Estimating Corporate Bond Market Volatility Using Asymmetric GARCH Models

Abstract

This study investigates the volatility of the Israeli corporate bond market, where corporate bonds are traded on a Limit Order Book (LOB) exchange with high retail trading activity. Using data from the Tel-Bond 20 and Tel-Bond 60 indices, we estimate various asymmetric GARCH models to capture the dynamics of bond returns. Our findings highlight a leverage effect, where negative shocks have a more significant impact on volatility than positive shocks, underscoring the importance of investor sentiment. The GJR model with a Student’s t-distribution best captures serial correlation, persistence of conditional volatility, and asymmetric volatility clustering. These results have significant implications for risk management, portfolio allocation, and regulatory policies, emphasizing the need for robust volatility forecasting models in transparent and active corporate bond markets.

1. Introduction

Forecasting volatility in financial markets remains a cornerstone of financial research, driven by its critical importance to asset pricing, portfolio management, and financial stability. While the role of investor sentiment in the volatility of stock returns has been extensively studied (Brown and Cliff 2004; Hadad and Kedar-Levy 2024; Verma and Verma 2007), empirical evidence on how sentiment-driven dynamics influence corporate bond market volatility remains limited. This gap is surprising given the rapid expansion of corporate debt markets globally and their rising exposure to behavioral trading activity, particularly from retail investors (Ederington et al. 2015; Liu et al. 2022).

Recent studies show varying degrees of sentiment’s impact on U.S. corporate bond yield (Bethke et al. 2017; Nayak 2010; Piñeiro-Chousa et al. 2022) and on European and Chinese yield spreads (Clayton et al. 2009; Lee 2019; Spyrou 2013), highlighting the potential impact of investors behavior on debt markets. Other research indicates that announcement shocks have a strong impact on bond market volatility dynamics, suggesting that bond investors incorporate news faster than other information (de Goeij and Marquering 2006). Intuitively, the sentiment effect in corporate bond markets affects the volatility of returns, given that these investors may be viewed as noise traders (Foucault et al. 2011; Kumar and Lee 2006; Yung and Nafar 2017). Indeed, an extant body of literature has found that changes in sentiment may be associated with changes in stock return volatility, often, but not always, with a positive sign (Brown and Cliff 2004; Baker and Wurgler 2006; Baker and Wurgler 2007). However, the impact on corporate bond markets is rarely discussed.

While knowledge about volatility behavior in corporate bond markets is scarce, financial literature acknowledges its economic implications. Volatility in bond markets significantly impacts investors and the broader economy. For investors, increased volatility translates to higher uncertainty and risk, potentially leading to greater returns or losses (Attarzadeh and Balcilar 2022; Bai et al. 2021). High volatility can affect bond pricing, making it difficult for investors to predict future prices and yields accurately (Pham and Cepni 2022). This uncertainty necessitates sophisticated risk management strategies to hedge against potential losses. Additionally, volatility in corporate bonds impacts portfolio allocation decisions, leading investors to shift their preferences toward more stable assets during volatile periods (Dick-Nielsen et al. 2012; Friewald et al. 2012).

Economically, volatility in the bond market can influence the cost of borrowing for corporations (Turkmen Muldur et al. 2019). Higher volatility can lead to increased risk premiums, raising the cost of issuing new debt (Bai et al. 2021). This, in turn, can affect corporate investment and growth strategies. Moreover, significant fluctuations in bond prices can affect financial market stability, influencing the decisions of policymakers (Abudy and Shust 2023). Given that corporate bond trading has been on the rise, particularly after the sub-prime crisis (Choi and Kim 2018; Graham et al. 2015), it is crucial to investigate bond market volatility for efficient economic stability and investor confidence.as

In this paper, we explore if well-established volatility forecasting models in stock markets are equally applicable to corporate bonds. Intuitively, sentiment effects on corporate bond returns would be more evident in markets with high participation rates of individual, as retail-sized investors are more prone to sentiment (Baker and Wurgler 2007). The Israeli corporate bond market provides an ideal setting for analyzing volatility behavior, as it is characterized by depth, high transparency, and high retail trading activity (Abudy and Wohl 2018; Gur-Gershgoren et al. 2020). Unlike the worldwide practice where corporate bonds are traded on a separate over-the-counter (OTC) market, bonds in the Tel Aviv Stock Exchange (TASE) are traded on a limit order book (LOB) trading system like stocks. This market design increases price discovery efficiency but may also intensify intraday volatility due to frequent order updates and the limited ability of institutional investors to provide stabilizing liquidity (Abudy and Shust 2023).

Recent evidence suggests that corporate bonds traded on the TASE exhibit stronger volatility transmission and contagion effects compared to traditional OTC markets, primarily due to TASE’s unique LOB mechanism. Abudy and Wohl (2018) show that corporate bonds on TASE display higher liquidity and narrower bid-ask spreads than their U.S. OTC counterparts, attributing these advantages to the high participation of retail investors and the centralized, transparent nature of the LOB structure. (Hadad and Kedar-Levy 2024) further demonstrate that changes in investor sentiment, particularly among retail participants, significantly influence conditional return volatility in both bond and stock markets, with the effects varying in magnitude and direction depending on prevailing market conditions. (Hadad 2025) provides further evidence that the LOB mechanism facilitates cross-asset contagion between stocks and bonds, especially during periods of market stress, highlighting how sentiment-driven trades in a retail-dominated environment can elevate systemic risk. These findings underscore the importance of studying volatility dynamics in the Israeli corporate bond market, where investor behavior and platform design jointly shape financial stability.

This study aims to fill the existing gap. We specifically focus on assessing various forecasting models for their efficacy in capturing the unique volatility features of the Israeli corporate bond market. Using the TASE bond indices, Tel-Bond 20 and Tel-Bond 60, we test and analyze the forecasting performance of GARCH, EGARCH, GJR, and APARCH models, along with normal distribution density and asymmetric Student’s t-distribution density. The outcomes of this research have broad implications, ranging from risk management and asset pricing to regulatory policy decisions.

Among the various GARCH models tested, we find that the GJR Student-t model is the most effective in capturing the unique characteristics of the Israeli corporate bond market. This model successfully captures asymmetric volatility clustering and high autocorrelation, providing more accurate volatility forecasts. The results also highlight the significant impact of investor sentiment on bond market volatility, with negative news leading to higher volatility than positive news. These findings suggest that behavioral frictions among retail investors are a major source of volatility in Israel’s corporate bonds market, with price reactions more pronounced following negative sentiment, consistent with noise trader models (Foucault et al. 2011; Kumar and Lee 2006).

Our findings advance the literature on volatility forecasting in bond markets by highlighting that platforms with LOB mechanisms and high retail engagement display distinct volatility dynamics. Given the rising role of non-institutional investors in global bond markets and the trend toward transparent, electronic trading (Abudy and Shust 2023; Hadad 2025), these results have important implications for market design and financial stability. Specifically, they highlight that platform structure and investor composition can significantly amplify volatility through sentiment-driven trading. Accordingly, our study underscores the importance of applying well-established volatility models to underexplored market microstructures, where behavioral factors and market transparency jointly shape risk transmission and systemic resilience.

The study proceeds as follows: Section 2 provides a review of the theoretical impact of retail trading activity on the volatility of returns, and summarize the findings on corporate bond returns. Section 3 presents the data and the methodology of the models used in the study. Section 4 describes the estimation procedures and presents the forecasting results and comparisons. Section 5 concludes.

2. Literature Review

Theories of behavioral finance argue that asset prices may be affected by investors’ psychological attributes. Shifts in sentiment, such as overconfidence, optimism, and wishful thinking, can significantly impact asset prices and their volatility (Barberis et al. 1998; Black 1986; Kyle 1985). Investor sentiment, broadly defined as investors’ beliefs about future cash flows and investment risks (Baker and Wurgler 2006), may not be justified by fundamental news or facts. Moreover, it is costly and risky to bet against sentimental investors, meaning rational investors or arbitrageurs are not as aggressive in forcing prices back to fundamental values ls (Baker and Wurgler 2007). Thus, price anomalies may form when sentiment investors over (under) estimate return and underestimate (overestimate) risk, hence investing more on the risky (safer) asset cause a mispricing of the asset in relative to its risk-based fundamental (Baker and Wurgler 2006, 2007). Hence, noise traders, who often exhibit irrational behavior, potentially can contribute to increased market volatility, imposing higher risks on rational arbitrageurs and ultimately affecting asset prices (Foucault et al. 2011; Huerta-Sanchez and Escobari 2018).

Extensive research has documented empirical evidence of the impact of investor sentiment on stock return volatility. Lee et al. (2002) found that changes in sentiment are inversely correlated with the conditional volatility of U.S. stock market indexes. Yu and Yuan (2011) showed that high sentiment periods correlate with a positive tradeoff between the mean and variance of U.S. stock returns, suggesting greater influence of sentiment-driven investors. Verma and Verma (2007) demonstrated that sentiment-driven retail investors significantly impact stock return volatility, with bullish sentiment having a greater effect than bearish sentiment. Other studies confirm that investor sentiment plays a significant role in international market volatility (Baker et al. 2012; Feldman and Liu 2017; Gong et al. 2022), highlighting that noise trading can contribute to increased market volatility.

Focusing on bond markets, fewer studies document the impact of shocks on U.S. Treasury bond volatilities, showing significant increases in bond market volatility on announcement days, which quickly subside as news is incorporated into prices (Christiansen 2000; Jones et al. 1998). Li and Engle (1998) demonstrate that news announcement shocks impact the volatility of U.S. Treasury bond futures, with volatility responding asymmetrically to these shocks. Other studies also document asymmetries in bond return volatilities (Cappiello et al. 2006; de Goeij and Marquering 2004, 2006). Additionally, Piazzesi (2005) shows that Federal Open Market Committee (FOMC) announcements are important determinants of bond market volatility, suggesting that bond investors underreact to information, implying that irrational behavior potentially impacts volatility in the debt market as well. Similarly, studies in other Asian markets have suggested the possibility of a leverage effect in bond yield volatility, implying for potential sentiment-driven impact on bond returns (BM et al. 2023; Mukherjee 2019; Rath 2023; Tan and Tian 2009). However, empirical evidence on volatility asymmetries in corporate bond markets remains limited, highlighting a clear gap in the literature that our study seeks to address.

If volatility is priced in the bond market, an anticipated increase in volatility would result in a higher required return in corporate bonds, which are generally perceived as more risky by investors (Acharya and Pedersen 2005; Dick-Nielsen et al. 2012; Friewald et al. 2012). Despite the well-documented impact of sentiment on stock and government bond market volatility, the influence of sentiment on corporate bond markets has received less attention. The few exceptions include studies that show U.S. bond yield spreads co-vary with sentiment, similar to stocks (Nayak 2010; Spyrou 2013). Specifically, information uncertainty and information asymmetry are found to be prices U.S corporate bonds yields spreads after controlling for credit ratings (Lu et al. 2010). Additionally, U.S. bonds appear underpriced (with high yields) during pessimistic periods and overpriced (with low yields) when optimism reigns (Nayak 2010). Similarly, Bethke et al. (2017) document that U.S. corporate bond investors exhibit a flight-to-quality when sentiment is low. In fact, the sentiment effect in U.S corporate bonds is found to spilled over from the stocks markets, through the activity of investors involved in capital structure arbitrage (Huang et al. 2015). This pattern is also evident in international corporate bond pricing and liquidity, which are generally affected by the same factors as the U.S. market (Goldstein and Namin 2023; Rath 2023). While these findings suggest that sentiment can indeed affect bond markets, they do not consider the impact on corporate bond volatility, which evidently changes over time in a pattern similar to stocks (Reilly et al. 2000). To the best of our knowledge, no prior study has examined sentiment-driven volatility in corporate bond markets within emerging economies—particularly using asymmetric GARCH models—thereby providing new insights into the behavioral dynamics of credit markets. Given the recent evidence of sentiment effects in bond pricing, more research is needed to fully understand this dynamic (Goldstein and Namin 2023).

Focusing on the Israeli market, several studies document a significant presence of retail trading activity, indicating that retail investors play a crucial role in enhancing market liquidity and efficiency (Abudy and Shust 2023; Abudy and Wohl 2018; Hadad 2025; Hadad and Kedar-Levy 2024). Hadad and Kedar-Levy (2024) show that the high presence of retail-sized investors has a positive impact on corporate bond returns and volatility. Using an EGARCH (1,1) model on the TA-35 (stock) Index and the Tel-Bond-20 (bond) Index returns, they found that changes in market sentiment proxies, reflecting changes in risk expectations and investor sentiment, largely explain movements in the conditional volatility of both stock and bond market returns. This implies that investors in both stocks and corporate bonds should consider sentiment in their investment decisions, and that both asset classes may be attractive for speculative investors. However, their study focuses solely on the EGARCH (1,1) model and does not test several GARCH models to estimate market volatility efficiently.

Given the above-mentioned literature, we extend volatility models used in stock markets to corporate bonds. This pattern is particularly evident in the Israeli market, characterized by high retail trading activity (Abudy and Shust 2023; Hadad and Kedar-Levy 2024). Alberg et al. (2008) found that asymmetric GARCH models, such as EGARCH and GJR, were effective in capturing volatility dynamics in the Israeli stock market. Similarly, we aim to develop a volatility model that captures well-known stylized facts about conditional volatility, such as persistence, mean-reverting behavior, and asymmetric impacts of negative versus positive return innovations. We examine the estimation performance of various models, including GARCH, EGARCH, GJR, and APARCH, using different density functions: normal distribution and Student’s t-distribution. Our focus is on the Tel-Bond 20 and Tel-Bond 60 indices, which reflect the Israeli corporate bond market. This methodology allows us to assess the suitability of different volatility forecasting models and emphasize the role of investor sentiment in explaining volatility patterns.

3. Data

The Israeli corporate bond market is one of the most liquid and transparent globally, supported by the LOB trading mechanism and dominated by active retail participation. According to the TASE 2024 Annual Report, the total market capitalization of corporate bonds reached approximately US$114.7 billion, including US$69.1 billion in CPI-linked bonds and US$45.6 billion in non-linked bonds.1 The average daily trading volume was US$0.23 billion, marking a 6% increase from 2023. Notably, the Israeli public purchased US$4.7 billion in corporate bonds in 2024, offsetting net sales by foreign investors (US$1.9 billion) and long-term institutional investors (US$2.7 billion). These figures highlight the prominent role of retail investors and the distinctive exchange-based bond trading environment, making TASE an ideal setting for examining sentiment-driven volatility dynamics in the bond market.

Our data consist of 738 daily observations of the Tel-Bond 20 Index prices and 738 daily observations of the Tel-Bond 60 Index prices, covering the period from 1 July 2019, to 30 June 2022. These data were obtained from the TASE website2. The rationale for choosing this specific dataset lies in its relevance for modeling volatility during critical economic periods, notably the COVID-19 pandemic and the inflation period of 2022 in Israel. During the COVID-19 pandemic, the corporate bonds market in Israel experienced significant volatility due to economic uncertainty and market disruptions (Hadad and Kedar-Levy 2024). Additionally, during inflationary times, bonds generally exhibit higher volatility (Campbell et al. 2020; Kang and Pflueger 2015), making this period particularly valuable for studying volatility patterns.

The Tel-Bond 20 and Tel-Bond 60 indices are key benchmarks in the Israeli corporate bond market, representing the top 20 and 60 fixed-coupon and CPI-linked corporate bonds, respectively, based on market capitalization. According to the TASE website, the threshold conditions for inclusion in these indices require a minimum rating from Israeli rating companies “Midroog” and “Maalot” of A or A3. Bonds meeting these criteria are included in the index, while those falling below the exit rating are removed. Typically, the rating is based on the average market value. The entry and exit ratings for the Tel-Bond 20 index are 16 and 24, respectively. As for the Tel-Bond 60 index, no specific entry and exit ratings have been determined due to its inclusion of a broader range of companies. These indices offer a comprehensive view of market behavior, making them excellent choices for studying volatility.

Table 1 shows descriptive statistics for the daily log returns of the Tel-Bond 20 and Tel-Bond 60 indices, calculated as the natural logarithm of the ratio of consecutive prices. The mean and median returns for both indices are close to zero, indicating that the average daily returns are quite small. The standard deviation is slightly higher for the Tel-Bond 20 index (0.0039) compared to the Tel-Bond 60 index (0.0036), suggesting that the Tel-Bond 20 is slightly more volatile. This result may be associated with the concentration of higher market capitalization bonds in the Tel-Bond 20 index, leading to greater price movements due to higher trading volumes and more significant investor reactions to market news and events. In particular, retail investors, who are more prone to sentiment-driven trading (Baker and Wurgler 2007; Edwards et al. 2007), may have a larger impact on the Tel-Bond 20 index, leading to increased volatility compared to the broader and more diversified Tel-Bond 60 index.

Table 1. Descriptive statistics.

Index	Mean	Median	Max	Min	Std. Dev.	Skewness	Kurtosis
Tel-Bond-20	0.000012	0.0002	0.0340	−0.0258	0.0039	0.6443	25.6379
Tel-Bond-60	0.000015	0.0002	0.0312	−0.0234	0.0036	0.5124	26.3665

Notes: This table presents descriptive statistics of the daily log-returns for the Tel-Bond-20 and Tel-Bond-60 indices traded on the Tel Aviv Stock Exchange (TASE) during the sample period.

The maximum and minimum values show that the Tel-Bond 20 index has had both higher peaks and deeper troughs than the Tel-Bond 60 index. Both indices show excess kurtosis and positive skewness, indicating that there are more extreme positive returns than extreme negative returns. The higher skewness in the Tel-Bond 20 index (0.6443) compared to the Tel-Bond 60 index (0.5124) suggests that the Tel-Bond 20 index has experienced more frequent large positive returns, which may be associated with sentiment-driven trading activity. The high kurtosis values for both indices indicate that the returns distribution has heavy tails and sharp peaks compared to a normal distribution, suggesting the presence of outliers in the time series.

Table 2 reports the results of the Jarque–Bera test for normality (Jarque and Bera 1980) and the ARCH LM test for stationarity. The Jarque–Bera test results are highly significant for both index returns, leading to the rejection of the null hypothesis of normality. The ARCH LM test results are also highly significant for both indices, indicating that the returns of both indices exhibit volatility clustering, implying that periods of high volatility tend to cluster together. This pattern can also be attributed to retail-trading activity, which is known to induce such clustering in financial markets. Unreported Augmented Dickey–Fuller (ADF) (Dickey and Fuller 1981) and Phillips Perron (Phillips and Perron 1988) stationarity tests results also show high significance, suggesting stationarity in the time series. Overall, these findings support the necessity of employing GARCH modeling to capture the volatility dynamics accurately in our corporate bond indices.

Table 2. Jarque–Bera and LM test.

Index	Jarque–Bera	LM Statistic
Tel-Bond-20	15,788.27 ***	59.1581 ***
Tel-Bond-60	16,798.75 ***	91.8002 ***

Notes: *** indicates significance at the 1% level. Jarque–Bera tests for normality; LM tests for ARCH effects. Both reject the null hypotheses, supporting the use of GARCH models.

4. Methodology

In this section we succinctly describe the GARCH models used in the study. We use the daily returns of both indices and model the mean equation as follows:

$$r_t=\mu +{\epsilon }_t,$$

(1)

where $r_t$ represents the return of our corporate bond index at time $t$, $\mu$ is a constant term, and ${\epsilon }_t$ the error term used to model the conditional volatility in the various GARCH models.

First, we implemented the GARCH model, which imposes nonlinear restrictions (Park 2002). This model improves upon the Auto-Regressive Conditional Heteroskedasticity (ARCH) models by Engle (1982) by adding a more flexible lag structure. The GARCH (1,1) variance estimation process is given by:

$${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2,$$

(2)

where $\alpha$, $\beta$ and ω = ${\alpha }_0$ are the model parameters.

While the GARCH model is effective, it has limitations such as not accounting for the sign (positive or negative) of delayed innovations and exhibiting excess kurtosis of the residuals (Park 2002). To overcome these limitations, Bollerslev (1987) proposed using Student’s t-distribution, which can capture conditional leptokurtosis separately from conditional heteroskedasticity. Several empirical studies have employed the generalized t-distribution to capture the skewness and leverage effects of daily returns and to address kurtosis and skewness limitations (Hansen 1994; Harris et al. 2004). However, the GARCH model does not handle the asymmetric effect, potentially biasing its estimation of conditional volatility (Villar-Rubio et al. 2023).

To handle the asymmetric effect, we explore nonlinear asymmetric models, including the Exponential GARCH (EGARCH) (Nelson 1991), the GJR model by (Glosten et al. 1993) and the Asymmetric Power ARCH (APARCH) model (Ding et al. 1993). These models consider the magnitudes and signs of shocks to conditional variance and explain the leverage effect (Lama et al. 2015). Given the asymmetric response of investors to good and bad news (Barberis et al. 1998; Hadad and Kedar-Levy 2024; Verma and Verma 2007), these models may be more suitable to model the volatility.

We apply the EGARCH model (Nelson 1991), which identifies asymmetric effects on conditional volatility. This model ensures that conditional variance remains positive by specifying it in logarithmic form, thus avoiding restrictions on the model’s coefficients (López-Cabarcos et al. 2021; Piñeiro-Chousa et al. 2022). The logarithmic conditional variance of the corporate bond index is modeled using the EGARCH (1,1) model:

$$\log \left({\sigma }_t^2\right)=\omega +\alpha \left[{\displaystyle \frac{\left|{\epsilon }_{t-1}\right|}{{\sigma }_{t-1}}}-{\displaystyle \frac{\sqrt{2}}{\pi }}\right]+\gamma \left({\displaystyle \frac{{\epsilon }_{t-1}}{{\sigma }_{t-1}}}\right)+\mathsf{\beta }log\left({\sigma }_{t-1}^2\right),$$

(3)

where ${\sigma }_t^2$ is the conditional variance of the error term of the mean equation of the index, ${\epsilon }_{t-1}$ is the first-order lag from (1), and $\alpha$ represents the symmetric effect of the general autoregressive model. $\omega$ is a constant and $\gamma$ coefficient captures the asymmetric effect of innovations on the volatility of the index returns, when $\gamma <0$ negative news generates higher volatility than positive news. $\beta$ measures the stationary of the conditional volatility.

However, the EGARCH model has several limitations, such as capturing the leverage effect depending on the signs of the parameters and being highly sensitive to initial values. Additionally, the log-transformation in EGARCH models ensures positivity of the conditional variance but assumes a multiplicative effect, which might not always align with the actual data characteristics (Alberg et al. 2008). To address this, Glosten et al. (1993) extended the standard GARCH model by incorporating asymmetric effects of positive and negative shocks on volatility. For one lag in the return of the corporate bond index and variance, its conditional variance is modeled using the GJR (1,1) model as follows:

$${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\gamma I_{t-1}{\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2,$$

(4)

where $I_{t-1}$ is an indicator function that is equal to 1 if ${\epsilon }_{t-1}<0$ and zero otherwise, and $\omega ,\alpha ,\beta ,\gamma$ are the model parameters, while $\alpha ,\mathsf{\beta }\ge 0$, and $\omega >0$. The term $\gamma I_{t-1}{\epsilon }_{t-1}^2$ adds an extra component to the variance equation when the lagged error term is negative, capturing the phenomenon where negative shocks have a larger impact on volatility than positive shocks of the same magnitude (Glosten et al. 1993). While the GJR model effectively captures asymmetry through the indicator function, it does not allow for varying the power of the shocks. Furthermore, the GJR-GARCH model does not explicitly account for heavy-tailed distributions of returns, which are evident in time series of returns (Denes et al. 2023; Opschoor et al. 2018).

To address this issues, we also utilize the APARCH(1,1) model of Ding et al. (1993), which provides a more flexible approach by allowing for different powers of the absolute returns and asymmetry, by introducing a power parameter ($\delta$) that can be optimized to better fit the data. Our APARCH(1,1) model is given by

$${\sigma }_t^{\delta }=\omega +{\alpha }_1{(\left|{\epsilon }_{t-1}\right|-{\gamma }_1{\epsilon }_{t-1})}^{\delta }+{\beta }_1{\sigma }_{t-1}^{\delta },$$

(5)

where $-1<\gamma <1$, $\delta >0$ and $\omega ,\alpha ,\gamma ,$ $\beta$ and $\delta$ are parameters to be estimated. The APARCH model provides a more flexible framework for modeling asymmetric effects and can better handle heavy-tailed distributions and varying volatility patterns, making it a more comprehensive tool for analyzing financial time series. Furthermore, Alberg et al. (2008) show that fat-tail distributions are better suited for modeling returns in the Israeli market, underscoring the necessity of the APARCH (1,1) model.

Since our returns time series deviate from the normality assumption, a Maximum Likelihood (ML) method is employed to estimate the models’ parameters. We use the quasi-maximum likelihood estimator (QMLE) to obtain maximum likelihood estimates of the various GARCH model parameters. This method maximizes the Gaussian log-likelihood function of the multivariate normal distribution and results in consistent and asymptotic normality of the estimated parameters (Allen and McAleer 2018; Asai et al. 2021). Additionally, we utilize the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization algorithm for estimation to select the best-fitted model. This algorithm minimizes the likelihood function (Mahmood and Khan 2020). Following Alberg et al. (2008), we apply the QMLE method under the normal and Student’s t-distributions to select the best-fitted model.

Lastly, we select the optimal model by measuring several standard criteria to determine the most adequate specification. The best model was selected using the Akaike Information Criterion (AIC), Schwarz Information Criterion (SIC), and Hannan–Quinn Criterion (HQIC) statistics. These information criteria indicate how well the estimated model fits the data compared to other models (Shittu and Asemota 2009), allowing us to better capture volatility clustering behavior for efficient risk management strategies and better-informed portfolio allocation decisions.

5. Results

Table 3 summarizes the results of the GARCH (1,1) model on the Tel-Bonds indices daily returns.

Table 3. GARCH (1,1) model results.

Estimation Results of the Variance Equation:		${\mathit{\sigma }}_{\mathit{t}}^2=\mathit{\omega }+\mathit{\alpha }{\mathit{\epsilon }}_{\mathit{t}-1}^2+\mathit{\beta }{\mathit{\sigma }}_{\mathit{t}-1}^2$
Variable	Normal		Student’s t
	Tel-Bond-20	Tel-Bond-60	Tel-Bond-20	Tel-Bond-60
$\omega$	2.43 × 10−7 ***	2.04 × 10−7 ***	2.50 × 10−7 ***	2.23 × 10−7 ***
$\alpha$	0.201152 ***	0.199625 ***	0.223436 ***	0.231301 ***
$\beta$	0.779118 ***	0.781201 ***	0.765361 ***	0.758634 ***
Adjusted R-squared	0.040581	0.040827	0.039660	0.039538
Log likelihood	3411.521	3478.030	3424.784	3495.102
Akaike Info Criterion	−9.256851	−9.437582	−9.290173	−9.481255
Schwarz Criterion	−9.225593	−9.406324	−9.252663	−9.443745
Hannan–Quinn Criterion	−9.244796	−9.425527	−9.275707	−9.466788

Notes: The table reports GARCH(1,1) estimates under both normal and Student’s t error distributions. *** indicate significance at the 1% level.

The GARCH coefficients are positive and highly significant, indicating that current volatility is highly sensitive to past information, both errors ($\alpha$) and volatility ($\beta$). The $\alpha$ coefficient, representing model errors, is approximately 0.2, while $\beta$ coefficient, representing past conditional variance, is approximately 0.8. These results suggest that current volatility of bond returns is more influenced by past volatility than by past shocks. Furthermore, the information criteria (AIC, SBIC, and HQIC) have minimal values for Student’s t-distribution, indicating that it is a better fit model. These results align with BM et al. (2023), who noted the superiority of Student’s t-distribution for modeling financial time series with heavy tails.

Table 4 show estimation results for the EGARCH (1,1) model. The EGARCH model results, particularly with Student’s t-distribution, indicate that all coefficients are highly significant. The preference for Student’s t-distribution, supported by lower AIC, SIC, and HQIC values, aligns with Yong et al. (2021), who identified this distribution as optimal for EGARCH (1,1) modeling of other Asian markets during periods of market turbulence. For this distribution, we document a highly significant $\alpha$ coefficient, indicating the symmetric effect of past errors on volatility, with values as approximately of 0.3. The $\beta$ coefficient, which is around 0.9, indicates long-term persistence in bond market volatility.

Table 4. EGARCH (1,1) model results.

Estimation Results of the Variance Equation: $\mathit{log}\left({\mathit{\sigma }}^2\right)=\mathit{\omega }+\mathit{\alpha }\left[\frac{\left|{\mathit{\epsilon }}_{\mathit{t}-1}\right|}{{\mathit{\sigma }}_{\mathit{t}-1}}-\frac{\sqrt{2}}{\mathit{\pi }}\right]+\mathit{\gamma }\left(\frac{{\mathit{\epsilon }}_{\mathit{t}-1}}{{\mathit{\sigma }}_{\mathit{t}-1}}\right)+\mathit{\beta }\mathit{l}\mathit{o}\mathit{g}\left({\mathit{\sigma }}_{\mathit{t}-1}^2\right)$
Variable	Normal		Student’s t
	Tel-Bond-20	Tel-Bond-60	Tel-Bond-20	Tel-Bond-60
$\omega$	−11.80657 ***	−11.58546 ***	−0.680550 ***	−0.721949 ***
$\alpha$	1.107948 ***	1.153876 ***	0.321036 ***	0.334494 ***
$\beta$	0.055067	0.093176	0.963587 ***	0.961363 ***
$\gamma$	−0.019139	−0.103186	−0.101851 ***	−0.092911 ***
Adjusted R-squared	0.039736	0.042865	0.042187	0.041403
Log likelihood	3260.014	3336.649	3425.896	3495.751
Akaike Info Criterion	−8.842429	−9.050678	−9.290478	−9.480302
Schwarz Criterion	−8.804919	−9.013168	−9.246717	−9.436540
Hannan–Quinn Criterion	−8.827963	−9.036212	−9.273601	−9.463424

Note: The table reports EGARCH(1,1) estimates for Tel-Bond-20 and Tel-Bond-60 indices returns under both normal and Student’s t error distributions. $\gamma$ captures asymmetry (leverage effects). *** indicate significance at the 1% level.

The asymmetry term shows a negative $\gamma$ coefficient around −0.1 for both Tel-Bond 20 and Tel-Bond 60 returns, implying that negative shocks have a more significant impact on volatility than positive shocks of the same magnitude, demonstrating a leverage effect. This is consistent with findings by Hadad and Kedar-Levy (2024) in the context of the Israeli corporate bonds market during the COVID-19 pandemic. These results underscore the importance of accounting for asymmetry in volatility modeling, as negative news tends to amplify volatility more than positive news, reflecting the behavioral biases of investors.

Table 5 summarizes the results of the GJR (1,1) model on the indices daily returns. The GJR model results, particularly with Student’s t-distribution, indicate that the coefficients are positive and highly significant. The $\beta$ coefficient of approximately 0.8 indicates the persistence of conditional volatility. The positive $\gamma$ coefficient confirms the presence of a leverage effect in the Israeli bond market during the COVID-19 period, indicating that negative shocks have a larger impact on volatility than positive shocks of the same magnitude. This is especially evident when using Student’s t-distribution, where the α coefficient remains highly significant, unlike in the normal distribution where it is insignificant.

Table 5. GJR (1,1) model results.

Estimation Results of the Variance Equation: ${\mathit{\sigma }}_{\mathit{t}}^2=\mathit{\omega }+\mathit{\alpha }{\mathit{\epsilon }}_{\mathit{t}-1}^2+\mathit{\gamma }{\mathit{I}}_{\mathit{t}-1}{\mathit{\epsilon }}_{\mathit{t}-1}^2+\mathit{\beta }{\mathit{\sigma }}_{\mathit{t}-1}^2$
Variable	Normal		Student’s t
	Tel-Bond-20	Tel-Bond-60	Tel-Bond-20	Tel-Bond-60
$\omega$	2.60 × 10−7 ***	2.23 × 10−7 ***	2.62 × 10−7 ***	2.32 × 10−7 ***
$\alpha$	0.072221	0.080696 *	0.087298 **	0.104001 **
$\gamma$	0.194331 ***	0.184904 ***	0.203293 ***	0.189128 ***
$\beta$	0.798634 ***	0.794868 ***	0.784243 ***	0.775620 ***
Adjusted R-squared	0.042801	0.042593	0.041522	0.040910
Log likelihood	3419.856	3485.336	3430.154	3499.254
Akaike Info Criterion	−9.276783	−9.454718	−9.302048	−9.489822
Schwarz Criterion	−9.239273	−9.417208	−9.258286	−9.446060
Hannan–Quinn Criterion	−9.262317	−9.440252	−9.285170	−9.472945

Notes: This table reports GJR-GARCH(1,1) estimates for Tel-Bond-20 and Tel-Bond-60 indices return under both Normal and Student’s t distributions. γ captures asymmetric effects of negative shocks (leverage effects).***, ** and * indicate significance at the 1%, 5% and 10% level, respectively.

The superior performance of Student’s t-distribution for the GJR model is further underscored by the minimal AIC, SIC, and HQIC values, highlighting its better fit for the data. This finding is consistent with Alberg et al. (2008), who found that models assuming a Student’s t-distribution provide a better fit for financial time series data in the Israeli stock market. This underscores the importance of using a distribution that can capture the heavy tails and excess kurtosis often observed in financial returns, making Student’s t-distribution a robust choice for modeling bond market volatility during periods of market turbulence.

Finally, we estimated the conditional variance of the bond yield indices using the Asymmetric Power ARCH (APARCH) model. This model, which has the ability to generate many ARCH models by varying the parameters, couples the flexibility of a varying exponent with the asymmetry coefficient. Table 6 summarizes the results of the APARCH (1,1) model.

Table 6. APARCH (1,1) model results.

Estimation Results of the Variance Equation: ${\mathit{\sigma }}_{\mathit{t}}^{\mathit{\delta }}=\mathit{\omega }+\sum _{\mathit{i}=1}^{\mathit{q}}{\mathit{\alpha }}_{\mathit{i}}{(\left|{\mathit{\epsilon }}_{\mathit{t}-\mathit{i}}\right|-{\mathit{\gamma }}_{\mathit{i}}{\mathit{\epsilon }}_{\mathit{t}-\mathit{i}})}^{\mathit{\delta }}+\sum _{\mathit{j}=1}^{\mathit{p}}{\mathit{\beta }}_{\mathit{j}}{\mathit{\sigma }}_{\mathit{t}-\mathit{j}}^{\mathit{\delta }}$
Variable	Student’s t
	Tel-Bond-20	Tel-Bond-60
$\omega$	2.29 × 10−8	4.29 × 10−8
$\alpha$	0.166892 ***	0.181514 ***
$\gamma$	0.263686 ***	0.236297 **
$\beta$	0.764342 ***	0.762282 ***
$\delta$	2.398791 ***	2.273031 ***
Adjusted R-squared	0.041453	0.04960
Log likelihood	3430.362	3499.351
Akaike Info Criterion	−9.299896	−9.487368
Schwarz Criterion	−9.249883	−9.437354
Hannan–Quinn Criterion	−9.280608	−9.468080

Note: This table presents the estimated parameters of the Asymmetric Power ARCH (APARCH) model for the Tel-Bond-20 and Tel-Bond-60 indices return using Student’s t distribution. γ and δ capture the leverage effect and the power term, respectively, enhancing model flexibility *** and ** indicate significance at the 1% and 5% level, respectively.

For both indices, convergence could not be reached with the APARCH model and a normal distribution. Therefore, we used Student’s t version of the model. The results show that the coefficients $\alpha$ and $\beta$ are positive and statistically significant. The power coefficient $\delta$ is positive and significant as well, and not equal to 2, establishing that it is not a standard GARCH model (Ding et al. 1993). The asymmetry coefficient $\gamma$ is positive and significant, indicating the existence of a leverage effect where negative news increases volatility more than positive news of the same magnitude. The significant α coefficient implies that past errors significantly affect current volatility, while the high β coefficient indicates long-term volatility persistence.

Overall, the results show the presence of a leverage effect, where negative shocks have a more significant impact on volatility than positive shocks. This effect is consistently observed across the EGARCH, GJR and APARCH models, indicating that investor sentiment and reactions to negative news play a crucial role in driving bond market volatility. These findings highlight the importance of considering asymmetry and heavy-tailed distributions in modeling financial time series, especially in markets with significant retail trading activity.

The economic implications of these findings are significant, suggesting that investors should be cautious during periods of market stress and that portfolio allocation strategies should account for potential increases in volatility. We attribute these findings to the evolution of the unique trading platform, which improved market practices and trading behavior. This insight is crucial for developing more robust risk management strategies and ensuring a resilient financial system

Lastly, to test the accuracy of volatility forecasting among the different models with distribution assumptions, we compare several standard criteria: AIC, SBIC, HQIC and the Log-Like value. Given that the results across all models indicate the superior fit of Student’s t-distribution over the normal distribution, we focus on comparing Student’s t-distribution models to better capture the volatility dynamics of the Tel-Bond 20 and Tel-Bond 60 indices. Results for Tel-Bond 20 and Tel-Bond 60 are presented in Table 7.

Table 7. Comparison between the models for the Tel Bond 20.

	Student’s t GARCH	Student’s t EGARCH	Student’s t GJR	Student’s t APARCH
Tel-Bond-20
AIC	−9.290	−9.290	−9.302	−9.299
Schwarz	−9.253	−9.247	−9.258	−9.225
Hannan–Quinn	−9.276	−9.274	−9.285	−9.280
Tel-Bond-60
AIC	−9.481	−9.480	−9.490	−9.487
Schwarz	−9.444	−9.437	−9.446	−9.437
Hannan–Quinn	−9.467	−9.463	−9.473	−9.468

Notes: This table compares the performance of the four volatility models for the Tel-Bond-20 index and Tel-Bond 60 index returns. All models are estimated with Student’s t-distribution. Lower AIC, Schwarz, and Hannan–Quinn information criteria indicate better the model fit.

The comparison of the different models for the Tel-Bond 20 and Tel-Bond 60 indices, as summarized in Table 7, demonstrates that the GJR model with a Student’s t-distribution provides the best fit for the data. For the Tel-Bond 20 index, the GJR model outperformed other models with an AIC of −9.302, an SBIC of −9.258, and an HQIC of −9.285. Similarly, for the Tel-Bond 60 index, the GJR model achieved the lowest AIC of −9.490, an SBIC of −9.446, and an HQIC of −9.473. These results indicate that the GJR model with a Student’s t-distribution better captures the volatility dynamics of both indices compared to the GARCH, EGARCH, and APARCH models.

While these findings are consistent with previous studies documenting the outperformance of the GJR model in estimating stock returns volatility (Liu and Hung 2010), they contrast with findings for the Israeli market. Specifically, Alberg et al. (2008) found the EGARCH model to be the most successful for measuring conditional variance and forecasting the Israeli stock indices. Furthermore, Hadad and Kedar-Levy (2024) documented that the EGARCH(1,1) model had the best fit for the Tel-Bond 20 index returns. However, it should be noted that these studies primarily focused on outdated data. For instance, the sample in Hadad and Kedar-Levy (2024) ranges from 2000 to 2019, thus not considering the significant impact of COVID-19 and the rise in inflation, which evidently impact the Israeli corporate bonds market. Given that economic uncertainty and inflationary times exhibit higher volatility in corporate bonds (Bethke et al. 2017; Campbell et al. 2020; Engelberg et al. 2018; Kang and Pflueger 2015), this should favor the GJR results.

6. Conclusions

This study examines conditional volatility in the Israeli corporate bond market using GARCH family models. Our findings reveal that the GJR-GARCH model with a Student’s t-distribution most effectively captures the asymmetric volatility behavior of bond returns, highlighting the presence of a leverage effect and sentiment-driven volatility. These findings reflect that negative shocks lead to disproportionately higher volatility, highlighting increased sensitivity to downside risk. These dynamics are especially relevant in transparent and retail-dominated markets like Israel, where information is rapidly incorporated into prices. Such volatility asymmetry may distort risk premia, widen spreads, and impair market functioning during stress episodes, and thus accurate volatility modeling becomes essential for risk management, asset pricing, and regulatory oversight.

These findings are particularly relevant for the Israeli market, characterized by high levels of transparency and significant retail participation. Prior work shows that market transparency enhances liquidity and narrows bid-ask spreads (Bessembinder and Maxwell 2008; Cici et al. 2011; Goldstein et al. 2007), but also increases market responsiveness to new information. In such settings, retail-driven sentiment plays a larger role in shaping price volatility. Indeed, Baker and Wurgler (2006, 2007) highlight how retail sentiment intensifies volatility and exacerbates deviations from fundamentals—especially in less institutionalized markets. This supports recent findings by Goldstein and Namin (2023) and Wang (2023) that bond markets with active retail investors exhibit more pronounced volatility in response to sentiment shifts.

Our results reinforce the suitability of ARCH-type models for bond volatility analysis, in line with Reilly et al. (2000), and underscore the importance of incorporating asymmetry in volatility forecasts for transparent, sentiment-sensitive markets. Using appropriate models such as GJR-GARCH can improve risk management, enhance price discovery, and support financial stability initiatives in such environments.

Nonetheless, the study is limited to a univariate GARCH framework, subjective model selection, and a sample period that reflects a turbulent economic time (the COVID-19 pandemic), which may not capture the full diversity of market conditions and could limit the generalizability of the findings to more stable periods. Moreover, the localized nature of the Israeli corporate bond market, despite its unique structural strengths, further constrains the broader applicability of our results. Future research should extend the analysis across different corporate bond sectors within Israel, consider multivariate and regime-switching approaches, and evaluate how platform design and investor composition affect volatility patterns. Crucially, we recommend applying this framework to other emerging corporate bond markets with similar structural traits, namely high transparency and strong retail investor presence, to enable comparative insights into sentiment-driven volatility dynamics. Such extensions can guide both local and international policymakers in designing interventions that mitigate systemic risk under uncertainty.