In this paper we examine which Brownian Subordination with drift exhibits the symmetry property introduced by Fajardo and Mordecki (2006b). We obtain that when the subordination results in a Lévy process, a necessary and sufficient condition for the symmetry to hold is that drift must be equal to -1/2.Time Changed, Subordination, Symmetry

Ernesto Mordecki

José Fajardo

English

Research Papers in Economics

 
Economics Research Group 
IBMEC Business School – Rio de Janeiro 
http://professores.ibmecrj.br/erg 
 
 
 
 
IBMEC RJ ECONOMICS DISCUSSION PAPER 2008-02 
Full series available at http://professores.ibmecrj.br/erg/dp/dp.htm. 
 
 
 
 
 
SYMMETRY AND TIME CHANGED BROWNIAN 
MOTIONS 
 
 
 
José Fajardo 
Ernesto Mordecki 
 
 
 Symmetry and Time Changed Brownian
Motions∗
Jos´ e Fajardo† and Ernesto Mordecki‡
March 7, 2008
Abstract
In this paper we examine which Brownian Subordination with drift
exhibits the symmetry property introduced by Fajardo and Mordecki
(2006b). We obtain that when the subordination results in a L´ evy
process, a necessary and suﬃcient condition for the symmetry to hold
is that drift must be equal to -1/2.
Key Words: Time Changed, Subordination, Symmetry.
1 Introduction
Asset returns have been studied during many years, one of the main ﬁnd-
ings is the presence of many small jumps in a ﬁnite time interval. To deal with
that fact more realistic jump structures have been suggested, as for exam-
ple the Generalized Hyperbolic (GH) model of Eberlein, Keller, and Prause
(1998), the Variance-Gamma (VG) model of Madan, Carr, and Chang (1998)
and the CGMY model of Carr, Geman, Madan, and Yor (2002). All that
models are included in a huge family called L´ evy processes.
∗J. Fajardo Thanks CNPq for ﬁnancial support
†IBMEC Business School, Rio de Janeiro - Brazil, e-mail: pepe@ibmecrj.br
‡Centro de Matem´ atica, Facultad de Ciencias, Universidad de la Rep´ ublica, Montev-
ideo. Uruguay. e-mail: mordecki@cmat.edu.uy.
1By a result due to Monroe (1978), we know that every semimartingale can be
written as a time-changed Brownian motion. As a consequence many L´ evy
processes can be represented as Time-changed Brownian Motion. This fact
is very useful for the pricing of multiasset derivatives, since we can correlate
assets by correlating the Brownian motions.
On the other hand, it is not easy to express explicitly the time-change used.
As for example in the case of CGMY process introduced by Carr, Geman,
Madan, and Yor (2002) and the Meixner process, introduced by Grigelionis
(1999) and Schoutens (2002). The time-changes used for these processes have
been obtained recently by Madan and Yor (2006).
In this paper, based on this explicit time-changes, we study how does the
symmetry concept, introduced by Fajardo and Mordecki (2006b), works in
the CGMY and Meixner models and also in other L´ evy processes obtained
by Subordinating Brownian motion with drift.
The paper is organized as follows: in Section 2, we introduce Time-changed
Brownian Motion. In Section 3, we describe the market model. In Section 4,
we describe symmetry and obtain our main result. In last sections we have
the conclusions and an appendix.
2 Time-Changed Brownian Motion
Let X = (X1,...,Xd) be a d-dimensional L´ evy process respect to the
complete ﬁltration F = {Ft,t ≥ 0}, this process is deﬁned on the probability
space (Ω,F,P), in other words X is a c` adl` ag process with independent and
stationary increments.
We know by the L´ evy-Khintchine formula that the characteristic function
of Xt, φXt(z) ≡ EeizXt = exp(tψ(z)) where the characteristic exponent ψ is
given by:
ψ(z) = i(b,z) −
1
2
(z,Σz) +
Z
I Rd
³
e
i(z,y) − 1 − i(z,y)1{|y|≤1}
´
Π(dy), (1)
where b = (b1,...,bd) is a vector in I Rd, Π is a positive measure deﬁned on
I Rd\{0} such that
R
I Rd(|y|2∧1)Π(dy) is ﬁnite, and Σ = ((sij)) is a symmetric
2nonnegative deﬁnite matrix, that can always be written as Σ = A′A (where
′ denotes transposition) for some matrix A.
Now let t  → Tt, ,t ≥ 0, be an increasing c´ adl´ ag process, such that for each
ﬁxed t, Tt is a stopping time with respect to F. Furthermore, suppose Tt
is ﬁnite and positive P − a.s., ∀t ≥ 0 and Tt → ∞ as t → ∞. Then {Tt}
deﬁnes a random change on time, we can also impose ETt = t.
Now let Xt = Wt be a Brownian motion. Then, consider the process Yt
deﬁned by:
Yt ≡ XTt, t ≥ 0,
this process is called Time-changed Brownian Motion. Using diﬀerent time
changes Tt, we can obtain a good candidate for the underlying asset return
process. We know that if Tt is a L´ evy process we have that Y would be
another L´ evy process1. A more general situation is when Tt is modelled by
a non-decreasing semimartingale:
Tt = at +
Z t
0
Z ∞
0
yµ(dy,ds) (2)
where a is a drift and µ is the counting measure of jumps of the time change.
Now we can obtain the characteristic function of Yt:
φYt(z) = E(e
iz′XTt) = E
³
E
³
e
iz′Xu | Tt = u
´´
If Tt and Xt are independent, then:
φYt(z) = LTt(ψ(z)), (3)
where LTt is the Laplace transform of Tt. So if the Laplace transform of
T and the characteristic exponent of X have closed forms, we can obtain a
closed form for φYt, as we show in the next examples.
In this way we can obtain the distribution of Yt for every t and in this way
we can price some derivatives.
1See (Cont and Tankov (2004, Th. 4.2 pag. 108))
32.1 Subordinators
We say that Tt is a Subordinator if it is a L´ evy Processes with non decreasing
trajectories. As a consequence, trajectories take positive values almost sure.
These properties are necessary in order to make a time change, a desired fact
for a Time-changed and the choose of a L´ evy process will allow us to obtain
as a result a very good candidate to model asset returns.
2.1.1 Stable subortination
Now let Tt be a α−Stable with zero drift and α ∈ (0,1), that is a L´ evy
process with L´ evy measure given by:
ρ(x) =
A
x1+α, x > 0,
we can compute the Laplace transform of Tt :
LTt(z) = A
Z ∞
0
ezx − 1
xα+1 dx = −
AΓ(1 − α)
α
(−z)
α,
Let Xt be a symmetric β−stable process, that is using eq. (1) we have
ψ(z) = −B|z|
β,
where A and B are positive constants. Then Using eq. (3), we have that
Yt = XTt has characteristic exponent given by
φYt(z) = LTt(ψ(z)) = −C|z|
βα
where C =
ABαΓ(1−α)
α . That is Yt is a βα−Stable symmetric process. If Xt
be a Brownian Motion, i.e. β = 2, then Yt would be a 2α−Stable symmetric
process. As α < 1, we have that Yt will be a process with heavy tails, which
is an stylized fact of the majority of the observed asset returns.
2.1.2 Tempered subordination
Assume that Tt has L´ evy measure given by
ρ(x) =
Ce−λx
x1+α 1x>0,
4where C,λ > 0 and α ∈ (0,1). Then, we have
LTt(z) = CΓ(−α)[(λ − z)
α − λ
α]
Now let Xt be a Time-Changed Brownian motion with drift µ, i.e. Xt =
Ttµ + σW(Tt). Then, using eq. (1), we have:
ψ(z) = −
z2
2
σ
2 + iµz,
and using eq. (3), we have that Yt = XTt has characteristic exponent given
by
φYt(z) = LTt(ψ(z)) = CΓ(−α)
·
(λ +
z2
2
σ
2 − iµz)
α − λ
α
¸
3 Market Model
Consider a Time-changed Brownian market where we have a riskless asset,
with price process denoted by B = {Bt}t≥0, with
Bt = e
rt, r ≥ 0,
where we take B0 = 1 for simplicity, and a risky asset, with price process
denoted by S = {St}t≥0,
St = S0e
Yt, S0 = e
y > 0. (4)
Where Yt is a time-changed Brownian Motion with a drift, where the time
change is an independent Subordinator. Denote by (b,σ,ν) the character-
istics of the time-changed Brownian with drift process2. Also, we assume
that the stock pays dividends, with constant rate δ ≥ 0, and we assume that
the probability measure P is the chosen equivalent martingale measure. In
other words, prices are computed as expectations with respect to P, and the
discounted and reinvested process {e−(r−δ)tSt} is a P–martingale.
In order to this condition be satisﬁed, we need that
E
£
e
−(r−δ)tSt
¤
= S0,∀t
2Here we assume conditions to guarantee that this process is a L´ evy process, see Ap-
pendix.
5In other words, E(eYt) = e(r−δ)t. That means that the characteristic exponent
of Y must satisfy:
ψY(1) = (r − δ).
So to avoid arbitrage opportunities we have to restrict our attention to time-
changed Brownian process such that the exponential process eYt−(r−δ)t be a
P–martingale.
4 Symmetry
Consider a Time-changed Brownian market described above with driving
process characterized by (b,σ,ν). Now, consider a market model with two
assets, a deterministic savings account ˜ B = { ˜ Bt}t≥0, given by
˜ Bt = e
δt, r ≥ 0,
and a stock ˜ S = {˜ St}t≥0, modelled by
˜ St = Ke
˜ Yt, S0 = e
x > 0,
where ˜ Y = {˜ Yt}t≥0 is a L´ evy processes with characteristics under ˜ P given by
(˜ b, ˜ σ, ˜ ν). This market is the dual market in Fajardo and Mordecki (2006b).
Observe, that in the dual market (i.e. with respect to ˜ P), the process
{e−(δ−r)t ˜ St} is a martingale.
It is interesting to notice, that in a market with no jumps the distribution
(or laws) of the discounted (and reinvested) stocks in both the given and
dual markets coincide. It is then natural to deﬁne a market to be symmetric
when this relation hold, i.e. when
L
¡
e
−(r−δ)t+Yt | P
¢
= L
¡
e
−(δ−r)t−Yt | ˜ P
¢
, (5)
meaning equality in law. Fajardo and Mordecki (2006b) derived the charac-
teristics of the dual process e Yt, In particular they obtained that a necessary
and suﬃcient condition for (5) to hold is
ν(dx) = e
−xν(−dx). (6)
6This ensures ˜ ν = ν, and from this follows b − (r − δ) = ˜ b − (δ − r), giving
(5), as we always have ˜ σ = σ.
Now as we have assumed that Yt = Ttµ + W(Tt). If we denote the L´ evy
measure of Tt by ρ(dy). Then, we know by Sato (1999)[Th. 30.1] that L´ evy
measures are related by
ν(dx) =
·Z ∞
0
1
√
2πy
e
−
(x−µy)2
2y ρ(dy)
¸
dx,
we can express this relationship as
ν(dx) = e
µxf(x)dx,
where
f(x) =
Z ∞
0
1
√
2πy
e
− 1
2y(x2+µ2y2)ρ(dy),
this density is even, i.e. f(x) = f(−x).
Proposition 1. A Time-Changed Brownian Market is symmetric if and only
if the drift is equal -1/2.
Proof. By condition (6) we have that a market is symmetric iﬀ
e
µxf(x)dx = e
−x[e
−µxf(−x)],
from here µ = −1/2.
As an application of this Proposition we have the following
Corollary 4.1. a) The CGMY Market model will be symmetric if and
only if
G − M = −1.
b) The Meixner Market Model will be symmetric if and only if
2b + a = 0
Proof. Since, Madan and Yor (2006) have obtained the explicitly represen-
tations of CGMY model and Meixner Model as a Time-changed Brownian
motion with drift G−M
2 and b
a, respectively. The result follows.
Moreover, in Carr, Geman, Madan, and Yor (2002) the drift is estimated
under the market risk neutral measure and in the majority of cases the drift
is negative and less that -0.5. Also, Schoutens (2001) estimates the values
of a and b and obtain similar evidence. It give us evidence against market
symmetry.
75 Conclusions
In a Time-changed Brownian market we have shown that market will be
symmetric if and only if the drift is equal to -1/2.
Since, Time-Changed Brownian motion allow us to model correlations by
correlating the Brownian motions. Another important application that can
be address with duality techniques is the pricing of bidimensional derivatives
in a Time-Changed Brownian context as is done by Fajardo and Mordecki
(2006a) for the case of L´ evy processes.
6 Appendix
The following Theorem is taken from Cont and Tankov (2004)[Th. 4.3, Pag.
113] and it gives conditions for a Time-Changed Brownian motion with drift
to be a L´ evy process.
Theorem 6.1. Let ν be a Levy measure and µ ∈ I R. There exists a L´ evy
process Yt with L´ evy measure ν such that Yt = W(Zt) + µZt for some subor-
dinator Zt and some Brownian motion Wt independent from Z, if and only
if the following conditions are satisﬁed:
1. ν is absolutely continuous with density ν(x)
2. ν(x)e−µx = ν(−x)eµx
3. ν(
p
(x))e
−µ
√
(x) is a completely monotonic function on (0,1)
A function f : [a,b] → I R is called completely monotonic if all derivatives
exist and (−1)k
dkf(u)
duk > 0, ∀k ≥ 1.
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