Proceedings of the IConSSE FSM SWCU 2015, pp. MA.48–52 ISBN: 978-602-1047-21-7
SWUP
MA.48
Box–Cox realized asymmetric stochastic volatility model
Didit B. Nugroho
a
and Takayuki Morimoto
b a
Department of Mathematics, Satya Wacana Christian University
b
Department of Mathematical Sciences, Kwansei Gakuin University
Abstract
This study proposes a class of non-linear realized stochastic volatility model by applying the Box–Cox BC transformation, instead of the logarithmic transformation, to the
realized estimator. The proposed models are fitted to daily returns and realized kernel of six stocks: SP500, FTSE100, Nikkei225, Nasdaq100, DAX, and DJIA using an Markov
Chain Monte Carlo MCMC Bayesian method, in which the Hamiltonian Monte Carlo HMC algorithm updates BC parameter and the Riemann manifold HMC RMHMC
algorithm updates latent variables and other parameters that are unable to be sampled directly. Empirical studies provide evidence against log and raw versions of realized
stochastic volatility model.
Keywords Box–Cox transformation, HMC, MCMC, realized stochastic volatility
1. Introduction
Volatility of financial asset returns, defined as the standard deviation of log returns, has played a major role in many financial applications such as option pricing, portfolio
allocation, and value-at-risk models. In the context of SV model, very closely related studies of joint models has been proposed by Dobrev Szerszen 2010, Koopman M. Scharth
2013, and Takahashi et al. 2009. Their model is known as the realized stochastic volatility RSV model.
Recently, the Takahashi et al. 2009s model has been extended by Takahashi et al. 2014 by applying a general non-linear bias correction in the realized variance RV measure.
This study extends the asymmetric version of Takahashi et al. 2014s models, known as the realized asymmetric SV R-ASV model, by applying the BC transformation to the realized
measure. Here we focus on the realized kernel RK measure that has some robustness to market microstructure noise. The realized kernel data behave very well and better than any
available realized variance statistic Goncalves and Meddahi 2011.
Adding an innovation to obtain an RSV model substantially increase the difficulty in parameter estimation. Nugroho Morimoto 2015 provided some comparison of
estimation results between HMC-based samplers and multi-move Metropolis--Hastings sampler provided by Takahashi et al. 2019 and Takahashi et al. 2014, where the RMHMC
sampler give the best performance in terms of inefficiency factor. Therefore, this study employs HMC-based samplers methods to estimate the parameters and latent variables in
our proposed model when draws cannot be directly sampled.
2. Box-Cox R-ASV model
The basic discrete-time RSV model includes three stochastic processes describing the dynamics of returns, logarithmic RV, and logarithmic squared volatility. In this framework,
the conditional volatility of returns not only depends on the returns but also on the realized
D.B. Nugroho, T. Morimoto
SWUP
MA.49
measure of variance. The logarithmic transformation of RV is known to have better finite sample properties than RV. Goncalves and Meddahi 2011 provided a short overview of this
transformation and suggested that the log transformation can be improved upon by choosing values of BC parameter other than zero corresponding to the log transformation. Motivated
by this result, we apply the BC transformation to the RV measure to obtain a class of R-ASV model called the BCR-ASV model that takes the following form:
ü = exp -
› ”
ℎ .Y 7 =
¼
+ ℎ +
…
ℎ = m + † ℎ − m + ×
ℎ ~ •m,
Û
[ ”
o—
”
– N
K O
K P
, 1
where corr Y , ×
= , represents normal distribution, and 7 is the BC transformation for realized RV defined as
7 = 7 ü , G = ß
\ ]
o À
, if G ≠ 0, log ü
, if G = 0. 2
Notice that this transformation contains the log transformation for RV when G = 0 and the
raw statistic when G = 1 as special cases.
As in previous studies, in order to adopt a Gaussian nonlinear state space form where the error terms in both return and volatility equations are uncorrelated, we introduce a
transformation ×
= Y + Ý1 − ,
where ~ 0,1 and corr Y ,
= 0. A reparameterization of the model 1 yields ℎ
= m
¼
+ † ℎ − m
¼
+ Æü exp-−
› ”
ℎ . + T ,
ℎ ~ •m
¼
,
Û
[ ”
o—
”
–, where
Æ = and
T = 1 − .
3. MCMC simulation in the BCR-ASV model
