Realized Non-Linear Stochastic Volatility
Model with Leverage Effect and Generalized
Student’s
t
-Error Distributions
Didit B. Nugroho1,2 and Takayuki Morimoto1
1Department of Mathematical Sciences, Kwansei Gakuin University, Japan 2Department of Mathematics, Satya Wacana Christian University, Indonesia
International Conference on Econometrics January 11–12, 2014, Kyoto University
Abstract
1.1 Introduction: LRSV Model
Stochastic volatility (SV) model is among the most important tools for modeling volatility of a financial time series.
Recently,Takahashi et al. (2009) have proposed a leveraged realized
stochastic volatility (LRSV) model which utilizes returns and realized variance (RV) simultaneously:
A natural generalization of the above linear specification is to allow assuming a non-Gaussian error distribution which accommodates flex-ible skewness and heavy-tailedness.
1.2 Introduction: Generalized Student’s
t
-Distribution
A distribution for modeling skewed and heavy-tailed data is the
non-central Students t-distribution (NCT) proposed by Johnson et al.
(1995):
G(ν,µ) = (µ+V)√Zν, V∼ N(0, 1), Zν∼ IG(ν/2,ν/2),
where V and Z are independent and ν is the number of degree of
freedom.
An alternative distribution is the generalized hyperbolic (GH) skew
Student’st-distribution (SKT) proposed byNakajima and Omori (2012),
G(ν,β) =β(Zν−ν/(ν−2)) +
√ ZνV.
When µ,β = 0, the distributions reduce to a central Student’s t
1.3 Introduction: Power Transformations
Manly (1976)introduced a family of exponential transformations (ET):
PET(x,λ) =
John and Draper (1980)proposed the so-called modulus transforma-tion (MT):
Recently,Yeo and Johnson (2000)proposed
2.1 LRNSV model
We now introduce the leverage realized non-linear stochastic volatility
(LRNSV) model with skewed Student’st-distribution expressed as
Rt = exp
1 2ht
Gt(ν,µ,β),
logRVt = β0+β1ht+σut, ut iid∼ N(0, 1),
ht+1 = α+φ(Pt−α) +τηt+1, ηt+1 iid∼ N(0, 1),
h1 ∼ N α,τ2/ 1−φ2
,
2.2 MCMC Methods: HMC and RMHMC
Markov chain Monte Carlo (MCMC) is developed in two steps: 1 construct a Markov chain by Gibbs sampler when draws can be directly sampled and either Hamiltonian Monte Carlo (HMC) or Riemann Manifold HMC when draws cannot be directly sampled, 2 employ Monte Carlo methods for summarizing the posterior
distribution of parameter.
HMC-based methods are based on Hamiltonian dynamics system:
H(θ,ω) =−L(θ) +1
2log
n
(2π)D|M|o+1
2ω′M−1ω,
where L(θ) is the logarithm of the joint probability distribution for
the parameters θ∈RD,Mis the covariance matrix, andω∈RDis
the independent auxiliary variable.
In the RMHMC sampling,Mdepends on the variableθand is chosen
to be the metric tensor, i.e.
2.2 MCMC Methods (Cont’ed)
The full algorithm for HMC or RMHMC can then be summarized in the following three steps.
(1) Randomly draw a sample momentum vector ω∼ N(ω|0,M).
(2) Run the leapfrog algorithm for NL steps with step size∆τ to
generate a proposal (θ∗,ω∗)according to the Hamiltonian equations
dθ
over a fictitious time τ.
(3) Accept (θ∗,ω∗)with probability
3. Data: Returns and Realized Volatility Measures
The series of return are given by
Rt=100×(pt−pt−1), t=1, ...,T,
where pt denotes the log closing price for dayt.
Standard RV over a time interval of one day (under an ideal market
condition or no microstructure noise),Andersen et al. (2001):
RVt=100×
∑
Nk=t2 ptk−ptk−1 2,
whereptkdenotes the log-price at thek’th observation in dayt. In this
study, we consider RVs 1-min adjusted byHansen and Lunde (2005).
We estimate the proposed model using TOPIX data: 1 from January 2004 to December 2007 and
3. Data (Cont’ed)
Table: Descriptive statistics of daily returns in the TOPIX data sets.
JB LB(8)
Mean SD Skewness Kurtosis
(Normality) (Autocorr.) Period: 2004/1/6– 2011/12/30
−0.019 1.479 −0.413 11.24 5611.14 (No) 9.03 (No) Period: 2004/1/6– 2007/12/30
4.1 Results: Symmetrical & Skewness Parameter Estimates
Table: Summary of the posterior sample of the symmetrical (µ) and skewness (β) parameters for the model adopting RV 1-min.
Statistic Period Parameter
Mean (SD) 90%HPD IACT
4.2 Results: Power Parameter Estimates
Table: Summary of the posterior sample of the power parameterλfor the TOPIX 2004–2007 data set.
Statistic Parameter Model
Mean (SD) 90%HPD IACT
4.2 Results (Cont’ed)
Table: Summary of the posterior sample of the power parameterλfor the TOPIX 2004–2011 data set.
Statistic Parameter Model
Mean (SD) 90%HPD IACT
4.3 Results: Model Selection
Table: Log marginal likelihood for various LRSV and LRNSV models evaluated in the TOPIX data set.
4.4 Results: Sensitivity of Priors
5. Conclusion & Extension
Based on the empirical results, we conclude: In particular,
The model with leverage effect and SKT distribution performs the best among the four return error distribution specifications.
The modulus transformation best fitted the lagged volatility for the returns data having a very high kurtosis but worst fitted for the returns data having a small kurtosis.
In general, the non-linear specification model (RNSV) outperforms the linear specification model (RSV).
Performance of the λ posterior simulations showed considerable ro-bustness for priors with very diffused distributional behaviour.
Some References
Andersen, T.G., Bollerslev, T., Diebold, F.X., and Labys, P.(2001). The distri-bution of exchange rate volatility.Journal of the American Statistical Association,96(453), 42–55.
Hansen, P. R. and Lunde, A.(2005). A forecast comparison of volatility models: Does anything beat a GARCH(1,1).Journal of Applied Econometrics,20(7), 873-889.
John, J.A. and Draper, N.R.(1980). An alternative family of transformations.Applied Statistics,29, 190–197.
Johnson, N.L., Kotz, S., Balakrishnan, N.(1995).Continuous Univariate Distributions, John Wiley & Sons.
Manly, B.F.(1976). Exponential data transformation.The Statistician,25, 37–42.
Nakajima, J. and Omori,Y.(2012). Stochastic volatility model with leverage and asym-metrically heavy-tailed error using GH skew Student’st-distribution.Computational Statistics and Data Analysis,56, 3690–3704.
Takahashi, M., Omori, Y., and Watanabe, T.(2009). Estimating stochastic volatility models using daily returns and realized volatility simultaneously.Computational Statistics and Data Analysis,53, 2404–2426.
