Efficient Frontiers and Volatility Spreads of Convertible Bonds: Evidence from the Chinese Financial Market

Yutong Wu

School of Professional Study, Columbia University, New York, USA.

DOI: 10.4236/me.2025.168058                

Abstract

A convertible bond is a financial instrument which can be considered as a straight bond with embedded options. As option pricing theories underwent significant advancements, theories of convertible bond valuation have correspondingly advanced. This study examines how convertible bonds improve portfolio efficiency and analyzes whether these instruments are mispriced in Chinese financial market. Efficient frontiers of both the two-asset and three-asset portfolio are depicted through simulation. The results indicate that after adding the CSI Convertible Bond Index, the performance of efficient frontier improves compared with the condition in which only the CSI 300 Index and the CSI Aggregate Bond Index are included. Using the Black-Scholes Model and the GARCH (1,1) Model, this study calculates the implied volatilities and the GARCH volatilities of 942 convertible bonds respectively and analyzes their equal-weighted spreads. Moreover, the equal-weighted spread is persistently positive over recent years, which implies that convertible bonds in China appear to be underpriced.

Keywords

Convertible Bond, Efficient Frontier, Implied Volatility, GARCH Volatility, Equal-Weighted Spread

1. Introduction

1.1. Definition of the Convertible Bond

The convertible bond is the most common type of convertible securities. It can be decomposed into a straight bond and embedded option components. The characteristics of bond allow it to provide periodic coupon payments for investors, while the embedded option components create the potential upside of equity participation.

Given the structural similarity to straight bonds, coupon rates and maturity dates are decisive in the valuation of convertible bonds. For the embedded option components, various option pricing models can be applied based on specific attributes, including factors such as call provisions and conversion privileges. Theoretically, a convertible bond can be valued only if the call and the conversion strategies can be followed by the issuer and the investors respectively.

1.2. The Situation of Convertible Bonds in China

Globally, convertible bonds are employed as strategic instruments for financing, but Chinese convertible bond market exhibits distinct features. With the unique market concept, convertible bonds in China are often perceived more as equity-linked instruments than hybrid securities. For most corporations, issuing convertible bond is equal to a method of equity financing, indicating a preference for early conversion by bondholders to common shares at a high conversion price as fast as possible. The redemption at maturity is rare, which is typically five years. From the perspective of holders, they usually pay more attention to the movements of underlying stock prices rather than using convertible bonds as instruments for hedging or arbitrage.

China has recently experienced an issuance explosion of convertible bonds. As a result, convertible bonds become a popular instrument for capital-raising. However, special policy mandates of Chinese financial market make the situation of convertible bonds unique and complex. For example, the restriction of shorting in China is extremely strict, including financial policies such as high transaction fee, restricted transaction types, and limited trading quantities. These policies constrain the opportunities of arbitrage for convertible bonds gigantically.

Therefore, Chinese convertible bond market faces many potential challenges. At the beginning of the 21^st century, convertible bonds in China were widely acknowledged as highly underpriced. In recent years, low liquidity and trading frictions persistently impair pricing efficiency, while the investment structure is transforming dramatically with the influence of the pandemic. The convertible bond market has deviated significantly from the conditions required by the Efficient Market Hypothesis (EMH).

1.3. Research Questions and the Overview of Methods

Given the conditions of Chinese convertible bond market and the valuation models of convertible bonds, this study focuses on two research questions. 1) To what extent do convertible bonds enhance portfolio efficiency when incorporated into a multi-asset framework? 2) Are convertible bonds in China mispriced relative to risk-based valuation models?

The definition of the efficient frontier is put forward by Markowitz (1952). It depicts portfolio combinations that achieve maximum returns for given risk levels or minimum variances for specified expected returns. To address the first research question, this study chooses three indices from the financial market in China, representing stocks, bonds, and convertible bonds, respectively. The CSI 300 Index (stock index) and the CSI Aggregate Bond Index (bond index) are used to build a two-asset efficient frontier first. Then, the CSI Convertible Bond Index (convertible bond index) is incorporated to construct another three-asset efficient frontier. This study sets a series of expected returns and calculates the corresponding minimum variances to generate efficient frontier plots. The results show that the inclusion of convertible bonds can shift the efficient frontier outward, thereby improving the performances of portfolios.

The systematic analysis of option pricing models started in 1973, when Black and Scholes (1973), and Merton (1973) put forward a practical pricing model for European options. The Black-Sholes Model, also known as the Black-Scholes-Merton Model, assumes that stock prices follow a Geometric Brownian Motion (GBM) and employs stochastic calculus and partial differential equation in the formulation. The beginning of the convertible bond pricing model analysis was the research from Brennan and Schwartz (1977), who applied the Black-Scholes equation to convertible bonds and related the valuation to the total value of the corporation. This approach partially establishes the framework of single-factor models, while an alternative approach links the price to the underlying stock price movements. Based on the theories of single-factor models, two-factor models are developed soon by adding additional factors, such as the interest rate.

Other option pricing models are also applied to the valuation of convertible bonds. Considering varied provisions and restrictions, some researchers choose to classify convertible bonds as American options. Therefore, methods such as Monte Carlo simulation and binomial tree model are frequently employed through valuations, particularly in China.

For the second part of analysis, this study calculates the implied volatilities based on the Black-Scholes Model, and the GARCH volatilities by GARCH (1,1) Model, for all the tradable convertible bonds in China from January 2016 to February 2024. Then, the implied volatilities are subtracted from the GARCH volatilities to compute volatility spreads. The analysis of the equal-weighted spread shows a positive average suggesting that convertible bonds in China are probably undervalued, or investors are excessively pessimistic about their future returns.

1.4. Contributions of the Study

This study contributes to the following areas. First, the empirical scope of the study covers a total of 942 convertible bonds over an 8-year period, which is more comprehensive than comparable research in this field. Therefore, it can provide a more general description for Chinese convertible bond market. Second, this study establishes a connection between convertible bonds and the efficient frontier theory. Existing literatures rarely analyze or explore the application of convertible bonds in portfolio management, while this study provides a framework which can support more specific analyses. Third, this study focuses on the spread-based pricing inefficiency test using implied and realized volatilities, which is an innovative approach among other convertible bond valuation literatures. Besides, prior studies focus either on pricing models or investor behavior; this study uniquely links asset allocation efficiency with pricing anomalies in one coherent framework. Fourth, this study is meaningful for the formulation of convertible bond policies in China. Regulatory measures should be applied to switch the largely underpriced condition and improve market liquidity.

The remainder of this paper is organized as follows. Section 2 presents the literature review. Section 3 discusses the special policies for convertible bonds in China. Section 4 describes the data used in this study and their characteristics. Section 5 introduces the methodologies. Section 6 provides the empirical results, and Section 7 concludes.

2. Literature Review

Markowitz (1952) first put forward the theory of the efficient portfolio, which comprises portfolios that would achieve an optimal expected return with the same variance or have the minimum variance with the same level of expected return. Based on the efficient portfolio theory, Sharpe (1964) proposed the Capital Asset Pricing Model (CAPM), which is the fundamental of modern asset management theory. As for the application of convertible bonds in asset management, only few literatures mentioned and just provided some theoretical suggestions. Focusing on the convertible bonds in China, Tian and Chen (2020) established a pricing model for convertible bonds based on the Black-Sholes Model and the Gailai-Schneller model, then demonstrated the possibility of applying convertible bonds in portfolio management.

For the theories of option valuation, Black and Scholes (1973) believed that the European option follows a geometric Brownian motion with a constant volatility and can be traded continuously. Based on these assumptions, they developed the Black-Scholes Model, representing a breakthrough in option valuation. Merton (1973) developed the Black-Scholes Model by incorporating the condition of a continuous dividend payment. Cox and Ross (1976) advanced a generalized option pricing framework by extending the Black-Scholes Model using several diffusion processes. Brennan and Schwartz (1977) applied the Black-Scholes Model to convertible bonds. They assumed a dual option structure for both the investors and the issuers, establishing a pricing model focused on the time of maturity and the value of the corporation. This was the beginning of the systematic analysis for convertible bond pricing models. Ingersoll Jr. (1977) established a single-factor model based on the Black-Scholes Model. He characterized the convertible bond as a compound option whose price was related to the value of the corporation. Brennan and Schwartz (1979, 1980b) introduced a dual-interest-rate model which included both the instantaneous interest rate and the long-term interesting rate as valuation factors for bonds with embedded options and confirmed that the model had the potential to improve the portfolio management for bonds. Brennan and Schwartz (1980a) assumed that the prices of convertible bonds depend on both the total value of the corporation and the interest rate. This was an idea of two-factor models. Cox et al. (1985) put forward a general equilibrium model named Cox-Ingersoll-Ross (CIR) Model, which applied the non-arbitrage pricing principle and was related to the term structure of the interest rate.

Recently, there are various types of pricing models for convertible bonds based on different methodological approaches. The foundations of such models can be the Monte Carlo method, the GARCH Model, etc. Tsiveriotis and Fernandes (1998) proposed a pricing model which took the credit risk into consideration. Thus, the model could be more accurate when modelling with several embedded options. Kariya and Tsuda (2000) established a statistical model combining a time-dependent Markov model for calculating the value of the bond, and the Black-Scholes Model for estimating the value of embedded options. Wang and Li (2011) used the Monte Carlo method to capture the interest of the equity part and applied the GARCH model to simulate the volatility of the convertible bond. Zhu et al. (2018) used the incomplete Fourier transform method and the Black-Scholes Model to derive two pricing integral equations for the puttable convertible bonds. Dai et al. (2022) proposed an equity-price-tree-based pricing model considering factors such as the implied corporation value, the stochastic equity-price volatility, and the dilution effect. Tan et al. (2022) tried to solve the complexity of the underlying stock return processes by applying a financial time-series generative adversarial networks (FinGAN) to fit the risk-neutral condition and preserve the characteristics of convertible bonds. Lin and Zhu (2022) established an integral equation using Green’s function and an incomplete Fourier transform under the Black-Scholes Model and tried to price the callable-puttable convertible bonds.

Literatures about the convertible bonds in China also have a considerable volume, but most of the decisive studies emerged in the early 21^st century. Jiang and Zhang (2002) argued that the convertible bonds in China should be classified as American options to which applying the Black-Scholes Model and the Monte Carlo method were inappropriate. Therefore, they employed the binomial tree pricing model and included several provisions to test the accuracy of convertible bond pricing in China. Conversely, Zheng and Lin (2004) believed that as the underlying stocks in China were non-dividend-paying, the convertible bonds could be considered as European options. To establish a practical pricing model, they first used the GARCH model to simulate the volatility, then combined the binomial tree pricing method, the Monte Carlo method, and the finite difference method to price the convertible bonds. Lai et al. (2005) applied the credit risk pricing model which was proposed by Tsiveriotis and Fernandes to analyze the condition of convertible bonds in China. They especially considered the downward adjustment policy and used the binomial tree pricing method to price numerically. Wen and Qiu (2009) implemented the Monte Carlo method to estimate the prices and included the downward adjustment policy, while the result showed that the convertible bonds in China were highly underpriced. Zhang et al. (2011) expanded the Monte Carlo method based on the Total Least Squares (TLS) to a Quasi-Monte Carlo method which was tested to have a better performance in the accuracy of pricing. Based on the CSI Convertible Bond Index and the CSI 300 Index, Ni et al. (2024) applied a GED-GARCH model to analyze the correlation between the convertible bond market and the stock market in China. Cheng (2024) used the Black-Scholes Model to estimate the theoretical prices of convertible bonds in China and compared them with the actual values to analyze the reasons of deviation. Zheng et al. (2025) compared the accuracies of five different pricing models and concluded that long-short portfolios with cross-sectional pricing errors could earn an excess return and generate a positive Alpha.

3. The Special Policies of Convertible Bond in China

3.1. Downward Adjustment Policy

Beyond theoretical considerations, convertible bond pricing in China is heavily shaped by regulatory policies, which we now examine. The downward adjustment policy is a unique item among the Chinese convertible bond market regulatory framework. This mechanism offers issuers the right to lower the conversion price of a convertible bond when its underlying stock’s closing price is lower than the 85% of the current conversion price for at least 15 trading days within any consecutive 30 trading days.

The adjusted conversion price must not be lower than the higher of the average trading price of the corporation’s stock in the 20 trading days before the shareholders’ meeting and the average trading price of the corporation’s stock on the trading day before the adjustment. Furthermore, the adjusted conversion price must not be lower than the most recent audited net asset value per share or the par value of the stock. If the corporation adjusted the conversion price within 30 trading days, then the calculation should be based on the pre-adjustment conversion price before the adjustment and use the post-adjustment conversion price after the adjustment.

3.2. Call Provision

Issuers possess call provisions addressing two situations: the maturity redemption and the conditional redemption. For the case of the maturity redemption, a corporation should redeem all outstanding unconverted convertible bonds within 5 trading days after the maturity of this convertible bond.

For the case of the conditional redemption, when either of the following conditions happens during the conversion period of the convertible bond, a corporation will have the right to redeem all or part of the outstanding unconverted convertible bonds at the bond’s par value plus accrued interest: 1) The stock’s closing price is not lower than 130% of the prevailing conversion price (including 130%) for at least 15 trading days within any consecutive 30 trading days. 2) The outstanding balance of the unconverted convertible bond falls below CNY 30 million.

3.3. Case Study Summary

This study analyzes 20 convertible bonds across 4 sectors: energy, heavy industry, medicine, and banking. The result shows that 14 of the corporations employed conditional redemption provisions. 3 corporations redeemed the convertible bond at maturity, and 3 corporations triggered the downward adjustment policy. All the 5 energy corporations chose the conditional redemption. In the heavy industry sector, 4 corporations employed the conditional redemption while 1 triggered and executed the downward adjustment policy. 3 corporations in the medicine industry chose the conditional redemption, while 2 corporations triggered the downward adjustment policy but declined to execute. In the banking sector, 2 corporations employed the conditional redemption and 3 chose to redeem at maturity. Table 1 shows the detailed information for all 20 corporations.

Table 1. The detailed information of the 20 corporations in the case study. This table shows the detailed information of the 20 corporations in the case study, including the name of their convertible bond, the sector they belong to, the date of issuance of the convertible bond, and the final condition. “Downward (Executed)” means the corporation triggered the downward adjustment policy and announced to execute, and “Downward (Unexecuted)” means the corporation triggered the downward adjustment policy but chose not to execute.

Name	Sector	Date of Issuance	Condition
Chuantou CB	Energy	12/2/2019	Conditional redemption
Dibei CB	Energy	11/21/2019	Conditional redemption
Mengdian CB	Energy	1/9/2018	Conditional redemption
Heneng CB	Energy	5/8/2019	Conditional redemption
Jiuzhou CB	Energy	9/12/2019	Conditional redemption
Linggang CB	Heavy Industry	5/13/2020	Downward (Executed)
Huaikuang CB	Heavy Industry	1/13/2020	Conditional redemption
Huayu CB	Heavy Industry	7/10/2019	Conditional redemption
Shengtun CB	Heavy Industry	3/31/2020	Conditional redemption
Taihua CB	Heavy Industry	1/11/2019	Conditional redemption
Xiandai CB	Medicine	4/30/2019	Conditional redemption
Zhende CB	Medicine	1/14/2020	Conditional redemption
Liuyao CB	Medicine	2/24/2020	Downward (Unexecuted)
Yifeng CB	Medicine	6/22/2020	Downward (Unexecuted)
Fuxiang CB	Medicine	3/29/2019	Conditional redemption
Suyin CB	Banking	4/3/2019	Conditional redemption
Guangda CB	Banking	4/5/2017	Maturity redemption
Zheshang CB	Banking	3/28/2019	Conditional redemption
Zhongxin CB	Banking	3/19/2019	Maturity redemption
Changzheng CB	Banking	4/11/2018	Maturity redemption

The exclusive concentration of maturity redemptions within the banking sector can be seen as a signal of the relative stability in Chinese banking system. Therefore, these institutions don’t have a strong preference for convertible bond financing as a capital instrument.

For the corporations triggering the downward adjustment policy, one belongs to the heavy industry, while the other two represent the medicine sector. The optimal strategy for issuers in China is to convert all the convertible bonds at a relatively high conversion price as soon as possible. Only when the stock price drops dramatically which may cause a significant put pressure for the corporation, will the issuer execute the downward adjustment policy to make sure that the value of the convertible bond remains above the put price. These 3 corporations all declined to execute the adjustment when they met the condition for the first time, which suggested that they thought the drop of the stock price was still acceptable. However, Linggang convertible bond, the one from the heavy industry, decided to execute the adjustment upon the second trigger, whereas the medicine corporations announced that they would not adjust their prices during the next six months despite triggering the policy again. Linggang’s strategic reversal was probably because the corporation evaluated the put pressure they faced again and judged it would be risky if they kept the original conversion price. As the 3 convertible bonds remain outstanding, the ultimate situations of their redemptions are undetermined.

Among 20 bonds, 70% used conditional redemption, 15% chose maturity redemption, and 15% triggered the downward adjustment policy. 60% banking institutions in this sample selected maturity redemption, while heavy industry and medical issuers more frequently exercised conditional redemption or the downward adjustment right. This supports the thesis that sector-specific norms dominate issuer strategies.

4. Data

4.1 Data Description

For the first part of the analysis, the sample includes the stock price data of the CSI 300 Index, the CSI Aggregate Bond Index, and the CSI Convertible Bond Index, from January 2005 to February 2024. These data were used to build the efficient frontier by calculating the optimal portfolios with the minimum standard deviations under varied expected returns. Table 2 shows the descriptive statistics for the 3 indices. Table 3 is their correlation matrix, and Table 4 lists the annualized returns and annualized volatilities for the 3 indices from 2005 to 2023. The tendencies of the annualized returns and annualized volatilities for the 3 indices are presented in Figure 1.

Figure 1. Tendencies of the annualized return and the volatility for the three indices. This figure shows the tendencies of the annualized return and the annualized volatility for the three indices. Compared with the CSI 300 Index and the CSI Convertible Bond Index, the CSI Aggregate Index was very stable during the research period, which had no obvious fluctuation in both return and volatility. However, the trends of the CSI 300 Index and the CSI Convertible Bond Index were quite similar. The fluctuations of returns and the high volatilities around 2008 were due to the global financial crisis, while the high volatility in 2015 was because of the stock market crash in China and the influence of the RMB exchange rate mechanism reform.

Table 2. Descriptive statistics of the three indices. This table shows the summary statistics of the three indices. “Size” means the number of the sample. “Mean” and “Standard Deviation” are both annualized based on their daily returns.

	Size	Mean	Standard Deviation	Maximum Drawdown	Skewness	Kurtosis
CSI Aggregate Bond	4617	8.03%	0.1745	0.5202	−1.5872	34.1983
CSI 300	4617	10.18%	0.2576	0.7230	−0.4993	4.2174
CSI Convertible Bond	4617	4.81%	0.0154	0.0428	1.5790	28.0981

Table 3. The correlation matrix of the three indices. This table shows the correlation matrix of the CSI Convertible Bond Index, the CSI 300 Index (for stocks), and the CSI Aggregate Bond Index (for bonds) based on their daily returns.

	CSI Convertible Bond	CSI 300	CSI Aggregate Bond
CSI Convertible Bond	0.0305	0.0301	0.0000
CSI 300	0.0301	0.0664	−0.0002
CSI Aggregate Bond	0.0000	−0.0002	0.0002

Table 4. Annualized returns and standard deviations of the three indices. This table lists the annualized return and the annualized standard deviation of the three indices based on their daily returns from 2005 to 2023. “return” means the annualized return, and “SD” means the standard deviation.

	CSI Convertible Bond (return)	CSI 300 (return)	CSI Aggregate Bond (return)	CSI Convertible Bond (SD)	CSI 300 (SD)	CSI Aggregate Bond (SD)
2005	8.98%	−4.01%	11.85%	0.0578	0.2138	0.0136
2006	32.04%	84.84%	2.68%	0.0910	0.2225	0.0104
2007	76.41%	106.72%	−2.61%	0.2736	0.3680	0.0137
2008	−37.09%	−95.65%	15.13%	0.1937	0.4860	0.0324
2009	38.22%	70.44%	−1.25%	0.2469	0.3252	0.0179
2010	−4.22%	−7.53%	3.28%	0.1235	0.2519	0.0159
2011	−14.67%	−31.68%	5.84%	0.0954	0.2064	0.0220
2012	4.71%	11.76%	3.52%	0.0573	0.2033	0.0118
2013	−1.65%	−6.38%	−1.12%	0.1004	0.2222	0.0152
2014	52.46%	48.21%	10.58%	0.1747	0.1947	0.0149
2015	−26.70%	3.05%	8.63%	0.4971	0.3996	0.0132
2016	−9.33%	−1.61%	2.13%	0.1138	0.2107	0.0145
2017	0.97%	21.31%	−0.28%	0.0768	0.1020	0.0096
2018	−2.47%	−30.82%	8.97%	0.1001	0.2141	0.0116
2019	25.56%	36.60%	4.75%	0.1106	0.1983	0.0083
2020	5.88%	27.29%	3.08%	0.1184	0.2276	0.0158
2021	17.27%	−5.41%	5.78%	0.0889	0.1856	0.0083
2022	−10.28%	−22.38%	3.62%	0.1014	0.2041	0.0109
2023	−0.69%	−13.46%	5.21%	0.0602	0.1355	0.0082

This study also compared the Sharpe ratios under different simulated circumstances. For the first simulation, this study simulated several risk-free rates to calculate optimal asset allocations for both the two-asset and three-asset portfolios. The second simulation set the risk-free rate to be 4.8%, which was similar to the average return of the CSI Aggregate Bond Index and compared the statistics of the optimal portfolio with other portfolios. Table 5 and Table 6 show the results from the first and second simulation, respectively.

For the second part of analysis, the sample includes the implied volatilities, the GARCH volatilities, and the equal-weighted spreads of 942 convertible bonds that was issued in China, from January 2016 to February 2024. After cleaning data and deleting missing values, there were 432,564 pairs of data left (N = 432,564). The summary statistics of these three variables are in Table 7.

Table 5. Optimal Sharpe ratios with vs. without convertible bonds under different risk-free rates. This table lists the optimal Sharpe ratios with or without convertible bonds under different risk-free rates. The study chooses 4.8% as the benchmark, and compares it with the results under 2%, 3%, 4%, and 5%, respectively. The table also represents the return, the standard deviation, and the percentages of each index in the portfolio.

Rf Rate	4.8%		2%		3%		4%		r5%
CSI Convertible Bond	26%	/	0%	/	1%	/	1%	/	40%	/
CSI 300	33%	45%	1%	1%	1%	2%	2%	3%	60%	100%
CSI Aggregate Bond	41%	55%	98%	99%	98%	98%	96%	97%	0%	0%
Sum	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
Return	7.43%	7.23%	4.88%	4.88%	4.90%	4.90%	4.97%	4.96%	9.31%	10.18%
SD	0.1208	0.1163	0.0154	0.0154	0.0156	0.0156	0.0165	0.0164	0.2075	0.2576
Sharpe ratio	0.2176	0.2092	1.8669	1.8659	1.2212	1.2197	0.5904	0.5873	0.2076	0.2009

Table 6. Different Sharpe ratios under the 4.8% risk-free rate. This table changes the proportions among the three indices under a risk-free rate of 4.8%, and represents the return, the standard deviation, and the Sharpe ratio of different portfolios. The “Optimal” column shows the data of the optimal Sharpe ratio in this simulation. “Random 1” to “Random 4” show the data of random proportions.

	Optimal	Random 1	Random 2	Random 3	Random 4
CSI Convertible Bond	26%	30%	20%	20%	30%
CSI 300	33%	40%	30%	40%	30%
CSI Aggregate Bond	41%	30%	40%	40%	40%
Sum	100%	100%	100%	100%	100%
Return	7.43%	7.92%	6.58%	7.60%	7.38%
SD	0.1208	0.1434	0.1040	0.1290	0.1189
Sharpe ratio	0.2176	0.2175	0.1713	0.2169	0.2173

Table 7. Descriptive Statistics of the three variables. This table shows the descriptive statistics of the implied volatility, the GARCH volatility, and the equal-weighted spread.

	Mean	Median	Minimum	Maximum	Standard Deviation
Implied Volatility	35.42%	33.07%	0.00%	155.78%	0.1853
GARCH Volatility	48.10%	47.92%	7.13%	3182.05%	0.1033
Equal-weighted Spread	15.69%	16.67%	0.17%	29.46%	0.0703

4.2. Data Sources

The data used in this study were obtained from the eastmoney.com, including the stock price data of the 3 indices and the price data of the convertible bonds. Corporate policies and the announcements for the case study analysis were also searched and collected from the eastmoney.com.

5. Methodology

5.1. Efficient Frontier

This study draws efficient frontiers for both the two-asset and three-asset portfolios respectively based on the correlation matrix. The two-asset portfolio includes the CSI 300 Index and the CSI Aggregate Bond Index, while the three-asset portfolio includes the CSI Convertible Bond Index. For each portfolio, they fit Equation (1) to (3):

${\mu }_p={\displaystyle {\sum }_{i=1}^n{\omega }_i{\mu }_i}$ (1)

${\sigma }_p^2=w^T\sum w$ (2)

$1={\displaystyle {\sum }_{i=1}^n{\omega }_i}$ (3)

In these expressions, ${\mu }_p$ is the mean return of the portfolio; ${\sigma }_p^2$ is the variance of the portfolio; ${\mu }_i$ is the return of asset $i$ ; ${\omega }_i$ is the weight of asset $i$ in the portfolio; $w$ is the vector of asset weights, and $\text{Σ}$ is the variance-covariance matrix of the asset returns.

Specifically, for a 2-asset portfolio, the variance formula can also be given as Equation (4):

${\sigma }_p^2={\omega }_A^2\cdot {\sigma }_A^2+{\omega }_B^2\cdot {\sigma }_B^2+2\cdot {\omega }_A\cdot {\omega }_B\cdot Cov\left(A,B\right)$ (4)

where ${\omega }_A$ and ${\omega }_B$ are the weights of asset A and asset B. ${\sigma }_p^2$ , ${\sigma }_A^2$ , and ${\sigma }_B^2$ represent the variance of the portfolio, asset A, and asset B respectively. $Cov\left(A,B\right)$ is the covariance between asset A and asset B.

It is feasible to simulate a range of expected returns and compute the corresponding minimum-variance portfolios. The efficient frontier can then be drawn based on these paired results. A list of variable definitions is provided in Appendix A.

5.2. Implied Volatility

Following the classification of Zheng and Lin (2004) that convertible bonds in China can be considered as European options due to the non-dividend-paying condition of the underlying stocks, this study applies the Black-Scholes Model (1973) based on the prices of convertible bonds to calculate the implied volatilities. The formulation is given by Equation (5) to (7):

$C= S_{0}N\left(d_1\right)-K{e}^{-rT}N\left(d_2\right)$ (5)

$d_1=\frac{\ln \left(\frac{S_0}{K}\right)+\left(r+\frac{{\sigma }^2}{2}\right)T}{\sigma \sqrt{T}}$ (6)

$d_2=d_1-\sigma \sqrt{T}$ (7)

In these expressions, S₀ is the current price of the underlying stock; K is the strike price; r is the risk-free interest rate; T is the time to maturity; and $\sigma$ is the volatility of the convertible bond.

While the Black-Scholes Model assumes the European-style exercise and no credit risk, it can still capture the basic option characteristics of the embedded option components. Therefore, it is used to value the embedded option components of the convertible bond and considered as a benchmark to extract the implied volatilities for comparative purposes, as applied in Cheng (2024). To simplify the procedure of calculation, this study uses the government bond yield as the risk-free interest rate and assumes that the convertible bond has neither dividend payment nor the choice of conditional redemption provisions.

Implied volatilities are calculated through numerical inversion of the Black-Scholes Model using R software.

5.3. GARCH Volatility

To calculate the GARCH volatility, this study applies the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and sets the paraments p = 1 and q = 1, indicating that the conditional variance of returns has 1 lag of the squared residual and 1 lag of the past variance. The formula is given by Equation (8) to (9):

$r_t=\mu +{\epsilon }_t,{\epsilon }_t~N\left(0,{\sigma }^2\right)$ (8)

${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2$ (9)

In these expressions, $r_t$ is the log return of the underlying stock; ${\sigma }_t^2$ is the conditional variance of the underlying stock; $\mu$ is the conditional mean of the underlying stock; ${\epsilon }_t$ is the error term; $\omega ,\alpha ,\beta$ are parameters in the model.

Conditional variances are estimated recursively using the GARCH (1,1) process initialized by sample variance and estimated via maximum likelihood. The procedure is executed using R software.

5.4. Equal-Weighted Spread

The spread in this study is calculated as the difference between the GARCH volatility and the implied volatility, while the equal-weighted spread can represent the aggregate spread across the whole market in a time series. The formula is given by Equation (10):

${\text{EWS}}_t=\frac{i}{N_t}{\displaystyle {\sum }_{i=1}^{N_t}\left({\text{GARCH}}_{i,t}-{\text{Implied}}_{i,t}\right)}$ (10)

In this expression, ${\text{EWS}}_t$ is the equal-weighted spread of all the available convertible bonds on day t; ${\text{GARCH}}_{i,t}$ and ${\text{Implied}}_{i,t}$ are the GARCH volatility and the implied volatility for convertible bond $i$ on day t; $N_t$ is the total number of the convertible bonds which are available on day t.

6. Empirical Results

6.1. Portfolio Frontier Analysis

Figure 2. The efficient frontiers of the two-asset and three-asset portfolio. This figure presents the efficient frontiers of both the two-asset and three-asset portfolio, and the “Detail” figure shows the detail of the low-risk segment which is at the left bottom of the figure. It is obvious that the efficient frontier of the three-asset portfolio is “above” the two-asset portfolio, which indicates the performance of a three-asset efficient frontier to be better.

Given the average historical returns of 4.8% for the CSI Aggregate Bond Index, 8.0% for the CSI Convertible Bond Index, and 10.2% for the CSI 300 Index from 2025 to 2024, this study simulates a series of expected returns ranging from 5.0% to 10.0% in 0.2% increments and calculates to achieve the minimum possible variance for both the two-asset and the three-asset portfolio based on their daily returns. The two-asset portfolio includes the CSI 300 Index and the CSI Aggregate Bond Index, representing the performance of stocks and bonds respectively, while the three-asset portfolio additionally includes the CSI Convertible Bond Index. 26 pairs of data are used to plot the efficient frontier in each condition and the two frontiers are presented in Figure 2. The numeric results of the standard deviations are listed in Table 8.

Table 8. The minimum standard deviations under different expected returns. This table lists the minimum standard deviations for both the two-asset and the three-asset portfolio under different expected returns. It is obvious that the standard deviations of the two-asset portfolio are higher than those of the three-asset portfolio across the whole simulated range.

Expected Return	Two-Asset	Three-Asset
5.00%	0.0171	0.0170
5.20%	0.0231	0.0226
5.40%	0.0310	0.0300
5.60%	0.0396	0.0383
5.80%	0.0487	0.0469
6.00%	0.0579	0.0558
6.20%	0.0673	0.0647
6.40%	0.0767	0.0738
6.60%	0.0861	0.0829
6.80%	0.0957	0.0920
7.00%	0.1052	0.1011
7.20%	0.1147	0.1103
7.40%	0.1243	0.1195
7.60%	0.1339	0.1287
7.80%	0.1435	0.1379
8.00%	0.1531	0.1471
8.20%	0.1627	0.1563
8.40%	0.1723	0.1656
8.60%	0.1819	0.1748
8.80%	0.1915	0.1840
9.00%	0.2011	0.1933
9.20%	0.2107	0.2025
9.40%	0.2203	0.2121
9.60%	0.2300	0.2227
9.80%	0.2396	0.2342
10.00%	0.2492	0.2464

Figure 2 shows that the efficient frontier of the three assets consistently dominates the one of two assets, even in the low-risk segment at the left bottom part. It indicates that the inclusion of convertible bonds can expand the efficient frontier and improve portfolio diversification benefits by either providing a higher return at the same risk level or receiving a lower standard deviation with the same expected return. Additionally, the results support the notion that diversification is able to benefit the portfolio frontier across the whole range of risk.

6.2. Pricing Efficiency via Volatility Spread

The tendency of the equal-weighted spread from January 2016 to February 2024 is shown in Figure 3, and the annualized average equal-weighted spread is listed in Table 9 and visualized in Figure 4.

Figure 3. The tendency of the equal-weighted spread from 2016 to 2024. This figure shows the tendency of the equal-weighted spread which has an annual cycle that increases at the beginning of the year and falls at the end. The overall trend of the equal-weighted spread could be summarized as an initial increase followed by a decline. During 2016 to 2018, the equal-weighted spread increased steadily and stayed above the mean, but it started to fall in the middle of 2019 and kept below the mean value after 2021.

Figure 4. The bar chart of the annualized average equal-weighted spread. This figure shows the annualized average equal-weighted spread from 2016 to 2023. It is the visualization of Table 9. While the tendency is similar to that of Figure 3, which first rose from 2016 to 2018, then fell from 2019, and hit the lowest point in 2023.

Table 9. The annualized average equal-weighted spread. This table lists the annualized average equal-weighted spread from 2016 to 2023. Because the data for 2024 only include that in January, the average value of 2024 is excluded.

Year	Average Spread
2016	13.7054%
2017	17.0443%
2018	24.3592%
2019	23.5518%
2020	18.6864%
2021	15.9370%
2022	6.4979%
2023	6.0786%

Figure 3 and Figure 4 present a similar annual pattern, with increases observed in the early months of the year followed by year-end declines. As for the trend among the whole research period, the equal-weighted spread was far above the mean value from 2018 to 2020, which indicated a relatively low risk premium. In 2021, the equal-weighted spread dropped dramatically and stayed below the mean value from 2022 onward.

To examine whether there are statistically significant differences among different years, this study applies the ANOVA test on the equal-weighted spread, and the result is shown in Table 10, which presents that the differences among years are statistically significant at a 1% significant level. These annual variations suggest shifting investor expectations and market liquidity, potentially reflecting broader macroeconomic or regulatory shifts post-2019. To test the robustness, this study uses nine corporations from the case study and replaces the GARCH volatilities with the 30-day historical volatilities. The average equal-weighted spread remains positive, and the differences among years are statistically significant at a 1% significant level. The result of the ANOVA test is in Table 11.

Table 10. Significance test (ANOVA) for the equal-weighted spread across year. This table lists the results of the ANOVA test applied to the equal-weighted spread. “Df” means the degree of freedom. “Sum Sq” means the sum of square for years. “Mean Sq” means the mean of square of years, which is equal to Sum Sq/Df. “F value” means the result of the F test. “Pr(>F)” means the P value of the F test. Because the P value is almost zero, it means that the equal-weighted spread is statistically significant different among years.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Year	8	80,207	10,026	1149	<2e−16***
Residuals	1960	17,101	9	/	/

Significance codes: 0.01 “***”, 0.05 “**”, 0.1 “*”.

Table 11. Significance test (ANOVA) for the robustness check across year. This table lists the results of the ANOVA test applied to the equal-weighted spread between 30-day historical volatilities and implied volatilities of nine corporations from the case study. “Df” means the degree of freedom. “Sum Sq” means the sum of square for years. “Mean Sq” means the mean of square of years, which is equal to Sum Sq/Df. “F value” means the result of the F test. “Pr(>F)” means the P value of the F test. Because the P value is almost zero, it means that the equal-weighted spread is statistically significant different among years.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Year	7	107,447	15,350	242.5	<2e−16***
Residuals	1669	105,637	63	/	/

Significance codes: 0.01 “***”, 0.05 “**”, 0.1 “*”.

The implied volatility can present the market expectation of convertible bond prices, while the GARCH volatility is derived from historical prices, which reflects the objective risk of market. As the equal-weighted spread remains positive during the period of research, the expected volatility is consistently lower than the historical volatility. Thus, it implies that either the convertible bonds are substantially and persistently undervalued, or the risk of convertible bonds in China is underestimated. This result is consistent with the opinion of Wen and Qiu (2009) that Chinese convertible bonds are underpriced by 20% to 30%. In addition, it indicates that the Chinese convertible bond market exhibits significant inefficiency which is characterized by a low liquidity and a slow pricing speed.

Based on the equation of equal-weighted spread, while the persistent positive spread may signal undervaluation, it is also possible that investors rationally price latent risks are not captured by the model, such as illiquidity, redemption uncertainty, or conversion clause complexity. The Black-Scholes Model employed for implied volatility calculations assumes the European-style exercise and no credit risk. Other assumptions include no dividend or conditional redemption. Thus, the model might fail to capture the true dynamic of the convertible bonds. Future study aims to adjust the parameters in the GARCH model and apply methodological refinements to improve the stability and accuracy of the Black-Scholes Model.

7. Conclusion

As discussed in the previous sections, this study analyzes the influence of convertible bonds on the efficient frontier and the convertible bond pricing efficiency in China. Simulation generates the efficient frontiers for both the two-asset and three-asset portfolio, while the equal-weighted spread of the GARCH volatilities and the implied volatilities is analyzed to confirm the pricing condition.

The results show that, first, the inclusion of convertible bonds can increase the diversification of the portfolio, thereby improving the efficient frontier on a global range. Second, the equal-weighted spread of convertible bonds in China continuously stays above 0, and this anomaly is corrected relatively after the pandemic as evidenced by the decrease of negative risk premium. It might suggest three potential explanations: 1) the convertible bonds in China are highly underpriced, 2) the investors of convertible bonds are so pessimistic that they expect low future volatilities, 3) the transactions in the convertible bond market are inefficient.

The identification of persistent volatility spreads suggests that market participants and regulators should re-evaluate assumptions of risk neutrality in pricing, and policymakers might explore interventions to improve liquidity and valuation transparency. Policymakers can consider introducing convertible bond ETFs to improve the situation or revise pricing-mechanisms to enhance the liquidity. Issuers can increase investor confidence and pricing transparency by signaling clearer conversion or redemption policies or disclosing information about their repayment capacities. Portfolio managers can recommend convertible bond inclusion, but they should pay more attention to risk modelling and screen the liquidity and pricing anomalies of the convertible bonds more carefully, despite their potential diversification benefits. Further research is required to explore a more specific solution. Besides, this framework could be extended to other emerging convertible bond markets where valuation frictions and policy constraints prevail.

Appendix A: Variable Definitions

Variables	Definition
${\mu }_p$	The mean return of the portfolio.
${\mu }_i$	The return of asset i.
${\sigma }_p^2$	The variance of the portfolio.
${\omega }_i$	The weight of asset i in the portfolio.
$w$	The vector of asset weights.
$\text{Σ}$	The variance-covariance matrix of the asset return.
$Cov\left(A,B\right)$	The covariance between asset A and asset B.
$S_0$	The current price of the underlying stock.
$K$	The strike price.
$r$	The risk-free interest rate.
$T$	The time to maturity.
$\sigma$	The volatility of the convertible bond.
$r_t$	The log return of the underlying stock.
${\sigma }_t^2$	The conditional variance of the underlying stock.
$\mu$	The conditional mean of the underlying stock.
${\epsilon }_t$	The error term.
$EW{S}_t$	The equal-weighted spread of all the available convertible bonds on day t.
${\text{GARCH}}_{i,t}$	The GARCH volatility for convertible bond i on day t.
$Implie{d}_{i,t}$	The implied volatility for convertible bond i on day t.
$N_t$	The total number of the convertible bonds which are available on day t.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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