T ) , EQ
t [D(t, T ) 1], which is the same as the usual zero coupon bond price
definition.
• When t > T , P (t, T ) , EQ
t [1 D(T, t)] = D(T, t), where it is more convenient to regard D(T, t)
as the realized continuously­compounding factor known at time t beyond T , i.e., D(T, t) =
∫t
e
2
T
r(u) du
.
The Pricing of RFR Caps/Floors
An RFR (Risk­Free Rate) Cap/Floor is a new type of interest rate cap/floor derivative contract that
direct referencing those overnight alternative reference rate such as SOFR instead of LIBOR. It
is still a series of caplet/floorlet payments on a payment schedule as T = {Tγ+1 , Tγ+2 , · · · , Tη }.
Denote the day count fraction between two adjacent points Tj−1 and Tj as τj . For each given j
and the interest period [Tj−1 , Tj ), there are 2 distinct caplets/floorlets with strike K and paying
off at time Tj :
+
• A forward­looking with payoff (ω (Rj (Tj−1 ) − K)) ,
+
• A backward­looking with payoff (ω (Rj (Tj ) − K)) ,
where ω = 1 indicates a caplet and ω = −1 indicates a floorlet.
Meanwhile, Rj (t) is defined as the backward­looking forward rate that can be further written
as the classic simply­compounded forward rate formula using the extended zero­coupon bond
price P (t, T ). For each given j and the interest period [Tj−1 , Tj ), with the shorthand notation
Pj (t) = P (t; Tj−1 , Tj ), Rj (t) can be written as:
Rj (t) =
1
τj
Pj−1 (t)
−1 .
Pj (t)
A few more properties of the forward rate Rj (t),
• Is a martingale under the Tj ­forward measure.
• Is equal to the forward­looking rate at time Tj−1 .
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RFR Cap/Floor Formulae | September 2022
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• Is equal to the realized backward­looking rate at time Tj .
• Stops evolving (that is, it is fixed) after time Tj .
When time t is within the interest rate period [Tj−1 , Tj ), the forward rate Rj (t) “aggregates”
values of realized rate r(u) in the period of [Tj−1 , t), which means that
Rj (t) =
1
τj
D(Tj−1 , t)
−1 .
Pj (t)
The price of a forward­looking cap/floor is then
η
X
CAPFLOORF
γη (t) =
Tj
1 τj Pj (t) EQ
t
h
(ω (Rj (Tj−1 ) − K))
+
h
i
i
,
j=γ+1
while the backward­looking one is
CAPFLOORB
γη (t) =
η
X
Tj
1 τj Pj (t) EQ
t
(ω (Rj (Tj ) − K))
+
,
j=γ+1
where ω = 1 indicates a caplet and ω = −1 indicates a floorlet.
2.1
Under the Lognormal/Black Model
If we choose the dynamics of Rj (t) to be a lognormal process, since it is a martingale under the
Tj ­forward measure QTj , we can model it as a driftless process:
dRj (t) = σjLN Rj (t) gj (t) dW j (t),
where σjLN is the lognormal volatility of the forward rate Rj (t), and we often call it the caplet
volatility. gj (t) is the decay function, which is assumed to be a piece­wise linear deterministic
function defined as
gj (t) = min
!
+
(Tj − t)
,1 ,
Tj − Tj−1
such that
• gj (t) = 1 for t ≤ Tj−1 ,
• gj (t) is monotonically decreasing for Tj−1 < t < Tj ,
• gj (t) = 0 for t ≥ Tj .
With these assumptions, the pricing formulae of forward­looking and backward­looking cap­
s/floors are:
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η
X
CAPFLOORF
γη (t) =
=
j=γ+1
η
X
τj Pj (t) Black(K, Rj (t), ΣjLN,F
p
LN,F
τj P (t, Tj ) Black(K, Rj (t), σγη
Tj−1 − t, ω)
p
Tj−1 − t, ω),
(2.1)
j=γ+1
CAPFLOORB
γη (t) =
=
=
η
X
j=γ+1
η
X
j=γ+1
η
X
τj Pj (t) Black(K, Rj (t), ΣjLN,B (t)
p
LN,B,w
τj P (t, Tj ) Black(K, Rj (t), σγη
Tj − t, ω)
p
Tj − t, ω)
LN,B,w/o
τj P (t, Tj ) Black(K, Rj (t), σγη
Gj (t)
p
Tj − t, ω),
(2.2)
(2.3)
j=γ+1
where
ΣjLN,F = σjLN ,
ΣjLN,B (t) = σjLN Gj (t)
v
u
3
u
+
(Tj − t) − (Tj−1 − t)
u (Tj−1 − t)+
1
+
.
= σjLN t
(Tj − t)
3 (Tj − Tj−1 )2 (Tj − t)
Black(K, F, V, ω) is the Black formula defined as
Black(K, F, V, ω) = ω (F N(ω d+ ) − K N(ω d− )) ,
Z x
u2
1
√ e− 2 du,
N(x) =
2π
−∞
V2
F
ln K ± 2
d± =
.
V
LN,B,w/o
LN,F
LN,B,w
σγη
, σγη
, and σγη
are called the forward­looking lognormal volatility, the backward­
looking lognormal cap volatility with time­decay, and the backward­looking lognormal cap volatil­
ity without time­decay quoted in the RFR cap market, respectively.
ΣjLN,F is the forward­looking lognormal caplet volatility bootstrapped from the quoted cap
volatility. ΣjLN,B (t) is the backward­looking lognormal caplet volatility with time­decay, σjLN is
the backward­looking lognormal caplet volatility without time­decay, and the time­decay ad­
justment/factor is Gj (t). When it comes to backward­looking cap volatility bootstrapping, the
interpolation can be applied to either caplet volatility with time­decay ΣjLN,B (t) or without time­
decay σjLN .
Specifically, ΣjLN,B (t) is obtained based on the integration below:
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RFR Cap/Floor Formulae | September 2022
2
ΣjLN,B (t)
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2
Tj − s
(Tj − t) =
min
,1
ds
Tj − Tj−1
t

3 
+
2 
1 (Tj − t) − (Tj−1 − t)

+
= σjLN (Tj−1 − t) +
,
2
3
(Tj − Tj−1 )
2
σjLN
Z
Tj
v
u
3
u
+
(T
−
t)
−
(T
−
t)
u (Tj−1 − t)+
j
j−1
1
ΣjLN,B (t) = σjLN t
+
(Tj − t)
3 (Tj − Tj−1 )2 (Tj − t)

r

2 (Tj−1 −t)
σ LN √1
, t < Tj−1 ,
1
+
j
(Tj −t)
3
=
 LN 1 Tj −t

σj √3 Tj −Tj−1 ,
Tj−1 ≤ t ≤ Tj .
2.2
Under the Normal/Bachelier Model
Similarly, if we choose the dynamics of Rj (t) to be a normal driftless process
dRj (t) = σjN gj (t) dW j (t),
and with the same assumptions as in the previous section, we get
CAPFLOORF
γη (t)
=
=
η
X
j=γ+1
η
X
τj Pj (t) Bachelier(K, Rj (t), ΣjN,F
p
N,F
τj P (t, Tj ) Bachelier(K, Rj (t), σγη
Tj−1 − t, ω)
p
Tj−1 − t, ω),
(2.4)
j=γ+1
CAPFLOORB
γη (t) =
=
=
η
X
j=γ+1
η
X
j=γ+1
η
X
τj Pj (t) Bachelier(K, Rj (t), ΣjN,B (t)
p
N,B,w
τj P (t, Tj ) Bachelier(K, Rj (t), σγη
Tj − t, ω)
p
Tj − t, ω)
N,B,w/o
Gj (t)
τj P (t, Tj ) Bachelier(K, Rj (t), σγη
p
Tj − t, ω),
(2.5)
(2.6)
j=γ+1
where
ΣjN,F = σjN ,
ΣjN,B (t) = σjN Gj (t)
v
u
3
u
+
(T
−
t)
−
(T
−
t)
u (Tj−1 − t)+
j
j−1
1
.
= σjN t
+
(Tj − t)
3 (Tj − Tj−1 )2 (Tj − t)
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RFR Cap/Floor Formulae | September 2022
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Bachelier(K, F, V, ω) is the Bachelier formula defined as
Bachelier(K, F, V, ω) = V (ω d N(ω d) + N′ (ω d)) ,
Z x
u2
1
√ e− 2 du,
N(x) =
2π
−∞
1 − x2
′
N (x) = √ e 2 ,
2π
F −K
d=
.
V
N,B,w/o
N,F
N,B,w
σγη
, σγη
and σγη
are called the forward­looking normal cap volatility, the backward­
looking normal cap volatility with time­decay, and the backward­looking normal cap volatility
without time­decay quoted in the RFR cap market, respectively.
Similarly, ΣjN,B (t) can be expressed as
v
u
3
u
+
(Tj − t) − (Tj−1 − t)
u (Tj−1 − t)+
1
ΣjN,B (t) = σjN t
+
(Tj − t)
3 (Tj − Tj−1 )2 (Tj − t)

r

2 (Tj−1 −t)
σ N √1
t < Tj−1 ,
1
+
j
(T
−t)
j
3
=

σ N √1 Tj −t
Tj−1 ≤ t ≤ Tj .
j
3 Tj −Tj−1
ΣjN,F is the forward­looking normal caplet volatility bootstrapped from the quoted cap volatil­
ity. ΣjN,B (t) is the backward­looking normal caplet volatility with time­decay, σjN is the backward­
looking normal caplet volatility without time­decay, and the time­decay adjustment/factor is
Gj (t). When it comes to backward­looking cap volatility bootstrapping, the interpolation can be
applied to either caplet volatility with time­decay ΣjN,B (t) or without time­decay σjN .
References
[1] LYASHENKO, A., AND MERCURIO, F. Looking forward to backward­looking rates: A modeling
framework for term rates replacing LIBOR. SSRN (2020).
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Confidential Information. Copyright ©
2022 Numerix LLC. All rights reserved.
Numerix, the Numerix logo, and
CrossAsset are either registered
trademarks, or trademarks, of
Numerix LLC in the United States
and/or other countries.