PROS Didit BN, Takayuki M Box Cox realized fulltext
Proceedings of the IConSSE FSM SWCU (2015), pp. MA.48–52
ISBN: 978-602-1047-21-7
MA.48
Box–Cox realized asymmetric stochastic volatility model
Didit B. Nugrohoa and Takayuki Morimotob
a
Department of Mathematics, Satya Wacana Christian University
Department of Mathematical Sciences, Kwansei Gakuin University
b
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. (2009)'s 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. (2014)'s 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
SWUP
D.B. Nugroho, T. Morimoto
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
P
7 = ¼+ ℎ + …
K
(1)
ℎ " =m+† ℎ −m + × " ,
O
”
Û[
K
ℎ ~ •m, o—
”–
N
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,
(2)
log ü
, if G = 0.
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 " = m¼ + † ℎ − m¼ + Æü exp-−›”ℎ . + T^ " ,
where Æ =
ℎ ~ •m¼ ,
Û[”
–,
o—”
and T = 1 −
.
3. MCMC simulation in the BCR-ASV model
Let
= )ü +/# , ½ = )ü +/# , _ = )ℎ +/# , & = -G, ¼ , , … . , and & =
m, †, Æ, T . By Bayes' theorem, the joint posterior distribution of &, _ given R and V is
& , & , _| , ½ = & , & ×
|_ × ½|& , _ × _|& , ,
(3)
where & , & is a joint prior distribution on parameter & , & .
The MCMC method simulates a new value for each parameter in turn from its
conditional posterior distribution assuming that the current values for the other parameters
are true values. We employ the MCMC scheme as follows:
0) Initialize & , & , _ .
1) Given the current values, draw ¼ , , … , m, Æ, and T from their conditional posteriors
directly.
2) Given ¼ , , … , and ½, draw G by HMC.
3) Given m, Æ, T, _, and , draw † by RMHMC.
4) Given & , & , , and ½, draw _ by RMHMC.
For steps (1), (3), and (4), we implement the same sampling scheme proposed in Nugroho &
Morimoto (2014, 2015). In the following, we study the sampling scheme in step (2) only.
As far as the power parameter G ≠ 0 is only concerned to be sampled and by taking
the normal distribution ŽÀ , À for the prior of G, the logarithm of conditional posterior
distribution for G on the basis of Eq. (3) has the following expression:
SWUP
Box–Cox realized asymmetric stochastic volatility model
< G ∝−
∑/# È
Û”
B
MA.50
$\] o
À
−
¼
−
ℎÉ −
Ào•] ”
,
$]
which is not of standard form, and therefore we cannot sample G from it directly. In this case,
we implement the HMC algorithm for sampling G because the metric tensor required to
implement the RMHMC sampling scheme cannot be explicitly derived from the above
distribution.
4. Empirical results on real data sets
In this section, the BCR-ASV model and MCMC algorithm discussed in the previous
section are applied to the daily returns and RK data for six different international stock
indices: DAX, DJIA, FTSE100, Nasdaq100, Nikkei225, and SP500. These data sets were
downloaded from Oxford-Man Institute ``realised library" and range from January 2000 to
December 2013. Analyses are presented for the sampling efficiency, key parameters that
build the extension models.
We run the MCMC algorithm for 15,000 iterations but discard the first 5,000 draws.
From the resulting 10,000 draws which have passed the Geweke (1992)'s convergence test
for each parameter, we calculate the posterior mean, standard deviation (SD), and 95%
highest posterior density (HPD) interval. Summary of the posterior simulation results for G
are presented in Table 1.
Table 1. Summary of the posterior sample of parameter G in the BCR-ASV model for six
data sets from January 2000 to December 2013.
Statistic
DAX
DJIA
FTSE100
Nasdaq100 Nikkei225
SP500
Mean
–0.0406
0.1226
0.1572
0.0420
0.0503
0.1149
SD
0.0104
0.0099
0.0096
0.0096
0.0128
0.0094
LB
–0.0609
0.1022
0.1385
0.0242
0.0241
0.0968
UB
–0.0202
0.1409
0.1765
0.0617
0.0741
0.1340
Note: LB and UB denote lower and upper bounds of 95% HPD interval, respectively.
The posterior means of G are positive for DJIA, FTSE100, Nasdaq100, Nikkei225, and
SP500 and negative for DAX. In all cases, the 95% HPD interval of G exclude 0 (the log
transformation) or 1 (the raw version). Although not reported, we find that even 99% HPD
interval of G exclude 0 or 1, providing significant evidence to transform the RV series using
the BC transformation against both the logarithmic transformation and no transformation
cases.
To understand the effect of BC transformation, we present the marginal densities of
the transformed RK implied by G = 0 and the estimated G for each stock index in Figure 1.
The marginal density of the BC RV is obviously different from that of the log RV. When the
estimated G is a positive value, the left tail of the former is lighter than that of the latter,
while the right tail of former is heavier, and vice versa when the estimated G is a negative
value. Moreover the marginal density of the former has a heavier right tail than left tail (as
that of the latter) for both positive and negative values of G. Furthermore, the marginal
density of the former has a higher peak than, and a different location from, that of the latter.
SWUP
D.B. Nugroho, T. Morimoto
MA.51
Figure 1. Marginal densities of the log RV and BC-RV for six different international stock
indices.
5. Conclusions and extensions
This study presented an extension to the R-ASV models, in which the Box-Cox
transformation is applied to realized measure. The HMC and RMHMC algorithms were
proposed to implement the MCMC method for updating several parameters and the latent
variables in the proposed model, taking advantage of updating the entire latent volatility at
once. The model and sampling algorithms were applied to the daily returns and realized
kernels of six international stock indices. The estimation results showed very strong evidence
supporting the Box-Cox transformation of realized measure. The model can be extended by
assuming a non-Gaussian density of return errors.
References
Dobrev, D., & Szerszen, P. (2010). The information content of high-frequency data for estimating equity
return models and forecasting risk. Working Paper, Finance and Economics Discussion Series.
Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of
posterior moments. In J. M. Bernardo, J. O. Berger, A. P. Dawid, & A. F. M. Smith (Eds.), Bayesian
Statistics 4 (pp. 169–194 (with discussion)). Clarendon Press, Oxford, UK.
Goncalves, S., & Meddahi, N. (2011). Box-Cox transforms for realized volatility. Journal of
Econometrics, 160, 129–144.
Kim, S., Shephard, N., & Chib, S. (2005). Stochastic volatility: likelihood inference and comparison with
ARCH models. In N. Shephard, ed., Stochastic Volatility: Selected Readings (pp. 283–322). Oxford
University Press, New York.
Koopman, S.J., & Scharth, M. (2013). The analysis of stochastic volatility in the presence of daily
realized measures. Journal of Financial Econometrics, 11(1), 76–115.
Nugroho, D.B., & Morimoto, T. (2014). Realized non-linear stochastic volatility models with
asymmetric effects and generalized Student's t-distributions. Journal of The Japan Statistical
Society, 44, 83–118.
Nugroho, D.B., & Morimoto, T. (2015). Estimation of realized stochastic volatility models using
Hamiltonian Monte Carlo-based methods. Computational Statistics, 30(2), 491–516.
Takahashi, M., Omori, Y., & Watanabe, T. (2009). Estimating stochastic volatility models using daily
returns and realized volatility simultaneously. Computational Statistics and Data Analysis, 53,
2404–2426.
SWUP
Box–Cox realized asymmetric stochastic volatility model
MA.52
Takahashi, M., Omori, Y., & Watanabe, T. (2014). Volatility and quantile forecasts by realized
stochastic volatility models with generalized hyperbolic distribution. Working Paper Series
CIRJE-F-921, CIRJE, Faculty of Economics, University of Tokyo.
SWUP
ISBN: 978-602-1047-21-7
MA.48
Box–Cox realized asymmetric stochastic volatility model
Didit B. Nugrohoa and Takayuki Morimotob
a
Department of Mathematics, Satya Wacana Christian University
Department of Mathematical Sciences, Kwansei Gakuin University
b
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. (2009)'s 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. (2014)'s 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
SWUP
D.B. Nugroho, T. Morimoto
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
P
7 = ¼+ ℎ + …
K
(1)
ℎ " =m+† ℎ −m + × " ,
O
”
Û[
K
ℎ ~ •m, o—
”–
N
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,
(2)
log ü
, if G = 0.
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 " = m¼ + † ℎ − m¼ + Æü exp-−›”ℎ . + T^ " ,
where Æ =
ℎ ~ •m¼ ,
Û[”
–,
o—”
and T = 1 −
.
3. MCMC simulation in the BCR-ASV model
Let
= )ü +/# , ½ = )ü +/# , _ = )ℎ +/# , & = -G, ¼ , , … . , and & =
m, †, Æ, T . By Bayes' theorem, the joint posterior distribution of &, _ given R and V is
& , & , _| , ½ = & , & ×
|_ × ½|& , _ × _|& , ,
(3)
where & , & is a joint prior distribution on parameter & , & .
The MCMC method simulates a new value for each parameter in turn from its
conditional posterior distribution assuming that the current values for the other parameters
are true values. We employ the MCMC scheme as follows:
0) Initialize & , & , _ .
1) Given the current values, draw ¼ , , … , m, Æ, and T from their conditional posteriors
directly.
2) Given ¼ , , … , and ½, draw G by HMC.
3) Given m, Æ, T, _, and , draw † by RMHMC.
4) Given & , & , , and ½, draw _ by RMHMC.
For steps (1), (3), and (4), we implement the same sampling scheme proposed in Nugroho &
Morimoto (2014, 2015). In the following, we study the sampling scheme in step (2) only.
As far as the power parameter G ≠ 0 is only concerned to be sampled and by taking
the normal distribution ŽÀ , À for the prior of G, the logarithm of conditional posterior
distribution for G on the basis of Eq. (3) has the following expression:
SWUP
Box–Cox realized asymmetric stochastic volatility model
< G ∝−
∑/# È
Û”
B
MA.50
$\] o
À
−
¼
−
ℎÉ −
Ào•] ”
,
$]
which is not of standard form, and therefore we cannot sample G from it directly. In this case,
we implement the HMC algorithm for sampling G because the metric tensor required to
implement the RMHMC sampling scheme cannot be explicitly derived from the above
distribution.
4. Empirical results on real data sets
In this section, the BCR-ASV model and MCMC algorithm discussed in the previous
section are applied to the daily returns and RK data for six different international stock
indices: DAX, DJIA, FTSE100, Nasdaq100, Nikkei225, and SP500. These data sets were
downloaded from Oxford-Man Institute ``realised library" and range from January 2000 to
December 2013. Analyses are presented for the sampling efficiency, key parameters that
build the extension models.
We run the MCMC algorithm for 15,000 iterations but discard the first 5,000 draws.
From the resulting 10,000 draws which have passed the Geweke (1992)'s convergence test
for each parameter, we calculate the posterior mean, standard deviation (SD), and 95%
highest posterior density (HPD) interval. Summary of the posterior simulation results for G
are presented in Table 1.
Table 1. Summary of the posterior sample of parameter G in the BCR-ASV model for six
data sets from January 2000 to December 2013.
Statistic
DAX
DJIA
FTSE100
Nasdaq100 Nikkei225
SP500
Mean
–0.0406
0.1226
0.1572
0.0420
0.0503
0.1149
SD
0.0104
0.0099
0.0096
0.0096
0.0128
0.0094
LB
–0.0609
0.1022
0.1385
0.0242
0.0241
0.0968
UB
–0.0202
0.1409
0.1765
0.0617
0.0741
0.1340
Note: LB and UB denote lower and upper bounds of 95% HPD interval, respectively.
The posterior means of G are positive for DJIA, FTSE100, Nasdaq100, Nikkei225, and
SP500 and negative for DAX. In all cases, the 95% HPD interval of G exclude 0 (the log
transformation) or 1 (the raw version). Although not reported, we find that even 99% HPD
interval of G exclude 0 or 1, providing significant evidence to transform the RV series using
the BC transformation against both the logarithmic transformation and no transformation
cases.
To understand the effect of BC transformation, we present the marginal densities of
the transformed RK implied by G = 0 and the estimated G for each stock index in Figure 1.
The marginal density of the BC RV is obviously different from that of the log RV. When the
estimated G is a positive value, the left tail of the former is lighter than that of the latter,
while the right tail of former is heavier, and vice versa when the estimated G is a negative
value. Moreover the marginal density of the former has a heavier right tail than left tail (as
that of the latter) for both positive and negative values of G. Furthermore, the marginal
density of the former has a higher peak than, and a different location from, that of the latter.
SWUP
D.B. Nugroho, T. Morimoto
MA.51
Figure 1. Marginal densities of the log RV and BC-RV for six different international stock
indices.
5. Conclusions and extensions
This study presented an extension to the R-ASV models, in which the Box-Cox
transformation is applied to realized measure. The HMC and RMHMC algorithms were
proposed to implement the MCMC method for updating several parameters and the latent
variables in the proposed model, taking advantage of updating the entire latent volatility at
once. The model and sampling algorithms were applied to the daily returns and realized
kernels of six international stock indices. The estimation results showed very strong evidence
supporting the Box-Cox transformation of realized measure. The model can be extended by
assuming a non-Gaussian density of return errors.
References
Dobrev, D., & Szerszen, P. (2010). The information content of high-frequency data for estimating equity
return models and forecasting risk. Working Paper, Finance and Economics Discussion Series.
Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of
posterior moments. In J. M. Bernardo, J. O. Berger, A. P. Dawid, & A. F. M. Smith (Eds.), Bayesian
Statistics 4 (pp. 169–194 (with discussion)). Clarendon Press, Oxford, UK.
Goncalves, S., & Meddahi, N. (2011). Box-Cox transforms for realized volatility. Journal of
Econometrics, 160, 129–144.
Kim, S., Shephard, N., & Chib, S. (2005). Stochastic volatility: likelihood inference and comparison with
ARCH models. In N. Shephard, ed., Stochastic Volatility: Selected Readings (pp. 283–322). Oxford
University Press, New York.
Koopman, S.J., & Scharth, M. (2013). The analysis of stochastic volatility in the presence of daily
realized measures. Journal of Financial Econometrics, 11(1), 76–115.
Nugroho, D.B., & Morimoto, T. (2014). Realized non-linear stochastic volatility models with
asymmetric effects and generalized Student's t-distributions. Journal of The Japan Statistical
Society, 44, 83–118.
Nugroho, D.B., & Morimoto, T. (2015). Estimation of realized stochastic volatility models using
Hamiltonian Monte Carlo-based methods. Computational Statistics, 30(2), 491–516.
Takahashi, M., Omori, Y., & Watanabe, T. (2009). Estimating stochastic volatility models using daily
returns and realized volatility simultaneously. Computational Statistics and Data Analysis, 53,
2404–2426.
SWUP
Box–Cox realized asymmetric stochastic volatility model
MA.52
Takahashi, M., Omori, Y., & Watanabe, T. (2014). Volatility and quantile forecasts by realized
stochastic volatility models with generalized hyperbolic distribution. Working Paper Series
CIRJE-F-921, CIRJE, Faculty of Economics, University of Tokyo.
SWUP