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Financial Time Series Analysis of SV
Model by Hybrid Monte Carlo
Tetsuya Takaishi
Hiroshima University of Economics
Shanghai, 2008.09.18
Outline
Motivation
Stylized facts of financial data
Stochastic Volatility model
Bayesian inference
Markov Chain Monte Carlo
Hybrid Monte Carlo
Numerical simulations
Empirical results
Summary
Ｍｏｔｉｖａｔｉｏｎ
Financial time series (Stock price, Exchange rate etc.)
Model parameter estimations
•GARCH model
•Stochastic Volatility (SV) model
Maximum likelihood method
Markov Chain Monte Carlo (MCMC) method
MCMC method
•Metropolis method
•Hybrid Monte Carlo method
Global algorithm
(Bayesian inference)
efficiency?
correlation?
analyze the data with financial models
•Stylized facts for price returns
))(ln())(ln()( ttptptr
•price return
• fat tailed distribution
• volatility clustering
• absence of autocorrelations in return
• long time correlation in absolute return
• etc
Many empirical studies show some properties which can not be obtained from Gaussian noise
Stylized facts of financial data
Oct. 19 1987
Gopikrishnan et al.,cond-mat/9905305
price return
volatility
clustering
Fat-tailed distribution
Financial modeling
To analyze the financial data we use some models
which capture some of stylized facts.
•GARCH model
•Stochastic Volatility (SV) model
Volatility clustering
Fat-tailed distribution
estimate model parameters
popular model
Stochastic Volatility model
,ttty )1,0(~ Nt
),0(~ Nt
Volatility varies in time stochastically.
Thus it is not deterministic.
The volatility at t+1 is not determined from that at t.
ttt hh )( 1 )log(2
tth
time series ty
Volatility variable
How to find ,,
Taylor(1986)
? Bayesian inference by Markov Chain Monte Carlo
Bayesian inference
)()|()|( yfy
Bayes’ theorem)(
)()|()|(
yf
yfy
)|( y ：posterior distribution
)|( yf ：likelihood function
)( ：prior distribution
Probability distribution of θ
)(
)()|()( yfdyf
If there is some information on theta, then we use it for
.)( const
Bayes’ theorem tells us the probability distribution of theta
,,
dyZ
)|(1
Markov Chain Monte Carlo
n
i
in 1
1
)|( y
The value of the parameter is evaluated as an expectation value.
First, generate theta with probability distribution :
We obtain a set of theta ),,,,( 321 n
The generation of theta is performed by MCMC.
probability distribution
of theta
Numerical estimation by MCMC
Metropolis method
))(exp()|( fy
)5.0( d
Local update
uniform random number in [0,1]
)()( ffdh
))exp(,1min( dh
)()()()( PPPP
detailed balance condition
We want to generate theta with
•calculate
•accept with
•draw
Hybrid Monte Carlo
•Molecular dynamics simulations
•Metropolis accept/reject step
HMC is a global algorithm that can update all variables at once.
S. Duane , A.D. Kennedy , B.J. Pendleton, D. Roweth (1987)
HMC consists of two steps:
HMC may de-correlate fast variables updated in MCMC.
dgZ
))(exp(1
)exp(
))(2
1exp( 2
Hdpd
gpdpdZ
)(2
1 2 gpH
Partition function
we introduce momenta p conjugate to θ.
define
Hamiltonian
Hybrid Monte Carlo
This partition function does not change the value of <θ>.
Momenta have no dynamics.
This does not change the results.
))(exp()|( gy
dpdgpZ
))(2
1exp(
1 2
•Solve the Hamilton‘s equations of motion （Molecular dynamics
simulation） for all variables simultaneously.
),(),( pHpH
HHdH
•Metropolis accept/reject step
)1),min(exp( dH
In general, dh is not zero in
the numerical integration.
Hybrid Monte Carlo
Hamiltonian is conserved.
p
H
Hp
pp
Accept new theta with
This can be small.
22
2
2
1
2
1
2
1
22 );|()|()|( ntt
n
t
tt dddfyfyf
Zddddfyf ntt
n
t
tt /)();|()|( 22
2
2
1
2
1
2
1
22
Ex. The expectation value of phi is given by
Note that the number of integrals is n+3.
Hybrid Monte Carlo
Likelihood function of SV model
increases with n
Hybrid Monte Carlo
)()|()|( yfy posterior distribution
We have to integrate
volatilities
multiple integral
Numerical simulations
,ttty
ttt hh )( 1
)1,0(~ Nt
)1,(~ Nt
Generate artificial date (2000data) with the following parameters.
1.0,98.0,0.1
)log(2
tth
fat-tail
Metropolis
HMC
)log(2
1010 h
Metropolis
HMC
Metropolis
HMC
Metropolis
HMC
Autocorrelation function h10
small correlation
True
σ_
０．１
φ
0.98
μ
-1.0
h10
MCMC 0.1070(42) 0.9772(11) -1.064(64)
τ 300(60) 100(13) 1.2(1) 6.6(2)
The average values of the parameters are estimated from 20 independent
time series.
Numerical Simulations
1
)(2
1
t
tACFAutocorrelation time
τ 460(70) 130(16) 6.5(6) 150(15)
Metropolis
Empirical results
US$/MARK exchange rate 2 Jan 1980 to 31 May 1990
100]))(ln())1([ln()( riytyir
return
σ_ φ μ h10
0.0438(16) 0.9560(13) -.95197(78)
1400(500) 980(300) 5(1.3) 30(11)
MCMC results
US/MARK 2JAN1980 to 31MAY 1990
return
2
10h
Summary
•Hybrid Monte Carlo method is applied for MCMC of the
stochastic volatility model.
•HMC de-correlates volatility variables fast enough.
•On the other hand, the correlations of the parameters are not
well improved.
•HMC method improves the efficiency of MCMC partially.
•HMC method is an alternative method to perform MCMC simulations of SV model.
,ttty
,1
2
1
22
n
j
jtj
m
i
itit y
GARCH(m,n) model
)1,0(~ Nt
Likelihood function:
2
2
21 2exp
2
1),,|(
t
t
t
n
t
yyf
n
t t
tn
t
t
nyf
12
2
1
2
2
1ln
2
1)2ln(
2)|(ln
GARCH model
Maximize this function by a certain method.
Volatility varies in time
tytime series
volatility
Generalized Autoregressive
Conditional Heteroscedasticity
Bayesian inference
Ex. Gaussian time series , tty
)(2
)(exp
2
1)|(
2
2
21
tn
t
yy
2
0
2
0
2
02
)(exp
2
1)(
Assume sigma is known.
Take the following prior density
2
0
2
0
2
0
2
2
21 2
)(exp
2
1
2
)(exp
2
1)|(
t
n
t
yy
Then we obtain the posterior density,
Bayesian inference
2
2
2
0
2
0
2
0
2
2
21
2
)(exp
2
)(exp
2
1
2
)(exp
2
1)|(
k
yy t
n
t
2
0
2
2
0
0
2
11
n
n
x
k
2
0
2
2
1
1
nk
n
t
tyx1
Bayesian inference
When is given, how can we infer the value of μ?)|( y
dk
Z
dyZ
2
2
2
)(exp
1
)|(1
The value of μ is given as the expectation value of it.
2
0
2
2
0
0
2
11
n
n
x
k
x
In the limit of n
Maximum likelihood estimation
(MLE)
, tty
Ex：Gaussian time series
)1,0(~ Nt
Likelihood function:
2
2
21 2
)(exp
2
1),|(
t
n
t
yyf
n
t
tynn
yf1
2
2
2 )(2
1ln
2)2ln(
2)|(ln
The values of the parameters are given by maximizing the likelihood function.
n
t
tyn 1
22 )(1
Suppose we have n data and infer parameters of the model from n data.
variance
Gaussian noise with mean 0 and variance 1
This maximization is easy,
n
t
tyn 1
1 average
maximize
Markov Chain Monte Carlo
),,(
333222111
4321
The generation of theta is done in the Markov chain.
Typical algorithm to perform MCMC
Metropolis algorithm
When we have multi-parameter
Each parameter is updated sequentially.
Disadvantage of MCMC: data are correlated
1. chose a new value as candidate for theta
2. calculate
3. accept the new one with the probability
unless keep the old one.
4. return to 1.
Metropolis method
))(exp()|( fy )5.0( d
Local update
uniform random number in [0,1])()( ffdh
))exp(,1min( dh
)()()()( PPPP
detailed balance condition
generate theta with
Metropolis method
),,,( ),,,(
In the case of multi-parameter
)()( ffdh
All parameters could be updated at once…
))exp(,1min( dh
but in this case, the difference dh could be large,
This is large.acceptance ratio becomes small
22
2
2
1
2
1
2
1
22 );|()|()|( ntt
n
t
tt dddfyfyf
2
2
2
22
2exp
2
1)|(
t
t
t
tt
yyf
2
22
1
2
22
2
1
2
2
}]){ln()[ln(exp
2
1);|(
tt
t
ttf
multiple integral# of integrations is # of data
Likelihood function of SV model
Bayesian inference by Markov Chain Monte Carlo
Maximum likelihood method is not applicable.
,,
Hgg , , Poisson bracket
HggHL ,)(
)())(exp()( tgHtLttg
VT
xfLpLHL
))((2/)( 2
)()2/exp()exp()2/exp())(exp( 3ttTtVtTHtL
Leapfrog integrator
In general, we cannot solve this.
operator
g is x or p.
Simplectic integrator
Hamilton's equations of motion
2/)()2/()( :3
)2/()()( :2
2/)()()2/( :1
tttptttt
tttH
tpttp
ttpttt
θ
p
2/t
t
Δｔ is chosen such that the acceptance ratio takes 60～70％.
elementary step
repeat this step
Leapfrog integrator
t
t
optimal acceptance
０：initialize parameters
１：update φ、sigma_eta、μ
２：update volatilities h
３：return to 1Hybrid Monte Carlo
Numerical simulations
Gibbs samplerMetropolis method
Use artificial data with known parameters
•accumulate data
•evaluate parameters parameters of artificial data
Simulations
2
1222
exp)|(
C
yPT
T
t
tt hhhC2
2
1
2
1
2 )())(1(2
1where
Gamma distribution
Sigma_eta update
phi and mu are also easily updated.
Gaussian distribution
distribution of phi
