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170 7 Poisson Jumps

1. E



∗

(

,T)|



< ∞, g

∗

(

,t)=exp



−



r(X(s),s)ds



∗

(

,T −t).

2. E





(t)|dt



< ∞,

(t)=g

∗

(

,t)[ f

(

(t)) −1]

(X).

3. E





(t)|dt



1/2



< ∞,

(t)=g

∗

(

,t)

(t)v(X).

In the above ODEs, the term f

(

(t)) is responsible for the effects of jump com-

ponent in the ﬁnal solutions. Therefore, the concrete speciﬁcation of q determines

the concrete solution forms of

(t) and

(t). In the line with the above discussions,

we have the following choices for q in a one-dimensional setting.

1. q is constant: q is constant as in a simple jump in Section 7.2, and in particular,

may be equal to 1, f (z) reduces to the CF of a Poisson distribution,

(z;q = 1)=exp



−1)



, f

= e

2. q is Gaussian: Note that q is equivalent to ln(1+J) as in lognormal jumps. Hence,

q maybeassumedtonormallydistributedtoN(

), the “extended” Fourier

transform can be computed as

(z;q)=exp



( f

(z) −1)



, f

(z)=e

3. q is exponentially distributed: q is again equivalent to ln(1+ J) as in Pareto jump.

Consequently, q is exponentially distributed with a mean of

. The “extended”

Fourier transform takes a form of

(z;q)=exp



( f

(z) −1)



, f

(z)=

1 −z

Since the CF f

∗

(

,t) is the so-called discounted CF, the pricing formula based on

(

) and f

(

) as well as the alternative pricing formulas given in Chapter 4, can

not be applied directly. To value a European-style option, we deﬁne the underlying

stock price via X(t), namely

S(T)=e

X(T )

for a deterministic vector

∈ R

. A call option is then valued as follows:

= E



exp



−



r(X(s),s)ds



X(T )

−K]1

(

X(T )>K)



= E



exp



−



r(X(s),s)ds



X(T )

(

X(T )>K)



− KE



exp



−



r(X(s),s)ds



0X(T )

(

X(T )>K)



Deﬁne a function

In the previous sections in this chapter, we always specify the distribution of J instead of q.

Since q = ln(1 + J),wehave

= E[J]=E[e

] −1.

7.6 Afﬁne Jump-Diffusions 171

(y;X

,T)=E



exp



−



r(X(s),s)ds



X(T )

X(T ))<y



the call option may be expressed as

,−

(−lnK;X

,T) −K

0,−

(−lnK;X

,T).

Dufﬁe, Pan and Singleton derived a more detailed form for

(y;X

,T) based on

the inverse Fourier transform of f

∗

(

,t), which is given by

(y;X

,T)=

∗

(−iu

,T) −



∞

Im[ f

∗

(−iu

+ u

z,T)e

−izy

]

dz. (7.50)

An illustrative example proposed by Dufﬁe, Pan and Singleton is the Heston

model added by two correlated jumps. Denote X(t)=(G(t)=lnS(t),V(t)),the

detailed formulation of this special afﬁne jump-diffusion model reads



G(t)

V(t)





r −

m −

V(t)

[

−V(t)]





V(t)



σρ σ



1 −









, (7.51)

where W(t)=(W

(t),W

(t)) is an independent two-dimensional Brownian motion.

The jumps are driven jointly by two independent Poisson processes Y

and Y

, and

two correlated jumps size q =(q

). Denote f

) is the joint transform of

q. In order to ensure that G(t) is a martingale, m is required to be equal to E[J]=

E[e

] −1, that in turn is equal to f

(1,0) −1.

The quasi closed-form solutions for f

∗

(

,t) takes then the following form,

∗

(

,T)=e

−rT

(G(0)+rT)

Heston

(

) f

Jump

(

), (7.52)

where f

Heston

(

) is the second CF in the Heston model. As given in (3.16),

Heston

(

) has the the following solution,

Heston

(

)=exp



κθ

T)+H

(T)V

+ H

(T)



The detailed formulations for s

(T) and H

(T) can be found in Section 3.2.

Denote

(t)=s

+ H

(t), f

Jump

(

) may be calculated as follows:

Jump

(

)=exp



−i

mT +





(

(t)) −1





. (7.53)

Therefore, the ﬁnal solution of f

∗

(

,T) depends on the concrete form of f

)

that in turn depends on the concrete speciﬁcation of the joint distribution of q

and

. In the jump-diffusion models discussed in the previous sections, q

is speci-

ﬁed concretely while q

is left unspeciﬁed and is equal to nil. Therefore f

)

reduces to f

), and

(t) does not affect the jump transform f

172 7 Poisson Jumps

To incorporate some dependence between q

and q

, Dufﬁe, Pan and Singleton,

for instance, suggested the following nested transform,

[

)], (7.54)

with

)=exp(

1 −z

exp(

)

1 −z

−

These complicated speciﬁcations of jump sizes and their implied correlation may

be simply divided into three parts. The ﬁrst one is a normal (lognormal) jump given

by f

) for stock returns G(t = 1) (or S(t = 1)), the second one is an expo-

nential (Pareto) jump given by f

) for V (t = 1) (or e

V(t)

). The last transform

) determines the marginal joint distribution of the jumps in G(t) and

V(t). Conditional on a realization q

of jumps in V (1), the jump size in G(1) is nor-

mally distributed with a mean of

and a variance of

. With f

(z) having

been speciﬁed, we can integrate the term



(

(t))dt to obtain f

Jump

(

Chapter 8

evy Jumps

evy processes are referred to as a large class of stationary processes with indepen-

dent identical increments. Brownian motion and Poisson process can be regarded as

two special cases of L

evy process, and have only ﬁnite activity in a ﬁnite time inter-

val. In this chapter, we only consider L

evy processes with inﬁnity activity in a ﬁnite

time interval. With respect to jump event modeling in ﬁnance, compound Poisson

jumps discussed in Chapter 7 are appropriate for capturing rare and large jumps such

as market crashes, while L

evy processes with inﬁnity activity may be better suited

to describe many small jumps such as discontinuous information ﬂows and discrete

trading activities. To distinguish Poisson jump dynamics from jump dynamics with

inﬁnity activity, we label the latter jumps as L

evy jumps in this book.

In ﬁrst section, we introduce L

evy process and present the basic concepts of

evy processes, such as the L

evy measure and the L

evy-Khintchine theorem. In

Section 8.2, two popular L

evy models, the variance-gamma model and the normal

inverse Gaussian model, are examined. It can be shown that both L

evy processes

may be constructed by the subordination of a Brownian motion to gamma process

and inverse Gaussian process, respectively, that are called subordinators. A subor-

dinator may be intuitively interpreted as a stochastic time-change. Therefore the

option pricing models with the above L

evy processes are also referred to as stochas-

tic clock models. In Section 8.3, we consequently apply stochastic time-change to

general L

evy processes, even in the presence of a dependence between the stochas-

tic time and the L

evy process itself. To deal with such dependence conveniently, a

complex-valued measure is introduced. The resulting time-changed L

evy processes

are not necessarily L

evy processes, but have much richer spectra of distributions,

and therefore may describe asset dynamics better. In Section 8.4 and 8.5, we brieﬂy

discuss two special L

evy processes: the Barndorff-Nielsen and Shephard model and

the alpha log-stable model. Next, we summarize some empirical test results on vari-

ous L

evy processes and time-changed L

evy processes, their empirical performances

with respect to pricing and hedging are an important aspect for practical applications

of L

evy jump models. Finally in Section 8.7, some algorithms for generating L

evy

random numbers and some methods for simulating L

evy processes are given.

J. Zhu, Applications of Fourier Transform to Smile Modeling, Springer Finance,

DOI 10.1007/978-3-642-01808-4

 Springer-Verlag Berlin Heidelberg 2000, 2010

173

174 8 L

evy Jumps

8.1 Introduction

evy process is a natural extension of a Wiener process and a Poisson process,

and is deﬁned as follows:

Deﬁnition 8.1.1. L

evy process: A L

evy process {L(t),t  0} is a stochastic process

satisfying the following conditions,

1. For 0  t

 t

 ···  t

< ∞,L(t

),L(t

) − L(t

),··· ,L(t

) − L(t

n−1

) are

stochastically independent;

2. Its increments are time-homogenous, it means that L(t + s) −L(t),s > 0 is inde-

pendent of t;

3. For each ﬁxed

∈

,L(

,t) is continuous in t;

4. L(0)=0 almost surely.

By comparing the above deﬁnition of a L

evy process with the ones of a Wiener

process and a Poisson process, we can see that a L

evy process does not explicitly

deﬁne a concrete distribution of its increments. The conditions 1, 3, and 4 are the

same as for a Wiener process or for a Poisson process. Therefore, L

evy process con-

tents Wiener process and Poisson process as special cases and more other processes,

some of which are the focus of this chapter. In fact, as discussed later, Wiener pro-

cess and Poisson process are two important ingredients of L

evy process. So far, we

have discussed how to model ﬁnancial processes by using Wiener processes and

Poisson processes. Hence, applying L

evy processes to valuing ﬁnancial derivatives

is a self-understanding choice.

In more details, a L

evy process is a linear combination of a deterministic drift,

a Wiener process and a jump process (particularly Poisson process) where the de-

terministic drift, the Wiener process can be veriﬁed uniquely. The construction of

jumpsinaL

evy process is more involved and consists of many possible types that

could be characterized by the so-called L

evy measure. In this sense, different L

evy

measures generate different L

evy processes. A standard L

evy process can be ex-

pressed in the following form,

L(t)=

t +

W(t)+



R/{0}

[x −h(x)]

(x)dxds, (8.1)

where

is a deterministic increment, W(t) is a Wiener process and

(x) is a L

evy

measure. Roughly speaking, L

evy measure is a measure on R, and satisﬁes the fol-

lowing two conditions:

({0})=0 (8.2)

and



min(1,x

)

(x)dx < ∞. (8.3)

Note that the deﬁnition of L

evy measure

(x) does not requires



(x)dx < ∞,

an usual requirement for density measure. In fact, many L

evy measures are not

integrable.

8.1 Introduction 175

The function h(x) is an arbitrary truncation function. One of possible forms could

h(x)=x1

|x|<1

The truncation function h(x) ensures that the integral part in L

evy process is inte-

grable. Generally, we do not need a truncation function to deﬁne Poisson process

via the above formulation. By setting h(x)=0, we obtain



R/{0}

(x)dx =



R/{0}

g(x)dx =



R/{0}

xg(x)dx.

Here g(x) and

can be considered as size density function and Poisson intensity

parameter, respectively.



R/{0}

xg(x)dx is the cumulative jumps in a small interval

dt, and therefore, can be rewritten as



R/{0}

xg(x)dx = JdY,

which is the formulation for compound Poisson jump used before. In this simpliﬁed

form, L

evy measure becomes a joint measure for jump size and arrival rate, and

evy process reduces to a jump-diffusion process

dL(t)=

dt +

dW(t)+JdY.

In the previous chapter on Poisson jumps, we have observed that various ran-

dom jump processes are rather moderate in the sense: (1) they do not have inﬁnitely

large jumps and (2) they do not have too “many small changeful” jumps so that the

quadratic variation does not exists. To ensure that a L

evy process is a “reasonable”

process, inﬁnite large jumps and too “many small changeful” jumps should be elim-

inated from L

evy process. Mathematically, we have the following deﬁnitions for

these two irrational cases:

1. “Too large” jumps: L

evy process L(t) is regarded to have “too large” jumps, if



|x|>1

|x|

(x)dx = ∞.

2. Too “many small changeful” jumps: L

evy process L(t) is regarded to have too

“many small changeful” jumps, if



|x|<1

(x)dx = ∞.

It is easy to see in this two cases that a L

evy process does not allow that the large

jumps (|x|> 1) have inﬁnite variation, at the same time, does not allow that the small

jumps (|x| < 1) have inﬁnite quadratic variation. In fact, introducing a truncation

function h(t) inaL

evy process is to eliminate these two irrational cases from L

evy

process, and is equivalent to the following condition,

176 8 L

evy Jumps



R/{0}

∧1)

(x)dx < ∞. (8.4)

This means that a L

evy process must have a ﬁnite quadratic variation.

evy process is considered to have an inﬁnite activity, if



R/{0}

(x)dx = ∞,

otherwise, to have a ﬁnite activity. If we do not take the Wiener process into account,

and



R/{0}

(|x|∧1)

(x)dx = ∞,

evy process exhibits an inﬁnite variation, otherwise, exhibits a ﬁnite variation.

For instance, a Poisson process has a ﬁnite activity and a ﬁnite variation, this is the

reason why we think the Poisson process is a rather moderate L

evy process.

The L

evy-Khintchine theorem describes a L

evy process in terms of characteristic

function. Since the characteristic function is the key to establish the distribution of

evy random variable at a certain time, the L

evy-Khintchine theorem is important

in the context of the option valuation with L

evy process.

Theorem 8.1.2. L

evy-Khintchine theorem: Any L

evy process

L(t)=

t +

W(t)+



R/{0}

[x −h(x)]

(x)dxds

has the following characteristic function

f (

,t)=E[e

L(t)

]=e

(

,t)

, (8.5)

where

(

,t)=i

φμ

t −

t +



R/{0}

−1−i

h(x)]

(x)dx.

The term

(

,t) is called the L

evy exponent.

According to the L

evy-Khintchine theorem, any L

evy process is characterized

uniquely by the triple (

), that is referred to as the L

evy triple. Additionally,

The characteristic function of a L

evy process has the following properties:

1. The characteristic function of a L

evy process comprises the characteristic func-

tion of a Brownian motion with drift and the characteristic function of jump

components, namely, f (

,t)= f

(

) f

Jump

(

) with

(

)=i

φμ

t −

and

Jump

(



R/{0}

−1−i

h(x)]

(x)dx.

8.2 Stochastic Clock Models 177

2. L

evy process is inﬁnitely divisible,

f (

,L,t)= f

(

,L/n,t).

This property implies the self-similarity or the stability property of a L

evy pro-

cess.

There are various option pricing models based on L

evy processes which in turn

are described by various L

evy measures, for instance, the variance-gamma process.

Since a L

evy measure is generally a function, we can actually propose an inﬁnite

number of L

evy processes. For practical applications, only L

evy measures allowing

for an analytical closed-form solution for the L

evy exponent could be of interest.

As long as the characteristic function of a particular L

evy process is analytically

known, we can apply the Fourier inversion to obtain the corresponding distribution,

and therefore also the pricing formula for European-style options. Carr, Geman,

Madan and Yor (2001) discussed various L

evy processes and their applications in

option pricing. Cont and Tankov (2004) provided a detailed mathematical treat-

ment of L

evy process in the connection with option theory. Schoutens, Simons and

Tistaret (2004) compared different L

evy processes and the associated model risks.

Carr and Wu (2004) extended stochastic time L

evy process with non-zero corre-

lation between stochastic time and the underlying L

evy process. Huang and Wu

(2004) examined the speciﬁcation problems with respect to time-changed L

evy pro-

cesses. Wu (2006) presented an excellent overview on the ﬁnancial modeling with

evy process. For more detailed discussions on L

evy process, please see Bertoin

(1996), Sato (1999) and Schoutens (2003).

In most ﬁnancial literature, however, L

evy processes are implicitly referred to the

ones characterized only by inﬁnite activity, even excluding Poisson jumps discussed

perviously. To avoid any confusion in this chapter, we refer to all jumps excluding

Poisson jumps in L

evy process as L

evy jumps in following.

8.2 Stochastic Clock Models

In this section, we discuss the so-called stochastic clock model, a commonplace

for a range of L

evy models that are generated by subordinating a Brownian motion

to another process. A subordinator is a process L(t) such that for all t,t > 0, we

have L(t) > 0. This means that a subordinator just maps a positive time to another

positive time, and virtually serves as a clock changing time stochastically. So any

model underlying this mechanism can be classiﬁed to a stochastic clock model.

The intuition behind the subordination of a Brownian motion with drift, which is

usually applied to model the dynamics of asset returns, is that the stochastic clock

may be considered as a cumulative measure of economic activity. If the clock runs

faster than usually, i.e., L(t) > t, larger variances in a stochastic clock model could

be generated than in the corresponding Gaussian model. In contrast, if the clock runs

slower than usually, i.e., L(t) < t, we could expect smaller variances in a stochas-

178 8 L

evy Jumps

tic clock model. In a similar manner, negative skewness and excess kurtosis could

be also generated by applying a sound subordinator. As discussed in the section on

Poisson jumps, diffusion processes such as Brownian motion do not content large

movements enough so that the fat tail of the distribution of most asset returns can

not be properly recovered. The fact that the generated subordinated processes in

stochastic clock models may be mathematically equivalent to pure jump L

evy pro-

cess with inﬁnite activity, supports the potential capacity of stochastic clock models

in capturing leptokurtic feature of asset returns.

In the Black-Scholes world, the CF of log return x(T) is given by

f (

)=E[e

x(T)

Now we use a subordinator L(T) to randomize T. Denoting



T = L(T) and calculat-

ing f (

) under the Gaussian law leads to

f (

)=E[e

φμ



T−



]=e

(

)



, (8.6)

with

(

)=i

φμ

−

as the exponent of the CF of x(1). Therefore, the ex-

pected value

E[e

(

)



]

may be considered as the moment-generating function of



T with a parameter

(

On the other hand, we assume the moment-generating function of the subordinator

L(T) takes the following form

E[e

uL(T)

]=e

(u)

, (8.7)

where

(u) is the exponent of the moment-generating function of the L

evy sub-

ordinator. By setting u =

(

), we obtain the ﬁnal form of f (

) under the joint

distribution law of the Brownian motion and the L

evy subordinator as follows

f (

)=e

(

))

, (8.8)

which is comprised of two nested moment-generating functions. If both e

(u)

and

(

) have analytical solutions, the CF f (

) of the time-changed Brownian mo-

tion has also an analytical one. This result is especially important if we search for

tractable option pricing formulas of stochastic clock models. The procedure that

subordinating a Brownian motion B(t) to L(t) and deriving the corresponding CF is

called the Bochner procedure.

Note that

(u) is the exponent of a moment-generating function, not of char-

acteristic function of L(t). To keep the notations consistent, we can also express

the moment-generating function f (

) with the exponent of a characteristic function.

Denote the CF of L(T) as

E[e

L(T)

]=e

(

)

here

(u) is then the desired exponent. It follows immediately

8.2 Stochastic Clock Models 179

f (

)=e

(−i

(

))

. (8.9)

In following subsections, we discuss two popular stochastic clock models: the

variance-gamma model and the normal inverse Gaussian model. The implied dis-

tributions in both models are two special cases of the generalized hyperbolic distri-

bution. For more detailed applications of the generalized hyperbolic distribution in

ﬁnance, see Eberlein, Keller and Prause (1998).

8.2.1 Variance-Gamma Model

In this section we consider a popular L

evy jump model called the variance-gamma

model. This model was initialized by Madan and Milne (1991), and developed by

Madan, Carr and Chang (1998). The basic idea of the variance-gamma model is to

randomize the deterministic time length of a Brownian motion by a gamma distribu-

tion. More strictly, the Brownian motion is subordinated to a gamma process. Since

the original Brownian motion has a variance of one in a time unit, the variance of

the changed Brownian motion is then gamma-distributed. Therefore, this model is

referred to as the variance-gamma model.

Denote B(t) as a Brownian motion with drift,

B(t)=B(t;

t +

W(t)

and denote

(t;

) as a gamma process with a mean of

and a variance of

an unit time length. The variance-gamma process is constructed as

VG(t;

)=B(

(t;1,

);

), (8.10)

where it is clear that the time t in the Brownian motion is replaced by

(t;1,

This gamma process has a mean of one, and insures that the time-changed Brownian

motion still has an expected time length of t. Since the gamma distribution is deﬁned

in the positive real axis, the variance-gamma process is well-deﬁned.

In the variance-gamma model, the risk-neutral asset price S(t) is represented as

a martingale-adjusted form,

S(t)=S

exp(rt −mt +VG(t;

)), (8.11)

where m will compensate for the dynamics of VG(t;

) in terms of expectation,

and is equal to

m = lnE[expVG(1;

)].

Deﬁne x(t)=lnS(t), we may express S(t) in a form of stochastic process,

dx(t)=(r −m)dt + dVG(

). (8.12)