Journal of Mathematical Finance, 2011, 1, 58-62
doi:10.4236/jmf.2011.13008 Published Online November 2011 (http://www.SciRP.org/journal/jmf)
Maximum Quasi-Likelihood Estimation in Fractional Levy
Stochastic Volatility Model
Jaya Prakasah Narayan Bishwal
Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, USA
Received July 13, 2011; revised August 12, 2011; accepted August 22, 2011
Abstract
Usually asset price process has jumps and volatility process has long memory. We study maximum quasi-
likelihood estimators for the parameters of a fractionally integrated exponential GARCH, in short FIECO-
GARCH process based on discrete observations. We deal with a compound Poisson FIECOGARCH process
and study the asymptotic behavior of the maximum quasi-likelihood estimator. We show that the resulting
estimators are consistent and asymptotically normal.
Keywords: Stochastic Volatility, Long Memory Process, Jump Process, Fractional Levy Process, Poisson
Process, Ornstein-Uhlenbeck Process, Gaussian Quasi-Likelihood, Consistency, Asymptotic
Normality
1. Introduction
Parameter estimation in GARCH models has been studi-
ed extensively in view of their wide applications. In or-
der to incorporate long memory in the model we look at
the continuous time counterpart. Most of the financial
transactions are recorded these days with ultra high fre-
quency which are also called tick-by-tick data. The most
important feature of these data are irregular spacing of
observation time points. If one aggregates the data up to
fixed time intervals, there is loss of information which
should be avoided. Hence it is natural to model price
processes in continuous time. Also volatility clusters on
high level. This long-run volatility persistence is not taken
into account by GARCH models. Levy processes con-
tribute non-normality and jumps in the observed part of
the model and fractional processes contribute long me-
mory in unobserved part in the model. Based on discrete
observations of the price process we estimate all the un-
known parameters of the price and the volatility process
by the maximum quasi-likelihood estimation. We use
Gaussian quasi-likelihood as a contrast function. Maxi-
mum quasi-likelihood estimator is an application of Kal-
man filter in the context of maximum likelihood estima-
tion. Even if the data are non-normal, we use the Gaus-
sian log-likelihood.
Non-negative Ornstein-Uhlenbeck processes were first
studied by Wolfe [1] who obtained it as a limit of an au-
toregressive process with positive innovations. These days
non-negative Ornstein-Uhlenbeck processes have recei-
ved lot of attention in view of their applications as inter-
est rate and stochastic volatility models, see Barndorff-
Nielsen and Shephard [2,3]. Gaver and Lewis [4] studied
estimation in first order autoregressive gamma process.
Davis and McCormick [5] studied estimation for first or-
der autoregressive process with positive or bounded in-
novations. Neilsen and Shephard [6] studied likelihood
analysis of a first order autoregressive model with expo-
nential innovations. Brockwell, Davis and Yang [7] stu-
died parameter estimation of non-negative Levy driven
Ornstein-Uhlenbeck processes. Jongbloed, van der Meulen
and van der Vaart [8] studied nonparametric estimation for
Levy driven Ornstein-Uhlenbeck processes.
Parameter estimation in Gaussian Ornstein-Uhlenbeck
process was extensively studied in Bishwal [9]. To ac-
count for the strong persistence in volatility, which
sometimes is observed in empirical data, we study the
FIECOGARCH (fractionally integrated exponential con-
tinuous time generalized autoregressive conditional het-
eroscedastic) process. The log-price process has jumps
and the log-volatility process has long memory. The long
memory effect introduced in the log-volatility process,
propagates to the volatility process. Thus the model here
captures the two stylized facts of long memory and jumps.
Continuous time long memory process in volatility re-
ceived some attention, see Comte and Renault [10] where
they used fractional Ornstein-Uhlenbeck process as the
stochastic volatility process and Comte, Coutin and Renault

59 J. P. N. BISHWAL
[11] where they used fractional CIR square root process
as the stochastic volatility process. Marquardt [12] studi-
ed fractional Levy processes and applied it to long me-
mory moving average processes. Marquardt [13] studied
multivariate fractionally integrated CARMA processes.
Bender and Marquardt [14] studied stochastic calculus
for convoluted Levy processes. Haug and Czado [15]
studied structural properties of an exponential continuous
time GARCH process. Haug and Czado [16] studied the
structural properties of FIECOGARCH model. Czado and
Haug [17] used an exponential autoregressive conditional
duration model to describe the dependence structure in
durations of ultra high frequency financial data. Haug,
Kluppelberg, Lindner and Zapp [18] studied method of
moment estimation in the COGARCH(1,1) model. CO-
GARCH model is suitable for irregularly spaced obser-
vation times. Straumann and Mikosch [19] studied quasi-
maximum likelihood estimation in conditionally hetero-
scedastic time series model. Haug and Czado [20] stud-
ied quasi-maximum likelihood estimation and prediction
in the compound Poisson ECOGARCH(1,1) model.
2. Preliminaries
Fractional Levy Process is a generalization of fractional
Brownian motion and is defined as
 
1/21/2
,
1
=d
1
2



 






HH
Ht s
R
MtssMt
H
,
where



0.5,1,= max,0,= max,0

H
uuu u




 and
is a Levy process on with

,
t
Mt



2
11
=0, <EMEM 
and without Brownian component.
Following are the properties of fractional Levy process:
1) The covariance of the FLP is given by



2
,,
22 2
1
,=
22 1sin
Ht Hs
HH H
EL
Cov MMHH
tsts


 



2)
H
M
is not a martingale. For a large class of Levy
processes,
H
M
is neither a semimartingale.
If the Levy process M is of finite activity, then the cor-
responding fractional Levy process
H
M
is of finite va-
riation. In the case when M is not of finite activity, the
corresponding fractional Levy process is not a semimar-
tingale.
3)
H
M
is Hölder continuous of any order

less
than 1
2
H.
4)
H
M
has stationary increments.
5)
H
M
is symmetric.
6)
H
M
is self similar.
7)
H
M
has infinite total variation on compacts.
8)
H
M
is not self similar.
9)
H
M
has long memory.
10) Fractional Levy Ornstein-Uhlenbeck (FLOU).
Process:
The FIMA (fractionally integrated moving average)
process is defined as
 
=d
 

t
HH
Yt gtuLut,
where
 
3
2
0
1
=d
1
2




 


H
t
H
gtgtss st
H
,
and the kernel g is the kernel of a short memory moving
average process.
The process


H
Yt can be written as
 
,
=d,
 .



t
HH
YtgtuMt
u
We assume the following conditions on the kernel g:


.
(A1)


=0gt for all (causality). <0t
(A2)


ct
g
tCe

 for some constants and
(short memory).
>0C>0C
FIMA process is stationary and is infinite divisible.
Consider the kernel




0,
=ts

g
tse Its





 then



3
2
(0, )
0
=d
1
2







 


H
ts
H
gte Itssst
H
, .
Note that

 
,
,=d

,



Ht H
R
VgtuLut
is the fractional Levy Ornstein-Uhlenbeck (FLOU) pro-
cess satisfying the fractional Vasicek model
,, ,
d=dd, .

 
Ht HtHt
VVtMt
3. Main Results
The FIECOGARCH process is given by ,, d= d
H
tHt
YX
t
L
,
,=V
H
t
Ht
Xe
 
,,
d= dd
,
H
tHtH
VVtM
t


where t is a Levy process independent of ,
L
H
t which
is a fractional Levy process of integration (Hurst) para-
meter
M


0.5,0.75H.

60 J. P. N. BISHWAL
We estimate ,

 and

by the maximum quasi-
likelihood method and study the asymptotic properties of
the estimators.
Equivalently, ,
,
d=dd
XHt
H
tt
Yte  L
,
,,
d= dd
H
tHt
XXtM
Ht

.
The log-volatility process is strictly stationary and the
covariance function is of the form
when .
  

21
22
,,1
log, log 

H
Ht hHt
CovC hh
The autocorrelation function of log-volatility and vo-
latility process decay at a hyperbolic rate.
The autocovariance function of the price returns over a
time interval of length >0r

 
,,, ,
,
==d,
r
HtHtHt rHss
trt
SSS Ltr 


>0
d.


is given by






22 2
22
,, 1,
,=() , 





hr
rr r
HtHt hHts
h
Cov SSELCov Ss

We will treat the special case of Compound Poisson
FIECOGARCH process first. We will estimate the pa-
rameters by maximum quasi-likelihood estimation me-
thod.
A compound Poisson FIECOGARCH process is driven
by the Levy process L with Levy symbol



2
0,1/
=1
2
iux
LR
u
ue

  
d
x ,
where 0,1

 is a normal distribution with mean 0 and
variance 1

. This means that L is the sum of a standard
Brownian motion W and a compound Poisson process
=1
=,
Nt
tk
k
JZt
0
where is an independent Poisson process with in-
tensity
t
N
>0

and jump times



kkZ
T

0,1/
. The Poisson pro-
cess is also independent from the i.i.d. sequence of jump
sizes with ZN

k
ZkZ
1

. The Levy process
M in this case is given by

=1
=,
Nt
tkk
k
MZZCtt  
>0
with

0,1/
2
:=d =.
R
Cx x



 

The stationary log-volatility is of the form


1
2
,,
log=d,> 0.
tats
Ht Hs
eMt 




The log-price process is given by
0
=1
=,0,  

t
k
N
tTk
k
SZtS=0.
We observe at n consecutive jump times S
01 1
0=<<<< <,
 nn
TTTTTn
over the time interval


0,T. The state process
H
X
has
then the following autoregressive representation



1
1
1
1
,,
=1
,
=
d
=1
i
ii
Ti
ik
Ti
Tii
i
ii
i
T
HT HT
N
TT
kk
kN
Ts
T
TT
HTi i
XeX
eZZ
eCs
C
eXZ Ze
















 




 




where 1 :=,= 1, 2,,



iii
TTTin and
11== . i
TT
i
NN
i
Thus
 
1
2
,
log =1
ii
ii
TT
HT T
C
eX e

 


 


since

1
,,
=1
i
ii
T
HT HT
C
XX e





 as i
T
X
 does not
include the jump at time and its corresponding jump
size is
i
T
i
Z
.
The leverage effect depends on the sign of

 . We
will also identify the leverage parameter

.
The compound Poisson process has finite activity. The
observation of the log-price process is given by
1
,,
=1
==
i
ki
NT
TTkHTHT
ik
SZS 

.
i
i
Z
We study the parameter estimation in two steps. The rate

of the Poisson process N can be estimated given the
jump times i, therefore it is done at a first step. Since
we observe total number of jumps n of the Poisson pro-
cess N over the T intervals of length one, the MLE of
T

is given by
ˆ:= .
n
n
T

Theorem 3.1 We have
ˆ .. as
 
nasn 





1ˆ0,1 as. 




HD
n
nNee
U
n
Proof. Let be the number of jumps in the interval
i


1,ii. Then are i.i.d. Poisson distri-
buted with parameter
,=1,2, ,Uin
i

. Since is continuous, we have 


 

 ,
=,.. =
i
1
iH
00 ,2,1,
I
UISasi n. Note that
 




1..
,11
00
=1
1==0=
 


nas
Hi
i
ISEIU PUeasn
n
C
opyright © 2011 SciRes. JMF

61 J. P. N. BISHWAL
LLN and CLT and delta method applied to the se-
quence
 

1
,
0,= 1, 2,,
Hi
I
Si n give the results.
The CLT result above allows us to construct confi-
dence interval for the jump rate

. A


100 1%   con-
fidence interval for

is given by
11
22
11 11
,






nn
qq
TnTTn


T
where
12
q

 is the 12








-quantile of the standard nor-
mal distribution.
To estimate the remaining parameters

,,

 , we use
the quasi maximum likelihood (QML) estimation proce-
dure as developed by Straumann [21] in conditionally
heteroscedastic time series models. Since the conditional
distributions of the returns is unknown, we will use the
QML approach by choosing the Gaussian quasi-likeli-
hood as a contrast function.
The conditional log-likelihood has the representation




2
,
2
2
=1 =1
,:= log2
2
1log .
2
i
i
i
i
H
T
nn
HT
T
ii
T
n
LS
S

 










The quasi log-likelihood function for


=,,,

 

ˆn
given and the MLE

12
12
,,,
:=,,, 
 n
n
T
TT
HHT HTHT
SSS S

is
defined as





2
,
=1
2
,
2
=1 ,
1
ˆ
ˆ
,:= log,
2
1
ˆˆ
2ˆ,
i
i
i
i
n
HnHTn
i
T
nHT
i
ˆ
H
Tn
L
S
n










where the estimates of the volatility
are given by
2
,,= 1, 2,,
 
i
HT in

,
 

1
2
,,
ˆ
ˆ,:=exp ,
=1,2, ,

 




i
ii
T
HT nHT
eX CT
n
and given the parameters

and

the estimates of
the state process X are given by the recursion


1
,
,
,
ˆˆ
=ˆ,
ˆ.
ˆ,
i
i
ii
i
i
i
i
HT
T
HT T
T
HT
T
S
XeX
SCT






 

 


Note that


2
=, 0,1
π


EWW N.
Here the approximation for
small is

1z e

zz
used. The recursion needs a starting value ,0
ˆ
H
X
which
will be the mean value zero of the stationary distribution
of X.
MQLE of

is defined as

ˆˆ
:= arg,.
max





nn
LS
The parametric estimator of volatility at jump times
is given by
01 1
0=<<<< <,
nn
TT TTTnZ




1
ˆ
2
,,
ˆˆˆˆˆ
ˆˆ
,:=exp, ,
=1,2, ,
i
ii
T
n
HTn nnHTn ni
eX CT
in

 





which uses the MQLE ˆn

.
Let


0H
F

be the Fisher information of the log-price
process where 0

is the true parameter. The following
theorem gives the strong consistency and the asymptotic
normality of the MQLE.
Theorem 3.2
0
ˆ
a).. as
 
nasn 




1/21
00
ˆ
b)0, as  



D
nH
nNFn
Proof. The discrete time process process is strong
mixing with geometric rate and since strict stationarity of
S
2

implied strict stationarity of is also ergodic.
Hence application of the Birkoff ergodic theorem and
central limit theorem for strongly mixing process proves
the theorem.
S
Remark. We considered the case


0.5,0.75H

1/2
lognn
21H
n
.
For the case , the asymptotic distribution of
the MQLE will be normal with the rate and
the asymptotic distribution of the MQLE will be non-
Gaussian Rosenblatt distribution with rate
=0.75H

. This
shows the different behavior of the MQLE in different
parts of the Hurst space even in the stationary case. In
the nonstationary case, the limit distribution will be
mixture normal. Thus not only the original parameter
space, but also the Hurst space plays an important role in
obtaaining the limit distribution of the MQLE in long
memory models.
4. Conclusions
We studied parameter estimation in a FIECOGARCH
model from discrete data. We obtained the consistency
and asymptotic normality of the maximum quasi-likelihood
estimator. Here we assumed H to be known. An im-

J. P. N. BISHWAL
62
portant problem is estimation of H, for instance based on
second order quadratic variation and remains to be in-
vestigated.
5. References
[1] A. J. Wolfe, “On a Continuous Analogue of the Stochas-
tic Difference Equation Xn = ρXn + Bn,” Stochastic Proc-
esses and their Applications, Vol. 12, No. 3, 1982, pp. 301-
312. doi:/10.1016/0304-4149(82)90050-3
[2] O. E. Barndorff-Nielsen and N. Shephard, “Non-Gaus-
sian Ornstein-Uhlenbeck-Based Models and Some of Their
Uses In Financial Economics (with Discussion),” Journal
of the Royal Statistical Society, Series B, Vol. 63, No. 2,
2001, pp. 167-241. doi:/10.1111/1467-9868.00282
[3] O. E. Barndorff-Neilsen and N. Shephard, “Normal Mo-
dified Stable Processes,” Theory of Probability and Ma-
thematical Statistics, Vol. 65, 2002, pp. 7-20.
[4] D. P. Gaver and P. A. W. Lewis, “First Order Autore-
gressive Gamma Sequences and Point Processes,” Ad-
vances in Applied Probability, Vol. 12, No. 3, 1980, pp.
727-745. doi:/10.2307/1426429
[5] R. A. Davis and W. P. McCormick, “Estimation for First
Order Autoregressive Processes with Positive or Bounded
Innovations,” Stochastic Processes and Their Applica-
tions, Vol. 31, No. 2, 1989, pp. 237-250.
doi:/10.1016/0304-4149(89)90090-2
[6] B. Neilsen and N. Shephard, “Likelihood Analysis of a First
Order Autoregressive Model with Exponential Innovations,”
Journal of Time Series Analysis, Vol. 24, No. 3, 2003, pp.
337-344. doi:/10.1111/1467-9892.00310
[7] P. J. Brockwell, R. A. Davis and Y. Yang, “Estimation
for Non-Negative Levy Driven Ornstein-Uhlenbeck Proc-
esses,” Journal of Applied Probability, Vol. 44, No. 4,
2007, pp. 977-989. doi:/10.1239/jap/1197908818
[8] G. Jongbloed, F. H. van der Meulen and A. W. van der
Vaart, “Nonparametric Inference for Levy Driven Orn-
stein-Uhlenbeck Processes,” Bernoulli, Vol. 11, No. 5,
2005, pp. 759-791. doi:/10.3150/bj/1130077593
[9] J. P. N. Bishwal, “Parameter Estimation in Stochastic
Differential Equations,” Springer-Verlag, Berlin, 2008.
doi:/10.1007/978-3-540-74448-1
[10] F. Comte and E. Renault, “Long Memory in Continuous-
Time Stochastic Volatility Models,” Mathematical Fi-
nance, Vol. 8, No. 4, 1998, pp. 291-323.
doi:/10.1111/1467-9965.00057
[11] F. Comte, L. Coutin and E. Renault, “Affine Fractional
Stochastic Volatility Models with Application to Option
Pricing,” Recherche, Vol. 13, No. 1993, 2010, pp. 1-35.
[12] T. Marquardt, “Fractional Levy Processes with Applica-
Tion to Long Memory Moving Average Processes,” Ber-
noulli, Vol. 12, No. 6, 2006, pp. 1009-1126.
doi:/10.3150/bj/1165269152
[13] T. Marquardt, “Multivariate FICARMA Processes,” Jour-
nal of Multivariate Analysis, Vol. 98, No. 9, 2007, pp.
1705-1725. doi:/10.1016/j.jmva.2006.07.001
[14] C. Bender and T. Marquardt, “Stochastic Calculus for
Convoluted Levy Processes,” Bernoulli, Vol. 14, No. 2,
2007, pp. 499-518. doi:/10.3150/07-BEJ115
[15] S. Haug and C. Czado, “An Exponential Continuous
Time GARCH Process,” Journal of Applied Probability,
Vol. 44, No. 4, 2007, pp. 960-976.
doi:/10.1239/jap/1197908817
[16] S. Haug and C. Czado, “Fractionally Integrated ECO-
GARCH Process,” Sonderforschungsbereich, Vol. 386,
No. 484, 2006.
[17] C. Czado and S. Haug, “An ACD-ECOGARCH(1,1)
Model,” Journal of Financial Econometrics, Vol. 8, No.
3, 2010, pp. 335-344. doi:/10.1093/jjfinec/nbp023
[18] S. Haug, C. Kluppelberg, A. Lindner and M. Zapp,
“Method of Moment Estimation in the COGARCH(1,1)
Model,” Econometrics Journal, Vol. 10, No. 2, 2007, pp.
320-341. doi:/10.1111/j.1368-423X.2007.00210.x
[19] D. Straumann and T. Mikosch, “Quasi-Maximum Like-
lihood Estimation in Conditionally Heteroscedastic Time
Series: A Stochastic Recurrence Equations Approach,”
Annals of Statistics, Vol. 34, 2006, pp. 2449-2495.
doi:/10.1214/009053606000000803
[20] S. Haug and C. Czado, “Quasi Maximum Likelihood Esti-
mation and Prediction in the Compound Poisson ECO-
GARCH(1,1) Model,” Sonderforschungsbereich, Vol.
386, No. 516, 2006.
[21] D. Straumann, “Estimation in Conditionally Heterosce-
dastic Time Series Models,” Lecture Notes in Statistics,
Vol. 181, Springer-Verlag, Berlin, 2005.