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302 11 Libor Market Model with Stochastic Volatilities

11.4.3 The Wu and Zhang Model

As shown above, both the ABR model and the Piterbarg model incorporate the

Heston’s stochastic variance dynamics into to LMM, but are restricted to zero-

correlation case. The nice feature of a correlated stochastic volatility model to gen-

erate down-sloping skews does not show to advantage in their models. The volatility

skews in both models are generated via various local volatility formulations. Addi-

tionally, Fourier transform and CF do not play a key role in expressing the ﬁnal

option pricing formulas for caplets and swaptions. Wu and Zhang (2006) proposed

an extended LMM with stochastic volatility, which ﬁlls these two gaps and allows

for (1) a non-zero correlation between the dynamics of Libors and volatility, and (2)

applies extensively the technique of Fourier transform to derive pricing formulas.

The stochastic processes for Libors L

(t), 1  j M, and the corresponding vari-

ance process V(t) in Wu and Zhang’s model under the risk-neutral measure Q take

the following forms,

(t)



V(t)q

(t) ·[

(t)



V(t)dt + dW (t)],

dV(t)=

(

−V(t))dt +



(V(t)dZ(t), (11.57)

where W(t) is a multi-dimensional independent Brownian motion. q

(t) is a de-

terministic vector function with the same size as W(t). The stochastic variance V(t)

follows a mean-reverting square root process where Z(t) is a one-dimensional Brow-

nian motion that is correlated with W(t) in the following way

< q

(t), dW (t) > dZ(t)

(t)|

(t)dt. (11.58)

It is clear that the correlation coefﬁcient

(t) is time-dependent, precisely, depen-

dent on the deterministic volatility function q

(t). Hence, this correlation can not be

given explicitly, or at least, is difﬁcult to be used to control volatility skews when

we note that q

(t) is initially used to control volatility term structure.

(t) is also a

deterministic vector function and has a special form

j−1

(t) −

(t)=

(t)

1 +

(t)

(t).

It can be shown easily that such special construction of

(t) eases the change of

measure. Particularly, with the help of

(t), the Libor processes and variance pro-

cess under the respective forward measure Q

j+1

are given in a simpler form,

(t)



V(t)q

(t) ·dW(t),

dV(t)=[

κθ

−(

σξ

(t))V(t)]dt +



(V(t)dZ(t), (11.59)

11.4 Incorporating Stochastic Volatility 303

with

(t)=

∑

h=1

(t)

1 +

(t)

(t)|

(t)

∑

h=1

(t)

1 +

(t)

< q

(t),W(t) > Z(t). (11.60)

It is easy to verify that the disappearance of the term

(t) in the process L

(t)

and the appearance of the new term

(t) in the process V(t) are the results of the

change of measure from Q to Q

j+1

.As

(t) is a time-dependent function, the usual

methods for deriving CFs given in Section 3.2 are difﬁcult to use. Wu and Zhang

(2006) then used the freezing technique and approximated the

(t) by freezing the

forward Libors at the origin,

(t) ≈

∑

h=1

(0)

1 +

(0)

(t)|

(t).

It follows immediately the reformulation for dV(t),

dV(t)=

[

−

∗

(t)V (t)]dt +



(V(t)dZ(t)

with

∗

(t)=1 +

(t).

In spite of the freezing of L

(t) to their values at the origin, the time-dependent na-

ture of the Wu and Zhang model does not change due to the terms q

(t). Denote

x(t)=ln(L

(t)/L

(0)), According to the Feynman-Kac theorem, the moment gen-

erating function f (

,t)=E[e

x(T

)

] of x(t) satisﬁes the Kolmogorov’s backward

equation that we have used once in the Heston model,

∂

[

−

∗

(t)V ]

∂

−

(t)|



∂

−

∂



∂

σρ

(t)V |q

(t)|

∂

= 0, (11.61)

subject to the boundary condition

f (

,t = 0)=e

This PDE can be solved explicitly as in the Heston model if it is assumed that the

functions q

(t) and

(t) are piecewise constant on the grid {t

= 0,t

,··· ,t

}. The solution takes the following form

f (

)=exp(H

)+H

), (11.62)

where H

) and H

) are expressed by recursive formulas,

304 11 Libor Market Model with Stochastic Volatilities

h+1

)=H

κθ



+ d

)

−2ln



1 −c



h+1

)=H

+ d

−

)](1−e

)

(1−c

)

with the following parameters

κξ

∗

) −

σρ

)



−

)(

−

+ d

−

)

−d

−

)

(11.63)

Having the closed-form solution for the moment generating function f (

), we can

construct two CFs under the Delta measure and the forward measure Q

j+1

, under

which two exercise probabilities are deﬁned,

(

)= f (−i+

) and f

(

)= f (i

). (11.64)

Zhang and Wu expressed the pricing formula for caplet in terms of the moment

generating function f (

), which reads

CPL

(0)=

B(0,T

j+1

)[L

(0)F

−KF

], (11.65)

where F

and F

are given by



∞

Im[e

ln(L

(0)/K)

] f (1+ i

)

and



∞

Im[e

ln(L

(0)/K)

] f (i

)

It can be veriﬁed easily that these two probabilities F

,k = 1, 2, may be rewritten in

the usual forms throughout this book in terms of CFs, These are



∞



(

)

exp(i

ln(L

(0)/K))



, k = 1,2.

To value swaptions, Wu and Zhang used the same approach as in the models

of Andersen and Brotherton-Ratcliffe (2002) and Piterbarg (2003), and approxi-

mated S

n,m

(t) with a sum of weighted Libors, and kept the structure of the swap rate

process identical to the structure of Libors. Finally, Wu and Zhang arrived at the

following approximated dynamics for swap rate and the corresponding stochastic

variance,

11.4 Incorporating Stochastic Volatility 305

n,m

(t)

n,m

(t)



V(t)

m−1

∑

k=n

(0)[q

(t) ·dW

(t)]



V(t)q

n,m

(t)dU

(t), (11.66)

dV(t)=

(

−

n,m

(0)V(t))dt +



V(t)dZ

(t),

where dU

and dZ

are the Brownian motions under the swap measure, and are

correlated with

n,m

∑

m−1

k=n

(0)|q

(t)|

∑

m−1

k=n

(0)q

(t)|

The weights

(0) and the drift adjustment

n,m

(0) for dV(t) are given by

(0)=

∂

n,m

(0)L

(0)

∂

(0)S

n,m

and

n,m

(0)=1+

n,m

(0)

m−1

∑

k=n

B(0,T

k+1

)

(0).

With these approximated processes for S

n,m

(t) and V(t), all results including the

pricing formula for caplet may be directly applied to swaptions with the correspond-

ing replacements. Namely, we replace L

with S

n,m

, T

with T

, x

with x

n,m

, q

with

n,m

, and

with

n,m

11.4.4 The Zhu Model

In this subsection we present an extended LMM of Zhu (2007), which incorpo-

rates stochastic the volatility under the market forward measure Q

j+1

of each Li-

bor L

(t). Unlike the above LMMs with stochastic variance, Zhu assumed that

stochastic volatility v

(t) is governed by a mean-reverting Ornstein-Uhlenbeck pro-

cess under the forward measure Q

j+1

, as in Sch

obel and Zhu (1999). In more de-

tails, the processes for Libors L

(t), 1  j  M, and the corresponding volatilities

(t), 1  j  M, are given by

(t)

= v

(t)dW

(t),

(t)=

(

−v

(t))dt +

(t)dW. (11.67)

The Wiener process dW is correlated with the Wiener process dW

of the Libor

(t), i.e., dW

dW =

dt, which means that stochastic volatility v

(t), unlike in

Wu and Zhang’s model (2006), has its own special correlation with L

(t). The cor-

relation

is independent of any other deterministic and stochastic parameters and

has the same ﬁnancial meaning as in usual stochastic volatility models for equity

306 11 Libor Market Model with Stochastic Volatilities

derivatives. Furthermore, all stochastic volatilities v

(t), j  1, are driven by the

same Wiener process W(t). At ﬁrst glance, this assumption seems to be a restric-

tion. But if we look at the smile pattern of caplets in details, we can ﬁnd that the

smile structures of different caplets are mainly determined by the ﬂexible volatility

correlations

, and are less driven by the randomness of the volatility.

The second formulation of the Zhu model is to specify stochastic variance V(t)

to be a mean-reverting square-root process as in the Heston model,

(t)



(t)dW

(t),

(t)=

(

−V

(t))dt +

(t)



(t)dW. (11.68)

As indicated clearly by the speciﬁcations for the dynamics of Libors and the

corresponding stochastic volatilities, the above speciﬁcations of the LMM have the

following favorite features.

1. The Wiener processes of Libor and its stochastic volatility are individually corre-

lated. This correlation is free and could be different for all Libors. The individual

speciﬁcation of the correlation between Libor and its volatility is important since

it admits a ﬁne control of smile patterns and could signiﬁcantly improve the ca-

pability for a better ﬁtting to market volatility data.

2. The Zhu model admits the different formulations for stochastic volatility. Al-

though only mean-reverting Ornstein-Uhlenbeck process and mean-reverting

square root process are proposed to model stochastic volatility and variance re-

spectively. other interesting speciﬁcations discussed early in this book may be

incorporated.

3. The pricing formula via the Fourier transform for caplets can be given in a

straightforward way and takes the similar form to the well-known pricing so-

lution for equity options without any approximation or the freezing technique.

4. A main contribution of the Zhu model is to derive the CFs of swaptions within an

uncorrelated (orthogonal) framework where Libors are represented by some in-

dependent factors, namely the principal components. Due to the independence of

factors, the CF of a swap rate can be simply approximated to be a product of the

CFs of each factors. The pricing formula for swaption in the Zhu model coincides

with the formulas in the existing stochastic models, and can be implemented in a

very efﬁcient way.

5. Finally, by introducing some deterministic nesting functions for the parameters

of stochastic volatility process, the model becomes very parsimonious. As shown

later, the same type of parameter of all stochastic volatility processes is nested by

a deterministic function. Hence, this model is called a nested stochastic volatility

LMM. To illustrate the idea of nesting, we consider all mean level parameters

in the stochastic volatility processes. Obviously, in a pricing environment with

20 Libors there are 20 mean level parameters. A nesting function for

is to pa-

rameterize these 20 parameters with a few parameters. As a parsimonious model,

the calibration performs also efﬁciently and quickly.

11.4 Incorporating Stochastic Volatility 307

To keep the number of parameters of stochastic volatility processes in a manage-

able range, Zhu introduced some deterministic functions to connect

, v

(0)

and

respectively. In particular, Zhu assumed the following term structure for

, v

(0) and

= y

;

,a) :=

= y

;

) :=

+(b

+ b

(0)=y

) := v+(c

+ c

, (11.69)

= y

;

,d) :=

where he attempted to capture different forms of the caplet volatility structure

by specifying the mean levels

and the spot volatilities v

(0) with a popular

ABCD parametric function for the term volatility of caplets (Brigo and Mercurio

(2006)). With these speciﬁcations we only need four functions with 11 parameters

{

,a,

,d} for

, v

(0) and

, plus some parameters

for the correlation

between L

(t) and v

(t), to specify the stochastic volatility

processes for all Libors completely. The parameters of the stochastic processes for

Libors are nested by certain deterministic functions that allow for a parsimonious

but still reasonable formulation of stochastic volatility. Additionally, the correla-

tion coefﬁcients

may also be nested by a deterministic function. With a nesting

function for

, the total number of parameters for this stochastic volatility LMM

can be again reduced remarkably. Zhu suggested a nesting function for

with the

following function

= y

)=e

(1−e

), −1 < e

< 1. (11.70)

If e

lies in interval (−1,0) and e

is slightly negative, the short-term Libors have a

near zero correlation with their volatilities while the correlation between the long-

term Libors with their volatilities will converge to e

. With the given model setup

and the nesting functions, the extended LMM embraces a well-known ABCD para-

metric LMM as special case. Setting

(0)=

→ 0,

leads a degenerated volatility process v

(t)=v

(0), and the Libor process converges

to the LMM with a humped parametric volatility structure which is discussed exten-

sively in Brigo and Mercurio (2006).

It is worth comparing Zhu’s model with Wu and Zhang’s model in terms of the

relative volatility structure. As shown previously in Wu and Zhang’s model, the

relative volatility level of two Libors is given by

a(t)=

(t)



V(t)

(t)



V(t)

(t)

(11.71)

308 11 Libor Market Model with Stochastic Volatilities

with q

(t) as a deterministic function describing the term structure of volatility. Ob-

viously the relative volatility level a(t) is also a deterministic function, and therefore

displays a ﬁxed pattern which is certainly an unfavorable feature. In contrast, the rel-

ative volatility level in the extended LMM is deﬁned by

a(t)=



(t)



(t)

, (11.72)

which is a stochastic process. From the point of view of modeling, the speciﬁcation

of the stochastic volatility in the Zhu model is more ﬂexible in describing the relative

dynamics of the volatilities of two Libors.

In the following, we only take the mean-reverting Ornstain-Ulenbeck process to

illustrate Zhu’s model. The dynamics of Libor L

(t) and its volatility v

(t) given

in (11.67) are associated only with the forward measure Q

j+1

. For the purposes of

simulation and pricing exotic interest rate products, we have to change the processes

of L

(t) and v

(t) under an unique measure such as the spot measure or the terminal

measure Q

M+1

. Generally under an arbitrary forward measure Q

k+1

, the stochastic

processes for Libors and volatilities take the following forms

(t)=L

(t)[

(t)+v

(t)dW

(t)=[

−

(t)]dt +

dW, (11.73)

with



(t)=v

(t)

∑

h=k+1

(t)

, j > k,

(t)=

∑

h=k+1

(t)

, j > k,



(t)=0, j = k,

(t)= 0, j = k,



(t)=−v

(t)

∑

h= j+1

(t)

, j < k,

(t)= −

∑

h= j+1

(t)

, j < k.

The proof follows the usual ways as used extensively in other LMMs. The valuation

of caplets here is straightforward, and Zhu’s extended LMM admits a closed-form

pricing formula for caplet and ﬂoorlet via the Fourier transform under each forward

measure as in the Wu and Zhang model, which reads

CPL

(0)=

B(0,T

j+1

)[L

(0) ·F

−K ·F

] (11.74)

with



∞



(

))

exp(−i

lnK)



, k = 1,2.

The functions f

(

),k = 1,2, is the corresponding CFs under two martingale mea-

sures dQ

j+1

∗

and Q

i+1

, which are related via the Radon-Nikodym derivative

11.4 Incorporating Stochastic Volatility 309

j+1

∗

j+1

(t)

(0)

where

(t)

(0)

deﬁnes an exponential martingale according to the process of L

(t).For

(t) as a mean-reverting Ornstein-Uhlenbeck process, the exact forms of f

(

) may

be expressed by

(

)=e

lnL

(0)

(

), (11.75)

where f

(

),k = 1,2, are given in (3.32) and (3.36). Similarly, for V

(t) as a mean-

reverting square root process, the exact forms of f

(

) may be written as by

(

)=e

lnL

(0)

Heston

(

), (11.76)

where f

Heston

(

),k = 1, 2, are given in (3.16).

To prepare for the valuation of swaptions, we need a reformulation of LMM via

the principle component analysis (PCA). Note that the correlation matrix

= {

}

givenin(11.22) is positive semi-deﬁnite. It is known that a positive semi-deﬁnite

matrix has the following decomposition

= U



, (11.77)

where U is an eigenvector matrix with the eigenvectors of

as column vector.

the eigenvalue matrix with the eigenvalues of

as diagonal element,

⎛

⎜

⎝

⎞

⎟

⎠

Assume that {

}

1jM

is a non-increasing sequence. The ﬁrst few dominate

components are often called principal components. By using the principal compo-

nents, we can rewrite the Brownian motion W

with D independent standard Brow-

nian motions Z

≈

∑

l=1



, 1  j  M, 1  l  D. (11.78)

Setting W

into the Libor process yields

(t)=L

(t)



(t)dt + v

(t)

∑

l=1





. (11.79)

Now we denote

as the correlation between Z

and dW. To preserve the correlation

between W

and dW, the following condition on Z

and dW must be fulﬁlled,

310 11 Libor Market Model with Stochastic Volatilities

Corr <

∑

l=1



,dW > =

∑

l=1



dt (11.80)

∑

l=1

dt =

dt, (11.81)

where



. Setting D = M leads to a linear equation system with M un-

known variables

∑

l=1

. 1  j, l  M, (11.82)

which can be solved for

by a standard numerical procedure, for example, Gaus-

sian elimination. For a factor representation of L

(t) we need only

,1  l  D,

and reformulate the dynamics of of the Libors and theirs stochastic volatilities,

(t)=L

(t)v

(t)

∑

l=1

(t)=[

−

(t)]dt +

dW, (11.83)

dW(t)dZ

(t)=

dt. 1  l  D.

The factor representation (11.83) for the extended stochastic volatility LMM are

based on the ﬁrst D principal components, not whole components. Additionally, a

correction of variance to the original value is important when we implement an exact

Monte-Carlo simulation and the approximated swaption pricing formula.

As in other extended LMMs, the process for the swap rate is approximated in a

lognormal form as a sum of weighted Libors, and takes the following form

n,m

(t)

n,m

(t)

m−1

∑

k=n

(0)v

(t)dW

(t), (11.84)

where the weights

(0) is identical to the one in the previous models. Setting the

factor representation (11.83)ofL

(t) into the above approximated swap rate process

yields

n,m

(t)

n,m

(t)

∑

l=1

m−1

∑

k=n

(t)dZ

(t)

with

(0)

In order to simplify the expression of the swap rate process and makes its struc-

ture more transparent, we summarize the terms

∑

m−1

k=n

(t) with a single process.

By applying It

o’s lemma, we can show that given the mean-reverting Ornstein-

Uhlenbeck volatility process v

(t) in (11.67), the sum v

(t)=

∑

m−1

k=n

(t) with

the constant weights q

should be governed by an other mean-reverting Ornstein-

11.4 Incorporating Stochastic Volatility 311

Uhlenbeck process,

(t)=

(

−v

(t))dt +

dW (11.85)

with

(0)=

m−1

∑

k=n

(0),

≈

∑

m−1

k=n

(0)

m−1

∑

k=n

∑

m−1

k=n

Finally, we rewrite the swap rate process in SMM by using D independent Wiener

processes with its individual stochastic volatility process v

(t),

n,m

(t)

n,m

(t)

∑

l=1

(t)dZ

(t) (11.86)

with

(t)=

m−1

∑

k=n

(t).

Based on the simple process for swap rate S

n,m

(t) in (11.86), we are able to

derive the pricing formula for swaptions by applying CFs. Let X(t)=lnS

n,m

(t) −

lnS

n,m

(0),wehave

dX(t)=

∑

l=1



−

)

(t)dt + v

(t)dZ

(t)]. (11.87)

Although v

(t), 1  l  D, are correlated with each other, but Z

(t), 1  l  D, are

not. This independence leads to

Corr < v

(t)dZ

(t), v

(t)dZ

(t) >= 0, 1  l, k  D, l = k.

Additionally, by It

o’ rule the cross terms (v

)

(t)dt ·(v

)

(t)dt, (v

)

(t)dt ·v

(t)dZ

(t)

have at least an order of (dt)

, and do not deliver a dominate effect for the dynam-

ics of X(t). The correlation between v

(t) and Z

(t) is given by dW(t)dZ

(t)=

where

has been calculated by the factorization procedure.

Using the independence of v

(t)dZ

(t) and neglecting the second-order effect of

cross terms, Zhu derived the approximated characteristic functions of X(t) under the

swap measure Q

. For some complex number p, the moment-generating function for

X(t) is given by

(p, T )=E



exp



pX(T)



= E



exp



−

∑

l=1



)

(t)dt +

∑

l=1



(t)dZ

(t)

