arXiv:1001.0024v1  [q-fin.CP]  30 Dec 2009November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
Journal of Circuits, Systems, and Computers
c/circlecopyrtWorld Scientiﬁc Publishing Company
BAYESIAN INFERENCE OF STOCHASTIC VOLATILITY MODEL
BY HYBRID MONTE CARLO
Tetsuya Takaishi†
Hiroshima University of Economics,
Hiroshima 731-0192 JAPAN
†takaishi@hiroshima-u.ac.jp
Received (Day Month Year)
Revised (Day Month Year)
Accepted (Day Month Year)
The hybrid Monte Carlo (HMC) algorithm is applied for the Bay esian inference of the
stochastic volatility (SV) model. We use the HMC algorithm f or the Markov chain Monte
Carloupdates of volatility variables of the SV model. First we compute parameters of the
SV model by using the artiﬁcial ﬁnancial data and compare the results from the HMC
algorithm with those from the Metropolis algorithm. We ﬁnd t hat the HMC algorithm
decorrelates the volatility variables faster than the Metr opolis algorithm. Second we
make an empirical study for the time series of the Nikkei 225 s tock index by the HMC
algorithm. We ﬁnd the similar correlation behavior for the s ampled data to the results
from the artiﬁcial ﬁnancial data and obtain a φvalue close to one ( φ≈0.977), which
means that the time series has the strong persistency of the v olatility shock.
Keywords : Hybrid Monte Carlo Algorithm, Stochastic Volatility Mode l, Markov Chain
Monte Carlo, Bayesian Inference, Financial Data Analysis
1. Introduction
Many empirical studies of ﬁnancial prices such as stock indexes, ex change rates
have conﬁrmed that ﬁnancial time series of price returns shows va rious interesting
properties which can not be derived from a simple assumption that th e price re-
turns follow the geometric Brownian motion. Those properties are n ow classiﬁed
as stylized facts1,2. Some examples of the stylized facts are (i) fat-tailed distribu-
tion of return (ii) volatility clustering (iii) slow decay of the autocorre lation time
of the absolute returns. The true dynamics behind the stylized fac ts is not fully
understood. In order to imitate the real ﬁnancial markets and to understand the
origins of the stylized facts, a variety of models have been propose d and examined.
Actually many models are able to capture some of the stylized facts3-14.
In empirical ﬁnance the volatilityis an important value to measurethe risk. One
of the stylized facts of the volatility is that the volatility of price retu rns changes
in time and shows clustering, so called ”volatility clustering”. Then the histogram
of the resulting price returns shows a fat-tailed distribution which in dicates that
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2Authors’ Names
the probability of having a large price change is higher than that of th e Gaussian
distribution. In order to mimic these empirical properties of the vola tility and to
forecast the future volatility values, Engle advocated the autore gressive conditional
hetroskedasticity (ARCH) model15where the volatility variable changes determin-
istically depending on the past squared value of the return. Later t he ARCH model
is generalized by adding also the past volatility dependence to the vola tility change.
This model is known asthe generalizedARCH(GARCH) model16. The parameters
of the GARCH model applied to ﬁnancial time series are conventionally determined
by the maximum likelihood method. There are many extended versions of GARCH
models, such as EGARCH17, GJR18, QGARCH19,20models etc., which are de-
signed to increase the ability to forecast the volatility value.
The stochastic volatility (SV) model21,22is another model which captures the
propertiesofthevolatility.IncontrasttotheGARCHmodel,thevo latilityoftheSV
model changes stochastically in time. As a result the likelihood functio n of the SV
model is given as a multiple integral of the volatility variables. Such an in tegral in
general is not analytically calculable and thus the determination of th e parameters
of the SV model by the maximum likelihood method becomes diﬃcult. To o vercome
this diﬃculty in the maximum likelihood method the Markov Chain Monte Ca rlo
(MCMC) method based on the Bayesian approach is proposed and de veloped21. In
the MCMC of the SV model one has to update not only the parameter variables
but also the volatility ones from a joint probability distribution of the p arameters
and the volatility variables. The number of the volatility variables to be updated
increases with the data size of time series. The ﬁrst proposed upda te scheme of
the volatility variables is based on the local update such as the Metro polis-type
algorithm21. It is however known that when the local update scheme is used for
the volatility variables having interactions to their neighbor variables in time, the
autocorrelationtime ofsampledvolatilityvariablesbecomeslargeand thusthe local
update scheme becomes ineﬀective23. In order to improve the eﬃciency of the local
update method the blocked scheme which updates several variable s at once is also
proposed23,24. A recent survey on the MCMC studies of the SV model is seen in
Ref.25.
In our study we use the HMC algorithm26which had not been considered
seriously for the MCMC simulation of the SV model. In ﬁnance there ex ists an
application of the HMC algorithm to the GARCH model27where three GARCH
parameters are updated by the HMC scheme. It is more interesting to apply the
HMC for updates of the volatility variables because the HMC algorithm is a global
update scheme which can update all variables at once. This feature of the HMC
algorithm can be used for the global update of the volatility variables which can not
be achieved by the standard Metropolis algorithm. A preliminary stud y28shows
that the HMC algorithmsamplesthe volatilityvariableseﬀectively.In t his paperwe
give a detailed description of the HMC algorithm and examine the HMC alg orithm
with artiﬁcial ﬁnancial data up to the data size of T=5000. We also ma ke an
empirical analysis of the Nikkei 225 stock index by the HMC algorithm.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
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2. Stochastic Volatility Model
The standard version of the SV model21,22is given by
yt=σtǫt= exp(ht/2)ǫt, (1)
ht=µ+φ(ht−1−µ)+ηt, (2)
whereyt= (y1,y2,...,yn) represents the time series data, htis deﬁned by ht= lnσ2
t
andσtiscalledvolatility.Wealsocall htvolatilityvariable.Theerrorterms ǫtandηt
are taken from independent normal distributions N(0,1) andN(0,σ2
η) respectively.
We assume that |φ|<1. When φis close to one, the model exhibits the strong
persistency of the volatility shock.
For this model the parameters to be determined are µ,φandσ2
η. Let us use θ
asθ= (µ,φ,σ2
η). Then the likelihood function L(θ) for the SV model is written as
L(θ) =/integraldisplayn/productdisplay
t=1f(ǫt|σ2
t)f(ht|θ)dh1dh2...dhn, (3)
where
f(ǫt|σ2
t) =/parenleftbig
2πσ2
t/parenrightbig−1
2exp/parenleftbigg
−y2
t
2σ2
t/parenrightbigg
, (4)
f(h1|θ) =/parenleftBigg
2πσ2
η
1−φ2/parenrightBigg−1
2
exp/parenleftbigg
−[h1−µ]2
2σ2η/(1−φ2)/parenrightbigg
, (5)
f(ht|θ) =/parenleftbig
2πσ2
η/parenrightbig−1
2exp/parenleftbigg
−[ht−µ−φ(ht−1−µ)]2
2σ2η/parenrightbigg
. (6)
As seen in Eq.(3), L(θ) is constructed as a multiple integral of the volatility vari-
ables. For such an integral it is diﬃcult to apply the maximum likelihood me thod
which estimates values of θby maximizing the likelihood function. Instead of using
the maximum likelihood method we perform the MCMC simulations based o n the
Bayesian inference as explained in the next section.
3. Bayesian inference for the SV model
From the Bayes’ rule, the probability distribution of the parameter sθis given by
f(θ|y) =1
ZL(θ)π(θ), (7)
whereZis the normalization constant Z=/integraltext
L(θ)π(θ)dθandπ(θ) is a prior disti-
bution of θfor which we make a certian assumption. The values of the paramete rs
are inferred as the expectation values of θgiven by
/an}bracketle{tθ/an}bracketri}ht=/integraldisplay
θf(θ|y)dθ. (8)
In general this integral can not be performed analytically. For tha t case, one can
use the MCMC method to estimate the expectation values numerically .November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
4Authors’ Names
In the MCMC method, we ﬁrst generate a series of θwith a probability of
P(θ) =f(θ|y). Letθ(i)= (θ(1),θ(2),...,θ(k)) be values of θgenerated by the MCMC
sampling. Then using these kvalues the expectation value of θis estimated by an
average as
/an}bracketle{tθ/an}bracketri}ht=1
kk/summationdisplay
i=1θ(i). (9)
The statistical error for kindependent samples is proportional to1√
k. When the
sampled data are correlated the statistical error will be proportio nal to/radicalbigg
2τ
kwhere
τis the autocorrelation time between the sampled data. The value of τdepends
on the MCMC sampling scheme we take. In order to reduce the statis tical error
within limited sampled data it is better to choose an MCMC method which is able
to generate data with a small τ.
3.1.MCMC Sampling of θ
For the SV model, in addition to θ, volatility variables htalso have to be updated
sincetheyshouldbeintegratedoutasinEq.(3).Let P(θ,ht)be thejointprobability
distribution of θandht. ThenP(θ,ht) is given by
P(θ,ht)∼¯L(θ,ht)π(θ), (10)
where
¯L(θ,ht) =n/productdisplay
t=1f(ǫt|ht)f(ht|θ). (11)
For the prior π(θ) we assume that π(σ2
η)∼(σ2
η)−1and for others π(µ) =π(φ) =
constant.
The MCMC sampling methods for θare given in the following21,22. The prob-
ability distribution for each parameter can be derived from Eq.(10) b y extracting
the part including the corresponding parameter.
•σ2
ηupdate scheme.
The probability distribution of σ2
ηis given by
P(σ2
η)∼(σ2
η)−n
2−1exp/parenleftbigg
−A
σ2η/parenrightbigg
, (12)
where
A=1
2{(1−φ2)(h1−µ)2+n/summationdisplay
t=2[ht−µ−φ(ht−1−µ)]2}.(13)
Since Eq.(12) is an inverse gamma distribution we can easily draw a value
ofσ2
ηby using an appropriate statistical library in the computer.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
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•µupdate scheme.
The probability distribution of µis given by
P(µ)∼exp/braceleftbigg
−B
2σ2η(µ−C
B)2/bracerightbigg
, (14)
where
B= (1−φ2)+(n−1)(1−φ)2, (15)
and
C= (1−φ2)h1+(1−φ)n/summationdisplay
t=2(ht−φht−1). (16)
µis drawn from a Gaussian distribution of Eq.(14).
•φupdate scheme.
The probability distribution of φis given by
P(φ)∼(1−φ2)1/2exp{−D
2σ2η(φ−E
D)2}, (17)
where
D=−(h1−µ)2+n/summationdisplay
t=2(ht−1−µ)2, andE=/summationtextn
t=1(ht−µ)(ht−1−µ).(18)
In order to update φwith Eq.(17), we use the Metropolis-Hastings
algorithm30,31. Let us write Eq.(17) as P(φ)∼P1(φ)P2(φ) where
P1(φ) = (1−φ2)1/2, (19)
P2(φ)∼exp{−D
2σ2η(φ−E
D)2}. (20)
SinceP2(φ) is a Gaussian distribution we can easily draw φfrom Eq.(20).
Letφnewbe a candidate given from Eq.(20). Then in order to obtain the
correct distribution, φnewis accepted with the following probability PMH.
PMH= min/braceleftbiggP(φnew)P2(φ)
P(φ)P2(φnew),1/bracerightbigg
= min/braceleftBigg/radicalBigg
(1−φ2new)
(1−φ2),1/bracerightBigg
.(21)
In addition to the abovestep we restrict φwithin [−1,1]to avoida negative
value in the calculation of square root.
3.2.Probability distribution for ht
The probability distribution of the volatility variables htis given by
P(ht)≡P(h1,h2,...,hn)∼ (22)
exp/parenleftBig
−/summationtextn
i=1{ht
2+ǫ2
t
2e−ht}−[h1−µ]2
2σ2
η/(1−φ2)−/summationtextn
i=2[ht−µ−φ(ht−1−µ)]2
2σ2
η/parenrightBig
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Thisprobabilitydistributionisnotasimplefunction todrawvaluesof ht.Aconven-
tional method is the Metropolis method30,31which updates the variables locally.
There are several methods21,22,23,24developed to update htfrom Eq.(22). Here
we use the HMC algorithm to update htglobally. The HMC algorithm is described
in the next section.
4. Hybrid Monte Carlo Algorithm
Originallythe HMCalgorithmis developedforthe MCMCsimulationsofthe lattice
QuantumChromoDynamics(QCD) calculations26. Amajordiﬃcultyofthe lattice
QCDcalculationsistheinclusionofdynamicalfermions.Theeﬀectoft hedynamical
fermions is incorporated by the determinant of the fermion matrix. The computa-
tional work of the determinant calculation requires O(V3) arithmetic operations29,
whereVis the volume of a 4-dimensional lattice. A typical size of the volume is
V >104. The standard Metropolis algorithm which locally updates variables do es
not work since each local update requires O(V3) arithmetic operations for a deter-
minant calculation,which results in unacceptable computational cos t in total. Since
the HMC algorithm is a global update method, the computational cos t remains in
the acceptable region.
The basic idea of the HMC algorithm is a combination of molecular dynamic s
(MD) simulation and Metropolis accept/reject step. Let us conside r to evaluate the
following expectation value /an}bracketle{tO(x)/an}bracketri}htby the HMC algorithm.
/an}bracketle{tO(x)/an}bracketri}ht=/integraldisplay
O(x)f(x)dx=/integraldisplay
O(x)elnf(x)dx, (23)
wherex= (x1,x2,...,xn),f(x) is a probability density and O(x) stands for an
function of x. First we introduce momentum variables p= (p1,p2,...,pn) conjugate
to the variables xand then rewrite Eq.(23) as
/an}bracketle{tO(x)/an}bracketri}ht=1
Z/integraldisplay
O(x)e−1
2p2+lnf(x)dxdp=1
Z/integraldisplay
O(x)e−H(p,x)dxdp. (24)
whereZis a normalization constant given by
Z=/integraldisplay
exp/parenleftbigg
−1
2p2/parenrightbigg
dp, (25)
andp2stands for/summationtextn
i=1p2
i.H(p,x) is the Hamiltonian deﬁned by
H(p,x) =1
2p2−lnf(x). (26)
Note that the introduction of pdoes not change the value of /an}bracketle{tO(x)/an}bracketri}ht.
In the HMC algorithm, new candidates of the variables ( p,x) are drawn by
integrating the Hamilton’s equations of motion,
dxi
dt=∂H
∂pi, (27)
dpi
dt=−∂H
∂xi. (28)November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
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In general the Hamilton’s equations of motion arenot solved analytic ally. Therefore
wesolvethemnumericallybydoingthe MDsimulation.Let TMD(∆t) beanelemen-
tary MD step with a step size ∆ t, which evolves ( p(t),x(t)) to (p(t+∆t),x(t+∆t)):
TMD(∆t) : (p(t),x(t))→(p(t+∆t),x(t+∆t)). (29)
Any integrator can be used for the MD simulation provided that the f ollowing
conditions are satisﬁed26
•area preserving
dp(t)dx(t)dx=dp(t+∆t)dx(t+∆t). (30)
•time reversibility
TMD(−∆t) : (p(t+∆t),x(t+∆t))→(p(t),x(t)). (31)
The simplest and often used integrator satisfying the above two co nditions is
the 2nd order leapfrog integrator given by
xi(t+∆t/2) =xi(t)+∆t
2pi(t)
pi(t+∆t) =p(t)i−∆t∂H
∂xi
xi(t+∆t) =xi(t+∆t/2)+∆t
2pi(t+∆t). (32)
In this study we use this integrator.The numericalintegration is pe rformedNsteps
repeatedly by Eq.(32) and in this case the total trajectory length λof the MD is
λ=N×∆t.
At the end of the trajectory we obtain new candidates ( p′,x′). These candidates
are accepted with the Metropolis test, i.e. ( p′,x′) are globally accepted with the
following probability,
P= min{1,exp(−H(p′,x′))
exp(−H(p,x))}= min{1,exp(−∆H)}, (33)
where∆Histhe energydiﬀerencegivenby∆ H=H(p′,x′)−H(p,x). Sinceweinte-
grate the Hamilton’s equations of motion approximately by an integra tor, the total
Hamiltonianisnotconserved,i.e.∆ H/ne}ationslash= 0.Theacceptanceorthe magnitudeof∆ H
is tuned by the step size ∆ tto obtain a reasonable acceptance. Actually there ex-
ists the optimal acceptance which is about 60 −70%for 2nd order integrators32,33.
Surprisingly the optimal acceptance is not dependent of the model we consider. For
the n-th order integrator the optimal acceptance is expected to be32∼exp/parenleftbigg
−1
n/parenrightbigg
.
We could also use higher order integrators which give us a smaller ener gy dif-
ference ∆ H. However the higher order integrators are not always eﬀective sin ce
they need more arithmetic operations than the lower order integra tors32,33. The
eﬃciency of the higher order integrators depends on the model we consider. ThereNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
8Authors’ Names
also exist improved integrators which have less arithmetic operation s than the con-
ventional integrators34.
For the volatility variables ht, from Eq.(22), the Hamiltonian can be deﬁned by
H(pt,ht) =n/summationdisplay
i=11
2p2
i+n/summationdisplay
i=1{hi
2+ǫ2
i
2e−hi}+[h1−µ]2
2σ2η/(1−φ2)+n/summationdisplay
i=2[hi−µ−φ(hi−1−µ)]2
2σ2η,(34)
wherepiis deﬁned as a conjugate momentum to hi. Using this Hamiltonian we
perform the HMC algorithm for updates of ht.
5. Numerical Studies
In order to test the HMC algorithm we use artiﬁcial ﬁnancial time ser ies data
generatedbythe SVmodel with a setofknownparametersand per formthe MCMC
simulations to the artiﬁcial ﬁnancial data by the HMC algorithm. We als o perform
the MCMC simulations by the Metropolis algorithm to the same artiﬁcial data and
compare the results with those from the HMC algorithm.
Using Eq.(1) with φ= 0.97,σ2
η= 0.05 andµ=−1 we have generated 5000
time series data. The time series generated by Eq.(1) is shown in Fig.1. From those
data we prepared 3 data sets: (1)T=1000 data (the ﬁrst 1000 of the time series),
(2)T=2000data (the ﬁrst 2000ofthe time series)and (3) T=5000 (the whole data).
To these data sets we made the Bayesian inference by the HMC and M etropolis
algorithms.Preciselyspeakingboth algorithmsareusedonlyfor the MCMC update
of the volatility variables. For the update of the SV parameters we u sed the update
schemes in Sec.3.1.
For the volatility update in the Metropolis algorithm, we draw a new can didate
of the volatility variables randomly, i.e. a new volatility hnew
tis given from the
previous value hold
tby
hnew
t=hold
t+δ(r−0.5), (35)
whereris a uniform random number in [0 ,1) andδis a parameter to tune the
acceptance. The new volatility hnew
tis accepted with the acceptance Pmetro
Pmetro= min/braceleftbigg
1,P(hnew
t)
P(hold
t)/bracerightbigg
, (36)
whereP(ht) is given by Eq.(22).
The initial parameters for the MCMC simulations are set to φ= 0.5,σ2
η= 1.0
andµ= 0. The ﬁrst 10000 samples are discarded as thermalization or burn -in
process. Then 200000samples are recorded for analysis. The tot al trajectory length
λof the HMC algorithm is set to λ= 1 and the step size ∆ tis tuned so that the
acceptance of the volatility variables becomes more than 50%.
First we analyze the sampled volatility variables. Fig.2 shows the Mont e Carlo
(MC) history of the volatility variable h100fromT= 2000 data set. We take h100
as the representative one of the volatility variables since we have ob served theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
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0 1000 2000 3000 4000 5000t-6-4-20246yt
Fig. 1. The artiﬁcial SV time series used for this study.
50000 55000 60000
Monte Carlo history-2-10123h100HMC
50000 55000 60000
Monte Carlo history-2-10123h100Metropolis
Fig. 2. Monte Carlo histories of h100generated by HMC (left) and Metropolis (right) with
T= 2000 data set. The Monte Carlo histories in the window from 5 0000 to 60000 are shown.
similar behavior for other volatility variables. See also Fig.3 for the sim ilarity of the
autocorrelation functions of the volatility variables.
AcomparisonofthevolatilityhistoriesinFig.2clearlyindicatesthatth ecorrela-
tion of the volatility variable sampled from the HMC algorithm is smaller th an that
from the Metropolis algorithm. To quantify this we calculate the auto correlation
function (ACF) of the volatility variable. The ACF is deﬁned as
ACF(t) =1
N/summationtextN
j=1(x(j)−/an}bracketle{tx/an}bracketri}ht)(x(j+t)−/an}bracketle{tx/an}bracketri}ht)
σ2x, (37)
where/an}bracketle{tx/an}bracketri}htandσ2
xare the average value and the variance of xrespectively.
Fig.3 shows the ACF for three volatility variables, h10,h20andh100sampled
by the HMC. It is seen that those volatility variables have the similar co rrelation
behavior. Other volatility variables also show the similar behavior. Thu s hereafter
we only focus on the volatility variable h100as the representative one.
Fig.4 compares the ACF of h100by the HMC and Metropolis algorithms. It
is obvious that the ACF by the HMC decreases more rapidly than that by theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
10Authors’ Names
0 20 40 60 80t0.010.11ACFh10
h20
h100
Fig. 3. Autocorrelation functions of three volatility vari ablesh10,h20andh100sampled by the
HMC algorithm for T= 2000 data set. These autocorrelation functions show the si milar behavior.
0 100 200 300 400 500t0.010.11ACFHMC
Metropolis
Fig. 4. Autocorrelation function of the volatility variabl eh100by the HMC and Metropolis
algorithms for T= 2000 data set.
Metropolis algorithm. We also calculate the autocorrelation time τintdeﬁned by
τint=1
2+∞/summationdisplay
t=1ACF(t). (38)
The results of τintof the volatility variables are given in Table 1. The values in
the parentheses represent the statistical errors estimated by the jackknife method.
We ﬁnd that the HMC algorithm gives a smaller autocorrelation time tha n the
Metropolis algorithm, which means that the HMC algorithm samples the volatility
variables more eﬀectively than the Metropolis algorithm.
Next we analyze the sampled SV parameters. Fig.5 shows MC histories of the
φparameter sampled by the HMC and Metropolis algorithms. It seems t hat both
algorithms have the similar correlationfor φ. This similarity is also seen in the ACF
in Fig.6(left), i.e. both autocorrelation functions decrease in the sim ilar rate with
timet. The autocorrelation times of φare very large as seen in Table 1. We also
ﬁnd the similar behavior for σ2
η, i.e. both autocorrelation times of σ2
ηare large.
On the other hand we see small autocorrelations for µas seen in Fig.6(right).
Furthermore we observe that the HMC algorithm gives a smaller τintforµthanNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
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φ µ σ2
η h100
true 0.97 -1 0.05
T=1000 HMC 0.973 -1.13 0.053
SD 0.010 0.51 0.017
SE 0.0004 0.003 0.001
2τint 360(80) 3.1(5) 820(200) 12(1)
Metropolis 0.973 -1.14 0.053
SD 0.011 0.40 0.017
SE 0.0005 0.003 0.0013
2τint 320(60) 10.1(8) 720(160) 190(20)
T=2000 HMC 0.978 -0.92 0.053
SD 0.007 0.26 0.012
SE 0.0003 0.001 0.0009
2τint 540(60) 3(1) 1200(150) 18(1)
Metropolis 0.978 -0.92 0.052
SD 0.007 0.26 0.011
SE 0.0003 0.003 0.0009
2τint 400(100) 13(2) 1000(270) 210(50)
T=5000 HMC 0.969 -1.00 0.056
SD 0.005 0.11 0.009
SE 0.0003 0.0004 0.0007
2τint 670(100) 4.2(7) 1250(170) 10(1)
Metropolis 0.970 -1.00 0.054
SD 0.005 0.12 0.008
SE 0.00023 0.0011 0.0005
2τint 510(90) 30(10) 960(180) 230(28)
Table 1. Results estimated by the HMC and Metropolis algorit hms.SDstands for Standard
Deviation and SEstands for Statistical Error. The statistical errors are es timated by the jackknife
method. We observe no signiﬁcant diﬀerences on the autocorr elation times among three data sets.
that of the Metropolis algorithm, which means that HMC algorithm sam plesµ
more eﬀectively than the Metropolis algorithm although the values of τintforµ
take already very small even for the Metropolis algorithm.
The values of the SV parameters estimated by the HMC and the Metr opolis
algorithms are listed in Table 1. The results from both algorithms well r eproduce
the true values used for the generation of the artiﬁcial ﬁnancial d ata. Furthermore
for each parameter and each data set, the estimated parameter s by the HMC and
the Metropolis algorithms agree well. And their standard deviations a lso agree
well. This is not surprising because the same artiﬁcial ﬁnancial data, thus the same
likelihood function is usedfor both MCMC simulationsby the HMC and Met ropolis
algorithms. Therefore they should agree each other.November 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
12Authors’ Names
40000 45000 50000
MC history0.940.950.960.970.980.991φ
HMC
40000 45000 50000
MC history0.940.950.960.970.980.991φ
Metropolis
Fig. 5. Monte Carlo histories of φgenerated by HMC (left) and Metropolis (right) for T= 2000
data set.
0 1000t0.010.11ACFHMC
Metropolis
0 100 200 300t0.0010.010.1 ACFHMC
Metropolis
Fig. 6. Autocorrelation functions of φ(left) and µ(right) by the HMC and Metropolis algorithm
forT= 2000 data set.
6. Empirical Analysis
In this section we make an empirical study of the SV model by the HMC algorithm.
The empirical study is based on daily data of the Nikkei 225 stock inde x. The
sampling period is 4 January 1995 to 30 December 2005 and the numbe r of the
observations is 2706. Fig.7(left) shows the time series of the data. Letpibe the
Nikkei 225 index at time i. The Nikkei 225 index piare transformed to returns as
ri= 100ln( pi/pi−1−¯s), (39)
where ¯sis the average value of ln( pi/pi−1). Fig.7(right) shows the time series of
returns calculated by Eq.(39). We perform the same MCMC sampling b y the HMC
algorithm as in the previous section. The ﬁrst 10000 MC samples are d iscarded and
then 20000 samples are recorded for the analysis. The ACF of samp ledh100and
sampled parameters are shown in Fig.8. Qualitatively the results of t he ACF are
similar to those from the artiﬁcial ﬁnancial data, i.e. the ACF of the v olatility and
µdecrease quickly although the ACF of φandσ2
ηdecrease slowly. The estimated
values of the parameters are summarized in Table 2. The value of φis estimated to
beφ≈0.977. This value is very close to one, which means the time series has th e
strong persistency of the volatility shock. The similar values are also seen in theNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
Instructions for Typesetting Manuscripts (Condensed Titl e for the Paper) 13
HMC φ µ σ2
η h100
0.977 0.52 0.020
SD 0.006 0.13 0.005
SE 0.001 0.0016 0.001
2τint560(190) 4(1) 1120(360) 21(5)
Table 2. Results estimated by the HMC for the Nikkei 225 index data.
050010001500200025003000t10000150002000025000
Nikkei 225 Index
050010001500200025003000t-505rt
Fig. 7. Nikkei 225 stock index from 4 January 1995 to 30 Decemb er 2005(left) and returns(right).
0 20 40 60t0.010.11ACFh100
0 200 400 600800 1000t0.010.11ACFφ
ση2
µ
Fig. 8. Autocorrelation functions of the volatility variab leh100(left) and the sampled parameters
(right).
previous studies21,22.
7. Conclusions
We applied the HMC algorithm to the Bayesian inference of the SV mode l and
examined the property of the HMC algorithm in terms of the autocor relation times
of the sampled data. We observed that the autocorrelation times o f the volatility
variables and µparameter are small. On the other hand large autocorrelation times
are observed for the sampled data of φandσ2
ηparameters. The similar behavior
for the autocorrelation times are also seen in the literature22.
From comparison of the HMC and Metropolis algorithms we ﬁnd that th e HMCNovember 10, 2018 20:49 WSPC/INSTRUCTION FILE svJCSC3
14Authors’ Names
algorithmsamplesthevolatilityvariablesand µmoreeﬀectivelythantheMetropolis
algorithm. However there is no signiﬁcant diﬀerence for φandσ2
ηsampling. Since
the autocorrelation times of µfor both algorithms are estimated to be rather small
the improvement of sampling µby the HMC algorithm is limited. Therefore the
overall eﬃciency is considered to be similar to that of the Metropolis a lgorithm.
By using the artiﬁcial ﬁnancial data we conﬁrmed that the HMC algor ithm cor-
rectly reproduces the true parameter values used to generate t he artiﬁcial ﬁnancial
data. Thus it is concluded that the HMC algorithm can be used as an alt ernative
algorithm for the Bayesian inference of the SV model.
If we are only interested in parameter estimations of the SV model, t he HMC
algorithm may not be a superior algorithm. However the HMC algorithm samples
thevolatilityvariableseﬀectively.ThustheHMC algorithmmayservea saneﬃcient
algorithm for calculating a certain quantity including the volatility varia bles.
Acknowledgments.
The numerical calculations were carried out on SX8 at the Yukawa In stitute for
Theoretical Physics in Kyoto University and on Altix at the Institute of Statistical
Mathematics.
Note added in proof. After this work was completed the author noticed a sim-
ilar approach by Liu35. The author is grateful to M.A. Girolami for drawing his
attention to this.
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