Zhu J. Applications of Fourier Transform to Smile Modeling: Theory and Implementation

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180 8 L

evy Jumps

We apply the Bochner’s procedure to obtain the CF of the variance-gamma

model. The CF of the gamma process

(t;1,

) is known as

gamma

(

)=(1 −i

φβ

)

−

. (8.13)

It follows then

f (

)= f

gamma

(−i

(

)) =



1 −i

φμβ



−

(8.14)

with

(

)=i

φμ

−

The term m adjusting the variance-gamma process to be a martingale is then identi-

cal to f (−i;t = 1), and is equal to

m =

ln[1 −

μβ

−

With this adjustment of the drift, we can easily verify that the discounted asset price

in the variance-gamma process is a martingale, namely,

−r(T −t)

E[S(T)|F

]=S(t)exp(−m(T −t)+VG(T −t;

)) = S(t).

The L

evy measure of the variance-gamma process is given by

|x|

exp(−

|x|



−1

+ c

) (8.15)

with

c =

From the above L

evy measure we can derive the following properties of the

variance-gamma model:

1. The variance-gamma process exhibits inﬁnite activity, or equivalently exhibits

inﬁnite jump frequency, since it exists



(x)dx = ∞.

2. The variance-gamma process exhibits ﬁnite variation due to the following prop-

erty



|x|<1

(x)dx < ∞.

3. The ﬁrst four central moments of the variance-gamma process are given by

)t,

8.2 Stochastic Clock Models 181

=(2

+ 3

μβσ

)t,

=(3

βσ

+ 12

+ 6

+(3

+ 6

βσ

+ 3

It follows immediately that the parameter

controls the sign of the skew-

ness. Therefore, a negative

should be expected for modeling an asset process.

Moveover, if

= 0, the excess kurtosis is equal to 3(1+

Now we calculate the CFs for the option pricing formula.

(

)=E[g

(T)e

x(T)

]=E[e

−(x

+rT)+(1+i

)x(T)

]

= e

+rT)−(1+i

)mT



1 −(1 + i

)

μβ

−

(1+i

)



= e

+rT)

(

) (8.16)

and

(

)=E[e

x(T)

]=e

+rT)−i



1 −i

φμβ

βφ



= e

+rT)

(

). (8.17)

It is very convenient to derive the pricing formula based on the above two CFs within

the framework given in Chapter 2. The numerical Fourier inversion of the both CFs

share the same implementation as in stochastic volatility models, and present no

large challenge. However, the analytical formula given by Madan, Carr and Chang

(1998) is based on a modiﬁed Bessel function, and therefore is difﬁcult to imple-

ment. We do not discuss this complicated formula here.

8.2.2 Normal Inverse Gaussian Model

The normal inverse Gaussian (NIG) model suggested by Barndorff-Nielsen (1998)

is very similar to the variance-gamma model, except for that the deterministic time

length of a Brownian motion in the NIG model is randomized by an inverse Gaus-

sian distribution. Therefore, the time-changed Brownian motion in the NIG process

is distributed according to inverse Gaussian law. Denote IG(t;

) as an inverse

Gaussian distribution for a time length of t where

is mean and

is variance. An

inverse Gaussian random variable can be originated to be the ﬁrst passage time for

aﬁxedlevelb,b > 0, of a Brownian motion with a drift

.Sowehave

In the context of the normal inverse Gaussian model, we set b to be the time length

t of the subordinated Brownian motion B(t;

). The CF of an inverse Gaussian is

182 8 L

evy Jumps

given by

(

)=exp(t[

−



−2i

]). (8.18)

The NIG process NIG(t;

) can be expressed as

NIG(t;

)=B(IG(t;t/

,t/

);

). (8.19)

This means that the time t is in fact replaced by the ﬁrst passage time of an other

Brownian motion hitting t. Correspondingly, to construct a risk-neutral process for

asset S(t), we need an adjustment for drift and arrive at the following form,

S(t)=S

exp(rt −mt + NIG(t;

))

dx(t)=(r −m)dt + dNIG(t;

)) (8.20)

with

m = lnE[expNIG(1;

)].

According to the Bochner’s procedure, it is easy to obtain the CF of the NIG process,

NIG

(

)= f

(−i

(

)) = exp



−



−2

]



. (8.21)

It follows immediately

m =

−



−

−2

The L

evy measure implied in the NIG process is given by

NIG

(x)=



(

−1

+ c

)

K(|x|)

|x|

(8.22)

with

c =

where K(·) is the modiﬁed Bessel function of the second kind (see Carr-Geman-

Madan-Yor (2001)). From the given L

evy measure we can conclude

1. The NIG process has inﬁnite activity since it exists



NIG

(x)dx = ∞. This prop-

erty is identical to the one of variance-gamma process.

2. The NIG process has also inﬁnite variation due to the property



|x|<1

NIG

(x)dx = ∞.

8.3 Time-Changed L

evy Process 183

The two CFs of x(T) can be calculated as follows:

(

)=E[g

(T)e

x(T)

]=E[e

−(x

+rT)+(1+i

)x(T)

]

= e

+rT)−(1+i

)mT

exp



−



−

(1+i

)

−2

(1+i

)]



= e

+rT)

NIG

(

) (8.23)

and

(

)=E[e

x(T)

]

= e

+rT)−i

exp



−



−2

]



= e

+rT)

NIG

(

). (8.24)

The valuation of European-style options with the NIG process follows then the same

way as with the variance-gamma process.

8.3 Time-Changed L

evy Process

We has seen that the variance-gamma process and the NIG process result from a

subordination of a Brownian motion to a gamma process and an inverse Gaussian

process, respectively. In this section, we discuss how to construct a process by sub-

ordinating a general L

evy process with a (general) subordinator.

As indicated above, a subordinator is deﬁned as a L

evy process that changes a

positive time to another stochastic positive time. Since a clock generates always an

increasing time, the new cumulative stochastic time must be also increasing. Gen-

erally speaking, a subordinator is not necessarily a L

evy process so that such a

stochastic clock can be deﬁned. For example, a mean-reverting square root process

that has been discussed early in details in the Heston model, may be used to map

a positive time to another positive time if a suitable parameter set is chosen. As in

stochastic clock models such as the variance-gamma model, the stochastic clock

here may be interpreted as a driver for instantaneous business activity. A higher

activity rate generates higher volatility in a time-changed process, and a smaller ac-

tivity rate leads to smaller volatility in a time-changed process. Hence, a stochastic

time-change is often called instantaneous activity rate, in order to emphasis the sim-

ilar functionality of the stochastic time-change in a time-changed L

evy process to

the stochastic volatility in a diffusion process with respect to smile modeling.

As we can see in the CF of the mean-reverting square root process given in the

Heston model, a mean-reverting square root process is just an It

o process, not a L

evy

process since its CF is not inﬁnitely divisible, a necessary condition that L

evy pro-

cess must satisfy. This indicates that a subordinator may not be necessarily a L

evy

process. Except for the mean-reverting square root process, a so-called Gamma

184 8 L

evy Jumps

Ornstein-Uhlenback process, ﬁrst introduced by Barndorff-Nielsen and Schephard

(2001), is a L

evy process and can also serve as a subordinator.

In the following, two cases will be examined with respect to the correlation

between the L

evy process and the subordinator. The ﬁrst case is a case with un-

correlated time-change where the changed L

evy process and the substantiator are

independent of each other. Due to the independence, the distribution law of the sub-

ordinator can be different from the distribution law of the underlying L

evy process.

In the second case, the changed L

evy process is correlated with its subordinator.

Due to the very nature of correlation or dependence, the distribution type of the

subordinator is usually identical to the distribution type of the L

evy process.

8.3.1 Uncorrelated Time-Change

Denote L(t) as a L

evy process again, particularly it could be the variance-gamma

process or the NIG process. Due to the inﬁnite divisibility, the CF of a L

evy process

L(t) can be expressed in the following form

f (

)=E[e

L(T)

]=e

(

)

= e

∗

(

, (8.25)

which is essentially an extension of (8.6), and is identical to (8.9).

(

) may be

regarded as the CF exponent of L(1),

E[e

L(1)

]=e

∗

(

)

For a particular subordinator Y (t) that is uncorrelated with L(t), we deﬁne a time-

changed L

evy process as follows

X(t)=L(Y(t)).

By applying the Bochner’s procedure, we may obtain the CF of the generated pro-

cess X(t),

E[e

X(t)

]==E[e

X(Y(t))

]=E[e

∗

(

)Y (t)

]

= E[e

i(−i

∗

(

))Y (t)

]=e

(−i

∗

(

))

, (8.26)

with

(

) as the CF exponent of the subordinator process Y(t). Note that we

are using

= −i

∗

(

) as the characteristic parameter in the application of the

CF e

(

)

of Y(t).As

is deﬁned in the real domain, −i

∗

(

) is generally not

real-valued, but complex-valued. For example, let L(t) be a Brownian motion with

drift. Its CF exponent is i

φμ

. The resulting

is then

φμ

−

i.In

this sense, we have extended the deﬁnition domain of

to the complex plane un-

consciously. In fact, if a characteristic parameter is deﬁned in the complex plane,

the Fourier transform is referred to as the generalized Fourier transform. We will

discuss the generalized Fourier transform more in details later.

8.3 Time-Changed L

evy Process 185

Based on the time-changed process X(t), the corresponding asset process S(t) is

assumed to take the following form,

S(t)=S

exp(rt −m(t)+X(t)) (8.27)

with a drift compensator,

m(t)=lnE[exp(X(t)].

In the following, we will consider two cases: In the ﬁrst case, the subordinator Y (t)

is a mean-reverting square root process. In the second case, the L

evy process L(t)

is time-changed by a Gamma Ornstein-Uhlenbeck process. Finally, we will follow

the Carr and Wu’s approach (2004) to relax the restriction of the zero-correlation

between L(t) and Y(t), and explore the correlation case between L(t) and Y(t) by

introducing the generalized Fourier transform where a leverage-neutral measure and

the corresponding complex-valued change of measure play a key role in construct-

ing the generalized time-changed L

evy process.

Time-Changed By Square Root Process

We address ﬁrst brieﬂy how we may enhance the variance-gamma model and the

NIG model if we use a mean-reverting square root process to stochastically change

the time in the variance-gamma model and the NIG model again, as suggested

by Carr, Geman, Madan and Yor (2001). The stochastic clock deﬁned by a mean-

reverting square root process y(t) is

Y(t)=



y(u)du, (8.28)

where

dy(t)=

(

−y(t))dt +



y(t)dW (t).

TheCFofy(t) is known in the Heston model and has the following form,

(

)=H

(t)y

+ H

(t).

The functions H

(t) and H

(t) are deﬁned in the Heston model. Carr, Geman, Madan

and Yor constructed a so-called stochastic volatility L

evy process

as follows:

X(t)=L(Y(t)). (8.29)

From the CF of X(t), especially of Y (t), we can easily verify that X(t) is no longer

evy process since the resulting CF is not inﬁnitely divisible.

The name stochastic volatility L

evy process is a bit misleading. The term “stochastic volatility”

is generic for a mechanism generating volatility smile.

186 8 L

evy Jumps

Two particular processes of L(t), the variance-gamma process and the NIG pro-

cess can be employed to Y (t) to demonstrate how to enhance the existing L

evy

processes.

Variance-gamma process:

The new process XVG(T )=X

(t) can be expressed as

XVG(t)=VG(Y (t);

= 1).

The corresponding CF of X

(t) takes a nested form

XVG

(

;

)=exp(

(−i

∗

(

))).

Here

is set to 1 in the variance-gamma process. As argued by Carr, Geman, Madan

and Yor, the effective

in the time-changed process XVG(t) should be set to 1/y(0).

NIG Process:

Subordinating the NIG process to Y(t) yields the new process XNIG(t)=

NIG

(t),

XNIG(t)=NIG(Y (t);

,1,

)

and the corresponding CF of X

NIG

(t)

XNIG

(

;

)=exp(

(−i

∗

NIG

(

))).

Here y(0) may be suggested to be equal to the parameter

in the NIG model.

Time-Changed By Gamma-OU Process

The gamma Ornstein-Uhlenback process, suggested by Barndorff-Nielsen and Schep-

hard (2001), takes the following form,

dy(t)=−

y(t)dt + dZ(t), (8.30)

where Z(t) is the so-called background driving L

evy process for y(t).

Here we

consider a special construction where Z(t) is a compound Poisson process

Z(t)=

N(t)

∑

n=1

with N(t) as a Poisson counting process and x

as an independent and identical

exponentially distributed sequence. Hence, the gamma Ornstein-Uhlenback process

As shown in Barndorff-Nielsen and Schephard (2001), the background driving L

evy process

could be any L

evy process that satisﬁes the technical condition E[ln(1+ |Z(1)|)] < ∞.

8.3 Time-Changed L

evy Process 187

can be alternatively expressed as

dy(t)=−

y(t)dt + xdN(t). (8.31)

This presentation of the compound Poisson process agrees with the one in Chap-

ter 7.

It can be shown that y(t) then follows a marginal gamma distribution. Due to the

formal similarity between the process (8.30) and the Gaussian Ornstein-Uhlenbeck

process, the process given in (8.30) is referred to as gamma Ornstein-Uhlenback

process, in short, gamma-OU process. Particularly, we assume that y(t) follows a

marginal gamma process with a mean of

and a variance of

, and examine

how to use Y (t)=



y(u)du to subordinate another L

evy process L(t).Asshownin

Schoutens, Simons and Tistaert (2004), the CF of Y (t) is given by

GammaOU

(

)=exp



φγ

−

φκ

−



c −i

φγ



= e

(

)

(8.32)

with

(1−e

−

), c =

Variance-gamma process:

The new process XVG(T )=X

(t) can be constructed as

XVG(t)=VG(Y(t);

The corresponding CF of X

(t) takes the following form

XVG

(

;

)=exp(

(−i

∗

(

))).

NIG Process:

Subordinating the NIG process to the gamma-OU process Y (t) yields the new

process XNIG(t)=X

NIG

(t),

XNIG(t)=NIG(Y(t);

)

and the corresponding CF of X

NIG

(t)

XNIG

(

;

)=exp(

(−i

∗

NIG

(

))).

188 8 L

evy Jumps

8.3.2 Correlated Time-Change

Generalized Fourier Transform

As shown above, two subordinators, the mean-reverting square root process and the

gamma Ornstein-Uhlenbeck process, are not correlated with the underlying L

evy

process. In this section, we relax the assumption of zero correlation and accommo-

date a non-zero correlation between the subordinator Y(t) and the underlying L

evy

process L(t). In the context of pure jumps, we refer to this correlation as a leverage

effect that generates the asymmetry of the distribution of the time-changed L

evy

process.

To deal with leverage effect properly, we extend formally the real-valued param-

eter

inaCFE[e

−i

x(t)

] to the complex plane although we have done it through

this chapter unconsciously. Denote D ⊆ C

as the domain where the CF with

well-deﬁned. Such a CF E[e

−i

x(t)

] with

∈ D, is called the generalized Fourier

transform according to Tichmarsh (1975). Recall the expression for the CF of the

time-changed L

evy process X(t) in (8.26),

E[e

X(t)

]=e

(−i

(

))

, (8.33)

that is only valid if the stochastic time Y(t) is independent of L(x), but no longer

valid if a non-zero correlation between Y(t) and L(x) is present. To evaluate the CF

in the case with the correlation, we explain an approach of Carr and Wu (2004) in

following.

Denote

(

) as the characteristic exponent of a Fourier transform such that

E[e

x(t)

]=e

(

)

Next, we introduce the Laplace transform that is deﬁned by

(

) ≡ E[e

−

x(t)

]

with

as the Laplace parameter.

Carr and Wu (2004) showed that evaluating the generalized Fourier transform of

the time-changed L

evy process X(t) under the risk-neutral measure Q is equivalent

to evaluating the Laplace transform of the random time Y(t) under a complex-valued

measure Q

. That in turn “can be used to reduce the the problem of ﬁnding the

characteristic function of X(t) in the original economy into the problem of ﬁnding

it in an artiﬁcial economy that devoid of the leverage effect”. More mathematically,

a new equivalent complex-valued measures Q

with respect to Q may be deﬁned by

(t)=

(t) (8.34)

with a complex-valued martingale

CarrandWuused−

(

) as the characteristic exponent to introduce complex-valued measure.

As a consequence, some results and equations here seem different from the original ones in Carr

and Wu (2004).

8.3 Time-Changed L

evy Process 189

(t)=exp



X(t) −Y (t)

(

)



. (8.35)

This new measure Q

is called the leverage-neutral measure. It can be shown that

(t) is a well-deﬁned complex-valued Q-martingale, and is dependent on the value

. Under the leverage-neutral measure Q

,wehave

X(t)

(

) ≡ E



X(t)



= E



−Y (t)[−

(

)]



≡ g

Y(t)

(−

(

)), (8.36)

where g

Y(t)

(−

(

)) denotes the Laplace transform of Y(t) under the complex-

valued measure Q

. By introducing a new notation

∗

(

)=−

(

), the above

relation may be expressed as



X(t)



= E



−Y (t)

∗

(

)]



⇒ f

X(t)

(

)=g

Y(t)

(

∗

(

)).

This means that through this special change of measure, we in fact change a Fourier

transform under Q to a Laplace transform under Q

. In many cases where a leverage

effect between the random time and the original L

evy process is present, it is more

convenient to compute the Laplace transform g

Y(t)

(

∗

(

)) under the measure Q

for f

X(t)

(

). In more details, the calculation of g

Y(t)

(

∗

(

)) may be performed in

analogy to the zero bond pricing,

Y(t)

(

∗

(

)) = E



−



∗

(

)y(t)dt



. (8.37)

Obviously, if y(t) is a diffusion process, the Feynman-Kac theorem may be directly

applied to solve the above expected value, as we have done repeatedly in stochastic

volatility models and short rate models.

Similarly to the Dol

eans exponential of a diffusion process, for any complex

value

,theDol

eans exponential of time-changed L

evy process X(t) is given by

E (

X(t)) = exp



X(t)+Y(t)

∗

(−i

)



which has an expected value of one. The complex-valued martingale M

(t) is then

a special case of E (

X(t)), namely

(t)=E (i

X(t)).

Because correlation or dependence is a very nature of the processes of a same

type, in order to specify a concrete measure for a correlation or dependence, it is

reasonable to assume that the L

evy process L(t) and the time-change process y(t) are

governed by a same type of distribution laws or L

evy measures. Therefore, if a L

evy

process is purely continuous, i.e., is a Brownian motion, non-zero correlation can

only be generated by a continuous component. Similarly, if a L

evy process is purely

jump, non-zero correlation can only be generated by a pure jump. In particular,

a correlated time change of a L

evy process with ﬁnite (inﬁnite) activity may be

only achieved by a subordinator with ﬁnite (inﬁnite) activity. In following, we will