Volatility Analysis of Returns of Financial Assets Using a Bayesian Time-Varying Realized GARCH-Itô Model

2. Realized GARCH-Itô Model

The realized GARCH-Itô model first introduced by Song et al. (2020) presents a significant advancement in volatility analysis by integrating high frequency financial data with continuous-time jump-diffusion processes. This hybrid model extends the united GARCH-Itô framework of Kim and Wang (2016) by incorporating realized volatility (RV) and jump variation (JV) as innovations. Thereby, Song et al. (2020) filled the gap between discrete-time econometric models and continuous time stochastic processes. It is particularly effective to account dynamic change of volatility in financial markets including high frequency data analysis whereas market noises, price jumps and intraday volatility patterns exist. Realized GARCH-Itô model is formulated as a stochastic differential equation. The log stock price process $X_t$ for $t ϵ R_{+}$ follows:

$${dX}_t={\mu }_{t}dt+{\sigma }_t\left(\theta \right){dB}_t+L_t{d\mathsf{\Lambda }}_t$$

(1)

Where ${\mu }_t$ is the drift, ${\sigma }_t\left(\theta \right)$ is the instantaneous volatility, $B_t$ is a Brownian motion, $L_t$ is jump sizes, and ${\mathsf{\Lambda }}_t$ is a counting process for jumps. At integer times n, the conditional variance is:

$${\sigma }_n^2\left(\theta \right)=\omega +\gamma {\sigma }_{n-1}^2\left(\theta \right)+\alpha {\int }_{n-1}^n{\sigma }_s^2\left(\theta \right)ds+\beta {\int }_{n-1}^n{L}_s^{2}d{\mathsf{\Lambda }}_s$$

(2)

Where $\theta =\left(\omega ,\alpha ,\beta ,\gamma \right)$, ${\int }_{n-1}^n{\sigma }_s^2\left(\theta \right)ds\approx {RV}_n$, ${\int }_{n-1}^n{L}_s^{2}d{\mathsf{\Lambda }}_s\approx {JV}_n$, and the integrated volatility over [n – 1, n] is:

$${\int }_{\left[n-1\right]}^n{\sigma }_t^2\left(\theta \right)dt=h_n\left(\theta \right)+D_n$$

(3)

With

$$h_t\left(\theta \right)={\omega }^g+\gamma h_{t-1}\left(\theta \right)+{\alpha }^g.{RV}_{t-1}+{\beta }^g{JV}_{t-1}$$

(4)

Where $h_t$ is the latent conditional variance at time t, ${\omega }^g$ is the baseline volatility, ${\alpha }^g$ is lagged realized volatility (${RV}_{t-1}$), ${\beta }^g$ is lagged jump variation (${JV}_{t-1}$), $\gamma$ is lagged variance of latent volatility, and $\theta$= (${\omega }^g$,${\alpha }^g$,${\beta }^g, \gamma )$ parameters are assumed constant over time following a normal distribution.

The quasi-likelihood function used for static parameters in this model is stated as:

$${\widehat{L}}_{n,m}^{GH}\left(\theta \right)=-{\sum }_{i=1}^n\left[\log \left({\widehat{h}}_i\left(\theta \right)\right)+{\displaystyle \frac{{RV}_i}{{\widehat{h}}_i\left(\theta \right)}}\right]$$

(5)

Where ${\widehat{h}}_i\left(\theta \right)={\sum }_{l=1}^{i=1}{\gamma }^{i-1}{\omega }^g+{\alpha }^g{RV}_{i-l}+{\beta }^g{JV}_{i-l}+{\gamma }^{i-1}{h}_1\left(\theta \right)$, and $h_1\left(\theta \right)={\displaystyle \frac{{\omega }^g+{\beta }^g+{\lambda }_{\omega ,L}}{1-{\alpha }^g-\gamma }}$subject to ${\alpha }^g+\gamma <1$.

3. Bayesian Time-Varying GARCH-Itô Model

The Bayesian Time-varying GARCH-Itô (BtvGARCH-Itô) model extends the Generalized unified GARCH-Itô model by incorporating time-varying paramters and Bayesian inference to study dynamic volatility in high-frequency financial markets. This section delivers the BtvGARCH-Itô model specification which is objected to effectively estimate the latent volatility and its time-varying components using observed realized volatility, jump variation impact and lagged variance within a Bayesian inference framework. This model is suitable to financial applications of high-frequency data analysis where volatility dynamics swing continuously. The BtvGARCH-Itô model defines the latent volatility process {$h_t{\}}_{t=1}^N$ as follows:

For t=1

$$h_1={\displaystyle \frac{{\omega }_1+{\beta }_1{\lambda }_{\omega ,L}}{\mathrm{m}\mathrm{a}\mathrm{x}(e,1-{\alpha }_1-{\gamma }_1}}, for small e>0$$

(6)

Where $e>0$ is a small constant ensuring positivity, and ${\lambda }_{\omega , L}$ is a constant derived from the empirical distribution of jump variation ${JV}_t$.

For t=2,…,N

$$h_1={\omega }_t+{\alpha }_t{RV}_{t-1}+{\beta }_t{JV}_{t-1}+{\gamma }_t{h}_{t-1}, for \mathrm{t}=2,\dots ,\mathrm{N}$$

(7)

Where ${RV}_t$ is the realized volatility, ${JV}_t$ is the jump variation, and ${\omega }_t^g,{\alpha }_t^g,{\beta }_t^g,{\gamma }_t\in \left(\mathrm{0,1}\right)$ are time-varying parameters satisfying ${\alpha }_t+{\gamma }_t<1.$The time-varying parameters shifts via a Gaussian random walk in logit space:

$${\theta }_t^{raw}={\theta }_{t-1}^{raw}+{\sigma }_{\theta }{\acute{\theta }}_t,{\theta }_t=in{v}_{logit\left({\theta }_t^{raw}\right)}={\displaystyle \frac{1}{1+\exp \left(-{\theta }_t^{raw}\right)}}$$

(8)

Where $\theta \in \left\{\omega ,\alpha ,\beta ,\gamma \right\},$ and ${\tilde{\theta }}_t~N\left(\mathrm{0,1}\right)$. The measurement equation is:

$${RV}_t~N\left(h_t,{\sigma }_{RV}^2\right),{\sigma }_{RV}>0$$

(9)

To complete the Bayesian specification, we assign prior distributions as follows:

Initial coefficients:

$${\omega }_0~N\left(0.3,{0.1}^2\right),{\alpha }_0~N\left(0.1,{0.05}^2\right),{\beta }_0~N\left(0.05,{0.02}^2\right),{\gamma }_0~N\left(0.6,{0.05}^2\right)$$

(9)

$${\theta }_0~N\left({\mu }_{\theta },{\tau }_{\theta }^2\right)$$

$${\sigma }_{\theta }~C^{+}\left(0,b\right)$$

$${\sigma }_{RV}~C^{+}\left(0,c\right)$$

$${\tilde{\theta }}_t~N\left(\mathrm{0,1}\right),t=1,\dots ,N$$

Where $C^{+}\left(0,b\right)$ is the half-Cauchy distribution adjusted at 0 with scale b, and (${\mu }_{\theta }, {\tau }_{\theta })$ are prior mean and standard deviation hyperparameters.

Innovation standard deviations:

$${\sigma }_{\theta }~Half-Cauchy\left(\mathrm{0,0.1}\right),{\sigma }_{RV}~Half-Cauchy\left(\mathrm{0,0.2}\right)$$

(10)

Random walk innovations:

$${\tilde{\theta }}_t~N\left(\mathrm{0,1}\right),t=1,\dots ,N,,\theta \in \left\{\omega ,\alpha ,\beta ,\gamma \right\}$$

(11)

The Bayesian posterior of this model can be written as:

$$p\left({\theta }_{1:N},h_{1:N},{\sigma }_{RV}|{RV}_{1:N},{JV}_{1:N}\right)\propto {\prod }_{t=1}^{N}N\left({RV}_t|h_t,{\sigma }_{RV}^2\right).p(h_{1:N}\left|{\theta }_{1:N},{RV}_{1:N}\right).p\left({\theta }_{1:N}\right).p\left({\sigma }_{RV}\right)$$

(12)

We provide the implementation details of the BtvGARCH-Itô model in RStudio software in the supplementary material.

4. Simulation Study

4.1. Data Generation Process

We simulated synthetic data for N=100 and N=200 time points to evaluate the model’s finite sample performance. The jump variation process JV was generated as independent draws from a uniform distribution on [0, 0.2] with the median value denoted by ${\lambda }_{\omega , L}$. The time-varying parameters ${\omega }_{g,t}$,${\alpha }_{g,t}$,${\beta }_{g,t} and{\gamma }_t$ were defined to vary smoothly over time according to sinusoidal functions to mimic realistic dynamics. The latent volatility $h_t$ was computed recursively as follows:

$${\omega }_{g,t}=0.3+0.1sin\left({\displaystyle \frac{2\pi t}{N}}\right),{\alpha }_{g,t}=0.1+0.05\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{4\pi t}{N}}\right),$$

$${\beta }_{g,t}=0.05+0.03\sin \left({\displaystyle \frac{6\pi t}{N}}\right),{\gamma }_t=0.6+0.05cos\left({\displaystyle \frac{2\pi t}{N}}\right)$$

(14)

With ${JV}_t~N\left(\mathrm{0,0.2}\right),{\lambda }_{\omega ,L}=median\left({JV}_t\right)$ and

$$h_1={\displaystyle \frac{{\omega }_{g,1}+{\beta }_{g,1}.{\mathsf{\lambda }}_{{\omega }_L}.}{\max \left({10}^{-6},1-{\alpha }_{g,1}-{\gamma }_1\right)}}$$

(15)

$$h_t={\omega }_{g,t}+{\gamma }_t{h}_{t-1}+{\alpha }_{g,t}{RV}_{t-1}+{\beta }_{g,t}{JV}_{t-1},{RV}_t~N\left(h_t,{0.1}^2\right)$$

(16)

4.2. Simulation Results

This section provides the estimated simulation results from GARCH-Itô and BtvGARCH-Itô models for two simulated data sample sizes. Table 1 shows the posterior means of static parameters, posterior intervals, R hat estimates and effective sample sizes. Figures 1 to 10 present the posterior time varying estimates of means and 95 percent incredible intervals of parameters. Table 2 reports model prediction accuracy metrics for two models. Overall, these simulation results demonstrate that the BtvGARCH-Itô is capable of more accurately estimating volatilities under realistic synthetic dataset. Besides, this model can be used to estimate both static and time varying coefficients thus, it supports practical application to volatility analysis in empirical financial markets.

Table 3. Model Accuracy Metrics: GARCH-Itô vs. Bayesian Time-varying GARCH-Itô.

	N=100						N=200
Parameter	GARCH-Itô			BtvGARCH-Itô			GARCH-Itô			BtvGARCH-Itô
	MAE	MSE	RMSE	MAE	MSE	RMSE	MAE	MSE	RMSE	MAE	MSE	RMSE
$\omega$	0.1133	0.0128	0.1133	0.0432	0.00242	0.0492	0.1385	0.0191	0.1385	0.0469	0.0028	0.0529
$\alpha$	0.0632	0.0039	0.0632	0.0289	0.0011	0.0339	0.0442	0.0019	0.0442	0.0346	0.0019	0.0440
$\beta$	0.0499	0.0024	0.0499	0.0198	0.0004	0.0218	0.0499	0.0024	0.0499	0.0188	0.0004	0.0212
$\gamma$	0.0072	0.0000	0.0071	0.0440	0.0033	0.0575	0.1413	0.0199	0.1413	0.0419	0.0023	0.0489
$h_t$	0.0568	0.0085	0.0922	0.0299	0.0012	0.0348	0.0535	0.0072	0.0848	0.0192	0.006	0.0253

Notes: Table 3 provides the forecast accuracy of both original GARCH-Itô and BtvGARCH-Itô models based on three error metrics: Mean Absolute Error (MAE), Mean Squared Error (MSE) and Root Mean Squared Error (RMSE). Accuracy is assessed for each key parameter and for the latent volatility using simulated samples N = 100 and N=200. These results revealed that the BtvGARCH-Itô model achieves lower errors compared to original GARCH-Itô.

Table 3. Model Accuracy Metrics: GARCH-Itô vs. Bayesian Time-varying GARCH-Itô.

	N=100						N=200
Parameter	GARCH-Itô			BtvGARCH-Itô			GARCH-Itô			BtvGARCH-Itô
	MAE	MSE	RMSE	MAE	MSE	RMSE	MAE	MSE	RMSE	MAE	MSE	RMSE
$\omega$	0.1133	0.0128	0.1133	0.0432	0.00242	0.0492	0.1385	0.0191	0.1385	0.0469	0.0028	0.0529
$\alpha$	0.0632	0.0039	0.0632	0.0289	0.0011	0.0339	0.0442	0.0019	0.0442	0.0346	0.0019	0.0440
$\beta$	0.0499	0.0024	0.0499	0.0198	0.0004	0.0218	0.0499	0.0024	0.0499	0.0188	0.0004	0.0212
$\gamma$	0.0072	0.0000	0.0071	0.0440	0.0033	0.0575	0.1413	0.0199	0.1413	0.0419	0.0023	0.0489
$h_t$	0.0568	0.0085	0.0922	0.0299	0.0012	0.0348	0.0535	0.0072	0.0848	0.0192	0.006	0.0253

Notes: Table 3 provides the forecast accuracy of both original GARCH-Itô and BtvGARCH-Itô models based on three error metrics: Mean Absolute Error (MAE), Mean Squared Error (MSE) and Root Mean Squared Error (RMSE). Accuracy is assessed for each key parameter and for the latent volatility using simulated samples N = 100 and N=200. These results revealed that the BtvGARCH-Itô model achieves lower errors compared to original GARCH-Itô.

Figures 1–10. Time-varying posterior means of parameters estimated by BtvGARCH-Itô Model. Notes: Figures 1 to 10 illustrate the time varying posterior means of parameters, omega (${\omega }_{g,t})$, alpha (${\alpha }_{g,t}$), Beta (${\beta }_{g,t}$), Gamma (${\gamma }_t)$ and latent variance or factor ($h_t$) estimated from the BtvGARCH-Itô model for simulated datasets with time lengths t=100 and t=200. The solid lines present true parameter values that being used to generate the data and dashed lines represent the time-varying posterior mean estimates.

Figures 1–10. Time-varying posterior means of parameters estimated by BtvGARCH-Itô Model. Notes: Figures 1 to 10 illustrate the time varying posterior means of parameters, omega (${\omega }_{g,t})$, alpha (${\alpha }_{g,t}$), Beta (${\beta }_{g,t}$), Gamma (${\gamma }_t)$ and latent variance or factor ($h_t$) estimated from the BtvGARCH-Itô model for simulated datasets with time lengths t=100 and t=200. The solid lines present true parameter values that being used to generate the data and dashed lines represent the time-varying posterior mean estimates.

5. Empirical Study

This section provides the empirical application of BtvGARCH-Itô model to two financial assets namely S&P500 and Bitcoin using one-minute intraday returns time series data. The study sample periods for S&P500 ranges from 2023-01-03 to 2023-12-31 and for Bitcoin covers from 2023-01-01 to 2025-01-01. S&P500 is considered to be highly liquid equity stock and Bitcoin (BTC) as a volatile and speculative digital asset. The selection of these two markets is to reflect different market structures. This model assesses its capability to model latent volatility dynamics across two markets.

For the case of S&P500 (Table 4), it shows a strong empirical support for the model’s effectiveness in studying main features of volatility dynamics in a stable financial market. The static posterior intercept (${\omega }_g$) which sets the baseline level of conditional volatility is estimated at $4.17\times {10}^{-5}$. The estimated average volatility level is low which is consistent with the relatively stable U.S. equity market during the study period. The static posterior coefficient of realized volatility (${\alpha }_g$) is 0.0479. This value indicates a moderate impact of realized measures on volatility dynamics in the well-regulated market. The static posterior means of jump variation (${\beta }_g$) is 0.0324, which assesses the influence of discontinuous price jumps on current volatility. The small positive value revealed that price jumps in S&P500 has a small effect on the overall volatile. The static posterior means estimated for lagged variance ($\gamma$) is 0.598 which measures the persistence of latent volatility.

For the case of Bitcoin (Table 5), the static intercept posterior means for parameter (${\omega }_g$) is $2.11{\times 10}^{-4}$. The estimated average volatility level is much larger than in S&P500 index during the study period. This indicates how Bitcoin remains a speculative and risk-sensitive asset. The static posterior coefficient of realized volatility (${\alpha }_g$) is 0.0642 which also exceeds that of the S&P500. This finding highlights Bitcoin’s high volatility reactivity to incoming market information. The static posterior means of jump variation (${\beta }_g$) is 0.00245 which is substantially lower than S&P500 estimates. This result suggests that persistent price jumps contribute least to latent volatility. One possible reason is that the baseline and RV already absorb most of high frequency variation. The static posterior means estimated for lagged variance ($\gamma$) is 0.573 which is slightly less than in the equity market S&P500. However, the larger baseline volatility in BTC highlights the adaptability of the model to differentiate the nature of high risk and stable financial markets.

While the estimated static posterior means of parameters present fixed volatility of the market, it proves overly restrictive in high-frequency data and continuous changing financial environments. This model addresses this by allowing parameters vary over time to more accurately examine dynamic nature of investor behaviour, risk perceptions and volatility transmission mechanisms. The time-varying estimates of parameters including latent volatility are provided in Figures 11-15 for S&P500 and Figures 16-20 for Bitcoin. With this method on modeling volatility persistence and shifts, investors can benefit from value-at-risk estimation and derivative pricing. The evidence from both assets supports the conclusion that while static models reveal the overall market behaviour, time-varying models report unexpected shifts in volatility drivers that affect market-risk. In sum, the empirical evidence validates the efficacy of BtvGARCH-Itô model in analysing dynamic volatility structures across both traditional and digital financial assets.

Figures 11–15. Time-varying posterior means and 95% credible intervals of volatilities of S&P500 (2023-01-03 to 2024-12-31). Notes: Figures 11–15 present the time-varying posterior means (solid lines) and 95% credible intervals (shaded areas) for all parameters estimated from the BtvGARCH-Itô model using 1-minute close prices of the S&P500.

Figures 11–15. Time-varying posterior means and 95% credible intervals of volatilities of S&P500 (2023-01-03 to 2024-12-31). Notes: Figures 11–15 present the time-varying posterior means (solid lines) and 95% credible intervals (shaded areas) for all parameters estimated from the BtvGARCH-Itô model using 1-minute close prices of the S&P500.

Figures 16–20. Time-varying posterior means and 95% credible intervals of volatilities of Bitcoin (2023-01-01 to 2025-01-01). Notes: Figures 16–20 present the time-varying posterior means (solid lines) and associated 95% credible intervals (shaded areas) for key volatility-related parameters estimated from the BtvGARCH-Itô model using 1-minute close prices of Bitcoin (BTC).

Figures 16–20. Time-varying posterior means and 95% credible intervals of volatilities of Bitcoin (2023-01-01 to 2025-01-01). Notes: Figures 16–20 present the time-varying posterior means (solid lines) and associated 95% credible intervals (shaded areas) for key volatility-related parameters estimated from the BtvGARCH-Itô model using 1-minute close prices of Bitcoin (BTC).

6. Conclusions

This paper develops and empirically validates a Bayesian Time-Varying Realized GARCH-Itô model, a novel framework that integrates discreate-time GARCH dynamics with continuous-time Itô processes and time-varying coefficients within Bayesian inference. This model advanced volatility modeling because it allows model parameters to vary over time. Unlike original GARCH-Itô models that assume static coefficients, this BtvGARCH-Itô model enables joint inference on both static and time-varying posterior distributions. Besides, this model offers a more flexible and accurate classification of volatility shifts through the use of latent stochastic components estimated for financial markets. This new model adds a significant contribution to the field of financial econometrics and financial volatility analysis.

Simulation studies conducted for sample sizes N = 100 and N = 200 determine how well this model recover both static and time-varying parameters. In particular, the Bayesian posterior estimates closely track the true latent volatility path. The estimated simulation results confirmed that the BtvGARCH-Itô model grasps greater in-sample fit and out-of-sample forecast accuracy. This is evident in its closer alignment with true parameter values and lower error metrics (MAE, MSE, RMSE) compared to the original GARCH-Itô specification (Song et al., 2020). This model is also capable of tracking volatility shifts and identifying distinct volatility spikes linked to market shocks in the empirical application of volatility analysis of S&P500 intraday returns. Besides, its application to Bitcoin returns demonstrates that the BtvGARCH-Itô can be applied to markets with heavy tails and frequent jumps.

Overall, this model offers a statical tool for risk management and portfolio allocation in modern financial markets. It unifies time-varying parameters of realized volatility and jump variation within a full Bayesian posterior framework.

The BtvGARCH-Itô is just the first step and several directions remain open. In financial markets, assets move together so that extending this model to a multivariate setting would benefit further for volatility spillovers and co-jumps which are essential for portfolio risk assessment and cross-asset dynamics for investors. In practical applications, fixed bandwidth might limit flexibility thus, allowing data-driven bandwidths could improve the accuracy of time-varying coefficient estimates. As fat tails exist in financial returns, incorporating heavy tailed or skewed distributions, for example, Student-t distribution may better catch extreme events or price jumps.