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312 11 Libor Market Model with Stochastic Volatilities
=
D
∏
l=1
E
S
exp
−
p
2
T
0
(v
s
l
)
2
(t)dt + p
T
0
v
s
l
(t)dZ
l
(t)
=
D
∏
l=1
f
∗
l
(p, T ) (11.88)
with
f
∗
l
(p, T )=E
exp
−
p
2
T
0
(v
s
l
)
2
(t)dt + p
T
0
v
s
l
(t)dZ
l
(t)
.
The moment-generating function f
X
(p, T ) is then approximated by a product of the
moment-generating function f
∗
l
(
φ
, p) of each factor v
s
l
(t)dZ
l
(t). The concrete func-
tions of f
∗
l
(
φ
, p) are given in the Sch
¨
obel-Zhu model (see Section 3.3). If we specify
the stochastic variance V
j
(t) to be a mean-reverting square root process as in the He-
ston model, the function f
∗
l
(
φ
, p) is determined correspondingly (see Section 3.2).
Applying the moment-generating function, we define two CFs with two different p,
f
s
1
(
φ
)=exp
i
φ
lnS
n,m
(0)
f
X
(−i +
φ
,T
n
), (11.89)
f
s
2
(
φ
)=exp
i
φ
lnS
n,m
(0)
f
X
(i
φ
,T
n
). (11.90)
The pricing formula for a payer swaption via CFs is given by
SW P(0)=A
n,m
(0)[S
n,m
(0) ·F
1
−K ·F
2
] (11.91)
with
F
k
=
1
2
+
1
π
∞
0
Re
f
s
k
(
φ
))
exp(i
φ
lnK)
i
φ
d
φ
, k = 1,2.
The receiver swaption takes the similar form by applying the put-call parity,
SW R(0)=A
n,m
(0)[K ·(1−F
1
) −S
n,m
(0) ·(1 −F
2
)]. (11.92)
The numerical implementation is identical to other stochastic volatility models ex-
cept that we have to integrate the CFs of each factor into a single CF. This pricing
formula for swaption is numerically very efficient and admits a quickly pricing and
calibration. For a practical application, five factors are usually sufficient to cover
most dynamics of a yield curve reflected in swaptions.
11.4.5 The Belomestny, Matthew and Schoenmakers Model
Belomestny, Matthew and Schoenmakers(BMS, 2007) proposed an extended LMM
with a high-dimensional specially structured system of mean-reverting square vari-
ance processes. Additionally, the model of BMS involves a deterministic volatility
term structure q
i
(t) as in the ABR model and the Piterbarg model, and a propor-
11.4 Incorporating Stochastic Volatility 313
tion factor
α
that links the dynamics of standard LMM and the added dynamics
of stochastic volatilities. In particular, the BMS model
5
has the following structure
under some measure,
dL
j
(t)
L
j
(t)
=(···)dt +
1 −
α
2
[q
j
(t) ·dW(t)] +
α
[
β
j
(t) ·dU(t)],
dU
k
=
V
k
(t)dZ
k
(t), 1 k D, (11.93)
dV
k
(t)=
κ
k
(1−V
k
(t))dt +
σ
k
V
k
(t)(
ρ
k
Z
k
(t)+
1 −
ρ
2
k
dZ
∗
k
(t),
(11.94)
where Z and Z
∗
are two independent D-dimensional uncorrelated standard Brown-
ian motions, and both also independent of W that is a M-dimensional uncorrelated
standard Brownian motions. Both q
j
(t) ∈ R
N
and
β
j
∈ R
D
are deterministic vector
functions. The parameter
α
describes the leverage effect between Brownian mo-
tions W and U, and may also be interpreted as a perturbation to standard LMM. If
α
reduces to zero, this model degenerates to a standard LMM. In a more compact
manner, the above system can be reformulated in a vector-valued stochastic process
of dimension M driven by (N + 2D) standard Brownian motions W , that takes the
form
dL
j
(t)
L
j
(t)
=(···)dt +
Γ
j
·dW (11.95)
with
Γ
j
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
√
1 −
α
2
q
j1
√
1 −
α
2
q
j2
·
·
√
1 −
α
2
q
jN
αβ
j1
√
V
1
·
·
αβ
jD
√
V
D
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, W =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
dW
1
dW
2
·
·
dW
N
dZ
1
·
·
dZ
D
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
which shows the high dimensionality of the BMS model.
To value caplets and floorlets as well as swaptions, we need, however, the dy-
namics of Libors and volatilities under the respective forward measure, that can be
obtained via the standard change of measures. The system of whole dynamics given
in (11.93) under the terminal measure Q
M+1
reads
5
We consider here a slightly simplified BMS model with which Belomestny, Matthew and Schoen-
makers actually worked.
314 11 Libor Market Model with Stochastic Volatilities
dL
j
(t)
L
j
(t)
=
1 −
α
2
[q
j
(t) ·dW(t)] +
α
D
∑
k=1
β
jk
V
k
(t)dZ
k
(t),
dV
k
(t)=
κ
∗
k
(
θ
∗
k
−V
k
(t))dt +
σ
k
V
k
(t)
ρ
k
Z
k
(t)+
1 −
ρ
2
k
dZ
∗
k
(t)
(11.96)
with
κ
∗
k
=
κ
k
−
ασ
k
ρ
k
M
∑
i= j+1
τ
i
L
i
(0)
1 +
τ
i
L
i
(0)
β
ik
and
θ
∗
k
=
κ
k
/
κ
∗
k
.
In order to value swaptions, Belomestny, Matthew and Schoenmakers applied the
same approach as used in other extended LMMs and obtain the following process
for swap rate
dS
n,m
(t)
S
n,m
(t)
=
m−1
∑
i=n
ω
i
(0)
Γ
i
·dW
S
(11.97)
with
ω
i
(0)=
∂
S
n,m
(0)L
i
(0)
∂
L
i
(0)S
n,m
(0)
.
The process for variance V
i
(t) under swap measure is much more complicated. By
freezing the values of Libors L(t) at the values at the origin, we may obtain
dV
S
k
(t)=
κ
S
k
(
θ
S
k
−V
k
(t))dt +
σ
k
V
k
(t)
ρ
k
Z
S
k
(t)+
1 −
ρ
2
k
dZ
∗,S
k
(t), (11.98)
with
κ
S
k
=
κ
k
−
ασ
k
ρ
k
m−1
∑
h=n
τ
h
B(0,T
h
)
A
n,m
(0)
M
∑
i=h+1
τ
i
L
i
(0)
1 +
τ
i
L
i
(0)
β
ik
and
θ
S
k
=
κ
k
/
κ
S
k
.
The calibration of this LMM includes two sub-calibrations. The first one is a
pre-calibration to ATM caplet or ATM swaption market prices. This pre-calibration
is performed with the standard LMM having no stochastic volatility, and its goal is
to find q
j
(t), 1 j M. The second calibration will fit the smile pattern of caplet
or swaption volatilities by searching for appropriate parameter values of the volatil-
ity process V
j
(t) and the leverage parameter
α
. For the pre-calibration we refer the
readers to Section 11.2, or in the connection with the BMS model here, to Schoen-
makers (2005). For the second calibration, Belomestny, Matthew and Schoenmakers
imposed the following assumptions and restrictions:
1. To reduce the number of parameters, there are some restrictions for covariance
structure, from which a special equation for q
i
and
β
i
holds
11.4 Incorporating Stochastic Volatility 315
t
0
q
i
q
j
ds =
t
0
β
i
β
j
ds.
Note that q
i
is used in the BMS model to fit the market ATM volatilities of caps
and swaptions, and has been determined in the pre-calibration,
β
i
will be esti-
mated from q
i
.
2. The number M of Libors is equal to the number D of the Brownian motions
driving stochastic volatilities.
β
∈ R
M×D
is a squared upper triangular matrix of
a rank M. This means that the last Libor L
M
(t) depends only on the volatility
dynamics U
M
, and the Libor L
M−1
(t) only on U
M
and U
M−1
, and so forth.
3. The spot variance V
k
(0) is equal to
θ
k
, namely V
k
(0)=1. The parameters left to
estimate are then
κ
,
σ
and
ρ
for each Libor.
Thanks to the above assumptions, and especially to the upper triangular structure
of
β
, a recursive strategy for the estimation of
κ
,
σ
,
ρ
and
α
from L
M
down to L
1
can be developed. For a more detailed calibration, please see Belomestny, Matthew
and Schoenmakers (2007). With the respect to the assumption (2), the volatility
dynamics of a Libor L
j
(t) are solely determined by the variance V
k
, j k M.As
a result, the Libor process can be rewritten as
dL
j
(t)
L
j
(t)
=
1 −
α
2
[q
j
(t) ·dW(t)] +
α
D
∑
k= j
β
jk
V
k
(t)dZ
k
(t). (11.99)
Now we consider the valuation of caplets via the Fourier transform in the BMS
model. As indicated in the above Libor process, each Libor process includes D − j
independent stochastic volatility processes V
k
(t), that in turn have only the deter-
ministic parameters, we face then a similar valuation problem as for swaptions in
the Zhu model. Belomestny, Matthew and Schoenmakers derived the following CF
for lnL
j
(t) under the forward measure Q
j+1
,
f (
φ
)=e
i
φ
lnL
j
(0)
f
∗
0
(
φ
)
D
∑
k= j
f
∗
k
(
φ
) (11.100)
where
f
∗
0
(
φ
)=exp
−
1
2
(1−
α
2
)(
φ
2
+ i
φ
)
T
j
0
|q
j
|
2
dt
and
f
∗
k
(
φ
)=exp(H
k,1
(T
j
)+H
k,2
(T
j
)V
k
(0))
with
H
k,1
(T
j
)=
κ
∗
k
θ
∗
k
σ
2
k
(a
k
+ d
k
)T
j
−2ln
1 −c
k
e
d
k
T
j
1 −c
k
and
H
k,2
(T
j
)=
[a
k
+ d
k
](1−e
d
k
T
j
)
σ
2
k
(1−c
k
e
d
k
T
j
)
.
316 11 Libor Market Model with Stochastic Volatilities
The parameters a
k
,b
k
and c
k
are given by
6
a
k
=
κ
k
−i
σ
k
ρ
k
αβ
jk
φ
,
d
k
=
a
2
k
−
α
2
β
2
jk
σ
2
k
(
φ
2
+ i
φ
),
c
k
=
a
k
+ d
k
a
k
−d
k
.
(11.101)
The deviation of the pricing formula for caplets/floorlets then follows the same
way as discussed above in the Zhang and Wu model, or in the Zhu model.
7
Particu-
larly, we have the two CFs for the exercise probabilities,
f
1
(
φ
)= f (−i+
φ
), f
2
(
φ
)= f (
φ
).
The derivation of the pricing formula for swaptions follows the same way by apply-
ing the process (11.98) for swap rate.
11.5 Conclusive Remarks
In this chapter, we have briefly addressed the standard LMM and shown that the
market pricing approach for caplets and floorlets via Black76 formula agrees with
the standard LMM where each Libor lives in its own stochastic world with its own
probability measure. This also highlights the nature of the high-dimensionality and
the mutual correlation of LMM. Since a swaption may be regarded as a correlation
product of Libors, the analytical valuation of swaption in LMM is never exact and
requires some ad-hoc approximations. However, a swaption can be valued analyt-
ically with a Black-like formula in Swap Market Model (SMM) in which a swap
annuity is used as numeraire. It can be shown that LMM and SMM do not agree
with each other.
The smile/skew effects in cap markets and swaption markets are evident, and ever
stronger than these ones in equity markets. To recover the implied volatility surface
of caps and swaptions, LMM, as equity option pricing model, should accommodate
stochastic volatility as the first choice for extension. However, adding stochastic
volatility dynamics to LMM increases dramatically the technical complexity and
the number of model parameters. In particular, the (semi-) analytical valuation of
swaptions in extended LMMs will encounter more problems. We have examined
five stochastic volatility LMMs. The first two models of Andersen and Brotherton-
Ratcliffe (2000), and Piterbarg (2003) are in fact a mixture of local volatility and
stochastic volatility. In their models, stochastic volatilities are uncorrelated with the
6
The index j is suppressed in the following expressions for notation simplicity.
7
The pricing formula for caplets given by BMS, however, takes a different form which is similar
to Attari’s formula in Chapter 4.
11.5 Conclusive Remarks 317
corresponding Libors, and local volatilities are responsible for generating down-
slope skews. the Zhang and Wu model (2006), the Zhu model (2007) and the BMS
model (2007) allow not only stochastic volatility in LMM, but also a non-zero cor-
relation between stochastic volatility and its Libor. Consequently, these three mod-
els utilize intensively the Fourier transform as a powerful tool to derive caplet and
swaption pricing formulas.
Recently, SABR model has drawn a strong attention in smile modeling and is
combined with LMM. However, the technique to develop SABR model is beyond
the scope of Fourier transform, hence SABR model is not addressed in this book. To
capture the smile/skew effects, similarly to equity modeling, jump-diffusion model
may be an alternative way. For example, Glasserman and Kou (2003), Belomestny
and Schoenmakers (2006) proposed two special LMMs with jump components.
In spite of a number of vigorous efforts to search for a “better” LMM, not all valu-
ation issues with respect to smile/skew effects are satisfactorily solved. In particular,
recovering swaption smiles for all tenors and option maturities is still a challenge.
In other words, a global consistent calibration to swaption markets is a stress test for
all extended LMMs. From the point of view of practitioners, global consistent cali-
bration to swaption markets is of great importance for reliable and efficient hedging
of interest rate derivatives. For example, Bermudan swaptions, CMS spread options
and range accruals are inhomogeneous in terms of the underlying rate and the struc-
ture. As a result, any trading book of interest rate derivatives in investment banks
including such instruments may be exposed to the risks of these inhomogeneous
underlying rates. To hedge these risks efficiently in a portfolio, a global consistent
calibration is then necessary, and even crucial. A partial fitting of swaption smiles
may result in a non-agreement with Vega risks in a trading book. Additionally, pa-
rameter parsimoniousness is often an underestimated aspect. Every additional model
parameter increases model risk, and reduces the manageability of a model. There-
fore, a reasonable tradeoff between model parsimoniousness and model capability
of fitting markets should be achieved. Nevertheless, thanks to the Fourier transform
and CFs, as shown above, remarkable achievements of the smile modeling within
the LMM framework have been arrived.
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