Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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590 10 Mixed Processes
Here, (see (10.7.4)),
C
E
(S
0
,T)=
∞
n=0
e
−λT
(λT )
n
n!
e
Γ
n
T
BS (S
0
,r+ Γ
n
,σ,T)
and BS is the Black and Scholes function:
BS (S
0
,θ,σ,T)=S
0
N(d
1
) − Ke
−θT
N(d
2
) ,
Γ
n
= −δ − φλ +
n ln(1 + φ)
T
,
d
1
=
ln(S
0
/K)+(θ +
σ
2
2
)T
σ
√
T
,d
2
= d
1
− σ
√
T.
This approximation was obtained by Bates [59], for the put, in a more
general context in which 1 + φ is a log-normal random variable. This means
that his results could even be used with positive jumps for an American call.
However, as shown in Subsection 10.7.2, in this case the differential equation
whose solution is the American option approximation value, takes a specific
form just below the exercise boundary. Unfortunately, there is no known
solution to this differential equation. Positive jumps generate discontinuities
in the process on the exercise boundary, and therefore the problem is more
difficult to solve (see Chapter 11).
11
L´evy Processes
In this chapter, we present briefly L´evy processes and some of their applica-
tions to finance. L´evy processes provide a class of models with jumps which
is sufficiently rich to reproduce empirical data and allow for some explicit
computations.
In a first part, we are concerned with infinitely divisible laws and in
particular, the stable laws and self-decomposable laws. We give, without
proof, the L´evy-Khintchine representation for the characteristic function of
an infinitely divisible random variable.
In a second part, we study L´evy processes and we present some martingales
in that setting. We present stochastic calculus for L´evy processes and changes
of probability. We study more carefully the case of exponentials and stochastic
exponentials of L´evy processes.
In a third part, we develop briefly the fluctuation theory and we proceed
with the study of L´evy processes without positive jumps and increasing
L´evy processes.
We end the chapter with the introduction of some classes of L´evy processes
used in finance such as the CGMY processes and we give an application of
fluctuation theory to perpetual American option pricing.
The main books on L´evy processes are Bertoin [78], Doney [260], Itˆo
[463], Kyprianou [553], Sato [761], Skorokhod [801, 802] and Zolotarev [878].
Each of these books has a particular emphasis: Bertoin’s has a general
Markov processes flavor, mixing deeply analysis and study of trajectories,
Sato’s concentrates more on the infinite divisibility properties of the one-
dimensional marginals involved and Skorokhod’s describes in a more general
setting processes with independent increments. Kyprianou [553]presentsa
study of global and local path properties and local time. The tutorial papers
of Bertoin [81] and Sato [762] provide a concise introduction to the subject
from a mathematical point of view.
Various applications to finance can be found in the books by Barndorff-
Nielsen and Shephard [55], Boyarchenko and Levendorskii [120], Cont and
Tankov [192], Overhaus et al. [690], Schoutens [766]. Barndorff-Nielsen and
M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial
Markets, Springer Finance, DOI 10.1007/978-1-84628-737-4
11,
c
Springer-Verlag London Limited 2009
591
592 11 L´evy Processes
Shephard deal with simulation of L´evy processes and stochastic volatility,
Cont and Tankov with simulation, estimation, option pricing and integro-
differential equations and Overhaus et al. (the quantitative research team of
Deutsche Bank) with various applications of L´evy processes in quantitative
research, (as equity linked structures and volatility modeling); Schoutens
presents the mathematical tools in a concise manner.
The books [554]and[53] contain many interesting papers with application
to finance. Control theory for L´evy processes can be found in Øksendal and
Sulem [685].
11.1 Infinitely Divisible Random Variables
11.1.1 Definition
In what follows, we denote by x y the scalar product of x and y, two elements
of R
d
,andby|x| the euclidean norm of x.
Definition 11.1.1.1 A random variable X taking values in R
d
with dis-
tribution μ is said to be infinitely divisible if its characteristic function
ˆμ(u)=E(e
iu X
) where u ∈ R
d
, may be written for any integer n as the
n
th
-power of a characteristic function μ
n
, that is if
μ(u)=(μ
n
(u))
n
.
By a slight abuse of language, we shall also say that such a characteristic
function (or distribution function) is infinitely divisible. Equivalently, X is
infinitely divisible if
∀n, ∃(X
(n)
i
,i≤ n, i.i.d.) such that X
law
= X
(n)
1
+ X
(n)
2
+ ···+ X
(n)
n
.
Example 11.1.1.2 A Gaussian variable, a Cauchy variable, a Poisson vari-
able and the hitting time of the level a for a Brownian motion are examples
of infinitely divisible random variables. Gamma, Inverse Gaussian, Normal
Inverse Gaussian and Variance Gamma variables are also infinitely divisible
(see Examples 11.1.1.9 for details and Appendix A.4.4 and A.4.5 for
definitions of these laws). A uniformly distributed random variable, and more
generally any bounded random variable is not infinitely divisible. The next
Proposition 11.1.1.4 will play a crucial rˆole in the description of infinitely
divisible laws.
Definition 11.1.1.3 A L´evy measure on R
d
is a positive measure ν on
R
d
\{0} such that
R
d
\{0}
(1 ∧|x|
2
) ν(dx) < ∞,
i.e.,
11.1 Infinitely Divisible Random Variables 593
|x|>1
ν(dx) < ∞ and
0<|x|≤1
|x|
2
ν(dx) < ∞.
In Feller [343], Lukacs [605], Sato [761] and Zolotarev [878], the reader will
find more properties of infinitely divisible random variables, as well as the
proofs of the following results.
Proposition 11.1.1.4 (L´evy-Khintchine Representation.) If X is an
infinitely divisible random variable with characteristic function μ, there exists
auniquetriple(m, A, ν) where m ∈ R
d
, A is a positive matrix and ν is a
L´evy measure such that
μ(u) = exp
iu m −
1
2
u Au +
R
d
(e
iu x
− 1 − iu x1
{|x|≤1}
)ν(dx)
(11.1.1)
Sketch of the Proof: The L´evy-Khintchine representation formula may
be obtained by proving that an infinitely divisible random variable is the limit
in law of random variables of the form
Z
(n)
+
N
(n)
1
k=1
Y
(n)
k
where Z
(n)
is a Gaussian r.v. and
N
(n)
1
k=1
Y
(n)
k
a compound Poisson process
(see Section 8.6) evaluated at time t = 1. The characteristic function for a
compound Poisson process is given in Proposition 8.6.3.4. See Bertoin [78]or
Sato [761] for a complete proof of this representation formula.
When
R
d
\{0}
1
{|x|≤1}
|x|ν(dx) < ∞, we can write (11.1.1)inthereduced
form
μ(u) = exp
iu m
0
−
1
2
u Au +
R
d
(e
iu x
− 1)ν(dx)
, (11.1.2)
where m
0
= m −
R
d
x1
{|x|≤1}
ν(dx). We shall use this reduced form
representation whenever possible.
When
R
d
1
{|x|>1}
|x|ν(dx) < ∞,itispossibletowrite(11.1.1)intheform
μ(u) = exp
iu ˜m −
1
2
u Au +
R
d
(e
iu x
− 1 − iu x)ν(dx)
,
where ˜m = m +
|x|>1
xν(dx). In that case, by differentiation of the Fourier
transform μ(u) with respect to u,
594 11 L´evy Processes
E(X)=−iμ
(0) = ˜m = m +
|x|>1
xν(dx) .
If the r.v. X is positive, the L´evy measure ν is a measure on ]0, ∞[ with
]0,∞[
(1 ∧ x)ν(dx) < ∞ (see Proposition 11.2.3.11). It is more natural, in
thiscase,toconsider,forλ>0, E(e
−λX
) the Laplace transform of X,and
now, the L´evy-Khintchine representation takes the form
E(e
−λX
) = exp
−
λm
0
+
]0,∞[
ν(dx)(1 − e
−λx
)
.
Remark 11.1.1.5 The following converse of the L´evy-Khintchine represen-
tation holds true: any function ϑ of the form
ϑ(u) = exp
iu m − u Au +
R
d
(e
iu x
− 1 − iu x1
{|x|≤1}
)ν(dx)
where ν is a L´evy measure and A a positive matrix, is a characteristic function
which is obviously infinitely divisible (take m
n
= m/n, A
n
= A/n, ν
n
= ν/n).
Hence, there exists μ, infinitely divisible, such that ϑ = μ.
Warning 11.1.1.6 Some authors use a slightly different representation for
the L´evy-Khintchine formula. They define a centering function (also called
a truncation function) as an R
d
-valued measurable bounded function h such
that (h(x) − x)/|x|
2
is bounded. Then they prove that, if X is an infinitely
divisible random variable, there exists a triple (m
h
,A,ν), where ν is a
L´evy measure, such that
μ(u) = exp
iu m
h
−
1
2
u Au +
R
d
(e
iu x
− 1 − iu h(x))ν(dx)
.
Common choices of centering functions on R are h(x)=x1
{|x|≤1}
,asin
Proposition 11.1.1.4 or h(x)=
x
1+x
2
(Kolmogorov centering). The triple
(m
h
,A,ν) is called a characteristic triple. The parameters A and ν do not
depend on h;whenh is the centering function of Proposition 11.1.1.4,wedo
not indicate the dependence on h for the parameter m.
Remark 11.1.1.7 The choice of the level 1 in the common centering function
h(x)=1
{|x|≤1}
is not essential and, up to a change of the constant m,any
centering function h
r
(x)=1
{|x|≤r}
can be considered.
Comment 11.1.1.8 In the one-dimensional case, when the law of X admits
a second order moment, the L´evy-Khintchine representation was obtained by
Kolmogorov [536]. Kolmogorov’s measure ν corresponds to the representation
exp
ium +
R
e
iux
− 1 − iux
x
2
ν(dx)
.
11.1 Infinitely Divisible Random Variables 595
Hence, ν(dx)=1
{x=0}
ν(dx)/x
2
and the mass of ν at 0 corresponds to the
Gaussian term.
Example 11.1.1.9 We present here some examples of infinitely divisible laws
and give their characteristics in reduced form whenever possible (see equation
(11.1.2)).
• Gaussian Laws. The Gaussian law N(a, σ
2
) has characteristic function
exp(iua − u
2
σ
2
/2). Its characteristic triple in reduced form is (a, σ
2
, 0).
• Cauchy Laws. The Cauchy law with parameter c>0 has the character-
istic function
exp(−c|u|) = exp
c
π
∞
−∞
(e
iux
− 1)x
−2
dx
.
Here, we make the convention
∞
−∞
(e
iux
− 1)x
−2
dx = lim
→0
∞
−∞
(e
iux
− 1)x
−2
1
{|x|≥}
dx .
The reduced form of the characteristic triple for a Cauchy law is
(0, 0,cπ
−1
x
−2
dx) .
• Gamma Laws. If X follows a Γ (a, ν) law, its Laplace transform is, for
λ>0,
E(e
−λX
)=
1+
λ
ν
−a
=exp
−a
∞
0
(1 − e
−λx
)e
−νx
dx
x
.
Hence, the reduced form of the characteristic triple for a Gamma law is
(0, 0, 1
{x>0}
ax
−1
e
−νx
dx).
• Brownian Hitting Times. The first hitting time of a>0 for a Brownian
motion has characteristic triple (in reduced form)
0, 0,
a
√
2π
x
−3/2
1
{x>0}
dx
.
Indeed, we have seen in Proposition 3.1.6.1 that E(e
−λT
a
)=e
−a
√
2λ
.
Moreover, from Appendix A.5.8
√
2λ =
1
√
2Γ (1/2)
∞
0
(1 − e
−λx
)x
−3/2
dx ,
hence, using that Γ (1/2) =
√
π
E(e
−λT
a
) = exp
−
a
√
2π
∞
0
(1 − e
−λx
) x
−3/2
dx
.
596 11 L´evy Processes
• Inverse Gaussian Laws. The Inverse Gaussian laws (see Appendix
A.4.4) have characteristic triple (in reduced form)
0, 0,
a
√
2πx
3
exp
−
1
2
ν
2
x
1
{x>0}
dx
.
Indeed
exp
−
a
√
2π
∞
0
dx
x
3/2
(1 − e
−λx
) e
−ν
2
x/2
=exp
−
a
√
2π
∞
0
dx
x
3/2
(e
−ν
2
x/2
− 1) + (1 − e
−(λ+ν
2
/2)x
)
=exp(−a(−ν +
ν
2
+2λ)
is the Laplace transform of the first hitting time of a for a Brownian motion
with drift ν.
11.1.2 Self-decomposable Random Variables
We now focus on a particular class of infinitely divisible laws.
Definition 11.1.2.1 A random variable is self-decomposable (or of class
L)if
∀c ∈]0, 1[, ∀u ∈ R, μ(u)=μ(cu)μ
c
(u) ,
where μ
c
is a characteristic function.
In other words, X is self-decomposable if
for 0 <c<1, ∃X
c
such that X
law
= cX + X
c
where on the right-hand side the r.v’s X and X
c
are independent. Intuitively,
we “compare” X with its multiple cX, and need to add a “residual” variable
X
c
to recover X.
We recall some properties of self-decomposable variables (see Sato [761]for
proofs). Self-decomposable variables are infinitely divisible. The L´evy measure
of a self-decomposable r.v. is of the form
ν(dx)=
h(x)
|x|
dx
where h is increasing for x>0 and decreasing for x<0.
Proposition 11.1.2.2 (Sato’s Theorem.) If the r.v. X is self-decomposa-
ble, then X
law
= Z
1
where the process Z satisfies Z
ct
law
= cZ
t
and has
independent increments.
11.1 Infinitely Divisible Random Variables 597
Processes with independent increments are called additive processes in
Sato [761], Chapter 2. Self-decomposable random variables are linked with
L´evy processes in a number of ways, in particular the following Jurek-Vervaat
representation [499] of self-decomposable variables X, which, for simplicity,
we assume to take values in R
+
. For the definitions of L´evy processes and
subordinators, see Subsection 11.2.1.
Proposition 11.1.2.3 (Jurek-Vervaat Representation.) A random vari-
able X ≥ 0 is self-decomposable if and only if it satisfies X
law
=
∞
0
e
−s
dY
s
,
where (Y
s
,s ≥ 0) denotes a subordinator, called the background driving
L´evy process (BDLP) of X.
The Laplace transforms of the random variables X and Y
1
are related by
the following
E(exp(−λY
1
)) = exp
λ
d
dλ
ln E(exp(−λX))
.
Example 11.1.2.4 We present some examples of self-decomposable random
variables:
• First Hitting Times for Brownian Motion. Let W be a Brownian
motion. The random variable T
r
=inf{t : W
t
= r} is self-decomposable.
Indeed, for 0 <λ<1,
T
r
= T
λr
+(T
r
− T
λr
)
= λ
2
T
r
+(T
r
− T
λr
)
where
T
r
=
1
λ
2
T
λr
law
= T
r
,and
T
r
and (T
r
− T
λr
) are independent, as a
consequence of both the scaling property of the process (T
a
,a ≥ 0) and
the strong Markov property of the Brownian motion process at T
λr
(see
Subsection 3.1.2).
• Last Passage Times for Transient Bessel Processes. Let R be a
transient Bessel process (with index ν>0) and
Λ
r
=sup{t : R
t
= r}.
The random variable Λ
r
is self-decomposable. To prove the self-decompo-
sability, we use an argument similar to the previous one, i.e., independence
for pre- and post-Λ
r
processes, although Λ
r
is not a stopping time.
Comment 11.1.2.5 See Sato [760], Jeanblanc et al. [484] and Shanbhag and
Sreehari [783] for more information on self-decomposable r.v’s and BDLP’s.
In Madan and Yor [613] the self-decomposability property is used to construct
martingales with given marginals. In Carr et al. [152] the risk-neutral process
is modeled by a self-decomposable process.
598 11 L´evy Processes
11.1.3 Stable Random Variables
Definition 11.1.3.1 A real-valued r. v. is stable if for any a>0, there exist
b>0 and c ∈ R such that [μ(u)]
a
= μ(bu) e
icu
. A random variable is strictly
stable if for any a>0, there exists b>0 such that [μ(u)]
a
= μ(bu) .
In terms of r.v’s, X is stable if
∀n, ∃(β
n
,γ
n
), such that X
(n)
1
+ ···+ X
(n)
n
law
= β
n
X + γ
n
where (X
(n)
i
,i≤ n) are i.i.d. random variables with the same law as X.
For a strictly stable r.v., it can be proved, with the notation of the
definition, that b (which depends on a)isoftheformb(a)=ka
1/α
with
0 <α≤ 2. The r.v. X is then said to be α-stable.For0<α<2, an α-stable
random variable satisfies E(|X|
γ
) < ∞ if and only if γ<α. The second order
moment exists if and only if α = 2, and in that case X is a Gaussian random
variable, hence has all moments.
A stable random variable is self-decomposable and hence is infinitely
divisible.
Proposition 11.1.3.2 The characteristic function μ of an α-stable law can
be written
μ(u)=
⎧
⎨
⎩
exp(imu −
1
2
σ
2
u
2
), for α =2
exp (imu − γ|u|
α
[1 − iβ sgn(u)tan(πα/2)]) , for α =1, =2
exp (imu − γ|u|(1 + iβ ln |u|)) ,α=1
where β ∈ [−1, 1] and m, γ, σ ∈ R.Forα =2, the L´evy measure of an α-stable
law is absolutely continuous with respect to the Lebesgue measure, with density
ν(dx)=
c
+
x
−α−1
dx if x>0
c
−
|x|
−α−1
dx if x<0 .
(11.1.3)
Here, c
±
are positive real numbers given by
c
+
=
1
2
(1 + β)
αγ
Γ (1 − α)cos(απ/2)
,
c
−
=
1
2
(1 − β)
αγ
Γ (1 − α)cos(απ/2)
.
Conversely, if ν is a L´evy measure of the form (11.1.3), we obtain the
characteristic function of the law on setting β =(c
+
− c
−
)/(c
+
+ c
−
) .
11.2 L´evy Processes 599
For α = 1, the definition of c
±
can be given by passing to the limit: c
±
=1±β.
For β =0,m=0,X is said to have a symmetric stable law. In that case
μ(u)=exp(−γ|u|
α
) .
Example 11.1.3.3 A Gaussian variable is α-stable with α = 2. The Cauchy
law is stable with α = 1. The hitting time T
1
=inf{t : W
t
=1} where W is
a Brownian motion is a (1/2)-stable variable.
Comment 11.1.3.4 The reader can refer to Bondesson [108] for a particular
class of infinitely divisible random variables and to Samorodnitsky and Taqqu
[756] and Zolotarev [878] for an extensive study of stable laws and stable
processes. See also L´evy-V´ehel and Walter [586] for applications to finance.
11.2 L´evy Processes
11.2.1 Definition and Main Properties
Definition 11.2.1.1 Let (Ω, F, P) be a probability space. An R
d
-valued
process X such that X
0
=0is a L´evy process if
(a) for every s, t ≥ 0 ,X
t+s
− X
s
is independent of F
X
s
,
(b) for every s, t ≥ 0 the r.v’s X
t+s
− X
s
and X
t
have the same law,
(c) X is continuous in probability, i.e., for fixed t, P(|X
t
− X
u
| >) → 0
when u → t for every >0.
It can be shown that up to a modification, a L´evy process is c`adl`ag (it is in
fact a semi-martingale, see Corollary 11.2.3.8).
Example 11.2.1.2 Brownian motion, Poisson processes and compound Pois-
son processes are examples of L´evy processes (see Section 8.6). If X is a
L´evy process, C amatrixandD a vector, CX
t
+ Dt is also a L´evy process.
More generally, the sum of two independent L´evy processes is a L´evy process.
One can easily generalize this definition to F-L´evy processes, where F is
a given filtration, by changing (a) into:
(a’) for every s, t, X
t+s
− X
s
is independent of F
s
.
Another generalization consists of the class of additive processes that satisfy
(a) and often (c), but not (b). Natural examples of additive processes are
the processes (T
a
,a ≥ 0) of first hitting times of levels by a diffusion Y ,a
consequence of the strong Markov property.
A particular class of L´evy processes is that of subordinators: