Is the Euro STOXX 50® volatility index (VSTOXX®) a significant forecaster for financial stress in the Euro area?

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Universiteit van Amsterdam Author: Fabio Scaramelli Supervisor: L.Zhao

IS THE EURO STOXX 50® VOLATILITY INDEX

(VSTOXX®) A SIGNIFICANT FORECASTER FOR

FINANCIAL STRESS IN THE EURO AREA?

Bachelor Economics & Business Student ID: 10004498

This paper’s aim is to foresee disruptive event in the economy in the Euro area by using the EURO STOXX 50® VOLATILITY INDEX (VSTOXX®) as stress measure. The VSTOXX is assumed to have a causal relationship with the stock market volatility. Stock market volatility is correlated with the (non)economic activities. Therefore, VSTOXX is able to serve as an early warning indicator for negative impacts on the economy. The causal relationship between VSTOXX and the economy is examined by constructing multiple regression models for testing the relationship of implied volatility with realized stock market volatility. The empirical analysis reveals that implied volatility has less predictive power than the commonly used for financial markets, especially stock markets, historical volatility. However, the empirical results show that by using realized volatility as explanatory variable is inefficient. Instead, results indicate that both implied and realized volatility should be implemented to forecast future realized volatility.

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2 Introduction

The European economy is currently experiencing a period of financial stress. This stress is led by an unexpected increase of financial losses, a lack of confidence and an increase of uncertainty in the European economy. The high level of unemployment rate, the great amount of debt in several sectors in the economy and the financial imbalances are indicators of the severeness of this matter. This has led international investors to discriminate more strongly among emerging market economies, capital flows to countries with sizeable external imbalances and domestic weaknesses to dry up. For this reason, there is a great need for constructing and finding better models and indicators for disruptive events, in order to be prepared and to mitigate their effects.

In the past recoveries, policymakers would base their strategy mainly on the change on economic factors; the key economic factors include labor costs, interest rates, government policy, taxes and management. These factors need to be taken into account when determining the current and expected future value of a business or investment portfolio. However, these economic factors can give mixed signals, and because of that it might be difficult for policymakers to make the right policy decision. Further, the investors’ attitude towards the financial market is not always derived from these economic fundamentals. Therefore, it might be of significant value to find an indicator, which is influenced not only by economic factors but also by non-economic factors. Stock market volatility is assumed to fit this criterion: the research done by (Beetsma & Giuliodori, 2014) has proven that the stock market volatility significantly affects the macro-economy, i.e., the level of volatility on the financial market is correlated with economic activities.

A different research conducted by Campbel, Lettau, Malkiel, & Xu, (2001) discusses that

market volatility is related to structural change in the economy. This research suggests that elevated stock market volatility reflects an enhanced uncertainty about future cash flows and discount rates, which is a result from fundamental changes in the economy that depress GDP

growth. Moreover, Campbel, Lettau, Malkiel, & Xu, (2001) show that stock market volatility

has significant predictive power for real GDP growth. According to the results of this research, realized market volatility provides relevant information about future economic activities.

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To forecast future realized volatility, the focus will be on the volatility index that is jointly developed by Goldman Sachs and Deutsche Börse AG. The EURO STOXX 50® VOLATILITY INDEX (VSTOXX®) is one of the products of Eurex with the highest trading volume, which is based on the implied volatility derived from EURO STOXX 50® Index Options.

“The VSTOXX Indices are based on EURO STOXX 50 realtime options prices and are designed to reflect the market expectations of near-term up to long-term volatility by measuring the square root of the implied variance across all options of a given time to expiration” –website of STOXX.

Whether the implied volatility is a significant forecaster of future realized volatility will be tested in this paper, done by testing the level of correlation between the EURO STOXX 50®

VOLATILITY INDEX (VSTOXX 30 days) and the realized volatility of the EURO STOXX

50 INDEX NET RETURN. The EURO STOXX 50 INDEX NET RETURN is European’s

blue chip index, which covers approximately 60% of the free floating market capitalization of Europe. Net return refers to the assumption that all cash (including dividend) distributions are reinvested after deduction of withholding taxes.

Volatility is a measure of the level of uncertainty in the financial market. An approach to

estimate volatility is by computing implied volatility through Black-Scholes formula. Implied

volatility indices are aligned to future development and based on transparent market prices, these indices are attractive for investors. Implied volatility is expected to capture both uncertainty about the fundamental values of assets and uncertainty about the behavior of (other) investors (Hakkio & Keeton, 2009).

Unfortunately the majority of the academic literature, which tests the level of forecasting power of implied volatility on future realized volatility, focuses predominantly on the US markets while leaving the European markets largely untouched, i.e., most research do not test the application of the EURO STOXX 50® VOLATILITY INDEX as a significant forecaster of future realized volatility, but instead focuses on the CBOE VOLATILITY INDEX® (VIX®), the volatility index of the S&P 500. That being said, the results established from these researches, using the VIX as indicator, are considered relevant and fundamental for this paper.

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Bearing this in mind, previous research conducted by Christensen & Prabhala, (1996)

concludes that for a longer period, implied volatility does effectively forecast future realized volatility; in fact the level of significance outperforms the most common alternative method for forecasting future volatility: the historical volatility. These results are verified by Blair, Poon, & Taylor, (2000) and Corrado & Miller, (2005).

Similar to prior studies, this paper tests the forecast quality of the EURO STOXX 50® VOLATILITY INDEX (VSTOXX) 30 days, extracted from the data available from

Bloomberg. The methodology which Corrado & Miller, (2005) formulated will be applied

further in the research of this paper; it is a method which avoids overlapping data

(overlapping data causes estimators to be biased and inefficient)i. This will enable us to

construct a series of data covering one implied and one realized volatility for each time period. The focus will be on two closely related topics between implied volatility and forecast future volatility: 1) the contemporaneous relationship between relative changes in implied volatility and changes in realized volatility, and 2) the possible relationship between implied volatility and future realized volatility.

This paper is organized as follows: In the next section an analysis will be by provide in form of a literature review on related studies. In section 3 we discuss the data acquisition process and some statistical properties. In section 4, we expand on the causal relationship between realized volatility and financial stress. Furthermore, we analyze and discuss the data using the methodology employed by Corrado and Miller. In section 5 we provide the detailed conclusion derived from this analysis and we conclude this paper with a summary conclusion.

i

Overlapping samples are characterized by “maturity mismatch”, in which the time-period is not equivalent to

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5 Literature Review

The role of stress measurement has become a widely researched topic, particularly since the disruption on the financial market, begin 2008, that has been negatively effecting Europe on different sectors and scales. This section of the paper is to acknowledge the broad view of the different methods applied in researches that test the ability of (composite) indicators to forecast financial stress. In closure, we address the methods alternatively to implied volatility for forecasting future realized volatility.

Nearly all academic researches obtained from academic literature have constructed a

composite financial stress indicator (FSI), which is comprised of several underlying variables.

For example, the article conducted by Aboura & van Roye, (2013) developed a financial

stress index for France by creating a real-time financial stress indicator, consisted of 17 financial variables of different market segments. According to this report, the financial variables used were considered of minor relevance to be included in macroeconomic models, prior to the financial crisis. The results of the research show contrary evidence in terms of relevance; they conclude that the FSI is correlated with highly stressful economic events.

Therefore,this indicator can be used as an early warning signal of market risk in the French

financial sector. A similar method is formed by (Jakubík & Slačík, 2013), the commonality between the two articles, besides the similar methods, is considered to be the complexity given the interaction between the independent macroprudential variables. Even though both studies conclude that the econometric models constructed can be used as a tool to detect potential financial imbalances.

On the contrary to previous articles, Estrella & Mishkin, (1998) are convinced that

policy makers can benefit greatly by just examining a few well-chosen indicators for forecasting future U.S. recessions. First, it is beneficial to test one or a few explanatory variables at a time, so that one can be reasonably sure that the results one gets are accurate and valid. A second reason for looking at simple financial indicators is the potential problem of overfitting. Overfitting generally occurs when the model is excessively complex and associated with poor predictive ability. A third reason for looking at single financial indicators is because it is a simple and quick method. The results show that overfitting is considered to be a significant problem in macroeconomic. Moreover, a few simple indicators, such as interest rates and stock prices, have equal or greater predictive power than complex models.

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Regards to implied volatility, the article conducted by Lewis & Day (1993) tested the accuracy of the two methods: implied volatility and the generalized autoregressive conditional heteroskedasticity model, the latter known as GARCH-method. In short, the complex GARCH-model is a model often employed in modelling financial time series, which displays time-varying volatility. They have proven that the GARCH-method is implied with incremental information that is not incorporated in the option price framework. Nonetheless, their research provides evidence that neither the GARCH-method nor the simple-historical volatility is a superior indicator to forecast near-term volatility compared to implied volatility. However, Canina & Figlewski (1993) have proven contrary evidence to Lewis & Day. They conclude that the implied volatility derived from the S&P 100 Index options (OEX option), the most liquid option-market of the United States of America, to be a poor indicator of future realized volatility. They are firmly stating that the demand and supply of the option-market is influenced by exogenous variables, which are not impounded in the option-pricing model. According to Canina and Figlewski (1993) to compute the most significant forecaster of future volatility both implied volatility and historical volatility should be used as explanatory variables to derive a significant estimate of future realized volatility.

Bandopadhyaya & Jones (2008) conducted a research using data from January 2, 2004 until April 11, 2006, that focuses on two measures of investor sentiment: the volatility index (VIX) and the put/call ratio (PCR). In order to determine which measurement is significant for explaining short-term movements in the financial market, S&P 500. According to the results PCR is a better indicator than the VIX. However, this research does not discuss the predictive power of the put/call ratio or the VIX, on future realized volatility.

From the interpretation of the various papers, we conclude that the opinions differ with respect to forecasting future volatility. On one hand, implied volatility is considered to be a weak explanatory variable for realized volatility; on the other hand, it is proven to be a significant indicator for forecasting future realized volatility. Thus, no general conclusion has been established.

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7 Data Sources and Volatility Measure

Data Sources

The data is acquired primarily from Bloomberg. Bloomberg is a platform used mainly to collect information on business and financial markets. The secondary data source is STOXX Limited. STOXX Limited is a leading index specialist specified on the European market. The data consists of index prices and option-implied volatilities. Index prices are acquired from the EURO STOXX 50 Index, noted as the following tickersymbol: SX5T. EURO STOXX 50 Volatility Index (VSTOXX) is the measure used for implied volatility, noted as V2TX.

The EURO STOXX 50 Index is a stock market index of the Eurozone developed by STOXX Ltd. Its goal is to provide an index representative, representing the fifty most profitable, reliable and qualified corporations in supersectors. The index covers 50 stocks from 12 Eurozone countries: Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, the Netherlands, Portugal, and Spain (STOXX, 2014).

In 2005, the Eurex Exchange introduced the volatility index, noted as The EURO STOXX 50 Volatility Index. The EURO STOXX 50 volatility index is an implied volatility derived from the EURO STOXX 50 index options. This volatility derivative index measures the market participants’ sentiment of short-term volatility in the Eurozone.

The data sample timespan of SX5T and V2TX is from January 1999 through December 2013. The data will be further implemented for testing and analyzing purposes.

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Volatility Measure

The volatility measure used in this study is the sample standard deviation of daily index continuous return of the EURO STOXX 50 Index, which serves as the dependent variable for this study. Daily index return, also known as rate of return, is computed by using the ordinary return method (I). In finance it is common to use the log-return method for mathematical convenience, such as time-additive. However, in this study the ordinary return method is chosen for simplification and accuracy.

Monthly standard deviation is computed by applying the standard deviation sample formula (III) to each month, with daily index returns as degree of freedom. As final, to compute

annualized index realized volatility, noted as VOLM, monthly standard deviation is multiplied

by the square root of the average trading days from 1999 through 2013. According to statistical calculations, there are on average 257 trading days in a year.

(III)

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9 Data Summary Statistics

The data on the EURO STOXX 50 index and the EURO STOXX 50 volatility index distributed from 1999 through 2013 is partitioned into 108-month. Figure 1 provides a

graphical display of the time-serie of realized volatility (VOLM) and implied volatility

(VSTOXXM-1).

FIGURE 1

EURO STOXX 50 realized vs implied volatility

In table 1 a descriptive statistics summary is provided of the variables, VOLM and

VSTOXXM-1. Interpreting the results of table 1 it reveals that VOLM and VSTOXXM-1 are

highly leptokurtic and skewed. In addition to the results in table 1, figure 2 and 3 display the sample-distribution of implied volatility and realized volatility.

Table 1

Descriptive Statistics

Obs Mean Std. Dev. Min Max Kurtosis Skewness

VSTOXX 180 26,27886 9,053604 12,70157 65,05569 4,73654 1,233569 VOL 180 21,9084 11,55622 7,423763 78,33401 6,84128 1,771056 0,00% 10,00% 20,00% 30,00% 40,00% 50,00% 60,00% 70,00% 80,00% 90,00% jan -9 9 se p-9 9 m ei -0 0 jan -0 1 se p-0 1 m ei -0 2 jan -0 3 se p-0 3 m ei -0 4 jan -0 5 se p-0 5 m ei -0 6 jan -0 7 se p-0 7 m ei -0 8 jan -0 9 se p-0 9 m ei -1 0 jan -1 1 se p-1 1 m ei -1 2 jan -1 3 se p-1 3 Vola tilit y Realized volatility Implied volatility

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Skewness measures the lack of symmetry of a distribution (Stock& Watson, 2012). Figure 2 and 3 display a non-normal distribution of the sample, both figures are asymmetric to the left, also referred as skewed right. Kurtosis is a numerical measure to indicate the height and sharpness of the peak of a distribution, the reference standard is a normal distribution which has a kurtosis level of three. The figures 2 and 3 display high and sharp peaks indicating that the distribution is more clustered around the mean, rather than a lot of modest differences from the mean. To confirm, the results in table 1 reveal kurtosis levels greater than 3; to be precise implied volatility indicates a kurtosis level of 4,73654 and realized volatility indicates a kurtosis level of 6,84128.

The Wilcoxon sign test of matched pairs is used to test the equality of matched pairs of

observations, by comparing the mean values of VOLM and VSTOXXM-1. The null hypothesis

is that the median of the differences is zero; no further assumptions are made about the distributions. The Wilcoxon test measures a p-value of 0, 00000. This value indicates that both sample means are significantly non-equivalent from each other. In absolute value, implied volatility and realized volatility differ by 4,36046.

According to (STOXX Ltd), implied volatility tends to be higher than realized volatility, due to the fact that investors buying protection in the form of options for their portfolios have to pay a risk premium to the protection seller.

FIGURE 2 Histogram (VOLm) FIGURE 3 Histogram (VSTOXXM-1) 0 .0 1 .0 2 .0 3 .0 4 .0 5 De ns it y 0 20 40 60 80 Vol 0 .0 2 .0 4 .0 6 De ns it y 0 20 40 60 Vstoxx

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However, by taking the log-transformation of realized volatility and implied volatility, which can be applied due to the fact that volatility can’t be negative, different results are found. Table 2 below reveals intriguing results on the log-transformed variables. The first thing that catches the eye is the low level of skewness, the values indicate that the distribution of both samples are almost perfectly symmetric. Furthermore, presuming the rule of thumb on kurtosis the values indicate no leptokurtic distribution.

Table 2

Descriptive statistics

Figure 4 and 5 display the distribution of the log-implied volatility and the log-realized volatility; both figures clearly visualize a stronger log-normal distribution. Furthermore, by

measuring the degree of correlation, the log-variables, Ln(VOLM) and Ln(VSTOXXM-1),

reveal a slightly stronger correlation (0,1075) compared to the original variables, VOLM and

VSTOXXM-1, that measure a correlation degree of 0.0961. Besides the level of correlation, the

R-squared is also greater for the log transformation model (0,0116;0,0092). These results indicate that the log-transformation model fits the data significantly better compared to the non-transformed model.

In conclusion, log-transformation brings the skewness and kurtosis of their volatility data closer to that of a normal distribution. Moreover, the statistical properties of a log- transformation model makes the data to appear more closely to the statistical inferences of this study. Therefore, the log-transformed variables will be exclusively applied in this study.

Obs(%) Mean(%) Std(%) Min (%) Max(%) Skewness(%) Kurtosis(%)

Ln(VSTOXX) 180 3.215973 0.3203825 2.541728 4.175244 0.333418 2.936665 Ln(VOL) 180 2.974784 0.4606414 2.004691 4.360982 0.4231523 2.974693 FIGURE 4 Histogram Ln(VOLM) FIGURE 5 Histogram Ln(VSTOXXM-1) 0 .5 1 1. 5 De ns it y 2.5 3 3.5 4 4.5 LnVstoxx 0 .2 .4 .6 .8 1 De ns it y 2 2.5 3 3.5 4 4.5 lnVol

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12 Methodology

In this section a framework is developed for further analysis of monthly volatility forecast. Furthermore, testable hypothesis will be conducted.

Model Framework

In order to test if implied volatility is a significant forecaster for financial stress, a regression is performed on a multiple fitting model. The data is used to study the evolution of variables over time and to forecast future values of concerning variables. The data set contains observation on two variables(VSTOXX;VOL) to optimize the significance of forecasting power, the data will be divided in two time-periods, a time-period indicating movements before the crisis, from 1999 through 2009, The other time period indicating movement after

the crisis, from 2010 through 2013. The independent variables are EURO STOXX 50

Implied volatility index prices and the dependent variables are the EURO STOXX 50 NET

RETURN realized volatility prices. To split the data set in two time-periods as mentioned previously, a dummy variable (D1,i) is applied. The dummy variable will interact with the continuous variable,VSTOXXM-1,i.. The differential effect of the interaction has important implications for the interpretation of the statistical model. As final, implied volatility will be compared to historical volatility to test if at times historical volatility and implied volatility significantly deviate from each other. Equation (1) and (2) represent the models on which the regression is performed.

ln(VOLM)= αm+β1 D1

,i

+ β2ln(VSTOXXM-1)+ β3(D1

,i

*ln(VSTOXXM-1))+ εm (1)

ln(

VOLM)= αm+ π1D1

,i

+ π2ln(VOLM-1)+ π3(D1

,i

* ln(VOLM-1)) + εm

(2)

M=1,2,3….,180

where

D1 = 1 before the crisis, 1999-2009

D2= 0 after the crisis, 2009-2013

Comment [WU1]: Still, what does “i”

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According to the conclusions drawn from the literature, two testable hypotheses are constructed; one on the forecasting power of implied volatility and one on the comparison of historical volatility versus implied volatility. The first hypothesis is constructed to conclude whether the EURO STOXX 50® Volatility is a significant forecaster for financial stress. According to the literature, the opinions differ with respect to forecasting future volatility. On one hand, implied volatility is considered to be a weak explanatory variable; on the other hand, it is proven to be a significant indicator for forecasting future realized volatility. The coefficient on the interaction term between implied volatility and the dummy variable for the

crisis-period (D1) should be significant. More formally stated:

H0: Implied volatility is a significant forecaster of future realized volatility (β2+β3≠0) H1: Implied volatility is not a significant forecaster of future realized volatility (β2+β3=0) The second hypothesis is constructed to conclude which model forecasts future realized volatility better, the historical volatility or implied volatility.

H0: Historical volatility is a significantly stronger forecaster than implied volatility (π2+π3>

β2+β3)

H1: Historical volatility is not a significantly stronger forecaster than implied volatility

(π2+π3<β2+β3)

Test-assumption: the tests executed on the regression model uses a critical value of 10% for significance level.

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14 Results

In this section, the results of the time-series regression models are analyzed and discussed. At the end of this section, some concluding remarks are made.

-the ordinary least squares assumptions of a multiple regression model are presumed to be met in this study.

Regression analysis

Regression analysis on realized volatility using implied volatility as explanatory variable.

A regression is performed on the log-transformed multiple linear regression model including the lagged implied volatility variable, noted as LnVstoxx, a dummy variable (D_1) and the interaction term between the lagged implied volatility and the dummy, noted as D_2. This regression is performed on 180 observations, every observation indicating a month in the time-period from 1999 through 2013. The results on the first regression are as follows:

(1) LnVol D_1 4.268*** (0.604) LnVstoxx 1.273*** (0.124) D_2 -1.341*** (0.186) _cons -1.076*** (0.398) N 180 R2 _0.1314 Prob > F 0.0000

Standard errors in parentheses * p<0.05, ** p<0.01, *** p<0.001

Figure 6: The estimated parameters on a multiple regression model revealing strong significance for all three parameter; D_1, LnVstoxx and D_2.

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In detail, figure 6 displays the estimated parameters including the standard error and significance values. The heteroskedastic-robust standard errors method is used on the regression, according to the book Introduction to Econometrics (Stock & Watson, 2012) one should use heteroskeskedastic-robust, if the standard errors of the regression differ from heteroskedastic-robust to homoskedastic-only. In “Appendix A” the estimated parameters on the linear regression model assuming homoskedasticity is displayed, detailing that the standard errors indeed differ from heteroskedasticity to homoskedasticity.

However, in figure 7 the distribution of the residual is plotted, analyzing the spread of the distribution indicates no-heteroskedasticity. According to figure 7 the variance of the

residuals conditioned on the explanatory variable, LnVSTOXXM-1, is approximately constant.

Implicating that the estimated parameter is a best linear conditionally unbiased estimator, denoted as BLUE (Stock & Watson, 2012). To be certain, the heteroskedasticity test also known as Breusch-Pagan and Cook-Weisberg test is executed. The test measures a p-value of 99,30% exceeding the critical value of 10%, proving that the null-hypothesis cannot be rejected; which states that the variance of the residuals, conditioned to the lnVstoxx’s, is constant. Due to the different results on heteroskedasticity and homoskedasticity, this study will regress using heteroskedasticity-robust standard errors.

Figure 7: The distribution shows an even envelope of residuals, the width of the envelope is considerably constant for the independent variable, LnVstoxxM-1. -1 -.5 0 .5 1 1. 5 Re s id ua ls 2.5 3 3.5 4 4.5 LnVstoxx

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Regardless of the confusion in terms of using heteroskedasticity or homoskedasticity, the estimated parameters of the regression model are established to be strongly significant. The p-values significantly indicate that the parameters deviate from zero. The statistical interaction

term D_2 and LnVstoxx, in equation(I) noted as

(D1

,i

*ln(VSTOXXM-1)) and

ln(VSTOXXM-1),

indicate that during the time-period before the crisis implied volatility had a significantly negative effect on realized volatility, i.e., from 1999 through 2009 a percentage increase of 1% of the ln(

VSTOXXM-1)

, is associated with a percentage increase

of 1%*0.067334 (

β2+

β3

)ii for the ln(VOLM). Having this said, the model also shows a

considerable low value for R-squared (0,1314), which indicates how well the regression line fits the real data points. In appendix B the fitted line of Ln(Vol) on Ln(Vstoxx), before the crisis and after the crisis, is plotted using a linear regression.

To strengthen these results a F-statistic is used to test joint hypothesis about regression coefficients; null hypothesis is that the coefficients on the 2th and 3th regressors is zero; that is, LnVstoxx=0 and D_2=0. The F-test measures a p-value of 0,000. The p-value is smaller than the critical value of 10%, therefore the null hypothesis is rejected. Concluding that the

parameters, β2 and β3, deviate from zero. In appendix C: the formula and calculation of

F-test.

In order to get a better idea on the relationship between implied volatility and realized volatility, the regression model is expanded by implementing multiple lagged-variables on the implied volatility. The Equation (II) is as follows:

ln(VOLM)= αm+ β1D1 + β2ln(VSTOXXM-1) + β3(D1*ln(VSTOXXM-1)) + β4 ln(VSTOXXM-2) + β5(D1*ln(VSTOXXM-2)) + β6ln(VSTOXXM-3) + β7(D1*ln(VSTOXXM-3)) + εm

(II)

M=1,2,3….,180

where

D1 = 1 before the crisis, 1999-2009

D2= 0 after the crisis, 2009-2013

For D1=1 you have ln(VOLM,i) = (αm + β1) + (β2+ β3)*(ln(VSTOXXM-1)) + εi

For D1=0 you have ln(VOLM,i) = αm + β2*ln(VSTOXXM-1) + εi

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(2) LnVol D_1 3.429*** (0.605) LnVstoxx 1.796*** (0.175) D_2 -2.134*** (0.339) LnVstoxx_1 -.581 (0.184) D_3 0.588 (0.462) LnVstoxx_2 -.113 (0.136) D_4 0.464 (0.304) _cons -.522 (0.395) N 180 R2 _0.1736 Prob > F 0.0000

Figure 8:The estimated parameters on a multiple regression model revealing strong insignificance for the parameters:

LnVstoxx_1, D_3, LnVstoxx_2, D_4.

Figure 8 shows great insignificance among the parameters: LnVstoxx_1, D_3, LnVstoxx_2, D_4 concluding that expanding the model by adding additional lagged-implied variables and interaction terms between lagged implied variables and dummy variables, does not improve the forecasting power on realized volatility.

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Regression analysis on realized volatility using multiple lagged-realized volatility variables as explanatory variable.

A similar regression as to the previous regression using implied volatility will be generated including lagged realized volatility variables, noted as lnVol_1, a dummy variable(D_1) and a interaction term between the lagged volatility variable and the dummy variable, noted as B_2. This regression is performed on 180 observations, every observation indicating a month in the time-period from 1999 through 2013. The results on the regression are as follows:

(3) LnVol D_1 -.762** (0.370) LnVol_1 0.510*** (0.115) B_2 0.254** (0.124) _cons 1.458*** (0.340) N 180 R2 _0.5306 Prob > F 0.0000

Figure 9: The estimated parameters on the multiple regression model reveal strong significance; D_1, LnVol_1

and B_2. ‘the heteroskedasticity-standard errors method is applied’

Figure 9 displays the estimated parameters of the linear regression model. The p-values of all three parameters are smaller than the critical value of 10%, indicating that the parameters are significant, i.e., the estimated parameters significantly deviate from zero.

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The statistical interaction term B_2 and lnVol_1, in equation(II) noted as D1,i*ln(VolM-1) and

ln(VolM-1), indicate that during the time-period before the crisis, the ln(VolM-1) had a

significantly positive effect on realized volatility, i.e., from 1999 through 2009 a percentage

increase of 1% ,of the ln(VolM-1), is associated with a percentage increase of 1%*

0,7641554(π2+π3)iii

for the VOLM. In appendix D the fitted line of Ln(VolM,i) on Ln(Vol

M-1,i), before the crisis and after the crisis, is plotted using a linear regression.

The difference between this linear regression model with the previous linear regression model

is the level of R-squared. This model indicates a R-squared level of 0.5306(R2) which is

significantly greater than 0,1314(R2). In other words, the regressor, Ln(VolM-1), is a better

predictor for realized volatility than ln(VSTOXXM-1). On the contrary, the parameters of the

first regression model are slightly more significant than the parameters of the second regression model, but not greater than the critical value.

To be certain that the less significant coefficients,π2 and π3, deviate from zero a F-test is

performed. The F-test measures a p-value of 0,1222;(12,22%). The p-value exceeds the

critical value of 10%; the null hypothesis cannot be rejected. Concluding that one of the two parameters, π2 and π3, is equal to zero.

iii_{For D}

1=1 you have ln(VOLM,) = (αm + π1) + (π2+ π3)*(ln(VSTOXXM-1)) + εi

For D1=0 you have ln(VOLM,) = αm + π2*ln(VSTOXXM-1) + εi

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(2) LnVol D_1 -.818* (0.492) LnVol_1 0.490*** (0.140) B_2 0.253 (0.165) LnVol_2 0.064 (0.192) B_3 -.035 (0.219) LnVol_3 -.044 (0.140) B_4 0.057 (0.162) _cons 1.359*** (0.466) N 180 R2 0.5349 Prob > F 0.0000

Figure 10:The estimated parameters on the multiple regression model show strong insignificance:D_1 lnVol_1

B_2 lnVol_2 B_3 lnVol_3 B_4iv.

Figure 10 reveals the results of using multiple lagged variables to incorporate feedback over time on a multiple regression model. By analyzing the results, the same conclusion can be drawn as from the previous multiple regression model. Most of the parameters, excluding D_1 and LnVol_1, greatly exceed the critical level of 10%, indicate that these parameters are insignificant. Furthermore, adding multiple lagged variables to the model increases the level

of R-squared with 0,0043(R2). This minor increase of determination and great level of

insignificance implies that including multiple lagged variables in to the regression does not significant increase the power of predicting.

iv

ln(VOLM)= αm + π 1D1 + π 2ln(VOLM-1) + π 3(D1*ln(VOLM-1)) + π 4 ln(VOLM-2) + π 5(D1*ln(VOL M-2)) + π 6ln(VOL M-3) + π7(D1*ln(VOLM-3))

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Regression analysis on realized volatility using lagged implied and realized volatility variables as explanatory variables.

The regression will include a dummy variable, lagged variables and interaction terms between the lagged variables and the dummy variable. The difference now between this regression with the previous ones is that implied volatility and realized volatility is incorporated in one model. The results are as follows:

(3) LnVol D_1 1.941*** (0.478) LnVstoxx 1.470*** (0.160) D_2 -1.496*** (0.183) LnVol_1 -.182** (0.099) B_2 0.945*** (0.109) _cons -1.161*** (0.375) N 180 R2 0.6167 Prob > F 0.0000

Figure 11: The estimated parameters of the linear regression model using lagged implied and realized volatility variables as explanatory variables. ‘The heteroskedasticity-standard errors method is applied’

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The linear regression model displayed above reveals the estimated parametersv including the

p-values, standard errors and confidence interval. The measured p-values indicate strong significance among all parameters, i.e., the p-values do not exceed the critical value of 10% indicating that the coefficients significantly deviate from zero.

In order to be certain that all parameters are significant a F-test is applied. However, this time

one restriction is executed. Due to the fact that all parameters except for one (LnVol_1) are

strongly significant, p-value of 0,000. The results of the F-test reveal that the variable,

LnVol_1, is significant. Concluding that ln(VOLM-1) is a significant predictor for the

dependent variable ln(VOLM).

To conclude, the parameters of the linear regression model are not only significant, they are also good at explaining the values of the dependent variable. Which is interpreted from the level of R-squared(0,6167) of the regression model. At a mathematical level, 1% of

ln(VSTOXXM-1,i) and ln(VOLM-1,i), is associated with a increase of 1%* -0,025367(θ2+ θ3)+

1% *0,7631888(θ4+ θ5) for ln(VolM,i).

v

ln(VOLM)= αm + θ1D1,i + θ2ln(VSTOXXM-1) + θ3(D1,i*ln(VSTOXXM-1)) + θ4ln(VOLM-1) + θ5(D1,i*ln(VOLM-1)) + εi

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23 Conclusion

The empirical analysis of this study is based on the forecasting power of the EURO STOXX 50® VOLATILITY INDEX (VSTOXX®) corresponding to the realized volatility of the EURO STOXX 50 NET RETURN INDEX (SX5T). Realized volatility is assumed to be a significant indicator to address stress in the European financial market. Previous studies have expressed strong disagreements on whether implied volatility is a significant forecaster. Several work indicate implied volatility to be biased and inefficient.

In this study, the results of the multiple linear regression models reveal that implied volatility is less significant in terms of how good the regressors are in predicting the real data compared to realized volatility. However, the F-test executed on implied volatility and realized volatility indicate that implied volatility is unbiased and provides significantly more efficient forecasts than realized volatility.

Further regression analysis reveals that the highest regression R-squared value is obtained when implied and realized volatility are both incorporated in the regression as explanatory variable. This concludes that both implied volatility and historical volatility should be used to derive a significant estimate of future realized volatility.

Whether these results would also hold in other global financial markets have to be concluded from further research on this topic.

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24 Bibliography

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25 Appendix

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B. The best fitted line for the observed data with “LnVstoxx” as explanatory variable. - Before crisis (1999 – 2009) - after crisis (2010 – 2013) 2 2. 5 3 3. 5 4 2.8 3 3.2 3.4 3.6 3.8 LnVstoxx

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C. the formula and calculation of F-test

SSR stands for Sum of Squares of Residuals. Residual is the difference between the actual y and the predicted y from the model. Therefore, the smaller SSR is, the better the model is.

q: number of restriction (the number of independent variables are dropped). In this case, q=3. k: number of independent variables

q: numerator degrees of freedom n-k-1: denominator degrees of freedom

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D. The best fitted line for the observed data with “LnVol” as explanatory variable.

- Before the crisis (1999-2009)

- After the crisis (2010-2013)

2 2. 5 3 3. 5 4 2 2.5 3 3.5 4 lnVol_1

Fitted values lnVol

2 2. 5 3 3. 5 4 4. 5 2 2.5 3 3.5 4 4.5 lnVol_1

Is the Euro STOXX 50® volatility index (VSTOXX®) a significant forecaster for financial stress in the Euro area?