FORECASTING THE DUTCH STOCK INDEX VOLATILITY

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FORECASTING THE DUTCH STOCK INDEX VOLATILITY

An analysis of implied volatility and other forecasting models

By

TOMAS MARSMAN (1738992)

Supervisor: Dr. Peter Smid

University of Groningen Faculty of Economics and Business

Master Thesis MSc Finance

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Abstract

The information content of implied volatility (VAEX) is tested and the forecasting performance of multiple volatility forecasting models is compared on the AEX for the one and three month forecast horizon over the period January 3, 2000 to December 31, 2012. Firstly, our findings support our hypothesis that implied volatility provides the most accurate forecasts for both the and forecast horizon both including and excluding crises. In addition to this, we find that implied volatility is a biased and inefficient forecaster of future volatility for both forecast horizons. Lastly, we find that for the one month forecast horizon HISVOL(66), EWMA and GJR-GARCH provide incremental information beyond implied volatility and for the three month forecast horizon none of the forecasting models provide incremental information.

Jel Classification: C22; C53; G17

Keywords: ARCH models; Forecasting; Future Volatility; Implied Volatility; Stock Index

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3 Table of Contents 1. Introduction 4 2. Literature Review 8 2.1 Theoretical Review 8 2.2 Empirical Review 10 3. Data 14

3.1 Index Returns and Future volatility 14

3.2 Implied Volatility 16

3.3 Descriptive statistics 17

4. Methodology 21

4.1 Time series volatility forecasting models 21

4.1.1 Historical Volatility (HISVOL) 21

4.1.2 EWMA 23

4.1.3 GARCH(1,1) 24

4.1.4 GJR-GARCH 25

4.2 Volatility Forecasts based on Options: Implied volatility 25

4.3 Regressions 26

4.4 Forecast Evaluation 27

5. Empirical Results 28

5.1 Root Mean Square Error and Coefficient of Determination 28

5.2 Conventional regression analysis 31

5.3 Incremental regressions 32

5.4 Tests for Robustness 35

6. Conclusions and Directions for Further Research 38

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1. Introduction

Risk is one of the most important concepts in finance. Finance in essence deals with the tradeoff between risk and expected return. The standard deviation of a security’s return is often used as a measure of risk; standard deviation also is a measure for volatility. Hull (2012) defines volatility as a measure of the uncertainty of the return to be realized on an asset or portfolio. Forecasting volatility is an important concept of financial markets and has been studied by many researchers and academics. In the financial world, asset pricing theory suggests that risk and return are highly related concepts and in most asset pricing models market volatility affects expected returns of all financials assets. Due to this, forecasting volatility is valuable across the entire financial world. In addition, volatility is the most important input in valuing derivatives. Accurate volatility estimates are of significance for many finance professionals and academics in asset pricing, risk management and asset allocation. For these reasons, we study the volatility forecasting accuracy of several forecasting models on the AEX Index volatility for different time horizons two crises.

Today, there are even derivative securities on the volatility of underlying assets called volatility derivatives; examples include variance swaps, volatility swaps, correlation swaps, Chicago Board Options Exchange Market Volatility Index (VIX) futures and options and others. According to Carr and Lee (2009) the first volatility derivative was a variance swap originated in 1993. The popularity of these volatility derivatives in the financial world has skyrocketed in recent times, the volume of trades in futures and options on the VIX have been soaring and reached new highs in spite of a relatively calm market (http://www.tradersmagazine.com/news/cboe-options-based-vix-popular-110967-1.html). The payoff of such a volatility derivative depends on the volatility or other measure of risk of the underlying asset. Volatility is crucial for these derivatives because not only is the tradable product the volatility on the underlying asset but there also is volatility on the tradable product.

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5 According to Hull (2012) implied volatility is the volatility implied from an option price using Black-Scholes or other option models. Implied volatilities are forward looking, are used to monitor the market opinion of the volatility of an asset and are seen as the markets forecast of future volatility. On the contrary, historical volatility is a volatility estimated from historical data and, thus, estimating volatility based on historical volatility is backward looking that ignores recent news and data. Intuitively historical volatility is a good predictor due to the past being a good predictor of the future and it being simple. Some consider historical volatility to be a better forecaster than implied volatility due to many option models including the Black-Scholes failing to perform well as a result of not only measurement errors but also fundamental errors in the model. Nonetheless, it is widely believed that the implied volatility is the market’s most accurate prediction and is believed to dominate historical volatility in forecasting future volatility (Jorion, 1995, Pong et al., 2004 and others). This is supported but also contradicted both by theoretical and empirical evidence and is further discussed in the literature review. Many researchers have tested the performance of numerous volatility forecasting models and have found contradictory results. Highlighting the importance and attention to this subject, Poon and Granger (2003) give an overview of all the 93 published and working papers that have studied the forecasting of volatility, with many contradictory results.

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6 line should be flat. However, our aim is to test which forecasting model indeed is the most accurate forecaster of future volatility since theoretically and empirically there are arguments in favor of both historical forecasting models and volatility forecasts based on option models.

Many papers have studied the forecasting ability of in particular the Chicago Board Options Exchange Market Volatility Index (VIX) also known as the S&P 500 implied volatility and often being called the ‘fear index’, as a forecast of future stock market volatility, including Fleming (1998), Day and Lewis (1992). When the CBOE introduced the VIX in 1993 it was firstly based on at-the-money S&P 100 Index Options but was soon updated to be based on the at-the-money S&P 500 Index Options. According to Whaley (1993), Fleming et al. (1995) and Fleming (1998) the VIX is a real-time measure of the expected stock market volatility. The AEX Volatility Index (VAEX) is also a real-time measure of the expected stock market volatility. The VIX represents the markets forecast of stock market volatility over the next 30 calendar days implied from at-the-money S&P 500 Index Options (Whaley, 1993).

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7 future volatility on the AEX Total Return Index? This is done by extending the formula for testing the informational content of implied volatility by adding an extra term for the different historical forecasting models. If the coefficient of the time series forecast is significant this means that this time series forecasting model does have incremental information beyond the informational content of the implied volatility. In addition to this, we perform an F-test to check the joint hypothesis if the implied volatility contains all information and if the time series forecast contains no information about the future volatility on the AEX.

This research does not only intend to test whether implied volatility is an unbiased and efficient predictor of future volatility but, in addition to this I also test the predictive power of simple historical volatility, exponential-weighted average historical volatility, and of the time-series models GARCH(1,1) and GJR-GARCH. To test the three research questions the daily data from the AEX Total Return Index and the AEX Volatility Index over the period from January 3, 2000 to December 31, 2012 collected from Datastream is used. From this data the different times series volatility forecasts, the implied volatility and the future volatility for the one and three month forecast horizons are constructed.

Our research is different from previous studies and papers due to it focusing on the Dutch stock market index, the AEX, and the sample period from 2000 to 2013 which includes two financial crises, the dot-com bubble of 2000 and the credit crunch of 2007, that both greatly impacted the worldwide stock markets. This is of significance due to an increase in correlation of volatility during financial crises (Poon and Granger, 2003). Franses and Ghijssels (1999) and Franses and Van Dijk (1996) previously focused on volatility forecasting on stock indices including the Dutch stock market, however, these papers differ from our research due to our study including the implied volatility as a stock market volatility estimate and focusing on recent data including financial crises. The results of our research can be used by finance professionals and scholars to make the most accurate volatility forecasts on the AEX, price derivatives, hedge portfolios, calculate value-at-risk and forecast volatility with the ever present uncertainty of another crisis emerging.

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8 provides our empirical results and checks for robustness. Section 6 presents our conclusions and directions for future research.

2. Literature review

2.1 Theoretical Review

As previously mentioned, volatility forecasting is an important part of the financial markets and has been under continuous consideration by many researchers as can be seen from the vast amount of literature concerning forecasting volatility (Poon and Granger, 2003). Black and Scholes (1973) put forward an option valuation formula, where the value of an option on an asset will depend on the current underlying asset price, the strike price, the current risk-free rate, the time to expiration and the volatility on the underlying asset. Latané and Rendleman (1976) was the first paper that used the Black and Scholes model to implicitly derive the volatility in option contracts by using the variables in the formula including the price of the option as inputs to determine the volatility. It is theorized and tested that investors can make good forecasts of return variability and that these estimates are indeed the market’s expectation of volatility. Nonetheless, the assumptions of the Black-Scholes formula and market efficiency are never met completely otherwise all options on a given stock should have the same volatility.

The renowned efficient markets hypothesis (EMH) as put forward by Fama (1970) theorizes that all currently available information is already fully reflected in the prices of stocks. Bearing in mind this theory, the Black-Scholes model to implicitly calculate volatility should result in more accurate volatility forecasts than using historical volatility that ignores recent news and data. To put it differently, volatility estimates based on historical data do not fully reflect all currently available information. Market efficiency does not suggest that volatility forecasts based on index options are identical to future volatility, nevertheless, it should result in option based volatility forecasts approximating future volatility more closely than historical volatility forecasts given there are no forecasting and model errors which is almost never the case.

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9 various options will result in a more accurate representation of the market’s estimation of 30-day volatility. This is in accordance with the method to calculate the VIX and VAEX and is different from other papers which calculate implied volatility from just a few options using the Black-Scholes model.

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10 To conclude, relevant theory provides contradictory results, some supporting the theoretical superiority of implied volatility, others the superiority of historical volatility. However, more relevant theory is found in support of the theoretical advantage of implied volatility over historical volatility models in forecasting future volatility. Additionally, most relevant theory supports the use of the VIX over simply calculating implied volatility from an index option.

2.2 Empirical Review

Numerous studies have investigated market volatility prediction using implied volatility as well as ARCH class models and time-series models. Prior studies on the S&P 100 index have found conflicting results on the informational content of implied volatility and time-series models. Empirical results appears to support the superiority of implied volatility over historical volatility and conditional variance of time-series models in forecasting future volatility (e.g., Fleming, 1998; Blair et al., 2001; Christensen and Prabhala, 1998; and Jorion, 1995). However, there are numerous papers that find simple historical volatility or conditional variance based on GARCH and GJR-GARCH models to dominate implied volatility (e.g., Canina and Figlewski, 1993; Lamoureux and Lastrapes, 1993; Pong et al., 2003; and Ederington and Guan, 2002). Despite the extensive number of papers written on this subject no unanimity exists on which model is the best forecaster for volatility. As previously mentioned, Poon and Granger (2003) investigated all 93 papers that at the time of writing studied forecasting performance of volatility. They determined that implied volatility had better results than simple historical volatility models in 76% of the studies. The forecasting performance of implied volatility outperformed that of GARCH models in 94% percent of studies. GARCH models and simple historical volatility were rather similar in accuracy with simple historical volatility methods being superior in 56% of the studies. In addition to this Poon and Granger (2003) state that GARCH models that include asymmetric volatility like EGARCH and GJR-GARCH perform better than GARCH(1,1).

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11 on the one hand that implied volatility may contain incremental information over time-series models and on the other hand that conditional volatilities of GARCH and EGARCH models contain incremental information relative to implied volatility. Ederington and Guan (2002) find that no model forecasts better than naïve models and that implied volatility is a biased forecast of future volatility. Nonetheless, implied volatility estimates outperform the times series forecasts when the bias is taken account for. Canina and Figlewski (1993) conclude that implied volatility lacks the informational content in recent observed volatility; it has virtually no correlation with, and is a poor forecast of, future volatility, which results in worse results than historical volatility. Pong et al. (2003) compare the volatility forecasting ability of different methods in the currency market, with volatility periods ranging from one day to three months ahead, and find historical volatility to be superior and have significant incremental information over implied volatility. They found that historical volatility forecasts result in superior accuracy over implied volatility due to the use of high frequency returns (5-minute and 30-minute data). In addition to this they found that GARCH, the autoregressive–moving-averagemodel (ARMA) and the autoregressive fractionally integrated moving average model (ARFIMA) all have incremental information over implied volatility for the one day and one week forecast horizons. Nonetheless, implied volatilities are found to have the most accurate volatility forecasts and contain the most information in the one month and three month horizon.

Lamoureux and Lastrapes (1993) assume informational efficiency of markets and that asset pricing models correctly price options. Using both in-sample and out-of-sample tests, they reject their hypothesis that time series forecasts using publicly available historical information has no predictive power and cannot improve the market’s variance forecast. In addition to this, they conclude that implied volatility is a biased and inefficient estimator of future volatility. However, despite these results against the predictive ability of implied volatility, the option model does have incremental information that is not contained in the historical price for forecasting future volatility over a 90 to 180 day horizon.

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12 investigates the performance of implied volatility of the S&P 100 as a forecast for future stock market volatility and concludes that there is sufficient evidence to support empirical use of implied volatility as a forecast (proxy) of the future stock market volatility in spite of the fact that it is a biased forecast. Implied volatility dominates historical volatility in terms of forecasting power and, despite being an upward biased forecast, contains relevant information concerning future volatility. Harvey and Whaley (1992) test the forecasting power of implied volatility of S&P 100 index options as a forecast of market volatility and find enough evidence to reject their hypothesis that volatility changes are unpredictable on a daily basis. They find empirical evidence that there is high correlation between weighted implied standard deviations and future standard deviations. Moreover, implied standard deviations are a better estimator of future standard deviations than historical standard deviations (Latané and Rendleman, 1976).

Christensen and Prabhala (1998) find that implied volatility dominates historical volatility in forecasting future volatility and find implied volatility neither to be an inefficient, nor a biased forecast for S&P 100 Index Options. They attribute the differing results from previous studies to longer time series resulting in lower sampling frequency and the use of nonoverlapping data. Jorion (1995) examines the forecasting ability and information content of implied standard deviation (ISD) on foreign currency futures and finds that ISDs outperform statistical time-series models, even when the parameters of the GARCH models are estimated from the entire sample period. However, ISDs are biased forecasts of future volatility.

As previously mentioned, Franses and Van Dijk (1996) and Franses and Ghijsels (1999) focus on predicting stock market volatility using forecasting models including GARCH models like GARCH(1,1) and GJR-GARCH on several European stock market indices including the Dutch stock market. These studies compared and investigated which GARCH models had the most accurate forecasts and found contradictory results. Franses and Ghijssels (1996) found that the forecasting accuracy of GARCH can be significantly improved due to the exclusion of additive outliers that normally can lead to biased estimates. On the other hand, the forecasting accuracy of the QGARCH significantly improves on the linear GARCH model according to Franses and Van Dijk (1996).

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13 forecaster of future volatility in addition to being an unbiased and efficient forecaster, followed by GJR-GARCH, GARCH(1,1), EWMA and simple historical volatility. The first hypothesis is deduced from our first research question: which volatility forecasting model provides the most accurate forecasts of future volatility on the AEX Total Return Index?

H1: Implied volatilities provide the most accurate forecasts of future volatility of the AEX Total Return Index.

Implied volatility is expected to dominate the other time series forecasting models concerning their informational content as in accordance with previous literature. The second research question, whether implied volatility contains all information about the future volatility of the AEX Total Return Index leads to the following hypothesis:

H2: Implied volatility is an unbiased and efficient forecast of future volatility on the AEX Total Return Index.

We expect implied volatility to contain all information regarding future volatility. Lastly, the third hypothesis is inferred from the third research question: do time series volatility forecasting models provide incremental information beyond implied volatility about the future volatility of the AEX Total Return Index?

H3: Time series volatility forecasting models do not contain incremental information beyond implied volatility about the future volatility of the AEX Total Return Index.

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3. Data

Our empirical analysis contains two types of data: index returns and implied volatilities. The AEX Volatility Index data are available from January 3, 2000 and the AEX Index data are available from January 3,1983 both using Datastream. The data I use is over the 13 year period from January 3, 2000 to December 31, 2012 for AEX Index Options. I use daily closing prices in forecasting daily volatility.

Forecast horizons are also of significance when comparing the accuracy of forecasts. For our research objective of determining the most accurate volatility forecasting models for asset, pricing, risk management and asset allocation the intermediate forecast horizons are more important than short (one day) and long term forecasts (one year). Hence, I use the one-month and three-month forecast horizons. Overlapping data results in a number of problems despite increasing the power of statistical conclusions as there are more forecasts. It results in serial correlation in the prediction errors which need to be taken into consideration. Due to our sample period consisting of 13 years of data, I have chosen to only use non-overlapping forecasts as it eliminates the overlapping data problem and still retains a sufficient number of forecasts.

3.1 Index Returns and Future volatility

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15 outperforms daily index returns, includes incremental information over these daily returns which is, nonetheless, almost completely incorporated by implied volatility (Blair et al., 2001). Additionally, the increase in data frequency also improves the performance of ARCH model forecasts (Blair et al., 2001). According to Andersen, Bollerslev, Diebold and Labys (2003) the availability of data for increasingly shorter return horizons has enabled the emphasis to move from estimating using quarterly and monthly frequencies to daily and even intraday as forecasting performance has increased with the inclusion of more data. High-frequency volatility is highly predictable and the information of high-frequency data is also useful for forecasting at longer horizons, such as monthly or quarterly. Due to the unavailability of high-frequency intraday data, I focus on the easily accessible daily index returns, daily implied volatilities and calculate future volatility with squared daily returns. Despite general claims as in Koopman, Jungbacker and Hol (2005) that squared daily returns provide a poor approximation of actual volatility and that the incorporation of high-frequency intraday data also improves the forecasting performance in both short and long horizons (Andersen, Bollerslev, Diebold and Labys, 2003), I use daily squared returns as a proxy for actual volatility as in Day and Lewis (1992) and Fleming (1998) which is defined in equation 1.

An important part of volatility forecasting is identifying an appropriate proxy for the future volatility. The future standard deviation for day follows the methods and procedures of Hull (2012) and Poon and Granger (2003). Hereafter the future standard deviation for n days is called the future volatility (FV) and is defined by

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Where n is the number of days, ū is the mean return, which represents the daily returns on the financial asset (index), so including dividends, in other words, it represents the total returns. Equation 2 contains the measure of the annualized future volatility (FV).

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16 3.2 Implied Volatility

In contrast to Day and Lewis (1992), Fleming (1998), Harvey and Whaley (1998) and others I do not use option data to derive the implied volatility from the Black-Scholes model. Many papers including Blair, Poon and Tailor (2001) and Becker, Clements and White (2007) use the VIX as the implied volatility of the S&P 500 index options. In accordance with these and many other studies, in our research, we use the AEX Volatility Index as the implied volatility of the AEX index options.

The Chicago Board Options Exchange (CBOE) created the VIX of the S&P 500 to measure the market’s expectation of one month ahead volatility. The AEX Volatility Index on AEX index options is the estimate of implied volatility used in this paper. The VAEX is composed to measure the market’s expectation of 30-day ahead AEX volatility just as the VIX. It is derived from at-the-money Index Call and Put options available via NYSE Liffe (http://www.nyxdata.com/Data-Products/Euronext-Volatility-Indices).

The VIX is constructed in such a way as to eliminate the many problems in calculating the implied volatility from option valuation models including measurement error and smile effects. The VIX eliminates these problems by calculating the weighted average of implied volatilities of eight at-the-money call and put options that are close to expiration, making it a more accurate measurement of implied volatility (Blair et al., 2001). The VAEX is modeled and calculated in the same way as the VIX (http://www.efinancialnews.com/story/2007-07-24/euronext-launches-volatility-indices). The formula used by the CBOE for calculating the VIX is the following (http://www.cboe.com/micro/vix/vixwhite.pdf).

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17 Additionally, without adjusting the equation for calculating the VAEX the AEX Volatility Index takes dividends into account, in spite of the fact that the difference between the real implied volatility and the implied volatility ignoring dividends is constant (Christensen and Prabhala, 1998). The VIX is the measure of market expectation over 22 trading days (one month). The forecasting accuracy of VIX increases when the forecast horizon approaches the one month (Blair et al., 2001). As previously mentioned, one of the forecast horizons of interest in our research is the one month horizon.

3.3 Descriptive statistics

The data in our sample cover two financial crises that heavily impacted the value of the stock indices worldwide, the first being the dot-com bubble of 2000 and the second being the credit crunch of 2007. Therefore, the tables, charts and numbers in our descriptive statistics will include large dispersions in index prices and volatility over the sample period and these crises will be easily observable.

Table 1 displays the descriptive statistics of the AEX Total Return Index, The AEX Daily Log Returns, the AEX Annualized Daily Historical Volatility and AEX Volatility Index for the entire sample from January 2000 until January 2013. Figures 1 through 4 show charts of these AEX data.

Figures 1 and 2 display the AEX Total Return Index prices and the AEX Daily Returns measured at the end of each day over the period from 1/3/2000 until 1/2/2013. Figures 3 and 4 show the AEX Annualized Daily Historical Volatility calculated by equation 2 from end-of-day AEX Total Return Index prices and the AEX Volatility Index prices measured at the end of each day over the period from 1/3/2000 until 1/2/2013.

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Table 1: Descriptive Statistics

Descriptive statistics for the AEX Total Return Index, the AEX Daily Log Returns, the AEX Annualized Daily Historical Volatility and the AEX Volatility Index over the period from 1/3/2000 until 1/2/2013.

AEX Total Return Index

AEX Daily Returns

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19 Figure 1: The AEX Total Return Index prices measured at the end of each day over the period from 1/3/2000 until 1/2/2013. The x-axis represents the date and the y-axis represent the close of the AEX in Euro.

Figure 2: The AEX Daily Returns measured at the end of each day over the period from 1/3/2000 until 1/2/2013. The x-axis represents the date and the y-axis represent the daily log returns.

400 500 600 700 800 900 1,000 1,100 1,200 1,300 1/3/ 2000 1/1/ 2001 1/1/ 2002 1/1/ 2003 1/1/ 2004 1/3/ 2005 1/2/ 2006 1/1/ 2007 1/1/ 2008 1/1/ 2009 1/1/ 2010 1/3/ 2011 1/2/ 2012

AEX Total Return Index

-12% -8% -4% 0% 4% 8% 12% 1/3/ 2000 1/1/ 2001 1/1/ 2002 1/1/ 2003 1/1/ 2004 1/3/ 2005 1/2/ 2006 1/1/ 2007 1/1/ 2008 1/1/ 2009 1/1/ 2010 1/3/ 2011 1/2/ 2012

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20 Figure 3: The AEX Annualized Daily Historical Volatility calculated using equation 4 assuming ū = 0 from end-of-day AEX Total Return Index prices over the period from 1/3/2000 until 1/2/2013. The x-axis represents the date and the y-axis represents the annualized daily volatility on the AEX.

Figure 4: The AEX Volatility Index prices measured at the end of each day over the period from 1/3/2000 until 1/2/2013. The x-axis represents the date and the y-axis represents the level of the VAEX.

0% 40% 80% 120% 160% 200% 1/3/ 2000 1/1/ 2001 1/1/ 2002 1/1/ 2003 1/1/ 2004 1/3/ 2005 1/2/ 2006 1/1/ 2007 1/1/ 2008 1/1/ 2009 1/1/ 2010 1/3/ 2011 1/2/ 2012

AEX Annualized Daily Historical Volatility

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 1/3/ 2000 1/1/ 2001 1/1/ 2002 1/1/ 2003 1/1/ 2004 1/3/ 2005 1/2/ 2006 1/1/ 2007 1/1/ 2008 1/1/ 2009 1/1/ 2010 1/3/ 2011 1/2/ 2012

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4. Methodology

4.1 Time series volatility forecasting models

These models base their forecasting estimations on historical data, including simple historical volatility estimates and ARCH class models.

The timeline for testing the one month forecast horizon is provided below in figure 5. The black line represents the parameters for the ARCH class models starting with parameters over the first year and each month being updated by adding the newest month. The red and blue lines are used for making estimates of HISVOL, EWMA, GARCH(1,1), GJR-GARCH and VIX. The green line is the forecast period.

Figure 5: Timeline for testing the one month forecast horizon

Figure 6 presents the timeline for testing the volatility forecasting models over the three month forecast horizon. The black line again represents the parameters for GARCH(1,1) and GJR-GARCH starting with parameters over the first year and each period the parameters are updated by adding the latest three month period. The red and blue lines are the periods used for making estimates of HISVOL, EWMA, GARCH(1,1), GJR-GARCH and VIX. The green line is the forecast period.

FV 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month

1 month HISVOL EWMA GARCH(1,1) GJR-GARCH VIX FV 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month 1 month

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22 Figure 6: Timeline for testing the three month forecast horizon

4.1.1 Historical Volatility (HISVOL)

Using historical volatility is the simplest method of forecasting volatility. The forecast of volatility for future periods in the historical volatility model is the past variance (or standard deviation) over a specified period. It is a simple but useful measure despite the fact that as previously mentioned other forecasting models often provide more accurate forecasts (Fleming, 1998; Blair et al., 2001; and Christensen and Prabhala, 1998).

An important part of volatility forecasting is identifying an appropriate proxy for historical volatility. The historical standard deviation for day t follows the methods and procedures of Hull (2012) and Poon and Granger (2003). Hereafter the historical standard deviation is called the Historical Volatility (HISVOL) and is calculated by

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Where n is the number of days, ū is the mean return, which represents the

daily returns on the financial asset (index), so including dividends, in other words, it represents

FV

3 months 3 months 3 months 3 months

3 months HISVOL EWMA GARCH(1,1) GJR-GARCH VIX FV

3 months 3 months 3 months 3 months 3 months

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23 the total returns. Equation 4 contains the measure of the annualized historical volatility (HISVOL).

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4.1.2 EWMA

The exponentially weighted moving average (EWMA) model is a forecasting method where exponential weighting and historical volatilities are used to estimate volatility and more weight is placed on recent historical volatilities. The variance equation for the EWMA model is

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24 4.1.3 GARCH(1,1)

The generalized autoregressive conditional heteroscedasticity (GARCH) model is a forecasting method where the volatility is mean reverting and is as the previous models a time series model. Bollerslev (1986) and Taylor (1986) developed the GARCH model which is an extension of the regular ARCH model as developed by Engle (1982). ARCH class models are a useful tool in case of heteroscedasticity and volatility clustering in financial time series. The variance equation for the GARCH (1,1) model is

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Where is the conditional variance due to it being the one-step ahead volatility estimate based on historical data, γ, α and β assign weights to the long-term variance , the most recent daily log return and the volatility estimate for day n - 1 and must meet the following condition, . When γ = 0, the GARCH(1,1) is simply the EWMA. Another way of phrasing, to forecast the volatility on day n using the data at the end of day n -1, the following equation is used,

(7) To forecast the volatility on day n + t using the end of day n - 1 data the following equation is applicable as found in Hull (2012)

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25 reverts back to the long-term average variance. GARCH (P,Q) is an extension of the GARCH(1,1) but GARCH(1,1) is the most widely used in financial time series (Poon and Granger, 2003) and rarely a higher order model is estimated in academic finance papers (Brooks, 2012). Problem with GARCH is that it doesn’t capture leverage effects or asymmetric volatility, which results in negative shocks having larger impact on volatility than positive shocks (Liu and Hung, 2010).That is why in the next sub section GJR-GARCH is explained which does take care of the possible asymmetries.

4.1.4 GJR-GARCH

The GJR-GARCH model was developed by and named after Glosten, Jagannathan and Runkle (1993) and is an extension of the GARCH model with a single additional term. The additional term, γ, is the asymmetry term due to negative shocks having a larger impact than positive shocks on the volatility as a result of financial leverage effects.

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Where , if and if 0. If there is a

leverage effect. The GJR-GARCH forecast of future volatility over the given forecast horizon is calculated in Eviews 7 by calculating the variance for each day for the forecast period of days. The variance of the days is averaged, next it is annualized and the square root is taken. The problems of GARCH with respect to asymmetric volatility have been captured by asymmetric GARCH models including EGARCH and GJR-GARCH. EGARCH and GJR-GARCH have several benefits over GARCH(1,1). Nonetheless, GARCH(1,1) is the most commonly used in financial time series.

4.2 Volatility Forecasts based on Options: Implied volatility

Black and Scholes (1973) proved that option prices could be derived using long and short positions in options and their underlying assets and formulated a valuation formula that derives the value of an option. The Black-Scholes-Merton formulas for calculating the prices of European call and put options are the following as in Hull (2012)

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26 and (11) Where (12) (13)

Where is the price of the European call option, is the price of the European put option, is the value of the underlying asset at time zero, is the strike price, is the continuously compounded risk-free rate, is the time to option maturity, is the volatility of the underlying asset and is a cumulative normal probability distribution for a standard normal distribution. The volatility estimate is an essential component for all option pricing models, however, in the Black-Scholes-Merton model the volatility can be determined implicitly given the price of the option and all other variables

4.3 Regressions

In this section we describe the regressions performed to test our hypotheses. The first hypothesis is tested by performing a regression analysis to assess and compare the different forecasting models. The second hypothesis is tested by performing ordinary least squares (OLS) regression to evaluate the information content of implied volatility. The following regression is used

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27 an F-test if VAEX is and unbiased estimator of future volatility. Lastly, we check if VAEX is an efficient estimator of future volatility by testing for autocorrelation and heteroscedasticity.

The final regression is for testing the incremental information of the other forecasting models beyond implied volatility by regressing the future volatility on the VAEX and one of the different historical forecasting models (HIS). This tests our third hypothesis. The following equation is used to test this

(15) Where are the different forecasting models based on historical volatility.

4.4 Forecast Evaluation

This section evaluates the usefulness and accuracy of the volatility forecasting methods. There are many popular forecasting evaluation measures but I have decided to only include Root Mean Square Error (RMSE). Poon and Granger (2003) researched all papers on forecasting volatility and most were found to use Mean Squared Error (MSE) or Root Mean Square Error (RMSE) in evaluating forecast performance. RMSE is defined as

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A coefficient of determination, R2, is used by many studies, so this is also included in our forecast evaluation. Mean Absolute Error (MAE) and Theil-U Statistic are used as robustness checks. The following equations are used to calculate MAE and the Theil-U statistic

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28

5. Empirical Results

This section evaluates the information content of the different volatility estimators and tests the three set out hypotheses. The results are presented and the forecasts of the different volatility forecasting models are compared. This is done, firstly, by assessing which forecasting method is best by comparing among other things their coefficients of determination ( ) and by measuring and comparing the prediction errors by means of the frequently used Root Mean Square Error ( ). In this the way the first hypothesis is tested. In addition to this, the second hypothesis is tested by evaluating the informational content of implied volatility. And, lastly, we check if other forecasting models based on historical volatility possess some incremental information over implied volatilities. In accordance with our first hypothesis we expect implied volatility to provide best forecasts followed by GJR-GARCH, GARCH(1,1), EWMA and historical volatility, respectively. We expect implied volatility to be and unbiased and efficient forecast of future volatility and expect the forecasting models based on historical volatility to provide no incremental information over VAEX. These are the tests for the second and third hypothesis.

5.1 Root Mean Square Error and Coefficient of Determination

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29 Table 2: Root Mean Square Error

N = One month N = Three months

HISVOL(22) 0.0722(4) 0.0721(3) HISVOL(66) _0.0700(3) _0.0746(5) EWMA 0.0732(5) 0.0716(2) GARCH(1,1) 0.0745(6) 0.0740(4) GJR-GARCH 0.0695(2) 0.0764(6) VAEX 0.0644(1) 0.0657(1)

For the one month forecasts HISVOL(22), EWMA and VAEX consist of 154 observations, HISVOL(66) consists of 51 observations, GARCH(1,1) and GJR-GARCH consist of 142 observations. For the three month forecast horizon the HISVOL(22), HISVOL(66), EWMA and VAEX contain 51 observations, GARCH(1,1) and GJR-GARCH 47 observations.

For the one month forecast horizon the VAEX performs best with a RMSE of 0.0644. This is only approached by second best GJR-GARCH and third best three month simple historical volatility (HISVOL(66)) with values of 0.0695 and 0.700, respectively. Followed by HISVOL(22) (0.0722), EWMA (0.0732) and the disappointing GARCH(1,1). However, in contrast to our expectations EWMA and one month simple historical volatility (HISVOL(22)) perform second (0.0716) and third best (0.0721) for the three month forecast, whereas, the GARCH (1,1) and GJR-GARCH perform less than expected with 0.0740 and 0.0764, respectively.

By ranking the results we can see which forecast method yields the best results. As expected, overall VAEX has the best performance with first places in both forecast horizons, followed by EWMA with a fifth and second place and HISVOL(22) with fourth and third place. GJR-GARCH is fourth overall with a second and sixth place, HISVOL(66) is fifth with third and fifth places and GARCH(1,1) is last with sixth and fourth place. GARCH(1,1) and GJR-GARCH disappoint heavily as they were expected to perform best after implied volatility and theoretically being an upgrade over the simple historical volatility forecasts.

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30 (Fleming, 1998; Blair et al., 2001; Christensen and Prabhala, 1998; and Jorion, 1995) the VAEX has the highest R2 for all the different forecast horizons. The VAEX which is calculated in the same way as the VIX was expected to dominate the one month (N=22) forecast horizon due to the VIX being the measure of market expectation over 22 trading days and forecasting accuracy increases when the forecasting horizon approaches one month (Blair, Poon and Taylor, 2001).

Table 3: Coefficient of Determination (R²) of volatility forecasts on Future Volatility

N = One month N = Three months

HISVOL(22) 0.543(3) 0.381(4) HISVOL(66) 0.525(6) 0.338(6) EWMA _0.530(5) _0.389(2) GARCH(1,1) 0.542(4) 0.389(2) GJR-GARCH 0.602(2) 0.349(5) VAEX _0.637(1) _0.485(1)

At the one month forecast horizon the VIX indeed does have the highest coefficient of determination of 0.637. In addition to this, as expected the GJR-GARCH performs second best with an R2 of 0.602, followed by HISVOL(22) with 0.543. In contrast to expectations, GARCH(1,1) performs below expectation with an R2 of 0.542, only outperforming EWMA and HISVOL(66), 0.530 and 0.525, respectively. The three month forecast horizon yields different results, with the exception of VAEX with an R2 of 0.485. However, GARCH(1,1) and EWMA perform second best with a value of 0.389 followed by HISVOL(22) with 0.381. GJR-GARCH again performs below expectations with an R2 of 0.349 only outperforming HISVOL(66) with a coefficient of determination of 0.338.

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31 5.2 Conventional regression analysis

In addition to using RMSE and R2 to compare the different forecasting models, we assess the information content of implied volatility using the regression from equation 14. This equation enables us to examine whether implied volatility contains some information, is an unbiased and efficient estimator of future volatility. Multiple hypotheses are tested from equation 14. Firstly, we test if implied volatility holds any information about future volatility. If , the VAEX contains some information about the future volatility. In Table 4 the results from the OLS regression are displayed. For the one month forecast horizon, the estimate of

and is significantly different from the null hypothesis of . Hence, we are able to reject that and we can conclude that implied volatility contains some information about future volatility. For the three month forecast horizon the regression coefficient is also significantly different from the null of but the regression coefficient drops to . Thus, the null hypothesis is rejected and we conclude that implied volatility contains some information about future volatility.

Table 4: Information content of Implied Volatility

α0 αVAEX R² DW F-Statistic N = One month -0.0195 0.7566* 0.637 1.547 12840.07* (-1.5434) (16.3195)* N = Three months 0.0157 0.5825* 0.485 2.055 3986.51* (0.6616) (6.7992*) * = p-value < 0.01 ** = p-value < 0.10

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32 but not significant at 10%. So we are unable to reject that . For the three month forecast the F-statistic (2, 49) is also significant at p = 0.01 but the F-statistic has dropped to 3986.51. Again we are able to reject the null hypothesis that and , accordingly, implied volatility is not an unbiased and efficient forecast of future volatility for the three month forecast horizon. The intercept is different from zero but not significant at 10%, so we cannot reject the null hypothesis that .

Lastly, we check if there is covariance between the errors terms in other words by using the Durbin-Watson statistic we check for autocorrelation of the residuals to see if implied volatility is an efficient estimator of future volatility. For the one month forecast horizon the Durbin-Watson statistic is not significantly different from two, meaning that there is no autocorrelation in the residuals. Additionally, we check for heteroscedasticity using the white’s test for heteroscedasticity. If there is heteroscedasticity the regressions are done with heteroscedasticity consistent coefficient variances in other words adjusted standard errors are used to ensure stability of the regression results. There is significant heteroscedasticity present for both forecast horizons. Thus, implied volatility is an inefficient estimator of future volatility. For the three month forecast horizon, again the Durbin-Watson statistic is not significantly different from two. The error terms have no relation with the previous error terms and the results of this test are presented in Table 4. For the three month forecast volatility forecast implied volatility is also an inefficient forecaster.

5.3 Incremental regressions

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33 Table 5: Incremental information of the HIS forecasting models for N = One month

α0 αVAEX αHIS Adjusted R² DW F-Statistic

VAEX vs HISVOL(22) -0.0206 (-1.4632) 0.7785 (6.3292)* -0.0251 (-0.1894) 0.632 1.535 22.088* VAEX vs HISVOL(66) -0.0512 (-3.2433)* 1.1326 (9.0465)* -0.4201 (-2.7275)* 0.817 1.677 29.410* VAEX vs EWMA -0.0358 (-2.5018)** 1.0922 (7.1564)* -0.3139 (-2.3048)** 0.644 1.509 71.799* VAEX vs GARCH(1,1) -0.0217 (-1.5181) 0.8546 (6.0260)* -0.0989 (-0.7595) 0.632 1.501 56.333* VAEX vs GJR-GARCH -0.0111 (-0.8361) 0.4903 (4.5865)* (2.7400)* 0.2909 0.649 1.723 28.864* * = p-value < 0.01 ** = p-value < 0.05

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34 models the results are significant at 1%, so we reject the joint hypothesis for HISVOL (22), HISVOL(66), EWMA, GARCH(1,1) and GJR-GARCH. The accompanying F-statistics are presented in Table 5. Lastly, the Durbin Watson test statistics are checked and there is little evidence of autocorrelation of the residuals so the null of no autocorrelation is not rejected. Additionally, we check for heteroscedasticity using the white’s test for heteroscedasticity. If there is heteroscedasticity adjusted standard errors are used to ensure stability of the regression results.

Table 6: Incremental information of the HIS forecasting models for N = Three months

α0 αVAEX αHIS Adjusted R² DW F-Statistic VAEX vs HISVOL(22) 0.0096 (0.3451) 0.6659 (3.1537)* -0.0855 (-0.4328) 0.466 1.992 17.560* VAEX vs HISVOL(66) 0.0117 (0.4845) 0.7393 (3.8561)* -0.2157 (-0.9145) 0.473 1.931 9.826* VAEX vs EWMA -0.0006 (-0.0214) 0.8343 (3.1977)* -0.2152 (-1.0218) 0.475 1.893 39.585* VAEX vs GARCH(1,1) 0.0113 (0.4101) 0.7014 (2.8898)* -0.1112 (-0.5335) 0.463 1.990 31.338* VAEX vs GJR-GARCH 0.01557 (0.5949) 0.6129 (3.3935)* -0.0335 (-0.2028) 0.460 2.036 24.343* * = p-value < 0.01 ** = p-value < 0.10

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35 incremental information on future volatility beyond implied volatility. The values for the regression coefficient are all significant at 1%. Subsequently, the null hypothesis

is rejected and VAEX does contain information about future volatility. The intercept is not significant at 10% in all the regressions, so we cannot reject the null hypothesis that . By means of an F-test I am able test the joint hypothesis , and . As presented in Table 6, the F-Statistics of all the forecasting models are significant at 1%. Thus, we reject the joint hypothesis for HISVOL (22), HISVOL(66), EWMA, GARCH(1,1) and GJR-GARCH.

For the three month forecast horizon the results support our hypothesis that the time series volatility forecasting models do not contain incremental information beyond implied volatility. However, for the one month forecast horizon multiple of the historical forecasting models do contain significant incremental information beyond implied volatility.

5.4 Tests for Robustness

The last part of the analysis is checking the robustness of our results. This is done by checking the results using other measures of forecast errors than the RMSE. This is done by using the Mean Absolute Error (MAE) and the Theil’s Inequality Coefficient (Theil-U). Moreover, we test for robustness by checking whether excluding the data for the two financial crises produces different results.

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36 Table 7: Mean Absolute Error, Theil's Inequality Coefficient , Root Mean Square Error and Coefficient of Determination

MAE Theil-U RMSE R² N = One month HISVOL(22) 0.0472(4) 0.1877(4) _0.0722(4) _0.543(3) HISVOL(66) 0.0441(2) 0.1849(3) _0.0700(3) _0.525(6) EWMA 0.0462(3) 0.1906(6) 0.0732(5) _0.530(5) GARCH(1,1) 0.0486(5) 0.1899(5) 0.0745(6) _0.542(4) GJR-GARCH 0.0495(6) 0.1763(2) 0.0695(2) 0.602(2) VAEX 0.0410(1) 0.1662(1) 0.0644(1) 0.637(1) N = Three months HISVOL(22) 0.0498(4) 0.1993(3) 0.0721(3) 0.381(4) HISVOL(66) 0.0495(3) 0.2068(5) 0.0746(5) 0.338(6) EWMA 0.0488(2) 0.1979(2) 0.0716(2) _0.389(2) GARCH(1,1) 0.0501(5) 0.2005(4) 0.0740(4) 0.389(2) GJR-GARCH 0.0530(6) 0.2077(6) 0.0764(6) 0.349(5) VAEX 0.0431(1) 0.1805(1) 0.0657(1) 0.485(1)

The results for the Theil’s U-statistic are shown in Table 7 and the lowest coefficient for each forecast horizon is highlighted. The Theil Inequality coefficient was developed by Theil in 1966 as a measure to calculate the deviation between predictions and actual values. For the one month and three forecast horizon the results are in accordance with RMSE, R2, MAE, with our predictions and previous literature as the implied volatility outperforms all other forecasting models. The Theil U-statistics of the VAEX are 0.1662 and 0.1805, respectively. Once again EWMA and HISVOL(66) perform best after implied volatility. GARCH(1,1) has the second worst performance only outperforming the again disappointing results of GJR-GARCH.

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37 The previous regressions took into account the entire sample period including the two large crises of 2000 and 2007. It could be hypothesized that stock market crashes due to crises that are unlikely to ever occur again should be eliminated from the dataset as an unusual outlier if inclusion of these crises impact our conclusions. In the next table the results of the analysis of the different forecasting models excluding the data of the financial crises are presented. So the data of the years 2000, 2001, 2007 and 2008 are left out. As one can see from the results in table 8 all the forecasting models provide more accurate results than when the financial crises are included in the model with the sole exception of the R2 of HISVOL(66) for the one month forecast horizon which decreased. This is in accordance with Poon and Granger (2003) who state that the correlation of volatility increases during financial crises. Thus, the volatility is higher during crises.

Table 8: Mean Absolute Error, Theil's Inequality Coefficient, Root Mean Square Error and Coefficient of Determination excluding crises

MAE Theil-U RMSE R²

N = One month HISVOL(22) 0.0414(3) 0.1650(4) _0.0607(4) _0.630(4) HISVOL(66) 0.0466(6) 0.2114(6) 0.0776(6) 0.430(6) EWMA 0.0389(2) 0.1636(3) 0.0602(3) 0.636(3) GARCH(1,1) 0.0417(4) 0.1665(5) 0.0612(5) 0.624(5) GJR-GARCH 0.0460(5) 0.1631(2) 0.0600(2) 0.638(2) VAEX 0.0375(1) 0.1523(1) 0.0562(1) _0.682(1) N = Three months HISVOL(22) 0.0440(4) 0.1789(3) 0.0621(3) 0.524(3) HISVOL(66) 0.0473(6) 0.2051(6) _0.0705(6) _0.387(6) EWMA 0.0425(3) 0.1785(2) 0.0619(2) 0.527(2) GARCH(1,1) 0.0421(2) 0.1798(4) 0.0624(4) 0.520(4) GJR-GARCH 0.0457(5) 0.1835(5) 0.0636(5) 0.501(5) VAEX 0.0357(1) 0.1521(1) 0.0532(1) 0.650(1)

For the one month forecasts HISVOL(22), EWMA, GARCH(1,1), GJR-GARCH and VAEX consist of 107 observations and HISVOL(66) consists of 36 observations. For the three month forecast horizon the HISVOL(22), HISVOL(66), EWMA, GARCH(1,1), GJR-GARCH and VAEX all contain 36 observations.

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38 followed by HISVOL(22). In contrast to results including the crises, GJR-GARCH performs better and is ranked fourth followed by GARCH(1,1). Last place finish is HISVOL(66), which performs worst for all forecast evaluation measures over both forecast horizons. Despite the increase in accuracy of the volatility forecasts by the different forecasting models the outcomes hardly change with the exclusion of the data of the two crises.

6. Conclusions and Directions for future research

The main contribution and crucial question of this paper was if the VAEX is an unbiased and efficient predictor of the future volatility on the Dutch stock market index over different forecast horizons and to compare its results to other well-known forecasting models, simple historical volatility (one month and three months), EWMA, GARCH(1,1) and GJR-GARCH.

Our first hypothesis is not rejected, implied volatility provides the most accurate forecasts of future volatility on the AEX Total Return Index. The VAEX outperforms the other forecasting models for both the N = One month as well as the N = Three months forecast horizon. Implied volatility has the lowest RMSE for both forecast horizons and the highest R2 for each forecast horizon of all the different forecasting models. The robustness checks do not contradict these results with implied volatility having the lowest MAE and Theil’s Inequality coefficients of all the different forecasting models for both forecast horizons. The results and conclusions barely change when the crises are removed from the sample.

Our results are in accordance with, among others, Poon and Granger (2003), Fleming (1998), Blair et al. (2001), Christensen and Prabhala (1998) and Jorion (1995): as hypothesized, VAEX has better results than simple historical volatility (one and three months), EWMA, GARCH(1,1) and GJR-GARCH.

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39 Return Index. In the one month forecast horizon HISVOL(66), EWMA and GJR-GARCH contain significant incremental information beyond implied volatility. For the three month forecast horizon none of the historical volatility forecasting models have a regression coefficient that is significantly different from zero at 10%. Hence, none of the models contain incremental information on future volatility beyond implied volatility. However, in spite of the fact that in the three month forecast horizon none of the historical volatility forecasting models contains any information beyond implied volatility, implied volatility does not contain all information about future volatility. The regression coefficient of implied volatility is significantly different from one.

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40

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