SSE 50 ETF Options

Tian Yue

A thesis submitted in fulﬁlment of the requirements for the degree of Doctor of Philosophy in Finance at the Department of Accountancy and Finance Otago Business School, University of Otago Dunedin, New Zealand

December 2019 i

Abstract

This thesis studies the newly established exchange-traded-fund (ETF) option in China: the Shanghai Stock Exchange (SSE) 50 ETF option. In Chapter2, we examine the applicability one-dimensional diffusion model and delta-hedged gain in the SSE 50 ETF option market. Launched on 9 February 2015, the SSE 50 ETF option is the first equity option listed in mainland China. We show that the one-dimensional diffusion model does not apply in the SSE 50 ETF option market, which is consistent with the finding in the major worldwide option markets. The delta-hedged gain in the SSE 50 ETF option is mainly negative and it is mainly driven by the volatility related factors in the market. In Chapter3, we examine the implied volatility smirk in the SSE 50 ETF option market. We quantify the implied volatility smirk by following Zhang and Xiang (2008) and we find that the implied volatility smirk existed in the SSE 50 ETF option market. However, the implied volatility smirk in the SSE 50 ETF option market exhibits a right-skewed pattern, which is different from the left-skewed shape in other major option markets. We further analyze the right-skewed implied volatility curve by using several investor sentiment proxies in the market and find that the right- skewed implied volatility curves are strongly related to liquidity based sentiment proxies. Lastly, in Chapter4, we construct a volatility index in China (CNVIX) with the SSE 50 ETF option and analyze the empirical properties of the volatility index. We find that the leverage effect existed in the CNVIX and its underlying asset returns. We further test the volatility risk premium (VRP) based on the CNVIX and find that the VRP could forecast the return of SSE50 ETF over a 30-day horizon. ii

Acknowledgements

Firstly, I would like to thank my primary supervisor: Professor Jin E. Zhang for his guidance and support in my PhD journey. He has a comprehensive plan for his PhD students and is willing to support them when they have difficulties in their studies. He carefully designed training on both research and teaching, which helped me to complete my PhD journey smoothly. Secondly, I am grateful of my co-supervisors: Dr. Eric Tan, Dr. Zheyao Pan, Dr. Sebastian A. Gehricke and Dr. Xinfeng Ruan, for their help in methodology, computer program design/implementation and in revising/polishing the working papers. With their patience and guidance in my research, I was able to finish my PhD thesis on time. Thirdly, I would like to thank the faculty and staff members in the Department of Accountancy and Finance at the University of Otago, including, but not limited to, Professor Timothy Falcon Crack, Professor I M Premachandra, Professor David Lont, Associate Professor Ivan Diaz-Rainey, Associate Professor Ros Whiting, Dr. Helen Roberts, Dr. Xing Han, Dr. Duminda Kuruppuarachchi, Dr. Sriyalatha Ku- marasinghe, Dr. Marinela Finta, Stephen Hall-Jones, Margaret Grundy and Mari- anne Lown. I would also like to thank seminar and conference participants for their helpful comments particularly, Professor Hamish Anderson, Professor Ben Marshall, Professor Bart Frijns, Professor Hai Lin, Professor Nuttawat Visaltanachoti, Profes- sor Robert I. Webb, Professor Jiling Cao and Associate Professor Ryan McKeon and our PhD group meeting participants Dr. Fang Zhen, Nhu Nguyen, Jianhui Li, Wei Guo, Wei Lin, Pakorn Aschakulporn, Isabella Yoon, Zisha Zhang and Jansson Ford. Fourthly, I am so grateful to my lovely parents and their unconditional support and love throughout my life. After I became a father I realised how much effort you have put into raising me up, I love you! Last but not the least, I want to thank my dearest wife, Xiaolan Jia, for your unconditional love, support and sacrifice during my studies for the PhD. You have devoted most of your time to our family so that I could concentrate on my research. We will achieve a bright future with our lovely daughter, Tracy, together. iii

Contents

Abstracti

Acknowledgements ii

1 Introduction1

1.1 Background...... 2 1.2 Structure of this PhD thesis...... 4 1.3 Contribution of this PhD thesis...... 7

2 The Chinese Equity Index Option Market8

2.1 Introduction...... 8 2.2 Overview of the Chinese equity index options market...... 12 2.3 Data...... 15 2.3.1 Summary statistics of the Chinese options market...... 17 2.3.2 The basic properties of the one-dimensional diffusion model.. 22 2.3.3 Option’s BS model price (with historical volatility) and their market price...... 23 2.3.3.1 The BS mdoel and market price of the SSE 50 option. 25 2.3.3.2 The implied volatility and market volatility in the Chinese options market...... 26 2.4 Empirical results...... 32 2.4.1 The perfect correlation in the Chinese option market...... 32 iv

2.4.2 Monotonicity in the Chinese option market...... 33 2.4.2.1 Monotonicity type I violation...... 33 2.4.2.2 Monotonicity type II violation...... 37 2.4.2.3 Monotonicity type III violation...... 38 2.4.2.4 Monotonicity type IV violation...... 38 2.4.3 The delta-hedged gain analysis...... 41 2.4.3.1 The delta-hedged portfolio...... 41 2.4.3.2 The delta-hedged portfolio...... 44 2.4.3.3 The delta-hedged gain with the SSE 50 options.... 46 2.4.3.4 The regression analysis of non zero delta-hedged gain 46 2.5 Conclusion...... 53 2.6 Limitations and future research...... 53 2.7 Appendix...... 54 2.7.1 The dividend adjustment in the Chinese option market..... 54 2.7.2 The derivation of the delta hedged gain in a two-dimensional call option...... 55

3 How do Chinese Options Traders “Smirk” on China: Evidence from SSE 50 ETF Options 58

3.1 Introduction...... 58 3.2 Background of SSE 50 index ETF and options market...... 62 3.3 Data...... 68 3.4 Methodology...... 70 3.4.1 The calculation of implied volatility...... 72 3.4.2 Quantifying the IV curve...... 72 3.5 Empirical Results...... 74 3.5.1 The quantiﬁed IV curves...... 74 v

3.5.2 The constant maturity IV factors...... 82 3.5.3 Sub sample analysis...... 86 3.6 Determinants of the IV curve...... 89 3.6.1 Stationarity test of sentiment proxies...... 91 3.6.2 Regression analysis...... 96 3.7 Conclusion...... 104 3.8 Appendix...... 105 3.8.1 Put-to-Call ratio (PCR):...... 105 3.8.2 Turnover ratio:...... 105 3.8.3 Short-selling and Margin trading ratio:...... 106

4 The Volatility Index and Volatility Risk Premium in China 107

4.1 Introduction...... 107 4.1.1 Contributions...... 112 4.2 Data...... 114 4.3 Methodology...... 115 4.3.1 The approximation errors in the VIX...... 117 4.3.2 Smooth interpolation-extrapolation in the SSE 50 ETF options market...... 120 4.4 Empirical results...... 125 4.4.1 Relationship of the model-free volatility index and the underlying asset...... 126 4.4.2 The asymmetric return-volatility relationship in the CNVIX and the SSE 50 ETF...... 131 4.4.3 CNVIX-based VRP...... 134 4.4.4 VRP and return predictability in China...... 142 4.4.4.1 Model-free VRP measure...... 142 vi

4.4.4.2 Alternative model-based VRP measure...... 146 4.5 Conclusion...... 147 4.6 Appendix...... 148 4.6.1 Interpolation-extrapolation extended option dataset in less liquid options market...... 148 4.6.1.1 Calculate the implied forward price of the underlying asset...... 148 4.6.1.2 Calculate the implied volatility (IV) of option..... 148 4.6.1.3 Construct the IV curve function...... 149 4.6.1.4 The interpolation within the option chain’s strike range149 4.6.1.5 The extrapolation beyond the option chain’s strike range...... 150

5 Conclusion 151

Bibliography 153 vii

List of Figures

2.3.1 Call and put options’ market and BS price comparison by moneyness group...... 30 2.3.2 Chinese put and call warrants’ market and Black-Scholes (BS) price.. 31

3.2.1 SSE 50 and SPY ETF options’ trading volume, value and open interest 65 3.2.1 SSE 50 and SPY ETF options’ trading volume, value and open interest (continued)...... 66 3.2.1 SSE 50 and SPY ETF options’ trading volume, value and open interest (continued)...... 67 3.3.1 The SSE 50 index and the SSE 50 ETF index tracking in our data sample...... 69 3.5.1 Fitted IV curves on 8 May 2015...... 77 3.5.2 Fitted IV curves on 17 February 2016...... 78 3.5.3 Fitted IV curves on 2 May 2017...... 79 3.5.4 Mean implied volatility (IV) curves...... 81 3.5.5 Constant maturity mean implied volatility (IV) curves...... 85 3.5.6 Time series of IV curve factors...... 87 3.6.1 Investor sentiment proxies in China...... 92 3.6.1 Investor sentiment proxies in China (continued)...... 93

4.3.1 Extend the option strikes by the implied volatility function...... 122 4.3.2 Extension of out-of-the-money (OTM) dataset...... 123 viii

4.3.3 Time series plot of the CNVIX and iVX...... 126 4.4.1 Time series plot of the CNVIX and the SSE 50 ETF...... 128 4.4.2 Quantile regression estimate across quantiles: response variable to CNVIX changes...... 135 4.4.3 Time series plot of the CNVIX, realized volatility (RV) and volatility risk premium (VRP) of the SSE 50 ETF...... 138 ix

List of Tables

1.2.1 Thesis Chapters...... 6

2.2.1 The trading rules in the SSE 50 options...... 14 2.3.1 The summary statistics of the SSE 50 index, ETF and risk free rate... 16 2.3.2 Moneyness and maturity subgroup deﬁnition...... 18 2.3.3 The summary statistics number of strikes of the SSE 50 option chains. 19 2.3.4 SSE 50 options’ trading summary statistics...... 21 2.3.5 Comparison of options’ market price and Black-Scholes (BS) model price...... 27 2.3.6 Comparison of options’ Black-Scholes (BS) model implied volatility (IV) and historical volatility (HV)...... 29 2.4.1 The correlation table of SSE 50 options’ price, Black-Scholes (BS) model price and underlying asset price change...... 34 2.4.2 The SSE 50 ETF options’ type I violation ratio...... 36 2.4.3 The SSE 50 ETF options’ type II violation ratio...... 39 2.4.4 The SSE 50 ETF options’ type III violation ratio...... 40 2.4.5 The SSE 50 ETF options’ type IV violation ratio...... 42 2.4.6 The delta-hedged gain (DHG) with call options...... 47 2.4.7 The delta-hedged gain (DHG) with put options...... 48 2.4.8 The regression analysis of delta-hedged gain (DHG) with call options 51 2.4.9 The regression analysis of delta-hedged gain (DHG) with put options. 52 x

3.2.1 Summary of leading exchange-traded-fund (ETF) in China (2017)... 63 3.4.1 Summary of the SSE 50 ETF options market...... 71 3.5.1 Summary of fitted implied volatility (IV) coefficients...... 75 3.5.2 Summary of interpolated term structure...... 84 3.5.3 Summary of fitted implied volatility (IV) coefficients in sub samples. 88 3.6.1 Stationary test of investor sentiment proxies...... 94 3.6.2 Correlation of investor sentiment proxies in China...... 95

3.6.3 Regression analysis of 30-day constant maturity coefﬁcient: Level (γ0) 97

3.6.4 Regression analysis of 120-day constant maturity coefﬁcient: Level (γ0) 98

3.6.5 Regression analysis of 30-day constant maturity coefﬁcient: Slope (γ1) 100

3.6.6 Regression analysis of 120-day constant maturity coefﬁcient: Slope (γ1) 101 3.6.7 Regression analysis of 30-day constant maturity coefﬁcient: Curva-

ture (γ2)...... 102 3.6.8 Regression analysis of 120-day constant maturity coefﬁcient: Curva-

ture (γ2)...... 103

4.1.1 Summary statistics of volatility indexes...... 110 4.1.2 The Top 10 Stocks by weight in FXI and SSE 50 ETF...... 113 4.3.1 Summary of option strikes in SSE50, SPX and FXI ETF option markets 119 4.3.2 Trading days without model based volatility index...... 125 4.3.3 The correlations of volatility indexes...... 127 4.4.1 Leverage effect regression of CNVIX and VXFXI...... 132 4.4.2 Leverage effect: quantile regression analysis...... 136 4.4.3 Summary statistics of variables in VRP regression...... 140 4.4.4 Correlation table of variables in VRP regression...... 141 4.4.5 Return (overlapping) and VRP predictability from SSE 50 ETF options 143 xi

4.4.6 Return (non-overlapping) and VRP predictability from SSE 50 ETF options...... 145 1

Chapter 1

Introduction

I am a professional trader of equity, fixed income and derivatives in the China/US market with nearly 10-years of experience. With increased trading experience and assets under management, I became extremely aware of the importance of mitigat- ing the risk associated with investments. I have used a short-selling position on stock index futures as insurance against systematic risk since 2010, when the China Finan- cial Futures Exchange (CFFE) introduced the China Stock Index (CSI) 300 index futures. However, the notional value of stock index futures is too large for precise risk hedging. In 2013, I learned that investors could use put options as an effect/precise hedging tool to downside risk. However, after brief research into the put option market, I found that its price is modeled by mathematical formulas like the Black and Scholes(1973) formula. After I finished my Master of Business at Massey University (MU) in 2015, I decided to further study the theory in financial risk management, especially in the derivatives markets. After I enrolled in the PhD program at the University of Otago (UO), I attended Professor Zhang’s two core papers in the finance program at UO: FINC 306 Deriva- tives and FINC 405 Mathematical Finance. The studies in these two papers further enhanced my theoretical/practical understanding of the derivatives market. Profes- sor Zhang also invited me to join his papers as a tutor, and the fulfilment of tutorials Chapter 1. Introduction 2

greatly improved my teaching skills in derivatives markets. Previously, I thought the mathematical theoretical framework of option is not the crucial part of successful options trading, as the option price will be ultimately determined by the underlying asset. When I read a highly cited paper, Bakshi and Kapadia(2003a) I found that investors could achieve superior profit in a side-ways market by harvesting option premiums with a delta-hedged option portfolio. Tradi- tional long/short investments in the equity market could not achieve any profit in a balanced/side-ways market. The distinguished characteristics in the option markets inspired my strong research interest in the option market. Nowadays, the revenue from the delta-gamma-vega hedged option portfolio contributes more than half of the profit in my overall investment. Based on my previous experience in the Chinese options market, Professor Zhang suggested that we could do some empirical research on the newly established Shang- hai Stock Exchange (SSE) 50 exchange-traded-fund (ETF) option market in China. Then my supervisors and I agreed on our research topic, SSE 50 ETF Options Market. Along with the process of my research, I have studied the one-dimensional diffusion model, delta-hedged gain, implied volatility smirk, volatility index and volatility risk premium in SSE 50 ETF options.

1.1 Background

Financial options are relatively new products in the capital market when compared with traditional equity securities. The exchange-listed option first began to be offi- cially traded at the Chicago Board Options Exchange (CBOE) in 1973. Equity options were not available to investors in China, until the SSE launched its first equity option: the SSE 50 ETF on 9 February 2015. Before the launch of the option market in China, equity warrants were available to investors. The warrants Chapter 1. Introduction 3

are similar to the options, in that only institutional investors could issue/write new warrants in the market. Due to the investors’ gambling behavior and “T+0” intraday trading settlement of warrants, huge bubbles were found in the warrants market in China (see, e.g., Xiong and Yu, 2011; Chang, Luo, Shi, and Zhang, 2013b; Tang and Wang, 2013). In Chapter2, we follow Bakshi, Cao, and Chen(2000), Bakshi and Kapadia(2003a) and Chang et al.(2013b) and empirically test the SSE 50 ETF option market which filled the literature gap in the Chinese equity option market. The implied volatility “smile” or “smirk” in the market reflects the investors’ risk preference. Rubinstein(1985) first found the symmetrical implied volatility smile curve in the US equity option market. Bates(1991) finds that after the 1987-crash, investors were willing to pay more on the put options compared with call options with the same moneyness, which shifted the symmetrical implied volatility smile to the left-skewed smirk shape. The dynamics related to the implied volatility shape provides a starting point for a more realistic option pricing model. Chapter3 focuses on documenting the implied volatility smirk in the SSE 50 ETF option market and analyzes the cause for the shape of the implied volatility curve, which fills the literature gap on the implied volatility curve in China. A volatility index derived from the option markets becomes one of investor’s “fear gauges” and market sentiment proxy (Whaley, 2000). The CBOE launched its volatility index (VIX) based on the implied volatility of S&P 100 index options in 1993. Investors build various trading strategies based on the VIX. Later, the CBOE modified its VIX from being model-based to model-free based on the pioneer work of Breeden and Litzenberger(1978), Demeterfi, Derman, Kamal, and Zou(1999) and Britten-Jones and Neuberger(2000). The availability of VIX provides a foundation for research on volatility trading. Chapter4 constructs a model-free volatility index in China (CNVIX) with the SSE 50 ETF options and empirically tests the relationship between the CNVIX and its underlying asset. Chapter 1. Introduction 4

1.2 Structure of this PhD thesis

Chapter2 to Chapter4 in this thesis comprise three related papers. Table 1.2.1 sum- marizes the details of the three chapters and the contributions. After the decision on studying the derivatives market in China for my PhD. I searched the literature on the derivatives and option market. I quickly became aware on the lack of research related to the equity option market in China. Since the first equity option, the SSE 50 ETF option, was listed in the SSE on 9 February 2015, to the best of our knowledge, there has been no published paper on this newly established option market in 2016. In our study in Chapter2, The Chinese Equity Index Option Market, Professor Zhang, Dr Tan and I found that the one-dimension diffusion model (for example, the Black and Scholes(1973) model) is not applicable to the SSE 50 ETF option market. Four assumptions of the one-dimension diffusion model and the violation ratios are much higher than in the developed option markets. The delta-hedged gains in the SSE 50 ETF call (put) option are not zero, which is caused by volatility-related factors. This chapter was presented at the 2018 New Zealand Finance Colloquium and is currently under review by a journal for publication. After we finished our paper in Chapter2, my supervisors suggested we target the implied volatility in the option market in the next chapter. In our Chapter3, How do Chinese Options Traders “Smirk” on China: Evidence from SSE 50 ETF Options, we fo- cused on documenting the implied volatility curve in the Chinese option market. We quantify and analyze the dynamics of the implied volatility curve in the SSE 50 ETF option market. We analyzed the cause of the right-skewed implied volatility curve with several investor sentiment proxies and found that liquidity factors contribute to the shape of implied volatility curve. In December 2018, I finished this chapter Chapter 1. Introduction 5

with my supervisors and the chapter was presented at the 2019 New Zealand Fi- nance Colloquium and 2019 Derivatives Markets Conference and is currently under review by a journal for publication. In Chapter4, we explored the model-free implied volatility index in China by using the SSE 50 ETF options. Since the Chicago Board Options Exchange launched its volatility index (VIX) in 1993, investors use the VIX as one of the most important sentiment proxies in the market. With the availability of equity options in China, we construct a model-free volatility index in China with the SSE 50 ETF options by following the CBOE’s VIX methodology. We analyze the time-series properties of the China volatility index and inspect the volatility risk premium based on the model- free volatility index in China. In October 2018, I ﬁnished this chapter jointly with my supervisors, The Volatility Index and Volatility Risk Premium in China, and this chapter has been accepted for presentation by the 2020 New Zealand Finance Colloquium. The three chapters discussed in this section are as follows:

1 Yue, Tian, Jin E. Zhang and Eric Tan, 2017, The Chinese equity Index Option Market. Presented at 2018 New Zealand Finance Colloquium. [Chapter2].

2 Yue, Tian, Sebastian A. Gehricke, Jin E. Zhang and Zheyao Pan, 2018, How do Chinese Options Traders “Smirk” on China: Evidence from SSE 50 ETF Options. Presented at the 2019 New Zealand Finance Colloquium and 2019 Derivatives Markets Conference. [Chapter3].

3 Yue, Tian, Xinfeng Ruan, Sebastian A. Gehricke, Jin E. Zhang 2019. The volatility Index and Volatility Risk Premium in China. Accepted for presentation at the 2020 New Zealand Finance Colloquium. [Chapter4]. Chapter 1. Introduction 6 Status Presented atZealand 2018quium, Finance and New publication. submitted Collo- for Presented at 2019 Derivatives Markets Conference and 2019 New Zealandloquium, Finance and Col- submittedpublication. for Accepted for presentation at 2020 New ZealandColloquium. Finance Contribution of candidate Undertook option dataand collection processing, methodologysign, de- program implementation, results analysis andthors writing. provided guidance Coau- onture, litera- methodology design, support on theoretical techniques,skills, writing submitting and revisingpaper. the Undertook option dataand collection processing, methodologysign, de- program implementation, results analysis andthors writing. provided support on Coau- methodology, programwriting skills, submitting implementation, and revising the paper. Undertook option dataand collection processing, methodologysign, de- program implementation, results analysis andthors writing. provided Coau- supportture, on methodology design, litera- submit- ting and revising the paper. Thesis Chapters Table 1.2.1: Authors Tian Yue, JinZhang E. andTan Eric Tian Yue, Sebas- tian A. Gehricke, Jin E. Zhang and Zheyao Pan Tian Yue, Xinfeng Ruan, Sebastian A. Gehricke and Jin E. Zhang Working paper The Chinese Equity In- dex Options Market How do Chinesetions Op- traderson “Smirk” China?from SSE Evidence 50tions ETF Op- The Volatilityand Index Premium Volatility in China Risk 2 3 4 Chapter Chapter 1. Introduction 7

1.3 Contribution of this PhD thesis

Chapter2 contributes to the literature by introducing the newly established option market in China. We provide the first analysis of the SSE 50 ETF option with the one-dimensional diffusion model. We find that the one-dimensional diffusion model does not apply to the option market in China. We further analyze the delta-hedged gain in the SSE 50 ETF option market and find that the non-zero delta-hedged gain is related to the time-varying volatility in the option market. In Chapter3, we provide the first research on the implied volatility smirk in the SSE 50 ETF option market. We extend the methodology developed by Zhang and Xiang(2008) to quantify the shape of the implied volatility curve in the option market in China. We study the time series and term structure of the implied volatility curve in the SSE 50 ETF options. We also analyze the cause of the right-skewed implied volatility curve with investor sentiment proxies. Lastly, in Chapter4, the first contribution is that this is the first paper to construct the model-free volatility index with interpolation-extrapolation extended SSE 50 option dataset in China. Based on the model-free index, the second contribution we make is that this chapter is the first to empirically test the leverage effect and volatility risk premium in China. 8

Chapter 2

The Chinese Equity Index Option Market

This chapter is a joint work with Jin E. Zhang and Eric Tan. It was presented at the 2018 New Zealand Finance Colloquium, 7-9 February 2018, Massey University, Palmerston North, New Zealand.

2.1 Introduction

Financial options are contingent claims with non-linear pay-offs from underlying asset, which are primarily traded in markets as an effective risk management tool. The index options which use indexes as their underlying assets are the most actively traded options in derivatives market. For example, the S&P 100 index option which was ﬁrst developed in the Chicago Board Options Exchange (CBOE) in 1983, was the ﬁrst exchange-traded stock index option. Since then, more exchanged index options (for example, the S&P 500, and NASDAQ 100 options) have been listed, and such index options provide investors with effective risk management tools. This paper explores the equity index option market in China, which was newly developed on 9 February 2015. Currently, China is the world’s second largest market in terms of economic entity and its capital market size. The capital market in China Chapter 2. The Chinese Equity Index Option Market 9

is growing exponentially along with its economic development. The Shanghai Stock Exchange (SSE), the first stock exchange in China, was established in 1990. SSE listed the first derivatives in China: the Treasury futures (TF) on 28 December 1992. However, the excess short-selling and manipulation by large brokers in the market, lead to the crash of the TF market. The over speculation and manipulation in the TF market resulted in the closure of the market on 18 May 1995 by the China Securities Regulatory Commission (CSRC). While the CSRC realized that innovation in the derivatives market would enhance economic development, the potential abuses and manipulations would be devastating to the capital market. Since then, there are no exchanged derivatives products made available to investors in China. In 2005, the SSE and the Shenzhen Stock Exchange (SZSE) introduced the call (put) equity warrants on some pilot blue-chip stocks for the split share structure re- form. The warrants market in China soon became one of the most liquid warrants markets for the following reasons. Firstly, China adopts a “T + 1 trading rule”, which prevents investors from selling stocks bought on the same day “T + 0 trading” intraday trading). Guo, Li, and Tu(2012) find that the “T+1” trading rule, which prevents investors from intraday trading and it could reduces the China A-Share’s volatility and trading volume. While the warrants derivatives are traded with a “T+0” rule, which allows investors to sell their warrants intraday. Secondly, the trading in warrant has wider price fluctuation range compared with the trading range in stock (daily range in −/ + 10%). Third, the transaction fee is lower in warrants and tax is exempted from warrants trading. As investors take advantage of the warrants’ “T+0” trading rule, lower transaction costs and high leverage in warrants market, result in huge bubbles in the warrants market in China. Fung, Zhang, and Zhao(2009) find Chinese investors having lack of basic understanding in the warrants derivatives, which leads to over speculation and excessive trading in the warrants market. Xiong and Yu(2011) analyse the put warrants Chapter 2. The Chinese Equity Index Option Market 10

market in China from 2005 to 2008 by using the Chinese warrants data to test the bubble theory and provide evidence of the bubbles in the Chinese warrants market. Zhou, Wu, and Yang(2011) further analyse the Chinese warrants market with high-frequency intraday warrants trading data, and they find the bubbles relate to trading volume, turnover, and outstanding number of warrants in the market. Tang and Wang(2013) study the pricing error in the warrants market and find that the mispricing is due to short-selling restrictions. Chang et al.(2013b) examine all 30 expired put (call) warrants (from 2005 to 2008) with violation tests in one-dimensional diffusion model. They find that irrational investment in the put warrants creates huge bubbles in the market. Powers and Xiao(2014) document the mispricing in the Chinese warrants market, they find that the put (call) warrants are on average above (below) their model price and attribute the mispricing to transaction tax and firms’ P/E ratios. Due to the speculation and immature investors’ excessive trading in warrants market, the CSRC decided to close the warrants market in 2011. The options are the new derivatives in the Chinese market and the literature on the Chinese options is relatively scarce. O’Neill, Wang, and Liu(2016) derive a state price volatility (SPV) based volatility index for the Chinese stock market and show that the SPV volatility index could forecast the realized volatility (RV) of the Shang- hai Composite Index (SHCI). Huang, Liu, Zhang, and Zhu(2018) construct three Chinese volatility indexes with the SSE50, China Stock Index 300 (CSI 300), and FXI (iShares China Large-Cap ETF) options by following CBOE’s standard methodology and document the term structure of volatility risk premium across the three markets. Huang, Tong, and Wang(2019) analyze the popular volatility models in term of pricing performance with SSE 50 options and they find that newly developed model with realized measures outperformed conventional GARCH (Generalized Autore- gressive Conditional Heteroskedasticity) type models. Classical option pricing theory is based on the assumption that the underlying Chapter 2. The Chinese Equity Index Option Market 11

asset follows a one-dimensional diffusion process. The one-dimensional diffusion model is also known as the local (deterministic) volatility model, which is the fundamental assumption of classical option pricing theory. Black and Scholes(1973), Merton(1973), and the Cox and Ross(1976) with constant elasticity of variance are classical examples of a one-dimensional diffusion model. In this paper, we mainly follow the methodology proposed by Bakshi et al.(2000) to test the Chinese option market in the properties of a one-dimensional diffusion model. There are three common basic properties for the model: (1) Monotonicity: the call (put) options are monotonically increase (decrease) with the upward movement of the underlying assets; (2) Perfect correlation: the underlying asset’s price is the only source of uncertainty for its corresponding options; (3) Redundancy: options can be replicated by using risk-free asset and their underlying assets. We examine the one-dimensional diffusion model’s fundamental properties in the newly developed derivatives market in China in order to answer the following testable hypothesis: Hypothesis 1: The newly developed option market share similar behaviour of the previous closed Chinese warrants market, bubbles existed in the market. Hypothesis 2: The most widely used one-dimensional diffusion model, the Black- Scholes (BS) model is not ideal for dynamic and precise hedging in the new options market in China. The short-selling and margin trading are banned in China until 31 March 2010, and there are no down-side risk hedging products for investors in the Chinese capital market. Investors’ excessive demand on the put warrants create huge bubbles in the Chinese capital market (Xiong and Yu, 2011). Beber and Pagano(2013) ﬁnd that when compared with the stocks without options in the short-selling restricted market, stocks with options have better efﬁciency. The CSRC are very cautious about the development of short-selling and product with implied short-selling (for example, the long put warrant position). Even after the short-selling was permitted in Chapter 2. The Chinese Equity Index Option Market 12

2010, investors ﬁnd it is very difﬁcult to borrow securities from their brokers for their short-selling positions. Similar to the put warrant bubbles in the Chinese warrant market, our hypothesis 1 indicates that investors are taking long position in put options as the alternative short-selling in the short-selling restricted market, and the price of put options will be higher than their intrinsic values when compared to call options. The rest of our paper is organised as follows. Section 2.2 provides an overview of the Chinese equity index options market. Section 2.3 describes our data. Section 2.4 reports empirical results. Section 2.5 concludes the paper.

2.2 Overview of the Chinese equity index options market

The SSE 50 stock index, which includes the 50 largest listed companies in the SSE and it was published on 1 Jan 2004. The SSE 50 option’s underlying asset is the SSE 50 ETF (exchange-traded-fund), which is the largest ETF that replicates the SSE 50 index in China. The SSE launched the virtual option trading (ﬁve stocks and one index options) on 8 Nov 2013 and the SSE 50 options on 9 February 2015. Currently, it is very hard for investors to participate in the options market, since the SSE sets high entry requirements to the options market. These requirements include: (1) Investors should have at least one month’s virtual option trading experience; (2) Investors should have trading experience in margin trading and short- selling in the stock market; (3) Investors need to pass option trading tests in 3 levels; (4) Investors must provide evidence of fund of CNY 500,000 (at least) in their accounts.1 Those requirements in the option market restricted the investors’ access to the options market in China compared with the previous equity warrants market.2

1CNY 500,000 is about USD 77,800 by using the average exchange rate (USD/CNY = 6.42) in 2015. 2Investors have access to the Chinese equity warrants market by default and there are no similar entry requirements and tests in the options market. Chapter 2. The Chinese Equity Index Option Market 13

The SSE 50 option chains are adjusted with the following rules: At the inception of an option chain, there are four out-of-the-money (OTM) options, one at-the- money (ATM) option and four in-the-money (ITM) options with the strike interval of 50-points (100-points strike interval, if the SSE 50 index is above 3,000 points) in the SSE 50 index. Along with the movements of SSE 50 index, new OTM, ATM and ITM options will be added on the option chain to maintain at least four OTM, ITM options and one ATM options in the option chain. The SSE adjusts the number of option strikes according to the index’s movement and investors could only trade options listed in the option chains. 3 The SSE 50 ETF, which is valued at one thousandth of the underlying SSE 50 Index. For example, when the SSE 50 index is at 1,900 points, the SSE 50 ETF’s unit net asset value (NAV) is approximately CNY 1.900. The settlement style of SSE 50 option is the European style with delivery of SSE 50 ETF. Investors need to pass corresponding level of tests and submit the proof of fund to access option trading. Table 2.2.1 reports the detailed trading rules of the SSE 50 option.

3At the inception of each option chain, there are ﬁve strikes (two OTM, ITM and one ATM options) in each option chain, the SSE has updated the initial number of strikes to nine (four OTM, ITM and one ATM options) at the inception of each option chain since 4 January 2018. Chapter 2. The Chinese Equity Index Option Market 14

Table 2.2.1: The trading rules in the SSE 50 options

This table reports the trading rules in the SSE 50 options.

Option issuer: Institutional, retail investors.

Exchange: Shanghai Stock Exchange (SSE).

Underlying asset: SSE 50 Index Exchange Traded Fund (ETF).

Exercise and assignment type: European style.

Maturity term: The current, next month and two months in

March-June-September-December cycle.

Maturity date: The fourth Wednesday in each mature month or speciﬁed by SSE.

Settlement date: The fourth Thursday in each mature month or speciﬁed by SSE.

Settlement type: Delivery of the underlying ETF.

Strikes: Four ITM, OTM and one ATM options in each chains’ inception.

Units of exercise: 10,000 shares of SSE 50 ETF fund per contract.

Standard Commission: CNY 20 per contract.

Market Makers: Eight pilot security companies.

Trading privilege:

Level I: Covered call only (short call backed by the underlying asset).

Level II: Level I and long option position.

Level III: Level I, II and short (naked) option position.

Trading Hours: 9:30 to 15:00 (Time in Beijing, China), Monday to Friday. Chapter 2. The Chinese Equity Index Option Market 15

2.3 Data

In this section, we introduce the details of the Chinese option dataset and summary statistics. Then we report all options’ BS model price with historical volatility and their implied volatility (IV). The dataset employed in our paper is downloaded from the WIND ﬁnancial terminal, which is one of the largest ﬁnancial data provider in China.4 The dataset includes 876 expired put and call options from 9 February 2015 to 30 June 2017, the dataset includes all options’ daily closing, open, high, low, settlement price, trading volume (number of contracts), trading value (in CNY) and open interest (number of contracts). Table 2.3.1 reports the summary statistics of the SSE 50 index, ETF and the risk free rate (SHIBOR: Shanghai Interbank Offered rate) in our dataset.

4The Wind financial database is the most comprehensive and powerful tool for financial pro- fessionals who need the most complete information on Chinese stocks, bonds, funds, futures, RMB rates, and the economy. The Wind financial terminal’s website is: https://www.wind.com.cn/en/ about.html Chapter 2. The Chinese Equity Index Option Market 16 The summary statistics of the SSE 50 index, ETF and risk free rate Table 2.3.1: VariablesSSE50 Index closing priceIndex’s log return (%)Index’s trading volume 582 (million) 2389.777 582SSE50 ETF 313.603 closing price 57.988 1912.721 582ETF’s log 3458.707 return N (%) 56.083 2175.397 0.014ETF’s trading mean 2337.215 volume 9.219 (million) 582 2466.006 1.778SHIBOR overnight 582 rate 377.459 (%) 2.390 sd 1.503 644.392 -9.852 22.825 582 982.662 0.296 4.903 min 582 7.547 34.364 77.855 0.035 1.886 2.092 9146.831 70.074 -0.530 1.786 max 174.927 3.427 0.489 0.051 2.108 285.099 -9.982 p25 655.096 2.198 1.027 8.428 0.616 8.267 4.189 2.338 3.464 p50 -0.509 -0.929 26.555 2.459 1.901 0.000 p75 9.476 2.010 1.449 skewness 0.614 kurtosis 2.296 4.877 -0.636 0.388 9.696 3.907 Table 2.3.1 reports the summaryOffered Rate) statistics from of 9 February the 2015 SSE to 50 30 June index, 2017. ETF and the risk free rate (SHIBOR: Shanghai Inter Bank Chapter 2. The Chinese Equity Index Option Market 17

2.3.1 Summary statistics of the Chinese options market

We divide the full option dataset into twelve subgroups by options’ moneyness and time of maturity. If a call option’s strike price is equal to the spot price of the underlying asset, it is classified as an ATM option. If a call option’s strike price is higher or lower than the spot price of the underlying asset, it is classified as an OTM or ITM option. As it is hard for the underlying asset’s closing price to fall exactly at an option’s strike price, in order to define the moneyness of an option, we define ATM groups by their strike variation around +/-3% to its closing price. When the underlying asset’s price experiencing volatile changes, options with adjacent strike prices will shift in different moneyness groups; however, they share similar properties. The ATM moneyness groups are formed with +/-3% strike variation to include similarity in options. We categorize all option’s maturity length into three terciles, the options in first, second and third terciles are categorized into short, medium and long term options correspondingly.5 We have detailed definition of option sub-groups in Table 2.3.2. Table 2.3.3 reports the number of strikes in the SSE 50 option chains in our sample from 9 February 2015 to 30 June 2017.

5The ﬁrst tercile includes options with days to maturity in less than 55 days, the second tercile includes options with days to maturity from 55 days to 117 days, the third tercile includes options with days to maturity higher than 117 days. Chapter 2. The Chinese Equity Index Option Market 18

Table 2.3.2: Moneyness and maturity subgroup deﬁnition

Table 2.3.2 reports the category and deﬁnition of moneyness and maturity-term in our sample. K is the option’s strike price and St is the underlying asset’s price at date t. In our sample the tercile 1 includes all the options with maturity less than 55 days; tercile 2 includes all options with maturity from 55 days to 117 days and tercile 3 includes all options with maturity higher than 117 days.

SSE 50 options’ subgroups

Category Name Deﬁnition

1 Deep in-the-money (DITM) call K/St < 0.9

2 Deep in-the-money (DITM) put 1.1 < K/St

3 In-the-money (ITM) call 0.9 ≤ K/St < 0.97

4 In-the-money (ITM) put 1.03 < K/St ≤ 1.1

5 At-the-money (ATM) call and put 0.97 ≤ K/St ≤ 1.03

6 Out-of-the-money (OTM) call 1.03 < K/St ≤ 1.1

7 Out-of-the-money (OTM) put 0.9 ≤ K/St < 0.97

8 Deep-out-of-the-money (DOTM) call 1.1 < K/St

9 Deep-out-of-the-money (DOTM) put K/St < 0.9 10 Short term option option with initial maturity days in tercile 1 11 Middle term option option with initial maturity days in tercile 2 12 Long term option option with initial maturity days in tercile 3 Chapter 2. The Chinese Equity Index Option Market 19 days 120 > days 120 < days 180 > days 180 − 90 days 90 − 30 days 30 < The summary statistics number of strikes of the SSE 50 option chains Full Table 2.3.3: ObsMeanMedianMinMax 2155Mean trading volume 14 276,952Mean open 12 interest 578 192,397 778,462 14 5 84,681 429,202 14 734 31 261,964 13 15,912 6 12 31 508 124,025 6,376 16 5 41,167 31 15 266,372 335 701,403 12,322 10 1511 89,747 5 31 8 644 14 23 13 5 12 31 11 5 24 5 Table 2.3.1 reports the summaryFebruary 2015 statistics to of 30 number June of 2017. strikes in the SSE 50 option chains (date-maturity group) from 9 Chapter 2. The Chinese Equity Index Option Market 20

Table 2.3.4 reports the total, average daily trading volume, value and their percentage in total trading volume in our sample period. We ﬁnd that the trading in the option market increased steadily as more investors and institutions participated in the option market. Trading in call options takes about 55% of total trading volume and put option which takes around 45% in total trading volume. The underlying asset SSE 50 ETF, pays discrete cash dividend to adjust its replication of the SSE 50 index. we use an option’s maturity time window length to deﬁne its expected dividend yield. According to historical dividend payment record, we match each option’s dividend yield time window to the dividend payment history table. For those options which have the maturity time window that spanned any ex-dividend date, we use the matched annualized dividend yield otherwise we set the dividend yield to zero. After each dividend payment, all existing options’ strike price and exercise units will be adjusted to maintain their notional value unchanged after the dividend payment. Meanwhile, new standard options will be created on the dividend payment date. The adjusted options have non-standard strike price and exercise units when compared with standard options (50-points strike interval and 10,000 exercise units per contract). The non-standard option chains will not be adjusted along with the movement of underlying asset and they are less actively traded compared with the standard options. We provide the introduction to the option dividend adjustment trading rule in the appendix. Chapter 2. The Chinese Equity Index Option Market 21

Table 2.3.4: SSE 50 options’ trading summary statistics

Table 2.3.4 reports summary statistics of options: the total trading volume, value, the average daily trading volume and the average percentage of put and call options in total option trading volume from 9 February 2015 to 30 June 2017.

SSE 50 options’ trading statistics

Trading Trading vol- Trading value Quarter/Year Average daily trading volume days ume (CNY, million) Q1 2015 32 777,251 741 24,289 Q2 2015 62 3,688,659 5,210 59,495 Q3 2015 64 8,006,545 9,730 125,102 Q4 2015 61 10,645,207 7,964 174,512 Q1 2016 59 12,812,480 8,975 217,161 Q2 2016 66 13,750,269 6,781 208,337 Q3 2016 64 20,318,042 10,563 317,469 Q4 2016 60 31,955,933 16,860 532,598 Q1 2017 59 27,641,743 11,505 468,504 Q2 2017 60 30,255,661 11,496 504,261 SSE 50 call options’ trading statistics Trading Trading vol- Trading value Average daily % in total vol- Quarter—Year days ume (CNY, million) volume ume Q1 2015 32 425,136 513 13,286 54.7% Q2 2015 62 2,096,407 3,332 33,813 56.83% Q3 2015 64 4,603,151 4,034 71,924 57.49% Q4 2015 61 5,991,532 4,762 98,222 56.28% Q1 2016 59 7,071,559 4,094 119,857 55.19% Q2 2016 66 7,398,973 3,680 112,106 53.81% Q3 2016 64 11,778,482 7,130 184,038 57.97% Q4 2016 60 18,859,171 11,075 314,319 59.02% Q1 2017 59 15,776,912 6,784 267,405 57.08% Q2 2017 60 17,518,601 7,505 291,976 57.90% Chapter 2. The Chinese Equity Index Option Market 22

Table 2.3.4 (continued): SSE 50 options’ trading statistics

SSE 50 put options’ trading statistics

Trading Trading vol- Trading value Average daily % in total vol- Quarter/Year days ume (CNY/million) volume ume Q1 2015 32 352,115 227 11,004 45.30% Q2 2015 62 1,592,252 1,877 25,681 43.17% Q3 2015 64 3,403,394 5,695 53,178 42.51% Q4 2015 61 4,653,675 3,202 76,290 43.72% Q1 2016 59 5,740,921 4,881 97,304 44.81% Q2 2016 66 6,351,296 3,100 96,232 46.19% Q3 2016 64 8,539,560 3,432 133,430 42.03% Q4 2016 60 13,096,762 5,785 218,279 40.98% Q1 2017 59 11,864,831 4,720 201,098 42.92% Q2 2017 60 12,737,060 3,990 212,284 42.10%

2.3.2 The basic properties of the one-dimensional diffusion model

Before we proceed to the empirical results of the Chinese option market, we brieﬂy review the one-dimensional diffusion model and its basic assumptions. Assume a non-dividend paying underlying asset’s price, St follows the one-dimensional diffusion model:

dSt = µ(S, t)Stdt + σ(S, t)StdBt, (2.1) where the drift term, µ(S, t) and volatility term, σ(S, t) are determined by the underlying asset price, St at time t, only, the Bt is a standard Brownian motion. Option pricing models that assume the underlying asset follows a one-dimensional diffusion model share the following three fundamental properties: monotonicity, perfect correlation with the underlying asset, and redundancy replication. The Black and Scholes(1973) model with constant volatility and the Cox and Ross(1976) constant elasticity of variance model are the most famous one-dimensional diffusion process models. The monotonicity property in contingent claims has been proven Chapter 2. The Chinese Equity Index Option Market 23

by Bergman, Grundy, and Wiener(1996) who shows that under the assumption that the underlying asset follows a one-dimensional process (with restricted stochastic volatility), the option’s delta is bounded by both an upper and lower boundary (Call option’s delta [0,1], put option’s delta [-1,0]). Furthermore, if an option’s payoff is convex (concave), the option’s price is a convex (concave) function of the underlying asset’s price. The perfect correlation property in the model means that the underlying asset is the only source of variation of derivatives and the option’s price will be perfectly correlated with the underlying asset. The redundancy property in the model means that an option can always be exactly dynamically replicated with a combination of its underlying asset and risk free bond.

2.3.3 Option’s BS model price (with historical volatility) and their market price

We use the following variables to calculate an option’s BS model price. (1) Risk free rate (r) We use SHIBOR (Shanghai Interbank Offered Rate) as the risk-free rate in our Black-Scholes model price calculation. The SHIBOR is the reference rate based on the interest rates at which banks offer to lend unsecured funds to other banks in the Shanghai wholesale (interbank) money market. There are eight SHIBOR rates, with maturities vary from overnight to one year. The SHIBOR are calculated from rates quoted by 18 banks, eliminating the two highest and lowest in all of rates, and then averaging the remaining 14 rates. We use the SHIBOR overnight rate to match each option’s risk-free rate. Bakshi et al.(2000) use the policy rate as derivatives’ risk-free rate, however, the policy rate in China stay unchanged (1.5% p.a.) since 2015, we use the SHIBOR as the risk-free rate. We also run the robustness test by using the policy rate as the risk-free in our calculation.6 6We re-run our test with policy rate as the risk free rate, the result are consistent with the main result calculation with SHIBOR rate. The interest rate could be stochastic in the option pricing model, Chapter 2. The Chinese Equity Index Option Market 24

(2) Underlying asset’s volatility (σ) The historical volatility (HV) of each options are computed as the standard deviation of log return of the underlying asset in a moving window (depending on the option’s maturity). s Pi−τ 2 (ri − r¯) √ σ = i × 242, (2.2) i n − 1 where the τ is the option’s time to maturity in days, ri is the log return of the underlying asset at day i, r¯ is the mean return, n is the number of observations in the window. For an option with a maturity of 20 days, we use the underlying asset volatility based on its previous 20 days and we annualized the volatility by 242 √ trading days in a year with 242.7 For the short term options with initial time to maturity in less than 14 days, the estimation would be unreliable, they are ﬁltered out (296 observations) in our BS model price calculation. (3) Underlying asset’s daily spot price (S) We use the underlying asset: SSE 50 ETF’s daily closing price as the spot price in our calculation. The Shanghai stock and option exchange close at 15:00 Beijing time in each trading day. (4): Underlying asset’s dividend yield (δ) We use the maturity time window of each options to match their corresponding dividend yields. For those options with maturity time window that span any dividend payment date in the underlying asset, we use the underlying asset’s actual annualized dividend yield as options’ dividend yield. For the other options, we use 0% as their dividend yield. (5) Time to maturity (τ) for example, Bakshi, Cao, and Chen(1997) and etc, however found by Bakshi et al.(1997), the stochastic interest rates are not that important in pricing or hedging with options. For brevity, we did not report the results in our paper. 7There are about 242.4 trading days per year in China, depending on the Chinese public holidays (from 2010 to 2015) Chapter 2. The Chinese Equity Index Option Market 25

Time to maturity is an option’s remaining days to its maturity as percentage of a year (remaining days to maturity/365)%. (6) Option’s strike price (K) The strike is the price that the call (put) option holders could buy (sell) the underlying asset.

2.3.3.1 The BS mdoel and market price of the SSE 50 option

In this section, we report the detailed results of the options’ BS model price and compare them with their market price (settlement price). We analyse the differences in the following aspects: (1) The options’ daily BS model price and their market closing price. (2) The pricing errors (ξ) of options, which is one of the measures in the derivatives’ “bubble size”, we deﬁne the pricing errors by following Tang and Wang(2013), the ξ of an option as the percentage of absolute difference between the market price and its BS model price:

Ot − BSt ξt = , (2.3) Ot where the Ot and BSt are the market and BS model option price, respectively. (3) The implied volatility (IV) is calculated from the BS model and the estimated historical volatility (HV) is calculated from the underlying asset. In Table 2.3.5, we report all options’ average market and BS model price across the full sample and sub-groups. An option’s BS model price is calculated from the six variables in Section 3.3, then we calculate the average difference between the market price and the BS model price. We ﬁnd that in the Chinese options market, the call options’ average market price is signiﬁcantly lower than their average BS model price in full sample and subgroups. The put options’ average market price is higher than their average BS model price in the full sample, DITM, ITM, Short and Medium Chapter 2. The Chinese Equity Index Option Market 26

term options. The average pricing error in call (put) is 10.05% (2.25%), however, the “bubbles” in the index put option is much smaller than the put warrants in terms of average pricing error.8 The short-selling and margin trading in stocks are banned in China until 31 March 2010, however, after short-selling was permitted in China investors find that it is difficult to borrow stocks for short-selling. Chang, Luo, and Ren(2014) find that short-selling has improved the price efficiency and reduced stock return volatility in China. Derivatives like the put warrants provide an alternative short-selling tool for investors. Investors have long position in either put warrants or put options as the hedging position to downside risk. Similar to the put warrants bubbles, which documents by Xiong and Yu(2011) and Tang and Wang(2013) in the Chinese warrant market. They find some put option investors trading put options at a high premium above their intrinsic values when compared to call options.9

2.3.3.2 The implied volatility and market volatility in the Chinese options market

An option’s IV is the implied BS model volatility calculated from the market price of the option’s market price. The IV is considered as the market’s expectation of the volatility of the underlying asset and it could be used to quote an option instead of its premium.

8Powers and Xiao(2014) report the average BS model pricing errors in their paper (Table 4), they ﬁnd that the average BS model pricing error in Chinese call (put) warrants is 26.8% (99.40%). 9The last equity warrant (Changhong CWB1 warrant) in China had expired on 18 August 2011, since then warrants are not available for investors in China. Chapter 2. The Chinese Equity Index Option Market 27

Table 2.3.5: Comparison of options’ market price and Black- Scholes (BS) model price

Table 2.3.5 reports the options’ average market price, average BS model price and their pricing errors (ξ) across full sample and moneyness subgroups: at-the- money (ATM), in-the-money (ITM), deep-in-the-money (DITM), out-of-the-money (OTM), deep-out-of-the-money (DOTM) and options’ maturity term groups in Short/Medium/Long.

Average mar- Average BS Average pric- Average t-statistic Obs ket price model price ing error difference

Panel A: Call option

Full Sample 0.1695 0.1866 10.05% -0.0170 -55.12 29,349

DITM 0.4663 0.4842 3.83% -0.0178 -23.70 5,237

ITM 0.2010 0.2293 14.10% -0.0283 -42.47 6,496

ATM 0.1023 0.1251 22.24% -0.0228 -34.26 6,466

OTM 0.0655 0.0752 14.87% -0.0097 -13.73 5,469

DOTM 0.0366 0.0404 10.55% -0.0039 -6.31 5,681

Short 0.1341 0.1401 4.52% -0.0061 -24.66 13,781

Medium 0.1806 0.2005 11.00% -0.0199 -29.52 7,122

Long 0.2180 0.2505 14.93% -0.0325 -41.41 8,446

Panel B: Put option

Full Sample 0.2021 0.1975 2.25% 0.0046 18.83 29,349

DITM 0.6124 0.5764 5.88% 0.0360 56.16 5,681

ITM 0.2159 0.2063 4.49% 0.0097 19.09 5,469

ATM 0.1115 0.1160 4.05% -0.0045 -9.45 6,466

OTM 0.0610 0.0714 16.99% -0.0104 -23.90 6,496

DOTM 0.0293 0.0345 17.84% -0.0052 -14.22 5,237

Short 0.1681 0.1614 3.99% 0.0067 26.81 13,781

Medium 0.2154 0.2094 2.76% 0.0059 10.76 7,122

Long 0.2463 0.2464 0.05% -0.0001 -0.23 8,446 Chapter 2. The Chinese Equity Index Option Market 28

We calculate all options’ IV with their daily closing price and Table 2.3.6 reports the average IV for our full sample, maturity and moneyness sub-groups. On the one hand, we find that the average IV of call options is significantly lower than its average annualized historical volatility. On the other hand, the average implied volatility in put options is significantly higher than its average historical volatility, which are all consistent with the findings reported in Table 2.3.5. The IV could be used to evaluate the price level of options across moneyness/maturity groups in different markets. The CBOE VIX (Volatility Index) which represents the market’s expectation of the 30-day forward-looking volatility is about 18% from 2010 to 2015. The average IV in the call (put) options is 30.94% (38.18%) which is lower than the average IV in the equity warrants market (142.36%). 10 From Figure 2.3.1 and Figure 2.3.2, we observe that the randomly selected options’ IV levels (right axis) are much lower than the IV levels of the equity warrants and especially in those OTM put warrants (Figure 2.3.2.c and d) with huge bubbles. In Figure 2.3.1, we plot the options’ daily market price, BS model price, intrinsic price and IV of the ATM, ITM and OTM at maturity day. We can see that the two figures’ results are consistent with the full sample test in Table 2.3.5, with the market prices of put (call) being higher (lower) than their BS model price. The put options’ IV levels are higher than call options (except the ITM group), which indicate that investors willing to pay higher price on put options. We also plot four expired Chinese warrants in Figure 2.3.2 for comparison.11 The underlying assets in two derivatives are different, however, the highest IV level in OTM warrant is over 1,300% (Figure 2.3.2.b) which is almost 15 times of the highest IV level in OTM put option (Figure 2.3.1.d).

10Tang and Wang(2013) report the average IV of Chinese warrants is 142.36% (Table 1 of their paper). 11The 2006 - 2007 Bullish market in China caused most of the put warrants to be so deep out-of-the- money that they were almost certain to expire worthless. We report only ITM and OTM call warrants and OTM put warrants in the Figure 2.3.2. Chapter 2. The Chinese Equity Index Option Market 29

Table 2.3.6: Comparison of options’ Black-Scholes (BS) model implied volatility (IV) and historical volatility (HV)

Table 2.3.6 reports the results of the volatility comparison between options’ HV and BS model IV and normalized volatility error in absolute difference over IV. The results are reported by full sample, moneyness subgroups: at-the-money (ATM), in- the-money (ITM), deep-in-the-money (DITM), out-of-the-money (OTM), deep-out- of-the-money (DOTM) and maturity-term groups in Short, Medium and Long.

Volatility Average Average IV Average HV t-statistic Obs error (IV-HV) Panel A: Call option Full Sample 0.3094 0.3203 3.51% -0.0109 -8.81 24,503 DITM 0.4495 0.3638 19.06% 0.0857 16.25 2,590 ITM 0.2539 0.3364 32.51% -0.0825 -37.29 4,636 ATM 0.2204 0.2801 27.12% -0.0598 -39.26 6,283 OTM 0.2588 0.2729 5.41% -0.0140 -8.22 5,407 DOTM 0.4395 0.3777 14.07% 0.0619 18.37 5,587 Short 0.3463 0.3051 11.90% 0.0412 17.59 10,456 Medium 0.2899 0.3300 13.85% -0.0401 -20.44 6,061 Long 0.2759 0.3327 20.61% -0.0568 -41.07 7,986 Panel B: Put option Full Sample 0.3818 0.3232 15.35% 0.0586 38.83 28,625 DITM 0.6718 0.3785 43.65% 0.2932 48.14 5,378 ITM 0.3231 0.2743 15.09% 0.0487 28.98 5,316 ATM 0.2681 0.2788 3.98% -0.0107 -8.27 6,373 OTM 0.2858 0.3125 9.32% -0.0266 -18.54 6,410 DOTM 0.3997 0.3842 3.88% 0.0155 5.99 5,148 Short 0.4249 0.3104 26.96% 0.1146 38.17 13,070 Medium 0.3560 0.3322 6.69% 0.0238 13.58 7,110 Long 0.3367 0.3355 0.37% 0.0012 1.07 8,445 Chapter 2. The Chinese Equity Index Option Market 30

Figure 2.3.1: Call and put options’ market and BS price comparison by moneyness group

Figure 2.3.1 reports call and put options’ price comparison between the market and their calculated Black-Scholes price, their implied volatility in six options for at-the- money (ATM), in-the-money (ITM) and out-of-the-money (OTM) groups at their maturity day. The option’s name include its strike price and maturity date, for example, Call2400 Oct 2015 indicates a call option with strike 2400 in SSE 50 index and maturity in October 2015.

ATM Call2250 Oct2015 ATM Put2250 Oct2015 0.40 60 0.60 110 Market price (L) Market price (L) 100 0.35 BS price (L) BS price (L) Intrinsic value (L) 50 0.50 Intrinsic value (L) 90 Implied volatility (R) Implied volatility (R) 0.30 80 40 0.40 0.25 70

60 0.20 30 0.30 50 Price (CNY) Price (CNY) 0.15 40 20 0.20 Implied volatility (%) Implied volatility (%) 30 0.10

10 0.10 20 0.05 10

0.00 0 0.00 0 Aug 31 Sep 14 Sep 28 Oct 12 Oct 26 Nov 09 Sep 01 Sep 15 Sep 29 Oct 13 Oct 27 Nov 10 Date 2015 Date 2015 (a) ATM call option (b) ATM put option

OTM Call2400 Oct2015 OTM Put2200 Oct2015 0.10 60 0.60 120 Market price (L) Market price (L) 0.09 BS price (L) BS price (L) Intrinsic value (L) 50 0.50 Intrinsic value (L) 100 0.08 Implied volatility (R) Implied volatility (R)

0.07 40 0.40 80 0.06

0.05 30 0.30 60

Price (CNY) 0.04 Price (CNY) 20 0.20 40 0.03 Implied volatility (%) Implied volatility (%)

0.02 10 0.10 20 0.01

0.00 0 0.00 0 Oct 10 Oct 13 Oct 16 Oct 19 Oct 22 Oct 25 Oct 28 Oct 31 Aug 31 Sep 14 Sep 28 Oct 12 Oct 26 Nov 09 Date 2015 Date 2015 (c) OTM call option (d) OTM put option

ITM Call2100 Oct2015 ITM Put2400 Oct2015 0.60 0.40 60 120 Market price (L) Market price (L) BS price (L) 0.35 BS price (L) 0.50 Intrinsic value (L) Intrinsic value (L) 50 Implied volatility (R) 100 Implied volatility (R) 0.30

0.40 40 80 0.25

0.30 0.20 30 60

Price (CNY) Price (CNY) 0.15 0.20 20 40 Implied volatility (%) Implied volatility (%) 0.10

0.10 20 10 0.05

0.00 0 0.00 0 Aug 28 Sep 11 Sep 25 Oct 09 Oct 23 Nov 06 Oct 13 Oct 16 Oct 19 Oct 22 Oct 25 Oct 28 Oct 31 Date 2015 Date 2015 (e) ITM call option (f) ITM put option Chapter 2. The Chinese Equity Index Option Market 31

Figure 2.3.2: Chinese put and call warrants’ market and Black- Scholes (BS) price

This ﬁgure reports price and implied volatility comparison of four Chinese warrants in in-the-money (ITM) and out-of-the-money (OTM) groups. ITM Call warrant ANSC (An Shan Steel Company) JEC1, OTM Call warrant COSCO (China Ocean Shipping Company) CWB1, OTM Put warrant PSV (Pangang Group Vanadium Ti- tanium & Resources Co., Ltd) PGP1 and WULIANGYE (Wuliangye Yibin Company limited) YGP1. ITM Call Warrant ANSC JEC1 OTM Call Warrant COSCO CWB1 10.00 350 20.00 1400 Market price (L) Market price (L) 9.00 BS price (L) 18.00 BS price (L) 300 1200 Intrinsic value (L) Intrinsic value (L) 8.00 Implied volatility (R) 16.00 Implied volatility (R)

7.00 250 14.00 1000

6.00 12.00 200 800 5.00 10.00 150 600

Price (CNY) 4.00 Price (CNY) 8.00 Implied volatility (%) Implied volatility (%) 3.00 100 6.00 400

2.00 4.00 50 200 1.00 2.00

0.00 0 0.00 0 Jan 2006 Apr 2006 Jul 2006 Oct 2006 Jan 2007 Apr 2008 Jul 2008 Oct 2008 Jan 2009 Apr 2009 Jul 2009 Date Date (a) ITM call warrant (b) OTM call warrant

OTM Put Warrant PSV PGP1 OTM Put Warrant WULIANGYE YGP1 4.00 550 10.00 1000 Market price (L) Market price (L) 500 9.00 900 3.50 BS price (L) BS price (L) Intrinsic value (L) Intrinsic value (L) Implied volatility (R) 450 8.00 Implied volatility (R) 800 3.00 400 7.00 700 2.50 350 6.00 600

2.00 300 5.00 500

Price (CNY) 250 Price (CNY) 4.00 400 1.50

200 Implied volatility (%) 3.00 300 Implied volatility (%) 1.00 150 2.00 200 0.50 100 1.00 100

0.00 50 0.00 0 Jan 2006 Apr 2006 Jul 2006 Oct 2006 Jan 2007 Apr 2007 Jul 2006 Jan 2007 Jul 2007 Jan 2008 Date Date (c) OTM put warrant (d) OTM put warrant Chapter 2. The Chinese Equity Index Option Market 32

2.4 Empirical results

2.4.1 The perfect correlation in the Chinese option market

We construct the correlation table among the four asset prices: the daily change of market option price, the daily change of BS model option price, the daily change of the underlying asset price and the daily change of Shanghai Composite Index (SHCI) level. We compute the correlation coefficients between each other from the times series of daily price changes. Table 2.4.1 reports the correlation table of call and put options. The prefect correlation assumption in the one-dimensional diffusion models indicate the correlation coefficient of price changes between underlying asset and call (put) options to be 1 (-1). The over all correlation between the price change of call (put) and the price change of the underlying asset is 0.6882 (-0.7657), which indicates the imperfect correlation between each other. Especially, the correlation in the DOTM call (put) options group is only 0.5855 (-0.5392). Our results on the correlation between price changes in the BS model and the underlying asset are also imperfect and only the DITM options have correlation coefficients close to perfect correlation. The imperfect correlation between the price change of option and underlying asset indicates that investors are trading some other risk factors in options rather than the underlying risk only. The previous put warrants’ average correlation with underlying asset is -0.081 only (Xiong and Yu, 2011, Table 2 and Chang et al., 2013b Table 4). The low correlation between the warrants and underlying price changes making it is very difficult to argue whether the warrants investors are hedging the risks in the underlying stocks with the put warrants. The relative larger correlation coefficient in the Chi- nese equity option markets suggests the options could be used to hedge the risk Chapter 2. The Chinese Equity Index Option Market 33

in the underlying asset. Especially, those DITM call (put) options with correlation coefﬁcients close to 1 (-1).

2.4.2 Monotonicity in the Chinese option market

Monotonicity is one of the fundamental model properties of the one-dimensional diffusion model. It describes the price movement behaviour between options and its underlying asset. The call and put option price should move together with/opposite to the movement of their underlying asset. The test for monotonicity in the Chinese option market is grouped into eight subcategories by following the methodology of (Bakshi et al., 2000). We have the following variables in our violation ratio calculation: C is a call option’s market price, we use the daily closing price as a call option’s market price. ∆C is the difference of a call option’s closing price to the previous day. P is a put option’s market price, we use the daily closing price as put options’ market price. ∆P is the difference of a put option’s closing price to the previous day. S is the underlying asset price, which is the SSE 50 ETF’s daily closing price. ∆S is the difference of the underlying asset’s closing price to the previous day.

2.4.2.1 Monotonicity type I violation

Type I violation: Call and put price moves in the opposite and same direction with the underlying asset. For call options: ∆S ∗ ∆C < 0 (Sub Type A: ∆S > 0, ∆C > 0 or Sub Type B: ∆S < 0, ∆C > 0). For put options: ∆S ∗ ∆P > 0 (Sub Type A: ∆S > 0, ∆P > 0 or Sub Type B: ∆S < 0, ∆P < 0). A violation data matrix is formed with two new variables to inspect the type I violation: A1 = ∆S ∗ ∆C and A2 = ∆S ∗ ∆P . Chapter 2. The Chinese Equity Index Option Market 34

Table 2.4.1: The correlation table of SSE 50 options’ price, Black- Scholes (BS) model price and underlying asset price change

Table 2.4.1 reports the correlation table of the time series of daily price changes of the options’ closing price (∆O), the underlying asset’s closing price (∆S), the daily BS model price change (∆BS) and the daily Shanghai Composite Index level change (∆SHCI). The results are grouped by full sample, moneyness subgroups: at-the-money (ATM), in-the-money (ITM), deep-in-the-money (DITM), out-of-the- money (OTM), deep-out-of-the-money (DOTM) and options’ maturity groups in Long, Medium and Short.

∆O ∆BS ∆O ∆O Obs ∆S ∆S ∆SHCI ∆BS Panel A: Call option Full Sample 0.6882 0.8370 0.6056 0.8299 28,944 DITM 0.8081 0.9933 0.7390 0.8127 5,233 ITM 0.8603 0.9841 0.7700 0.8770 6,383 ATM 0.8701 0.9740 0.7839 0.8932 6,330 OTM 0.8391 0.9376 0.7470 0.8612 5,323 DOTM 0.5855 0.7252 0.5198 0.7423 5,675 Short 0.6854 0.7632 0.5974 0.8989 13,663 Medium 0.7080 0.8884 0.6236 0.7978 6,985 Long 0.6861 0.9013 0.6122 0.7562 8,296 Panel B: Put option Full Sample -0.7657 -0.8539 -0.7371 0.8893 28,944 DITM -0.8834 -0.9843 -0.8460 0.8952 5,675 ITM -0.8770 -0.9734 -0.8223 0.8929 5,323 ATM -0.8620 -0.9658 -0.7931 0.8815 6,330 OTM -0.7780 -0.9251 -0.7216 0.8275 6,383 DOTM -0.5392 -0.7239 -0.5186 0.7052 5,233 Short -0.7827 -0.8309 -0.7414 0.9311 13,663 Medium -0.7316 -0.8433 -0.7327 0.8208 6,985 Long -0.7828 -0.9131 -0.7536 0.8552 8,296 Chapter 2. The Chinese Equity Index Option Market 35

The violation ratio of type I in the call option:

Days with negative A1 V iolation ratio = T otal trading days

The violation ration of type I in the put option:

Days with positive A2 V iolation ratio = T otal trading days

Table 2.4.2 reports the average type I violation ratio in full sample, options’ moneyness and maturity sub groups. When the underlying price increased (decreased), the call (put) price decreased quite offen and the full call (put) options’ type I violation ratio is about 14.84% (17.58%). This violation is persistent across options’ moneyness and maturity groups, for example, in call options (Panel A), the violation ratio increases from 5.52% (DITM) to 26.64% (DOTM) and increases from 14.42% (Short) to 15.83% (Long). The observed type I violation ratio indicates that investors are trading some other risk factors to the underlying asset risk only. The overall type I violation ratio in call (put) options market is much lower than the closed warrants market (18.94% and 44.86%, in Table 3, Chang et al., 2013b). Chapter 2. The Chinese Equity Index Option Market 36

Table 2.4.2: The SSE 50 ETF options’ type I violation ratio

Table 2.4.2 reports the type I violation ratio (VR) of call and put options in each sample groups. The results are grouped by full sample, moneyness subgroups: at-the- money (ATM), in-the-money (ITM), deep-in-the-money (DITM), out-of-the-money (OTM), deep-out-of-the-money (DOTM) and options’ maturity groups in Long (L), Medium (M) and Short (S). The type I violation in call option:

∆S × ∆C < 0 (∆S > 0, ∆C < 0 or ∆S < 0, ∆C > 0).

The type I violation in put option:

∆S × ∆P > 0 (∆S > 0, ∆P > 0 or ∆S < 0, ∆P < 0).

Violation type I Obs VR Obs (S) VR Obs (M) VR Obs (L) VR

Panel A: Call option

Full Sample 29,497 14.84% 13,929 14.42% 7,122 14.50% 8,446 15.83%

DITM 5,237 5.52% 2,441 2.50% 1,489 6.92% 1,307 9.56%

ITM 6,533 9.57% 3,097 5.62% 1,462 10.60% 1,974 14.99%

ATM 6,509 13.81% 3,118 12.60% 1,430 13.57% 1,961 15.91%

OTM 5,527 18.98% 2,542 21.36% 1,272 16.12% 1,713 17.57%

DOTM 5,691 26.64% 2,731 30.65% 1,469 25.60% 1,491 20.32%

Panel B: Put option

Full Sample 29,497 17.58% 13,929 18.08% 7,122 15.47% 8,446 18.52%

DITM 5,691 13.48% 2,731 12.60% 1,469 14.70% 1,491 13.88%

ITM 5,527 11.20% 2,542 8.10% 1,272 10.30% 1,713 16.46%

ATM 6,509 15.58% 3,118 15.07% 1,430 13.08% 1,961 18.20%

OTM 6,533 21.55% 3,097 24.12% 1,462 17.17% 1,974 20.77%

DOTM 5,237 26.29% 2,441 30.81% 1,489 21.29% 1,307 23.57% Chapter 2. The Chinese Equity Index Option Market 37

2.4.2.2 Monotonicity type II violation

Type II violation: Call (put) option prices do not change given that the underlying asset’s price changed. When ∆S 6= 0, but ∆C = 0 or ∆P = 0. A violation data matrix is formed with two new variables to inspect the type II violation: A1 = ∆S ∗ ∆C and A2 = ∆S ∗ ∆P . We select the option data by excluding the ﬁve trading days with zero change in the underlying asset’s price in our sample. The type II violation in call option:

Number of trading days with zero A1 V iolation ratio = T otal call option trading days

The type II violation in put option:

Number of trading days with zero A2 V iolation ratio = T otal put option trading days

Table 2.4.3 reports the average type II violation ratio in full sample, options’ moneyness and maturity sub groups. The type II violations are rare, we ﬁnd that when the underlying price changed, the call (put) options’ price stay the same and the overall type II violation ratio is only about 2.22% (2.69%). The type II violation is also rare in the equity warrants market Chang et al.(2013b) ﬁnd the average type II violation ratio is 0.69% in call warrants and 1.31% in put warrants in Table 3 of their paper. The violation rate is the lowest in the DITM moneyenss group and the highest in the DOTM group. The violation rate in both call and put options decreases monotonically with the increases of options’ time to maturity from short to long. The type II violation indicates that the market changes regularly, however, the options’ price not sensitive to the changes in the underlying, especially those DOTM options’ with small deltas and option premium. The type II violation is related to bid-ask bounce, Chapter 2. The Chinese Equity Index Option Market 38

however, the WIND database does not have option’s bid and ask price, so we leave this for future research.

2.4.2.3 Monotonicity type III violation

Type III violation: Given that call or put option prices changed, however, the underlying asset’s price stayed unchanged: ∆S = 0, but ∆C or ∆P 6= 0. In our dataset, there are only five trading days with 0% change in the underlying asset. We take these five trading days with 0% change into our sub-sample for type III violation analysis. The type III violation ratio in call (put) option are very high, which are all near 100%. Since the observations in type III sample is only about 2% of the full sample and the tick size in the option market (CNY 0.0001) which is smaller than the tick size in the underlying asset (CNY 0.001). Sometimes, the option price could change with the option’s tick size though the underlying asset price unchanged. The equity warrants and their underlying assets have same tick size and Chang et al.(2013b) find that the type III violation is only 3.58% (3.27%) in call (put) warrants in Table 3 of their paper.

2.4.2.4 Monotonicity type IV violation

Type IV violation: The over-adjusts in option price compared with the changes in the underlying asset price.

∆C For call option: ∆S > 1 while ∆S 6= 0. ∆P For put option: ∆S < −1, while ∆S 6= 0. In our sample, there are only ﬁve trading days with 0% change in the underlying asset. We exclude these ﬁve trading days’ data from our whole sample for the type IV violation analysis. Table 2.4.5 reports the type IV violation of SSE 50 options. The type IV violations occurs in 16.07% and 14.61% in call and put options of the full sample. The type IV violation ratio increase monotonically along with the increase Chapter 2. The Chinese Equity Index Option Market 39

Table 2.4.3: The SSE 50 ETF options’ type II violation ratio

Table 2.4.3 reports the type II violation ratio (VR) of call and put options in each sample groups. The results are grouped by full sample, moneyness subgroups: at-the- money (ATM), in-the-money (ITM), deep-in-the-money (DITM), out-of-the-money (OTM), deep-out-of-the-money (DOTM) and options’ maturity groups in Long (L), Medium (M) and Short (S).

The type II violation in call/put option:

∆S 6= 0, but ∆C = 0, or ∆P = 0.

Violation type II Obs VR Obs (S) VR Obs (M) VR Obs (L) VR

Panel A: Call option Full Sample 28,923 2.22% 13,691 3.93% 6,958 1.02% 8,274 0.39% DITM 5,146 0.08% 2,398 0.13% 1,467 0.00% 1,281 0.08% ITM 6,413 0.22% 3,047 0.26% 1,429 0.14% 1,937 0.21% ATM 6,384 0.55% 3,063 0.52% 1,398 0.79% 1,923 0.42% OTM 5,412 2.81% 2,497 4.89% 1,242 1.45% 1,673 0.72% DOTM 5,568 7.83% 2,686 14.48% 1,422 2.81% 1,460 0.48%

Panel B: Put option Full Sample 28,923 2.69% 13,691 4.72% 6,958 1.16% 8,274 0.62% DITM 5,568 0.18% 2,686 0.22% 1,422 0.14% 1,460 0.14% ITM 5,412 0.33% 2,497 0.44% 1,242 0.16% 1,673 0.30% ATM 6,384 0.69% 3,063 0.78% 1,398 0.57% 1,923 0.62% OTM 6,413 3.62% 3,047 6.20% 1,429 1.89% 1,937 0.83% DOTM 5,146 9.21% 2,398 17.35% 1,467 2.86% 1,281 1.25% Chapter 2. The Chinese Equity Index Option Market 40

Table 2.4.4: The SSE 50 ETF options’ type III violation ratio

Table 2.4.4 reports the type III violation ratio (VR) of call and put options in each sample groups. The results are grouped by full sample, moneyness subgroups: at-the-money (ATM), in-the-money (ITM), deep-in-the-money (DITM), out-of-the- money (OTM), deep-out-of-the-money (DOTM) and options’ maturity groups in Long (L), Medium (M) and Short (S).

The type III violation in call and put option:

∆S = 0, ∆C 6= 0 or ∆P 6= 0.

Violation type III Obs VR Obs (S) VR Obs (M) VR Obs (L) VR

Panel A: Call option Full Sample 574 92.33% 238 86.97% 164 93.90% 172 98.26% DITM 91 86.81% 43 88.37% 22 77.27% 26 92.31% ITM 120 90.83% 50 80.00% 33 100.00% 37 97.30% ATM 125 99.20% 55 98.18% 32 100.00% 38 100.00% OTM 115 96.52% 45 95.56% 30 93.33% 40 100.00% DOTM 123 86.99% 45 71.11% 47 93.62% 31 100.00%

Panel B: Put option Full Sample 574 96.17% 238 91.60% 164 98.78% 172 100.00% DITM 123 95.12% 45 86.67% 47 100.00% 31 100.00% ITM 115 100.00% 45 100.00% 30 100.00% 40 100.00% ATM 125 97.60% 55 96.36% 32 96.88% 38 100.00% OTM 120 95.00% 50 88.00% 33 100.00% 37 100.00% DOTM 91 92.31% 43 86.05% 22 95.45% 26 100.00% Chapter 2. The Chinese Equity Index Option Market 41

value in the money, type IV violation is the highest in the DITM groups and the lowest in the DOTM groups. From four type of violations tests in the SSE 50 options reported above, we find that the options market in China do not support the monotonicity property of the one-dimensional diffusion models. The results are consistent with the Hypothesis 1, the new options market share some properties in the closed equity warrants market in China. However, the ”bubbles” in the equity warrants market does not exist in the equity options market. For example, the average correlation coefficient between the price change of equity warrant puts and underlying is almost to zero, while the average coefficient between the price of change of equity put and underlying asset is -0.76. The average IV in call (put) options is about 31% (38%) and 142% in equity warrants.

2.4.3 The delta-hedged gain analysis

In section 3, we ﬁnd that the one-dimensional diffusional model monotonicity assumption does not hold in the newly established equity options market in China. In this section, we implement deeper analysis on the redundancy property of the one- dimensional diffusion model and explore whether there are any other risk factors in addition to the underlying asset risk only. If the one-dimensional diffusional model holds in the options market, the sole uncertainty source hedged portfolio should earn a risk-free return. The portfolio with the delta-hedged position and its risk-free cost should have zero delta-hedged gain.

2.4.3.1 The delta-hedged portfolio

The delta-hedged portfolio can be constructed with call (put) options. (1) The portfolio with a long position in call option, hedged by corresponding short position (call option’s delta number of shares) in the underlying asset. (2) The portfolio with a Chapter 2. The Chinese Equity Index Option Market 42

Table 2.4.5: The SSE 50 ETF options’ type IV violation ratio

Table 2.4.5 reports the type IV violation violation ratio (VR) of call and put options in each sample groups. The results are grouped by full sample, moneyness subgroups: at-the-money (ATM), in-the-money (ITM), deep-in-the-money (DITM), out-of-the- money (OTM), deep-out-of-the-money (DOTM) and options’ maturity groups in Long (L), Medium (M) and Short (S).

The type IV violation in call and put option:

∆C ∆P ∆S 6= 0, ∆S > 1 or ∆S < −1.

Violation type IV Obs VR Obs (S) VR Obs (M) VR Obs (L) VR

Panel A: Call option Full Sample 28,923 16.07% 13,691 16.82% 6,958 15.09% 8,274 15.66% DITM 5,146 36.59% 2,398 39.53% 1,467 35.11% 1,281 32.79% ITM 6,413 23.95% 3,047 29.67% 1,429 20.29% 1,937 17.66% ATM 6,384 11.64% 3,063 11.43% 1,398 9.94% 1,923 13.21% OTM 5,412 5.91% 2,497 3.16% 1,242 5.31% 1,673 10.46% DOTM 5,568 3.00% 2,686 0.82% 1,422 2.81% 1,460 7.19%

Panel B: Put option Full Sample 28,923 14.61% 13,691 15.24% 6,958 15.54% 8,274 12.81% DITM 5,568 32.81% 2,686 34.66% 1,422 37.90% 1,460 24.45% ITM 5,412 23.10% 2,497 27.83% 1,242 22.14% 1,673 16.74% ATM 6,384 12.05% 3,063 11.98% 1,398 11.87% 1,923 12.27% OTM 6,413 4.52% 3,047 2.53% 1,429 5.25% 1,937 7.12% DOTM 5,146 1.77% 2,398 0.67% 1,467 1.77% 1,281 3.83% Chapter 2. The Chinese Equity Index Option Market 43

long position in put option, hedged with corresponding long position (put option’s absolute delta number of shares) in the underlying asset. After the position added its risk-free cost, the portfolio should earn zero delta-hedged gain. In order to analyze the excessive gain in the delta-hedged portfolio, we assume that there are no additional risk factors other than the underlying asset that are being traded in the option market, under the assumption that the underlying asset follows the diffusion in Eq (2.1), which is one-dimensional. If there are some other risk factors traded in option market, we denote them as X(t). An additional dimensional model such as Heston (1993)’s square root stochastic volatility model, the additional risk is the stochastic volatility factor. The two-dimensional diffusion model’s additional risk factor can be described as follows:

X dXt = θ(X, t)dt + η(X, t)dBt , (2.4)

X where Bt is a standard Brownian motion which is correlated to the underlying asset in correlation ρ. The call option in a two-dimensional diffusion model is C(St,Xt, t) a function of underlying asset, stochastic volatility and time. By Ito’sˆ lemma, we have the process for call options in a two-dimensional diffusion model:

∂C ∂C ∂C 1 ∂2C 1 ∂2C 1 ∂2C dC = + µS + θ + σ2S2 + η2 + σSηρ dt ∂t ∂S ∂X 2 ∂S2 2 ∂X2 2 ∂S∂X (2.5) ∂C ∂C + σS dB + η dBX , ∂S t ∂X t

The delta-hedged portfolio with call options follows a two-dimensional diffusion

∂C model which is deﬁned as Π = C − ∆S, where ∆ = ∂S . The delta-hedged portfolio Chapter 2. The Chinese Equity Index Option Market 44

is:

∂C ∂C ∂C 1 ∂2C 1 ∂2C ∂2C dΠ − rΠdt = + rS + θ + η2 + σ2S2 + σSηρ − rC dt ∂t ∂S ∂X 2 ∂X2 2 ∂S2 ∂S∂X ∂C + η dBX . ∂X t (2.6)

If there is only one risk factor in the options market, the hedged portfolio should have zero excess gain. On the other hand, if there are some other risk factors in the options, the underlying asset-hedged portfolio should have non-zero gains. We may come to the conclusion that other risk factors like Xt exist in the Chinese option market, as we illustrated above. We provide the derivation of two-dimensional diffusion delta-hedged gain portfolio in Appendix.

2.4.3.2 The delta-hedged portfolio

Following the methodology Bakshi and Kapadia(2003a) used in delta-hedged gain in the U.S. options market, we form SSE 50 option delta-hedged portfolio as follow:

(1) At initial time t, a call option is purchased at its closing price Ct. A put option is purchased at its closing price Pt. (2) The call (put) options are hedged with the underlying asset discretely until their expiration date: t + τ, where the τ is an option’s time to maturity. (3) The hedging ratio ∆ is calculated at the options’ closing price. It is ideal to hedge with continuously rebalancing:

Z t+τ Z t+τ Πt,t+τ = Ct+τ − Ct − ∆dS − r(C − ∆S)du, (2.7) t t where t is the current time and τ is the time to maturity of an option, Ct+τ is the option’s premium at maturity, Ct is the option’s premium at current time t, ∆ is Chapter 2. The Chinese Equity Index Option Market 45

the option’s Black-Scholes delta, S is the underlying asset’s price and r is the risk- free rate. It is unrealistic to simply ignore the large transaction cost in continuously trading, we implement the delta-hedged portfolio on daily basis. The daily rebalanced delta-hedged portfolio with call option: A long call option hedged by a short position (∆call number of shares) of its underlying asset.

N−1 N−1 X X τ Π = Max(S −K, 0)−C − ∆ (S −S )− r(C −∆S ) , (2.8) t,t+τ t+τ t tn tn+1 tn tn tn N n=0 n=0 where the St+τ is the underlying asset’s price at maturity, K is the strike price of an option, Ct is the call option’s price at time t, ∆tn is the option’s Black-Scholes delta at time tn. Stn is the underlying asset’s price at time tn, Ctn is the call option’s price at time tn and r is the risk-free rate. The daily rebalanced delta-hedged portfolio with put option: A long put option hedged by a long position (absolute ∆put number of shares) of its underlying asset.

N−1 N−1 X X τ Π = Max(K −S , 0)−P − ∆ (S −S )− r (P −∆S ) , (2.9) t,t+τ t+τ t tn tn+1 tn tn tn tn N n=0 n=0 where the St+τ is the underlying asset’s price at maturity, K is the strike price of an option, Pt is the put option’s price at time t, ∆tn is the option’s Black-Scholes delta at time tn. Stn is the underlying asset’s price at time tn, Ptn is the put option’s price at time tn and r is the risk-free rate. The delta-hedged portfolio is rebalanced on daily basis and we assume there are 242 trading days per year in the Chinese stock market. An option’s ∆ is calculated on

∂Ctn a daily basis with the Black-Scholes formula, ∆tn = = N(d1(Stn , tn)) ,where the ∂Stn S 1 √ tn N(·) is the cumulative normal distribution function. d1(Stn , tn) = [ln( )+ σtn,tn+τ τ K 1 2 (r + 2 σtn,tn+τ )τ]. Since at each options’ maturity date their time to maturity will be “ 0 ”, so the last trading day for the delta-hedge will be one day before each options’ Chapter 2. The Chinese Equity Index Option Market 46

maturity date.

2.4.3.3 The delta-hedged gain with the SSE 50 options

Table 2.4.6 reports the delta-hedged portfolios with call options. The pooled average delta-hedged gain in the full sample is not zero and 61.79% of them are negative. The delta-hedged gain is negative in all the moneyness and maturity groups. The average loss in each delta-hedged portfolios as measured in the initial option premium and underlying asset price is -0.07% or -1.42%. Table 2.4.7 reports delta-hedged portfolios with put options. The pooled average delta-hedged gain in the full sample is not zero and 63.25% of them are negative. The delta-hedged gain The average loss in each delta-hedged portfolios as measured in initial option premium and underlying asset price is -1.37% or -0.09% as measured in initial option premium or underlying asset price. The non-zero delta-hedged gain in both call and put options suggests that there are some additional risk factors to the underlying asset are unhedged. The average accumulated delta-hedged gain to the last trading day is signiﬁcantly negative in both call and put options.

2.4.3.4 The regression analysis of non zero delta-hedged gain

Beside the underlying asset risk, there are some additional risk factors contribute to the non-zero gain in the delta-hedge portfolio. We have the following potential explanatory variables to analyse the non-zero delta-hedged gain. Option’s Vega: option’s price sensitive to the underlying asset’s volatility. Option’s Gamma: option’s price sensitive to the change of delta. Option’s open interest change: the change in an option’s open interest. Trading volume (option/underlying asset): daily trading volume of option/underlying asset. Chapter 2. The Chinese Equity Index Option Market 47

Table 2.4.6: The delta-hedged gain (DHG) with call options

Table 2.4.6 reports the DHG in CNY with call options. The DHG/S0 measure the DHG with the percentage of initial underlying asset price S0. The DHG/C0 measures the DHG with the percentage of the initial call option premium C0. Negative DHG measures the percentage of DHG with a negative return in the sample. The results are grouped by full sample, moneyness subgroups: at-the-money (ATM), in- the-money (ITM), deep-in-the-money (DITM), out-of-the-money (OTM), deep-out- of-the-money (DOTM) and options’ maturity groups in Long, Medium and Short.

Obs DHG t-statistic DHG/S0 DHG/C0 Negative DHG Panel A: Average daily DHG Full Sample 28,539 -0.0018 -20.56 -0.07% -1.42% 61.79% DITM 5,144 -0.0005 -1.59 -0.02% -0.37% 50.52% ITM 6,297 -0.0021 -11.42 -0.09% -1.20% 58.38% ATM 6,256 -0.0025 -18.03 -0.10% -2.64% 66.14% OTM 5,261 -0.0021 -15.71 -0.08% -2.12% 68.62% DOTM 5,581 -0.0014 -11.08 -0.05% -0.61% 64.72% Short 13,258 -0.0020 -23.43 -0.08% -2.21% 66.13% Medium 6,985 -0.0016 -8.57 -0.07% -0.88% 59.33% Long 8,296 -0.0015 -7.07 -0.06% -0.61% 56.93%

Panel B: Average accumulated DHG to the last trading day Full Sample 405 -0.1252 -20.75 -4.95% -100.17% 96.30% Chapter 2. The Chinese Equity Index Option Market 48

Table 2.4.7: The delta-hedged gain (DHG) with put options

Table 2.4.7 reports the DHG in CNY with put options. The DHG/S0 measures the DHG with the percentage of initial underlying asset price S0. The DHG/P0 measures the DHG with the percentage of initial put option premium. Negative DHG measures the percentage of DHG with a negative return in the sample. The results are grouped by full sample, moneyness subgroups: at-the-money (ATM), in-the- money (ITM), deep-in-the-money (DITM), out-of-the-money (OTM), deep-out-of- the-money (DOTM) and options’ maturity groups in Long, Medium and Short.

Obs DHG t-statistic DHG/S0 DHG/P0 Negative DHG Panel A: Average daily DHG Full Sample 28,539 -0.0021 -21.40 -0.09% -1.37% 63.25% DITM 5,581 -0.0020 -4.84 -0.07% -0.77% 54.17% ITM 5,261 -0.0021 -10.32 -0.09% -1.15% 60.96% ATM 6,256 -0.0031 -22.33 -0.13% -2.52% 70.62% OTM 6,297 -0.0021 -18.25 -0.09% -1.60% 67.78% DOTM 5,144 -0.0010 -11.26 -0.05% -0.55% 60.96% Short 13,258 -0.0025 -22.33 -0.10% -1.98% 67.00% Medium 6,985 -0.0021 -9.15 -0.09% -1.09% 61.45% Long 8,296 -0.0014 -6.73 -0.06% -0.62% 58.79%

Panel B: Average accumulated DHG to the last trading day Full Sample 405 -0.1477 -27.75 -6.12% -96.40% 99.75% Chapter 2. The Chinese Equity Index Option Market 49

Trading value (option/underlying asset): daily trading value of option/underlying asset. The underlying asset’s daily log return. Shanghai stock index’s return: Shanghai composite index’s daily log return. Short-selling volume (underlying asset): daily short-selling volume to total trading volume of the underlying asset. Intra day volatility: The option’s/underlying asset’s estimated intra day volatility (σˆt), we calculate the intra day volatility by following Parkinson(1980):

s 2 Ht 1 σˆt = ln , (2.10) Lt 4 ln 2 where the Ht and Lt are the highest and lowest price of the option or underlying asset in the trading day t. We run panel regression analysis and keep the following signiﬁcant explanatory variables in our regression analysis in Table 2.4.8 and Table 2.4.9.

(1) DHG = α1 + β1Option V ega + 1,

(2) DHG = α2 + β2Intra day V olatility(U : Underlying) + 2,

(3) DHG = α3 + β3Underlying Return + 3,

(4) DHG = α4 + β4Option Gamma + 1,

(5) DHG = α5 + β5Option price change + 5,

(6) DHG = α6 + β6Short sell volume + 6,

(7) DHG = α7 + β1Option V ega + β2Intra day V olatility(U) + β3Underlying return +

β4Option Gamma + β5Option price change + β6Short sell volume + 7, Chapter 2. The Chinese Equity Index Option Market 50

where the DHG is the delta-hedged gain, Option V ega is the option’s Black-Scholes Greek Vega, Intra day V olatility(U) is the intra day volatility of the underlying asset, Underlying Return is the log return of the underling asset, Option Gamma is the option’s Black-Scholes Greek Gamma, Option price change is the change of an opiton’s premium, Short sell volume is the daily short-selling volume to total trading volume of the underlying asset and is the residual term. Table 2.4.8 reports the analysis of delta-hedge gain with call options. We find that the all the coefficients are significant at 1% level except the short-selling volume the R2 is the highest in the intra day volatility model. The result indicates that the volatility related risk and the variation of option price risk are the main source of non-zero delta-hedged gain with call options. 2.4.9 reports the analysis of delta-hedge gain with put options, all the coefficients of explanatory variables are significant at 1% level except the Option Vega. The result indicates that variation of of option, underling price risk and the intra day volatility risk are the main source of non-zero delta-hedged gain with put options. The one-dimensional diffusional models which assume the underlying asset’s price is the only source of uncertainty in the option price. We examine the assumption violation and then inspect the non-zero delta-hedge gain to illustrate the one- dimensional diffusional models are not suitable for precise hedging with the options in our Hypothesis 2. Option traders rely on the most widely used BS model (for example, most of the brokers’ trading platforms provide options’ Greeks based on the BS model). Investors can not achieve precise risk hedging in their portfolios and there are also some other risks traded in the Chinese option market. Investors need to update their hedging model to include more factors related to volatility for precise risk hedging, for example, using the Heston(1993) model which includes stochastic volatility. Chapter 2. The Chinese Equity Index Option Market 51

Table 2.4.8: The regression analysis of delta-hedged gain (DHG) with call options

Table 2.4.8 reports the regression analysis of daily DHG with call options. Option Vega (Gamma) is an option’s Black-Scholes model Vega (Gamma). Intra day Volatil- ity is the underlying asset’s intra day volatility. Underlying return is the log return of the underlying asset. Short-selling volume is the percentage of short-selling trading volume in total trading volume (underlying asset). Option price change is the change in daily closing price of an option. We report t-statistics in the brackets below the coefﬁcient estimate, ***,** and * indicate signiﬁcance at at 1%, 5% and 10%.

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Intercept α -0.0010*** -0.0011*** -0.0017*** -0.0012*** -0.0017*** -0.0021*** 0.0030 (-6.26) (-7.37) (-20.89) (-9.20) (-19.85) (-13.28) (1.36)

Option Vega β1 -0.0025*** -0.0008* (-5.56) (-1.74)

Intra day volatility β2 -0.0663*** -0.0969*** (-5.07) (-7.69)

Underlying return β3 -0.0018*** -0.0036*** (-42.39) (-67.34)

Option Gamma β4 -0.0005*** -0.0028*** (-5.23) (-3.20)

Option price change β5 0.0233*** 0.1857*** (8.10) (51.47)

Short-selling Volume β6 0.0017 -0.0013 (0.79) (-0.49) Fixed Effect Yes Yes Yes Yes Yes No Yes R2 0.0004 0.0070 0.0059 0.0003 0.0028 0.0002 0.1354 Chapter 2. The Chinese Equity Index Option Market 52

Table 2.4.9: The regression analysis of delta-hedged gain (DHG) with put options

Table 2.4.9 reports the regression analysis of daily DHG with put options. Option Vega (Gamma) is an option’s Black-Scholes model Vega (Gamma). Intra day Volatil- ity is the underlying asset’s intra day volatility. Underlying return is the log return of the underlying asset. Short-selling volume is the percentage of short-selling trading volume in total trading volume (underlying asset). Option price change is the change in daily closing price of an option. We report t-statistics in the brackets below the coefﬁcient estimate, ***,** and * indicate signiﬁcance at at 1%, 5% and 10%.

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Intercept α -0.0019*** -0.0030*** -0.0021*** -0.0016*** -0.0021*** -0.0026*** -0.0011*** (-10.12) (-17.44) (-21.80) (-10.33) (-21.59) (-18.24) (-5.17)

Option Vega β1 -0.0008 0.0001 (-1.46) (0.21)

Intra day Volatility β2 0.0978*** -0.1227*** (6.57) (-9.77)

Underlying return β3 0.0012*** 0.0074*** (24.29) (114.15)

Option Gamma β4 -0.0005*** 0.0001 (-4.39) (0.62)

Option price change β5 0.1253*** 0.4973*** (41.66) (120.81)

Short-selling Volume β6 0.0142*** 0.0050* (4.67) (1.87) Fixed Effect Yes Yes Yes Yes Yes Yes Yes R2 0.0001 0.0015 0.0203 0.0003 0.0569 0.0006 0.3570 Chapter 2. The Chinese Equity Index Option Market 53

2.5 Conclusion

In this paper, we examine the newly-developed Chinese index option market since 9 February 2015. We ﬁnd that the call (put) options’ calculated Black-Scholes prices are generally lower (higher) than their market prices. This suggests that investors are willing to pay more premium on put options to guard against downside risk. Furthermore, we implement the delta-hedge gain analysis to inspect risk factors as deeper study to analysis why the one-dimensional diffusional models are not applied to the Chinese options market. The non-zero delta-hedged gain in the option sample indicates that the option traders in the Chinese options market are trading at some other risk factors related to volatility rather than the underlying risk only. The empirical results suggest investors need to update the option pricing model to include more risk factors (for example, stochastic volatility risk) to implement precise hedging with options in China. We also compared the new options market with the previous equity warrants market. The result suggests that compared to the equity warrants market, the option market shows less “bubbles” in terms of lower implied volatility, higher correlation with the underlying, and lower pricing error.

2.6 Limitations and future research

This study analyses the newly established Chinese SSE 50 options market with one- dimensional diffusion option pricing model by four violation tests, compare the options’ market price to intrinsic value, and delta-hedge gain analysis. However, we can test more properties like jump diffusion in the option market with recent option pricing models. Future studies could also analyses the properties related to volatility and option behaviours in the SSE 50 options markets. Chapter 2. The Chinese Equity Index Option Market 54

2.7 Appendix

2.7.1 The dividend adjustment in the Chinese option market

After each dividend payment, all existing options’ strike price and exercise units will be adjusted to maintain their notional value unchanged after the dividend payment. Meanwhile, new standard options will be created on the dividend payment date. The dividend adjustment has no impacts on the notional value of the current position of the option. The adjustment date is the underlying asset’s ex-dividend date; the adjustment in options include the adjusted strike price, exercise units, trading code and option’s name. The dividend adjusted options with non-standard 50- points strike interval are called non-standard options. For the non-standard strike price option chain, no options will be added to chain along with the movement of the underlying. New standard option chains will be listed at the ﬁxed 50 points strike interval (or 100 points strike interval if the underlying index is over 3,000 points). There will be a total of 72 new standard put and call options listed in the option after the dividend adjustment: nine strikes (four OTM, ITM and one ATM options), four maturity terms (current month, the next month, two months in the March-June- September-December cycle.). The standard options have “standard” exercise unit of 10,000 shares of the SSE 50 ETF and with strike interval 50 points (or 100 points). As investors like to use standard options to manage ﬁnancial risk, the non-standard option’s average trading volume is lower than that of the standard option. The rules of option dividend adjustment:

(1) Adjusted number of exercise units = Pre-dividend exercise unit × Closing price before ex-dividend date Closing price before ex-dividend date − cash dividend paid

(2) Adjusted strike price = Pre-dividend strike price × Pre-dividend exercise unit Adjusted exercise unit Chapter 2. The Chinese Equity Index Option Market 55

(3) The 12th letter in the trading code will change from “M” to “A”, which means the option has been adjusted while the other trading code remains the same.

(4) An “A” letter will be added to the end of an adjusted option’s name, meaning that the option has been adjusted for a dividend payment. With the dividend adjustment in both the strike price and exercise units, the notional value stay unchanged. For example, in the ﬁrst dividend adjustment in the Chinese option market (the ex-dividend date 29 November 2016), the underlying asset paid a cash dividend of CNY 0.0530 per share. A standard option before adjustment with strike price CNY 2.2, exercise units 10,000 shares with maturity in January 2016:

10, 000 × 2.460 (1) The adjusted number of exercise units: = 10, 220 shares. 2.460 − 0.053 2.2 × 10, 000 (2) The adjusted strike price: = 2.153 (2.1526 round up to 2.153). 10, 220

(3) The adjusted trading code: 510050C1701A2153, the previous trading code: 510050C1701M2200.

(4) The notional value of the call option before dividend payment: CNY 2.2×10, 000 = CNY 22, 000. The notional value of the call option after the dividend adjustment: CNY 2.153 × 10, 220 ≈ CNY 22, 000. The closing price of underlying asset before the ex-dividend date: CNY 2.460.

2.7.2 The derivation of the delta hedged gain in a two-dimensional call option

In the one-dimensional diffusion model:

dSt = µ(S, t)Stdt + σ(S, t)StdBt, (2.7.1) Chapter 2. The Chinese Equity Index Option Market 56

where the drift term, µ(S, t) and the volatility term, σ(S, t) are determined by the underlying asset price, St, and time t, only, the Bt is a standard Brownian motion. If an additional standard Brownian motion Xt existed in our model as follow:

X dXt = θ(X, t)dt + η(X, t)dBt , (2.7.2)

X Bt is the another standard Brownian motion which is correlated to the underlying S in ρ. The θ is the drift term and η is determined by the Brownian motion from addition risk factor Xt.

The price of a call option, Ct, is a function of St, Xt and t:

Ct = Ct(St,Xt, t). (2.7.3)

The stochastic process of Ct can be derived:

∂C ∂C ∂C 1 ∂2C 1 ∂2C ∂2C dC = dt + dS + dX + (dS)2 + (dX)2 + dSdX ∂t ∂S ∂X 2 ∂S2 2 ∂X2 ∂S∂X ∂C ∂C ∂C 1 ∂2C = dt + (µSdt + σSdB ) + (θdt + ηdBX ) + [µ2S2(dt)2 ∂t ∂S t ∂X t 2 ∂S2 1 ∂2C + 2µS2dB dt + σ2S2(dB )2] + [θ2(dt)2 + 2θηdBX dt + η2(dBX )2] t t 2 ∂X2 t t ∂2C + [µSθ(dt)2 + µSηdtdBX + σSθdB dt + σSηρ(dB )2] ∂S∂X t t t ∂C ∂C ∂C 1 ∂2C = dt + (µSdt + σSdB ) + (θdt + ηdBX ) + σ2S2dt ∂t ∂S t ∂X t 2 ∂S2 1 ∂2C ∂2C + η2dt + σSηρdt 2 ∂X2 ∂S∂X ∂C ∂C ∂C 1 ∂2C 1 ∂2C 1 ∂2C = + µS + θ + σ2S2 + η2 + σSηρ dt ∂t ∂S ∂X 2 ∂S2 2 ∂X2 2 ∂S∂X ∂C ∂C + σS dB + η dBX , ∂S t ∂X t (2.7.4) Chapter 2. The Chinese Equity Index Option Market 57

2 X where the differential of higher order become fast zero, (dt) , dtdBt and dtdBt are 2 X 2 neglected, Ito’sˆ lemma [(dBt) = dt and d(Bt ) = dt] is used. The delta-hedged portfolio with call option can be implement with one long call

∂C position and short-selling of ∆ = ∂S share of underlying asset with risk free rate r as the cost. The portfolio is deﬁned as Π = C − ∆S and its delta-hedge gain is:

∂C ∂C dΠ − rΠdt = d(C − ∆S) − r(C − ∆S)dt = dC − (µSdt + σSdB ) − rCdt + r Sdt ∂S t ∂S ∂C ∂C ∂C 1 ∂2C 1 ∂2C 1 ∂2C = + µS + θ + σ2S2 + η2 + σSηρ dt ∂t ∂S ∂X 2 ∂S2 2 ∂X2 2 ∂S∂X ∂C ∂C ∂C ∂C + σS dB + η dBX − (µSdt + σSdB ) − rCdt + r Sdt ∂S t ∂X t ∂S t ∂S ∂C ∂C ∂C 1 ∂2C 1 ∂2C 1 ∂2C = + θ + rS + σ2S2 + η2 + σSηρ − rC dt ∂t ∂X ∂S 2 ∂S2 2 ∂X2 2 ∂S∂X ∂C + η dBX . ∂X t (2.7.5)

The delta-hedged portfolio with put options can be derived similarly. 58

Chapter 3

How do Chinese Options Traders “Smirk” on China: Evidence from SSE 50 ETF Options

This chapter is a joint work with Sebastian A. Gehricke, Jin E. Zhang and Zheyao Pan. It was presented at the 2019 New Zealand Finance Colloquium, 13-15 February 2019, Lincoln University, Christchurch, New Zealand; and 2019 Derivatives Market Conference, 8-9 August 2019, AUT, Queenstown, New Zealand.

3.1 Introduction

In this chapter, we adopt the methodology of Zhang and Xiang(2008) to quantify and analyze the implied volatility (IV) curves of the newly established Shanghai Stock Exchange (SSE) 50 exchange-traded-fund (ETF) option market in China. We ﬁnd that the IV curve usually resembles a right-skewed “smirk”, which is different from the commonly observed left-skewed smirk IV shape in the U.S. and other international options markets (e.g., Foresi and Wu, 2005). Further, we analyze the determinants of the changes in the IV curve shape and ﬁnd that it is driven by investor sentiment. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 59 SSE 50 ETF Options

The SSE 50 ETF option is the first equity option traded in the mainland China and its underlying asset is the SSE 50 ETF fund, which tracks the SSE 50 index. The SSE 50 index includes the 50 largest blue-chip stocks traded on the SSE, which constitute about 59% of the SSE’s market capitalization as of end of 2017. Since the launch of SSE 50 ETF options on 9 February 2015, investors are able to access volatility trading in the world’s largest emerging capital market. This options market has experienced a significant growth and now has a daily trading volume (value) close to 35% (60%) of the SPY (SPDR S&P 500 ETF) options, at the end of 2017 (see Figure 3.2.1). We quantify all the SSE 50 ETF option IV curves by following Zhang and Xiang’s (2008) methodology and find that they usually exhibit a right-skewed smirk shape. Furthermore, we report the time series dynamics of the quantified IV curve factors, which are also proportionally related to the risk neutral moments of SSE 50 ETF option returns (Zhang and Xiang, 2008). We further investigate whether the IV smirk factors, the level, slope and curvature, are driven by investor sentiment. This could be expected, as more optimistic investor sentiment should lead to a bullish options market, which is reflected by an increased slope of the IV curves. The IV, is the volatility which, matches the option’s market price when used in the option pricing model (usually the Black and Scholes, 1973 and Merton, 1973, hereafter, the BSM model) by using the market option price. It is therefore a forward- looking measure of expected volatility . In the BSM model, all options across different strikes are assumed to have the same volatility. If this assumption is true, using the market options data we would find a flat BSM IV curve across different strikes. However, Rubinstein(1985) constructs the IV curve from the U.S. equity options market and finds a “smile” shape instead. Option pricing models have been developed to reproduce option’s IV shape through implementing stochastic factors (e.g., Heston, 1993; Bakshi, Cao, and Chen, Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 60 SSE 50 ETF Options

1997; Pan, 2002; Papanicolaou and Sircar, 2014). The IV from the BSM model is widely used by industry practitioners and academics as a way of quoting options in a comparable manner, although it is understood that the BSM is not the correct model for accurately pricing options. Bates(1991) finds that the price of out-of-the-money (OTM) put options become unusually expensive during the year after the 1987-financial crisis. After 1987, the S&P 500 IV curve shape changed from the symmetric “smile” shape to a left skewed smirk shape (e.g., Rubinstein, 1994; Jackwerth and Rubinstein, 1996;A ¨ıt-Sahalia and Lo, 1998; Carr and Wu, 2003 and Foresi and Wu, 2005). The left skewed IV im- plies that OTM put options are more expensive than the corresponding OTM call options. There are several potential drivers of the asymmetric shape of the IV curve. Hentschel(2003) attributes the smirk shape to the measurement errors in options which violate the non-arbitrage principle. Bollen and Whaley(2004) find that the net buying pressure (defined as the difference of buyer-initiated orders and seller initiated orders) has an impact on the IV curve. Han(2007) attributes the IV shape to investor sentiment. Consistent with the findings in the US market, the left-skewed IV smirk pattern has also been documented in the international option markets (Pena, Rubio, and Serna, 1999; Foresi and Wu, 2005; Shiu, Pan, Lin, and Wu, 2010; Norden´ and Xu, 2012; Tanha and Dempsey, 2016). The only exception is Gemmill(1996), who shows that the IV smirk of the FTSE 100 index options in the United Kingdom skewed to the right rather than the left from 1 July 1985 to 31 December 1990, a period of market turmoil, indicating that option traders in the United Kingdom expected market recovery in the future. We find similar bullish option trader behaviour leading up to and during the 2015 stock market crash in China. There is a vast literature showing that the shape of IV curve has significant predictive power for the future return of an underlying asset (e.g., Dumas, Fleming, and Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 61 SSE 50 ETF Options

Whaley, 1998; Dennis and Mayhew, 2002; Dennis, Mayhew, and Stivers, 2006). Xing, Zhang, and Zhao(2010) use the measure in difference of IV between OTM options and at-the-money (ATM) call options to define an IV smirk factor, and find that this has significant cross-sectional predictive power for future equity returns. Later, Yan (2011) adopts the same measure and finds that the IV smirk is negatively correlated to the underlying stock. The literature on the derivatives market in China is scarce. Xiong and Yu(2011) document the put warrants bubbles in China and examine a set of bubbles they find that the bubbles related to short-selling constraints and heterogeneous beliefs. Chang et al.(2013b) analyze whether the Chinese warrants market shares some of the properties of options, and find there are huge bubbles in the put warrants market. Wang, Chen, Tao, and Zhang(2017) develop a state price dynamic factor model to forecast the Implied Volatility Surface (IVS) of the SSE 50 ETF options. 1 Huang et al.(2018) construct the volatility index in China and analyze its variance risk premium. In our Chapter2, we examine the SSE 50 ETF options and show that one dimensional diffusion model does not apply to the option market in China. Li, Yao, Chen, and Lee(2018) examine the momentum effect of the SSE 50 ETF options. Li, Gehricke, and Zhang(2019a) document the IV curve of iShares China Large-Cap (FXI) ETF options, which is listed in the US option market. Huang et al.(2019) examine popular volatility model in term of pricing performance with SSE 50 ETF options and they find that newly developed model with realized measures outperformed conventional GARCH type models. This paper provides a comprehensive analysis for the IV curves of SSE 50 ETF options, the first equity options market in China. We find that the SSE 50 ETF options’ average IV curve is a right-skewed smirk, different to other equity options

1Wang et al.(2017) has developed the cross sectional IVS model by following Goncalves and Guidolin(2006) which is different from Zhang and Xiang(2008). They ﬁnd that estimated left-skewed IVS in the sample from 9 Feb 2015 to 6 Feb 2016, which include two major market crashes in 2015 and 2016. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 62 SSE 50 ETF Options

markets. Further, we show this average shape is predominantly driven by the lead up to and impact of the 2015 ﬁnancial crisis periods after which it changes to an almost symmetric smile shape. Lastly, we ﬁnd that the investor sentiment drives the shape of the IV curves. The rest of this paper is organized as follows. Section 3.2 provides background of the SSE 50 ETF and options markets. Section 3.3 reports the data used in this study. Section 3.4 describes the methodology. In section 3.5, we present the documentation of the IV curves. Section 3.6 presents the sentiment proxies we constructed and their relationship to the IV curve shape though time. Lastly, Section 3.7 concludes the paper.

3.2 Background of SSE 50 index ETF and options market

On 2 January 2004, the SSE introduced the SSE 50 index, which is a capitalization- weighted index consisting of the 50 largest and most liquid stocks listed on the SSE. The SSE 50 index reflects the performance of the most influential blue chip stocks, which constitute more than 25% of the total market capitalization of the Chinese equity market. The Hua Xia fund management company launched its first ETF fund, the SSE 50 ETF, which tracks the SSE 50 index on 30 December, 2004. With the growth in popularity of passive ETF fund index investments (Gastineau, 2001; Poterba and Shoven, 2002), there are now 141 ETF funds with a total capitalization of CNY 232 billion in China (as at the end of 2017). The SSE 50 ETF fund is the largest ETF fund in China with a capitalization of CNY 38 billion (as at the end of 2017). Table 3.2.1 below presents a summary of the leading ETF funds in China. As can be seen the SSE 50 ETF is the largest, most liquid, oldest and the only ETF with an options market (as at the end of 2017). Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 63 SSE 50 ETF Options CSI 300 ETF CSI 500CSI ETF 300 Index CSI 500 Index SZSEETF GEB SZSE GEB In- dex SZSEETF SME SZSE SME In- dex SSE 180 Index SZSE 100dex In- Summary of leading exchange-traded-fund (ETF) in China (2017) Hua Xia EFund Hua An Hua Xia EFund Hua Tai Nan Fang 38,085310,029 4,0488,016 6,236825,740 20,070 12562,937 8,832 27,892 2,628 874 116 11,065 28,816 5,165 124,295 37,792 876 1,091 20,321 113,778 216,948 0 4,444 18,517 26,034 422,633 9,795 19,088 171,318 68,053 2,639 9,776 SSE 50 ETF SZSE 100 ETF SSE 180 ETF Table 3.2.1: Capitalization InceptionFundcompany management 23 Feb 2005 24 Apr 2006 18 May 2006 05 Sep 2006 09 Dec 2011 28 May 2012 15 Mar 2013 (CNY 000,000’s) Mean dailyvolume (000’s) trading Meanvolume (000’s) short-selling Mean dailyvalue (CNY 000’s) trading Mean margin trading value (CNY 000’s) Options Yes No No No No No No Underlying Index SSE 50 Index This table reports the market(value) statistics in of 2017. the leading(SZSE) These ETFs 100, leading SSE in ETFs’ 180, China SZSE underlying as Small-Medium-sized-EnterprisesIndex include: (SME), the (CSI) SZSE end 300 Growth-Enterprise-Board Shanghai indexes. (GEB) of Stock and 2017 China Exchange and Stock (SSE) their 50, average trading Shenzhen volume Stock Exchange Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 64 SSE 50 ETF Options

The SSE introduced the first exchange traded options in China, the SSE 50 ETF options on 9 February, 2015. Each SSE 50 ETF option contract is written on 10,000 shares of the SSE 50 ETF fund. The SSE 50 ETF options have four different maturity terms: the current month, the next month and the first two following two months out of the March-June-September-December cycle. The SSE 50 ETF options mature on the fourth Wednesday of their maturity month. For each option chain, a range of strikes are available on each trading day. On the initial trading day there are four OTM, one ATM and four ITM (in-the-money) call (put) options.2 The SSE sets a high entry barrier to the SSE 50 ETF option markets, which include capital requirement (at least CNY 500,000, approximately USD 71,000) and qualification tests.3 This means that SSE 50 ETF option traders are sophisticated and experienced traders. Figure 3.2.1 presents the daily trading volume, value and open interest of SSE 50 ETF options in comparison with SPY options, the most active ETF options in the U.S. options market.4 From Figure 3.2.1, we can see the liquidity of the SSE 50 ETF option market has increased significantly since its inception at February 2015 and its daily trading volume (value) is close to 35% (60%) of SPY option at the end of 2017. 2From 6 February 2015 to 2 January 2018, there were only two OTM, ITM and one ATM options listed on the option chain at the initial trading day of each option chain. 3There are three levels of trading privileges in SSE options: level 1 investors can open covered call and protective put position only, level 2 includes privileges in level 1 and investors can open long positions in call or put options, and level 3 includes all the privileges in previous levels and investors can open naked short-selling positions. 4SPY options’ trading value are not available in OptionMetrics, we use the mid of bid-ask quote price times total trading volume to approximate the SPY options’ daily trading value. We convert SPY’s trading value from USD to CNY with the average exchange rate in 2017: USD/CNY = 6.7518. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 65 SSE 50 ETF Options

Figure 3.2.1: SSE 50 and SPY ETF options’ trading volume, value and open interest

This ﬁgure reports the total daily trading volume, trading value and open interest in SSE 50 and SPY ETF options. The trading value of SPY options is converted to CNY with the mean USD/CNY exchange rate (USD/CNY = 6.7821) in 2017. SSE 50 ETF Option daily trading volume 1.8 All option 1.6 Call option Put option 1.4

1.2

0.8

0.6

0.4

0.2 Trading Volume # of contracts in (000,000)

0 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Trading Date (a) Daily trading volume (SSE50)

SPY ETF Option daily trading volume 12 All option Call option 10 Put option

2 Trading Volume # of contracts in (000,000)

0 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Trading Date (b) Daily trading volume (SPY) Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 66 SSE 50 ETF Options

Figure 3.2.1: SSE 50 and SPY ETF options’ trading volume, value and open interest (continued)

SSE 50 ETF Option daily trading value 1400 All option Call option 1200 Put option

1000

800

600

400 Trading value in CNY (000,000) 200

0 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Trading Date (c) Daily trading value (SSE50)

SPY ETF Option daily trading value 4000 All option 3500 Call option Put option

3000

2500

2000

1500

1000 Trading Value in CNY (000,000)

500

0 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Trading Date (d) Daily trading value (SPY) Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 67 SSE 50 ETF Options

Figure 3.2.1: SSE 50 and SPY ETF options’ trading volume, value and open interest (continued)

SSE 50 ETF Option daily Open interest 2 All option 1.8 Call option Put option 1.6

1.4

1.2

0.8

0.6

0.4

Open interest in # of contracts (000,000) 0.2

0 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Trading Date (e) Daily open interest (SSE50)

SPY ETF Option daily open interest 30 All option Call option 25 Put option

5 Open Interest in # of contracts (000,000)

0 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Trading Date (f) Daily open interest (SPY) Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 68 SSE 50 ETF Options

3.3 Data

Our sample period is from 9 February 2015 to 31 December 2017. The SSE 50 ETF option data are sourced from the WIND ﬁnancial terminal. The SPY ETF options data are downloaded from the OptionMetrics Ivy DB database. We follow the option data cleaning method used by Bakshi, Cao, and Chen(1997) and Zhang and Xiang(2008) to process our SSE 50 ETF options dataset (64,537 observations) as follows:

(1) Options with less than seven days to maturity are discarded (3,414 observations), since very short term options may introduce liquidity biases.

(2) Options with unsolvable implied volatility are discarded (1,762 observations).5

(3) Option contracts with price quotes lower than CNY 2 (as the minimum commission fee charged by SSE is CNY 2 per contract) are discarded (859 observations) to mitigate the impact of price discreteness.6

(4) Options violate the non-arbitrage principle are discarded (1,655 observations), and Eq (3.3.1) illustrates violations of the non-arbitrage bonds:

−r(T −t) −r(T −t) ct,T ≤ max(0,Ft,T e − Ke ), (3.3.1) −r(T −t) −r(T −t) pt,T ≤ max(0, Ke − Ft,T e ),

where K is the strike price, Ft,T is the implied forward price at time t with maturity T and r is the risk free rate.

Figure 3.3.1 presents the performance of the SSE 50 ETF fund, its benchmark index and the mean IV level from ATM call options.

5We set the maximum limit of the IV to 1,000% and the minimum limit to 0%: if the IV is higher than 1,000% or less than 0%, they are discarded from our sample. 6The SSE 50 ETF options’ minimum commission fee was CNY 2 per contract (10,000 shares) before 11 November 2016; SSE adjusted the minimum commission fee to CNY 1.3 after 11 November 2016. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 69 SSE 50 ETF Options

Figure 3.3.1: The SSE 50 index and the SSE 50 ETF index tracking in our data sample

The figure 3.3.1 (a) reports the performance of the SSE 50 index and the SSE 50 ETF’s replicated index during our sample period (9 February 2015 to 31 December 2017). We also annotate the sub-sample: the 2015 financial crisis from 15 June 2015 to 31 August according to the time window defined by (Han and Pan, 2017). The figure 3.3.1 (b) reports the mean implied volatility (IV) of the at-the-money (ATM) call options in the sample.

SSE 50 index and SSE 50 ETF fund's index tracking in our sample 3600

SSE50 Index 3400 SSE50 ETF implied Index

3200 2015 Financial crisis: 15 June 2015 to 31 August 2015

3000

2800

2600 Index level

2400

2200

2000

1800 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Date

SSE 50 ETF Option mean ATM Call Implied Volatility 0.9

0.8

0.7 2015 Financial crisis: 15 June 2015 to 31 August 2015

0.6

0.5

0.4 Implied Volatility

0.3

0.2

0.1

0 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Date Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 70 SSE 50 ETF Options

In Table 3.4.1, we report the trading activity of the SSE 50 ETF options by maturity groups. It shows that the majority of options have a time to maturity less than 180 days, and that options’ trading activities, such as trading volume, number of strikes and open interest, decrease with maturity.

3.4 Methodology

With the IV calculated from the market price of options using the BSM model, we document the IV curve from SSE 50 ETF options. We inspect whether the asymmetric IV pattern, known as the IV smirk (Pena et al., 1999; Foresi and Wu, 2005; Shiu et al., 2010) exists in the SSE 50 ETF options market. We first apply the methodology developed by Zhang and Xiang(2008) to quantify the IV curve, across all maturities in our sample. The Goncalves and Guidolin(2006) IVS model is an alternative IV model which includes the additional time to maturity term and it use the log return of IV rather than the IV level to construct the model. The Zhang and Xiang(2008) IV model is the “industry standard” method to quantify the IV smirk. They carefully designed the moneyness normalized with the square root of time to maturity can eliminate the effect of different maturities. The moneyness which built on the Wall Street standard, forms a natural link between the IV smirk’s level, slope and curvature to the risk-neutral standard deviation, skewness, and kurtosis. We further follow Li et al.’s (2019) methodology to calculate the constant maturity quantified IV factors and investigate these coefficients’ time series. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 71 SSE 50 ETF Options 120 > 120 < 250 − 180 180 − Maturity sub-group (days) 90 90 − 30 30 < Summary of the SSE 50 ETF options market Full sample Table 3.4.1: Option groupsMean number of strikesMedian number of strikesMean 13 trading volume 11 2,448Median trading volumeMean open interest 180,285 137,845Median open interest 13 517 13 128,127 585,467 100,828 564,579 66,705 881 13 44,803 11 14,381 343,367 9,879 629 311,250 15 6,976 205,873 15 173,717 3,468 105,071 421 89,052 9 41,885 168,244 8 21,417 130,476 13,065 6,417 1,632 505,314 465,716 86,972 13 816 12 61,015 11 10 This table reports thecontracts), daily open mean interest and (number medianmaturity of number term contracts) groups. of of The SSE sample each 50 period trading is ETF day. from option 9 The strikes, February results 2015 trading to are volume 31 reported December (number in 2017. of the option full sample and sub- Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 72 SSE 50 ETF Options

3.4.1 The calculation of implied volatility

The WIND ﬁnancial dataset provides SSE 50 ETF options IV based on the BSM model. However, WIND assumes a zero dividend yield in their IV calculation, while in fact the SSE 50 ETF will pay discrete cash dividends.7 Therefore, we use the implied forward price from the put-call parity, which contains the implied continuous dividend yield for the IV calculation. We calculate the IV of SSE 50 ETF options by inverting market options prices back to the BSM model with this implied forward price. The ATM strike price K is selected with the smallest absolute difference between the call and put option prices (with same strike and maturity). Based on the put-call parity, we can calculate the implied forward price from:

AT M r(T −t) AT M AT M Ft,T = Kt,T + e × (ct,T − pt,T ), (3.4.1)

AT M AT M where ct,T and pt,T are ATM option prices. With the calculation of implied forward price we can get a more precise IV than the WIND database, accounting for dividends.

3.4.2 Quantifying the IV curve

Zhang and Xiang(2008) develop an approach to quantifying the IV curve by fitting the second order polynomial function. They further show that the coefficients of the polynomial are proportional to the risk-neutral moments of the underlying asset’s return. Following Carr and Wu(2003), Zhang and Xiang(2008) and industry practice, we define moneyness as the logarithm of the strike price over the implied forward

7We checked WIND database’s support documents and contacted one of the WIND’s technical support staff, they conﬁrmed that WIND database uses zero dividend yield and closing price for options’ IV calculation. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 73 SSE 50 ETF Options

price, normalized by the volatility as follows:

ln(K/Ft,T ) ξ = √ , (3.4.2) σ¯ T − t where ξ is the moneyness of the option, t is the current time, T is the maturity date,

K is the strike price and Ft,T is the implied forward price. We use the linearly interpolated IV of two ATM call options with maturities closest to 30 days as the measure of 30-day constant volatility σ¯. With the deﬁnition of moneyness in Eq (3.4.2), we quantify the IV curve by ﬁtting the second order polynomial function given by

2 IV (ξ, t, T ) = α0 + α1ξ + α2ξ . (3.4.3)

We then convert the coefﬁcients α0, α1 and α2 to the dimensionless quantiﬁed IV factors through the following transformations:

α1 α2 γ0 = α0, γ1 = , γ2 = . α0 α0 resulting in the following quantiﬁed IV function:

2 IV (ξ, t, T ) = γ0(1 + γ1ξ + γ2ξ ). (3.4.4)

8 The first factor γ0 is the level, which is an estimate of the exact ATM IV. The pa- rameter γ1 captures the slope of quantified IV curve and γ2 captures its curvature. The solution of all three quantified IV curve coefficients in Eq (3.4.3) is estimated by

8The ATM we discussed earlier is where the moneyness level ξ is approximately zero and the exact ATM here is where the moneyness level ξ is zero. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 74 SSE 50 ETF Options

minimizing the trading volume-weighted mean squared error:

P 2 ξ V olume(ξi) × [IVmarket(ξi) − IV (ξi)] VWMSE = i , (3.4.5) P V olume(ξ ) ξi i where V olume(ξi) is the trading volume (number of option contracts), IVmarket(ξi) is the IV calculated from market price and IV (ξi) is the model IV. We estimate the IV function with OTM options only. The level, slope and curvature are approximately related to the risk-neutral volatility (σ), skewness (λ1) and kurtosis (λ2), respectively:

λ2 1 1 γ ≈ 1 − σ, γ ≈ λ , γ ≈ λ , (3.4.6) 0 24 1 6 1 2 24 2

as shown by Zhang and Xiang(2008).

3.5 Empirical Results

3.5.1 The quantiﬁed IV curves

In this section, we report and analyze the average shape of the quantified IV curves of the SSE 50 ETF options market. In Table 3.5.1, we report the summary statistics of the implied forward price, fitted IV curve level (γ0), slope (γ1) and curvature (γ2) factors, fit statistics and mean trading volume by maturity grouping. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 75 SSE 50 ETF Options

Table 3.5.1: Summary of ﬁtted implied volatility (IV) coefﬁ- cients

This table reports the ﬁtted result for the IV function:

2 IV(ξ)=α0+α1ξ+α2ξ , where ξ is the standard moneyness of an option. The estimated coefficient α0, α1 and α2 can be converted to the dimensionless coefficient γ0, γ1 and γ2 as reported in Section 5. We report the mean coefficients across four maturity groups.

Maturity sub-group (days) Full Sample < 30 30 − 90 90 − 180 > 180

Fˆt,T 2.4193 2.4334 2.4302 2.4154 2.3849

αˆ0 0.2413 0.2365 0.2384 0.2440 0.2493

αˆ1 0.0063 0.0028 0.0016 0.0097 0.0153

αˆ2 0.0103 0.0093 0.0060 0.0139 0.0153

γˆ0 0.2413 0.2365 0.2384 0.2440 0.2493

γˆ1 0.0300 0.0158 0.0163 0.0406 0.0602

γˆ2 0.0386 0.0454 0.0339 0.0413 0.0363 Standard Deviation

Fˆt,T 0.3243 0.3104 0.3097 0.3329 0.3551

αˆ0 0.1270 0.1487 0.1301 0.1151 0.1073

αˆ1 0.0527 0.0354 0.0664 0.0440 0.0482

αˆ2 0.0609 0.0097 0.0753 0.0431 0.0822

γˆ0 0.1270 0.1487 0.1301 0.1151 0.1073

γˆ1 0.1059 0.0637 0.1220 0.0967 0.1159

γˆ2 0.1268 0.0217 0.1246 0.0884 0.2207 Signiﬁcant of coefﬁcient at 5% level

αˆ0 99.96% 100% 100% 100% 99.76%

αˆ1 73.00% 58.80% 72.19% 85.69% 73.16%

αˆ2 73.98% 91.30% 78.09% 73.77% 44.42% R2 and Adjusted R2 Mean volume 4,753 11,890 4,887 1,263 922 Mean R2 0.9131 0.9226 0.9306 0.9024 0.8422 Mean Adj R2 0.8704 0.8836 0.8954 0.8624 0.7430 Option Groups 2,448 517 881 629 421 Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 76 SSE 50 ETF Options

From Table 3.5.1, we can see that the mean SSE 50 forward price in the full sample is 2.4193. The mean forward price decreases as maturity increases, from 2.4334 to 2.3849 for maturity less than 30 days and more than 180 days, respectively. The term structure of the implied forward price is therefore downward sloping. The standard deviation of the implied forward price is 0.3243 and it is increasing from 0.3104 to 0.3551.

The level factor (γˆ0 =α ˆ0), which estimates the exact ATM IV, is 0.2413, on average. The average level factor monotonically increases from 0.2365 to 0.2493. There- fore the term structure of the level factor is upward sloping on average.

For the slope factor (γ1), we can see that, on average, the IV curves are upward sloping for all maturity groups. This right-skewed IV curve indicates that the OTM call options’ IV level are higher than corresponding OTM put options. As maturity increases, the slope becomes steeper from 0.0158 to 0.0602. The slope coefficients are significant, at the 5% level, for 73% of IV curves overall. The proportion of significant slope coefficients is lowest in the shortest maturity group (58.80%), which may be due to the inclusion of the 2015 stock market crash in the sample.

For the curvature factor (γ2), on average, it is positive across all maturity groups, which means that the SSE 50 ETF options’ IV curves are convex, on average. The overall average curvature factor is 0.0386. The curvature coefficients are significant for 73.98% of the IV curves, and the proportion of significant coefficients decreases as maturity increases. It is also clear that the proportions of significant factors and R squared decrease with maturity, indicating that the option traders’ views on long-term options in the SSE 50 maybe less consistent. This may be due to a vast drop in liquidity as maturity increases, as revealed by the decrease in trading volume along maturities. We have randomly selected three trading days to inspect the SSE 50 option’s IV curve: 8 May 2015, 17 February 2016 and 2 May 2017, and present them in Figures Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 77 SSE 50 ETF Options

3.5.1, 3.5.2 and 3.5.3 respectively.

Figure 3.5.1: Fitted IV curves on 8 May 2015

This figure illustrates the fitted IV curves for four different maturity terms (May, June, September and December) on 8 May 2015. The stars in each sub-figures are the market IV of each options, the bars indicate each options’ trading volume and the solid lines are their fitted IV curves. The moneyness ξ is calculated from ξ = ln(K/F ) √ t,T ξ t T σ¯ T −t , where is the moneyness of the option, is the current time, is the maturity date, K is the strike price and Ft,T is the implied forward price, σ¯ is one month at-the-money (ATM) implied volatility.

Maturity in 19 days Maturity in 47 days 0.6 8000 0.6 8000

0.55 7000 0.55 7000

0.5 6000 0.5 6000

0.45 5000 0.45 5000

0.4 4000 0.4 4000

0.35 3000 0.35 3000 Trading Volume Trading Volume Implied Volatility Implied Volatility

0.3 2000 0.3 2000

0.25 1000 0.25 1000

0.2 0 0.2 0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 Moneyness Moneyness

(a) Maturity in 19 days (b) Maturity in 47 days

Maturity in 138 days Maturity in 229 days 0.45 8000 0.45 8000

7000 7000

6000 6000

0.4 0.4 5000 5000

4000 4000

3000 3000 Trading Volume Trading Volume Implied Volatility 0.35 0.35

2000 2000

1000 1000

0.3 0 0.3 0 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Moneyness Moneyness

(c) Maturity in 138 days (d) Maturity in 229 days Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 78 SSE 50 ETF Options

Figure 3.5.2: Fitted IV curves on 17 February 2016

This figure illustrates the fitted IV curves for four different maturity terms (February, March, June and September) on 17 February 2016. The stars in each sub-figures are the market IV of each options, the bars indicate each options’ trading volume and the solid lines are their fitted IV curves. The moneyness ξ is calculated from ln(K/F ) ξ = √ t,T ξ t T σ¯ T −t , where is the moneyness of the option, is the current time, is the maturity date, K is the strike price and Ft,T is the implied forward price, σ¯ is one month at-the-money (ATM) implied volatility.

Maturity in 7 days 104 Maturity in 35 days 104 0.8 2 0.6 2

1.8 1.8 0.55 0.7 1.6 1.6 0.5 1.4 1.4 0.6 0.45 1.2 1.2

0.5 1 0.4 1

0.8 0.8 0.35 Trading Volume Trading Volume Implied Volatility 0.4 Implied Volatility 0.6 0.6 0.3 0.4 0.4 0.3 0.25 0.2 0.2

0.2 0 0.2 0 -3 -2 -1 0 1 2 3 4 5 6 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Moneyness Moneyness

(a) Maturity in 7 days (b) Maturity in 35 days

Maturity in 126 days 104 Maturity in 224 days 104 0.4 2 0.4 2

0.39 1.8 0.39 1.8

0.38 1.6 0.38 1.6

0.37 1.4 0.37 1.4

0.36 1.2 0.36 1.2

0.35 1 0.35 1

0.34 0.8 0.34 0.8 Trading Volume Trading Volume Implied Volatility Implied Volatility 0.33 0.6 0.33 0.6

0.32 0.4 0.32 0.4

0.31 0.2 0.31 0.2

0.3 0 0.3 0 -0.5 0 0.5 1 1.5 2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Moneyness Moneyness

(c) Maturity in 126 days (d) Maturity in 224 days Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 79 SSE 50 ETF Options

Figure 3.5.3: Fitted IV curves on 2 May 2017

This figure illustrates the fitted IV curves for four different maturity terms (May, June, September and December) on 2 May 2017. The stars in each sub-figures are the market IV of each options, the bars indicate each options’ trading volume and the solid lines are their fitted IV curves. The moneyness ξ is calculated from ξ = ln(K/F ) √ t,T ξ t T σ¯ T −t , where is the moneyness of the option, is the current time, is the maturity date, K is the strike price and Ft,T is the implied forward price, σ¯ is one month at-the-money (ATM) implied volatility.

Maturity in 22 days 104 Maturity in 57 days 104 0.4 10 0.4 10

9 9 0.35 0.35 8 8 0.3 0.3 7 7 0.25 0.25 6 6

0.2 5 0.2 5

4 4 0.15 0.15 Trading Volume Trading Volume Implied Volatility Implied Volatility 3 3 0.1 0.1 2 2 0.05 0.05 1 1

0 0 0 0 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 Moneyness Moneyness

(a) Maturity in 22 days (b) Maturity in 57 days

Maturity in 148 days 104 Maturity in 239 days 104 0.2 10 0.2 10

0.18 9 0.18 9

0.16 8 0.16 8

0.14 7 0.14 7

0.12 6 0.12 6

0.1 5 0.1 5

0.08 4 0.08 4 Trading Volume Trading Volume Implied Volatility Implied Volatility 0.06 3 0.06 3

0.04 2 0.04 2

0.02 1 0.02 1

0 0 0 0 -1.5 -1 -0.5 0 0.5 1 1.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Moneyness Moneyness

(c) Maturity in 148 days (d) Maturity in 239 days Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 80 SSE 50 ETF Options

From these figures, we can see that the IV curves are usually upward sloping (except for some long-maturity option groups) and slightly convex.9 Consistent with the positive slope factor identified above, the IV curves are skewed to the right side. The right-skewed SSE 50 ETF options’ IV curves indicate that with the same distance from the ATM level, OTM call options have higher IV level than OTM put options. This pattern suggests that the SSE 50 ETF option traders are willing to pay more on call options because they are betting on the upward movement of the underling asset. This pattern is significantly different from other options traders worldwide, who are willing to pay more for put options as insurance (Bates, 1991; Rubinstein, 1994; Foresi and Wu, 2005). To the best of our knowledge, only Gemmill(1996) finds similar right-skewed IV curves in the FTSE 100 options market in the United Kingdom from 1 July 1985 to 31 December 1990.

9SSE will add new strikes to the option chain along with the movements of the underlying index to maintain at least four OTM, ITM and one ATM options in the option chain. The OTM put call option strikes are not symmetrically distributed. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 81 SSE 50 ETF Options

Figure 3.5.4: Mean implied volatility (IV) curves

This ﬁgure displays the ﬁtted IV curves from mean IV factors (level, slope and curvature) for the full sample and different maturity sub-groups from 9 February 2015 to 31 December 2017. Implied Volatility from mean coefficients 0.5 Full sample Less than 30 days 0.45 30 ~ 90 days 90 ~ 180 days Greater than 180 days

0.4

0.35

Implied Volatility 0.3

0.25

0.2 -4 -3 -2 -1 0 1 2 3 4 Moneyness Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 82 SSE 50 ETF Options

In Figure 3.5.4, we plot the mean IV curves across maturity groups. We can see that the IV curves are skewed to the right side, the IV curves with longer maturity become more convex and steeper.

3.5.2 The constant maturity IV factors

We have been examining the term structure of SSE 50 ETF option IV curves across different maturity groups. For each trading day, there are four IV curves with different maturities; these four maturity terms are not constant through time. To analyze the IV curves more accurately, we create the constant 30-day and 120-day quantiﬁed IV factors through linear interpolation. The constant maturity factors will enable us to study the term structure, time series and evolution of the term structure of the factors for the same horizon of option traders’ expectations. The desired constant maturity (τ) term’s quantiﬁed IV factors is interpolated from the two nearest maturity terms (τ1 and τ2):

τ τ1 τ2 Ft = wFt + (1 − w)Ft ,

τ τ1 τ2 γ0 = wγ0 + (1 − w)γ0 , (3.5.1) τ τ1 τ2 γ1 = wγ1 + (1 − w)γ1 ,

τ τ1 τ2 γ2 = wγ2 + (1 − w)γ2 ,

τ τ τ where the Ft is the implied forward at time t with the desired maturity τ, γ0 , γ1 τ and γ2 are quantiﬁed IV level, slope and curvature with the desired maturity τ. The desired constant term τ is calculated from

τ = wτ1 + (1 − w)τ2, (3.5.2) Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 83 SSE 50 ETF Options

where τ1 and τ2 are two nearest terms with maturity below and above the desired constant term. The interpolation weight w is calculated from

τ − τ2 w = . (3.5.3) τ1 − τ2

In 2015, investors opened their margin trading privilege and gained access to margin trading and high leverage products in the SSE and Shenzhen Stock Exchange (SZSE). The SSE composite index increased from 3,258 to 5,178 points or 58.93% in less than six months. Due to forced liquidation of high-leveraged margin positions (leverage-induced fire sales), the market crashed and the SSE index plunged from 5,178 to 2,850 points or 44.96% in 52 trading days. Bian, He, Shue, and Zhou (2018) find that the leverage-induced fire sales lead to abnormal price declines and subsequent reversal in stocks with a greater margin level. Han and Pan(2017) test the relation between futures-cash basis and liquidity during the 2015 financial crisis in China. To compare option traders’ expectations of market movements in different sample periods, following Han and Pan(2017), we split our sample into three sub-samples: pre-crisis, during crisis and post-crisis period. Table 3.5.2 reports the summary statistics for the constant maturity factors over the full sample and three sub-samples. The overall findings in interpolated maturity terms are consistent with the finding in Table 3.5.1: the IV of longer maturity terms is higher than short maturity terms and the IV functions slopes coefficients are positive.

During the financial crisis period, the mean IV level almost doubled compared with the full sample’s mean IV level. The IV function’s slope coefficients are still positive before, during and after the financial crisis, indicating that options traders are willing to pay more on OTM call options compared with OTM put options during the sub sample of 2015 financial crisis. The IV function’s slope coefficient is the Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 84 SSE 50 ETF Options

Table 3.5.2: Summary of interpolated term structure

This table reports the mean and standard deviation of the interpolated implied forward price and implied volatility curve factors for the two maturity terms: 30-day 2 and 120-day from the IV function: IV(ξ)=α0+α1ξ+α2ξ , where ξ is the moneyness, τ α0, α1, and α2 are the level, slope and curvature coefficient of IV function. F is the τ τ τ implied forward price with maturity τ, α0, α1 and α2 are interpolated IV function’s τ τ τ level, slope and curvature coefficients with maturity τ. γ0 , γ1 and γ2 are converted dimensionless implied volatility function’s level, slope and curvature coefficients with maturity τ. We also report the three sub-samples: during, before and after the financial crisis (FC) sub-sample period (15 June 2015 to 31 August 2015).

Full Sample Before FC During FC After FC Days 30 120 30 120 30 120 30 120 Panel A: Implied Forward Price Mean F τ 2.3908 2.3883 2.8573 2.8847 2.6154 2.6611 2.2734 2.2596 Standard deviation F τ 0.3001 0.3252 0.3608 0.3766 0.2764 0.3139 0.1524 0.1666 Panel B: Fitted coefﬁcients Mean τ α0 0.2512 0.2579 0.3633 0.3391 0.4945 0.4427 0.2024 0.2213 τ α1 0.0011 0.0095 0.0120 0.0327 0.0169 0.0576 -0.0032 -0.0008 τ α2 0.0082 0.0155 0.0158 0.0191 0.0079 0.0198 0.0062 0.0153 τ γ0 0.2512 0.2579 0.3633 0.3391 0.4945 0.4427 0.2024 0.2213 τ γ1 0.0107 0.0341 0.0205 0.0786 0.0315 0.1093 0.0059 0.0164 τ γ2 0.0395 0.0526 0.0424 0.0512 0.0210 0.0499 0.0401 0.0547 Standard deviation τ α0 0.1421 0.1178 0.0962 0.0755 0.1321 0.076 0.1075 0.1003 τ α1 0.0477 0.0492 0.0238 0.0461 0.0573 0.1066 0.0497 0.0341 τ α2 0.0251 0.0491 0.0153 0.0306 0.0357 0.0724 0.0268 0.0506 τ γ0 0.1421 0.1178 0.0962 0.0755 0.1321 0.076 0.1075 0.1003 τ γ1 0.0859 0.1089 0.0615 0.1101 0.0944 0.1975 0.0889 0.0901 τ γ2 0.0466 0.0974 0.0416 0.0785 0.0551 0.1286 0.0488 0.0989 Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 85 SSE 50 ETF Options

highest in the 30-day maturity sub-sample during the 2015 financial crisis, which reflects the options traders’ expectation of short term recovery. We plot the mean IV curves in the full sample and sub-sample period around the 2015 financial crisis using the constant maturity factors in Figure 3.5.5.

Figure 3.5.5: Constant maturity mean implied volatility (IV) curves

This figure shows the IV curve interpolated for two constant maturities, 30-day and 120-day. The sub-sample includes the trading days during the financial crisis from 15 June 2015 to 31 August 2015. We follow Han and Pan’s (2017) definition of time period in analyzing the pricing and efficiency of stock index futures during the 2015 financial crisis.

Interpolated Constant Term Implied Volatility Interpolated Constant Term Implied Volatility 1.1 1.1

30 day 30 day 1 1 120 day 120 day 0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5 Implied Volatility Implied Volatility 0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 Moneyness Moneyness

(a) Full sample (b) Sub sample: Financial crisis 2015

Interpolated Constant Term Implied Volatility Interpolated Constant Term Implied Volatility 1.1 1.1 30 day 30 day 1 120 day 1 120 day

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5 Implied Volatility Implied Volatility 0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 Moneyness Moneyness

(c) Sub sample: Before the crisis (d) Sub sample: After the crisis Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 86 SSE 50 ETF Options

We next study the time series of the IV curve factors from the interpolated constant maturity term. We plot the time series of the 30- and 120-day constant maturity IV curve in Figure 3.5.6. We can see that the ATM IV varies in a mean-reverting manner, the difference between the 30- and 120-day maturity level varies in small magnitude around zero. During the period of financial crisis, there are spikes in short-term 30-day maturity IVs, which cause a huge difference between 30-day and 120-day maturity term. Referring to the volatility index level during the financial crisis, the option traders were willing to pay more for short term options. The slope factor, presented in Figure 3.5.6 (c), also exhibits mean reversion. Both short-term and long-term option trading have a negative slope during the 2015 financial crisis; however, there is a huge difference between short and long-term investors. This difference indicates that the short-term options traders are willing to pay more on insurance than long term investors. The the mean difference between the curvature between long term and short term investor is small, however, during major market turmoils long term investors expect a rapid increase in IV of options.

3.5.3 Sub sample analysis

This section reports some robustness checks of our fitted result. Investors risk preference varies across bull, bear and side-way markets. Our sample covers two large market crashes: the margin trading financial crisis in 2015 and the “circuit-breaker” halt trading liquidity crisis in 2016. 10 Our quantified results indicates the average right-skewed IV curve pattern across different maturity terms. Table 3.5.3 reports the quantified IV curve on left, right-skewed slope coefficient. We find that 32.04% of our quantified IV curves with negative slope and 67.96% with positive slope. The mean IV curve is left-skewed and especially in the long maturity term groups.

10The SSE composite index plunged 43.34% from 15 June 2015 to 26 August 2015 and 25.67% from 31 December 2015 to 28 January 2016. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 87 SSE 50 ETF Options

Figure 3.5.6: Time series of IV curve factors

This ﬁgure reports the interpolated time series of ATM IV level, slope and curvature for 30- and 120-day maturity terms and their difference.

level in constant maturity term 30 and 120 day Difference in level 0.9 0.15 30 day 0.8 120 day 0.1

0.7 0.05

0.6 0

0.5 -0.05

level 0.4 -0.1

0.3 -0.15 120 day less 30 level 0.2 -0.2

0.1 -0.25

0 -0.3 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Date Date

(a) IV level in two terms (b) Difference in IV level

Slope in constant maturity term 30 and 120 day Difference in slope 1 0.6 30 day 120 day 0.4

0.5 0.2

0 -0.2

-0.4 Slope -0.5 -0.6

-0.8 120 day less 30 slope -1 -1

-1.2

-1.5 -1.4 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Date Date

Curvature in constant maturity term 30 and 120 day Difference in curvature 1.5 2 30 day 120 day 1.5 1

1 0.5

0.5

Curvature 0 0

-0.5 120 day less 30 curvature -0.5

-1 -1 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Date Date

(e) IV curvature in two terms (f) Difference in IV curvature Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 88 SSE 50 ETF Options

Table 3.5.3: Summary of ﬁtted implied volatility (IV) coefﬁ- cients in sub samples

This table reports the ﬁtted result for the IV function:

2 IV(ξ)=α0+α1ξ+α2ξ , where ξ is the standard moneyness of an option. The estimated coefﬁcients α0, α1 and α2 can be converted to the dimensionless coefﬁcients γ0, γ1 and γ2. The results are reported in sub sample with negative and positive slope across options’ maturity term.

Maturity sub-group (days) Coefﬁcients Full Sample <30 30-90 90-180 >180 Obs with negative slope (10,244)

αˆ0 0.288 0.274 0.276 0.304 0.333

αˆ1 -0.015 -0.013 -0.015 -0.017 -0.019

αˆ2 0.012 0.012 0.010 0.010 0.019

γˆ0 0.288 0.274 0.276 0.304 0.333

γˆ1 -0.043 -0.037 -0.044 -0.045 -0.051

γˆ2 0.046 0.056 0.045 0.031 0.055 Signiﬁcant of coefﬁcient at 5% level

αˆ0 100% 100% 100% 100% 100%

αˆ1 60.0% 56.3% 62.3% 69.3% 43.4%

αˆ2 74.7% 89.2% 76.5% 70.8% 43.4% Obs with positive slope (21,733)

αˆ0 0.219 0.209 0.212 0.228 0.229

αˆ1 0.016 0.012 0.012 0.018 0.023

αˆ2 0.010 0.011 0.011 0.010 0.008

γˆ0 0.219 0.209 0.212 0.228 0.229

γˆ1 0.063 0.046 0.052 0.071 0.089

γˆ2 0.047 0.065 0.060 0.039 0.017 Signiﬁcant of coefﬁcient at 5% level

αˆ0 99.9% 100% 100% 100% 99.7%

αˆ1 78.7% 64.1% 77.1% 90.4% 79.0%

αˆ2 73.9% 94.8% 79.1% 71.9% 44.1% Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 89 SSE 50 ETF Options

We further analysis our result with the sample period in Wang et al.(2017) (from 9 February 2015 to 6 February 2016). They use of SSE 50 ETF option’ IV to construct the IVS model by following Goncalves and Guidolin(2006). We find that only short maturity term option groups (less than 90-day) have negative slope coefficient which is consistent with Wang et al.(2017)’s finding and the other international option markets after the market crash. The long maturity term option groups have larger slope coefficients and the mean slope in the sub sample is positive. For brevity, we didn’t report the sub sample results in the paper.

3.6 Determinants of the IV curve

The dynamics of the SSE 50 ETF options IV curves described above may be related to investor sentiment, as options provide a great vehicle for leveraged investment. We consider the following sentiment proxies from the literature. Amihud(2002) finds that the market turn-over (the trading volume of the underlying asset to total listed number of shares) and the ratio of absolute market return to turnover are good predictors of future returns through a sentiment mechanism. Baker and Stein(2004) find higher turnover and increased liquidity predict lower future subsequent return, and this is the linked to investor sentiment. Therefore, we include the turnover ratio in our analysis as a proxy for investor sentiment. Whaley(2000) finds that the CBOE (Chicago Board Options Exchange) VIX (Volatil- ity Index), which is called the “investor fear gauge”, fits the name as a sentiment indicator: high levels of VIX are accompanied with market turmoils. However, the VIX constructed by options listed in China is not publicly available and our quantified level coefficient is the mean of ATM IV in the market which is similar to the first version of VIX.11 11The CBOE construct the first version of VIX with the mean IV of near-the-money S&P 100 options in 1993 by following Whaley(1993) and it has been renamed to VXO since 2003. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 90 SSE 50 ETF Options

Billingsley and Chance(1988), Pan and Poteshman(2006), Yang and Wu(2011) and Houlihan and Creamer(2018) find that the put-to-call ratio (PCR) is a good market sentiment proxy and can be used to forecast the direction of the market.12 Deuskar, Gupta, and Subrahmanyam(2008) inspect the two sentiment indicators VIX and PCR, and find PCR is a better explanatory variable for investors sentiment. A higher PCR ratio indicates investors purchased more put options, relative to call options, as insurance against expected market turmoil. Cremers and Weinbaum (2010) find that the deviations from put-call parity contains information on future abnormal underlying returns. Han(2007) analyses the impact of investor sentiment on the option price and he find that index option’s smile is steeper (flatter) when investors’ sentiment becomes more bearish (bullish). He use the three investor sentiment proxies: investor sentiment index from 150 news papers, trading activity of S&P 500 futures and Sharpe (2002) valuation errors of the S&P 500 index. The sentiment proxies constructed from stock index futures in China are not reliable as the trading volume sharply decreased due to the regulations on the stock index futures since the 2015 market crash (Han and Pan, 2017). The valuation error approach is also not feasible because the market level earnings forecast data is generally unavailable and lacks of quality in China (see Hung, Shi, and Wang, 2015; Han, Pan, and Zhang, 2019). Lamont and Stein(2004), Yang and Wu(2011) and Stambaugh, Yu, and Yuan (2012) find short selling and margin trade volume and open interest are contrarian indicators, and reflect investor sentiment in the market. We use several investors’ sentiment proxies on market liquidity and investor’s behaviour as independent variables in our analysis of IV curve.13

12Three versions of PCR ratios could be constructed from the daily trading volume, value and open interest of options. 13We provide details of the calculation of sentiment proxies in the appendix. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 91 SSE 50 ETF Options

Hentschel(2003) attributes the IV smirk shape to the violation of non-arbitrage principle, as we have already deleted all the options violating the non-arbitrage principle, the measurement errors are unlikely to be the driving force. Bollen and Wha- ley(2004) ﬁnd the net buying pressure (the difference between buyer-initiated and seller-initiate orders) has an impact on the IV curve. However, the WIND ﬁnancial database does not provide buyer and seller ID or demand and supply order data, limited by data, we leave this net buying pressure mechanism for future research.

3.6.1 Stationarity test of sentiment proxies

Before analyzing of the relationship between investor sentiment and IV factors, we plot the sentiment proxies in Figure 3.6.1 to inspect the existence of trends in the time series. Table 3.6.1 reports the stationarity test of all the sentiment proxies. We convert non-stationary proxies through the difference in log, and ﬁnd that PCROI is not stationary, while its ﬁrst difference log is stationary. Therefore, we use all stationary sentiment proxies in the following regression analysis. Since some of the sentiment proxies might be correlated to each other, to investigate multi-collinerity we build a correlation matrix of all the sentiment proxies. In Table 3.6.2, we can see the PCR volume and PCR value ratios are highly correlated (0.5488), when we estimate the multivariate regression we use only one or the other. The short and margin ratios are also highly correlated(0.5509), therefore, we exclude the margin ratio and include the short ratio in order to avoid multi-collinerity in our analysis. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 92 SSE 50 ETF Options

Figure 3.6.1: Investor sentiment proxies in China

This ﬁgure reports the time series of investor sentiment proxies in our analysis: put-to-call ratio (PCR) of trading volume, PCR of trading Value, PCR of open interest (OI), SSE 50 ETF turn-over ratio, SSE 50 Short ratio (Short selling volume/total trading volume), SSE 50 Margin ratio (Margin trading value/total trading value) and SSE 50 one day lagged return.

PCR Volume Index PCR Value Index 240 400

220 350 200 300 180 250 160

140 200

120

Index level /% Index level /% 150 100 100 80 50 60

40 0 Jan 2015Jul 2015 Jan 2016Jul 2016 Jan 2017Jul 2017 Jan 2018Jul 2018 Jan 2019 Jan 2015Jul 2015 Jan 2016Jul 2016 Jan 2017Jul 2017 Jan 2018Jul 2018 Jan 2019 Date Date

(a) PCR Volume Index (b) PCR Value Index

PCR Open Interest Index PCR Open Interest Log Difference Index 1.8 0.6

1.6 0.4

1.4 0.2

1.2 0 1 -0.2

Index level /% 0.8 Index level /%

-0.4 0.6

0.4 -0.6

0.2 -0.8 Jan 2015Jul 2015 Jan 2016Jul 2016 Jan 2017Jul 2017 Jan 2018Jul 2018 Jan 2019 Jan 2015Jul 2015 Jan 2016Jul 2016 Jan 2017Jul 2017 Jan 2018Jul 2018 Jan 2019 Date Date

(c) PCR Open Interest Index (d) PCR Open Interest Log Difference Index Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 93 SSE 50 ETF Options

Figure 3.6.1: Investor sentiment proxies in China (continued)

SSE 50 ETF Turn Over Index SSE 50 ETF ShortRatio 90 70

80 60

70 50 60

50 40

40 30 Index level /% Index level /% 30 20 20

10 10

0 0 Jan 2015Jul 2015 Jan 2016Jul 2016 Jan 2017Jul 2017 Jan 2018Jul 2018 Jan 2019 Jan 2015Jul 2015 Jan 2016Jul 2016 Jan 2017Jul 2017 Jan 2018Jul 2018 Jan 2019 Date Date

(e) SSE 50 Turn Over Index (f) SSE 50 ETF ShortRatio

SSE 50 ETF MarginRatio SSE 50 ETF Lagged Return 45 10

40 8

35 6

4 30 2 25 0 20 -2 Index level /% Index level /% 15 -4 10 -6

5 -8

0 -10 Jan 2015Jul 2015 Jan 2016Jul 2016 Jan 2017Jul 2017 Jan 2018Jul 2018 Jan 2019 Jan 2015Jul 2015 Jan 2016Jul 2016 Jan 2017Jul 2017 Jan 2018Jul 2018 Jan 2019 Date Date

(g) SSE 50 ETF MarginRatio (h) SSE 50 ETF Lagged Return Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 94 SSE 50 ETF Options

Table 3.6.1: Stationary test of investor sentiment proxies

This table reports the stationary test of all the sentiment proxies we constructed. Two stationary tests are Augmented Dickey-Fuller (ADF) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test. PCRVAL is the Put-to-Call ratio of trading value, PCRVOL is the Put-to-Call ratio of trading volume, PCROI is the Put-to-Call ratio of open interest, LNPCROI is the log difference PCROI, TurnOver is the turn over of the underlying asset, Lag Ret is the one day lagged return of the underlying asset, Short Ratio is the short selling volume to the trading volume, Mar- gin Ratio is the margin trading value to the total trading value. *, ** and *** denote 10%, 5% and 1% critical level of stationary test.

Stationary? ADF-stats Stationary? KPSS-stats

PCRVAL Yes -4.1224*** Yes 2.5267**

PCRVOL Yes -2.9981*** Yes 2.1622**

PCROI No -1.3082 No 0.0450

LNPCROI Yes -30.9290*** No 0.0000

TurnOver Yes -6.7926*** Yes 7.7917**

Lag Ret Yes -30.5993*** No 0.0594*

Short Ratio Yes -12.2181*** Yes 3.5927**

Margin Ratio Yes -8.6404*** Yes 1.6960** Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 95 SSE 50 ETF Options Correlation of investor sentiment proxies in China Table 3.6.2: PCRVOL PCRVAL LNPCROI TurnOver LagRet ShortRatio MarginRatio LagRet -0.2672 -0.3722 0.1400 -0.0477 1 PCRVAL 0.5488 1 PCRVOL 1 TurnOver -0.0801 -0.0211 -0.0313 1 LNPCROI 0.0139 -0.1759 1 ShortRatio -0.0396 -0.1871 0.0041 0.2957 0.0979 1 MarginRatio 0.0268 -0.1473 -0.0392 0.0159 0.0695 0.5509 1 This table reports thetrading correlation value, matrix PCRVOL is ofopen the the interest, Put-to-Call sentiment TurnOver ratio is proxies’ the ofis time turn trading the over series. volume, short of selling LNPCROI the volume is The underlyingThe to the asset, sample the PCRVAL period is Lag total log is trading Ret the difference from volume, is of 9 Put-to-Call Margin lagged February Put-to-Call Ratio return ratio 2015 is ratio of to of the of underlying 31 margin December asset, trading 2017 Short value and to Ratio it the has total 696 trading daily value. observations. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 96 SSE 50 ETF Options

3.6.2 Regression analysis

We ﬁrst estimate the univariate regressions to analyze the determinants of the IV curve in China given by

Yi = β0 + β1 ∗ Xj + i, (3.6.1)

where Yi is the dependent variable, which is the time series of quantified IV curve factors (level, slope or curvature). Xj is one of the following independent variables: PCR of trading volume (PCRVOL), PCR of trading value (PCRVAL), log difference of PCR of open interest (LNPCROI), turnover of underlying asset (TurnOver), the one day lagged return of underlying asset (LagRet), the short-selling ratio of underlying asset (ShortRatio) and the margin trading ratio of underlying asset (MarginRatio). We then combine all the variables, except the highly correlated ones to examine their joint effect. n X Yi = β0 + βj ∗ Xj + i, (3.6.2) j=1 where the highly correlated independent variables are separated in different analysis. We report the regression of investor sentiment proxies with the quantified IV curves’ factors (level, slope and curvature) in Table 3.6.3 to Table 3.6.8. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 97 SSE 50 ETF Options 0.0899*** (8.83) ) 0 0.1056*** (2.94) γ 0.0079 (0.08) 0.3045***(2.83) -0.2470*** -0.1003 (-3.02) (-1.29) -0.0108(-0.03) 0.6021** (2.13) 1.0369*** (3.89) 0.0120**(20.39) 0.0129*** (20.75) 0.0126*** (21.96) 0.1526(1.30) 0.0623 (0.70) 0.2082*** (2.49) Regression analysis of 30-day constant maturity coefficient: Level ( 0.0650*** (4.76) Table 3.6.3: 0.0001 0.0480 0.0038 0.4807 0.0000 0.0175 0.0000 0.503 0.5689 (-0.25) -0.0122 0.2571***(6.56) 0.1969*** 0.2473*** (15.81) 0.1854*** (37.41) 0.2475*** (32.75) 0.2346*** 0.2466*** (37.38) 0.1079*** (29.32) 0.1157*** (19.85) (3.70) (11.21) Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 6 7 2 3 4 5 1 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ β β β β β β β β 2 R ShortRatio MarginRatio PCRVAL LNPCROI TurnOver LagRet Intercept PCRVOL This table reports the regression analysisthe of multivariate investor regression sentiment of proxies’ the relationof following to the variables: the SSE level PCRVOL 50 (Put-to-Call of ETFone IV ratio option), in day of LNPCROI 30-day lagged trading (Log constant return difference volume maturity of of ofThe group, Put-to-Call the SSE value as ratio SSE 50 in described of 50 ETF in the open ETF), option), Eq parentheses interest ShortRatio are ( 3.6.2 ). PCRVAL of (Put-to-Call (the t-stats, Based SSE ratio *, short on 50 of ** selling ETF trading and volume/total option), value *** trading TurnOver denote (the volume) 10%, turn and 5% over MarginRatio and of (the 1% the margin significance SSE trading level. 50 value/total ETF), trading LagRet value). (the Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 98 SSE 50 ETF Options 0.0787*** (8.21) ) 0 0.1139*** (3.41) γ 0.0584 (0.74) 0.2105***(2.31) -0.2074*** -0.0700 (-2.66) (-0.93) -0.1818(-0.60) 0.3462 (1.32) 0.6904*** (2.75) 0.0085***(-15.60) 0.0093*** (16.03) 0.0090*** (16.51) 0.0735(0.76) 0.0253 (0.31) 0.1534* (1.95) Regression analysis of 120-day constant maturity coefficient: Level ( (5.45) 0.0613*** Table 3.6.4: 0.0013 0.0625 0.0013 0.3529 0.0008 0.0118 0.0012 0.3784 0.4464 Model 1 Model 2 Model 3(0.75) Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 0.2311***(7.10) 0.2073***0.0307 0.2550*** (20.20) 0.2109*** (46.27) 0.2553*** (40.04) 0.2464*** 0.2487*** (46.31) 0.1257*** (36.97) 0.1495*** (24.07) (4.64) (15.40) 3 4 5 6 7 2 0 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ β β β β β β β β 2 LNPCROI TurnOver R LagRet ShortRatio MarginRatio PCRVAL Intercept PCRVOL This table reports theEq regression ( 3.6.2 ). analysis of Based investor onratio sentiment the of proxies’ multivariate trading relation regression value of toSSE of the the 50 the following level ETF), SSE variables: of LagRet 50 PCRVOL impliedvalue/total (the ETF (Put-to-Call volatility trading one option), ratio value). in day LNPCROI of The 120-day lagged (Log trading value constant return difference volume in maturity of of the of group, parentheses the Put-to-Call SSE are SSE as ratio 50 t-stats, 50 of described ETF *, ETF), open in ** option), and ShortRatio interest PCRVAL (Put-to-Call *** of (the denote SSE short 10%, 50 selling 5% ETF volume/total and option), 1% trading significance TurnOver volume) level. (the and turn MarginRatio over (the of margin the trading Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 99 SSE 50 ETF Options

In Table 3.6.3 and Table 3.6.4, we can see that the PCRVAL, TurnOver and Short- Ratio significantly contribute to the higher IV level (exact ATM IV) in both 30 day and 120 day maturities. This is consistent with the negative relationship between TurnOver, IV, PCR, ShortRatio and asset returns (e.g., Dechow, Hutton, Meulbroek, and Sloan, 2001; Amihud, 2002; Giot, 2005; Cremers and Weinbaum, 2010). Table 3.6.5 and Table 3.6.6 show that the PCRVAL and LNPCROI contribute to the decreasing slope in the IV smirk, which means option investors in China purchased more put options as the insurance during market turmoils which consistent with other options markets (Foresi and Wu, 2005). Turnover is positively related to the IV slope and shows that, as sentiment improves, investors willing to pay more on bullish (e.g., long OTM calls) option positions. The table also shows that more short selling leads to a more positive IV slope, which is contrary to the expectation (Lamont and Stein, 2004; Stambaugh et al., 2012). However, the Chinese option market is restricted to qualified investors only, and the sentiment of ETF traders may be different to the option traders, especially, the long term option investors. Table 3.6.7 and Table 3.6.8 present the regression results on IVs’ curvature. We find that only the higher LNPCROI contribute to future lower IV curvature in the 30-day short term maturity group, while Lagged return, LNPCROI and Margin ratio affect IV curvature in the 120-day group. However, both of maturity groups’ R2 are quite small that means all the sentiment proxies could only provide minor explanation on the IV curvature. Overall, investor sentiment proxies related to the IV curve’s shape in China. Liq- uidity based sentiment proxies including Turn over, PCR ratios, and short-selling ratio exert the most significant influence on the IV curves. Our study further identi- fies investor sentiment’s impacts on the perspective of level, slope and curvature. Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 100 SSE 50 ETF Options -0.0526*** (-6.07) ) 1 -0.0423 (-1.43) γ 0.0473 (0.85) 0.1198*(1.88) 0.1062 (1.58) 0.0228 (0.34) -0.3691*(-1.72) -0.3072 (-1.32) -0.5899*** (-2.60) 0.0010**(2.06) 0.0006 (1.21) 0.0008 (1.58) -0.1531**(-2.23) -0.1328* (-1.83) -0.2179*** (-3.06) Regression analysis of 30-day constant maturity coefficient: Slope ( (-4.84) -0.0389*** Table 3.6.5: 0.0037 0.0496 0.0109 0.0094 0.0066 0.0078 0.0016 0.0319 0.1018 Model 1 Model 2 Model 3(-1.30) Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 0.0384(1.66)-0.0376 0.0393*** (5.35) 0.0092*** 0.0038 (2.37) 0.0091*** (0.82) 0.0038 (2.34) 0.0036 (0.81) 0.0348 (0.50) 0.0457*** (1.45) (5.21) 3 4 5 6 7 2 0 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ β β β β β β β β 2 LNPCROI TurnOver R LagRet ShortRatio MarginRatio PCRVAL Intercept PCRVOL This table reports the regression( 3.6.2 ). analysis Based of on investor the sentiment multivariate proxies’of regression relation trading of to value the the of following slope theETF), variables: of SSE LagRet PCRVOL implied (the 50 (Put-to-Call volatility one ETF ratio day in option), of lagged 30-daytrading LNPCROI return trading value). constant of (Log volume maturity the The difference of SSE group, value of SSE 50 as in Put-to-Call ETF), 50 the described ShortRatio ratio ETF parentheses in (the of option), are short Eq open PCRVAL t-stats, selling (Put-to-Call interest volume/total *, ratio trading of ** volume) SSE and and 50 *** MarginRatio denote ETF (the 10%, margin option), trading 5% TurnOver value/total (the and 1% turn significance over level. of the SSE 50 Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 101 SSE 50 ETF Options -0.0422*** (-4.63) ) 1 -0.0397 (-1.30) γ -0.0926 (-1.45) 0.2723***(3.71) 0.0626 (0.88) -0.0081 (-0.11) -0.3608(-1.46) -0.1615 (-0.67) -0.3775 (-1.58) 0.0052***(10.55) 0.005*** (9.29) 0.0051*** (9.90) -0.0903(-1.14) -0.1002 (-1.34) -0.1685** (-2.26) Regression analysis of 120-day constant maturity coefficient: Slope ( (-3.73) -0.0346*** Table 3.6.6: 0.0111 0.0302 0.0029 0.1996 0.0048 0.0299 0.0047 0.2083 0.2421 Model 1 Model 2 Model 3(-2.24) Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 0.0932***(3.54) 0.0622***-0.0741** 0.0353*** (7.33) 0.0081 (7.89) 0.0353*** (1.70) 0.0238*** 0.0454*** (7.91) 0.0382 (4.44) 0.0424*** (5.42) (1.54) (4.59) 3 4 5 6 7 2 0 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ β β β β β β β β 2 LNPCROI TurnOver R LagRet ShortRatio MarginRatio PCRVAL Intercept PCRVOL This table reports theEq regression analysis ( 3.6.2 ). of Based investor onratio sentiment the of proxies’ multivariate trading relation regression value to ofSSE of the the 50 the slope following ETF), SSE of variables: LagRet 50 implied PCRVOLvalue/total (the ETF volatility (Put-to-Call trading one option), in ratio value). day LNPCROI 120-day of The lagged (Log constant trading value return difference maturity volume in of of the of group, parentheses the Put-to-Call SSE as are SSE ratio 50 described t-stats, 50 of ETF in *, ETF), open ** option), and ShortRatio interest PCRVAL (Put-to-Call *** of (the denote SSE short 10%, 50 selling 5% ETF volume/total and option), 1% trading significance TurnOver volume) level. (the and turn MarginRatio over (the of margin the trading Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 102 SSE 50 ETF Options -0.0073 (-1.44) ) 2 γ 0.0154 (0.94) 0.0082 (0.27) -0.013(-0.37) 0.0099 (0.26) 0.0009 (0.02) -0.1896(-1.60) -0.1212 (-0.94) -0.1912 (-1.46) -0.0004(-1.50) -0.0004 (-1.39) -0.0004 (-1.55) -0.0810**(-2.12) -0.0649 (-1.61) -0.0763* (-1.86) Regression analysis of 30-day constant maturity coefficient: Curvature ( (-0.44) -0.002 0.0046 0.0004 0.0100 0.0050 0.0056 0.0003 0.0002 0.0194 0.0221 Model 1 Model 2 Model 3(1.43) Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 0.0221*(1.73) 0.0417***0.023 (10.02) 0.0403*** 0.0422*** (18.73) 0.0402*** (16.50) 0.0407*** (18.67) 0.0392*** 0.0299** (15.45) 0.0482*** (9.67) (2.23) (9.51) Table 3.6.7: 3 4 5 6 7 2 0 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ β β β β β β β β 2 LNPCROI TurnOver R LagRet ShortRatio MarginRatio PCRVAL Intercept PCRVOL This table reports the regression analysisEq of ( 3.6.2 ). investor sentiment Based proxies’ onratio relation the to of multivariate the trading regression curvature value of ofSSE of implied the volatility 50 the following in ETF), SSE variables: 30-day LagRet 50 constant PCRVOLvalue/total (the ETF maturity (Put-to-Call trading one group, option), ratio value). as day LNPCROI of described The lagged (Log trading in value return difference volume in of of the of parentheses the Put-to-Call SSE are SSE ratio 50 t-stats, 50 of ETF *, ETF), open ** option), and ShortRatio interest PCRVAL (Put-to-Call *** of (the denote SSE short 10%, 50 selling 5% ETF volume/total and option), 1% trading significance TurnOver volume) level. (the and turn MarginRatio over (the of margin the trading Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 103 SSE 50 ETF Options 0.0134 (1.32) ) 2 γ 0.0519 (1.56) -0.1143* (-1.83) -0.1136(-1.56) -0.1582** (-2.04) -0.1302* (-1.64) 0.5283**(2.20) 0.5787** (2.20) 0.5894** (2.21) -0.0001(-0.21) 0.0005 (0.86) 0.0003 (0.60) 0.1295*(1.68) 0.0807 (0.99) 0.1030 (1.24) (0.65) 0.0060 Regression analysis of 120-day constant maturity coefficient: Curvature ( 0.0024 0.0009 0.0063 0.0001 0.0108 0.0054 0.0075 0.0274 0.0259 Model 1 Model 2 Model 3(1.04) Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 0.0255(0.99)0.0339 0.0475*** (5.63) 0.0519*** 0.0528*** (11.87) 0.0519*** (10.13) 0.0569*** (11.89) 0.0648*** 0.0148 (10.69) 0.0447*** (7.91) (0.55) (4.36) Table 3.6.8: 3 4 5 6 7 2 0 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ β β β β β β β β 2 LNPCROI TurnOver R LagRet ShortRatio MarginRatio PCRVAL Intercept PCRVOL This table reports the regressionin analysis Eq of ( 3.6.2 ). investor Based sentiment onratio proxies’ the relation of multivariate to trading regression the of value curvature theSSE of of following 50 the variables: implied ETF), SSE volatility PCRVOL LagRet 50 (Put-to-Call in ratiovalue/total (the ETF 120-day of trading one constant option), trading value). maturity day LNPCROI volume The group, lagged (Log of value as return SSE difference in described 50 of of the ETF parentheses the Put-to-Call option), are SSE ratio PCRVAL t-stats, (Put-to-Call 50 of *, ETF), open ** and ShortRatio interest *** of (the denote SSE short 10%, 50 selling 5% ETF volume/total and option), 1% trading significance TurnOver volume) level. (the and turn MarginRatio over (the of margin the trading Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 104 SSE 50 ETF Options

3.7 Conclusion

In this paper, we document the implied volatility of the newly established SSE 50 ETF options market in China and quantify its IV curves by following the methodology developed by Zhang and Xiang(2008). We examine the time series and term structure of the quantified IV curve factor dynamics by maturity groups and by calculating constant maturity factors. We further examine the effect of the two market crashes in our sample on the shape of IV curves. Lastly, we examine investor sentiment as a determinant of the IV curve shape. We find that, on average, the IV curve for the SSE 50 ETF options market reflects a smirk shape which is skewed to the right. This is different to the common finding of the left-skewed IV curve in the United States (Rubinstein, 1985; Bates, 1991), in major international options markets (Foresi and Wu, 2005) and for US-traded China ETF options (Li et al., 2019a). However, a similar right skewed IV curve slope was found in the FTSE 100 options market during the 1987 financial crisis in the United Kingdom by Gemmill(1996), which indicates that options traders have a strong expectation of market recovery in the future. Overall, the level (exact ATM IV) factor decreases with maturity and becomes less volatile with longer maturities, which shows that the options traders expect the SSE 50’s volatility to be mean-reverting. The IV curves are, on average, upward sloping and become steeper as maturity increases. The IV curves have positive curvature, on average, and the IV curve becomes more convex as maturity increases. We further analyze the quantified IV curve factors by splitting the sample into: before, during and after the 2015 financial crisis period (15 June 2015 to 31 August 2015). We find that, during the financial crisis, the level (exact ATM IV) almost doubled from before the financial crisis. We observe the right-skewed IV smirk before the crisis, which becomes even more skewed to the right during the crisis, indicating Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 105 SSE 50 ETF Options

investors have strong confidence in the recovery of the SSE 50 index. However, after the crisis the IV curve become less skewed and forms an almost symmetric smile pattern. This story is very different to the IV curve of the international equity option markets, which has an even more left-skewed IV curve during the global financial crisis in 2008 (Guo, Gehricke, and Zhang, 2018; Li et al., 2019a). We further analyze if we can explain the variation in the IV factors using several investor sentiment proxies. We find that liquidity based sentiment proxies: turnover, the PCR ratios and the short sell ratio of underlying assets are related to the slope of IV in China. Overall, the shape of SSE 50 ETF IV curves is strongly related to the liquidity based investor sentiment.

3.8 Appendix

3.8.1 Put-to-Call ratio (PCR):

The PCR indicators are calculated from the total trading volume (number of contracts), value (CNY) and open interest (number of contracts) from the put and call options: Total trading volume of put option PCRVOL = , (3.8.1) Total trading volume of call option

Total trading value of put option P CRV AL = , (3.8.2) Total trading value of call option

Total open interest of put option PCROI = . (3.8.3) Total open interest of call option

3.8.2 Turnover ratio:

The turnover ratio of the underlying asset is calculated from daily trading volume (number of shares) and listed number of shares: Chapter 3. How do Chinese Options Traders “Smirk” on China: Evidence from 106 SSE 50 ETF Options

Daily trading volume T urnover = . (3.8.4) Listed number of shares

3.8.3 Short-selling and Margin trading ratio:

The short selling ratio is calculated from daily short-selling volume (number of shares) and daily trading volume (number of shares):

Daily short selling volume ShortRatio = . (3.8.5) Daily trading volume

The margin trading ratio is calculated from daily margin trading value (CNY) and daily trading value (CNY):

Daily margin trading value MarginRatio = . (3.8.6) Daily trading value 107

Chapter 4

The Volatility Index and Volatility Risk Premium in China

This chapter is a joint work with Xinfeng Ruan, Sebastian A. Gehricke and Jin E. Zhang. It has been accepted for presentation at the 2020 New Zealand Finance Col- loquium, 13-14 February 2020, AUT, Auckland, New Zealand.

4.1 Introduction

Option-based volatility indexes have become popular since the Chicago Board Op- tions Exchange (CBOE) launched its Volatility Index (VIX) based on the S&P 100 options in 1993 by following the methodology of Whaley(1993). The VIX reﬂects the expectation of 30-day ahead volatility, a barometer of investors’ sentiment in the U.S. and other international stock markets (Whaley, 2000; Ryu, 2012; Siriopoulos and Fassas, 2012; Bugge, Guttormsen, Molnar,´ and Ringdal, 2016 and Sensoy and Omole, 2018). Fleming, Ostdiek, and Whaley(1995) ﬁnd a strong negative correlation between the VIX and stock market returns, whereby the increases/decreases in VIX are associated with negative/positive returns in the S&P 100 index. In this paper, we construct the model free volatility index on the VIX methodology for the Chapter 4. The Volatility Index and Volatility Risk Premium in China 108

fast-growing Chinese derivatives market and evaluate its relationship with underlying equity returns. There were no option markets available in mainland China until 9 February 2015, when the Shanghai Stock Exchange (SSE) launched its first equity option contract, the SSE 50 exchange-traded fund (ETF) option. Later on, the SSE published its China Volatility Index (iVX) based on the CBOE(2003) methodology from 5 December 2016; however, the SSE ceased the iVX from 23 February 2018.1 There has been no official volatility index in mainland China since then. This paper proposes the model-free China Volatility Index (CNVIX) based on the SSE 50 ETF option dataset. We find a significant negative relationship between CNVIX and the underlying equity returns, especially during the recent financial crisis in China. With the availability of a model-free volatility index we further test the predictive power of the volatility risk premium (VRP) in China and find the VRP can predict the future SSE 50 ETF returns at the monthly horizon. The VIX has become one of the most followed market sentiment indicators in the world, it is known as the investor’s “fear gauge” (Whaley, 2000). The first version of the VIX was constructed with eight near-the-money CRR (Cox, Ross, Rubinstein, et al., 1979) model-based implied volatilities (IVs). In 2003, CBOE updated the VIX methodology in its CBOE(2003) VIX white paper, which describes the new model- free VIX with S&P 500 options portfolios by following on the pioneering work of Breeden and Litzenberger(1978), Demeterfi et al.(1999) and Britten-Jones and Neu- berger(2000). The model-based VIX has now been renamed to VXO. Fleming et al. (1995) and Whaley(2000) find that the VIX exhibits a strong negatively correlation with its underlying equity index most of the time and it becomes highly negatively correlated during market turmoil. Siriopoulos and Fassas(2012), Bugge et al.(2016)

1The SSE iVX’s methodology (Chinese version) is available at http://www.sse.com.cn/ assortment/options/neovolatility/c/4206989.pdf. Chapter 4. The Volatility Index and Volatility Risk Premium in China 109

and Sensoy and Omole(2018) ﬁnd a similar relation of volatility index to the underlying return in emerging markets. Later, CBOE published several other volatility indexes for other markets, for example, the Dow Jones Volatility Index (VXD), NASDAQ Volatility Index (VXN) and Russell 2000 Volatility Index (RVX). The CBOE also publishes a series of emerging market volatility indexes based on some ETF options, for example, the Emerging Markets ETF Volatility Index (VXEEM), China ETF Volatility Index (VXFXI) and Brazil ETF Volatility Index (VXEWZ). Table 4.1.1 reports the summary statistics of major volatility indexes in CBOE and China. Before the availability of the SSE 50 ETF options to investors in China, the equity warrants were the only contingent claim derivatives available to investors there. An equity warrant is similar to an option, however, only institutions could write or issue equity warrants in China. Currently, the stock market in China adopts a “T+1 trading rule”, which prevents investors from selling stocks bought on the same day “T+0 trading” intra-day trading). Guo et al.(2012) ﬁnd that the “T+1” trading rule, which prevents investors from intra-day trading and it could reduces the China A- share’s volatility and trading volume. While the warrants derivatives are traded with a “T+0” rule, which allow investors to sell their warrants intraday. All the investors have access to the equity warrant by default and some investors lack the basic knowledge in warrants derivatives. Due to the “T+0” trading rule of warrant markets and irrational trading, huge bubbles are created in the warrant market (e.g. Xiong and Yu, 2011; Tang and Wang, 2013, Chang et al., 2013b and Powers and Xiao, 2014). 2

2The equity warrant “bubbles” are documented by Xiong and Yu(2011); Tang and Wang(2013), Chang et al.(2013b) and Powers and Xiao(2014)). They ﬁnd the bubbles which are measured in term of average daily implied volatility of call (put) warrants is 167% (139%), some put warrant’s price are positively correlated to the underlying asset and average daily turnover of call (put) warrants is 83% (104%). Chapter 4. The Volatility Index and Volatility Risk Premium in China 110 R), China Black-Scholes model Summary statistics of volatility indexes Table 4.1.1: mean median p25 p75 min max sd skew kurt Obs R 21.53 19.74 13.69 27.46 5.84 58.84 10.13 1.03 4.00 907 CNVXO 22.60iVX 19.56 13.64 14.08 13.79 28.88 11.11 7.09 15.25 85.25 8.31 11.92 33.06 1.47 3.37 5.91 1.58 907 8.76 296 VIXVXO 14.73VXN 14.21VXD 13.53 17.52CNVIX 13.23 11.79 14.73VXFXI 16.34 23.69 10.74 16.17CNVIX 13.79 25.38 14.58 20.61 16.22 9.14 12.07 23.89 19.00 14.70 6.32 16.20 40.74 20.17 10.31 29.81 37.66 7.58 4.35 29.31 42.95 8.11 4.87 15.09 34.51 4.47 1.61 82.58 1.24 58.40 3.89 1.50 12.26 6.52 4.74 6.93 1.37 1.41 5.50 955 955 1.15 5.14 5.48 955 4.76 955 922 955 This table reports summary statisticsIndex for (VIX), the CBOE volatility S&P indexes: 100(VXD), Volatility CBOE China Index China Volatility (VXO), Index ETF CBOE Volatility (CNVIX), Index NASDAQbased CBOE (VXFXI), Volatility Volatility CNVIX S&P Index Index with (CNVXO) 500 (VXN), actual and Volatility CBOE option SSE75th DJIA only Volatility percentile Index Volatility (CNVXI (p75), Index (iVX). minimum This (min), table maximumsample (max), reports standard the period deviations mean, from (sd), median, 9 skewness 25th (skew) February percentileon and 2015 (p25), kurtosis 22 to (kurt). February 20 The 2018). November 2018 (The SSE iVX was published on 26 November 2016 and closed Chapter 4. The Volatility Index and Volatility Risk Premium in China 111

O’Neill et al.(2016) derive a state price volatility (SPV)-based volatility index for the Chinese stock market and show that the SPV volatility index could forecast the realized volatility (RV) of the Shanghai Composite Index (SHCI). Huang et al.(2018) construct three Chinese volatility indexes based on the SSE50 ETF, China Stock Index 300 (CSI 300) and iShares China Large-Cap ETF (FXI) options by following CBOE’s (2003) standard methodology and document the term structure of volatility across the three markets. Li, Yu, and Luo(2019b) use intraday iVX data to study the asym- metrical return-volatility relation of the iVX and SSE 50 ETF in China. Luo and Qin (2017) investigate the effect of volatility shocks on Chinese stock returns and find the volatility shocks have negative impact on the Chinese stock market. Currently, no official volatility index is available in mainland China, neither the SSE nor the Shenzhen Stock Exchange (SZSE). The CBOE launched the China-related volatility index, VXFXI on 21 March 2011; however, the index is based on options written on the FXI, which tracks the 50 largest Chinese stocks listed on the Hong Kong Stock Exchange (HKEX). Most of the blue-chip stocks in the FXI ETF are cross- listed in the HKEX and the SSE; Table 4.1.2 reports the top 10 stocks in both the FXI ETF and the SSE 50 ETF fund, six of the top 10 weighted stocks are cross-listed in both exchanges.3 The asymmetry of information, media coverage and trading rules between investors in mainland China and Hong Kong has often caused discount or premium on cross-listed stocks (e.g. Chakravarty, Sarkar, and Wu, 1998; Ding, Hou, Liu, and Zhang, 2018 Yuan, Zhou, and Li, 2018). Also, FXI options are traded in the U.S., and so do not necessarily reflect the expectations of Chinese investors. The SSE 50 ETF options provide direct volatility exposure to investors in the Chinese stock market, and therefore their prices reflect the expectations of Chinese option traders on Chinese equities, as shown by the difference in the IV curves of the FXI and SSE

3In total, six out of the top 10 stocks in FXI are cross-listed in HKEX, SSE and they take about 33.87% total weight in the FXI and about 36.26% total weight in the SSE 50 ETF as of 23 September 2019. Chapter 4. The Volatility Index and Volatility Risk Premium in China 112

50 ETF options markets (Li et al., 2019a, and our Chapter3).

4.1.1 Contributions

In this paper, we make the following contributions. First, we construct model-free CNVIX based on the interpolation-extrapolation option dataset. With the extended option dataset we can construct the model-free volatility index during the volatile trading days. The methodology could be applied to emerging option markets with low liquidity and less number of option strikes. The CNVIX could be used as a benchmark for both practitioners and academics about volatility in the Chinese market. We further evaluate the well known negative relationship between the volatility and equity index returns, which is the leverage effect in China.4 We ﬁnd that the leverage effect effect exists in the model-free volatility index in China, which is consistent with the U.S. market. Finally, we study the VRP in the Chinese market and test whether it can predict equity returns. We ﬁnd the VRP can predict the return of the underlying SSE 50 ETF at the daily, two-weekly overlapping return horizons and non-overlapping weekly, two-weekly return horizons. The rest of paper is organized as follows: Section 4.2 describes the option data sources, cleaning method and sample period. Section 4.3 describes the construction of the extended option data set and the CNVIX. Section 4.4 focuses on the leverage effect and the predictability of the underlying asset, using the CNVIX. Section 4.5 concludes.

4The leverage effect was first discussed by Black(1976) and Christie(1982) and refers to companies’ financial leverage (debt-to-equity ratio) when equity prices decline the companies’ financial leverage will increase correspondingly. The company with a high leverage becomes more risky and volatile in the future. There are negative relationship between the stock’s return and volatility. Chapter 4. The Volatility Index and Volatility Risk Premium in China 113 Total 56.86 Total 56.8 The Top 10 Stocks by weight in FXI and SSE 50 ETF Table 4.1.2: Panel B: SSE 50 HuaXia Shanghai Stock601318 Exchange 50 Index ETF (SSE) PING AN600519 INSURANCE (GROUP) OF CHINA LTD KWEICHOW MOUTAI600036 CO LTD CHINA MERCHANTS600276 BANK CO LTD JIANGSU HENGRUI601166 MEDICINE Financials CO LTD INDUSTRIAL AND600030 COMERICIAL BANK OF CHINA CITIC SECURITIES600887 CO LTD INNER MONGOLIA601328 YILI INDUSTRIAL Financials GROUP CO LTD BANK OF600016 COMMUNICATIONS CO LTD Consumer Staples China CHINA MINSHENG600000 Financials BANKING CORP Health LTD Care Consumer SHANGHAI Staples PUDONG DEVELOPMENT 17.05 BANK CO LTD China Financials China Yes China 3.05 4.47 Financials China China Financials 10.53 Financials 4.52 6.22 Yes China Yes 2.44 China China China 2.67 2.65 3.2 Yes Yes Yes Ticker NamePanel A: FXI iShare China Large-Cap ETF700 (HEX) 939 TENCENT HOLDINGS LTD2318 CHINA CONSTRUCTION BANK1398 CORP H PING AN INSURANCE (GROUP) CO941 OF CHINA INDUSTRIAL LTD AND COMMERCIAL BANK OF3690 CHINA CHINA MOBILE Financials LTD3988 MEITUAN Financials DIANPING883 BANK OF CHINA LTD Financials H2628 CNOOC LTD3968 CHINA LIFE INSURANCE LTD H Communication CHINA China MERCHANTS BANK LTD H China 8.03 China 6.47 Sector China 8.84 Yes Yes 8.89 Communication Yes Consumer Discretionary Financials Financials Financials China 4.78 Location China Weight % Cross-listed 5.25 Energy China China China 3.06 3.04 4.43 Yes Yes Yes China 4.07 This table reports the top 10funds stocks management. by weight in FXI and SSE 50 ETF fund as of 23 September 2019 from iShares and Huaxia Chapter 4. The Volatility Index and Volatility Risk Premium in China 114

4.2 Data

The CNVIX is constructed from the European-style SSE 50 ETF options data, which has been available since 9 February 2015. All the SSE 50 ETF options data are downloaded from the WIND ﬁnancial terminal (WFT), and all the U.S. options data are from Ivy DB OptionMetrics. We need the Black and Scholes(1973) and Merton(1973) model IV to construct the options’ IV function and the Black-Scholes model’s based volatility index (CN- VXO); however, the WFT terminal only provides options’ IV based on the assumption of a zero dividend yield and the constant risk free rate. We calculate options’ IV using the implied forward price as the underlying asset and the interpolated Shang- hai Interbank Offer Rate (SHIBOR) matched to each option’s maturity term as the risk free rate. In order to handle the random discrete dividend payments in the SSE 50 options, we calculate each day’s implied forward price and use the Black-Scholes model to calculate the options’ IV.

For the option price, Q(Ki), we use the closing price of the SSE 50 ETF options. We use the standard term structure of SHIBOR to interpolate the risk-free rate which matches each options’ maturity term each day.5 The implied forward price is derived from the put-call-parity using the at-the-money (ATM) option pair, where ATM is deﬁned as the strike price where the put and call prices are closest. The implied forward price accounts for the potential future dividend payments; therefore, we can set the dividend ratio to zero in the following calculations. We have applied the following data ﬁlters in our option dataset by following Carr and Wu(2008) and Chang, Christoffersen, and Jacobs(2013a):

• We discard options with maturity of less than seven days.

5The SHIBOR rate’s standard term structure with the following rates: overnight, one week, one month, three months, six months, nine months and one year. http://www.shibor.org/shibor/ web/html/index_e.html Chapter 4. The Volatility Index and Volatility Risk Premium in China 115

• We discard options with the closing price less than 2 CNY per contract.6

• We discard options with closing prices that violate no-arbitrage conditions.7.

• We discard in-the-money (ITM) options (call options: Kc < Ft and put options:

Kp > Ft).

• We discard non-standard options which have same strike and maturity with standard options. 8

4.3 Methodology

Following CBOE’s methodology for the VIX, we calculate the expected volatility estimate, σ, for two selected maturity terms the near-term and the next-near-term, which represent the ﬁrst options expiring longer than seven days and the next- closest maturity. We propose a robust volatility index for China, the CNVIX and we empirically test the leverage effect and explore whether the CNVIX-based VRP could foreast the return of underlying asset. The model-free volatility index is constructed with the model-free VIX methodology from Demeterﬁ et al.(1999) and Britten-Jones and Neuberger(2000) currently employed by the CBOE. The VIX represents the market’s risk-neutral expectation of the volatility of the underlying asset’s return over the next 30 days, given the current available information. 6The standard commission for a SSE option contract is 2 CNY per contract since 9 February 2015; and the SSE adjusted the standard commission to 1.3 CNY per contract after 26 October 2016. Trading options below the minimum commission are considered as market noise. 7 The lower call and put arbitrage restriction: Ct,T ≥ max[0,Ft,T −K] and Pt,T ≥ max[0,K−Ft,T ]. (T −t) The upper call and put arbitrage restriction: Ct,T ≤ Ft,T and Pt,T ≤ Ke . 8A standard SSE 50 ETF option has exercise units of 10,000 shares ETF per contract. A non-standard option’s strike and exercise units are adjusted for cash dividend payments of the underlying asset; after the adjustments, the notional value of options remain unchanged after the dividend payment. Chapter 4. The Volatility Index and Volatility Risk Premium in China 116

The volatility estimate, σ, is calculated by:

2 2 X ∆Ki 1 Fi σ2 = erτ Q(K ) − − 1 , (4.3.1) i τ K2 i τ K i i 0 where τ is the option’s time to maturity in years, r is the risk free rate, F is the implied forward price of the underlying asset, K0 is the ﬁrst strike price below the forward price, Ki is the strike price of the ith out-of-the-money (OTM) option, ∆Ki 1 is the strike interval 2 × (Ki+1 − Ki−1) and Q(Ki) is the closing price of the option with strike Ki. In order to estimate the volatility expectations, we need the implied forward price (F ) for each term, which is calculated as follows:

rτ Fi = Ki + e × (Pcall − Pput), (4.3.2)

where Ki is the option’s strike price, r is the risk-free rate, τ is the option’s time to maturity and Pcall and Pput are the call and put option premium, respectively. The strike price and option premium are selected from the put-call option pair, which has the minimum absolute difference in the option premium. We select the call (put) OTM option with a strike price higher (lower) than the implied forward price. All the ITM options are discarded for calculation. The ﬁnal one-month volatility estimate is based on a linear interpolation between near-term and next-near-term volatility as follows:

s 2 NT2 − N30 2 N30 − NT1 N365 CNVIX = 100 × T1σ1 + T2σ2 × , (4.3.3) NT2 − NT1 NT2 − NT1 N30 where T1 and T2 are the number of days in the near-term and next-near-term’s time to maturity in years, σ1 and σ2 are the near-term and next near-term’s estimated volatility, NT1 and NT2 are number of days to maturity of near-term and the next Chapter 4. The Volatility Index and Volatility Risk Premium in China 117

near-term options, N365 and N30 are the number of days in a year and a month.

4.3.1 The approximation errors in the VIX

Jiang and Tian(2007) quantify the truncation and discretization approximation errors in the VIX model-free methodology related to a limited number of strikes in the

SPX options. The CBOE(2003) methodology uses a ﬁnite strike range [KL,KU ] to approximate the inﬁnite range of strike prices:

Z K0 P (T,K) Z ∞ C(T,K) Z K0 P (T,K) Z KU C(T,K) 2 dK + 2 dK = 2 dK + 2 dK + δ, 0 K K0 K KL K K0 K (4.3.4) where C(T,K) and P (T,K) are the option premium of the call and put option with maturity T and strike K, K0 is the strike at the money, KL and KU are the minimum and maximum strike in the option chain. δ is the truncation error in approximation. The size of the truncation error’s downward bias in VIX is then given by:

Z KL Z ∞ 2 rT P (T,K) C(T,K) δT runcation = − e 2 dK + 2 dK . (4.3.5) T 0 K KU K

The truncation error varies over time as new strikes will be added to the option chain along with the movement of the underlying asset over the initial strike range [KL,KU ]. The newly added option strikes could reduce the truncation error in volatility index. The discretization errors arise in the procedure of numerical integration. The CBOE(2003) procedure applies the following numerical integration and calculates the VIX in Eq (4.3.1):

Z K0 Z KU P (T,K) C(T,K) X ∆Ki dK + dK = Q(T,K ) + δ, (4.3.6) K2 K2 K2 i KL K0 i i Chapter 4. The Volatility Index and Volatility Risk Premium in China 118

where C(T,K) and P (T,K) are the option premium of the call and put option with maturity T and strike K. K0 is the strike at the money, KL and KU are the minimum and maximum strike in the option chain. ∆K is the interval between two option strikes, Q(T,Ki) is the premium of OTM call or put options with maturity T and δ is the discretization error in approximation. The discretization error can be reduced by using a large number of option strikes, however, the actual partition used in the formula is based on listed option strikes only. Therefore, the error size of discretization errors is:

( ) Z K0 Z KU 2 X ∆Ki P (T,K) C(T,K) δ = erT Q(T,K ) − dK + dK . disc T K2 i K2 K2 i i KL K0 (4.3.7) There are a relatively abundant number of strikes in the SPX options market, which the CBOE uses to calculate the VIX index. However, according to the trading rules in the SSE 50 ETF option, there are only ﬁve option strikes (2 OTM, 2 ITM and 1 ATM option) at the inception of each option chain.9 Table 4.3.1 reports the mean number of strikes in SPX, FXI and SSE 50 ETF options. For each option terms in the SSE 50 ETF, the mean number of strikes is only about 10% and 30% of the mean number in SPX and FXI options, which the CBOE uses in calculating the VIX and VXFXI volatility index. Direct implementation of the CBOE(2003) methodology to calculate the CNVIX with the SSE 50 ETF options could lead to large truncation and discretization errors as the number of strikes is so limited.

9The SSE updated the SSE 50 ETF option trading rule on 2 January 2018; the option chains will have 4 OTM and ITM call (put) strikes at their initial trading day, along with the movement of the underlying asset and additional options will be listed to maintain at least 4 OTM and ITM options in each option chains. Chapter 4. The Volatility Index and Volatility Risk Premium in China 119 120 days < 120 days > 180 days > SPX option FXI ETF option SSE50 ETF option 30 days 30-90 days 90-180 days < Summary of option strikes in SSE50, SPX and FXI ETF option markets MaturityNo. ObsMean # of strikes Full sample Median # of strikesMinimum # of 13 strikes 11Maximum # 2,645 5 of strikesMean 31 trading volumeMean 388,549 open interest 14 705 13 6 945,124 268,807No. 31 Obs 888 13 117,647Mean # 11 of 511,484 strikesMedian # 5 of strikes 21,517 31 303,221Minimum 631 # of 126 strikes 15 111 15Maximum 159,987 # 22,922 36 of strikes 10,575Mean 272 trading 5 31 volume 421 62,843Mean 1,101,844 146 open 6,876 371,607 interest 9 145 8 84 660,895 830,472 13,158,509 19,172 272 6,434No. 166 1,837 Obs 23 5 335,760 170 5,268,259Mean 129,743 # of 14 strikes 3,926,149 49 13 3,709 272Median 65,734 # 808 of strikes 1,708,567 91 31Minimum 74 # of 44 5 strikes 40 2,255,534Maximum 11 39,455 # 10,028 16 5,903 of 45 strikes 215 10 Mean 9,960,175 97 trading volume 79 24 1,039,200Mean 95,420 3,198,334 84 open 3,219 interest 45 14,973 5 40 62,644 103 29 36 2,879,432 2,829 42,843 7,949 97 152 43 154 844,933 39 32,700 272 22 1,484 48 874,062 97 77 80 11,751 48 569,438 48 210 2,496 16 36 71 8,190 594,119 6,721 42 1,964,021 41 21 81,449 915,411 70 3,307 13,971 44 40 21 97 43 43 16 71 Table 4.3.1: This table reports the daily mean/medianperiod number from of 9 option February strikes, 2015 trading to volume 31 and December open 2017. interest in The groups results by are date-maturity reported date in for full the sample sample and sub maturity term groups Chapter 4. The Volatility Index and Volatility Risk Premium in China 120

4.3.2 Smooth interpolation-extrapolation in the SSE 50 ETF options market

Jiang and Tian(2007) suggest that using smooth interpolation-extrapolation method to increase the number of option strikes, which could improve the accuracy of the model-free approach. Later research adopted the smooth interpolation-extrapolation methodology in equity and index options market with relative limited number of strikes in the option chain to reduce the errors in the volatility index (e.g. Carr and Wu, 2008; Chang et al., 2013a). Practitioners and academics use the IV of an option, which is the volatility calculated from the options’ market price to value options across different moneyness and maturity term. By utilizing an IV function we can calculate the IV for those strikes not listed in the option chain. With IV from IV function we use Black-Scholes formula with an implied forward price to get a virtual option’s premium. Foresi and Wu(2005) and Zhang and Xiang(2008) proposed the methodology to quantify an option’s IV curve by a polynomial function of option’s moneyness, as given by:

2 IV (ξ) = γ0(1 + γ1ξ + γ2ξ ), (4.3.8) where ξ is an option’s moneyness deﬁned as:

ln(K/Ft,T ) ξ ≡ √ , (4.3.9) σ¯ T − t

where Ft,T is the implied forward price from ATM options at time t with maturity T , σ¯ is the interpolated 30-day average ATM IV and K is the option’s strike price. By following Carr and Wu(2008) and Chang et al.(2013a), we create virtual options within the actual strike range (interpolation) and beyond the actual strike range (extrapolation). For virtual options’ strikes within the range, we use the IV function Chapter 4. The Volatility Index and Volatility Risk Premium in China 121

proposed by Zhang and Xiang(2008) in Eq (4.3.8) to calculate virtual options’ IV, while for the virtual options strikes beyond the actual strike, we use the IV of the actual option at boundary, as these virtual options’ IV.10 We create an extra 100 option strikes within the actual strike range, and we delete those virtual options that have the same strikes as the existing options. We then use the SSE 50 ETF’s monthly standard deviation to deﬁne the range for the extra 100 virtual strikes, so that they cover eight standard deviations of underlying asset returns. We report the detailed interpolation-extrapolation methodology in Appendix. We select two random days: 9 February 2015 and 2 December 2015, to illustrate the interpolation-extrapolation methodology. Panels (a) and (c) in Figure 4.3.1 plot the actual IV of options and panels (b) and (d) in Figure 4.3.1 plot the additional virtual IV from the IV function in Eq (4.3.8). With interpolation-extrapolation from the IV function, we use Black-Scholes formula to calculate their option premium and discard options with our data ﬁlter described in Section 2.

10We delete all virtual options with negative IVs in Eq (4.3.8). Chapter 4. The Volatility Index and Volatility Risk Premium in China 122

Figure 4.3.1: Extend the option strikes by the implied volatility function

This ﬁgure reports two sample days’ actual and extended option implied volatility (IV) group (group 1: 9 February 2015 with maturity 25 March 2015 and group 2: 2 December 2015 with maturity 23 March 2016). The extended IV group with an extra 100 extrapolated IVs above and below actual IV boundary and 100 interpolated IVs within actual IV boundary. Chapter 4. The Volatility Index and Volatility Risk Premium in China 123

Figure 4.3.2 plots the density of the actual and extended option dataset by following the option data ﬁlter. The interpolation-extrapolation methodology extended options in the option chain, which could reduce the truncation and discretization errors in the volatility index.

Figure 4.3.2: Extension of out-of-the-money (OTM) dataset

This ﬁgure reports the density of the OTM dataset with implied volatility function. The number of options and their time to maturity are counted at their ﬁrst trading day. Chapter 4. The Volatility Index and Volatility Risk Premium in China 124

We also construct two alternative China volatility indexes, the Black-Scholes model based volatility index (CNVXO) and the volatility index based on SSE 50 ETF options without interpolation-extrapolation (CNVIX R). We construct the CNVXO by following the first version of the CBOE VIX (Whaley, 1993) now named VXO. Whaley’s (1993) methodology requires only eight near-the-money options’ implied volatilities to calculate the volatility index. However, in some extremely volatile trading days, near-the-money put or call options are not available in the option chains, therefore, the index cannot be calculated. For example, on 27 July 2015, the SSE 50 ETF plunged 9.14% and near-the-money put options are not available in the option chains. When we take the mean value of IV with a missing near-the-money put option’s IV on that day, the volatility index can not be calculated. There are only 907 observations in CNVXO and CNVIX R in 922 trading days in our sample. Table 4.3.2 reports the days without CNVXO and CNVIX R in our sample. We also compare the CNVIX with the SSE’s ceased volatility index iVX, and we find that our CNVIX is consistent with the iVX and the correlation between the two index is 0.98 in the period the iVX was reported. Figure 4.3.3 plot the time series of these two volatility indexes in China. We report the correlation table of the China volatility indexes in Table 4.3.3; the correlations among the four China-based volatility indexes are close to 1 and the CNVIX correlation to the U.S.-based China volatility index VXFXI is about 0.77. This verifies, firstly, that our CNVIX is a valid volatility index for China, and secondly, that such an index is necessary, as it portrays different information to the U.S.-based volatility indexes with correlation ranging from 0.45 to 0.77. Chapter 4. The Volatility Index and Volatility Risk Premium in China 125

Table 4.3.2: Trading days without model based volatility index

This table reports the trading days that some volatility indexes in China could not constructed due to limited number of out-of-the-money (OTM) options. The Black-Scholes model based volatility index (CNVXO) and model-free volatility index based on listed options only (CNVIX R) could not be constructed. The Return (%) is the log return of underlying asset, CNVIX is the model-free volatility index based on extended option dataset and RV is the realized volatility based on the log return of the underlying asset.

Date Return (%) CNVIX RV CNVXO CNVIX R 2015/6/26 -7.98 63.81 50.96 - - 2015/7/8 -6.67 67.36 63.80 - - 2015/7/27 -9.14 37.55 66.05 - - 2015/7/30 -2.57 43.26 60.38 - - 2015/8/20 -2.72 40.92 46.72 - - 2015/8/21 -3.68 39.49 48.12 - - 2015/8/24 -9.98 68.74 50.41 - - 2015/8/25 -7.87 82.58 57.89 - - 2015/11/27 -5.27 33.51 29.88 - - 2015/12/28 -2.98 26.28 28.66 - - 2016/1/4 -5.71 32.60 29.19 - - 2016/1/7 -6.04 37.03 36.49 - - 2016/1/11 -4.24 39.80 39.95 - - 2016/1/26 -5.18 35.85 43.37 - - 2016/2/25 -5.05 32.48 33.91 - -

4.4 Empirical results

In section 4.1 we analyze the relation between the CNVIX and the underlying asset. Section 4.2 documents the VRP in China, and Section 4.3 analyzes the predictability of VRP to the return of the underlying market. Chapter 4. The Volatility Index and Volatility Risk Premium in China 126

Figure 4.3.3: Time series plot of the CNVIX and iVX

This ﬁgure plots the time series of the CNVIX and SSE iVX from 9 February 2015 to 20 November 2018. The SSE iVX was published on 26 November 2016 and closed on 22 February 2018.

CNVIX and iVX 90 CNVIX 80 iVX

Volatility index level 30

0 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Date

4.4.1 Relationship of the model-free volatility index and the underlying asset

Figure 4.4.1 shows the time series of the CNVIX index and the underlying asset SSE 50 ETF from 9 February 2015 to 20 November 2018. From Figure 4.4.1, we can see that the CNVIX does usually move in the opposite direction to the underlying, and sometimes the CNVIX moves together with the underlying. There is a negative correlation between the CNVIX index level and SSE 50 ETF price level; when the CNVIX rises, the SSE 50 ETF falls, and vice versa. The negative correlation between the level of volatility index and the underlying could provide trading signals to investors. Chapter 4. The Volatility Index and Volatility Risk Premium in China 127

Table 4.3.3: The correlations of volatility indexes

Panel A in this table reports the correlations of time series of the daily closing price of volatility indexes: The China Volatility Index (CNVIX) we constructed in this paper, CBOE Volatility Index (VIX), CBOE S&P 100 Volatility Index (VXO), CBOE NASDAQ Volatility Index (VXN), CBOE DJIA Volatility Index (VXD) and CBOE China ETF Volatility Index (VXFXI). The sample period is from 9 February 2015 to 20 November 2018, with 922 observations in the CNVIX and 955 observations in the U.S. volatility indexes. Panel B in this table reports the correlations of the daily closing price time series of volatility indexes in China: The China volatility index (CNVIX) , the SSE volatility index (iVX), the China volatility index we constructed with listed option data only (CNVIX R), and the China model-based volatility index (CNVXO) we constructed from Black-Scholes implied volatilities. The sample period is from 9 Feb 2015 to 20 Nov 2018, there are 922 observations in CNVIX, 907 observations in CNVXO, 907 observations in CNVIX R and 296 observations in iVX (The SSE iVX was published on 26 November 2016 and closed on 22 February 2018).

CNVIX VIX VXO VXN VXD VXFXI Panel A: CNVIX’s correlation with major U.S. volatility indexes CNVIX 1.00 VIX 0.51 1.00 VXO 0.56 0.98 1.00 VXN 0.45 0.92 0.91 1.00 VXD 0.55 0.97 0.97 0.94 1.00 VXFXI 0.77 0.80 0.84 0.73 0.81 1.00 Panel B: CNVIX’s correlation with volatility indexes in China CNVIX iVX CNVIX R CNVXO CNVIX 1.00 iVX 0.98 1.00 CNVIX R 0.98 0.96 1.00 CNVXO 0.99 0.98 0.96 1.00 Chapter 4. The Volatility Index and Volatility Risk Premium in China 128

The fact that the CNVIX spikes during a market crash means it could become useful as a market barometer or “investor fear gauge”. From Figure 4.4.1, two sharp spikes occurred in the market crash in June 2015 and January 2016, the SSE 50 ETF declined about 33% and 24% in a month; however, the CNVIX reached to about 74% and 81% respectively. The CNVIX spike in January 2016; followed a market crisis from the induction of the ’Circuit Breaker’ meltdown trading mechanism in China. After the crash, the CNVIX decreased, and moves sideways around 18%, and the VIX’s mean level is 15% (from 2015 to 2018).

Figure 4.4.1: Time series plot of the CNVIX and the SSE 50 ETF

This ﬁgure displays the time series plot of the CNVIX and the SSE 50 ETF.

CNVIX and the SSE 50 ETF 100 3.6

CNVIX 90 2015 Financial 3.4 SSE 50 ETF Crisis 80 3.2

70 2018 US China Tradewar 3

60 2.8 2016 'Circuit Breaker' halt trading 50 Financial Crisis 2.6 SSE 50 ETF 40 CNVIX Volatility Index 2.4 30

2.2 20

10 2

0 1.8 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Chapter 4. The Volatility Index and Volatility Risk Premium in China 129

Volatility plays a vital role in asset pricing. The leverage effect in finance refers to the negative relationship between the volatility and equity’s return. The leverage effect was first discussed by Black(1976) and Christie(1982) and refers to companies’ financial leverage (debt-to-equity ratio) when equity prices decline the companies’ financial leverage will increase correspondingly. Schwert(1989) shows that volatility is an increasing function of leverage. Investors expect the stocks to become more volatile with the increase in the financial leverage, they close their position in the stocks, which leads to the negative relationship between volatility and return. How- ever, Schwert(1989) shows that the leverage itself is not sufficient for the negative relation to the volatility. Furthermore, the negative relation between volatility and equity return is asymmetric, which means with same magnitude of changes in positive and negative returns of underlying asset the negative returns have large impact on the volatility. Another explanation for the asymmetric relation is the volatility feedback effect. If the volatility is priced by the market, an increase in volatility is associated with the future expectation of returns, which results in a further decrease in the stock price (e.g. French, Schwert, and Stambaugh, 1987; Campbell and Hentschel, 1992; Fleming et al., 1995; Whaley, 2000; Bekaert and Wu, 2000; Wu, 2001). Previous studies show that there is negative relationship between volatility and equity returns (e.g. French et al., 1987; Campbell and Hentschel, 1992; Fleming et al., 1995; Bekaert and Wu, 2000; Bartram, Brown, and Waller, 2015; Choi and Richard- son, 2016). The volatility estimated with options exhibit a more pronounced asymmetry leverage effect (e.g. Bates, 2000, Whaley, 2000, Eraker, 2004; Whaley, 2009). In order to document the time series behavior of the leverage effect in the model- free volatility index in China we inspect the volatility indexes and their underlying return by splitting the positive and negative returns of their underlying asset. We Chapter 4. The Volatility Index and Volatility Risk Premium in China 130

use the following regression which can capture the asymmetric leverage effect in the index level and return:

+ − Indext = α + β0Indext−1 + β1Rt + β2Rt + t, (4.4.1)

+ − ∆Indext = α + β1Rt + β2Rt + t, (4.4.2) where Indext is the closing value of the volatility index at time t, Indext−1 is the

+ − lagged index. ∆Indext is the log return of volatility index. R (R ) is the positive (negative) return of underlying asset which is calculated from:

+ − Rt = max(Rt, 0),Rt = |min(Rt, 0)|,

where Rt is the daily log return of the underlying asset. If the leverage effect existed in the model-free volatility indexes, we should observe the negative (positive) value of β1 (β2) and in terms of an asymmetric relationship, and β2 should be higher than

β1. We test the stationarity of CNVIX, VXFXI and their log returns, we ﬁnd that both the index levels are not stationary, therefore, we use the index returns, which are stationary in our analysis. We would expect the volatility index to show a mean- reverting pattern, which is stationary, however, our sample is too short to catch the long cycle of volatility. 11 Fleming et al.(1995) ﬁnd that negative returns of the underlying are associated with larger changes in the index compared with positive returns. In order to inspect the asymmetric relationship between the volatility index and return, we report the

11The results for Eq (4.4.1) are not reproduced in the paper, as they are non-stationary, although practitioners use the volatility index level as one of their major technical indicator in trading. We ﬁnd the index level regression results also exhibit a similar leverage effect in the index return regression; they are available upon request. Chapter 4. The Volatility Index and Volatility Risk Premium in China 131

regression by the period before the 2015 financial crisis (9 February 2015 to 15 June 2015), during the crisis (15 June 2015 to 31 August 2015) and after the crisis (after 31 August 2015). Table 4.4.1 reports the regression results for the leverage effect. From the regression in panel A, we find that the return of the SSE 50 ETF has a significant positive β2 coefficient for the full sample and sub samples, except for the early stage since the inception of the option market. While the positive β1 is not significant in the full sample, it is smaller than β2, which indicates that the negative return of the underlying has a larger magnitude of impact on the volatility index. The negative relationship becomes more pronounced after the financial crisis in 2015. We have a similar finding in panel B when we model the volatility index with VXFXI. The VXFXI index return is negatively correlated to the underlying return, and the larger coefficient β2 of the negative return indicates the leverage effect existed in the VXFXI as well. The leverage effect is more pronounced for the underlying return during the financial crisis in 2015.

4.4.2 The asymmetric return-volatility relationship in the CNVIX and the SSE 50 ETF

Badshah(2013) implements the quantile regression to inspect the asymmetric return- volatility relation in several stock indexes. He ﬁnds that strong negative asymmetric return-volatility relation existed in stock index and volatility, and he ﬁnds that OLS (ordinary-least-square) regression underestimates (overestimates) the relation in the positive (negative) change of stock indexes. Chapter 4. The Volatility Index and Volatility Risk Premium in China 132

Table 4.4.1: Leverage effect regression of CNVIX and VXFXI

Table 4.4.1 reports the leverage effect regression for the index return in Eq (4.4.2) from 9 February 2015 to 20 November 2018 to get the slope coefﬁcients and their Newey and West(1987) t-statistics are reported in the parenthesis below. *,** and *** indicate signiﬁcance at the 10%, 5% and 1% levels.

Regression Coefﬁcients

Obs Constant R+ R− Adj R2 (%) Panel A: CNVIX index

∆CNVIX Full Sample 921 -1.27*** 0.16 2.26*** 14.25 (-4.31) (0.46) (5.88) Feb 2015 - Jun 2015 82 0.54 0.31 -0.59 0.79 (0.47) (0.75) (-0.95) Jun 2015 - Aug 2015 55 -3.80* -0.79 2.72*** 46.92 (-1.86) (-1.45) (3.42) Aug 2015 - Nov 2018 784 -1.63*** 0.84* 2.80*** 12.44 (-5.47) (1.82) (6.81)

Panel B: VXFXI index

∆VXFXI Full Sample 954 -1.60*** -0.46 3.43*** 37.19 (-6.61) (-1.27) (14.01) Feb 2015 - Jun 2015 86 -2.82*** 3.28*** 1.67*** 44.38 (-5.62) (6.42) (6.26) Jun 2015 - Aug 2015 55 -3.70*** -1.24*** 4.29*** 73.66 (-3.63) (-3.13) (6.24) Aug 2015 - Nov 2018 813 -1.29*** -0.94*** 3.45*** 39.23 (-6.30) (-4.29) (14.46) Chapter 4. The Volatility Index and Volatility Risk Premium in China 133

Our paper build on the previous literature by implementing quantile regression on the model-free volatility index and its undelrying asset return to inspect asymmetric return volatility relationship in China. Table 4.4.2 reports the quantile repres- sion in Eq 4.4.2 for the daily asymmetric relation between CNVIX changes and SSE 50 ETF returns. We run 9 sets of quantile (0.1, 0.2,... ,0.9) regressions to inspect the negative and positive returns’ coefficient estimate. By analysing the return to CN- VIX’s change across different quantiles, which will provide a more complete analysis of asymmetric-volatility relationship. The quantile regression results are presented in Table 4.4.2, while Figure 4.4.2 shows the positive and negative returns’ response to CNVIX changes across quantiles. Figure 4.4.2 plots the contemporaneous positive (sub-figure a) and negative (sub-figure b) response to the CNVIX change as the red dashed line. The OLS regression estimate are plotted as blue solid straight line across different quantiles. The x-axis indicates the quantile number, and the y-axis indicates the returns response to the change of CNVIX. In the OLS regression we find negative return’s response to CNVIX’s change is around 2.2 which is about 10 times larger than the response from positive return. The positive return negatively response to the change of CN- VIX from lower quantiles to medium quantile and positively response to the change of CNVIX in upper quantiles. The negative return positively response to the change of CNVIX across all the quantiles and the response coefficients are larger than the positive return’s response coefficients in upper quantiles above the median. Table 4.4.2 reports the quintile regression result that analysis the asymmetric return-volatility relationship in the model-free volatility index and return in China. The results are different from the OLS regression in Table 4.4.1. The response coefficients of both positive and negative return in medium quantile are close to the OLS regression coefficient. The response coefficients of the positive returns are significantly negative in first two lower quantiles and significantly positive in last two Chapter 4. The Volatility Index and Volatility Risk Premium in China 134

upper quantiles. While in the OLS regression the positive return’s response coefficient is 0.16 and not significant. The response coefficients of negative returns’ are positive across all quintiles and significant from the third quantiles. The results suggest that the negative returns are positive correlated with the change of CNVIX from the third quantiles. The positive returns’ response coefficients are significant negative in first lower quantiles and significant positive in last upper quantiles, and not significant in the middle quantiles. The larger and significant coefficients of negative returns in upper quintiles provide evidence of asymmetric return-volatility in China.

4.4.3 CNVIX-based VRP

Bakshi and Kapadia(2003a) and Bakshi and Kapadia(2003b) document the negative market volatility risk premium in index options and equity options, which indicates that the implied volatilities are higher than realized volatilities. Previous studies document that variables extracted from options’ implied volatility, for example, the volatility spread or the volatility smirk, can predict the return of the underlying asset (e.g. Bakshi and Madan, 2006; Bali and Hovakimian, 2009; Xing et al., 2010; DeMiguel, Plyakha, Uppal, and Vilkov, 2013). The option-price-implied information in these variables captures information not fully reﬂected in the market price of the underlying asset. Chapter 4. The Volatility Index and Volatility Risk Premium in China 135

Figure 4.4.2: Quantile regression estimate across quantiles: response variable to CNVIX changes

Figure 4.4.2 reports the quantile regression and OLS (Ordinary least squares) estimate of the CNVIX index return in Eq (4.4.2) from 9 February 2015 to 20 November 2018. Quantile 0.1 includes the smallest CNVIX index change, whereas quantile 0.9 includes the largest CNVIX index change.

CNVIX responses to the positive returns 4 OLS Regression 3.5 Quantile Regression 3

2.5

1.5

0.5 Response coefficient 0

-0.5

-1

-1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Quantile (a) CNVIX changes’ response to positive returns

CNVIX responses to the negative returns 4 OLS Regression 3.5 Quantile Regression 3

2.5

1.5

0.5 Response coefficient 0

-0.5

-1

-1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Quantile (b) CNVIX changes’ response to positive returns Chapter 4. The Volatility Index and Volatility Risk Premium in China 136 Leverage effect: quantile regression analysis Table 4.4.2: 0.44 0.63 1.38** 1.88*** 2.02*** 2.16*** 2.57*** 2.76*** 3.73* (1.27) (1.49) (2.15) (7.18) (9.22) (9.03) (4.68) (10.04) (1.71) Q(0.1) Q(0.2) Q(0.3) Q(0.4) Q(0.5) Q(0.6) Q(0.7) Q(0.8) Q(0.9) (-3.30) (-1.93) (-1.33) (-0.42) (0.88) (1.36) (1.37) (2.20) (1.99) (-13.47) (-10.97) (-8.67) (-6.90) (-5.59) (-1.60) (1.88) (5.51) (3.72) -1.26*** -1.18* -0.45 -0.23 0.16 0.24 0.43 1.04** 2.34** (%) 3.89 5.69 13.15 14.08 14.44 14.42 14.33 13.40 11.28 2 R + − Constant -0.06***R -0.04*** -0.03*** -0.02*** -0.01*** -0.00 0.01* 0.02*** 0.03*** R ObsPseudo 921 921 921 921 921 921 921 921 921 Table 4.4.2 reports theNovember leverage 2018. effect Quantile Q(0.1) quantile includesvolatility regression the index smallest for change. volatility the The index standard change, indextheses errors whereas return are below) quintile obtained Q(0.9) in are using includes computed Eq the thelevels. bootstrap for ( 4.4.2 ) largest method; each from and of the 9 robust the February t-statistics quantile (in 2015 estimates. paren- to *,** 20 and *** indicate signiﬁcance at the 10%, 5% and 1% Chapter 4. The Volatility Index and Volatility Risk Premium in China 137

Recent studies have argued that the aggregate stock return can be predicted by VRP over horizons ranging up to two quarters (e.g. Bollerslev, Tauchen, and Zhou, 2009; Drechsler and Yaron, 2010; Drechsler, 2013). Following the standard approach, we deﬁne the VRP as the difference of the model-free CNVIX, the ex ante risk neutral expectation of future return volatility over the next 30-day (t,t+30) time interval and the realized volatility (RV) over the previous 30-day (t-30,t) time interval (e.g. Bollerslev et al., 2009; Drechsler, 2013):

VRPt = CNVIXt − RVt, (4.4.3)

q 242 P20 2 where the RVt = 100 × 20 i=1 ri is the annualized 30-day (20 trading-day) realized volatility and r = ln Pi , which is the log return of the underlying asset i Pi−1 12 price (Pi) at day i. Figure 4.4.3 plots the time series of model-free VRP calculated from CNVIX. 12Depending the Chinese public holidays the stock exchange will adjust the trading days in China, the average number of trading days is 242.2 (2010 to 2015). Chapter 4. The Volatility Index and Volatility Risk Premium in China 138

Figure 4.4.3: Time series plot of the CNVIX, realized volatility (RV) and volatility risk premium (VRP) of the SSE 50 ETF

This ﬁgure displays the time series of the RV and CNVIX in China. The monthly (20-trading-days) RV is calculated by: v u 20 u240 X RV = 100 × t R2, 20 t t=1

R P R = ln Pt where the t is the log return of the underlying asset’s closing price t: t Pt−1 . The VRP is deﬁned as: VRPt = CNVIXt − RVt.

CNVIX SSE 50 RV and VRP 100 CNVIX SSE 50 RV VRP 80

20 Index level /%

-20

-40 Jan 2015 Jul 2015 Jan 2016 Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018 Jan 2019 Date Chapter 4. The Volatility Index and Volatility Risk Premium in China 139

Economically, the VRP can be viewed as the compensation paid in the SSE 50 ETF options market for selling insurance when delta-hedging with volatility risk exposure left. Investors could implement short (long) option delta-hedge portfolio to harvest volatility premium when positive (negative) VRP observed in the market. We calculate the VRP in China from Eq (4.4.3) and Table 4.4.3 reports the summary statistics variables in our VRP analysis. To ensure the prediction results are not affected by the model, we construct three control variables by following Bali and Hovakimian(2009) and Bollerslev et al.(2009). Firstly, the default spread (DFSP), which is defined by the difference of yields between the one-year BBB China Bond Index rated corporate bond and one-year AAA China Bond Index rated corporate bond. Secondly, the term spread (TERM), which is defined as the difference between the 10-year China Treasury Bond and one-month treasury bill. Lastly, the stochas- tically detrended risk-free rate (RREL), which is defined as the yield of one-month China treasury bill minus its one-year backward moving average. 13 We find that the average VRP for the CNVIX is 2.28. The mean of the implied and realized volatilities are 23.69 and 21.41. The positive mean VRP in the Chinese options market is consistent with the previous studies in the U.S. options market (e.g. Carr and Wu, 2008; Bollerslev et al., 2009). All the variables are highly persistent with the first order autocorrelations except the daily return. Prior to the regression analysis, we test whether or not the time series of regressors are stationary from Augmented Dickey-Fuller (ADF) test. We find that all the variables are stationary except DFSP.

13The yield curve information is downloaded from Wind Financial Terminal and China Central Depository & Clearing Co.,Ltd. Chapter 4. The Volatility Index and Volatility Risk Premium in China 140 . The *,** and *** denote ADF-test statistics and n 100 × 12 4 q = lags Summary statistics of variables in VRP regression Table 4.4.3: mean min max sd skew kurt p25 p50 p75 AR(1) ADF VRPRVCNVIXDaily Return (%)Weekly Return (%) 2.28Two-Weekly Return (%) 0.01 0.09 0.04Monthly 21.41 -32.66 23.69 Return (%) -9.98 24.69TERM -30.71 5.41 (%) -25.22 8.11 0.19 8.09 15.81DFSP 6.54 15.81 (%) 69.98 82.58 -29.53RREL 4.59 1.62 (%) 3.17 -1.52 13.73 12.26 21.88 -1.32 -0.81 -1.09 9.52 0.93 1.52 1.41 6.51 9.03 10.34 11.18 10.40 0.42 -0.35 5.19 5.48 -0.63 -0.57 -0.31 -1.20 8.83 -1.67 2.01 2.81 11.80 14.70 5.22 0.04 -1.13 0.18 0.42 12.14 16.65 0.41 20.61 4.93 -2.50 0.75 0.64 1.72 0.92 2.51 27.84 29.81 0.28 0.93 0.84 0.32 0.99 0.03 0.76 0.29 0.98 0.88 3.17 -5.99*** 3.83 -0.15 -3.04 -14.73*** -16.28*** 2.22 -2.78 -8.41*** 0.72 0.94 3.20 9.80 0.86 -4.75*** -0.52 10.30 1.16 -0.25 10.90 0.98 -0.13 0.99 0.97 -4.16*** -1.40 -6.28*** This table reports the summary statisticsand for control VRP, variables: RV, CNVIX, default overlapping return spreadthe of (DFSP), mean, the term minimum SSE (min), spread 50 maximum (TERM) ETF (max), andcentile standard from (p75), detrended deviations daily skewness risk-free (sd), to (skew) 25th rate Monthly , percentile (RREL). kurtosis (p25),2018. This (kurt) 50th and table percentile Augmented ﬁrst-order (p50), reports auto-correlation Dickey 75th AR(1) per- Fuller from 9 (ADF)based February test 2015 on statistics to 20 sample are November size reported for accordingindicate stationary stationary to at test.Schwert ( 2002 ): 10%, 5% We select and 1% the critical lags level. of stationary test Chapter 4. The Volatility Index and Volatility Risk Premium in China 141 TERM RREL DFSP Monthly Return Two- Weekly Return Weekly Return Daily Return Correlation table of variables in VRP regression Table 4.4.4: VRP RV CNVIX VRPRVCNVIXDaily Return 1.00 Weekly Return -0.05Two-Week Return 0.03 -0.45 -0.03 0.00Monthly Return -0.06 1.00 0.88 -0.15TERM 0.31 -0.26RREL -0.09 -0.18 1.00 -0.43 -0.30DFSP 1.00 0.48 -0.08 -0.32 0.32 0.10 1.00 0.39 0.22 0.63 0.01 0.19 0.40 0.43 1.00 -0.39 0.27 -0.06 0.67 -0.43 -0.01 -0.12 1.00 -0.01 -0.02 -0.16 -0.02 -0.05 -0.20 -0.03 -0.06 1.00 -0.04 -0.58 0.23 1.00 -0.50 1.00 This table reports thecontrol correlation variables: of default spread VRP, (DFSP), RV, CNVIX, term20 spread November overlapping (TERM) 2018. return and detrended of risk-free the rate (RREL) SSE from 50 9 February ETF 2015 from to daily to Monthly and Chapter 4. The Volatility Index and Volatility Risk Premium in China 142

Table 4.4.4 reports the correlation between the VRP and overlapping return from daily to monthly. The VRP is correlated with the return (daily to monthly), ranging from -0.05 to 0.31.

4.4.4 VRP and return predictability in China

Since the SSE 50 ETF option launched only four years ago, we have a limited sample for quarterly analysis like Bollerslev et al.(2009). We implement our forecast analysis in daily, weekly, two-weekly and monthly horizons with the 30-day VRP in the following regression:

3 X rt+1 = α + βV RP30,t + θjyj,t + t+1, (4.4.4) j=1 where rt+1 is the daily, weekly, two-weekly and monthly overlapping and non- overlapping returns of the underlying ETF at time t + 1. The overlapping returns are calculated based on intersected horizon windows, while the non-overlapping returns are calculated based on non-intersected horizon windows. The independent variables are VRP30,t, which is the 30-day VRP at time t; yj,t is the control variable j at time t and t+1 is the residual term.

4.4.4.1 Model-free VRP measure

We run the univarite regression of Eq (4.4.4) in different horizons to get the slope coefﬁcients and their robust t-statistics and in sample adjusted R2. We present the result of the regression in Eq (4.4.4) of the future overlapping and non-overlapping returns with different horizons in Table 4.4.5 and Table 4.4.6. Chapter 4. The Volatility Index and Volatility Risk Premium in China 143

Table 4.4.5: Return (overlapping) and VRP predictability from SSE 50 ETF options

This table reports the multivariate regressions of CNVIX-based VRP and three control variables: default spread (DFSP), term spread (TERM) and detrended risk-free rate (RREL) on overlapping returns of the SSE 50 ETF from daily to monthly in Eq (4.4.4) from 9 February 2015 to 20 November 2018, and their robust Hodrick(1992) t-statistics account for the overlapping are reported in the parenthesis. The *,** and *** denote 10%, 5% and 1% signiﬁcance level.

(1) (2) (3) (4) (5) (6) (7) (8)

Daily Return Weekly Return Two-Weekly Return Monthly Return

Constant -0.03 0.98 -0.11 2.82 -0.17 4.50 0.00 8.47

(-0.46) (1.58) (-0.48) (1.29) (-0.41) (1.05) (0.00) (1.27)

VRP 0.02* 0.02* 0.06 0.06 0.11* 0.10* 0.07 0.05

(1.70) (1.79) (1.47) (1.57) (1.65) (1.73) (0.55) (0.53)

TERM -0.45** -1.72** -3.97*** -9.13***

(-2.23) (-2.19) (-2.66) (-5.41)

DFSP -0.07 -0.18 -0.17 -0.16

(-1.22) (-0.83) (-0.41) (-0.26)

RREL -0.57** -1.57* -2.53 -5.31***

(-2.04) (-1.68) (-1.58) (-2.90)

Obs 922 922 918 918 913 913 903 903

Adj R2 (%) 0.35 1.02 1.53 4.54 2.21 10.42 0.38 22.64 Chapter 4. The Volatility Index and Volatility Risk Premium in China 144

Comparing columns (1) to (8) in Table 4.4.5, we find that the VRP can significantly predict the future daily and two-weekly overlapping returns of the SSE 50 ETF. When the VRP increase 1% the expected one-day ahead return on the SSE 50 ETF increases by 0.02%. The t-statistic of the VRP slope coefficient is 1.70, with a statistical significance at the 10% level and the adjusted R2 is only 0.35%. For the two-weekly ahead return of the SSE 50 ETF in column (5), we find that the VRP has a large slope coefficient of 0.11 and significant t-statistic of 1.65 with adjusted R2 of 2.21%. We find that the VRP can significantly predict the future daily and two-weekly returns, using the overlapping return. In order to verify that our results are not affected by model misspecification, we add a set of control variables, DFSP, TERM and RREL into the regression. All of our t-statistics report in the parenthesis below the coefficients are based on heteroskedasticity and serial correlation consistent standard errors that taking account of the overlapping effect by following Hodrick (1992). We inspect the VRP’s predictability in non-overlapping horizons in Table 4.4.6. From columns (1) to (8) in Table 4.4.6, we find that the VRP can significantly predict the future daily, weekly and two-weekly non-overlapping returns on the SSE 50 ETF. In column (3), we find that the slope coefficient of VRP is 0.08 and significant at 10% level with adjusted R2 of 2.51%. While in overlapping sample, the VRP does not have significant predictive power over ahead weekly return. In column (5), we find that the slope coefficient of VRP is 0.16 and significant at 5% level with adjusted R2 of 4.72%. The VRP could significantly predict the future weekly and two-weekly returns, using the non-overlapping returns. The degree of VRP’s predictability on non- overalpping weekly return is 0.08 and it is significant at 10% significance level with adjust R2 of 2.51%. The two-weekly regression indicates the VRP has predictive Chapter 4. The Volatility Index and Volatility Risk Premium in China 145

Table 4.4.6: Return (non-overlapping) and VRP predictability from SSE 50 ETF options

This table reports the multivariate regressions of CNVIX-based VRP and three control variables: default spread (DFSP), term spread (Term) and detrended risk-free rate (RREL) on non-overlapping returns of the SSE 50 ETF from weekly to monthly in Eq (4.4.4) from 9 February 2015 to 20 November 2018 and their Newey and West (1987) t-statistics are reported in the parenthesis. The *,** and *** denote 10%, 5% and 1% signiﬁcance level.

(1) (2) (3) (4) (5) (6) (7) (8)

Daily Return Weekly Return Two-Weekly Return Monthly Return

Constant -0.03 0.98 -0.25 2.81 -0.38 7.38 0.04 15.88

(-0.46) (1.58) (-1.00) (1.16) (-0.67) (1.26) (0.03) (1.18)

VRP 0.02* 0.02* 0.08* 0.09** 0.16** 0.16** -0.05 -0.01

(1.70) (1.79) (1.97) (2.24) (2.16) (2.42) (-0.28) (-0.06)

TERM -0.45** -1.91*** -4.05** -12.23***

(-2.23) (-2.61) (-2.54) (-2.89)

DFSP -0.07 -0.20 -0.54 -0.69

(-1.22) (-0.84) (-1.05) (-0.62)

RREL -0.57** -2.50*** -4.95** -7.84

(-2.04) (-2.70) (-2.34) (-1.49)

Obs 922 922 183 183 91 91 45 45

Adj R2 (%) 0.35 1.02 2.51 5.69 4.72 13.42 -2.23 24.30 Chapter 4. The Volatility Index and Volatility Risk Premium in China 146

power of 0.16 and it significant at 5% significant level with adjusted R2 of 4.72%. While the VRP could not forecast the non-overlapping return on monthly horizon. By following Bollerslev et al.(2009), we include the three control variables: DFSP, TERM and RREL to the multiple regressions, we find that the all the significant slope coefficients of VRP in univariate regression remain significant and their adjusted R2 and t-statistics increased slightly. After controlling for the macroeconomic variables, the slope coefficient only the slope coefficient of VRP in the regression over two- weekly overlapping return decreased slightly (from 0.11 to 0.10). The increased adjusted R2 in multivariate regression indicate that the set of macroeconomic variables have some information about the future return, especially, the TERM factors, which are significant across different return horizons. In summary, the model-free VRP in China can significantly predict daily and two-weekly overlapping return, weekly and two-weekly non-overleapping return. Our empirical finding suggest that the higher (lower) model-free VRP are associated with higher (lower) future return.

4.4.4.2 Alternative model-based VRP measure

Bollerslev et al.(2009) ﬁnd that the model-free variance measure contains more effective information on variance risk premium than the “old” Black-Scholes model based measure. By following Bollerslev et al.(2009), we also implemented the regression on the VRP calculated from Black-Scholes model-based volatility index CN- VXO. The results indicate that the model-based VRP also have predictive power on the return of underlying asset and the models’ adjusted R2 are slightly higher than the model-free volatility index based VRP. However, the two volatility indexes have different number of observations in our sample period and CNVXO is not available in some volatile trading days without ATM call or put options when market close (those days are reported in Table 4.3.2). We could not draw the conclusion whether Chapter 4. The Volatility Index and Volatility Risk Premium in China 147

the VRP based on model-free methodology has more information on future return than the Black-Scholes model-based VRP measure. For brevity, we did not report the analysis on CNVXO based VRP analysis in our paper and they are available upon request.

4.5 Conclusion

In this paper, we introduce the model-free volatility index in China, the CNVIX, based on the newly established SSE 50 ETF options market and analyze its properties from 9 February 2015 to 20 November 2018. We interpolate-extrapolate the option dataset with the IV function, which reduces the truncation and discretization errors in construction of the volatility index. Throughout the paper, we study the properties of the CNVIX, CBOE VXFXI and show that the volatility indexes share some characteristics, such as being negatively correlated with the underlying return. We further analyze the leverage effect between the volatility indexes and their underlying asset. We find a significant asymmetric leverage effect between the volatility indexes and their underlying returns. In the end, we analyze the model-free VRP in China. we find that the VRP can forecast the overalpping return of the underlying over daily and two-weely horizon. The results are consistent with previous literature on developed option markets, which confirmed that the model-free methodology is also applicable to emerging and less liquid option markets. Chapter 4. The Volatility Index and Volatility Risk Premium in China 148

4.6 Appendix

4.6.1 Interpolation-extrapolation extended option dataset in less liquid options market

We use interpolation-extrapolation methodology to extend the option dataset in less liquid options market. We calculate the option’s IV in interpolation by following Zhang and Xiang’s 2008 methodology. We use actual boundary IV as the IV in ﬂat extrapolation. We have the following detailed steps in the following subsections.

4.6.1.1 Calculate the implied forward price of the underlying asset

First, we use the European style put-call parity to calculate the options’ implied forward price: r(T −t) Ft,T = K + e (Ct,T − Pt,T ), (4.6.1)

where Ft,T is the option’s implied forward price at time t with maturity T , K is the option’s strike price, r is risk-free rate and Ct,T and Pt,T are the call and put option’s premium at time t with maturity T . The strike and option premiums are selected from put-call option pairs which have the minimum absolute difference in option premium.

4.6.1.2 Calculate the implied volatility (IV) of option

We use the Black-Scholes model with the implied forward as the underlying asset to calculate a virtual option’s premium.

−rt,T (T −t) −rt,T (T −t) ct = Ft,T e N(d1) − Ke N(d2), (4.6.2) −rt,T (T −t) −rt,T (T −t) pt = Ke N(−d2) − Ft,T e N(−d1), Chapter 4. The Volatility Index and Volatility Risk Premium in China 149

where Ft,T σ2 √ ln ( K ) + ( 2 )(T − t) d1 = √ , d2 = d1 − σ T − t, σ T − t where t is the current time, T is the maturity date, Ft,T denotes the implied forward price of the SSE 50 ETF option at time t with maturity T , K is the option’s strike price, rt,T is the interpolated risk free from the term structure of SHIBOR rate at current time t with maturity date T , and N(·) is the standard cumulative normal distribution. The SSE 50 ETF will pay discrete dividend for index tracking adjustments. We use implied forward price from put-call parity, which include investors’ expectation of future dividend payments to approximate the random discrete dividend payments in the underlying asset. Then we use the Black-Scholes mode to calculate an option’s IV.

4.6.1.3 Construct the IV curve function

We use the actual options’ IV to calculate the quantiﬁed level, slope and curvature in the IV curve function for each group of options (same trading day and maturity day) by following Zhang and Xiang(2008):

2 IV (ξ) = γ0(1 + γ1ξ + γ2ξ ), (4.6.3) where ξ is an option’s moneyness, deﬁned as the logarithm of the option’s strike divided by the implied forward price and normalized by the average IV of 30-day ATM options (σ¯): ln(K/Ft,T ) ξ ≡ √ . (4.6.4) σ¯ T − t

4.6.1.4 The interpolation within the option chain’s strike range

We use the Eq (4.6.3) to calculate the IV for the extra 100 virtual options interpolated within the strike range of each option chains. If a virtual option’s strike duplicated Chapter 4. The Volatility Index and Volatility Risk Premium in China 150

with any strikes listed in the option chain, it will be discarded. Virtual options with negative IVs calculated from Eq (4.6.3) will be discarded as well.

4.6.1.5 The extrapolation beyond the option chain’s strike range

We calculate the underlying asset’s previous 30-day standard deviation and extract the upward and downward boundary IV from each option chains. Then we examine whether the actual strike range in the option chain could cover a +\- 8 standard deviations price range of the underlying asset. If the actual strike range in option chain could cover the volatility range in a +\− 8 standard deviations of the underlying asset, extrapolation is not necessary. While if the actual strike range could not cover a +\- 8 standard deviations price range of the underlying asset, extrapolation is necessary. We use ﬂat extrapolation to create the extra 100 virtual options with strike above (below) the upper (lower) strike boundary for each group of options. We use the upper and lower boundary’s IV in the option chain as the constant IV (ﬂat-extrapolation). Lastly, we use the Black-Scholes model to calculate the virtual options’ premium in each option groups.14 The extended virtual options dataset is combined with the actual options dataset for the construction of the volatility index. Figure (4.3.1) illustrate how we extend option dataset based on two sample days.

14The Black-Scholes model is not the true option pricing model and it is merely used as a tool to provide transfer of IV to European style SSE 50 ETF option premium. 151

Chapter 5

Conclusion

This thesis studies the newly established SSE 50 ETF option market in China. In Chapter2, we find that the one-dimensional diffusion model does not apply to the SSE 50 ETF option market. The non-zero delta-hedged gain in the option sample indicates that the option traders in the Chinese option market are trading at some other risk factors related to volatility rather than the underlying risk only. In comparison with mature option markets such as the CBOE SPX option, the option market in China is restricted to domestic investors and has less efficiency. In Chapter3, we document the implied volatility curve in the SSE 50 ETF option market. We use the methodology of Zhang and Xiang(2008) to quantify the implied volatility curves of the SSE 50 ETF options. We find that the SSE 50 ETF option implied volatility curve is right-skewed, which is different to the usual left-skewed curves in other option markets. The right-skewed implied volatility curve indicates that the option investor is willing to pay a higher premium for the upward potential of the underlying asset on call options. We further analyze the several investor sentiment proxies’ impact on the implied volatility curves and find that liquidity-based proxies contribute to the right-skewed implied volatility curve in China. Lastly, in Chapter4 we construct a model-free volatility index in China (CNVIX) with the SSE 50 ETF option. We evaluate the CNVIX for its negative correlation with Chapter 5. Conclusion 152

the underlying asset and further analyze the leverage effect between the volatility indexes and their underlying returns. We find a significant asymmetric leverage effect in the CNVIX and the SSE 50 ETF returns. We also undertake a further analysis on the forecastability of the volatility risk premium (VRP) based on the model-free and Black-Scholes model, and find that the VRP could forecast the return of the underlying over daily and two-weekly overlapping horizon. 153

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