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The asymmetric and negative relationship between stock returns and

implied volatility: A behavioral explanation for European stocks.

Kit Yin Liu

10466460

University of Amsterdam, Amsterdam Business School

MSc. Finance, Duisenberg Honours Programme in Corporate Finance & Banking

July 2018

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Statement of Originality

This document is written by Kit Yin Liu who declares to take full responsibility for the contents of

this document.

I declare that the text and the work presented in this document is original and that no sources

other than those mentioned in the text and its references have been used in creating it.

The faculty of Economics and Business is responsible solely for the supervision of completion of

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Abstract

Empirical evidence in the current literature shows that there is a negative and asymmetric relationship between implied volatility and stock returns. However, no clear consensus about the mechanisms behind this relationship has been reached yet. This study adopts a behavioral framework to investigate this relationship for continental European stocks at the daily level. Using ten years of trading days data from the period December 2007 until December 2017, the results indicate that the behavioral concepts such as representativeness heuristic, affect heuristic, and loss aversion are important in this daily and contemporaneous relationship. Furthermore, the results show that this negative and asymmetric relationship depends on market circumstances. Bear market period, as characterized by negative return and high uncertainty, leads to a larger asymmetry for European stock markets.

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Table of content

Introduction

5

Literature review

9

The implied volatility index

9

Leverage effect

10

Volatility feedback effect

11

Behavioral effect

14

Data

16

Hypotheses and methodology

22

Hypotheses formulation

22

Methodology

24

Results

29

Main results

29

Robustness test results

36

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I. Introduction

The equity and option markets are two markets with different mechanics. Yet even with so distinct characteristics, financial markets as a whole is highly interconnected and integrated due to globalization effects and financial innovations. Day and Lewis (1992), suggest that informed market participants will first use the option markets since option prices represent traders’ expectations of the underlying asset. This information content in option prices, which is based on its implied volatility, contains information to efficiently forecast future volatility. The relationship between implied volatility and future volatility has therefore been extensively studied and confirmed by Lamoureux and Lastrapes (1993), Fleming (1998), Christensen and Prabhala (1998), Jiang and Tian (2005), and Diavatopoulos et al. (2008) among others.

The information content in option prices can also be used to examine the relationship between implied volatility and stock returns. Empirical evidence in the current literature shows that there is a negative and asymmetric relationship between implied volatility and stock returns. However, no clear consensus about the mechanisms behind this relationship has been reached yet. There are three competing explanations for the negative and asymmetric relationship between implied volatility and stock returns. Black (1976) is the first to develop the leverage effect theory, in which he states that a decrease in stock price leads to a lower value of the firm’s equity and therefore a higher financial leverage. As a result, stocks of the firm become riskier which drives up the volatility. This theory has been empirically confirmed by several studies including Christie (1982), Cheung and Ng (1992), and Duffee (1995). Another explanation is the volatility feedback effect by Campbell and Hentschel (1992). They argue that an increase in volatility will lead to an increase in the expected future stock returns, therefore stock prices will decrease in order to adjust to this expectation (Campbell and Hentschel, 1992). These two competing theories are the traditional explanations for the relationship behind implied volatility and stock returns. However, recent studies show that the traditional explanations are inconclusive and not adequate to fully

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explain the negative and asymmetric relationship in the short-term and daily environment. According to Hibbert et al. (2008), the leverage effect and volatility feedback effect theories are based on fundamental firm factors that is only reflected in the long run. Therefore, a behavioral explanation has been proposed by Hibbert et al. (2008). This behavioral explanation is first briefly introduced by Low (2004), in which he describes the asymmetric and nonlinear relationship between implied volatility and stock returns as a downward sloping reclined S-curve, similar to the loss aversion concept and value function of Tversky and Kahneman’s (1979). Hibbert et al. (2008) extend the work of Low (2004) and are the first to fully work out an empirical model for the behavioral explanation. They postulate that psychological biases of market participants are the main cause of the negative and asymmetric relationship between implied volatility and stock returns.

This study builds on the behavioral explanation proposed by Hibbert et al. (2008) and examines the contemporaneous relationship between implied volatility and stock returns in (continental) Europe. More specifically, this study focus on the stock market of The Netherlands (AEX 25), Germany (DAX 30), France (CAC 40) and the Eurozone (EuroStoxx 50). In addition, the stock market of U.S. (S&P 500) and U.K. (FTSE 100) will also be included for benchmarking purposes. The United States Oil Fund (USO) ETF will be added for a complimentary robustness check. The inclusion of a commodity ETF allows me to extend the behavioral framework to a slightly different security and see whether this relationship also holds outside the equity market. Furthermore, stock index return has been chosen to adopt a holistic view on this relationship. The volatility index of the underlying stock index will be used as a proxy for implied volatility since they represent market participants’ best estimate of the volatility of the underlying stock index over the next 30 days (Low, 2004). Volatility indices are also widely available and can therefore be seen as truly public information for every investor. In addition, volatility indices are much more common and well known among market participants as it is often commonly referred to as the “fear gauge” in the media. The period in this study covers ten years of trading days, from December 2007 until December 2017.

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Furthermore, this period will be separated into a bear market subsample and bull market subsample. The bear market subsample covers the period December 2007 until December 2012 and represents a period of relatively high volatility, high uncertainty and low economic growth. Conversely, the bull market subsample refers to the post crisis period, from January 2013 until December 2017, and is characterized by low volatility, low uncertainty and growing economic activity.

The main objective of this study is to examine the negative and asymmetric relationship between implied volatility and European stock markets during a bear market period and a bull market period. This dichotomy allows me to examine whether the relation between implied volatility and the stock market will be different under specific market circumstances. In order to answer this main objective, a behavioral framework similar to Hibbert et al. (2008) will be used. First, this study will test and verify the negative relationship between implied volatility and European stock markets. Second, the behavioral explanation and its psychological biases will be empirically tested and verified for European stock markets. Third, the asymmetric relationship between implied volatility and European stock markets will be empirically tested using a behavioral framework and the combined findings of Low (2004) and Hibbert et al. (2008). Lastly, the asymmetry will be examined in a bear market sample and bull market sample with concepts from the behavioral explanation.

This study contributes to the literature in several ways. First, this study focus on the continental European stock markets whereas previous studies on this topic focus predominantly on the U.S. stock market and to a lesser extent the U.K. stock market. However, the market model of continental Europe is vastly different compared to Anglo-Saxon countries. According to La Porta et al. (1997), the Anglo-Saxon market model is characterized by large capital markets, dispersed ownership, and strong legal rights for investors. Conversely, the continental Europe market model is often described as a market with small capital markets, concentrated ownership, and weak legal rights for investors. This dichotomy might also be reflected in the behavior of investors. Therefore, this study could give valuable insights about the

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differences in the relationship between implied volatility and stock market return of both market models. Second, this study is the first to examine the link between implied volatility and stock market return after the financial turmoil of 2008. There are no academic studies yet that have incorporated the financial crisis, European debt crisis, and the post-crisis period when investigating the relationship between implied volatility and stock market return. In addition, the recent globalization trends, financial innovations, and geopolitical events such as Brexit or trade wars might alter how investors look at this relationship compared to the pre-crisis period where many academic studies based their research on. Related to this, there is evidence to assume that risk aversion might have changed after the financial crisis. For example, Guiso et al. (2017) find that risk aversion increased substantially after the financial crisis. Since this study employs a behavioral framework on the relationship between implied volatility and the stock market, the period after the most recent financial crisis is an interesting one. Third, this study adds to the literature by studying the link between implied volatility and the stock market in a bull market and bear market period. Previous studies only consider one large period to investigate this relationship, but evidence from the behavioral effect theory suggests that this relationship is dynamic and changes accordingly with specific market conditions. Finally, this study extends limited previous research, combines most recent findings, and provides evidence that the behavioral effect theory is the most plausible explanation for the short-term and contemporaneous relationship between implied volatility and the stock markets.

The main empirical findings can be summarized along four dimensions. First and foremost, consistent with the existing literature, I find a highly significant and negative correlation between implied volatility and stock market return. Second, empirical findings indicate that contemporaneous stock market return is the most significant determinant for changes in implied volatility whereas the lagged stock market return is marginally significant at best. This finding is consistent with Hibbert et al. (2008) and provides further evidence for the behavioral effect theory as the most plausible explanation for the relationship between implied volatility and stock market return. In a similar way, consistent with Hibbert et al. (2008),

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the findings provide evidence for representativeness heuristic and affect heuristic biases under market participants. However, in contrast with Hibbert et al. (2008), the findings do not indicate that market participants suffer from extrapolation bias. Third, the findings of this study also reveal that there is asymmetry in the relationship between implied volatility and stock market return. This implies that market participants react differently when faced with negative return. More specifically, negative stock market returns will have a larger impact on changes in implied volatility than positive stock market returns. Lastly, the findings suggest that asymmetry is significantly higher in a bear market period for European stock markets. This is consistent with the concepts of loss aversion in the behavioral effect theory, which implies that market participants are more fearful in a bear market period. However, this finding is reversed for U.S. and U.K. stock markets as the asymmetry is higher in the bull market period.

The rest of this study is organized as follows. Section II reviews the literature and associated theories about the relationship between volatility and return. Section III discusses the data used in this study. Section IV describes the hypotheses and methodology. Section V presents the empirical results of this study. Section VI concludes this study.

II. Literature review

A. The implied volatility index

Many investors have been intrigued by the idea of having a continuously updated index that tracks the average level of option premiums since 1973, the start of listed option trading. Gastineau (1977) first postulate the idea of creating such an index with individual stock option prices. He combined the average implied volatility of fourteen at-the-money call options of the S&P 100 and created the predecessor of the volatility index. In 1993, the Chicago Board Options Exchange (CBOE) refined this idea and created the CBOE Volatility Index (VIX) that tracks the implied volatility of the S&P 100 over thirty days. This new method uses index options rather than individual stock options to allow for a greater emphasis on the

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market-wide systematic risk rather than only idiosyncratic risk of individual stocks. In addition, at-the-money put options were included to allow for a greater spectrum of market expectations (Whaley, 1993). In September 2003, the CBOE revamped the calculation and definition of the VIX. The volatility index is now based on the S&P 500 rather than the S&P 100 before September 2003. In addition, due to the popularity of out-of-money put options as portfolio insurance, the CBOE added out-of-money options in the construction of the VIX (Whaley, 2008). After the introduction of the VIX by the CBOE, many other exchanges created their own volatility index using the same methodology

B. Leverage effect

In the current literature, there is a prevalent negative relationship between volatility indices and stock indices that can be explained by three competing theories: leverage effect, volatility feedback effect, and behavioral effect. The leverage effect is the first explanation postulated by Black (1976) for the negative relationship between volatility and stock prices. This theory builds on the premise that stock return leads to changes in volatility. In his research, Black (1976) uses daily data of 30 Dow Jones Industrials stocks from 1964 to 1975 to study the relationship between stock returns and changes in volatility. Black (1976) argues that a decrease in a firm’s stock price leads to a lower value of the firm’s equity and therefore a higher financial leverage (i.e. debt-to-equity ratio) given that the debt outstanding is fixed. This increased leverage leads to more risk and therefore an increase in the volatility of the stock, giving rise to the leverage effect. This effect has been empirically confirmed by several studies including Christie (1982), Cheung and Ng (1992), and Duffee (1995). However, the validity of the leverage effect is not yet

conclusive according to Figlewski and Wang (2001). The authors also find a strong “leverage effect” with decreasing stock prices but argue that there are some empirical anomalies associated with this theory. Specifically, the leverage effect is much weaker or almost nonexistent when positive stock returns reduce leverage. Furthermore, the change in volatility associated with a given change in leverage is inconsistent and decays over time. The most important anomaly, however, is that there is no change in

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volatility when leverage changes due to outstanding debt or shares. The leverage effect is only observed with stock price changes (Figlewski & Wang, 2001). Since the leverage effect theory assumes leverage to be the main driver behind volatility changes, this finding is inconsistent with the results of Black’s (1976) leverage effect theory. In addition, Hasanhodzic & Lo (2011) also find that the leverage effect is not due to financial leverage. If leverage is indeed the main driver behind volatility changes, then firms with no debt outstanding should not experience any changes in volatility. Using data of equity-only financed companies from January 1972 to December 2008, Hasanhodzic & Lo (2011) conclude that there is an equally strong negative relationship between stock returns and volatility changes as for their debt-only financed counterparts. These results contradict the leverage effect and lead to another possible explanation for the negative relationship between stock prices and volatility. An explanation in which volatility is endogenously determined in equilibrium by a mix of feedback effects and market

circumstances: the volatility feedback effect.

C. Volatility feedback effect

The volatility feedback effect is based on the time-varying expected risk premium hypothesis as the link between stock returns and changes in volatility (French et al., 1987). In their research, the authors find evidence for a positive relationship between expected risk premium, defined as the expected return on a stock minus risk-free interest rate, and volatility of the stock. Furthermore, French et al. (1987) find strong evidence for a negative relationship between expected risk premium and stock market returns. Combining these two findings, French et al. (1987) argue that an increase in volatility implies an increase in the future expected risk premium, hence a decline in the current stock price. Haugen, Talmor and Torous (1991) also share this view. They find that changes in volatility lead to significant revisions in risk premium and hence changing stock prices. This time-varying risk premium explanation forms the foundation of the volatility feedback effect. It is important to note that under the volatility feedback theory, changes in volatility leads

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to changes in the stock price. The leverage effect theory, however, assumes that changes in stock price leads to changes in volatility.

The most important contribution to the volatility feedback theory is Campbell and Hentschel (1992), since earlier research only discuss the concepts of volatility feedback effect informally. Campbell and Hentschel (1992) are the first to fully work out a formal model of volatility feedback using a quadratic generalized autoregressive conditionally heteroskedastic (QGARCH) model. This model builds on the notion of time-varying expected risk premium and allows for negative skewness and excess kurtosis in the stock return data. They use monthly and daily data of U.S. stock returns over the period 1926 until 1988 and show that positive shocks to volatility leads to negative stock returns. Campbell and Hentschel (1992) argue that an increase in volatility will lead to an increase in expected future stock returns, therefore stock prices will decrease in order to adjust to this expectation. Campbell and Hentschel (1992) identify two important assumptions in the volatility feedback effect. The first assumption is that volatility is persistent in the sense that large pieces of news tend to be followed by more good news. The second assumption is the positive relationship between expected future stock return and volatility. For example, suppose there is a large piece of good news about future dividends for a firm. Due to the persistent nature of volatility, good news is usually followed by more good news in the future. This large piece of good news therefore increases the future expected volatility, which in turn increases the expected future stock returns. In order to adjust to the higher expected future stock returns, the current stock price needs to decrease. This adjustment in stock price diminish the positive stock price impact of good news about dividends. Now suppose there is a large piece of bad news about future dividends. Since volatility is persistent, this bad news will usually be followed by more bad news and therefore future expected volatility will increase. Once again, the stock price decreases but this time the adjustment in stock price amplifies the negative impact of the bad news. These two examples of volatility feedback effects explain why stock returns have excess kurtosis and negative skewness. According to Campbell and Hentschel (1992), only no news or a

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small piece of news will decrease future expected volatility and increase stock price. The volatility feedback effect therefore implies that future volatility is related to stock price movements (Campbell & Hentschel, 1992).

There are also conflicting empirical findings with regard to the volatility feedback effect. One of the main tenets of the volatility feedback effect is the positive relation between volatility and expected future return (Campbell and Hentschel, 1992). However, Fama and Schwert (1997), Turner et al. (1989), and Glosten et al. (1993) find evidence for a negative relation. This contradicting relation has important implication for the validity of the volatility feedback effect. In addition, the QGARCH model from Campbell and Hentschel (1992) only explains half of the observed relationship between volatility and stock return. Campbell and Hentschel (1992) state that a large proportion of the variance in stock return comes from sources other than changes in volatility and that the model is insufficient to explain this. In addition, Dennis et al. (2006) show that the return-volatility relationship is attributed to systematic market factors whereas the leverage effect and volatility feedback effect imply a return-volatility relationship that is based on firm fundamental factors. Furthermore, Hibbert et al. (2008) study the short-term dynamic relation between the S&P 500 (Nasdaq 100) index return and the VIX (VXN) and find that neither leverage effect nor the volatility feedback effect explains the result sufficiently. They argue that leverage and volatility feedback effects should be long term effects, which implies that the delayed relationship between return and volatility should be important. However, Hibbert et al. (2008) find that this relationship is contemporaneous rather than delayed. In addition, the time-varying risk premium concept in the volatility feedback effect should not exist in such short-term environment. Moreover, the leverage effect theory is also not applicable due to the long-term nature of the debt-to-equity ratio (Hibbert et al., 2008). Instead of the leverage effect and volatility feedback effect, the authors propose a behavioral approach that is associated with biases of market participants.

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D. Behavioral effect

Low (2004) is the first to study the relationship between option trader’s risk perception, and contemporaneous market conditions of the S&P 100 using a behavioral approach. Using daily data of the VIX as proxy for the perception of risk, he finds an asymmetric and nonlinear relationship between stock return and volatility. According to Low (2004), the asymmetry and nonlinearity is similar to the concept of loss aversion in Kahneman and Tversky (1979), where losses loom larger than gains. This relationship is best described as a downward sloping reclined S-curve, implying convexity in the region of negative returns and concavity in the region of positive returns (Low, 2004). Low (2004) describes this convexity as accelerating increases in the VIX (fear) and concavity as accelerating decreases in the VIX (exuberance). He finds that option trader’s risk perception is more sensitive to downside returns than upside returns, indicating that fear strikes quickly but exuberance builds slowly (Low, 2004).

Hibbert et al. (2008) extend the work of Low (2004) and investigate the contemporaneous relation between stock market return and implied volatility. Using daily and intraday data of the S&P 500 (Nasdaq 100) and VIX (VXN) in the period 1998 till 2006, they find a strong negative and asymmetric relationship between stock market return and implied volatility. Hibbert et al. (2008) argue that these shorter-term frequency (daily and intraday) relationship can be explained with specific behavioral concepts: representativeness heuristics, affect heuristics, and extrapolation bias.

The representativeness heuristic is a principle applied by market participants to make quick and sometimes even irrational judgements based on limited information (Tversky and Kahneman, 1974). Heuristics are mental shortcuts or rule of thumb that market participants use to make judgements and decisions. Market participants make judgements about risk and return based on representativeness heuristics. This principle implies that high return and low risk is often regarded as representative of a bull market, good investment, or a financially stable firm. Conversely, low return and high risk will therefore

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be associated with a bear market, bad investment, or a financially unstable firm. For example, suppose that market participants hold incorrect beliefs of a bear market based on recent declining price movements. Under representativeness heuristics, this recent price decline is representative of a bear market period. This lead to an increased demand in out-of-money puts due to fear, implying an increase in the option price and ultimately an increase in the implied volatility. The implication is that market participants view negative returns to be correlated with an increase in implied volatility.

Affect heuristic refers to a mental shortcut that allows market participants to make judgements or decisions faster based on emotions (Finucane et al., 2000). This judgement and decision making process is based on the emotional feeling (i.e. positive or negative) that market participants associate with a particular situation. Future decisions or judgements will depend on the emotional label that market participants attached to a similar situation in the past. Under affect heuristics, market participants associate a declining market with a negative feeling (i.e. fear, low benefits and high risk). Subsequently, market participants massively buy put options for portfolio insurance, leading to a rising implied volatility. On the other hand, positive price movements will lead to a less steep decrease in implied volatility due to the phenomenon of loss aversion. According to Kahneman and Tversky (1979), the value function is concave for gains and convex and steeper for losses, implying an asymmetric and negative relationship for return and implied volatility.

Lastly, extrapolation bias is related to the representativeness and affect heuristic. This concept refers to the tendency of market participants to overweight recent events as representative of the future. In addition, market participants often erroneously seek trends in past events to forecast the future (Hibbert et al., 2008). For example, a recent decline in the market will lead to market participants expecting more negative price movements in the future. To protect their portfolios, market participants will buy put options and implied volatility will increase.

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III. Data

The focus of this research is mostly on the leading stock markets in continental Europe, however, the U.S. and U.K. stock market will be included as well for benchmarking purposes. In addition, USO ETF will also be examined for robustness test. To sum up, the following stock indices and their underlying volatility index will be studied: AEX 25, DAX 30, CAC 40, Euro Stoxx 50, S&P 500, FTSE 100, and the USO ETF.

Daily data of the stock index and volatility index pairs have been collected from Thomson DataStream. The data set covers a period from December 2007 until December 2017, providing 2,611 daily observations. The choice for this sample period is to fully incorporate the latest globalization effects, political events, and financial innovations. In addition, the financial crisis of 2008 might have changed the risk aversion of investors. Since this study employs a behavioral framework on the relationship between implied volatility and stock return, the period after the most recent financial crisis is an interesting one. For example, Guiso et al. (2017) find that risk aversion increased substantially after the financial crisis. The authors posit fear as one of the potential mechanisms for this increase in risk aversion. Their findings are also consistent with Bordalo et al. (2012), in which they show that individuals tend to overweight the probability of salient returns. The fall of Lehman Brothers and stock market crash increased the salience of extreme negative return, increasing the subjective probability of these extreme negative return. As a result, the risk aversion of investors increased substantially after the financial crisis (Guiso et al., 2017).

The period used in this data will be further divided into a bear market and bull market subsample. According to the Quarterly National Accounts database of the Organisation for Economic Co-operation and Development (OECD), which measures quarterly growth rates of real GDP, northwestern Europe was in a recession from Q1 2008 until approximately Q1 2013. This recession was caused by the financial crisis of 2008 followed by the European sovereign debt crisis. Therefore, the period December 2007 until December 2012 has been chosen as a proxy for the bear market period. Conversely, the bull market period

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will cover the period from January 2013 until December 2017, where the uncertainty is low and the economy is growing. This dichotomy allows me to examine whether the relation between implied volatility and the stock market will be different under a bear and bull market. To conclude, I will examine the relationship between implied volatility and stock market in a bear market sample (December 2007 until December 2012) and a bull market sample (January 2013 until December 2017).

The stock index will be the variable of choice for the stock market to adopt a holistic view on the relationship between implied volatility and the stock market. Daily closing prices of the various stock indices have been retrieved and transformed into daily percentage return. Volatility index has been selected as a proxy for the implied volatility. Volatility indices are freely available, whereas individual option implied volatility data usually requires a subscription for many market participants. Volatility indices can therefore be seen as truly public information for every investor and thus it can be considered as possible trading signals. In addition, volatility indices are much more common and well known among market participants as it is often commonly referred to as the “fear gauge” in the media.

Table 1 reports the summary statistics and table 2 reports the correlation matrix of volatility indices over the sample period. Results from table 1 show that that the statistical properties of volatility indices are quite similar. The mean volatility index level ranges from 19.64 to 26.10, with the exception of the USO ETF volatility index having an almost twice as large mean. The same pattern can be observed for the standard deviation, which ranges from 8.66 to 10.28 with the exception of the USO ETF volatility index. This indicates that the USO ETF volatility index is more volatile compared to volatility indices of stock markets. The maximum value of all the volatility indices is clustered around October and November 2018, indicating substantial fear amongst market participants after the fall of Lehman Brothers and the start of the financial turmoil. Furthermore, a test for skewness indicates that all volatility indices are slightly positively skewed, implying a longer right-sided tail in the probability density function. Lastly, a test for kurtosis confirms the leptokurtic nature of volatility indices and the existence of many extreme values.

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Table 2 reports the Pearson correlation coefficient of the volatility indices. There is a high degree of correlation amongst European volatility indices, which is not surprising given the proximity of those markets. In fact, all volatility indices are highly correlated with each other.

Table 1: Descriptive statistics of volatility indices data

All volatility index data are retrieved from DataStream in the period December 2007 until December 2017. Daily closing levels have been used in this summary description.

Variable N Mean S.D. Minimum Maximum Kurtosis Skewness VStoxx 2,611 24.78 9.32 11.05 87.51 8.44 1.90 VFTSE 2,611 19.64 8.93 6.19 75.54 9.77 2.15 VDAX 2,611 23.31 9.15 10.89 83.23 9.82 2.15 VCAC 2,611 23.19 8.66 0.43 78.05 8.57 1.86 VAEX 2,611 22.17 10.15 5.77 81.22 8.54 2.03 VS&P 2,611 20.15 9.90 9.14 80.86 9.78 2.28 VOil 2,611 37.32 14.12 14.50 100.42 5.55 1.33

Table 2: Correlation matrix of volatility indices

This table reports the Pearson correlation coefficient of various volatility indices. VStoxx, VFTSE, VDAX, VCAC, VAEX, VS&P, and VOil represent respectively the volatility index of EuroStoxx 50, FTSE 100, DAX 30, CAC 40, AEX 25, S&P 500, and USO ETF.

VStoxx VFTSE VDAX VCAC VAEX VS&P VOil

VStoxx 1.00 - - - - VFTSE 0.96 1.00 - - - - - VDAX 0.98 0.95 1.00 - - - - VCAC 0.99 0.96 0.97 1.00 - - - VAEX 0.97 0.97 0.97 0.96 1.00 - - VS&P 0.94 0.97 0.93 0.93 0.96 1.00 - VOil 0.75 0.75 0.78 0.75 0.81 0.75 1.00

Table 3 reports the summary statistics of stock index returns over the sample period. The mean stock index return varies a lot, ranging from 0.003% per day for the EuroStoxx 50 to 0.030% for the S&P 500. The average return for the USO ETF is even negative, with a daily decrease of 0.045%. This negative daily return is not surprising considering that oil price decreased from almost $147 per barrel in May 2008 to $30 per barrel in January 2016. According to Baumeister and Kilian (2016), this decline in oil price is caused by the financial crisis, followed by the weakening of China’s economy and an increased supply of oil in 2016. Based on table 3, it can also be shown that USO ETF return is the most volatile, with a standard deviation of 2.19% compared to approximately 1.50% standard deviation for the various stock index return. The

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largest decline occurred in October 2008 for almost all stock indices, which is not surprising given that volatility indices reached its maximum in that same period. One stock index however, the EuroStoxx 50, experienced its largest decline of -8.62% on 23 June 2016, when U.K. voted for a Brexit. Furthermore, the high values for kurtosis indicate leptokurtic properties for all stock index return and the existence of extreme values. Table 4 reports the Pearson correlation coefficient of the various stock indices. There is still a high correlation amongst European stock indices. However, European stock indices are less correlated with the S&P 500 and USO ETF compared to volatility indices.

Table 3: Descriptive statistics of stock indices data

All stock index data are retrieved from DataStream in the period December 2007 until December 2017. Daily closing prices have been retrieved and transformed into daily closing return. Mean, standard deviation, minimum, and maximum variables are denoted in percentage change.

Variable N Mean (%) S.D.

(%) Minimum (%) Maximum (%) Kurtosis Skewness EuroStoxx 50 2,611 0.003 1.502 -8.617 11.002 8.990 0.127 FTSE 100 2,611 0.012 1.217 -8.849 9.839 11.467 0.076 DAX 30 2,611 0.029 1.438 -7.164 11.402 9.610 0.182 CAC 40 2,611 0.009 1.493 -9.037 11.176 9.581 0.189 AEX 25 2,611 0.012 1.395 -9.145 10.548 11.444 0.031 S&P 500 2,611 0.030 1.266 -9.035 11.580 14.738 -0.086 Oil 2,611 -0.045 2.189 -10.685 9.603 5.449 -0.006

Table 4: Correlation matrix of stock indices

This table reports the Pearson correlation coefficient of various stock indices. EuroStoxx 50, FTSE 100, DAX 30, CAC 40, AEX 25, and S&P 500 represent respectively the stock index of the Eurozone, U.K., Germany, France, Netherlands, and the U.S.

EuroStoxx 50 FTSE 100 DAX 30 CAC 40 AEX 25 S&P 500 Oil

EuroStoxx 50 1.00 - - - - FTSE 100 0.89 1.00 - - - - - DAX 30 0.95 0.85 1.00 - - - - CAC 40 0.98 0.90 0.93 1.00 - - - AEX 25 0.94 0.91 0.89 0.94 1.00 - - S&P 500 0.62 0.59 0.63 0.61 0.61 1.00 - Oil 0.34 0.37 0.33 0.34 0.37 0.44 1.00

The volatility index and stock index return variables are time series data and therefore run the risk of having a spurious regression. Following Dickey and Fuller (1981), the following Augmented Dickey-Fuller

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test for unit root needs to be conducted for every stock index and volatility index pair to eliminate a possible spurious regression:

𝑅𝑡= 𝛽0+ 𝑅𝑡−1+ ∑𝑝 𝑖𝑅𝑡−𝑖

𝑖=1 + 𝑢𝑡 (1)

Where 𝑅𝑡 is the daily stock index return, 𝑅𝑡−1 is the 1-day lagged stock index return and 𝑢𝑡 the error term.

The error term is assumed to be independently and identically distributed. Under the null hypothesis, 𝑅𝑡

has a stochastic trend and under the alternative hypothesis, 𝑅𝑡 is stationary. The lags p are determined by

the Aikake Information Criterion (AIC) and results in a lag of 1. Table 5 reports the estimated coefficients of the Augmented Dickey-Fuller test for the stock indices.

Table 5: Unit root test for stock index return

This table reports the Augmented Dickey-Fuller statistic for stock indices. The dependent variable is the first difference of the stock index return. Independent variables are the 1-day lagged stock index return and 1-day lagged first difference of the stock index return. For every specification, the irrelevant 1-day lagged first difference of the stock index return has been omitted in this table for clearer overview. Robust standard errors are in parenthesis. ***, **, * denotes significance at respectively the 1%, 5% and 10% level.

ADF-Statistic Unit root EuroStoxx 50 -27.35*** (0.039) NO FTSE 100 -27.28*** (0.037) NO DAX 30 -26.67*** (0.038) NO CAC 40 -27.27*** (0.039) NO AEX 25 -26.10*** (0.039) NO S&P 500 -28.37*** (0.038) NO Oil -28.36*** (0.035) NO

Results from table 5 indicate that all stock indices are stationary at the 1% significance level. To examine stationarity of volatility indices, the following Augmented Dickey-Fuller test for unit root has been conducted:

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𝐼𝑉𝑡 = 𝛽0+ 𝐼𝑉𝑡+ ∑ 𝑖𝐼𝑉𝑡−𝑖 𝑝

𝑖=1 + 𝑢𝑡 (2)

Where 𝐼𝑉𝑡 is the daily volatility index closing level, 𝐼𝑉𝑡−1 is the 1-day lagged volatility index closing level

and 𝑢𝑡 the error term. The error term is assumed to be independently and identically distributed with a

mean of zero. Under the null hypothesis, 𝐼𝑉𝑡 has a stochastic trend and under the alternative

hypothesis, 𝐼𝑉𝑡 is stationary. The lags p are determined by the Aikake Information Criterion (AIC) and

results in a lag of 1. Table 6 reports the estimated coefficients of the Augmented Dickey-Fuller test for the volatility indices. According to the results in table 6, all volatility indices have a unit root except for the USO ETF volatility index. The solution to a possible spurious regression problem caused by unit roots is to take the first difference or the percentage change. After transforming daily volatility index closing levels into daily percentage change, results from the same Augmented Dickey-Fuller test in equation (2) show that all volatility indices are indeed stationary. Therefore, from this section onwards, the volatility index variable refers to the daily percentage change in the volatility index.

Table 6: Unit root test for volatility indices

This table reports the Augmented Dickey-Fuller statistic for volatility indices. The dependent variable is the first difference of the volatility index level. Independent variables are the 1-day lagged volatility index level and 1the irrelevant 1-day lagged volatility index level has been omitted in this table for better overview. Robust standard errors are in parenthesis. ***, **, * denotes significance at respectively the 1%, 5% and 10% level.

ADF-statistic Unit root

VStoxx -0.71 (0.005) YES VFTSE -0.72 (0.005) YES VDAX -1.51 (0.004) YES VCAC -1.47 (0.006) YES VAEX -0.77 (0.004) YES VS&P 0.35 (0.004) YES VOil -5.43*** (0.004) NO

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IV. Hypotheses and methodology

A. Hypotheses formulation

This research will start by proposing a hypothesis to test whether the relationship between implied volatility and the stock index is indeed negative in the sample period. Based on the prevalent literature about the negative relationship between realized volatility and stock return, I also expect a highly significant and negative relationship between implied volatility and stock index return.

Hypothesis I: There is a negative relationship between stock index return and implied volatility. Hypothesis I is immediately followed by the test whether the behavioral effect adequately explains this negative relationship over the leverage effect and volatility feedback effect. According to Hibbert et al. (2008), the leverage effect and volatility feedback effect are long term effects, which implies a lagged relationship between stock index return and implied volatility. The leverage effect is based on the leverage ratio, which is constant on a daily basis. In addition, the time-varying risk premium concept in the volatility feedback effect should also not exist in such short-term environment (Hibbert et al., 2008). Therefore, hypothesis II focus on the contemporaneous relationship between stock index return and implied volatility.

Hypothesis II: Contemporaneous stock index return is the most significant factor in determining changes in the implied volatility.

If hypothesis II is correct, then there is evidence that the behavioral effect explanation is the main mechanism behind the relationship between stock index return and changes in the implied volatility. The acknowledgement of behavioral effect explanation as the main mechanism implies that the representativeness heuristic, affect heuristic, and extrapolation bias play a pivotal role in this relationship. Under the representativeness heuristic, market participants relate negative return as a representative of a bear market period. Subsequently, demand for out-of-money put options rises due to the desire of

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portfolio protection. Affect heuristic strengthens this channel by associating fear and displeasure with a bear market. The implication is that market participants view negative returns to be correlated with an increase in implied volatility. Related to the representativeness and affect heuristics, is the hypothesis for extrapolation bias by examining past changes in implied volatilities.

Hypothesis III: Past changes in implied volatility are important determinants of current changes in implied volatility.

If hypothesis III holds, then there is evidence to that assume that market participants are impaired by extrapolation bias when making judgements or decision. Extrapolation bias refers to the tendency of market participants to seek trends in past events to forecast the future (Hibbert et al., 2008). I argue that market participants expect changes in implied volatility to maintain in the future, implying a positive and significant relationship between current changes in implied volatility and lagged changes in implied volatility. The occurrence of extrapolation bias is consistent with the behavioral effect theory.

The next hypothesis examines the asymmetric relationship between implied volatility and stock index returns. Low (2004) describes this asymmetry as a downward sloping reclined S-curve, implying convexity in the region of negative returns and concavity in the region of positive returns. This convex relationship is an indication of fear amongst market participants, consistent with the representativeness heuristic, affect heuristic, and the concept of loss aversion in Kahneman and Tversky (1979).

Hypothesis IV: There is an asymmetric relationship between stock index returns and changes in implied volatility. Negative stock index returns have a much larger impact than positive stock index returns.

I argue that under representativeness heuristic, market participants view negative returns as a representative of a period characterized by low return and high risk. Subsequently, affect heuristic leads to market participants attaching a negative emotional label to this period of low return and high risk.

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Market participants will therefore make irrational judgements or decisions based on this negative emotion, which is predominantly fear. According to Low (2004), fear is defined as the convex relation between changes in implied volatility and (negative) stock index return. This convexity emerges from the loss aversion characteristics of market participants, where utility decreases more with losses compared to the same amount of gains (Kahneman and Tversky, 1979). Therefore if hypothesis IV holds, then representativeness heuristic, affect heuristic, and loss aversion concepts are indeed good explanations for the asymmetric relationship between stock index returns and changes in implied volatility.

The next hypothesis strives to answer the main research question and involves the study of a bear market subsample and bull market subsample as defined in section III. Following previous hypotheses, I expect that the asymmetric relationship between stock index returns and changes in implied volatility will be stronger in a bear market period. Fear is the dominant emotion in a bear market period, indicating a more pronounced asymmetry in a bear market period.

Hypothesis V: Changes in implied volatility during bear market periods will have a more pronounced effect compared to bull market periods.

If hypothesis V holds, then not only does fear cause the asymmetric relationship, it also reinforces the asymmetric relationship between stock index return and changes in implied volatility.

B. Methodology

The first stage of the methodology involves a simple ordinary least squares regression to test whether the relationship between stock index returns and changes in implied volatility is indeed negative as proposed by hypothesis I. The following model will be used for every volatility index and stock index pair.

𝐼𝑉𝑡= 𝛽0+ 𝛽1𝑅𝑡+ 𝑢𝑡 (3) Where 𝐼𝑉𝑡 represents the change in implied volatility at time t, 𝑅𝑡 is the underlying stock index return at

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relationship between changes in implied volatility and stock index return. Therefore, the coefficient 𝛽1 is

expected to be highly significant and negative.

Hypothesis II involves testing whether contemporaneous stock index return is the most important factor in determining changes in implied volatility. This hypothesis will be tested using the following autoregressive distributed lag (ADL) model for every volatility index and stock index pair:

(4)

Lag lengths p and q are determined by the Aikake Information Criterion and results in a lag length of 1 for both variables. The variable 𝐼𝑉𝑡 is the current change in implied volatility and 𝐼𝑉𝑡−1 the 1-day lagged

change in implied volatility. 𝑅𝑡 and 𝑅𝑡−1 represents respectively the contemporaneous and 1-day lagged

return in the underlying stock index. The error term 𝑢𝑡 is assumed to be independently and identically

distributed. Following Hibbert et al. (2008) conclusion, it is expected that the contemporaneous stock index return 𝑅𝑡 is the most significant component. The choice for an ADL(1,1) model has been made to

study the negative relationship in a dynamic environment. According to the literature, leverage effect, volatility feedback effect, and behavioral effect are all theories that are proposed as a possible explanation for the relationship between changes in implied volatility and stock index return. Leverage and volatility feedback effect are long term effects and should therefore be based on the lagged variables. However, in light of the behavioral effect, Hibbert et al. (2008) conclude that contemporaneous return is the most important factor in determining the negative relationship between changes in implied volatility and stock index return. This ADL(1,1) model encompass all three theories and is able to study this negative relationship in a dynamic environment. However, it is important to note that significance in the contemporaneous stock index return does not refute the leverage effect and volatility feedback effect completely. It simply means that within this short-term, daily, and contemporaneous environment,

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behavioral effect theory is the most plausible explanation for the negative relationship between changes in implied volatility and stock index return. It could be very well that the leverage effect and volatility feedback effect possess some long run multiplier effects. This will be illustrated by the following example. Consider the following ADL(1,1) model as in equation (4):

𝐼𝑉𝑡= α + β𝐼𝑉𝑡−1+ γ𝑅𝑡+ δ𝑅𝑡−1+ 𝜀𝑡 (5)

Where 𝐼𝑉𝑡 and 𝐼𝑉𝑡−1 represent respectively the current and 1-day lagged change in implied volatility. 𝑅𝑡

is the contemporaneous stock index return and 𝑅𝑡−1 is 1-day lagged stock index return. As usual, 𝜀𝑡

represents the error term and is assumed to be independently and identically distributed with a mean of zero. All variables are assumed to be stationary. Taking the unconditional expectation on the left and right hand side of the equation yields:

𝐸[𝐼𝑉𝑡] = α + β𝐸[𝐼𝑉𝑡−1] + γ𝐸[𝑅𝑡] + δ𝐸[𝑅𝑡−1] + 𝐸[𝜀𝑡] (6)

The expected value of the error term is zero since it assumed to be independently and identically distributed with a mean of zero. Under the assumption of stationarity, the expected values of changes in implied volatility and stock index return are constant over time:

𝐸[𝐼𝑉] = α + β𝐸[𝐼𝑉] + γ𝐸[𝑅] + δ𝐸[𝑅] (7)

Rearranging the formula above gives the following:

𝐸[𝐼𝑉] − β𝐸[𝐼𝑉] = α + γ𝐸[𝑅𝑡] + δ𝐸[𝑅] (8)

(1 − β)𝐸[𝐼𝑉] = α + (γ + δ)𝐸[𝑅] (9)

𝐸[𝐼𝑉] =(1−β)α +(γ+δ)(1−β)𝐸[𝑅] (10)

The short term multiplier effect is denoted by γ. The long term multiplier effect is given by (γ+δ)(1−β) and could possibly encompass the effects of leverage effect and volatility feedback effect over the long term.

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However, this research focus only on the short term and contemporaneous environment (i.e. daily data), thus significance in the contemporaneous stock index return as proposed in hypothesis II leads to the behavioral effect theory being the best explanation for the short term multiplier effect.

Hypothesis III tests whether market participants are impaired by extrapolation bias when making judgements or decisions. The methodology to test this hypothesis will follow the same ADL(1,1) model as defined in equation (4). Extrapolation bias refers to the tendency of market participants to seek trends in past events to forecast the future (Hibbert et al., 2008). This implies that past changes in implied volatility will be a significant factor in determining current changes in implied volatility. Therefore I expect 𝐼𝑉𝑡−1 to

be highly significant as well.

Hypothesis IV examines the asymmetric relationship between stock index return and changes in implied volatility. This asymmetry implies that negative stock index returns will have a larger impact on changes in implied volatility. In order to test this asymmetry, two partitions of stock index return will be created: a positive return partition in which days with negative stock index return are omitted and a negative return partition in which days with positive stock index return are omitted. The following model to test hypothesis IV will be used for every volatility index and stock index pair:

𝐼𝑉𝑡= 𝛽0+ 𝛽1𝑅𝑡++ 𝑢

𝑡 (12)

𝐼𝑉𝑡=0+1𝑅𝑡+ 𝑢

𝑡 (13)

Where 𝐼𝑉𝑡 represents the change in implied volatility at time t, 𝑅𝑡+ represents the positive stock index

return from the positive partition at time t and 𝑅𝑡− represents the negative stock index return from the

negative partition at time t. The error term is represented by 𝑢𝑡. Following hypothesis IV, I expect that the

coefficient 1 is more negative compared to 𝛽1. Next, this two partition regression will be modified to a

combined model to formally test whether the difference between 1 and 𝛽1 is indeed significant:

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In this model, 𝐷 is a dummy variable that equals one if the stock index return 𝑅𝑡 is negative at time t. The

interaction variable 𝐷𝑅𝑡 represents the asymmetric relationship between changes in implied volatility and

stock index return at time t. The coefficient of interest in this model is 𝛽3 which denotes the effect of a

negative stock index return compared to a positive stock index return. To be more specific, (𝛽1 + 𝛽3)

describes the total effect of a negative stock index return on changes in implied volatility whereas 𝛽1

describes the effect of a positive stock index return on changes in implied volatility. The advantage of this regression model is that significance of the difference between positive return and negative return can be formally verified. I expect that the coefficient 𝛽3 will be highly significant and negative, implying an

asymmetric relationship between stock index returns and changes in implied volatilities.

The last hypothesis examines the asymmetric relationship between stock index returns and changes in implied volatilities during a bear market sample and a bull market sample as defined in section III. The following regression model will be used to test this hypothesis for every volatility index and stock index pair:

𝐼𝑉𝑡𝐵𝑢𝑙𝑙 = 𝛽

0+ 𝛽1𝑅𝑡𝐵𝑢𝑙𝑙+ 𝛽2𝐷 + 𝛽3𝐷𝑅𝑡𝐵𝑢𝑙𝑙+ 𝑢𝑡 (15)

𝐼𝑉𝑡𝐵𝑒𝑎𝑟 =

0+1𝑅𝑡𝐵𝑒𝑎𝑟 +2𝐷 +3𝐷𝑅𝑡𝐵𝑒𝑎𝑟+ 𝑢𝑡 (16)

Where 𝐼𝑉𝑡𝐵𝑢𝑙𝑙 and 𝑅𝑡𝐵𝑢𝑙𝑙 represent respectively the change in implied volatility and stock index return

based on the bull market sample at time t. Conversely, 𝐼𝑉𝑡𝐵𝑒𝑎𝑟 and 𝑅𝑡𝐵𝑒𝑎𝑟 represent respectively the

change in implied volatility and stock index return based on the bear market sample at time t. Variable D represents a dummy variable that equals one if the stock index return is negative at time t. The error term in both equations are given by 𝑢𝑡. Based on hypothesis IV, it is expected that the asymmetric relationship

under a bear market period will be larger compared to a bull market period. This translates into the coefficient 3 under the bear market sample being more negative than the coefficient 𝛽3 under the bull

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IV. Results

A. Main results

Table 7 reports the estimated coefficients of equation (3). The 𝑅2 of this model is relatively high with

almost 30% - 60% of the variance in volatility index changes being explained by stock index return, except for the USO ETF pair and CAC 40 index pair. Moreover, the results show that the coefficient of stock index return is indeed negative and highly significant at the 1% level for all stock markets. This suggests that positive (negative) innovations in stock index return will induce a decrease (increase) in implied volatility. Furthermore, the results from table 7 also show that the S&P 500 and FTSE 100 are more negative than their European counterparts. This implies that stock index returns have a stronger impact on changes in implied volatility for the two Anglo-Saxon countries. Another observation is the relatively low impact of the USO ETF return on the oil volatility index. To conclude, the results in table 7 are consistent with hypothesis I and shows a negative relationship between stock index returns and changes in implied volatility.

Table 7: Hypothesis I results

The dependent variable is the daily percentage change in the volatility index. The independent variable, as denoted by 𝑅𝑡, is the daily return of the underlying stock index. Robust standard errors are in parentheses. ***,

** and * denotes significance at respectively the 1%, 5% and 10% level.

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

VStoxx VFTSE VDAX VCAC VAEX VS&P VOil 𝑅𝑡 -3.234*** -3.910*** -2.928*** -3.029*** -3.057*** -4.276*** -0.852*** (0.110) (0.196) (0.145) (0.143) (0.148) (0.190) (0.065) Constant 0.002** 0.003*** 0.002*** 0.005** 0.003** 0.004*** 0.001 (0.001) (0.001) (0.001) (0.003) (0.001) (0.001) (0.001) Observations 2,610 2,610 2,610 2,610 2,610 2,610 2,610 R-squared 0.570 0.424 0.512 0.099 0.322 0.528 0.145

Hypothesis II tests whether the contemporaneous stock index return is the most important factor in determining changes in implied volatility. If this is indeed the case, then behavioral effect is the most

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plausible theory in explaining the negative relationship in the short term. Table 8 reports the estimated coefficients of the ADL(1,1) model used in hypothesis II and equation (4).

Table 8: Hypothesis II and III results

The dependent variable is the daily percentage change in the volatility index. The independent variables are the daily stock index return (𝑅𝑡), the 1-day lagged stock index return (𝑅𝑡−1), and 1-day lagged percentage change in

the volatility index (𝐼𝑉𝑡−1). Robust standard errors are in parentheses. ***, ** and * denotes significance at

respectively the 1%, 5% and 10% level.

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

VStoxx VFTSE VDAX VCAC VAEX VS&P VOil 𝑅𝑡 -3.203*** -3.825*** -2.995*** -2.985*** -3.023*** -4.429*** -0.871*** (0.125) (0.226) (0.134) (0.153) (0.162) (0.219) (0.074) 𝑅𝑡−1 -0.171 -0.555** -0.255* -0.736 -1.386 -0.204 -0.131* (0.147) (0.265) (0.147) (0.481) (0.974) (0.205) (0.068) 𝐼𝑉𝑡−1 -0.012 -0.068* -0.003 -0.138 -0.349 -0.052 -0.130*** (0.033) (0.036) (0.031) (0.159) (0.269) (0.037) (0.049) Constant 0.002** 0.002** 0.003*** 0.006* 0.001 0.004*** -0.001 (0.001) (0.001) (0.001) (0.003) (0.001) (0.001) (0.001) Observations 2,087 2,087 2,087 2,087 2,087 2,087 2,087 R-squared 0.557 0.419 0.524 0.113 0.336 0.532 0.165

The results from table 8 show that contemporaneous stock index return is highly significant and negative at the 1% level. The 1-day lagged stock index returns are mostly insignificant or slightly significant for the FTSE 100 and DAX 30 at respectively the 5% and 10% level. For every volatility index and stock index pair, the contemporaneous return coefficient is larger and more negative. The results confirm that the contemporaneous stock index return is indeed the most important factor in determining changes in implied volatility. This implies that the leverage effect and volatility feedback effect, which are based on the lagged stock index returns, does not explain this relationship adequately. Instead, behavioral factors such as representativeness and affect heuristics are the most plausible explanations for this short term relationship. Therefore, hypothesis II is verified and the behavioral effect theory is the main mechanism behind the short term relationship between stock index return and changes in implied volatility. This finding is also consistent with Hibbert et al. (2008).

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Hypothesis III tests for the extrapolation bias and table 8 reports the results of this test. The results from table 8 show that the estimated coefficients of the 1-day lagged change in the volatility index are insignificant, with the exception of FTSE 100 and the USO ETF. According to Hibbert et al. (2008), market participants tend to find trends in past changes in implied volatility to forecast the future. This implies that changes in implied volatility should follow a trend and therefore past changes in implied volatility should be a significant factor in explaining current changes in implied volatility. Results from table 8, however, show that changes in implied volatility does not follow a trend for stock indices. Therefore, the extrapolation bias hypothesis is not supported by my results. This finding contrasts Hibbert et al. (2008), where they did find significant evidence for extrapolation bias. On the other hand, Padungsaksawasdi and Daigler (2014) also find inconsistency with regard to the extrapolation bias. They study the relationship between returns and implied volatility for several commodity ETFs and stock indices and find that there is no evidence for extrapolation bias in the stock market. However, Padungsaksawasdi and Daigler (2014) do find extrapolation bias in commodity ETFs, which is also consistent with my results. Table 9 shows that the 1-day lagged change in the volatility index of the USO ETF is highly significant at the 1% level, which is in stark contrast with the stock market indices. According to Padungsaksawasdi and Daigler (2014), this difference might be due to the less frequent revision in the option bid-ask spread for commodity ETFs when the price of the underlying ETF changes. Padungsaksawasdi and Daigler (2014) argue that the less frequent revision occurs because of market makers being reluctant to use delta hedging in commodity markets. Another possibility is that market makers tend to set the option bid-ask spread wider for commodities options (Padungsaksawasdi and Daigler, 2014). This argument is also supported by Grower and Thomas (2012), who show that illiquid options decreases the reliability of volatility indices.

Panel A and B of table 9 report the results of hypothesis IV. Panel A refers to the negative partition, where daily positive stock index returns are omitted from the sample. Conversely, panel B refers to the positive partition, where daily negative stock index returns are omitted from the sample.

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The dependent variable is the daily percentage change in the volatility index. The independent variable is the daily contemporaneous stock index return, as denoted by 𝑅𝑡− and 𝑅𝑡+ for respectively the negative and positive

partition. Panel A reports the estimated coefficients under a negative stock index return partition and panel B reports the estimated coefficients under a positive stock index return partition. Robust standard errors are in parentheses. ***, ** and * denotes significance at respectively the 1%, 5% and 10% level.

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

VStoxx VFTSE VDAX VCAC VAEX VS&P VOil

Panel A: Negative partition

𝑅𝑡− -3.775*** -4.425*** -3.399*** -3.166*** -3.200*** -4.306*** -1.500*** (0.110) (0.174) (0.112) (0.227) (0.138) (0.154) (0.077) Constant 0.001 0.005** 0.003* 0.009*** 0.008*** 0.014*** -0.008*** (0.002) (0.002) (0.002) (0.003) (0.002) (0.002) (0.002) Observations 1,322 1,306 1,256 1,302 1,295 1,244 1,363 R-squared 0.471 0.331 0.423 0.130 0.292 0.385 0.215

Panel B: Positive partition

𝑅𝑡+ -2.009*** -2.186*** -1.648*** -2.097*** -1.768*** -2.659*** -0.040*** (0.092) (0.144) (0.091) (0.419) (0.176) (0.109) (0.079) Constant -0.014*** -0.017*** -0.014*** -0.009 -0.015*** -0.017*** -0.012*** (0.001) (0.002) (0.001) (0.006) (0.002) (0.001) (0.002) Observations 1,333 1,388 1,425 1,356 1,366 1,459 1,352 R-squared 0.266 0.142 0.188 0.018 0.069 0.291 0.200

The objective of hypothesis IV is to examine the asymmetric relationship between stock index return and changes in implied volatility. If hypothesis IV holds, then there is evidence to assume that the representativeness heuristic, affect heuristic and the concept of loss aversion dictate this asymmetric relationship. A comparison of panel A and B shows us that 𝑅2 values are significantly higher under the

negative partition. This indicates that negative returns have a better fit in this asymmetric relationship between stock index returns and changes in implied volatility. All coefficients are also negative and significant at the 1% level, consistent with the results of hypothesis I. However, the magnitude of these coefficients are larger under the negative partition for all volatility index and stock index pairs. This implies that market participants react differently when faced with negative returns. The second part of this

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hypothesis is to test whether this difference is indeed significant. This model combines the two partition regressions and involves a dummy variable that equals one when it is a day with negative stock index return. Table 10 summarizes the results of this regression, with the dummy variable and contemporaneous stock index return omitted for a clearer table.

Table 9: Hypothesis IV results

The dependent variable is the daily percentage change in the volatility index. The independent variables are the daily stock index return, a dummy variable that equals 1 for a daily negative stock index return and an interaction variable. This regression is based on the full sample (Dec 2007 – Dec 2017). Standard errors are in parentheses. ***, ** and ** denotes significance at respectively the 1%, 5% and 10% level.

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

VStoxx VFTSE VDAX VCAC VAEX VS&P VOil 𝑅𝑡 -1.965*** -2.071*** -1.582*** -2.070*** -1.710*** -2.559*** 0.044 (0.105) (0.168) (0.104) (0.349) (0.165) (0.142) (0.081) 𝐷𝑢𝑚𝑚𝑦 0.016*** 0.025*** 0.019*** 0.019** 0.025*** 0.035*** 0.005* (0.002) (0.003) (0.002) (0.008) (0.003) (0.003) (0.003) 𝐷𝑢𝑚𝑚𝑦 ∗ 𝑅𝑡 -1.809*** -2.325*** -1.800*** -1.066** -1.468*** -1.659*** -1.605*** (0.148) (0.238) (0.150) (0.498) (0.231) (0.198) (0.113) Constant -0.015*** -0.019*** -0.015*** -0.009* -0.017*** -0.019*** -0.015*** (0.002) (0.002) (0.001) (0.005) (0.002) (0.002) (0.001) Observations 2,565 2,526 2,539 2,562 2,559 2,517 2,505 R-squared 0.601 0.459 0.553 0.103 0.347 0.573 0.217

Table 9 confirms previous findings under the two partitioned regression model. The 𝑅2 values are slightly

improved compared to the negative partition regression, indicating a better fit. The interaction variable is negative and highly significant at the 1% level for almost all volatility and stock index pair. Only the CAC 40 is significant at 5%, which is still a respectable level. These results imply that there is asymmetry in the relationship between changes in implied volatility and stock index return. Negative returns will induce a larger change in implied volatility compared to positive returns. The main drivers behind this asymmetry are the representativeness heuristic, affect heuristic and loss aversion concepts. These behavioral effects implies that market participants associate negative returns with a period of high risk and low return. Subsequently, affect heuristic reinforces this association by attaching a negative emotional label to this period. This leads to market participants being subject to fear when faced with negative returns. Under

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the concept of loss aversion, fear is defined as the convex relation between changes in implied volatility and (negative) stock index return (Low, 2004). The results in table 9 is consistent with Low (2004) and indicate that there is indeed a convex relationship for negative stock returns and changes in implied volatility. Therefore, hypothesis IV is verified.

Hypothesis V examines whether the asymmetry is different under a bear market period and bull market period. Bear market period covers the period December 2007 until December 2012. Conversely, the bull market period covers the period January 2013 until December 2017. This hypothesis follows the same regression as in hypothesis IV. The only difference here is that the sample is divided into two subsamples. Table 10 reports the results of hypothesis IV. It is expected that the asymmetry, as denoted by the interaction variable, will be larger under the bear market sample. During a bear market, market participants are more fearful when compared to a bull market. This implies that the interaction variable coefficient should be more negative. This is indeed the case for European stock indices, notably the DAX 30, AEX 25, CAC 40, and the broader EuroStoxx 50. However, for the FTSE 100 and S&P 500, the asymmetry is reversed since asymmetry is higher under the bull market period. There seems to be a difference between the asymmetric relationship of stock index returns and implied volatility in continental Europe and U.K. and U.S. This difference might arise due to the differences between continental Europe and Anglo-Saxon market models. Anglo-Saxon countries have a well-developed stock market in which market participants use for downside protection, i.e. they tend to buy put options more frequently to protect their portfolios. Bear market period is usually correlated with high implied volatility, leading to market participants unwillingly to bid it up higher through the purchase of put options (Giot, 2003). In a bull market period, the implied volatility tend to be low and put options are much cheaper. Negative returns in a bull market period induce fear rapidly due to loss aversion and motivates market participants to massively buy up put options as a protection for downside risk. This explanation is consistent with the findings of Low (2004), in which he explained that fear strikes quickly, but exuberance builds slowly. Continental European

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