CGMY and Meixner Subordinators are Absolutely Continuous with respect to One Sided Stable Subordinators

0.22 MB·English

by  Dilip Madan

CGMY and Meixner Subordinators are
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0 Absolutely Continuous with respect to One Sided
0
2
Stable Subordinators.

n
a
J
Dilip B. Madan

2
Robert H. Smith School of Business
1
Van Munching Hall

]
University of Maryland
R
P College Park, MD 20742
.
h Marc Yor
t
a Laboratoire de probabilit´es et Modeles al´eatoires
m
Universit´e Pierre et Marie Curie
[ 4, Place Jussieu F 75252 Paris Cedex

2
v August 16 2005
3
7
1
Abstract
1
0
WedescribetheCGMYandMeixnerprocessesastimechangedBrow-
6
nianmotions. TheCGMYusesatimechangeabsolutelycontinuouswith
0
respecttotheone-sidedstable(Y/2)subordinatorwhiletheMeixnertime
/
h changeisabsolutelycontinuouswithrespecttotheonesidedstable(1/2)
t subordinator.Therequiredtimechangesmaybegeneratedbysimulating
a
m the requisite one-sided stable subordinator and throwing away some of
the jumpsas described in Rosinski (2001).
:
v
i
X 1 Introduction
r
a
L´evy processes are increasingly being used to model the local motion of asset
returns,permitting theuseofdistributionsthatarebothskewedandcapableof
matching the high levels of kurtosis observed in factors driving equity returns.
By way of examples we cite the normal inverse Gaussian process (Barndorff-
Nielsen (1998)), the hyperbolic process (Eberlein, Keller and Prause (1998)),
and the variance gamma process (Madan, Carr and Chang (1998)). For the
valuation of structured equity products the importance of skewness is well rec-
ognized and has led to the development of local L´evy processes (See Carr, Ge-
man, Madan and Yor (2004)) that preserve skews in forward implied volatility
curves. Itisalsounderstoodfromthesteepnessofimpliedvolatilitycurvesthat
1
tail events have significantly higher prices than those implied by a Gaussian
distribution with the consequence that pricing distributions display high levels
of excess kurtosis.
Onasingleassetonemaysimulatethe L´evyprocesscalibratedtothe prices
of vanilla options to value equity structured products written on a single un-
derlier. Such a simulation (See Rosinski (2001)) may approximate the small
jumps using a diffusion process with the large jumps simulated as a compound
Poisson process where one uses the normalized large jump L´evy measure as
the density of jump magnitudes with the integral of the L´evymeasure over the
large jumps serving as the jump arrival rate. However, increasingly one sees
multiassetstructuresbeingtradedandthisrequiresamodelingofassetcorrela-
tions. Given marginal L´evy processes one could accomodate correlations if one
can represent the L´evy process as time changed Brownian motion. In this case
wecorrelatethe simulatedprocessesbycorrelatingthe Brownianmotionswhile
preserving the independent time changes for each of the marginal underliers.
It is therefore useful to have representations of L´evy processes as time
changed Brownianmotions. For some L´evy processes, like the variance gamma
processorthenormalinverseGaussianprocess,theseareknownbyconstruction
of the L´evy process via such a representation. For other L´evy processes, like
the CGMY process (Carr, Geman, Madan and Yor (2002), see also Koponen
(1995), Boyarchenko and Levendorskii (1999, 2000)) or the Meixner process
(Schoutens and Teugels (1998) see also Gregelionis (1999), Schoutens (2000),
and Pitman and Yor (2003)), the process is defined directly by its L´evy mea-
sure and it is not clear a priori whether the processes can be represented as
time changed Brownian motions. With a view to enhancing the applicability
of these processes, particularly with respect to multiasset structured products,
we develop the representations of these processes as time changed Brownian
motions.
Section 2 presents for completeness, some preliminary results on L´evy pro-
cesses that we employ in the subsequent development. In section 3 we develop
the CGMY process as a time changed Brownian motion with drift, where the
law of the time change is absolutely continuous over finite time intervals with
respect to that of the one sided stable Y/2 subordinator. The simulation of
CGMY as time changed Brownian motion is described in section 3. Section 4
developsthe time changefor the Meixnerprocessasabsolutelycontinuouswith
respect to the one-sided stable 1/2 subordinator. Simulation strategies for the
Meixnerprocessbasedonthese representationsaredescribedin Section5. Sec-
tion 6 reports on the simulation results using chi-squared goodness of fit tests.
Section 7 concludes.
2 Preliminary results on L´evy processes
WepresentthreeresultsfromthetheoryofL´evyprocessesthatwemakecritical
useofinoursubsequentdevelopment. ThefirstresultrelatestheL´evymeasure
ofaprocessobtainedonsubordinatingaBrownianmotiontothe L´evymeasure
2
of the subordinator. The second result establishes a criterion for the absolute
continuity of a subordinator with respect to another subordinator. The third
result presents the detailed relationship between the standard presentation of
the characteristicfunction ofa two sided jump and one-sidedjump stable L´evy
process and its L´evy measure. These are presented in three short subsections.
2.1 L´evy measure of a subordinated Brownian motion
Suppose the L´evy process X(t) is obtained by subordinating Brownian motion
with drift (i.e. the process θu+W(u), for (W(u),u 0) a Brownian motion)
≥
by an independent subordinator Y(t) with L´evy measure ν(dy). Then applying
Sato (1999) theorem 30.1 we get that the L´evy measure of the process X(t) is
given by µ(dx) where
∞ 1 (x−θy)2
µ(dx)=dx ν(dy) e− 2y . (1)
√2πy
Z0
2.2 Absolute Continuity Criterion for subordinators
Suppose we have two subordinators T = (T (t),t 0),T = (T (t),t 0).
A A B B
≥ ≥
The law of the subordinator T is absolutely continuous with respect to the
A
subordinatorT ,onfinitetimeintervals,justifthereexistsafunctionf(t)such
B
thattheL´evymeasuresν (dt),ν (dt)fortheprocessesT andT respectively
A B A B
are related by
ν (dt)=f(t)ν (dt) (2)
A B
and furthermore, (Sato (1999) Theorem 33.1)
∞ 2
ν (dt) f(t) 1 < . (3)
B
− ∞
Z0 (cid:16)p (cid:17)
2.3 Stable Processes
The Stable L´evy process (σ,α,β) = (X(t),t 0) with parameters (σ,α,β)
S ≥
( For details see DuMouchel (1973, 1975), Bertoin (1996), Samorodnitsky and
Taqqu(1998)Nolan(2001),Ito(2004))hasacharacteristicfunctioninstandard
form
E[eiuX(t)]=exp( tΨ(u))
−
where the characteristic exponent Ψ(u) is given by
πα
Ψ(u) = σα uα 1 iβsign(u)tan , α=1 (4)
| | − 2 6
(cid:16) 2 (cid:16) (cid:17)(cid:17)
= σ u 1+iβsign(u) log(u) , α=1.
| | π | |
(cid:18) (cid:19)
The parameters satisfy the restrictions, σ >0,0<α 2 and 1 β 1. The
≤ − ≤ ≤
one sided jump stable processes result when β = 1 and there are only positive
jumps or β = 1 in which case there are only negative jumps.
−
3
The L´evy density of the stable process is of the form
c c
p n
k(x)= 1 + 1 (5)
x1+α x>0 x1+α x<0
| |
and we have that
c c
p n
β = − . (6)
c +c
p n
It remains to express σ in terms of the parameters of the L´evy measure. In
the one sided case with only positive jumps we have
1
c Γ α Γ 1 α α
σ = p 2 − 2 (7)
2Γ(1+α)
" (cid:0) (cid:1) (cid:0) (cid:1)#
and more generally for the two sided jump case we have
1
c +c Γ α Γ 1 α α
σ = p n 2 − 2 . (8)
2 Γ(1+α)
" (cid:0) (cid:1) (cid:0) (cid:1)#
Conversely, c and c may be computed in terms of β and σ.
p n
3 CGMY as time changed Brownian motion
We wish to write the CGMY process in the form
X (t)=θY(t)+W(Y(t))
CGMY
for an increasing time change process given by a subordinator (Y(t),t 0)
≥
independent of the Brownian motion (W(s),s 0) .
≥
The characteristic function of the CGMY process is
(M iu)Y MY+
E[exp(iuXCGMY(t))]=(φCGMY(u))t =exp tCΓ(−Y)" (G−+iu)Y− GY #!
−
Thecomplexexponentiationisdefinedviathecomplexlogarithmwithabranch
cut on the negative real axis with polar coordinate arguments for the complex
logarithm restricted to the interval ] π,+π]. The CGMY process is defined
−
as a pure jump L´evy process by its L´evy measure
exp( Gx) exp( Mx)
k (x)=C − | | 1 + − 1 .
CGMY x1+Y x<0 x1+Y x>0
(cid:20) | | (cid:21)
On the other hand we have, in all generality, by conditioning on the time
change that
Y(t)
E eiu(θY(t)+W(Y(t)) = E exp iuθY(t) u2
− 2
h i (cid:20) (cid:18) (cid:19)(cid:21)
u2
= E exp iuθ Y(t)
− 2 −
(cid:20) (cid:18) (cid:18) (cid:19) (cid:19)(cid:21)
4
Take u(λ) to be any solution of
u2
λ= iuθ ;
2 −
(cid:18) (cid:19)
Then we have the Laplace transform of the time change subordinator as
E[e−λY(t)]=exp tCΓ( Y) (M iu(λ))Y MY +(G+iu(λ))Y GY
− − − −
(cid:16) h i(cid:17)
The solutions for u are:
u=iθ 2λ θ2
± −
p
where we suppose that θ2 <2λ.
We shall see that a good choice for θ , for sufficiently large λ, is
G M
θ = −
2
and in this case
2
G+M G M
M iu = +i 2λ −
− 2 s − 2
(cid:18) (cid:19)
G+M G M 2
G+iu = i 2λ − .
2 − s − 2
(cid:18) (cid:19)
It follows that the Laplace transform of the subordinator is
E[e λY(t)] = exp tCΓ( Y) 2rY cos(ηY) MY GY
−
− − −
r = √2λ(cid:0)+GM (cid:2) (cid:3)(cid:1)
2λ G M 2
η = arctan − −2
q G+M 
(cid:0) (cid:1)
2
 (cid:0) (cid:1) 
In the special case of G=M we have
E[e−λY(t)]=exp 2tCΓ( Y) 2λ+M2 Y/2cos Y arctan √2λ MY
− " M !!− #!
(cid:0) (cid:1)
3.1 The explicit time change for CGMY
We shall show that the time change subordinator Y(t) associated with the
CGMY process is absolutely continuous with respect to the one-sided stable
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Y/2subordinator andinparticularthatitsL´evymeasureν(dy) takesthe form
K
ν(dy) = f(y)dy
y1+Y2
(B2−A2)y B2yγY/2
f(y) = e− 2 E e− 2 γ1/2 (9)
(cid:20) (cid:21)
G+M
B =
2
CΓ Y Γ 1 Y
K = 4 − 4
" (cid:0)2Γ((cid:1)1+(cid:0)Y2) (cid:1)#
where γ ,γ are two independent gamma variates with unit scale parameters
Y 1
2 2
and shape parameters Y/2,1/2 respectively. Further we explicitly evaluate the
expectation in equation (9) in terms of the Hermite functions as follows.
B2yγY/2 Γ Y + 1 B2y Y2 B2y
E e− 2 γ1/2 = 2 2 2Y I Y,B2y,
Γ(Y)Γ(1) 2 2
(cid:20) (cid:21) (cid:0) 2(cid:1) (cid:18) (cid:19) (cid:18) (cid:19)
where
I(ν,a,λ)= ∞xν 1e ax λx2dx=(2λ) ν/2Γ(ν)h a
− − − − ν
Z0 − (cid:18)√2λ(cid:19)
andh (z)istheHermitefunctionwithparameter ν (seee.gLebedev(1972),
ν
− −
p 290-291).
3.2 Determining the time change for CGMY
For an explicit evaluation of the time change we begin by writing the CGMY
L´evy density in the form
eAx Bx G M G+M
− | |
k (x)=C , where: A= − ; B =
CGMY x1+Y 2 2
Henceforth, when we encounter a L´evy measure µ(dx) that is absolutely
continuous with respect to Lebesgue measure we shall denote its density by
µ(x). We now employ the result (1) and seek to find a L´evy measure of a
subordinator satisfying
eAx−B|x| ∞ 1 (x−θy)2
C = ν(dy) e− 2y
x1+Y √2πy
| | Z0
= ∞ν(dy) 1 e−x2y2−θ22y+θx
√2πy
Z0
We set θ = A and observe that the right choice for θ is (G M)/2 as
−
remarked earlier, and identify ν(dy) such that
e−B|x| ∞ 1 x2 θ2y
C = ν(dy) e−2y− 2 (10)
x1+Y √2πy
| | Z0
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We now recognize that the L´evy measure for the CGMY is (taking C =
Γ(Y)Γ(1 Y)
2 −2 , now), that of the symmetric stable Y L´evy process with L´evy
Γ(1+Y)
2
measure tilted as
kCGMY(x)=eAx−B|x|kStable(Y)(x).
We also know that
X (t)=B
Stable(Y) Y0(t)
whereY0(t)isthe onesidedstableY/2subordinator,independentofthe Brow-
nian motion (B ) .
u
We now write
X (t)=θY(1)(t)+W
CGMY Y(1)(t)
andwe seek to relatethe L´evymeasuresν(1) and ν(0) of the processesY(1) and
Y(0).
From the result (1) we may write
x2
µ (x) = ∞ν(0)(dy)e−2y
0 √2πy
Z0
(x−θy)2
µ (x) = ∞ν(1)(dy)e− 2y
1 √2πy
Z0
Hence we must have that
(x−θy)2 x2
∞ν(1)(dy)e− 2y =eAx−B|x| ∞ν(0)(dy)e−2y
√y √y
Z0 Z0
Taking θ =A, we get:
x2 A2y x2
∞ν(1)(dy)e−2y− 2 =e Bx ∞ν(0)(dy)e−2y
− | |
√y √y
Z0 Z0
We now use the well known fact that
e−B|x| = ∞du B e−B2u2−x22u
Z0 √2πu3
to write
∞ν(1)(dy)e−x2y2−A22y = ∞du B e−B2u2 ∞ν(0)(dy)e−x22(y1+u)
Z0 √y Z0 √2πu3 Z0 √y
ByuniquenessofLaplacetransformswegetthatforeveryfunctionf :R+
R+ →
A2y
∞ν(1)(dy)e− 2 f 1 = ∞du B e−B2u2 ∞ν(0)(dy) 1 f 1 +u
Z0 √y (cid:18)y(cid:19) Z0 √2πu3 Z0 √y (cid:18)y (cid:19)
7
or equivalently that, for every function g :R R
+ +
→
A2y
∞ν(1)(dy)e− 2 g(y) = ∞du B e−B2u2 ∞ν(0)(dy) 1 g y
Z0 √y Z0 √2πu3 Z0 √y (cid:18)1+uy(cid:19)
= ∞du B e−B2u2 u1 d s ν(0)(1−sus)g(s)
Z0 √2πu3 Z0 (cid:18)1−us(cid:19) 1 sus
−
= ∞du B e−B2u2 u1 ds ν(0)(q1−sus)g(s)
Z0 √2πu3 Z0 (1−su)2 1 sus
−
q
Hence it is the case that
ν(1)(y)e−A22y = y1 duBe−B2u2ν(0)(1−yuy)
2π(u(1 uy))3
Z0 −
= √y 1pdvBe−B22vyν(0)(1−yv)
2π(v(1 v))3
Z0 −
In particular we have p
1 dvBe−y2 Bv2−A2 ν(0)( y )
ν(1)(y)=√y (cid:16) (cid:17) 1−v
2π(v(1 v))3
Z0 −
p
Wenowintroducetheexplicitformofν (y)forourcasewhereitistheL´evy
0
density of the one-sided stable Y/2 subordinator,
K
ν (y)= .
0 y(Y2+1)
This gives the representation
K 1 dvBe−y2 Bv2−A2 (1 v)(Y2+1)
ν1(y) = yY2+1 Z0 (cid:16)2π(v(1(cid:17)−v−))3
K ∞ dw Bpe−y2(B2w−A2) 1 (Y2+1)
= 1
yY2+1 Z1 w2 2π(1(1 1))3 (cid:18) − w(cid:19)
w − w
q Y−1
= K ∞ dw Be−y2(B2w−A2) w−1 2
yY2+1 Z1 √2πw (cid:18) w (cid:19)
KBe−y2(B2−A2) ∞ dh yB2h hY2−1
= yY2+1 Z0 √2πe− 2 (1+h)Y2
8
3.2.1 Absolute Continuity relations
This subsection investigatesthe absolute continuity relationin generalbetween
two subordinated processes and the absolute continuity of the subordinators
as processes. It is easy to show that the laws of the CGMY process and the
symmetric stable Y process are locally equivalent, i.e. for each t,their laws, as
restricted to their past σ fields up to time t, are equivalent (from now
t
− F
on, as a slight abuse of language, we shall say of 2 such processes, that they
are equivalent). Now that we have identified these processes as subordinated
processes,welookfortheequivalenceinlawofthesubordinators. Indeedwefirst
observethatifthesubordinatorsareequivalentthenthesubordinatedprocesses
will be equivalent but the converse may not be true.
Indeed, consider two subordinators
T (t), T (t)
A B
such that the relation (2) between their L´evy measures holds for some function
f(t) for t>0.
WesupposetheabsolutecontinuityofT withrespecttoT orthecondition
A B
(3).
We also define the subordinated processes
X (t) = β
A TA(t)
X (t) = β
B TB(t)
where (β ) is a Brownian motion assumed to be independent of either T or
u A
T .
B
We have from the result (1) that at the level of L´evy measures µ ,µ for
A B
X ,X
A B
∞ e−x22t
µ (x) = ν (dt)
A A √2πt
Z0
∞ e−x22t
µ (x) = ν (dt)
B B √2πt
Z0
Thefollowingthenholdsasaconsequenceof(3),foreveryfunctionalF 0:
≥
E[F (T (s),s t)]=E[F(T (s),s t)φ(T (s),s t)]
A B B
≤ ≤ ≤
where
dP
φ(T (s),s t)= TA
B
≤ dP
(cid:18) TB(cid:19)t
As a consequence we deduce that, for every G 0:
≥
E[G(X (s),s t)]=E[G(X (s),s t)φ(T (s),s t)]
A B B
≤ ≤ ≤
Consequently we may write
E[G(X (s),s t)]=E[G(X (s),s t)ψ(X (s),s t)]
A B B
≤ ≤ ≤
9
where
ψ(X (s),s t)=E[φ(T (s),s t)(X (s),s t)]
B B B
≤ ≤ | ≤
This implies that we should have
µ (dx)=g(x)µ (dx)
A B
with
∞ 2
g(x) 1 µ (dx)< (11)
− B ∞
Z−∞(cid:16)p (cid:17)
We want to show that (3) implies (11).
Now we have explicitly that
−x2
ν (dt)e 2t
g(x) = A √t
R ν (dt)e−x22t
B √t
R −x2
ν (dt)f(t)e 2t
= B √t
R ν (dt)e−x22t
B √t
R
Let
−x2
ν (dt)e 2t
γ(x)(dt)= B √t
−x2
ν (dt)e 2t
B √t
and note that R
g(x)= γ(x)(dt)f(t)
Z
We then have
1
2
g(x) 1= γ(x)(dt)f(t) 1
− −
(cid:18)Z (cid:19)
p
and
( g(x) 1)2µ (dx)= ( g(x) 1)2 ν (dt)e−x22t dx
− B − B √2πt !
Z Z Z
p p
10