Relationships between the Dutch implied volatility index and Dutch stock index returns : are there sub period differences in asymmetry; Implied volatility, a leading indicator?

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Bachelor Thesis

Relationships Between the Dutch Implied Volatility

Index and Dutch Stock Index Returns

Are there sub period differences in asymmetry; Implied volatility, a leading

indicator?

29-06-2016

Joost Sisto (10650288)

Supervisor: Dr. L. Zou

Study programme: Economics and Finance

Specialization:

Economics and Finance

Number of credits thesis: 12

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Index

-‐ 1. Introduction 3

-‐ 2. Literature review 5

-‐ 2.1 Leverage hypothesis 5

-‐ 2.2 Volatility feedback hypothesis 7

-‐ 2.3 Behavioural approach 9

-‐ 3. Data and variable description 11

-‐ 3.1 AEX 11

-‐ 3.2 VIX 12

-‐ 3.2.1 Black & Scholes framework 12

-‐ 3.3 Sub periods 14

-‐ 4. Methodology 15

-‐ 5. Results 18

-‐ 5.1 Overall AEX-VIX relationship 18

-‐ 5.2 Testing for asymmetries 19

-‐ 5.3 Relevance of extreme measures of VIX 24

-‐ 5.4 VIX, a leading indicator? 26

-‐ 6. Summary and conclusions 28

-‐ References 29

Statement of Originality

This document is written by Student Joost Sisto 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 the work, not for the contents.

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1. Introduction

After the publication of the Black&Scholes (1972) paper on the valuation of option contracts and market efficiency, the trade in option contracts exploded. The active trade in these products yield a price per contract, as a consequence of supply and demand. Equating this market price against the option valuation formula Black&Scholes specified in their paper published in 1972, yields the ‘implied volatility’. The implied volatility is the market’s proxy of the standard deviation of the rate of return of the underlying stock.

The informational content of implied volatility is largely believed to be greater than historic volatility and stochastic volatility, meaning the implied volatility would be the best estimate of future volatility. For this reason the relationship between implied volatility and future volatility was extensively researched. Many studies confirm the superior informational content of implied volatility. For example the research of Harvey & Whaley (1991) rejects the hypothesis that volatility changes are unpredictable on a daily basis. Other studies reach contradictive conclusions. Research from Canina and Figlewski (1993) suggests implied volatility is no predictor of future volatility. The varying conclusions of 93 papers on this subject were summarised by Poon and Granger (2003), reaching the conclusion that implied volatility does have a superior informational content than historic- and stochastic volatility.

While most research focussed on the informational content of implied volatility regarding the prediction of future volatility, the relationship between implied volatility and market returns was less extensively reviewed. Also, most research on the relationship between implied volatility and market returns has been done using the S&P 100 and Nasdaq 100 indices (Giot 2005). The foremost reason is the availability of data; the Implied volatility indices on the S&P 100 and Nasdaq 100 are freely available on the CBEO website. Hibbert, Daigler and Dupoyet (2007) used the same datasets for their research, using a different theoretical vantage point. A strong asymmetric negative relation was found in all research regarding this topic. Negative innovations in implied volatility are found to be paired with relatively small positive innovations of market returns, where positive innovations in implied volatility are paired with greater negative innovations in market returns.

This papers aims to research the relationship between the Dutch implied volatility index and the AEX returns. During the research this broad statement was divided in four research questions:

Does the change in the Dutch implied volatility index explain the change in the AEX index returns?

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The analysis employs data from 2001 till 2015, providing the research with the data necessary to answer the research questions for a smaller market, and draw conclusions from differing and more recent market situations, for example the economic crisis of 2008. These are two innovations that contribute to past research.

Are there asymmetries in the results, and are the asymmetries dependent on the market situation?

I examine the return-volatility relationship during distinct sub periods, each representing a different state of financial welfare. The data from 2001-2007 represents the pre-crisis period. The data from 2008-2011 represents the crisis period, which is a period with extremely high volatility. Peaks of AEX index implied volatility reached 81,78 on the 16th_{of October 2008,}

which is over four times the mean volatility over the last 15 years. Past research hasn’t had the possibility of examining the relationship between implied volatility and market returns in such a high volatility market environment. The data from 2012-2015 represents the bear market, where markets slowly regain their financial health.

Is the return-volatility relationship dependent on extreme volatility?

The contemporaneous relationship between the market index returns and the corresponding implied volatility is believed to be weaker in a high-volatility trading environment (Giot 2005). This may not be true in times of severe financial distress, where the panics caused by the financial crisis of 2008 trigger large innovations in returns and implied volatility. The regression on this sub period will be done with the exclusion of extreme data points, possibly providing new insights in the difference between high volatility- and extreme volatility trading environments.

Are past changes in the Dutch implied volatility predictors for present market returns?

Peaks of high volatility are largely believed to be a signal for market bottoms (Giot 2005), meaning acquiring long positions after spikes of high implied volatility will yield positive returns. In the paper it was found extreme measures of change in implied volatility do indicate over bought or over sold markets. Thus after extreme positive innovations of implied

volatility, acquiring long positions in the S&P- or Nasdaq 100 indexes would yield positive returns. The same may hold true for the smaller AEX index. Due to the efficient market hypothesis, it is largely believed that such trading strategies do not necessarily yield positive returns. However, if data on implied volatility is not freely available to all investors, the efficient market hypothesis can be questioned.

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2. Literature review

The asymmetric negative relationship between returns and (implied) volatility was found with great regularity in past research, as pointed out by Bollerslev et. al (2007). The contribution to past literature of the research done by Bollerslev et. al was using 5-minute intraday data for their regression. Previous research was limited to daily or even weekly or monthly data. The regularity in the empirical findings still persists after their research. While past research has always agreed on the correlation found between returns and implied volatility, the

fundamental cause behind this relationship has not been agreed upon.

Early studies proposed that changes of financial leverage were the cause of the observed return-volatility relationship. Black (1976) postulated that a decrease in returns increases financial leverage, making stocks riskier and driving up volatility. This explanation is widely known as the “leverage hypothesis”. Another leading explanation for the observed return-volatility relationship was proposed by Campbell and Hentschel (1992): The volatility feedback hypothesis. Campbell and Hentschel use a framework that allow prices to be affected by changing expectations about dividend and required returns. More recently, Hibbert, Daigler and Dupoyet (2007) proposed a behavioural explanation for the observed relationship, explaining the observed return-volatility relationship in terms of sentiment and behaviour.

2.1 Leverage hypothesis

The first proposed explanation for the observed return-volatility relationship is based on a principle proposed by Modigliani and Miller’s in 1958. They account the value of the whole firm to the assets, and see securities, bonds and other financial instruments as ways of distributing ownership of the firm. This allowed the behavior of volatility to be tied to the degree of leverage of the firm. Based on the principle of Modigliani and Miller, Black (1973) concluded that if debt holders have a claim limited to the face value of the issued bonds, which they have, all variations in the value of the firm must be accounted to the equity. As a consequence fluctuation of the firms value and thus equity must have a direct impact on volatility, which is the standard deviation of the rate of return of the stock. Since the total value of a levered firm is greater than the amount of equity issued, and as mentioned above the equity bears the entire fluctuation of a firm’s total value, the proportional change in the value of the stock exceeds the proportional change of the entire firm. As a consequence the volatility of levered firms must exceed that of an unlevered firm. This is the foundation of the observed return-volatility relationship according to the leverage hypothesis. A negative innovation in returns causes the leverage of the firm to increase, holding debt constant. If the

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negative innovation in returns causes the firm to become more levered, the future equity volatility increases. The same must hold for positive innovations in returns. The leverage hypothesis thus explains how changes in returns cause changes in volatility.

The research done by Figlewski and Wang (2006) aims to test how much of the observed changes in (implied) volatility are actually caused by the leverage effect. The leverage hypothesis suggests that both positive and negative changes in returns should impact volatility equally, as both positive and negative changes in returns have equal but opposite effects on leverage. If the leverage hypothesis holds, one would expect to find a symmetrical return-volatility relationship. In the regression dummy variables for up and down market conditions were added, enabling to test for asymmetries between up- and down market situations.

For the EOX index it was found that a ten percent market drop corresponds to a 20.44 percent increase of (log) volatility. A ten percent market rise would also correspond to an increase of (log) volatility, of 10.76 percent. This clearly contradicts the leverage hypothesis; as for an increase in market returns there exists a ‘negative’ leverage effect. According to the leverage hypothesis, the size of the change in (log) volatility should be equal for both market situations, which is clearly not the case.

For individual stocks, the relationship was also found to be asymmetric between up- and down market situations. The down market coefficient is found to be highly significant and negative, the up market coefficients are much less significant and vary between -0.130 and 0.292. Clearly the observed results are not in line with the leverage hypothesis, as testing on positive returns produces weaker and differing correlations with volatility.

The leverage hypothesis suggests that changes of returns impact the volatility of the underlying stock. As a consequence changes of returns should impact implied volatilities and realized volatilities equally. Figlewski and Wang find the relationship of implied volatility and market returns to be strongly asymmetrical. In the case of falling stock prices, the observed elasticity was 5. This deeply exceeds the elasticity found for realized volatility, sharply contradicting the leverage hypothesis. In the case of rising stock prices, the observed relationship is weak or even not significant, contradicting the leverage hypothesis again. Furthermore the relationship between implied volatility and returns are found to be distinctly larger for the EOX index, compared to the individual stocks that compose the index. The leverage hypothesis offers no explanation for these observed differences.

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Financial leverage is a level variable. Therefore the amount of financial leverage should determine volatility, not the change in financial leverage. Because financial leverage is a level variable, a permanent change in financial leverage should cause permanent changes in volatility. Figlewski and Wang (2006) tested if permanent changes of financial leverage produce permanent changes in volatility. It was found permanent changes of leverage do not cause permanent changes in volatility.

A change in the amount of debt outstanding or a change in the valuation of shares impact leverage equally. Therefore the leverage effect should be equal for changes in debt and equity. Figlewski and Wang (2006) found that only changes in equity produce a significant change in volatility, and only in a down market situation.

Empirical findings strongly contradict the leverage hypothesis in many separate occasions; it seems highly likely other factors influence the return-volatility relationship.

2.2 Volatility feedback hypothesis

Campbell and Hentschel (1992) propose a different approach explaining the observed return-volatility relationship. Using a quadratic generalized autoregressive conditionally

heteroskedastic (QGARCH) model, they aim to explain the effect of innovations in volatility on stock price returns. This model suggests expectations of dividends and required returns explain the asymmetry found in the return-volatility relationship. The underlying concept is called “volatility feedback”. The volatility feedback hypothesis rests on two main pillars. The first is that pieces of positive news are followed by more pieces of positive news, in other words: the persistence of volatility. The same holds true for negative news. The realization of news thus causes a positive change in current and future volatility. The second pillar is the positive relationship between expected returns and future volatility, allowing prices to react negatively to increases in volatility.

Suppose there is positive news concerning future dividend or required returns. As a reaction to the good news logically share prices increase. Under the assumption that large pieces of good news are followed by more good news, expected future volatility increases. According to the second pillar of the volatility feedback hypothesis, this increases the expected returns en therefore lowers the share price. This dampens the positive effect of the good news. Suppose there is a realization of negative news, as a reaction share prices decline. Again, this causes expected future volatility to increase, this increases the expected returns en therefore lowers the share price. For negative returns, the volatility effect amplifies the effect

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of the negative news, providing an explanation for the observed negative return-volatility relationship and the asymmetry surrounding it.

Also the volatility feedback hypothesis has shortcomings. The QCARCH model proposed by Campbell and Hentschel (1992) explains around half of the observed return-volatility relationship. Therefore they conclude that much of the variance in returns must be accounted to other changes in expected returns not caused by changes in volatility. They conclude their model is not suited for explaining the “rest” of the observed relationship, still leaving the total cause behind observed relationship open for interpretation.

As pointed out by Bekaert and Wu (2000), there are conflicting empirical findings concerning the second pillar of the volatility feedback principle; the positive relationship between expected returns and future volatility. For example Campbell and Hentschel (1992) find this relationship to be positive, on the contrary, Turner, Startz and Nelson (1989) find the relationship to be negative. If the positive relationship does not hold, the causality underlying the volatility feedback hypothesis is invalid.

The volatility feedback hypothesis relies on economic processes that work at bigger time intervals, as is pointed out by Bollerslev et. al (2006). They define bigger time intervals as weekly or monthly, questioning that volatility feedback is a valid explanation for the observed return-volatility relationship using daily or intraday data. They propose the

QCARCH model needs to be reformulated in a more flexible continuous-time framework and tested against their empirical findings. As mentioned above, Bollerslev et. al (2007)

researched the observed return-volatility relationship, using daily and intraday data. The observed negative return-volatility relationship found in this paper cannot be explained by the volatility feedback hypothesis.

It is worth to point out the striking difference between the two explanations for the observed return-volatility relationship. The leverage hypothesis aims to explain how changes in returns cause changes in volatility. The volatility feedback hypothesis explains how changes in volatility cause changes in returns. The fundamental difference between the two theories lays in the causality.

Economists largely believe the volatility feedback principle offers a better explanation than the leverage hypothesis, still theory does not align perfectly with observations, and a large part of the observed relationship remains unexplained. For this reason the behavioural approach was suggested, as neither the leverage hypothesis nor the volatility feedback hypothesis fully explains the observed return-volatility relationship.

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2.3 Behavioural approach

Low (2004) first proposed a behavioural explanation to the observed return-volatility relationship. Hibbert et al. (2007) linked the behavioural explanation to empirical results, acquiring strong evidence supporting it. The behavioural approach is based on affect heuristics, in other words the association of good feelings with positive returns and bad feelings with negative returns. This may seem trivial, but it has a big influence on the way agents make decisions, and due to the social nature economics, it has an effect on prices, volatility etc. (Finucane et al. 2000).

The behavioural explanation for the observed asymmetry assumes agents give different ‘weights’ to positive and negative feelings. Negative feelings associated with negative returns have a bigger impact than positive feelings associated with positive returns. Let the agents be option traders. After negative news, more negative news is expected, as is assumed also in the volatility feedback hypothesis. This causes investors to expect bigger market declines after negative market innovations. Investors massively buy put options, protecting their outstanding positions from further losses. In other words investors tend to suffer from ‘fear’ and act accordingly. The excess demand drives up prices of put options, driving up the implied volatility. Note that because investors put higher weight on negative feelings, not as much call options will be traded after good news. Demand is relatively low and therefore option prices tend to be more stable in times of positive returns, equating to relatively low measures of implied volatility. This explains the asymmetry in the observed returns-volatility relationship. The technical cause of the reactions of implied volatility on option prices will be explained in the research method.

As agents are option traders, the behavioural explanation is based on measures of implied volatility. The leverage- and volatility feedback hypothesis have implications for all types of volatility. Observations for implied volatility strongly contradict the leverage- and volatility feedback hypothesis, the behavioural approach fits these observations better. For example Giot (2005) and Bollerslev et. al (2007) showed there is a strong negative

relationship for contemporaneous changes between implied volatility and index returns for the S&P 100 and Nasdaq 100 indexes. These changes are not explained by the leverage- and volatility feedback hypothesis, as the underlying economic process work relatively slowly. Furthermore the fact that negative returns are associated with greater changes in implied volatility compared to positive returns lies completely in line with the hypothesis that investors suffer from ‘fear’ and act accordingly.

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Hibbert et al. (2007) formulate the following hypothesis: ‘Contemporaneous return is the most important factor that determines changes in current implied volatility’. Due to the slow nature of the volatility feedback and leverage hypothesis, rejecting this hypothesis proofs the behavioral approach is a superior explanation. The hypothesis is rejected and therefore Hibbert et al. conclude that the leverage- and volatility feedback hypothesis are not the primary cause of the observed return-volatility relationship. For the contemporaneous relationship between index returns and measures of implied volatility, the behavioural approach offers the best explanation.

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3. Data and variable description

The analysis employs data from two sources. The daily values of the AEX index are obtained from the Euronext website, the reference exchange of the AEX index. This data is free and easily accessible for investors. The daily values of implied volatility on the AEX index (hereafter VIX) were obtained from the Bloomberg database. This data is not free and not easily accessible for investors. I acquired the data trough the DEGIRO, an online broker. The data of the AEX index and VIX cover a period from 2001 till 2015, a total of 3,836 trading days. 3.1 AEX: Observations 3,836 Mean 390.5921 Std. Dev. 86.50608 Variance 7483.302 Skewness .4724682 Kurtosis 2.573294

Exhibit 1: Descriptive statistics AEX, smallest values are represented in percentiles under 50%, largest values are represented in percentiles over 50%.

The descriptive statistics mentioned above are related to the daily closing levels of the AEX index. The AEX index is composed of the 25 companies with the largest market

capitalisations denoted on the reference exchange of Amsterdam.

In the analysis employs regressions on returns. The returns of the AEX are calculated as: ln 𝐴𝐸𝑋! − ln 𝐴𝐸𝑋!!! . Rather then conventional returns calculated as: !"#!!"# _!"# ,

logarithmic returns are used. Logarithmic returns tend to be normally distributed and are time consistent. Logarithmic returns covering multiple days, for example two- and three- day lag returns, also tend to be normally distributed; which is helpful for the analysis of lag returns.

Percentiles AEX 1% 199.25 25% 202.57 50% 368.985 75% 639.86 99% 642.29

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3.2 VIX: Observations 3,836 Mean 21.34485 Std. Dev. 10.19914 Variance 104.0224 Skewness 1.762843 Kurtosis 6.526979

Exhibit 2: Descriptive statistics VIX, smallest values are represented in percentiles under 50%, largest values are represented in percentiles over 50%.

The descriptive statistics mentioned above are related to measures of implied volatility derived from 20-day till expiration call options.

3.2.1 Black and Scholes model for option valuation

The Black and Scholes (1973) pricing formula for call options:

𝐶 = 𝑆_!𝑁 𝑑_! − 𝑋𝑁 𝑑_! 𝑒!!!! Where: 𝑑_!=!" !! ! ! !!! !! ! ! ! ! and 𝑑!= 𝑑!− 𝜎 𝑇

-‐ 𝐶 Denotes the current call option value. -‐ 𝑆_! Denotes the current stock price. -‐ 𝑋 Denotes the strike price (exercise price).

-‐ 𝑟_! Denotes the interest rate, annualized and continuously compounded on a risk free asset with the same maturity as the expiration date of the call option.

-‐ 𝑇 Denotes the time to expiration of the call option, specified in years.

-‐ 𝜎 Denotes the standard deviation of the annualized continuously compounded rate of return of the stock.

-‐ 𝑁 𝑑 Denotes the value of the accumulative normal density function

The 𝑁 𝑑 terms of the formula increase, as the option is more likely to expire in the money. ln !!

! Can be interpreted as an approximation of the percentage amount the call option is in

Percentiles VIX 1% 8.58 25% 8.78 50% 18.335 75% 71.55 99% 81.78

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the money. Where 𝜎 𝑇 can be interpreted as an adjustment of the moneyness of the option for the stock price volatility over the remaining life of the option. If the stock price volatility and the time till expiration are low, a currently in the money call option is more likely to expire in the money. Therefore the 𝑁 𝑑 terms can be seen as risk adjusted probabilities that the call option will expire in the money. 𝑇 Will approach zero near the end of the life of the option, as a consequence 𝑑_! will approach infinity or minus infinity depending on whether the option expires in the money or not, equating to values 𝑁 𝑑 of zero or one. If 𝑁 𝑑 is 1, it is sure the option will expire in the money, and thus the value of the option is 𝑆_!− 𝑋𝑒!!!!_{. If}

𝑁 𝑑 is 0, it is certain the option will expire out of the money and the value of the option is zero (Black & Scholes 1973). Based on these probabilities and the current ‘moneyness’ of the call option the Black and Scholes price for the call option is determined.

One of the assumptions enabling the Black-Scholes formula to calculate current values for call option is assuming a constant 𝑟_! and 𝜎 over the lifetime of the option. To estimate option prices accurately, the parameters of the formula need to be determined accurately. 𝑆_!, 𝑋, 𝑇 and 𝑟_! are agreed upon in the option’s contract or are easily observable or predictable over the short term. The standard deviation of the returns of the stock is not easily observable and unpredictable over the short term (Black & Scholes 1973). As a consequence of active trade in option contracts, the market’s approximation of 𝐶 is also observable. Using the market price of the call option, one can work backwards through the Black and Scholes formula and calculate the volatility implied by the market price of the call option, the implied volatility. The implied volatility in this analysis is obtained from 20-day till expiration call options on the AEX index.

The type of option (put or call) used is irrelevant as a consequence of the put-call parity. This is a no arbitrage argument; one can but a call option and add an investment in a risk free asset. One can buy the underlying stock, and buy a put option with the same exercise price. Both strategies produce an identical payoff pattern below the strike-price; both will yield (almost) the same implied volatility. The put-call parity is formulated in the following formula, where 𝑃 denotes the price of the put option.

𝐶 + 𝑋 1 + 𝑟! !

= 𝑆!+ 𝑃

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3.3 Sub periods

The sub periods can be categorized as follows:

Sub period Timespan Data points

1) Pre crisis 2/1/2001 – 31/12/2007 1,788 2) Crisis/high volatility 1/1/2008 – 31/12/2011 1,026 3) Post crisis 1/1/2012 – 31/12/2015 1,019

As mentioned in the paper of Ivashina & Scharfstein (2009), the seeds of the severe financial crisis of 2008 were sown in the credit boom ending in 2007. The decline starting in the end of 2007 caused the meltdown of mortgages and different types of securitized products. This let to the banking panic of 2008, with its peak at the forth quarter of 2008, where investment banks as Lehman Brother, Washington Mutual, Fannie Mea, Freddie Max and AIG failed. This clearly distinguishes 2007 as a pre-crisis year and 2008 as a crisis year. The Federal Reserve and Treasury of America kept track of on-going press releases concerning monetary policy to regain financial welfare. The final report was made 11th of April 2011, published on the official Federal Reserve website. This gives 2011 a clear preference for the final year of the crisis period. It is unclear if and when the financial crisis of 2008 exactly ended;

nevertheless it provides the research with a dataset clearly distinguishable as a high-volatility crisis period. As a consequence 2012 until 2015 is categorized as a post-crisis period.

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4. Methodology

The broad analysis of the return-implied volatility relationship is divided in four sub questions:

Q1: Does the change in the Dutch implied volatility index explain the change in the AEX index returns?

For examining the first sub question, a simple regression model is used to test the overall significance of the AEX-VIX relationship. According to theory and past research, a very significant negative relation is to be expected. The returns of the AEX are specified as 𝑟_!"#,!.

This is the dependent variable 𝑌! and is specified as: ln 𝐴𝐸𝑋! − ln 𝐴𝐸𝑋!!! . The change in

the VIX is specified as 𝑟!"#,!. This is the explanatory variable 𝑥! and is specified as:

ln 𝑉𝐼𝑋_! − ln 𝑉𝐼𝑋_!!! . Let 𝛽_! be a constant and 𝑒_! the error term. The setup of dependent and explanatory variables can be assumed to be constant throughout the research.

First stage regression:

(R1): 𝑌! = 𝛽!+ 𝛽!𝑥!+ 𝑒! H1: 𝐻0: 𝛽₁= 0 𝐻1: 𝛽₁< 0

The coefficient β₁will be examined using an ordinarily least squares regression with robust standard errors. Robust standard errors are used because of the high amount of

heteroscedasticity and serial correlation that is paired with the use of daily data. OLS and robust standard errors are used throughout the analysis.

Q2: Are there asymmetries in the results, and are the asymmetries dependent on the market situation?

The second stage of the analysis employs six independent regressions.

The first regression is done with the exclusion of data points with positive

innovations of the explanatory variable 𝑥!. Researching the AEX-VIX relationship in times of

growing index returns. These regressions are categorized with ‘a’. This is done for the three distinct sub periods.

The second regression is done with the exclusion of data points with negative

innovations of the explanatory variable 𝑥!. Researching the AEX-VIX relationship in times of

declining index returns. These regressions are categorized with ‘b’. This is done for the three distinct sub periods.

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Second stage regressions:

(R2a): 𝑌_!,!"!!!"#$#$ = 𝛽_!+ 𝛽_!𝑥_!+ 𝑒_! (R2b): 𝑌_!,!"#!!"#$#$ = 𝛽_!+ 𝛽_!𝑥_!+ 𝑒_!

(R2a): 𝑌!,!"#$#$ = 𝛽!+ 𝛽!𝑥!+ 𝑒! (R2b): 𝑌!,!"#$#$ = 𝛽!+ 𝛽!𝑥!+ 𝑒!

(R2a): 𝑌!,!"#!!!"#$#$ = 𝛽!+ 𝛽!𝑥!+ 𝑒! (R2b): 𝑌!,!"#$!!"#$#$ = 𝛽!+ 𝛽!𝑥!+ 𝑒!

“a’’ Categorized regressions; in times of growing index returns, are excepted to have a less pronounced relationship compared to the “b” categorized regressions; in times of declining index returns. According to the behavioural approach, declining stock markets cause ‘fear’ and therefore reactions in implied volatility are stronger. In all three sub periods I expect the coefficients to be more significant for ‘b’ type regressions; theory predicts a stronger relationship in down market situations. H2a Is formulated to confirm the asymmetry, rejecting the null hypothesis proves the observed asymmetry.

H2a: 𝐻₀: 𝛽_{1,𝑑𝑒𝑐𝑙𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠}= 𝛽_{1,𝑖𝑛𝑐𝑟𝑒𝑎𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠} 𝐻₁: 𝛽_{1,𝑑𝑒𝑐𝑙𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠}< 𝛽_{1,𝑖𝑛𝑐𝑟𝑒𝑎𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠}

The negative relationship is expected to be less pronounced in high volatility states of the market, so the coefficients of the R2a and R2b (crisis) regressions are expected to be less significant than R2a and R2b (pre-crisis) and R2a, and R2b (post-crisis) regressions. The coefficients of the regressions on the crisis sub period are expected to be less significant than the coefficients of the other sub periods. H2b is formulated to confirm the sub period

differences in the return-volatility relationship. Rejecting the null hypothesis confirms the observed relationship is less pronounced in times of high volatility.

H2b: 𝐻0: 𝛽1,𝑐𝑟𝑖𝑠𝑖𝑠= 𝛽1,𝑜𝑡ℎ𝑒𝑟 𝑠𝑢𝑏 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝐻1: 𝛽1,𝑐𝑟𝑖𝑠𝑖𝑠< 𝛽1,𝑜𝑡ℎ𝑒𝑟 𝑝𝑒𝑟𝑖𝑜𝑑𝑠

Q3: Are the results dependent on extreme data-points.

The crisis sub period provides the research the ability to compare high- and extreme volatility market situations. The potential difference is researched using two different regressions. The first on the complete sub period, including peaks of implied volatility as high as 81,78. This regression can be categorized as an ‘a’ regression. In the second regression levels of implied volatility higher than 50 are excluded from the analysis. This value is chosen, as it is more than twice the mean implied volatility. The dataset can be regarded a good representation of a high volatility trading environment. This regression can be categorized as a ‘b’ regression.

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Third stage regressions:

(R3a): 𝑌_{!,!"#$!%! !"#$%&#&%'} = 𝛽_!+ 𝛽_!𝑥_!+ 𝑒_! (R3b): 𝑌_{!,!!"! !"#$%&#&%'} = 𝛽_!+ 𝛽_!𝑥_!+ 𝑒_!

I expect the exclusion of the extreme data points to produce a significant difference in the coefficient 𝛽!. Rejecting the null hypothesis H3 confirms this expectation.

H3:

𝐻!: 𝛽!,!"#$!%! !"#$%&#&%' = 𝛽!,!!"! !"#$%&#&%' 𝐻!: 𝛽!,!"#$!%! !"#$%&#&%' ≠ 𝛽!,!!"! !"#$%&#&%'

Q4: Are past changes in the Dutch implied volatility predictors for present market returns?

The fourth sub question is examined using a multiple regression analysis. Again the returns of the AEX are specified as 𝑟_!"#,!. This is the dependent variable 𝑌! and is specified as:

ln 𝐴𝐸𝑋! − ln 𝐴𝐸𝑋!!! . The change in the VIX is specified as 𝑟!"#,!. This is the explanatory

variable 𝑥! and is specified as: ln 𝑉𝐼𝑋! − ln 𝑉𝐼𝑋!!! . Let 𝛽! be a constant and 𝑒! the error

term. By the inclusion of p, lag changes in implied volatility can be tested. The fourth stage regression includes a coefficient for contemporal returns, and coefficients for the one- till seven day lag changes in VIX. The first seven data points are excluded from the regression, as no values of implied volatility for 𝛽! correspond to the first seven entries of 𝑟!"#,!.

Forth stage regression

(R4):

𝑌_! = 𝛽_!+ 𝛽_!𝑥_!+ 𝛽_!𝑥_!!!+ 𝛽_!𝑥_!!!+ 𝛽_!𝑥_!!!+ 𝛽_!𝑥_!!!+ 𝛽_!𝑥_!!!+ 𝛽_!𝑥_!!!+ 𝛽_!𝑥_!!!+ 𝑒_!

The coefficient 𝛽_! is expected to be significantly different from 0, as suggested by theory, past research and (R1). H4a aims to test this and will most likely be rejected. Giot (2005) found that extreme peaks of implied volatility are indicators of oversold markets. For the analysis on the AEX index I expect to find significant coefficients for the lag changes of implied volatility, rejecting H4b will confirm this.

H4a: 𝐻0: 𝛽₁= 0 𝐻1: 𝛽₁< 0

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5. Results

5.1 - R1: The overall relationship between VIX- and AEX returns on the entire period

Y = rAEX Coefficient Rob.Std.Err. t-value P>|t| 95% C.I.

low

95% C.I. high

rVIX -.139997 .0047618 -29.40 0.000 -.149333 -.1306611

_constant -.000105 .0001713 -0.61 0.540 -.0004409 .0002309

Exhibit 3: Results of R1, tested with OLS using robust standard errors.

As expected a very large significant negative relationship was found between the implied volatility and the index returns. The t-value of -29.40 by far exceeds the 0.01 and 0.05 alpha boundary for significance. The p-value of approximately 0.000 confirms this. With the underlying theories provided one can confirm that changes in implied volatility explain changes in index returns for the Dutch index. The null hypothesis is rejected.

H1: 𝐻₀: 𝛽₁= 0 𝐻₁: 𝛽₁< 0 Number of obs. 3,835 F(1,3833) 864.35 Prob >F 0.0000 R-squared 3,835 Root MSE .01061

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5.2 - R2: Testing for asymmetries on distinct sub periods

Sub period 1: Pre-crisis

low

95% C.I. high

rVIX -.1150053 .0124765 -9.22 0.000 .1395002 -.0905103

_constant .0006952 .0006952 5.08 0.540 .0048954 .0048954

Exhibit 4: Results of R2a, on the pre–crisis sub period, test of growing index returns on negative innovations in VIX, tested with OLS using robust standard errors.

low

95% C.I. high

rVIX -.1191286 .0091934 -12.96 0.000 -.149333 -.1010761

_constant -.0040876 .0005789 -7.06 0.000 -.0004409 -.0029507

Exhibit 5: Results of R2b, on the pre-crisis sub period, test of declining index returns on positive innovations in VIX, tested with OLS using robust standard errors.

Number of obs. 715 F(1,3833) 84.97 Prob >F 0.0000 R-squared 0.2214 Root MSE .00994 Number of obs. 649 F(1,3833) 167.91 Prob >F 0.0000 R-squared 0.3314 Root MSE .00984

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Sub period 2: Crisis

low

95% C.I. high

rVIX -.1788741 .0203407 -8.79 0.000 -.218858 -.1388902

_constant .0023011 .0011121 2.07 0.039 .000115 .0044873

Exhibit 6: Results of R2a, on the crisis sub period, test of growing index returns on negative innovations in VIX, tested with OLS using robust standard errors.

low

95% C.I. high

rVIX -.1765447 .0158654 -11.13 0.000 -.2077358 -.1453536

_constant -.0029243 .0008319 -3.52 0.000 -.0045598 -.0012888

Exhibit 7: Results of R2b, on the crisis sub period, test of declining index returns on positive innovations in VIX, tested with OLS using robust standard errors.

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Sub period 3:Post- crisis

low

95% C.I. high

rVIX -.0893364 .0162948 -5.48 0.000 -.1213659 -.0573069

_constant .0035185 .0007902 4.45 0.039 .0019653 .0050716

Exhibit 8: Results of R2a, on the post-crisis sub period, test of growing index returns on negative innovations in VIX, tested with OLS using robust standard errors.

low

95% C.I. high

rVIX -.0730126 .0251993 -2.90 0.004 -.1225626 -.0234626

_constant -.0044253 .0013904 -3.18 0.002 -.0071592 -.0016913

Exhibit 9: Results of R2b, on the post-crisis sub period, test of declining index returns on positive innovations in VIX, tested with OLS using robust standard errors.

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Testing the asymmetry on the distinct sub periods produced differing outcomes. The t values, coefficients and R squared values per sub period are specified in the following table:

Regression type Coefficient t-value R squared

Pre-crisis, Growing index returns -.1150053 -9.22 0.2214 Pre-crisis, Declining index

returns

-.1191286 -12.96 0.3314

Crisis, Growing index returns -.1788741 -8.79 0.3798 Crisis, Declining index returns -.1765447 -11.13 0.4352 Post-crisis, Growing index

returns

-.0893364 -5.48 0.3039

Post-crisis, Declining index returns

-.0730126 -2.90 0.2866

Excibit 10: Results of regression 2a and b, for each sub period, tested with OLS using robust standard errors. For the pre-crisis and crisis sub periods, results are in line with expectations. The values of R2

are substantially higher in times of market declines and thus ‘fear’ of agents trading option contracts. Both the volatility feedback hypothesis and the behavioural approach give explanations for these findings. Judging from the t-values and the values of R2 _{it’s clear that}

down market situation are paired with higher changes of implied volatility. I reject the null hypothesis H2a for the pre-crisis and crisis sub period.

H2a: 𝐻0: 𝛽_{1,𝑑𝑒𝑐𝑙𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠}= 𝛽_{1,𝑖𝑛𝑐𝑟𝑒𝑎𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠} 𝐻1: 𝛽_{1,𝑑𝑒𝑐𝑙𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠}> 𝛽_{1,𝑖𝑛𝑐𝑟𝑒𝑎𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠}

For the post-crisis regression, the value of R2 _{is found to be smaller in the case of declining}

index returns. This finding is not in line with any of the theories proposed to explain the observed asymmetry. The leverage hypothesis does not assume any asymmetry; the volatility feedback hypothesis and the behavioural approach predict declining index returns to be paired with relatively large amounts of changes in VIX. For sub period three, the null hypothesis cannot be rejected.

Furthermore, the coefficients of the regressions on the crisis sub period are expected to be less significant than the coefficients of the other sub periods. Hibbert et al. (2007) found weaker results for their analysis on the Nasdaq data, and claim this could be directly related to the greater level of volatility found in that market. H2b is formulated to test this claim.

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H2b: 𝐻0: 𝛽_{1,𝑐𝑟𝑖𝑠𝑖𝑠}= 𝛽_{2,𝑜𝑡ℎ𝑒𝑟 𝑠𝑢𝑏 𝑝𝑒𝑟𝑖𝑜𝑑𝑠} 𝐻1: 𝛽_{1,𝑐𝑟𝑖𝑠𝑖𝑠}< 𝛽_{1,𝑜𝑡ℎ𝑒𝑟 𝑝𝑒𝑟𝑖𝑜𝑑𝑠}

As is visible in exhibit 10, the values of R2 _{in times of crises exceed both the value of R}2_for

the pre- and post crisis sub period. Clearly this finding is against the expectations based on the past findings of Hibbert et al. (2007); where a high volatility trading environment produced a less pronounced relationship between implied volatility and index returns. These findings will need further research. Null hypothesis H2b cannot be rejected.

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5.3 - R3: Testing for differences between extreme- and high volatility trading environments

low

95% C.I. high

rVIX -.1741653 .0064521 -26.99 0.000 -.1868269 -.1615038

_constant -.0003321 .0003341 -0.99 0.320 -.0009878 .0003236

Exhibit 11: Results of R3a, on the crisis sub period, with the exclusion of 40 data points with an implied volatility higher than 50, tested with OLS using robust standard errors.

low

95% C.I. high

rVIX -.1883539 .0073864 -25.50 0.000 -.2028481 -.1738596

_constant -.0004599 .0003652 -1.26 0.208 -.0011765 .0002568

Exhibit 12: Results of R3b, on the crisis sub period, without the exclusion of 40 data points with an implied volatility higher than 50, tested with OLS using robust standard errors

Number of obs. 986 F(1,3833) 728.64 Prob >F 0.0000 R-squared 0.6061 Root MSE .01053 Number of obs. 1,026 F(1,3833) 650.25 Prob >F 0.0000 R-squared 0.6045 Root MSE .0117

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The H3 hypothesis is formulated to test the relevance of excluding extreme measures of volatility, possible providing insight in the difference between high- and extreme volatility environments. Past research of for example (Giot 2005) and Hibbert et al. (2007) find the relationship to be weaker in times of high volatility. Based on past research, one would expect the regression with the exclusion of extreme data points to produce a more significant

outcome. The values of R2_{of regressions R3a and R3b are 0.6061 and 0.6045 respectively.}

These values are almost identical. The t values of both regressions are both extremely significant, and only differ 1.49 points.

H3: 𝐻0: 𝛽_{1,𝑒𝑥𝑡𝑟𝑒𝑚𝑒 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦}= 𝛽_{1,ℎ𝑖𝑔ℎ 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦} 𝐻0: 𝛽_{1,𝑒𝑥𝑡𝑟𝑒𝑚𝑒 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦}≠ 𝛽_{1,ℎ𝑖𝑔ℎ 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦}

The small difference in values of R2 _{and t statistics do not give reason to suspect the}

exclusion of extreme measures of implied volatility has a significant effect on the return-volatility relationship. A possible explanation for the small difference between R3a and R3b is the small quantity of excluded data points (40). Applying the same regression to for example the third and forth quarter of 2008 might produce a more significant difference in significance. Also the boundary for excluding data points is subjective, one might consider measures of implied of 45 to be extreme, and as a consequence exclude more data points in R3b. Based on these findings, H3 cannot be rejected.

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5.4 - R4: Testing for predictive powers of past changes in implied volatility, on index returns.

Y = rAEX Coefficients Rob.Std.Err. t-value P>|t| 95% C.I.

low 95% C.I. high rVIX -.1464607 .0045733 -32.03 0.000 -.155427 -.1374944 rVIXt-1 -.0299589 .0029737 -10.07 0.000 -.0357891 -.0241287 rVIXt-2 -.0125775 .0028198 -4.46 0.000 -.018106 -.0070491 rVIXt-3 -.0021765 .0029138 -0.75 0.455 -.0078892 .0035362 rVIXt-4 -.0089749 .0026894 -3.34 0.001 -.0142476 -.0037021 rVIXt-5 -.0004289 .0029804 -0.14 0.886 -.0062722 .0054143 rVIXt-6 -.0030521 .0030385 -1.00 0.315 -.0090094 .0029051 rVIXt-7 -.0048618 .0026452 -1.84 0.066 -.0100478 .0003243 _constant -.0000991 .0001678 -0.59 0.555 -.0004281 .0002299

Exhibit 13: Results of R4, on the entire period, testing for predictive power of past levels of implied volatility, tested with OLS using robust standard errors

Probably the most exciting outcome of this research is specified in Exhibit 13. In the fourth regression the lag changes of implied volatility are tested on index returns, answering the question most relevant for investors: Is acquiring long positions after negative innovations of implied volatility a way to generate positive returns.

The following hypothesis were proposed for R4:

H4a: 𝐻0: 𝛽10 𝐻1: 𝛽1< 0 H4b: 𝐻_!: 𝛽_!, 𝛽_!, 𝛽_!, 𝛽_!, 𝛽_!, 𝛽_!, 𝛽_! = 0 𝐻_!: 𝛽_!, 𝛽_!, 𝛽_!, 𝛽_!, 𝛽_!, 𝛽_!, 𝛽_!≠ 0 Number of obs. 3,828 F(1,3833) 148.85 Prob >F 0.0000 R-squared 0.5199 Root MSE .01038

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H4a is rejected, as expected by theory, past research and R1. The interesting part of the regression is the test on the intertemporal relationship between implied volatility and returns. As is visible in Exhibit 13, 𝛽!, 𝛽!, and 𝛽! are significant with an alpha of 0.01. This suggests

acquiring long positions after negative innovations of implied volatility generates a positive return on a one, two and four day timeframe. As mentioned before, this conclusion was not excluded from expectations, as data on implied volatility on the AEX index is not freely available to all investors, and Giot (2005) reached a similar conclusion using S&P 100 and Nasdaq 100 data. As this regression was done with one measure of implied volatility, the significance of the results can be questioned. Another shortcoming of R4 is the exclusion of transaction costs in the model. After subtracting transaction cost, returns may be equal to or even below 0. Hypothesis H4b is rejected, but drawing final conclusions from R4 is

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6. Summary and conclusions

This paper examines the relationship between index returns and implied volatility for the Dutch index. This relationship was researched in four distinct sub questions. At first, the overall significance of the return-implied volatility relationship was tested on the whole period, logically rejecting H1.

In the second stage of the research, a test on asymmetry was done on the distinct sub periods. For the pre-crisis and crisis period the findings confirm the expected asymmetry. For this period market declines are indeed paired with relatively larges changes of implied volatility. The regression on the post-crisis period produced a reverse asymmetry, where the return-volatility relationship proved to be more significant for positive index returns.

Furthermore the findings show a stronger relationship in a high volatility market environment. This will need further research.

In the third stage of the research, the relevance of extreme measures of volatility was tested. Little difference is the return-volatility relationship is found between high- and extreme volatility trading environments.

Finally, the significance of lag changes of implied volatility was tested in the last stage of the research, where multiple coefficients are found to be extremely significant. The difficulty of acquiring data on the implied volatility index on the AEX may be the direct cause of these results, as prices might not reflect all information available. As mentioned before, the lack of implementation of transaction costs and the use of one measure of implied pull the results into question.

In the analysis one measure of implied volatility is used; derived from the price of a 20-day till expiration near the money call option. For the S&P 500, a new VIX index was introduced. This index of implied volatility is derived from put and call options with differing strike prices. According to Hibbert et al. this increases the practical appeal of the VIX, and provide the research with a more robust measure of implied volatility as the data now includes the option implied volatility skew. This measure of implied volatility is not available for the Dutch index. If available, this measure of implied volatility would be preferred for the analysis.

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