Multi-curve modelling

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On HJM Model and Its Extension to Single-Currency-Multiple-
Curves Modelling
(Version 1 – 2017 Part I)
Chong Seng Choi
ARTICLE INFO ABSTRACT
Keywords:
Change-of-measure
Convexity adjustment
Credit crisis
Cross-currency derivatives
Discount curve
FRAs
Forward curve
Libor rates
No-arbitrage
OIS rates
Yield curve
Classical interest-rate models are developed to incorporate no-arbitrage conditions,
which allow one to hedge interest rate derivatives in terms of zero-coupon bonds.
As a result, forward rates with different tenors are related to each other by sharp
constraints, and these constraints between rates allowed the construction of a well-
defined forward curve. However, after the credit crunch in August 2007, the Libor
and OIS (risk-free) rates are separated by large basis spreads, which are no longer
considered negligible. As a result, FRA rates can no longer be replicated by the spot
Libor rates, and rates differing only for their payment frequency have started to
show large inconsistencies. The present market situation has thus necessitated the
development of single-currency-multiple-curves framework to take into account the
divergences between rates. This paper aims at reviewing the classical HJM model
and its extension to single-currency-multiple-curves valuation following a foreign-
currency analogy. The valuation of FRA is examined and a convexity adjustment
factor typical to the pricing of cross-currency derivatives is derived as a result.
1. Introduction
Classical interest-rate models are developed to incorporate
no-arbitrage conditions, which allow one to hedge interest
rate derivatives in terms of zero-coupon bonds. As a result,
forward rates with different tenors are related to each other
by sharp constraints, and these constraints between rates
allowed the construction of a well-defined forward curve.
However, after the credit crunch in August 2007, a
number of anomalies have emerged in that the market
quotes of forward rates and zero-coupon bonds started to
violate the no-arbitrage conditions embedded in these
classical models. In the pre-crisis environment, the Libor
and OIS (risk-free) rates were chasing each other closely,
the spreads between swap rates with the same maturity but
different payment lengths were negligible, and the market
quotes of FRA had the precise relationship with the spot
Libor rates that classical models predict. Therefore, the
pre-crisis standard market practice was to construct only
one unique yield curve and then compute on the same
curve the discount and forward rates.
Yet, in the post-crisis environment, the Libor and OIS
rates are separated by large basis spreads, which are no
longer considered negligible. As a result, FRA rates can
no longer be replicated by the spot Libor rates, and rates
differing only for their payment frequency have started to
show large inconsistencies. For example, as noted by
Mercurio (2009), a swap rate with semi-annual payments
1
For the rationale of choosing the HJM model, see
Cuchiero, Fontana and Gnoatto (2014), and Theis (2017).
based on the 6-month LIBOR can be different (and higher)
than the same-maturity swap rate but with quarterly
payments based on the 3-month LIBOR.
These divergences between rates thus suggest that
construction of a unique single yield curve for both
discounting and forwarding is possible only in the absence
of liquidity and counterparty risks. Indeed, the present
market situation has necessitated the development of
single-currency-multiple-curves framework to value
interest rate derivatives in that a unique curve is built for
discounting and separate curves corresponding to different
rate tenors, i.e. 1M, 3M,6M and 12M, are constructed for
forwarding.
This paper aims at reviewing the classical HJM1
model
and its extension to single-currency-multiple-curves
valuation following the foreign-currency analogy put
forward by Bianchetti (2009), Cuchiero, Fontana and
Gnoatto (2014), and Theis (2017). The valuation of FRA
is examined and a convexity adjustment typical to the
pricing of cross-currency derivatives is derived as a result.
Section 2 gives the mathematical notations used. Section
3 presents the derivations of the HJM model and Section
4 derives the dynamics of the forwarding curve under risk-
neutral and forward probability measures. Section 5
briefly discusses the implementation and estimation
procedure. Section 6 concludes.

 On HJM Model and Its Extension to Single-Currency-Multiple-Curves Modelling
Page 2 of 13
2. Mathematical Notation
Assuming a continuous-time trading economy with
trading interval [0, 𝑇∗
] for a fixed horizon 𝑇∗
> 0. The
uncertainty of the economy is defined on a probability
space (Ω, 𝔽, ℙ) with filtration 𝔽, which is assumed to be
the right-continuous and ℙ -completed version of the
filtration generated by the underlying d-dimensional
standard Brownian motion 𝑊.
Further, assuming there exists a continuum of non-
defaultable zero-coupon bonds with maturities 𝑇 ∈
[0, 𝑇∗], and let 𝐵(𝑡, 𝑇) denotes the price of the zero-
coupon bond at time 𝑡 ≤ 𝑇, 𝑡 ∈ [0, 𝑇], maturing at time
𝑇 ≤ 𝑇∗
. The bond price 𝐵(𝑡, 𝑇) is a strictly positive real-
valued adopted process with 𝐵(𝑇, 𝑇) = 1, and its partial
derivative with respect to maturity T, 𝜕 𝑇 ln 𝐵(𝑡, 𝑇), exits
for all 𝑇 ∈ [0, 𝑇∗
] . Lastly, let 𝔼ℙ(∙) denotes the
expectation under the probability measure ℙ.
Let 𝐹(𝑡, 𝑇, 𝑇 + 𝛿) be the forward rates at time 𝑡 < 𝑇 for
risk-free borrowing and lending at time 𝑇 maturing at time
𝑇 + 𝛿. Moreover, let 𝑓(𝑡, 𝑇) stands for the forward rate at
time 𝑡 ≤ 𝑇 for instantaneous risk-free borrowing and
lending at time 𝑇 ≤ 𝑇∗
. By taking the limit on 𝛿
𝑓(𝑡, 𝑇) = lim
𝛿→0
𝐹(𝑡, 𝑇, 𝑇 + 𝛿)
= − lim
𝛿→0
1
𝐵(𝑡, 𝑇 + 𝛿)
𝐵(𝑡, 𝑇 + 𝛿) − 𝐵(𝑡, 𝑇)
𝛿
= −
1
𝐵(𝑡, 𝑇)
𝜕𝐵(𝑡, 𝑇)
𝜕𝑇
= −
𝜕 ln 𝐵(𝑡, 𝑇)
𝜕𝑇
(Brigo and Mercurio, 2006:p.12)
The instantaneous forward rate 𝑓(𝑡, 𝑇) is thus given by
𝑓(𝑡, 𝑇) = − 𝜕 𝑇 ln 𝐵(𝑡, 𝑇) ∀ 𝑡 ∈ [0, 𝑇]
(1)
Integrating both sides and solving the above differential
equation yields the following formula for zero bonds
𝐵(𝑡, 𝑇) = exp (− ∫ 𝑓(𝑡, 𝑠)𝑑𝑠
𝑇
𝑡
) ∀ 𝑡 ∈ [0, 𝑇]
(2)
3. The Classical HJM Model
Previous interest rate models such as those of Vasicek
(1977), Cox, Ingersoll, and Ross (1985), and Hull and
White (1990) typically involved modelling directly the
dynamics of the short rate 𝑟 since all fundamental
quantities such as rates and bonds are readily defined,
which facilitated a convenient specification. Further, the
success of these models were mainly attributed to their
analytical tractability and the availability of explicit
2
The Vasicek and the Hull-White models are two
examples that produce negative rates with positive
probability due to their assumptions of a Gaussian
distribution.
3
The bond prices of most short-rate models such as those
of Vasicek (1977), Dothan (1978), Cox, Ingersoll, and
formulas for bonds and bond options (Brigo and Mercurio,
2006). Despite their advantages, these models generally
come with a major drawback in that the initial term
structure is endogenous rather than exogenous, meaning
that the initial term structure is an output of these models.
These endogenous term-structure models thus require
users to run optimization to find the set of model
parameters that fits as close as possible the initial curve to
the market curve (Brigo and Mercurio, 2006). However,
due to the small number of parameters in these short-rate
models, it is often difficult to calibrate an accurate initial
curve, and in some cases2
produce negative rates 𝑟𝑡 < 0.
In-light of the major disadvantage of most short-rate
models, one is thus in-need for a model that takes the
initial term structure as exogenously given. The first such
alternative was proposed by Ho and Lee (1986), who
modelled the behaviour of the entire yield curve in a
discrete setting instead of modelling the short-rate, which
represented only a single point on this curve (Musiela and
Rutkowski, 1997). Their intuition was then extended by
Heath, Jarrow and Mortion (HJM) (1992) in a continuous
time setting in that the instantaneous forward rates were
chosen to be the fundamental quantities to model. The
advantages of the HJM model lie not only in the
exogenous term-structure but also in the fact that nearly all
exogenous term-structure interest rate model can be
derived within this general framework (Brigo and
Mercurio, 2006). In particular, the HJM model links
together the dynamics of forward rates and zero-coupon
bond prices and assumes that the zero bonds evolve as a
continuous stochastic process3
. Therefore, the HJM model
is also a natural choice for single-currency-multiple-
curves modelling.
3.1 HJM Forward-Rate Dynamics
Heath, Jarrow and Morton (1992) assumes that for every
fixed 𝑇 ≤ 𝑇∗
, the dynamics of the instantaneous forward
rate satisfies the following stochastic process
𝑓(𝑡, 𝑇) = 𝑓(0, 𝑇) + ∫ 𝛼(𝑠, 𝑇)𝑑𝑠 + ∫ 𝜎(𝑠, 𝑇)𝑑𝑊(𝑠)
𝑡
0
𝑡
0
(3)
for all 𝑡 ∈ [0, 𝑇], where 𝑓(0, 𝑇) = 𝑓 𝑀(0, 𝑇): [0, 𝑇∗] is
the market instantaneous forward curve at time 𝑡 = 0, i.e.
initial term structure. 𝑓(0,∙): [0, 𝑇∗
] → ℝ is a Borel-
measurable function and 𝛼 ∶ 𝐶 × Ω → ℝ and 𝜎 ∶ 𝐶 ×
Ω → ℝ 𝑑
are some functions such that 𝐶 = {(𝑠, 𝑡)|0 ≤ 𝑠 ≤
𝑡 ≤ 𝑇∗}.
Moreover, for any maturity T, 𝛼(∙, 𝑇) is an adopted
process, and 𝜎(∙, 𝑇) = (𝜎1(∙, 𝑇), … , 𝜎 𝑑(∙, 𝑇)) is a vector of
adopted processes such that
Ross (1985), and Hull and White (1990) are derived by
solving the time-t conditional expectation 𝐵(𝑡, 𝑇) =
𝔼 𝑡 {𝑒𝑥𝑝 (− ∫ 𝑟(𝑢)𝑑𝑢
𝑇
𝑡
)} and thus are no stochastic (see
Brigo and Mercurio, 2006).

Page 3 of 13
∫ |𝛼(𝑠, 𝑇)|𝑑𝑠 + ∫ ‖𝜎(𝑠, 𝑇)‖2
𝑑𝑠 < ∞,
𝑇
𝑡
ℙ
𝑇
𝑡
− 𝑎. 𝑠
(Musiela and Rutkowski, 1997)
In this setup, the D-dimensional Brownian motion
determines the stochastic dynamics of the entire forward
rate term structure starting from the fixed initial term
structure 𝑓 𝑀(0, 𝑇).
3.2 HJM Bond Price Dynamics
For every fixed 𝑇 ≤ 𝑇∗
, assuming the dynamics of the
bond price 𝐵(𝑡, 𝑇) are determined by the following
expression
𝜕𝐵(𝑡, 𝑇) = 𝐵(𝑡, 𝑇)(𝑎(𝑡, 𝑇)𝑑𝑡 + 𝑏(𝑡, 𝑇)𝑑𝑊(𝑡))
(5)
With the above HJM assumptions and equations (2) and
(3), the discount bond process can be solved for in the
following steps
𝐵(𝑡, 𝑇) = exp (− ∫ 𝑓(𝑡, 𝑠)𝑑𝑠
𝑇
𝑡
)
= exp
(
−
(
∫ 𝑓(0, 𝑢)𝑑𝑢 + ∫ ∫ 𝛼(𝑠, 𝑢)𝑑𝑠𝑑𝑢
𝑡
0
𝑇
𝑡
𝑇
𝑡
+ ∫ ∫ 𝜎(𝑠, 𝑢)𝑑𝑊(𝑠)𝑑𝑢
𝑡
0
𝑇
𝑡 ))
By the Fubini’s standard and stochastic theorems,
∫ ∫ 𝛼(𝑠, 𝑢)𝑑𝑠𝑑𝑢
𝑡
0
𝑇
0
= ∫ ∫ 𝛼(𝑠, 𝑢)𝑑𝑢𝑑𝑠
𝑇
0
𝑡
0
= ∫ ∫ 𝛼(𝑠, 𝑢)𝑑𝑢𝑑𝑠 + ∫ ∫ 𝛼(𝑠, 𝑢)𝑑𝑢𝑑𝑠
𝑡
𝑠
𝑡
0
𝑇
𝑠
𝑡
0
= ∫ ∫ 𝛼(𝑢, 𝑠)𝑑𝑠𝑑𝑢 + ∫ ∫ 𝛼(𝑠, 𝑢)𝑑𝑠𝑑𝑢
𝑢
0
𝑡
0
𝑇
𝑢
𝑡
0
and the same holds for ∬ 𝜎(𝑠, 𝑢)𝑑𝑊𝑠 𝑑𝑢,
∫ ∫ 𝜎(𝑠, 𝑢)𝑑𝑊𝑠 𝑑𝑢
𝑡
0
𝑇
0
= ∫ ∫ 𝜎(𝑠, 𝑢)𝑑𝑢𝑑𝑊(𝑠)
𝑇
0
𝑡
0
= ∫ ∫ 𝜎(𝑠, 𝑢)𝑑𝑢𝑑𝑊𝑠 + ∫ ∫ 𝜎(𝑠, 𝑢)𝑑𝑢𝑑𝑊(𝑠)
𝑡
𝑠
𝑡
0
𝑇
𝑠
𝑡
0
= ∫ ∫ 𝜎(𝑢, 𝑠)𝑑𝑠𝑑𝑊𝑢 + ∫ ∫ 𝜎(𝑠, 𝑢)𝑑𝑢𝑑𝑊(𝑠)
𝑢
0
𝑡
0
𝑇
𝑢
𝑡
0
we thus have 𝐵(𝑡, 𝑇) = exp
(
− ∫ 𝑓(0, 𝑢)𝑑𝑢 − ∫ ∫ 𝛼(𝑢, 𝑠)𝑑𝑠𝑑𝑢
𝑇
𝑢
𝑡
0
𝑇
0
– ∫ ∫ 𝜎(𝑢, 𝑠)𝑑𝑠𝑑𝑊(𝑢)
𝑇
𝑢
𝑡
0
+
∫ (𝑓(0, 𝑢) + ∫ 𝛼(𝑠, 𝑢)𝑑𝑠
𝑢
0
+ ∫ 𝜎(𝑠, 𝑢)𝑑𝑊(𝑠)
𝑢
0
)
⏟
𝑑𝑢
𝑡
0
𝑓(𝑢, 𝑢) )
(6)
Moreover, the short rate 𝑟𝑢 at time 𝑢 can be derived
using the forward rate equation and taking the limit on 𝛿
over the time interval [𝑢, 𝑢 + 𝛿]
𝑟𝑢 = 𝑓(𝑢, 𝑢) = lim
𝛿→0
[𝛿−1
(
1 − 𝐵(𝑢, 𝑢 + 𝛿)
𝐵(𝑢, 𝑢 + 𝛿)
)]
= 𝑓(0, 𝑢) + ∫ 𝛼(𝑠, 𝑢)𝑑𝑠
𝑢
0
+ ∫ 𝜎(𝑠, 𝑢)𝑑𝑊(𝑠)
𝑢
0
(7)
(Heath, Jarrow and Morton, 1992)
Writing equation (6) in the form of an 𝐼𝑡𝑜̃ process, we
have 𝐵(𝑡, 𝑇) =
𝐵(0, 𝑇)
𝑒𝑥𝑝 (∫ 𝑟𝑢 𝑑𝑢 − ∫ ⋀(𝑢, 𝑇)𝑑𝑢 + ∫ Σ(𝑢, 𝑇)𝑑𝑊(𝑢)
𝑡
0
𝑡
0
𝑡
0
)
(8)
where
𝐵(0, 𝑇) = exp (− ∫ 𝑓(0, 𝑢)𝑑𝑢
𝑇
0
)
⋀(𝑢, 𝑇) = ∫ 𝛼(𝑡, 𝑠)𝑑𝑠,
𝑇
𝑡
Σ(𝑢, 𝑇) = ∫ 𝜎(𝑡, 𝑠)𝑑𝑠
𝑇
𝑡
The coefficients in equation (5) can be derived using
equation (8) and the 𝐼𝑡𝑜̃ formula to solve for 𝐵(𝑡, 𝑇) =
ln(𝐵(𝑡, 𝑇)).
In differential form, equation (8) equals
𝜕𝐵(𝑡, 𝑇) = 𝐵(𝑡, 𝑇)[(𝑟𝑡 − ⋀(𝑡, 𝑇)𝑑𝑡 + Σ(𝑡, 𝑇)𝑑𝑊(𝑡)]
and by the 𝐼𝑡𝑜̃ formula,
𝜕𝐵(𝑡, 𝑇)
𝐵(𝑡, 𝑇)
= 𝜕𝐵(𝑡, 𝑇) +
1
2
[𝜕𝐵(𝑡, 𝑇)]2
= (𝑟𝑡 − ⋀(𝑡, 𝑇) +
1
2
|Σ(𝑡, 𝑇)|2
) 𝑑𝑡 − Σ(𝑡, 𝑇)𝑑𝑊(𝑡)
(9)
Thus, we have
𝑎(𝑡, 𝑇) = 𝑟𝑡 − ⋀(𝑡, 𝑇) +
1
2
|Σ(𝑡, 𝑇)|2
and,
𝑏(𝑡, 𝑇) = − Σ(𝑡, 𝑇)
3.3 No-Arbitrage Condition
In the HJM setting, a continuum of bonds with different
maturities, 0 < 𝑇1
< 𝑇2
< ⋯ < 𝑇 𝑘
≤ 𝑇∗
, is available for
trade, therefore, one must ensure that there is no
opportunity for arbitrage by trading bonds with different
maturities.
From The First Fundamental Theorem of Asset Pricing,
the bond price process 𝐵(𝑡, 𝑇), 𝑡 ≤ 𝑇 ≤ 𝑇∗
, is arbitrage

Page 4 of 13
free if and only if there exits a probability measure ℙ̃ on
(Ω, 𝔽 𝑇∗̃) that is equivalent to ℙ such that for any maturity
𝑇 ∈ [0, 𝑇∗], the discounted bond price
𝐵(𝑡,𝑇)
𝐵𝑡
is a
martingale under ℙ̃.
Let us first introduce the saving account 𝐵𝑡, which is
also an adapted process defined on a filtered probability
space (Ω, 𝔽, ℙ). 𝐵𝑡 is the amount of cash accumulated up
to time 𝑡 by rolling over a risk-free bond and starting with
one unit of cash at time 0. Thus, the process for 𝐵𝑡 is given
by
𝐵𝑡 = exp (∫ 𝑓(𝑢, 𝑢)𝑑𝑢
𝑡
0
) ∀ 𝑡 ∈ [0, 𝑇∗
]
(10)
where
𝐵0 = 1, 0 < 𝐵𝑡 < +∞
By the 𝐼𝑡𝑜̃ quotient rule
𝜕(𝐵(𝑡, 𝑇)/𝐵𝑡)
𝐵(𝑡, 𝑇)/𝐵𝑡
=
𝜕𝐵(𝑡, 𝑇)
𝐵(𝑡, 𝑇)
−
𝜕𝐵𝑡
𝐵𝑡
+
𝜕[𝐵𝑡, 𝐵𝑡]
𝐵𝑡
2 −
𝜕[𝐵(𝑡, 𝑇), 𝐵𝑡]
𝐵(𝑡, 𝑇)𝐵𝑡
= (𝑟𝑡 − ⋀(𝑡, 𝑇) +
1
2
|Σ(𝑡, 𝑇)|2
) 𝑑𝑡 − Σ(𝑡, 𝑇)𝑑𝑊(𝑡) − 𝑟𝑡 𝑑𝑡
= (−⋀(𝑡, 𝑇) +
1
2
|Σ(𝑡, 𝑇)|2
(11)
where
𝜕𝐵𝑡 = 𝐵𝑡 𝑟𝑡 𝑑𝑡
(12)
From the above conditions, there must exist a
probability measure measure ℙ̃ on (Ω, 𝔽 𝑇∗̃ ) equivalent to
ℙ in order to ensure that there is no opportunity for
arbitrage. In a financial interpretation, the underlying
probability measure ℙ represents actual probability,
which is a subjective assessment of the future evolution of
the discounted bond price and therefore is not a martingale
under ℙ. On the other hand, the probability measure ℙ̃
reflects risk-neutral probability, which renders the
discounted bond price a martingale under ℙ̃ with zero drift.
Defining 𝑍(𝑡, 𝑇) =
𝐵(𝑡,𝑇)
𝐵𝑡
and rearranging the terms in
(11), we have
𝜕𝑍(𝑡, 𝑇)
𝑍(𝑡, 𝑇)
= − Σ(𝑡, 𝑇)[𝜃𝑡 𝑑𝑡 + 𝑑𝑊(𝑡)]
(13)
where
𝜃𝑡 =
−⋀(𝑡, 𝑇) +
1
2
|Σ(𝑡, 𝑇)|2
− Σ(𝑡, 𝑇)
(14)
is the market price of risk under the probability measure
ℙ. We can then proceed to define a new probability
measure ℙ̃ equivalent to ℙ on (Ω, 𝔽 𝑇∗̃ ) using the 𝑅𝑎𝑑𝑜𝑛
-𝑁𝑖𝑘𝑜𝑑𝑦́ 𝑚 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒
𝜕ℙ̃
𝜕ℙ
= ℇ 𝑇∗ (∫ 𝜃𝑡
∙
0
𝑑𝑊𝑡) ℙ − 𝑎. 𝑠.
for some predictable ℝ 𝑑
-valued process adopted to the
filtration 𝔽 𝑊
. The notation ℇ(∙) is the
𝐷𝑜𝑙𝑒́ 𝑎 𝑛𝑠 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙, which gives
ℇ 𝑇∗ (∫ 𝜃𝑡
∙
0
𝑑𝑊𝑡) = 𝑒𝑥𝑝 (∫ 𝜃 𝑢
𝑡
0
𝑑𝑊(𝑢) −
1
2
∫ |𝜃 𝑢|2
𝑡
0
𝑑𝑊(𝑢))
In view of Girsanov’s Theorem, the process
𝑊̃ (𝑡) = 𝑊(𝑡) + ∫ 𝜃 𝑢
𝑡
0
𝑑𝑢 ∀ 𝑡 ∈ [0, 𝑇∗
]
(15)
follows a d-dimensional Brownian motion under ℙ̃. Thus
we can change the probability measure of 𝑍(𝑡, 𝑇) from ℙ
to ℙ̃
𝜕𝑍(𝑡, 𝑇)
𝑍(𝑡, 𝑇)
= − Σ(𝑡, 𝑇)[𝜃𝑡 𝑑𝑡 + (𝑑𝑊̃ (𝑡) − 𝜃𝑡 𝑑𝑡)]
= − Σ(𝑡, 𝑇) 𝑑𝑊̃ (𝑡)
(16)
which is a martingale under ℙ̃.
Since equation (11) and (13) are equivalent, we can then
derive 𝛼(𝑡, 𝑇) in the forward process (3), under the
measure ℙ, by solving the following equation
(−⋀(𝑡, 𝑇) +
1
2
|Σ(𝑡, 𝑇)|2
= − Σ(𝑡, 𝑇)[𝜃𝑡 𝑑𝑡 + 𝑑𝑊(𝑡)]
−⋀(𝑡, 𝑇) +
1
2
|Σ(𝑡, 𝑇)|2
= − Σ(𝑡, 𝑇)𝜃𝑡
(17)
and by differentiating the equation with respect to T
𝜕 𝑇 [−⋀(𝑡, 𝑇) +
1
2
|Σ(𝑡, 𝑇)|2
] = 𝜕 𝑇[− Σ(𝑡, 𝑇)𝜃𝑡]
−𝛼(𝑡, 𝑇) + Σ(𝑡, 𝑇)𝜎(𝑡, 𝑇) = 𝜎(𝑡, 𝑇)𝜃𝑡
𝛼(𝑡, 𝑇) = 𝜎(𝑡, 𝑇)[Σ(𝑡, 𝑇) + 𝜃𝑡]
(18)
Thus, under the actual measure ℙ, we have
𝜕𝑓(𝑡, 𝑇)
𝑓(𝑡, 𝑇)
= (𝜎(𝑡, 𝑇)Σ(𝑡, 𝑇) + 𝜎(𝑡, 𝑇)𝜃𝑡)𝑑𝑡 + 𝜎(𝑡, 𝑇)𝑑𝑊(𝑡)
(19)
𝜕𝐵(𝑡, 𝑇)
𝐵(𝑡, 𝑇)
= (𝑟𝑡 − Σ(𝑡, 𝑇)𝜃𝑡)𝑑𝑡 − Σ(𝑡, 𝑇)𝑑𝑊(𝑡)
(20)

Page 5 of 13
𝜕𝑍(𝑡, 𝑇)
𝑍(𝑡, 𝑇)
= − Σ(𝑡, 𝑇)[𝜃𝑡 𝑑𝑡 + 𝑑𝑊(𝑡)]
(21)
If a term-structure model satisfies the HJM no-arbitrage
conditions then under the risk-neutral probability measure
ℙ̃ and by Girsanov’s Theorem in (15), we have
𝜕𝑓(𝑡, 𝑇)
𝑓(𝑡, 𝑇)
= 𝜎(𝑡, 𝑇)Σ(𝑡, 𝑇)𝑑𝑡 + 𝜎(𝑡, 𝑇)𝑑𝑊̃ (𝑡)
(22)
and the zero-coupon bond prices
𝜕𝐵(𝑡, 𝑇)
𝐵(𝑡, 𝑇)
= 𝑟𝑡 𝑑𝑡 − Σ(𝑡, 𝑇)𝑑𝑊̃ (𝑡)
(23)
and the discounted bond prices
𝜕𝑍(𝑡, 𝑇)
𝑍(𝑡, 𝑇)
= − Σ(𝑡, 𝑇) 𝑑𝑊̃ (𝑡)
(24)
Equations (19) - (24) are intuitive in that, under the risk-
neutral expectation, the bond price satisfies
𝐵(𝑡, 𝑇) = 𝐵𝑡 𝔼 𝑡
ℙ̃
(𝐵 𝑇
−1
|ℱ𝑡)
= 𝔼 𝑡
ℙ̃
(𝑒𝑥𝑝 (− ∫ 𝑓(𝑢, 𝑢)𝑑𝑢
𝑇
𝑡
)| ℱ𝑡)
Consequently, given any non-negative forward rate
process 𝑓 defined on a probability space (Ω, 𝔽, ℙ) and a
probability measure ℙ̃ on (Ω, 𝔽 𝑇∗̃) equivalent to ℙ, the
market price of risk 𝜃 must vanish under the risk-neutral
measure ℙ̃.
The HJM no-arbitrage conditions are best summarized
as follow
I. For any collection of maturities 0 < 𝑇1
< 𝑇2
<
⋯ < 𝑇 𝑘
≤ 𝑇∗
, there exit market prices of risk
𝜃 𝑖(∙ ; 𝒯) such that
−⋀𝑖(∙, 𝒯) +
1
2
|Σ 𝑖(∙, 𝒯)|
2
+ Σ 𝑖(∙, 𝒯)𝜃 𝑖(∙ ; 𝒯) = 0
which satisfies
∫ |𝜃 𝑖(𝑢 ; 𝒯)|
2
𝑇 𝑖
0
𝑑𝑢 < +∞ 𝑓𝑜𝑟 𝑖 = 1, … , 𝑘
and
𝔼ℙ
(ℇ 𝑇∗ (∫ 𝜃 𝑖(𝑢 ; 𝒯)
∙
0
𝑑𝑊 𝑖
(𝑢))) = 1
and
4
In a FRA contact, a fixed payment based on the strike K,
set at time 𝑡, is exchanged against a floating payment at
𝔼ℙ
(ℇ 𝑇∗ (∫ (Σ 𝑖(𝑢, 𝒯) + 𝜃 𝑖(𝑢 ; 𝒯))
∙
0
𝑑𝑊 𝑖
(𝑢))) = 1
for 𝒯 ∈ [𝑇1
, … , 𝑇 𝑘
] and 𝑖 = 1, … , 𝑘
II. There exits a probability measure ℙ̃ equivalent to ℙ
such that for any collection of maturities 0 < 𝑇1
<
𝑇2
< ⋯ < 𝑇 𝑘
≤ 𝑇∗
the discounted bond prices
𝑍(∙, 𝒯) are martingales under ℙ̃ with respect to
𝑓(∙, 𝒯).
III. The martingale measures in (II) are unique across
all bonds of all maturities 0 < 𝑇1
< 𝑇2
< ⋯ <
𝑇 𝑘
≤ 𝑇∗
where
 ℙ̃ is defined as ℙ̃ 𝑇1,…,𝑇 𝑘 for any maturity 0 <
𝑇1
< ⋯ < 𝑇 𝑘
≤ 𝑇∗
and ℙ̃ 𝑇 𝑖 is the unique
equivalent probability measure such that
𝑍(𝑡, 𝑇) is a martingale for all 𝑇 ∈ [0, 𝑇∗
] and
𝑡 ∈ [0, 𝑇 𝑖
]
 𝛼(𝑡, 𝑇) = ∑ 𝜎 𝑖(𝑡, 𝑇) [∫ 𝜎 𝑖(𝑡, 𝑢)𝑑𝑢
𝑇
𝑡
+ 𝜃 𝑖(∙𝑘
𝑖=1
; 𝒯)] for all 𝑇 ∈ [0, 𝑇∗
], 𝑡 ∈ [0, 𝑇] and for
any 𝒯 ∈ (0, 𝑇∗
],
Arbitrage-free bond pricing and term structure
movements (Heath, Jarrow and Merton, 1992:p. 83-87)
4. Post Liquidity Crisis
The direct result of the no-arbitrage conditions embedded
in HJM model and in other interest rate models is that they
link together the forward rates and zero-coupon bonds, and
thus allow one to price and hedge interest rate derivatives
on any single currency. In fixed income markets, the
underlying quantities of majority of interest-rate sensitive
products are the Libor rates 𝐿(𝑡, 𝑇, 𝑇 + 𝛿), where 𝛿 > 0
corresponds to the underlying rate tenors 1D, 1M, 3M, 6M,
and 12M. In the classical setting, the Libor rates are
associated with the forward rates 𝐹 under a forward
probability measure, namely
𝔼 𝑡
ℙ̃ 𝑇+𝛿
(𝐿(𝑇, 𝑇, 𝑇 + 𝛿)|ℱ𝑡) ≔ 𝐹(𝑡, 𝑇, 𝑇 + 𝛿)
(25)
and therefore the Libor rates can be replicated by buying
and selling zero bonds with maturities corresponding those
of the Libor rates, namely
𝐹(𝑡, 𝑇, 𝑇 + 𝛿) ≔ 𝛿−1
(
𝐵(𝑡, 𝑇)
𝐵(𝑡, 𝑇 + 𝛿)
− 1)
(26)
where 𝛿 is the year fraction between times 𝑇 and 𝑇 + 𝛿. In
this sense, the discounted bonds 𝐵(𝑡, 𝑇) and the forward
rates 𝐹(𝑡, 𝑇, 𝑇 + 𝛿) are computed using the spot Libor
quotes prevailing in the market at time 𝑡, 𝐿 𝑀
(𝑡, 𝑇). Taking
as example the pricing of FRA 4
(Forward Rate
maturity, 𝑇 + 𝛿, based on the spot Libor rate at time at
maturity 𝑇 for period [𝑇, 𝑇 + 𝛿].

Page 6 of 13
Agreement), the valuation of such contract, assuming a $1
notional, is given by
FRA (𝑡, 𝑇, 𝑇 + 𝛿, 𝐾) = 𝐵(𝑡, 𝑇 + 𝛿)𝛿(𝐾 − 𝐿(𝑡, 𝑇, 𝑇 + 𝛿))
(Brigo and Mercurio, 2006)
where 𝐾 is the strike that renders the contract fair at
initiation time 𝑡. Thus we have the time 𝑡 price of a FRA
equals to the strike 𝐾, giving
𝐾 = 𝔼 𝑡
ℙ̃ 𝑇+𝛿
(𝐿(𝑇, 𝑇, 𝑇 + 𝛿)|ℱ𝑡) = 𝛿−1
(
𝐵(𝑡, 𝑇)
𝐵(𝑡, 𝑇 + 𝛿)
− 1)
(27)
In this setting, Libor is not only the rate to which the
FRA is referenced but also the rate used to discount future
cash-flows. It is thus assumed that the risk in the interbank
lending market is negligible, and Libor rates can be
considered risk-free. As stressed by Bianchetti (2009), this
setting is known as single-currency-single-curve
approach in that a unique yield curve is constructed and
used to price and hedge interest rate derivatives on a given
currency.
Before the crisis in 2007, the OIS5
, commonly used as
the risk-free rate, and Libor rates were closely tracking
each other, and the market quotes of FRA rates had a
precise relationship with the spot Libor rates that the FRAs
are indexed to, i.e. equation (27). Yet, after the financial
crisis in 2007-2008 and the Eurozone sovereign debt crisis
in 2009-2012, the classical assumptions do not hold
anymore, with the basis spreads between the OIS and
Libor rates now larger than those in the past and are no
longer considered negligible. The main reasons behind
this phenomenon can be attributed to counterparty risk, the
risk of the counterparty defaulting on the contract
obligation, and liquidity risk, the risk of excessive funding
cost to finance the contract due to the lack of liquidity in
the market. Unfortunately, the increase in the basis spreads
does not create any arbitrage opportunities when
counterparty and liquidity risks are taken into account
(Mercurio, 2009). Therefore, the FRA rates can no longer
be replicated by the spot Libor rates, and the floating legs
differing only for the tenor 𝛿 are now represented by large
basis spreads Morini (2009). Indeed, with the counterparty
and liquidity risk in mind, Libor rates are not longer
considered risk-free and the OIS rates are now a more
appropriate candidate for discounting future cash-flows.
The market has thus evolved to a single-currency-
multiple-curves approach to take into account the different
dynamics represented by Libor rates with different tenors
and the discount rates that should be represented by a
unique separate discounting curve. From Ametrano and
Bianchetti (2013) Section 2.2, the corresponding approach
can be summarized as follows:
 Build a unique discounting curve using the OIS rates
5
According to Morini (2009), ‘an OIS, Overnight Index
Swap, is a fixed/floating interest rate swap with the
floating leg tied to a published index of a daily overnight
 Build multiple forwarding curves for the Libor rates
associated with difference tenors 𝛿
 Use the market quotes of multiple separated sets of
vanilla interest rate instruments with increasing
maturity to bootstrap the forwarding curves
 Compute on the forwarding curves the forward rates
 Compute on the discounting curve the discount rates
Post-crisis procedure for constructing multiple yield
curves (Ametrano and Bianchetti, 2013:p.7)
Returning to the FRA pricing equation, we know that
equation (27) does not hold anymore
𝔼 𝑡
ℙ̃ 𝑇+𝛿
(𝐿(𝑇, 𝑇, 𝑇 + 𝛿)|ℱ𝑡) ≠ 𝛿−1
(
𝐵(𝑡, 𝑇)
𝐵(𝑡, 𝑇 + 𝛿)
− 1)
because now 𝐿(𝑡, 𝑇, 𝑇 + 𝛿) > 𝐹(𝑡, 𝑇, 𝑇 + 𝛿). Adopting
the multi-curves approach and the framework noted above,
we have the floating leg of the FRA given by
𝑭𝑹𝑨 𝒇𝒍𝒐𝒂𝒕𝒊𝒏𝒈 (𝑡, 𝑇, 𝑇 + 𝛿)
= 𝛿𝐵 𝑑(𝑡, 𝑇 + 𝛿) [𝛿−1
(
𝐵 𝑓(𝑡, 𝑇)
𝐵 𝑓(𝑡, 𝑇 + 𝛿)
− 1)]
= 𝛿𝐵 𝑑(𝑡, 𝑇 + 𝛿)[𝐹𝑓(𝑡, 𝑇, 𝑇 + 𝛿)]
(28)
where 𝐵 𝑑(𝑡, ∙ ) is the discount factors computed from the
discounting curve and 𝐵 𝑓(𝑡, ∙ ) is the zero bond prices
computed from the forwarding curves. Using the
martingale property of forward rates, we therefore need to
compute equation (28) with the following expectation
𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
(𝐿(𝑇, 𝑇, 𝑇 + 𝛿)|ℱ𝑡) = 𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
(𝐹𝑓(𝑇, 𝑇, 𝑇 + 𝛿))
(29)
where 𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
is the expectation at time 𝑡 under the
discounting - 𝑇 + 𝛿 - forward neutral probability measure
ℙ̃ 𝑑
𝑇+𝛿
.
4.1 Multi-Curves Modelling
The single currency multi-curves modelling framework
was first proposed by Henrard (2007) who describes a
market in which each forward rate seems to act as a
separate underlying asset. Subsequently, other authors
have proposed several ways to go beyond the deterministic
basis spread assumption. Martinez (2009) derives the drift
conditions on a HJM model in the presence of a stochastic
basis but focuses only on risk-neutral measure. Mercurio
and Xie (2012) and Cuchiero, Fontana and Gnoatto (2014)
model the Libor-OIS spread directly as a stochastic
process.
While Cuchiero, Fontana and Gnoatto (2014) generalize
the HJM model, Mercurio (2010) extends the Libor
reference rate. Since an overnight rate refers to lending for
an extremely short period of time, it is assumed to
incorporate negligible credit or liquidity risk.’ (p.10)

Page 7 of 13
Market Model to incorporate an additive stochastic basis.
Moreover, OpenGamma (2012b) extends the Gaussian
HJM model to a multi-curves framework with
deterministic spread. Moreni and Pallavicini (2010) use a
HJM-LMM hybrid model in which the discounting curve
is described by a HJM dynamic while the forwarding
curves are modelled through a Libor Market approach. In
particular, Bianchetti (2009) and Theis (2017) recognizes
an analogy between FX pricing and the pricing of interest
rate derivatives when discounting is decoupled by
indexing (Morini, 2009), leading to a convexity adjustment
in the pricing equation.
In light of the simplicity of the foreign-currency
analogy examined by Bianchetti (2009) and Theis (2017),
we shall follow their approach and apply it to the HJM
model. The resulting single-currency-double-curves
framework can be easily extended to a multi-curves setting.
4.2 Foreign-Currency Analogy
Let us first consider the rationale behind this modelling
choice. As in Bianchetti (2009), and Cuchiero, Fontana
and Gnoatto (2014), one can associate the risky Libor rate
with artificial risky bonds 𝐵 𝑓(𝑡, 𝑇) issued by a bank
representative of the Libor panel. In this case, the pricing
procedure is consistent with the classical risk-free setting,
namely, 1+ 𝛿𝐿(𝑇, 𝑇, 𝑇 + 𝛿) =
1
𝐵 𝑓(𝑇,𝑇,𝑇+𝛿)
. Moreover, if we
interpret the risk-free bonds as domestic assets while the
artificial bonds as foreign assets expressed in unit of
foreign currency, we can introduce a spot “exchange” rate
𝑆𝑡, which allows us to link together the risk-free and the
artificial risky bonds. Applying this foreign-currency
analogy, we can proceed to derive a forward “exchange”
rate 𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) over the period [𝑡, 𝑇] for settlement
date 𝑇 + 𝛿 under the standard no-arbitrage argument in
the foreign exchange markets. Naturally, the quantity
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) can be interpreted as a multiplicative
forward interest rate spread between the risk-free and
risky bonds over the period [𝑇, 𝑇 + 𝛿]. As we shall see,
different interpretations of the quantity 𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)
would give different results for the single-currency-
double-curves framework. Nonetheless, in-light of the
above discussion, the foreign-currency analogy remains a
natural modelling choice.
Conversely, as in Theis (2017), we can consider 𝑆𝑡
6
the
price at time 𝑡 of a unit of currency borrowed on a rolling
basis at the risk-free rate and invested in the risky rate
starting from time 0. In this sense, the foreign-currency
analogy allows one to absorb the credit risk through the
exchange rate 𝑆𝑡 and therefore treating the artificial risky
bond process as continuous and continuously evolving
consistent with standard interest rate modelling practice.
Nevertheless, let us consider the framework of
Bianchetti (2009) and assume that there exit two different
interest markets 𝑀 𝑥 with 𝑥 = {𝑑, 𝑓} denoting
respectively the domestic (discounting) and foreign
(forwarding) market. Moreover, there are also two distinct
6
In Theis (2017), the quantity 𝑆𝑡 models the cumulative
credit risk of an investment in the risky bonds, and it is
saving accounts 𝐵𝑡
𝑥
, thus in the HJM setting we have the
following dynamics under the 𝑥 - risk-neutral measure
𝜕𝑓 𝑥(𝑡, 𝑇)
𝑓 𝑥(𝑡, 𝑇)
= 𝜎 𝑥(𝑡, 𝑇)Σ 𝑥(𝑡, 𝑇)𝑑𝑡 + 𝜎 𝑥(𝑡, 𝑇)𝑑𝑊̃𝑥(𝑡)
(30)
𝜕𝐵 𝑥(𝑡, 𝑇)
𝐵 𝑥(𝑡, 𝑇)
= 𝑟𝑡
𝑥
𝑑𝑡 − Σ 𝑥(𝑡, 𝑇)𝑑𝑊̃𝑥(𝑡)
(31)
𝜕𝐵𝑡
𝑥
𝐵𝑡
𝑥 = 𝑟𝑡
𝑥
𝑑𝑡
(32)
Moreover, a tradable asset in a foreign currency 𝑓 is
tradable in the domestic market through the exchange rate
𝑆. Thus, if 𝑋 𝑓
is the price process of the foreign asset,
𝑋𝑡
𝑓
𝑆𝑡 will be the price process of the foreign asset in
domestic currency. From Theis (2017), there must exist a
pre-visible process 𝜎𝑡
𝑠
such that the exchange rate process
can be postulated as follow
𝜕𝑆𝑡
𝑆𝑡
= (𝜇 𝑡 𝑑𝑡 + 𝜎𝑡
𝑠
𝑑𝑊̃ 𝑆(𝑡))
(33)
where the process 𝑆𝑡 is defined on a filtered probability
space (Ω, 𝔽, ℙ) with the filtration 𝔽 assumed to be the ℙ-
augmentation of the filtration generated by a d-
dimensional Brownian motion and ∫ |𝜎 𝑢
𝑠|2
𝑑𝑢 <
𝑡
0
∞.
Putting aside the foreign-currency analogy, the process
𝑆 can be viewed as a spot interest rate spread expressed
as a ratio between the short rates 𝑓 𝑑
(𝑡, 𝑡)/𝑓 𝑓
(𝑡, 𝑡). In
order to rule out arbitrage between investments in the
domestic (discounting) and foreign (forwarding) assets,
we shall refer to the no-arbitrage conditions in (I) and (II),
in particular
 There exits a probability measure measure ℙ̃
equivalent to ℙ on (Ω, 𝔽 𝑇∗̃ ) such that the the foreign
money account 𝐵𝑡
𝑓
when expressed in domestic
currency (𝐵𝑡
𝑓
𝑆𝑡) and discounted by the domestic
money account 𝐵𝑡
𝑑
is a martingale under the domestic
(discounting) risk neutral measure ℙ̃ 𝑑.
 There exits a probability measure measure ℙ̃
equivalent to ℙ on (Ω, 𝔽 𝑇∗̃ ) such that the the foreign
zero coupon bond 𝐵 𝑓(𝑡, 𝑇) when expressed in
domestic currency (𝐵 𝑓(𝑡, 𝑇)𝑆𝑡) and discounted by
the domestic money account 𝐵𝑡
𝑑
is a martingale under
the domestic (discounting) risk neutral measure ℙ̃ 𝑑.
Using the 𝐼𝑡𝑜̃ product and quotient rules, we have
𝜕(𝐵𝑡
𝑓
𝑆𝑡)/𝐵𝑡
𝑑
(𝐵𝑡
𝑓
𝑆𝑡)/𝐵𝑡
𝑑
=
𝜕𝐵𝑡
𝑓
𝐵𝑡
𝑓
+
𝜕𝑆𝑡
𝑆𝑡
−
𝜕𝐵𝑡
𝑑
𝐵𝑡
𝑑
+ 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑡𝑒𝑟𝑚𝑠⏟
=0
treated as an non-increasing function with 𝑆𝑡 > 0 for any
𝑡 and 𝑆0 = 1.

Page 8 of 13
= 𝑟𝑡
𝑓
𝑑𝑡 + 𝜇 𝑡 𝑑𝑡 + 𝜎𝑡
𝑠
𝑑 − 𝑟𝑡
𝑑
𝑑𝑡
= 𝜎𝑡
𝑠
(𝜃𝑡 𝑑𝑡 + 𝑑𝑊̃ 𝑆(𝑡))
where
𝜃𝑡 =
𝑟𝑡
𝑓
− 𝑟𝑡
𝑑
+ 𝜇 𝑡
𝜎𝑡
𝑠
is the market price of risk satisfying condition (I). By the
Girsanov’s Theorem in (15), we have
𝑊̃ 𝑆
𝑑
(𝑡) = 𝑊̃ 𝑆(𝑡) − ∫ 𝜃 𝑢
𝑡
0
𝑑𝑢 ∀ 𝑡 ∈ [0, 𝑇∗
]
and
𝜕(𝐵𝑡
𝑓
𝑆𝑡)/𝐵𝑡
𝑑
(𝐵𝑡
𝑓
𝑆𝑡)/𝐵𝑡
𝑑
= 𝜎𝑡
𝑠
𝑑𝑊̃ 𝑆
𝑑
(𝑡)
(34)
Multiplying equation (34) by 𝐵𝑡
𝑑
/𝐵𝑡
𝑓
and by the 𝐼𝑡𝑜̃
rules, we have the dynamics of the spot “exchange” rate
under the domestic (discounting) risk neutral measure ℙ̃ 𝑑
𝜕𝑆𝑡
𝑆𝑡
= (𝑟𝑡
𝑑
− 𝑟𝑡
𝑓
)𝑑𝑡 + 𝜎𝑡
𝑠
𝑑𝑊̃ 𝑆
𝑑
(𝑡)
= [𝑓 𝑑
(𝑡, 𝑡) − 𝑓 𝑓
(𝑡, 𝑡)]𝑑𝑡 + 𝜎𝑡
𝑠
𝑑𝑊̃ 𝑆
𝑑
(𝑡)
(35)
Similarly, for the foreign bond process, we have
𝜕(𝐵 𝑓(𝑡, 𝑇)𝑆𝑡)/𝐵𝑡
𝑑
(𝐵 𝑓(𝑡, 𝑇)𝑆𝑡)/𝐵𝑡
𝑑
=
𝜕𝐵 𝑓(𝑡, 𝑇)
+
𝜕𝑆𝑡
𝑆𝑡
+
𝜕[𝐵 𝑓(𝑡, 𝑇), 𝑆𝑡]
𝐵 𝑓(𝑡, 𝑇)𝑆𝑡
−
𝜕𝐵𝑡
𝑑
𝐵𝑡
𝑑 + 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑡𝑒𝑟𝑚𝑠⏟
=0
= 𝑟𝑡
𝑓
𝑑𝑡 − Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃ 𝑓(𝑡) + (𝑟𝑡
𝑑
− 𝑟𝑡
𝑓
𝑠
𝑑𝑊̃𝑠
𝑑
(𝑡)
− Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃ 𝑓(𝑡)𝜎𝑡
𝑠
𝑑𝑊̃𝑠
𝑑
(𝑡) − 𝑟𝑡
𝑑
𝑑𝑡
= −Σ 𝑓(𝑡, 𝑇) 𝜎𝑡
𝑠
𝜌𝑡
𝐵 𝑓,𝑠
𝑑𝑡 − Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃𝑓 (𝑡) + 𝜎𝑡
𝑠
𝑑𝑊̃𝑠
𝑑(𝑡)
(36)
where 〈𝑊̃ 𝑓, 𝑊̃ 𝑆
𝑑
〉 = 𝜌𝑡
𝐵 𝑓,𝑠
is the instantaneous correlation
between relative changes in 𝐵 𝑓(∙, 𝑇) and 𝑆. Moreover,
using the fact that the sum of two martingales under ℙ̃ 𝑑 is
a martingale under ℙ̃ 𝑑 , we can derive the Brownian
motion of the foreign bond process in the domestic
measure as
𝑊̃𝑓
𝑑
(𝑡) = 𝑊̃𝑓(𝑡) + ∫ 𝜎 𝑢
𝑠
𝜌 𝑢
𝐵 𝑓,𝑠
𝑡
0
𝑑𝑢 ∀ 𝑡 ∈ [0, 𝑇∗
]
7
From German et al. (1995), Musiela and Rutkowski
(2005), and Shreve (2004), the forward price of an asset is
Thus,
= [𝑟𝑡
𝑓
+ Σ 𝑓(𝑡, 𝑇)𝜎𝑡
𝑠
𝜌𝑡
𝐵 𝑓,𝑠
] 𝑑𝑡 − Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃ 𝑓
𝑑
(𝑡)
(37)
and the dynamics of the foreign (forwarding) forward
curve in domestic (discounting) measure can be derived by
solving equation (37) for ln 𝐵(𝑡, 𝑇) and by equation (1),
namely
𝑓 𝑓(𝑡, 𝑇) = − 𝜕 𝑇 ln 𝐵 𝑓(𝑡, 𝑇)
where
𝜕 ln 𝐵 𝑓(𝑡, 𝑇) = (𝑟𝑡
𝑓
+ Σ 𝑓(𝑡, 𝑇)𝜎𝑡
𝑠
𝜌𝑡
𝐵 𝑓,𝑠
−
1
2
|Σ 𝑓(𝑡, 𝑇)|
2
) 𝑑𝑡
− Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃ 𝑓
𝑑
(𝑡)
and
ln 𝐵 𝑓(𝑡, 𝑇) = ln 𝐵 𝑓(0, 𝑇) +
∫ (
(𝑟𝑢
𝑓
+ Σ 𝑓(𝑢, 𝑇)𝜎 𝑢
𝑠
𝜌 𝑢
𝐵 𝑓,𝑠
−
1
2
|Σ 𝑓(𝑢, 𝑇)|
2
) 𝑑𝑢
− Σ 𝑓(𝑢, 𝑇)𝑑𝑊̃𝑓
𝑑
(𝑢)
)
𝑡
0
and
𝑓 𝑓(𝑡, 𝑇) = 𝑓 𝑓(0, 𝑇) − ∫ (𝜎 𝑓(𝑢, 𝑇) ∫ 𝜎 𝑓(𝑢, 𝑠)𝑑𝑠 +
𝑇
𝑡
𝑡
0
𝜎 𝑓(𝑢, 𝑇)𝜎 𝑢
𝑠
𝜌 𝑢
𝐵 𝑓,𝑠
) 𝑑𝑢 + ∫ 𝜎 𝑓(𝑢, 𝑇)𝑑𝑊̃𝑓
𝑑
(𝑢)
𝑡
0
(38)
which is the instantaneous forward rate for the forwarding
curve under the discounting risk-neutral probability
measure ℙ̃ 𝑑. From Martinez (2009) and Grbac and
Runggaldier (2015), the HJM drift condition in equation
(38) under ℙ̃ 𝑑 can be derived directly by
𝜕 𝑇 [−⋀ 𝑓(𝑡, 𝑇) +
1
2
|Σ 𝑓(𝑡, 𝑇)|
2
] = 𝜕 𝑇[〈 𝜎𝑡
𝑠
, Σ 𝑓(𝑡, 𝑇)〉]
(39)
4.3 Forward Measure
As with many derivatives such as FX options, FRAs are
written on the forward rates instead of spot rates. Hence,
in order to price a forward contract under the no-arbitrage
framework, we need to apply the forward probability
measure7
. Recall that we are interested in calculating the
expectation in (29), namely
𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
(𝐿(𝑇, 𝑇 + 𝛿)|ℱ𝑡) = 𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
(𝐹𝑓(𝑇, 𝑇, 𝑇 + 𝛿)|ℱ𝑡)
which allows one to price the FRA in a single-currency-
double-curves framework at time 𝑡 for settlement date
a martingale under the forward probability measure
associated with the settlement date of the forward contract.

Page 9 of 13
𝑇 + 𝛿 using the artificial risky bonds 𝐵 𝑓(𝑡, ∙ ) and the
spot “exchange” rate 𝑆𝑡.
From Musiela and Rutkowski (2005), the forward
process of the zero bonds for the time interval [𝑇, 𝑇 + 𝛿]
is given by
𝐹𝐵(𝑡, 𝑇, 𝑇 + 𝛿) ≝
𝐵(𝑡, 𝑇)
𝐵(𝑡, 𝑇 + 𝛿)
∀ 𝑡 ∈ [0, 𝑇]
(40)
and the 𝑥 - 𝑇 + 𝛿 - forward prices are martingales under
the 𝑥 - 𝑇 + 𝛿 - forward measure ℙ̃ 𝑥
𝑇+𝛿
, 𝑥 = {𝑑, 𝑓}. By
𝐼𝑡𝑜̃ quotient rule, we have
𝐹𝐵
𝑥
(𝑡, 𝑇, 𝑇 + 𝛿) =
𝜕(𝐵 𝑥(𝑡, 𝑇)/𝐵 𝑥(𝑡, 𝑇 + 𝛿))
𝐵 𝑥(𝑡, 𝑇)/𝐵 𝑥(𝑡, 𝑇 + 𝛿)
=
𝜕𝐵 𝑥(𝑡, 𝑇)
𝐵 𝑥(𝑡, 𝑇)
−
𝜕𝐵 𝑥(𝑡, 𝑇 + 𝛿)
𝐵 𝑥(𝑡, 𝑇 + 𝛿)
+
𝜕[𝐵 𝑥(𝑡, 𝑇 + 𝛿), 𝐵 𝑥(𝑡, 𝑇 + 𝛿)]
(𝐵 𝑥(𝑡, 𝑇 + 𝛿))
2
−
𝜕[𝐵 𝑥(𝑡, 𝑇), 𝐵 𝑥(𝑡, 𝑇 + 𝛿)]
𝐵 𝑥(𝑡, 𝑇)𝐵 𝑥(𝑡, 𝑇 + 𝛿)
= (Σ 𝑥(𝑡, 𝑇 + 𝛿) − Σ 𝑥(𝑡, 𝑇))𝑑𝑊̃𝑥
𝑥(𝑡)
+(|Σ 𝑥(𝑡, 𝑇 + 𝛿)|2
− Σ 𝑥(𝑡, 𝑇)Σ 𝑥(𝑡, 𝑇 + 𝛿))𝑑𝑡
(41)
which we assumed, for simplicity, that the forward
volatilities (Σ 𝑥(𝑡, 𝑇 + 𝛿) − Σ 𝑥(𝑡, 𝑇)) are bounded.
The drift term in equation (41) is not equal zero because
𝑊̃𝑥
𝑥
(𝑡) is a Brownian motion under the 𝑥 - risk-neutral
measure instead of a Brownian motion under x - 𝑇 + 𝛿
forward measure ℙ̃ 𝑥
𝑇+𝛿
. Therefore, the 𝑅𝑎𝑑𝑜𝑛 -
𝑁𝑖𝑘𝑜𝑑𝑦́ 𝑚 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 gives
𝜕ℙ̃ 𝑥
𝑇+𝛿
𝜕ℙ̃ 𝑥
=
exp (− ∫ 𝑟𝑢
𝑥
𝑑𝑢
𝑇+𝛿
0
)
𝐵 𝑥(0, 𝑇 + 𝛿)
=
1
𝐵 𝑥(0, 𝑇 + 𝛿)𝐵 𝑇+𝛿
𝑥
= ℇ 𝑇+𝛿 (− ∫ Σ 𝑥(𝑢, 𝑇 + 𝛿)
∙
0
𝑑𝑊̃𝑥
𝑥
(𝑢)) ℙ̃ 𝑥 − 𝑎. 𝑠.
(42)
(Musiela and Rutkowski, 1997; Shreve, 2004)
which implies that moving to the 𝑥 - 𝑇 + 𝛿 - forward
measure on (Ω, 𝔽 𝑇+𝛿) amounts to changing the numeraire
from the saving account 𝐵 𝑇+𝛿
𝑥
to the zero-coupon bond
𝐵 𝑥(0, 𝑇 + 𝛿). Moreover, by Girsanov’s Theorem
𝑊̃𝑥
𝑥,𝑇+𝛿
(𝑡) = 𝑊̃𝑥
𝑥(𝑡) + ∫ Σ 𝑥(𝑢, 𝑇 + 𝛿)𝑑𝑢
𝑡
0
(43)
which gives the Brownian motion under the 𝑥 - forward
martingale measure for the settlement date 𝑇 + 𝛿
associated with the 𝑥 - 𝑇 + 𝛿 - maturity zero-coupon bond
as a numeraire.
From Musiela and Rutkowski (1997), the above
derivations satisfy the no-arbitrage condition if the
forward bond processes 𝐹𝐵(𝑡, 𝑇, 𝑇 + 𝛿) > 1, otherwise
the processes are said to satisfy the weak no-arbitrage
condition. Nevertheless, the 𝑥 zero bonds under the 𝑥 -
𝑇 + 𝛿 - forward martingale measure can be summarized as
follows
= (𝑟𝑡
𝑓
+ Σ 𝑓(𝑡, 𝑇)Σ 𝑓(𝑡, 𝑇 + 𝛿)) 𝑑𝑡
− Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃𝑓
𝑓,𝑇+𝛿
(𝑡)
(44)
𝜕𝐵 𝑑(𝑡, 𝑇 + 𝛿)
𝐵 𝑑(𝑡, 𝑇 + 𝛿)
= (𝑟𝑡
𝑑
+ |Σ 𝑑(𝑡, 𝑇 + 𝛿)|2)𝑑𝑡
− Σ 𝑑(𝑡, 𝑇 + 𝛿)𝑑𝑊̃𝑑
𝑑,𝑇+𝛿
(𝑡)
(45)
Before devolving into the dynamics of the foreign
(forwarding) curve, let us first consider a foreign exchange
forward contact written at time 𝑡 for settlement at time 𝑇.
By the interest rate parity, the interest rate differential
between the domestic and foreign economies must equal
to the difference between the forward exchange rate and
spot exchange rate. Therefore, we have the forward
exchange rate given by
𝐹𝑆(𝑡, 𝑇) = e(𝑟 𝑑−𝑟 𝑓)(𝑇−𝑡)
𝑆𝑡
=
𝐵 𝑑(𝑡, 𝑇)
𝑆𝑡 ∀ 𝑡 ∈ [0, 𝑇]
(46)
Forward FX-rate (Musiela and Rutkowski, 2005:p.155)
Applying this analogy to our case, we are interested in
deriving a forward process that is consistent with the
expectation in (29), namely a forward “exchange” rate
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) at time 𝑡 for the time interval [𝑡, 𝑇] and for
settlement date 𝑇 + 𝛿. Accordingly, one is naturally led to
look at the ratio
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) =
𝐵 𝑑(𝑡, 𝑇 + 𝛿)
𝑆𝑡 ∀ 𝑡 ∈ [0, 𝑇]
(47)
This derivation follows directly from the 𝑅𝑎𝑑𝑜𝑛 -
𝑁𝑖𝑘𝑜𝑑𝑦́ 𝑚 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 in (42) that under the domestic
(discounting) - 𝑇 + 𝛿 - forward measure, the forward
“exchange” rate associated to the numeraire 𝐵 𝑑(𝑡, 𝑇 + 𝛿)
is a martingale under ℙ̃ 𝑑
𝑇+𝛿
, assuming that 𝐵 𝑓(𝑡, 𝑇)𝑆𝑡 is a
domestic tradable asset. Yet, before deriving the dynamics
of equation (47), let us first change the measure of the spot
“exchange” rate 𝑆𝑡 under ℙ̃ 𝑑 to ℙ̃ 𝑑
𝑇+𝛿
𝜕(𝑆𝑡 𝐵𝑡
𝑓
)/𝐵 𝑑(𝑡, 𝑇 + 𝛿)
(𝑆𝑡 𝐵𝑡
𝑓
)/𝐵 𝑑(𝑡, 𝑇 + 𝛿)
=
𝜕𝑆𝑡
𝑆𝑡
+
𝜕𝐵𝑡
𝑓
𝐵𝑡
𝑓
−
𝜕𝐵 𝑑(𝑡, 𝑇 + 𝛿)
𝐵 𝑑(𝑡, 𝑇 + 𝛿)
+
[𝜕𝐵 𝑑(𝑡, 𝑇 + 𝛿)]2
(𝐵 𝑑(𝑡, 𝑇 + 𝛿))
2 −
𝜕𝑆𝑡 𝐵𝑡
𝑓
𝑆𝑡 𝐵𝑡
𝑓
𝜕𝐵 𝑑(𝑡, 𝑇 + 𝛿)
𝐵 𝑑(𝑡, 𝑇 + 𝛿)
= (𝑟𝑡
𝑑
− 𝑟𝑡
𝑓
𝑠
𝑑𝑊̃ 𝑆
𝑑
(𝑡) + 𝑟𝑡
𝑓
𝑑𝑡
− (𝑟𝑡
𝑑
+ |Σ 𝑑(𝑡, 𝑇 + 𝛿)|2)𝑑𝑡
+ Σ 𝑑(𝑡, 𝑇 + 𝛿)𝑑𝑊̃ 𝑑
𝑑,𝑇+𝛿
(𝑡)
+ |Σ 𝑑(𝑡, 𝑇 + 𝛿)|2
𝑑𝑡
+ 𝜌𝑡
𝐵 𝑑,𝑆
𝜎𝑡
𝑠
Σ 𝑑(𝑡, 𝑇 + 𝛿)𝑑𝑡

Page 10 of 13
= 𝜎𝑡
𝑠
(𝑑𝑊̃ 𝑆
𝑑
(𝑡) + {𝜌𝑡
𝐵 𝑑,𝑆
Σ 𝑑(𝑡, 𝑇 + 𝛿)} 𝑑𝑡)
𝑑,𝑇+𝛿
(𝑡)
we therefore have
𝑊̃𝑆
𝑑,𝑇+𝛿
(𝑡) = 𝑊̃ 𝑆
𝑑
(𝑡) + ∫ 𝜌 𝑢
𝐵 𝑑,𝑆
Σ 𝑑(𝑢, 𝑇 + 𝛿)
𝑡
0
𝑑𝑢
and the spot “exchange” rate 𝑆𝑡 under the domestic
(discounting) - 𝑇 + 𝛿 - forward measure
𝜕𝑆𝑡
𝑆𝑡
= (𝑟𝑡
𝑑
− 𝑟𝑡
𝑓
− 𝜌𝑡
𝐵 𝑑,𝑆
𝜎𝑡
𝑠
Σ 𝑑(𝑡, 𝑇 + 𝛿))𝑑𝑡 + 𝜎𝑡
𝑠
𝑑𝑊̃𝑠
𝑑,𝑇+𝛿
(𝑡)
(48)
Following from above, we can derive the forward
process for 𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) as
𝜕(𝑆𝑡 𝐵 𝑓(𝑡, 𝑇))/𝐵 𝑑(𝑡, 𝑇 + 𝛿)
(𝑆𝑡 𝐵 𝑓(𝑡, 𝑇))/𝐵 𝑑(𝑡, 𝑇 + 𝛿)
= (𝑟𝑡
𝑑
− 𝑟𝑡
𝑓
− 𝜌𝑡
𝐵 𝑑,𝑆
𝜎𝑡
𝑠
Σ 𝑑(𝑡, 𝑇 + 𝛿)) 𝑑𝑡
+𝜎𝑡
𝑠
𝑑𝑊̃𝑠
𝑑,𝑇+𝛿
(𝑡)
+ (𝑟𝑡
𝑓
+ Σ 𝑓(𝑡, 𝑇)Σ 𝑓(𝑡, 𝑇 + 𝛿)) 𝑑𝑡
𝑓,𝑇+𝛿
(𝑡)
− (𝑟𝑡
𝑑
+ |Σ 𝑑(𝑡, 𝑇 + 𝛿)|2)𝑑𝑡
+ Σ 𝑑(𝑡, 𝑇 + 𝛿)𝑑𝑊̃𝑑
𝑑,𝑇+𝛿
(𝑡)
+ |Σ 𝑑(𝑡, 𝑇 + 𝛿)|2
𝑑𝑡
− Σ 𝑓(𝑡, 𝑇)𝜎𝑡
𝑠
𝜌𝑡
𝐵 𝑓,𝑆
𝑑𝑡
+ Σ 𝑓(𝑡, 𝑇)Σ 𝑑(𝑡, 𝑇 + 𝛿)𝜌𝑡
𝐵 𝑑,𝐵 𝑓
𝑑𝑡
+ 𝜌𝑡
𝐵 𝑑,𝑆
𝜎𝑡
𝑠
Σ 𝑑(𝑡, 𝑇 + 𝛿)
= −Σ 𝑓(𝑡, 𝑇)( 𝑑𝑊̃𝑓
𝑓,𝑇+𝛿
(𝑡)
− {(Σ 𝑓(𝑡, 𝑇 + 𝛿) + Σ 𝑑(𝑡, 𝑇 + 𝛿)𝜌𝑡
𝐵 𝑑,𝐵 𝑓
− 𝜎𝑡
𝑠
𝜌𝑡
𝐵 𝑓,𝑆
) 𝑑𝑡}
𝑑,𝑇+𝛿
(𝑡)
+ 𝜎𝑡
𝑠
𝑑𝑊̃𝑠
𝑑,𝑇+𝛿
(𝑡)
which implies
𝜕ℙ̃ 𝑓
𝑑,𝑇+𝛿
𝜕ℙ̃
𝑓
𝑓,𝑇+𝛿
=
ℇ 𝑇+𝛿 (− ∫ {
Σ 𝑓(𝑢, 𝑇 + 𝛿)
+Σ 𝑑(𝑢, 𝑇 + 𝛿)𝜌 𝑢
𝐵 𝑑,𝐵 𝑓
−𝜎 𝑢
𝑠
𝜌 𝑢
𝐵 𝑓,𝑆
𝑑𝑊̃𝑓
𝑓,𝑇+𝛿
(𝑢)
}
∙
0
) ℙ̃ 𝑓
𝑓,𝑇+𝛿
− 𝑎. 𝑠.
(49)
and
𝑊̃𝑓
𝑑,𝑇+𝛿
(𝑡) = 𝑊̃𝑓
𝑓,𝑇+𝛿
(𝑡)
− ∫ (Σ 𝑓(𝑢, 𝑇 + 𝛿) + Σ 𝑑(𝑢, 𝑇 + 𝛿)𝜌 𝑢
𝐵 𝑑,𝐵 𝑓
− 𝜎 𝑢
𝑠
𝜌 𝑢
𝐵 𝑓,𝑆
)
𝑡
0
𝑑𝑢
(50)
where 𝑊̃𝑓
𝑑,𝑇+𝛿
(𝑡) is the Brownian motion of the foreign
(forwarding) bond under the domestic (discounting) - 𝑇 +
𝛿 - forward measure.
Hence, the foreign (reference) bond process under ℙ̃ 𝑑
𝑇+𝛿
is
= (𝑟𝑡
𝑓
− Σ 𝑓(𝑡, 𝑇) {Σ 𝑑(𝑡, 𝑇 + 𝛿)𝜌𝑡
𝐵 𝑑,𝐵 𝑓
− 𝜎𝑡
𝑠
𝜌𝑡
𝐵 𝑓,𝑆
}) 𝑑𝑡 − Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃𝑓
𝑑,𝑇+𝛿
(𝑡)
(51)
Solving equation (51) for ln 𝐵 𝑓(𝑡, 𝑇) and
differentiating with respect to 𝑇, 𝜕 𝑇 ln 𝐵 𝑓(𝑡, 𝑇), would
give the dynamics of the instantaneous forward rate
process for the forwarding curve 𝑓 𝑓( ∙ , 𝑇) under ℙ̃ 𝑑
𝑇+𝛿
.
However, such derivation is not necessary as we could
price the FRA in term of a quanto correction – a procedure
typical to the pricing of cross-currency derivatives.
4.3 Pricing Under Single-Currency-Double Curves
Having derived the 𝑅𝑎𝑑𝑜𝑛 - 𝑁𝑖𝑘𝑜𝑑𝑦́ 𝑚 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 in
equation (49), we can proceed to calculate the expectation
in (29). In a single-currency-double-curves framework,
this amounts to transforming a cash-flow on the
forwarding curve to the corresponding discounting curve
(Bianchetti, 2009). Thus, we have the Libor rate given by
the following
𝔼 𝑡
ℙ̃ 𝑑
𝑇+𝛿
(𝐿(𝑇, 𝑇 + 𝛿)) = 𝔼 𝑡
ℙ̃ 𝑑
𝑇+𝛿
(𝐹𝑓(𝑇, 𝑇, 𝑇 + 𝛿))
=
1
𝛿
(𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
[
𝐵 𝑓(𝑡, 𝑇 + 𝛿)
] − 1)
=
1
𝛿
(𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
[𝐹𝐵
𝑓
(𝑇, 𝑇, 𝑇 + 𝛿)] − 1)
(52)
From (41) and (50), under the domestic (discounting)
forward measure, we have the foreign (forwarding)
forward bond price given by
𝜕𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿)
= (Σ 𝑓(𝑡, 𝑇 + 𝛿) − Σ 𝑓(𝑡, 𝑇)) 𝑑𝑊̃𝑓
𝑓,𝑇+𝛿
(𝑡)
= (Σ 𝑓(𝑡, 𝑇 + 𝛿) − Σ 𝑓(𝑡, 𝑇))
{𝑑𝑊̃𝑓
𝑑,𝑇+𝛿
(𝑡) + (
Σ 𝑓(𝑡, 𝑇 + 𝛿) +
Σ 𝑑(𝑡, 𝑇 + 𝛿)𝜌𝑡
𝐵 𝑑,𝐵 𝑓
− 𝜎𝑡
𝑠
𝜌𝑡
𝐵 𝑓,𝑆
) 𝑑𝑡}
(53)
and
𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
(𝐹𝐵
𝑓
(𝑇, 𝑇, 𝑇 + 𝛿)) = 𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿) 𝑒𝑥𝑝
(
∫
(
(Σ 𝑓(𝑢, 𝑇 + 𝛿) − Σ 𝑓(𝑢, 𝑇))
{(
Σ 𝑓(𝑢, 𝑇 + 𝛿) + Σ 𝑑(𝑢, 𝑇 + 𝛿)𝜌 𝑢
𝐵 𝑑,𝐵 𝑓
−𝜎 𝑢
𝑠
𝜌 𝑢
𝐵 𝑓,𝑆
)}
)
𝑇
𝑡
𝑑𝑢
)⏟
𝑄𝑢𝑎𝑛𝑡𝑜 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛
(54)

Page 11 of 13
Hence,
𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
(𝐿(𝑇, 𝑇 + 𝛿)) = (1 + 𝛿 𝐹𝑓
𝑀
(𝑡, 𝑇, 𝑇 + 𝛿)) 𝑒𝑥𝑝
(
∫
(
(Σ 𝑓(𝑢, 𝑇 + 𝛿) − Σ 𝑓(𝑢, 𝑇))
{(
Σ 𝑓(𝑢, 𝑇 + 𝛿) + Σ 𝑑(𝑢, 𝑇 + 𝛿)𝜌 𝑢
𝐵 𝑑,𝐵 𝑓
−𝜎 𝑢
𝑠
𝜌 𝑢
𝐵 𝑓,𝑆
)}
)
𝑇
𝑡
𝑑𝑢
)
− 1
(55)
where 𝐹𝑓
𝑀
(𝑡, 𝑇, 𝑇 + 𝛿) denotes the Libor rate for period
{𝑇, 𝑇 + 𝛿}, observable at time t from market data.
As shown in the above derivations, the expectation of
𝐹𝐵
𝑓
(𝑇, 𝑇, 𝑇 + 𝛿) has a quanto correction, which depends
on the implied forward bond price volatilities and two
correlation terms. This means that the FRA can be priced
by discounting the cash-flow with the risk-free discount
rate and adding a convexity adjustment factor – a
procedure typical to the pricing of cross-currency products.
Furthermore, one could simplify the derivations and
obtain results similar to Bianchetti (2009) and Theis (2017)
by modelling the quantity 𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) directly, and
changing the measure of 𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿) to ℙ̃ 𝑑
𝑇+𝛿
.
Following from our foreign currency analogy and from
Cuchiero, Fontana and Gnoatto (2014), let us consider the
forward “exchange” rate for the time intervals [𝑡, 𝑇] and
[𝑡, 𝑇 + 𝛿]
𝐹𝑆(𝑡, 𝑇) =
and
𝐹𝑆(𝑡, 𝑇 + 𝛿) =
𝐵 𝑓(𝑡, 𝑇 + 𝛿)𝑆𝑡
𝐵 𝑑(𝑡, 𝑇 + 𝛿)
The forward interest rate spread8
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) can
be derived by
𝐹𝑆(𝑡, 𝑇 + 𝛿)
𝐹𝑆(𝑡, 𝑇)
=
𝐵 𝑓(𝑡, 𝑇 + 𝛿)𝑆𝑡
𝐵 𝑑(𝑡, 𝑇 + 𝛿)
From equation (46), we thus have
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿) = 𝐹𝐵
𝑑
(𝑡, 𝑇, 𝑇 + 𝛿)
(56)
Notice, however, that different dynamics can be
specified according to different interpretation of the
quantity 𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿). While Bianchetti (2009) and
Theis (2017) use an FX analogy, Henrard (2010),
Mercurio (2010), and Mercurio and Xie (2012) model it as
a stochastic basis spread. Here we shall continue with our
foreign exchange analogy and model 𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) as
forward “exchange” rate consistent with equation (47),
hence,
8
See Cuchiero, Fontana and Gnoatto (2014) Appendix B
for proof and derivation.
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) = 𝜎𝑡
𝑠
𝑑𝑊̃𝑠
𝑑,𝑇+𝛿
(𝑡) − Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃𝑓
𝑑,𝑇+𝛿
(𝑡)
𝑑,𝑇+𝛿
(𝑡)
(57)
which is a martingale under ℙ̃ 𝑑
𝑇+𝛿
. Combining the forward
“exchange” rate process into a single Brownian motion,
we have
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿) = 𝜎𝑡
𝐹𝑠
𝑑𝑊̃𝐹𝑠
𝑑,𝑇+𝛿
(𝑡)
(58)
where
𝜎𝑡
𝐹𝑠
𝑑𝑊̃𝐹𝑠
𝑑,𝑇+𝛿
(𝑡) = 𝜎𝑡
𝑠
𝑑𝑊̃𝑠
𝑑,𝑇+𝛿
(𝑡) − Σ 𝑓(𝑡, 𝑇)𝑑𝑊̃𝑓
𝑑,𝑇+𝛿
(𝑡)
𝑑,𝑇+𝛿
(𝑡)
(59)
From the no-arbitrage conditions, we know that the
process {𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿)} in (56) is a
martingale under ℙ̃ 𝑑
𝑇+𝛿
. Hence, by 𝐼𝑡𝑜̃ product rule
𝜕𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿)
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿)
=
𝜕𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)
+
𝜕𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿)
𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿)
+
𝜕𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)
𝐹𝑆(𝑡, 𝑇, 𝑇 + 𝛿)
𝜕𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿)
𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿)
= 𝜎𝑡
𝐹𝑠
𝑑𝑊̃𝐹𝑠
𝑑,𝑇+𝛿
(𝑡) + (Σ 𝑓(𝑡, 𝑇 + 𝛿) − Σ 𝑓(𝑡, 𝑇)) 𝑑𝑊̃𝑓
𝑓,𝑇+𝛿
(𝑡)
+ (Σ 𝑓(𝑡, 𝑇 + 𝛿) − Σ 𝑓(𝑡, 𝑇)) 𝜎𝑡
𝐹𝑠
𝜌𝑡
𝐵 𝑓,𝐹𝑠
𝑑𝑡
which gives
𝑊̃𝑓
𝑑,𝑇+𝛿
(𝑡) = 𝑊̃𝑓
𝑓,𝑇+𝛿
(𝑡) + ∫ 𝜎 𝑢
𝐹𝑠
𝜌 𝑢
𝐵 𝑓,𝐹𝑠
𝑡
0
𝑑𝑢
(60)
Furthermore, from equation (41) and (60), we can
derive
𝜕𝐹𝐵
𝑓
(𝑡, 𝑇, 𝑇 + 𝛿) = (Σ 𝑓(𝑡, 𝑇 + 𝛿) − Σ 𝑓(𝑡, 𝑇)) 𝑑𝑊̃𝑓
𝑓,𝑇+𝛿
(𝑡)
= (Σ 𝑓(𝑡, 𝑇 + 𝛿) − Σ 𝑓(𝑡, 𝑇)) {𝑑𝑊̃𝑓
𝑑,𝑇+𝛿
(𝑡)
− 𝜎𝑡
𝐹𝑠
𝜌𝑡
𝐵 𝑓,𝐹𝑠
𝑑𝑡}
and thus
𝔼 𝑡
ℙ̃
𝑑
𝑇+𝛿
(𝐿(𝑇, 𝑇 + 𝛿)) = (1 + 𝛿 𝐹𝑓
𝑀
(𝑡, 𝑇, 𝑇 + 𝛿)) 𝑒𝑥𝑝
(
− ∫ ((Σ 𝑓(𝑢, 𝑇 + 𝛿) − Σ 𝑓(𝑢, 𝑇)) 𝜎 𝑢
𝐹𝑠
𝜌 𝑢
𝐵 𝑓,𝐹𝑠
)
𝑇
𝑡
𝑑𝑢
⏟
𝑄𝑢𝑎𝑛𝑡𝑜 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 )
− 1
(61)

Page 12 of 13
In this derivation, the quanto correction depends on the
implied foreign forward bond price volatility, the implied
forward “exchange” rate volatility and the correlation
between them. Moreover, since in equations (57)-(59) we
have combined the forward “exchange” rate process into
a single Brownian motion, the forward volatility 𝜎𝑡
𝐹𝑠
can
be computed by
𝑉𝑎𝑟𝑡
ℙ̃
𝑑
𝑇+𝛿
(𝐹𝑆(𝑇, 𝑇, 𝑇 + 𝛿)) = ∫ (𝜎𝑡
𝐹𝑠
𝑑𝑊̃𝐹𝑠
𝑑,𝑇+𝛿
(𝑡))
2𝑇
𝑡
(𝜎𝑡
𝐹𝑠
𝑑𝑊̃𝐹𝑠
𝑑,𝑇+𝛿
(𝑡))
2
= (
𝜎𝑡
𝑠
𝑑𝑊̃𝑠
𝑑,𝑇+𝛿
(𝑡)
𝑑,𝑇+𝛿
(𝑡)
𝑑,𝑇+𝛿
(𝑡)
)
2
which gives
𝜎𝑡
𝐹𝑠
=
√{
(𝜎𝑡
𝑠)2 − 2𝜎𝑡
𝑠
Σ 𝑓(𝑡, 𝑇)𝜌𝑡
𝐵 𝑓,𝑆
+2𝜎𝑡
𝑠
Σ 𝑑(𝑡, 𝑇 + 𝛿)𝜌𝑡
𝐵 𝑑,𝑆
−2Σ 𝑓(𝑡, 𝑇)Σ 𝑑(𝑡, 𝑇 + 𝛿)𝜌𝑡
𝐵 𝑑,𝐵 𝑓
+|Σ 𝑓(𝑡, 𝑇)|2 + |Σ 𝑑(𝑡, 𝑇 + 𝛿)|2}
(62)
5. Implementation and Estimation
From the above equation, we can see that the implied
volatility of the forward “exchange” rate can be
calculated by integrating over the volatility functions. In
Bianchetti (2009) and Hunter (2017), the spot volatility 𝜎𝑡
𝑠
is taken to be zero as the spot exchange rate9
in a single-
currency-double-curves framework collapses to 1.
In the HJM model, the volatilities Σ 𝑓
and Σ 𝑑
are
stochastic and it would not be possible to estimate the
quanto correction in (54) and (61). Thus, one would need
to assume a Gaussian structure, in which Σ 𝑓
and Σ 𝑑
are
deterministic (Theis, 2017). Nevertheless, the parameters
of the qunato correction in equations (54) and (61) can be
extracted from market data. In particular, the forward
Libor rate volatilities can be derived from quoted cap/floor
options corresponding to the maturity date T and tenor 𝛿.
For 𝜎𝑡
𝐹𝑠
and 𝜌𝑡
𝐵 𝑓,𝐹𝑠
and for other tenors where there are no
options available, one shall resort to historical data
(Bianchetti, 2009). Moreover, using equation (62), one
could also bootstrap out a term structure for 𝜎𝑡
𝐹𝑠
with Σ 𝑓
and Σ 𝑑
extracted from market data. Conversely, since the
forward “exchange” rate in (46) is the exponential of
∫ 𝑓 𝑑(𝑡, 𝑢)𝑑𝑢 − ∫ 𝑓 𝑓(𝑡, 𝑢)𝑑𝑢
∙
𝑡
∙
𝑡
, its volatility and
correlation with the forwarding rates can be implied from
the behaviour of the spread (Theis, 2017).
Having introduced the OIS rates in Section 4, let us
consider the calculation of the OIS rates. An overnight
indexed swap (OIS) is a contract in which two
counterparties exchange a fixed rate 𝐾 with a floating
9
While the spot rate 𝑆𝑡 in Bianchetti (2009) is unspecified,
the spot rate in Morini (2009) using the same analogy is
modelled as survival probability.
compounded overnight rate (i.e. Fed Funds Effective Rate
for US Dollars and EONIA for Euros). From Brigo and
Mercurio (2006) and Cuchiero, Fontana and Gnoatto
(2014), the pricing formula with $1 notional is given by
OIS(𝑡, 𝑇, 𝑇 𝑛
, 𝐾) =
𝐵 𝑂𝐼𝑆(𝑡, 𝑇) − 𝐵 𝑂𝐼𝑆(𝑡, 𝑇 𝑛) − 𝐾𝛿 ∑ 𝐵 𝑂𝐼𝑆(𝑡, 𝑇 𝑖)
𝑛
𝑖=1
where 𝛿 in here is the payment frequency, typically 1-year
for the USD and EUR markets. 𝑇1
, … , 𝑇 𝑛
are the payment
dates and 𝑇 𝑖+1
− 𝑇 𝑖
= 𝛿 for every 𝑖 = 1, … , 𝑛 − 1 .
Moreover, the floating rate received at time 𝑇 𝑖+1
is set at
time 𝑇 𝑖
. Thus at every time 𝑇 𝑖
, one party pays 𝛿𝐾 while
the other party pays 𝛿𝑅 𝑂𝐼𝑆
(𝑇 𝑖
, 𝑇 𝑖+1
) , where
𝑅 𝑂𝐼𝑆
(𝑇 𝑖
, 𝑇 𝑖+1
) is the compounded OIS rate for the period
[𝑇 𝑖
, 𝑇 𝑖+1
] given by
𝑅 𝑂𝐼𝑆(𝑇 𝑖
, 𝑇 𝑖+1) =
𝛿−1
(∏ (1 + (𝑡 𝑗+1
− 𝑡 𝑗)𝑅 𝑂𝐼𝑆(𝑡 𝑗
, 𝑡 𝑗+1)) − 1
𝑘
𝑗=0
)
(63)
and 𝑇 𝑖
= 𝑡0
< 𝑡1
< ⋯ < 𝑡 𝑘
= 𝑇 𝑖+1
denotes the partition
of the period [𝑇 𝑖
, 𝑇 𝑖+1
] into 𝑘 business days and
𝑅 𝑂𝐼𝑆(𝑡 𝑗
, 𝑡 𝑗+1) denotes the respective OIS rates over the
interval [𝑡 𝑗
, 𝑡 𝑗+1
].
OIS compounded rate (Filipovic and Trolle, 2013:p.12)
Thus, the strike 𝐾 or the OIS rate that makes the swap
fair at inception date 𝑡 with maturity date 𝑇 and 𝑛
payment dates is given by
𝑂𝐼𝑆(𝑡, 𝑇, 𝑛) = 𝐾 =
𝐵 𝑂𝐼𝑆
(𝑡, 𝑇) − 𝐵 𝑂𝐼𝑆
(𝑡, 𝑇 𝑛
)
𝛿 ∑ 𝐵 𝑂𝐼𝑆(𝑡, 𝑇 𝑖)𝑛
𝑖=1
(64)
Taking as example one leg of the swap, i.e., one
payment. For 𝑛 = 1 and 𝑇 = 𝑇 𝑗
, equation (64) gives
𝑂𝐼𝑆(𝑡, 𝑇 𝑗
, 𝑛) = 𝐾 =
𝐵 𝑂𝐼𝑆
(𝑡, 𝑇 𝑗
) − 𝐵 𝑂𝐼𝑆
(𝑡, 𝑇 𝑗+1
)
𝛿𝐵 𝑂𝐼𝑆(𝑡, 𝑇 𝑗+1)
(Musiela and Rutkowski, 1997)
Thus, the term structure for the discount rates can be
bootstrapped from market quotes of OIS.
6. Conclusion
We have reviewed the derivations of the classical
single-currency-single-curve HJM model, and how the
credit crisis and the resulting inconsistencies between rates
have forced the market to evolve to a valuation framework
that incorporates multiple yield curves, namely, one for
discounting and one for each Libor tenor. Following a
foreign-currency analogy, we have also derived the

Page 13 of 13
dynamics of the discounting and forwarding curves. In
particular, we have examined the pricing of FRA under a
single-currency-double-curves valuation framework and a
convexity adjustment factor, typical to the pricing of
cross-currency derivatives, is derived as a result.
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