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190 8 L
´
evy Jumps
take a mean-reverting square root process and a gamma-OU process as two special
subordinators to show how to build up a leverage effect between L(t) and y(t), and
how to deal with such leverage effect by applying the technique of the leverage-
neutral measure.
Time-Changed By Square Root Process
We now consider a case where a mean-reverting square root process serves as a sub-
ordinator. As indicated above, a correlated time change of a Brownian motion can
be only generated by another Brownian motion. Hence, we apply a mean-reverting
square root process to time-change another Brownian motion that drives the dynam-
ics of asset price S(t).
We recall the Heston model again, and denote L(t)=W
1
(t) and y(t)=V (t),
dS(t)
S(t)
= rdt +
y(t)dL(t), (8.38)
dy(t)=
κ
(
θ
−y(t))dt +
σ
y(t)dW
2
(t),
dL(t)dW
2
(t)=
ρ
dt.
The new time-changed L
´
evy process X(t) is then given by
X(t)=L(Y (t)) =
t
0
y(s)dW
1
(s).
By the relation given in (8.36), the CF of X(t) under the risk-neutral measure Q
is equivalent to the Laplace transform of Y(t) under the complex-valued leverage-
neutral measure Q
φ
,
f
X(t)
(
φ
)=E
Q
e
i
φ
X(t)
= E
Q
φ
e
−
1
2
Y(t)
φ
2
,
where −
1
2
φ
2
is the characteristic exponent of a standard Brownian motion L(t).The
Radon-Nikodym derivative between the risk-neutral measure Q and the correspond-
ing leverage-neutral measure Q
φ
takes the following form,
dQ
φ
dQ
(t)=exp
i
φ
X(t)+
1
2
φ
2
t
0
y(s)ds
.
By setting
φ
= −i, we obtain the conventional Radon-Nikodym derivative
dQ
−i
dQ
(t)=exp
X(t) −
1
2
t
0
y(s)ds
.
Now we apply an extended Girsanov theorem under a complex-valued measure to
the process y(t), and obtain the following new Brownian motion,
8.3 Time-Changed L
´
evy Process 191
dW
Q
φ
(t)=−i
φ
y(t) < dW
1
(t), dW
2
(t) > +dW
2
(t).
It follows immediately a complex-valued process under Q
φ
,
dy(t)=[
κθ
−
κ
y(t)+i
φρσ
y(t)]dt +
σ
y(t)dW
Q
φ
(t), (8.39)
where the drift term includes a complex value i
φρσ
y(t). Under the measure Q
φ
,
computing
E
Q
φ
e
−
1
2
Y(t)
φ
2
= E
Q
φ
e
−
1
2
φ
2
t
0
√
y(s)ds
is then straightforward by using the Feynman-Kac theorem. More simply, we may
directly apply the results given in (3.16) to obtain the final analytical solution for
theCFofX(t).
Obviously, in the light of time change, the Heston model may be interpreted as a
time changed Brownian motion with drift where a mean-reverting square root pro-
cess serves as a subordinator. This is a new insight that we gain from the leverage-
neutral measure in the connection with a correlated time-changed Brownian motion.
A further question is then, weather this time-change approach may be extended to
other stochastic volatility models, for example, to the Sch
¨
oble and Zhu model where
stochastic volatility is specified to be a mean-reverting Ornstein-Uhlenbeck process?
Since a mean-reverting Ornstein-Uhlenbeck process is Gaussian and may take neg-
ative values, it is not appropriate to be a subordinator. However, if we relax and even
ignore the condition that a subordinator shall be a positive function, the above used
technique may still be applied to other stochastic volatility models.
Time-Changed By Gamma-OU Process
To illustrate a leverage effect between a jump L
´
evy process L(t) and its subordinator,
and to generate the related asymmetry in the time-changed pure L
´
evy process, we
consider a simplified Pareto model in Section 7.4 with a single Poisson counting
process. To void possible notational conflicts, we rewrite the asset process driven by
a compound Poisson process q
1
N(t) in the following form,
d lnS(t)=[r −
μ
−
1
2
v
2
]dt + vdW + q
1
dN(t) (8.40)
with identical exponentially distributed jump sizes q
1
. Therefore we have here
dL(t)=q
1
dN(t) or L(t)=
N(t)
∑
n=1
q
1
(n).
On the other hand, we specify the subordinator y(t) as a gamma-OU process,
dy(t)=[a −
κ
y
y(t)]dt + q
2
dN(t), (8.41)
192 8 L
´
evy Jumps
where N(t) is an identical Poisson process to the one specified in the asset price S(t),
thejumpsizeq
2
is exponentially distributed, and has a joint distribution with q
1
.
The joint distribution F(dq
1
,dq
2
) determines the dependence of q
1
and q
2
. Since
the both processes L(t) and y(t) share the same Poisson component, the leverage
effect comes solely from the correlated jump sizes.
Applying the time change to L(t) by Y (t) yields
X(t)=L(Y (t)) =
N(Y(t))
∑
n=1
q
1
(n).
Again by the relation given in (8.36), we should evaluate the following Laplace
transform under the leverage-neutral measure Q
φ
,
E
Q
φ
e
−
Ψ
∗
L
(
φ
)Y (t)
with
Ψ
∗
L
(
φ
)=
λ
1 −
1
1 −i
φμ
1
,
where
μ
1
is the mean of q
1
.
4
The Radon-Nikodym derivative for the change from Q
to Q
φ
is then given by
dQ
φ
dQ
(t)=exp
i
φ
X(t)+
Ψ
∗
L
(
φ
)
t
0
y(s)ds
.
To obtain a concrete form of Q
φ
, we consider the joint characteristic function
f (
φ
1
,
φ
2
) of q
1
and q
2
, which is defined by
f (
φ
1
,
φ
2
)=E
e
i
φ
1
q
1
+i
φ
2
q
2
=
R
+
R
+
e
i
φ
1
q
1
+i
φ
2
q
2
dF(dq
1
,dq
2
).
By setting
dF
∗
(dq
2
)=
R
+
e
i
φ
1
q
1
dF(dq
1
,dq
2
),
the joint characteristic function f (
φ
1
,
φ
2
) can be rewritten as
f (
φ
1
,
φ
2
)=
R
+
e
i
φ
2
q
2
dF
∗
(dq
2
)= f
∗
(
φ
2
),
which may be regarded as the CF of q
2
with the new distribution function F
∗
(dq
2
).
Hence, F
∗
(dq
2
) defines a new measure that is identical to Q
φ
1
under which the
expected value of q
2
may be calculated as if it is independent of q
1
. Formally, the
leverage-neutral measure Q
φ
1
implies the following equation
dF
Q
φ
1
(dq
2
)=
R
+
e
i
φ
q
1
dF(dq
1
,dq
2
). (8.42)
4
For a detailed calculation of
Ψ
∗
L
(
φ
), please see Section 7.4.
8.4 The Barndorff-Nielsen and Shephard Model 193
It follows
f
X(t)
(
φ
)=E
Q
φ
1
e
−
Ψ
∗
L
(
φ
)Y (t)
=
R
+
e
−
Ψ
∗
L
(
φ
)Y (t)
dF
Q
φ
1
(dq
2
).
Carr and Wu (2002) showed that the CF of X(t) still admits an exponential-affine
solution as follows:
f
X(t)
(
φ
)=e
α
(t)+
λβ
(t)
. (8.43)
Applying the ODEs (7.48) and (7.49) of the affine jump-diffusion model given in
Section 7.6,
α
(t) and
β
(t) are determined by the solutions of the ODEs,
β
(t)=−
Ψ
∗
L
(
φ
)+
κ
y
β
(t)+
λ
[ f
Q
φ
1
2
(
β
(t)) −1],
α
(t)=−a
β
(t),
where f
Q
φ
1
2
(
β
(t)) is the marginal moment-generating function of q
2
under Q
φ
1
,
f
Q
φ
1
2
(
β
(t)) =
R
+
e
β
(t)q
2
dF
Q
φ
1
(dq
2
).
8.4 The Barndorff-Nielsen and Shephard Model
Barndorff-Nielsen and Schephard (BNS, 2001) proposed an asset price which is
a mixture of a Brownian motion and a L
´
evy jump. Let x(t)=lnS(t), their mixed
process of x(t) is given by
dx(t)=
r −
λ
−
1
2
β
v
2
(t)
dt +v(t)dW(t)+
ρ
dZ(t), (8.44)
dv
2
(t)=−
κ
v
2
(t)dt + dZ(t), (8.45)
where dW (t) and dZ(t) are not correlated. Z(t) is not a Brownian motion, instead
aL
´
evy process describing the jumps of instantaneous asset returns. Additionally,
stochastic variances v
2
(t) is also driven by Z(t) only, and follows a gamma OU pro-
cess. This is different from the most stochastic volatility models such as the Heston
model. The parameter
ρ
is usually negative and links the inverse movements of the
jumps in volatility and the jumps in price, and should not be interpreted as a cor-
relation coefficient, but a leverage effect between volatility and asset price. In fact,
some practical calibrations deliver a larger negative number for
ρ
(see Schoutens,
Simons and Tistaert (2004)). Particularly, Z(t) in the BNS model is specified to fol-
low a gamma process with a mean of
α
and a variance of
β
in an unit time interval.
The density function of a gamma distribution is
194 8 L
´
evy Jumps
ν
(z)=
c
α
c
Γ
(
α
c)
e
−cz
z
α
c−1
, c =
α
β
, z > 0,
with
Γ
(·) as a gamma function. The discounted asset prices are martingale only if
λ
=
κ
R
e
ρ
z
ν
(dz)=
καρ
c −
ρ
.
The CFs in the BNS model can be derived as follows:
f
1
(
φ
)=E[e
−(x
0
+r)T +(1+i
φ
)x(T)
]
= exp
i
φ
x
0
+ i
φ
(r −
λ
)T −
1
2
γ
(
φ
2
−i
φ
)v
2
0
× exp
α
c −g
2
(1+i
φ
)
κ
g
2
(1+i
φ
)T + cln
c −g
1
(1+i
φ
)
b −(1 + i
φ
)
ρ
= e
i
φ
(x
0
+rT)
f
BNS
1
(
φ
) (8.46)
and
f
2
(
φ
)=E[e
i
φ
x(T)
]
= exp
i
φ
x
0
+ i
φ
(r −
λ
)T −
1
2
γ
(
φ
2
+ i
φ
)v
2
0
× exp
α
c −g
2
(
φ
)
κ
g
2
(
φ
)T + cln
c −g
1
(
φ
)
b −i
φρ
= e
i
φ
(x
0
+rT)
f
BNS
2
(
φ
) (8.47)
with
g
1
(u)=iu
ρ
−
1
2
γ
(u
2
+ iu),
g
2
(u)=iu
ρ
−
1
2
κ
(u
2
+ iu).
The function
γ
is already defined before in (8.32). Note that the BNS’s original
pricing formula for European-style options is based on f
2
(
φ
) that is essentially the
CF under the original risk-neutral measure. In fact, as mentioned in BNS, Z(t) can
also be driven by other L
´
evy pure jumps. Possible candidates for Z(t) could be the
normal inverse Gaussian process and the variance-gamma process given above.
8.5 Alpha Log-Stable Model
Alpha stable distribution is essentially a distribution family with four parameters.
Specifying these parameters to particular values leads to various well-known dis-
tributions such as normal distribution, L
´
evy distribution and Cauchy distribution.
8.5 Alpha Log-Stable Model 195
Denote the alpha stable distribution as L(x,
α
,
β
,
ν
,c) with four parameters. These
parameters can be roughly interpreted as follows:
•
α
∈ (0,2] is a measure for kurtosis, and determines the weight of the tails of the
distribution.
•
β
∈[−1,1] is a measure for skewness. Positive and negative values of
β
generate
positive and negative skew respectively. If
β
is equal to zero, the distribution is
symmetric.
•
μ
∈ R is responsible for location, and characterizes the mean of the distribution.
• c ∈ R
+
scales the distribution, and expresses the dispersion of the distribution.
Let x ∼L(
α
,
β
,
ν
,c), the CF of x is given by
f (
φ
)=exp
i
φμ
−|c
φ
|
α
(1−i
β
sign(
φ
)A
(8.48)
with
A =
tan
1
2
απ
if
α
= 1
−
2
π
ln|
φ
| if
α
= 1
.
Although the parameters
α
and
β
may be considered as the measures for kurtosis
and skewness, respectively, the moments E[L
k
],k
α
, of the alpha stable distri-
bution are not defined, and hence the kurtosis and skewness do not exist. If incre-
ments of a process are characterized by alpha stable law, it will, as expected, exhibit
the properties such as self-similarity and infinite-divisibility. Except for the case of
α
= 2 where alpha-stable distribution reduces to the normal distribution, the alpha
stable distribution is leptokurtic and heavy-tailed. Due to its ability of capturing lep-
tokurtic property of assets returns, the alpha stable process has been recognized by
many economists and mathematicians to model and analyze empirical financial data
(see Mandelbrot (1967), Voit (2003)), and recently to price options (see Popava and
Ritchken (1998), McCulloch (1996), and, Hurst, Platen and Rachev (1999)). How-
ever, the undesired properties of the alpha stable process is its infinite moments,
therefore also infinite price moments. As a result, call price can not be soundly eval-
uated under an alpha stable process without any restriction.
A simple version of the restricted alpha stable processes is the so-called finite
moment log-stable process. This special alpha stable process initialized by McCul-
loch (1996) and developed by Carr and Wu (2003a) is specified with the maximum
negative skewness
β
= −1. The intention of forcing the maximum negative skew-
ness to an alpha stable process, as argued by Carr and Wu, is twofold: Firstly, for
the case of
β
= −1, the process still retains the property of unrestricted alpha sta-
ble process allowing for fat tails, even for long maturities, and keeping asymmetric
structure of the distribution unchanged. Especially, it generates a down-sloping skew
as observed in typical equity option markets.
5
Secondly, although the moments of
returns in the case of
β
= −1 are infinite, the prices have finite moments. Conse-
quently, option prices with finite moment log-stable process are bounded.
5
The down-sloping skew of implied volatilities in equity markets is typical, but is not the single
smile pattern only. In many cases, we can often observe strong symmetric smile for short-term
volatilities.
196 8 L
´
evy Jumps
More in details, under the risk-neutral measure, the SDE of a finite moment log-
stable process for stock price reads
dx(t)=(r −m)dt + dL(t;
α
,
σ
),
α
∈ (1,2), (8.49)
where alpha stable distribution L(t;
α
,
σ
) has a location parameter
μ
equal to
zero, and a scaling parameter c equal to
σ
. Namely, L(t;
α
,
σ
) is equivalent to
L(t,
α
,−1,0,
σ
). The finite moment logs-table process given above is rather par-
simonious, and characterized by two free parameters
α
and
σ
only. Since
σ
is a
scaling parameter, The SDE of (8.49) can be rewritten as
dx(t)=(r −m)dt +
σ
dL(
α
,1).
The drift compensator m enforcing the stock process to be a martingale can be easily
verified as
m = ln[ f (
φ
= −i)] = −
σ
α
sec
απ
2
. (8.50)
Finally we can derive the CFs under two different martingale measures Q
1
and Q
2
,
f
1
(
φ
)=E[g
1
(T)e
i
φ
x(T)
]=E[e
−(x
0
+rT)+(1+i
φ
)x(T)
]
= exp
i
φ
(x
0
+ rT) −(1 + i
φ
)mT −((1+ i
φ
)
σ
)
α
T sec
απ
2
= e
i
φ
(x
0
+rT)
f
ALS
1
(
φ
) (8.51)
and
f
2
(
φ
)=E[e
i
φ
x(T)
]
= exp
i
φ
(x
0
+ rT) −i
φ
mT −(i
φσ
)
α
T sec
απ
2
= e
i
φ
(x
0
+rT)
f
ALS
2
(
φ
). (8.52)
The option pricing formula is then given correspondingly.
8.6 Empirical Performance of Various L
´
evy Processes
In the above sections we have examined a number of L
´
evy processes that are ap-
plied to capture the non-normality of assets returns, and consequently used to price
European-style options. We have seen that various L
´
evy processes can be con-
structed in different ways. One powerful and illustrative way is the subordination
of a L
´
evy process by another process. This procedure may be illustrated by random
time-changing, and corresponds to nesting the CFs of two involved processes. With
this method, we have formulated the variance-gamma process and the NIG process
as well as their four extended processes subordinated to a mean-reverting square-
root process and a gamma OU process respectively. Another way to construct L
´
evy
8.6 Empirical Performance of Various L
´
evy Processes 197
process is the direct specification of the distribution of the increments of the process,
for example, the gamma OU process in BNS (2001) and the finite moment logstable
process in Carr and Wu (2003a).
In this section we summarize some empirical results for these L
´
evy processes.
Particularly, comparative testings with respect to pure diffusion models and Poisson
jump models are carried out in some empirical studies. Some stylized results are
given in following.
1. The variance-gamma process and the NIG process have the similar performance
in pricing options. The test results based both on the statistic inference and on
the calibration (i.e. pure optimization) deliver the mean-square errors of a same
order. The equivalent performance should not be surprising since the marginal
distributions of both processes belong to the generalized hyperbolic distribution,
see Carr, Geman, Madan and Yor (2001).
2. Whenever the variance-gamma process and the NIG process are subordinated to a
same process, for example, to a mean-reverting square root process, they exhibit
the similar pricing performance, as shown in Schoutens, Simons and Tistaert
(2004).
3. With respect to the valuation of exotic options, as reported by Schoutens, Simons
and Tistaert (2004), the paths dynamics of stochastic volatility models such as the
Heston model are more changeful than theses of L
´
evy jump models. This is in-
dicated by the fact that the prices of lookback options and down-and-in barrier
options with the Heston model are significantly higher than the prices with L
´
evy
jump models although all models are equally good at the valuation of European-
style options. This result seems somehow surprising since it is always argued in
financial literature that L
´
evy jumps are born to generate fat tails and negative
skews in a distribution, and therefore would be also able to produce more ex-
treme events. One conclusion that we can draw from the large price deviations
between stochastic volatility models and L
´
evy jump models in the valuation of
path-dependent options, is that a good fitting of the terminal distribution for a
given maturity to market data for various models does not imply a same local
behavior of prices in these models. So far, to my best knowledge, there is no lit-
erature discussing the hedging performance of different L
´
evy jump models, and
carrying out the corresponding comparative studies to stochastic volatility mod-
els, in order to answer what type models may be preferred to in the valuation of
exotic options.
4. An empirical study of Huang and Wu (2004) investigated what type of jump
structure best describes the return and process movements within time changed
L
´
evy processes by using the market data of S&P index options. Their main result
is that the variance-gamma model and the finite moment logstable model outper-
form Poisson jump models. In other words, a L
´
evy process incorporating hight
frequency jumps may be better than the one incorporating low frequency jumps
in capturing the behavior of options. Therefore, Huang and Wu recommended
that an appropriate pricing model should incorporate L
´
evy jump components.
5. The finite moment log-stable model with the maximum negative skew is designed
to capture the down-sloping pattern (smirk) of volatility. However. due to its in-
198 8 L
´
evy Jumps
flexible skew structure, it is impossible for finite moment log-stable model to
generate a symmetric volatility smile that happens often for short maturity op-
tions in an indecisive market.
8.7 Monte-Carlo Simulation
8.7.1 Generating Random Variables
The much broader applications of a pricing model in practice is to value and hedge
exotic structures consistently to plain-vanilla options. For this purpose, an efficient
Monte-Carlo simulation for L
´
evy jump models is an important issue for practition-
ers. Glasserman (2004), Schoutens, Simons and Tistaert (2004) have discussed how
to simulate a pure jump process. As a nature of L
´
evy process, the key to this simu-
lation is to produce random increments at each time step by generating the desired
random variables according to the marginal distribution law. For example, if we
want to simulate a NIG process, we need to generate the NIG random variables at
each time step, this is in analog to simulating a Brownian motion where normal ran-
dom variables are generated. As indicated in the construction of some L
´
evy jump
processes, the pure jump random variables are a composition of two or more random
variables of different distribution laws. Therefore, it makes sense here to address
first how to generate random variables of some basic distributions, they are random
variables of Poisson distribution, gamma distribution, inverse Gaussian distribution,
and alpha stable distribution.
Poisson Distribution
A Poisson distribution is defined on the set of non-negative integers, and its density
function for N = k is given by
p(N = k)=e
−
λ
λ
k
k!
, k = 0,1,2,··· , (8.53)
where
λ
is the mean and controls the intensity of events recorded in an unit time. It
is easy to see that
p(N = k + 1)=p(N = k)
λ
k + 1
. (8.54)
The cumulative Poisson distribution function F(k) is then the sum of p(N = j) over
all integers up to k,
F(k)=
k
∑
j=0
p(N = j)=
k
∑
j=0
e
−
λ
λ
j
j!
. (8.55)
8.7 Monte-Carlo Simulation 199
Given the cumulative distribution function, we can apply the inverse transform
method to generate Poisson random variables.
6
Namely, we first generate an uni-
form random variable U, and then find a x such that
x = k, if F(k) U < F(k + 1).
This idea implies the following detailed algorithm for generating Poisson random
variables.
1. Set j = 0, p = p( j = 0)=e
−
λ
,F = p,
2. Generate an uniform random variable U ∼Unif[0,1],
3. While U > F, then set p = p
λ
j+1
,F = F + p, j = j + 1,
4. Finally set x = j.
where in Step 3 we have utilized the property (8.54) of Poisson distribution to cal-
culate the cumulative distribution F(k) such that k is the critical integer satisfying
the condition F(k) U < F(k + 1).
Gamma distribution
As in the variance-gamma model, we assume that x has a gamma distribution
γ
(
α
,
β
) with a mean of
α
and a variance of
β
.Thevalueb =
β
/
α
is the scal-
ing parameter for the gamma distribution. This means that the random number
y,y ∼
γ
(
α
/b,
β
/b
2
) and the random number x,x ∼
γ
(
α
,
β
) have the same distribu-
tion. Therefore, we need only to consider a gamma random variable x,x ∼
γ
(
α
,
α
)
with a scaling parameter of one. As discussed in Fishman (1996), Glasserman
(2000), a two-case strategy based on the so-called ratio-of-uniforms method may
be an efficient way to generate gamma distributed random variables, the two cases
that must be considered are
α
> 1 and
α
1.
Case 1: a 1.
1. Set a =(
α
+ e)/e with e = exp(1),
2. Generate U
1
∼Unif[0,1] and U
2
∼Unif[0,1], then compute d = aU
1
,
3. If d 1, then compute z = d
1/
α
.IfU
2
< exp(−z), accept z,
4. Otherwise if d > 1, then compute −ln((a −d)/al pha).IfU
2
z
α
−1
, accept z,
5. If no appropriate z is found, repeat steps 2, 3, 4.
Case 2: a > 1.
1. Set a =(
α
−
1
6
α
)/(
α
−1), c =
2
α
−1
,
2. Generate U
1
∼Unif[0, 1] and U
2
∼Unif[0, 1], then compute d = a
U
2
U
1
and z =
(
α
−1)d,
3. Compute y = cU
1
+ d −(c + 2)+1/d.Ify 0, accept z,
4. Otherwise compute y = clnU
1
−lnd +(d −1).Ify 0, accept z,
5. If no appropriate z is found, repeat steps 2, 3, 4.
6
More about inverse transform method for generating random variables, see Glasserman (2004).